mathematical modeling in cellular immunology: t cell ... · mathematical modeling in cellular...

Mathematical modeling in cellular immunology:T cell activation and parameter estimation

by

Omer Dushek

A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philosophy

in

The Faculty of Graduate Studies

(Mathematics)

The University Of British Columbia

December, 2008

c© Omer Dushek 2008

ii

Abstract

A critical step in mounting an immune response is antigen recognition by T cells. This step proceeds by pro-

ductive interactions between T cell receptors (TCR) on the surface of T cells and foreign antigen, in the form

of peptide-major-histocompatibility-complexes (pMHC),on the surface of antigen-presenting-cells (APC). Anti-

gen recognition is exceedingly difficult to understand because the vast majority of pMHC on APCs are derived

from self-proteins. Nevertheless, T cells have been shown to be exquisitely sensitive, responding to as few as 10

antigenic pMHC in an ocean of tens of thousands of self pMHC. In addition, T cells are extremely specific and

respond only to a small subset of pMHC by virtue of their specific TCR.

To explain the sensitivity of T cells to pMHC it has been proposed that a single pMHC may serially bind mul-

tiple TCRs. Integrating present knowledge on the spatial-temporal dynamics of TCR/pMHC in the T cell-APC

contact interface, we have constructed mathematical models to investigate the degree of TCR serial engagements

by pMHC. In addition to reactions within clusters, the models capture the formation and mobility of TCR clusters.

We find that a single pMHC serially binds a substantial numberof TCRs in a TCR cluster only if the TCR/pMHC

bond is stabilized by coreceptors and/or pMHC dimerization. In a separate study we propose that serial en-

gagements can explain T cell specificity. Using Monte Carlo simulations, we show that the stochastic nature of

TCR/pMHC interactions means that multiple binding events are needed for accurate detection of foreign pMHC.

Critical to our studies are estimates of TCR/pMHC reaction rates and mobilities. In the second half of the

thesis, we show that Fluorescence Recovery After Photobleaching (FRAP) experiments can reveal effective diffu-

sion coefficients. We then show, using asymptotic analysis and model fitting, that FRAP experiments can be used

to estimate reaction rates between cell surface proteins, like TCR/pMHC. Lastly, we use FRAP experiments to

investigate how the actin cytoskeleton modulates TCR mobility and report effective reaction rates between TCR

and the cytoskeleton.

iii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . ii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xv

Co-Authorship Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Models of T cell activation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 4

1.1.1 The serial engagement hypothesis . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 4

1.1.2 The kinetic proofreading model . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 6

1.1.3 Combining kinetic proofreading and serial engagement . . . . . . . . . . . . . . . . . . . 6

1.1.4 The equilibrium model . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 7

1.2 The immune synapse is the site of T cell activation . . . . . .. . . . . . . . . . . . . . . . . . . 7

1.3 T cell receptor clusters in T cell activation . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9

1.4 Modeling relies on parameter estimates . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 10

1.4.1 FRAP experiments reveal protein diffusion coefficients . . . . . . . . . . . . . . . . . . . 11

1.4.2 FRAP can determine binding constants . . . . . . . . . . . . . .. . . . . . . . . . . . . 11

1.5 Brief chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 12

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Contents iv

2 pMHC dynamics in TCR microclusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 20

2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 22

2.2.1 Four state escape time formulation from a mobile TCR cluster . . . . . . . . . . . . . . . 22

2.2.2 Mean escape time and reaction rates determine the number of hits . . . . . . . . . . . . . 24

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 25

2.3.1 TCR/pMHC binding events in a mobile TCR cluster . . . . . .. . . . . . . . . . . . . . 25

2.3.2 Coreceptors augment TCR/pMHC interactions . . . . . . . .. . . . . . . . . . . . . . . 30

2.3.3 T cell stimulation is correlated to pMHC transport to cSMAC . . . . . . . . . . . . . . . 32

2.3.4 Potential effects of pMHC dimers on the APC . . . . . . . . . .. . . . . . . . . . . . . . 33

2.3.5 TCR/pMHC binding events with multiple TCR clusters . .. . . . . . . . . . . . . . . . . 35

2.3.6 Effects of multiple stationary clusters . . . . . . . . . . .. . . . . . . . . . . . . . . . . 35

2.3.7 Effects of multiple mobile clusters . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 39

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 42

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

3 T cell receptor clustering is essential for accurate ligand discrimination . . . . . . . . . . . . . . . 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 51

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 52

3.2.1 Simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 52

3.2.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 54

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 55

3.3.1 T cells cannot make a direct and reliable estimate ofKD . . . . . . . . . . . . . . . . . . 55

3.3.2 Multiple binding events are required to determine theTCR/pMHC reaction rates . . . . . 56

3.3.3 TCR enrichment impacts pMHC detection . . . . . . . . . . . . .. . . . . . . . . . . . . 57

3.3.4 TCR enrichment with coreceptors improves pMHC detection for weakly binding pMHC . 58

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 59

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61

4 Improving parameter estimation for cell surface FRAP data . . . . . . . . . . . . . . . . . . . . . 64

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 64

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 65

4.2.1 Experimental protocols for cell surface FRAP . . . . . . .. . . . . . . . . . . . . . . . . 65

Contents v

4.2.2 An improved protocol for cell surface FRAP . . . . . . . . . .. . . . . . . . . . . . . . 68

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 69

4.3.1 Error estimates for common FRAP protocols . . . . . . . . . .. . . . . . . . . . . . . . 69

4.3.2 Utility of the 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 72

4.3.3 Practical recommendations for cell surface FRAP experiments . . . . . . . . . . . . . . . 76

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 76

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

5 Analysis of membrane-localized binding kinetics with FRAP . . . . . . . . . . . . . . . . . . . . . 80

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 80

5.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 82

5.2.1 Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 82

5.2.2 Reduced models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 83

5.2.3 Evaluation of theoretical FRAP recovery curves . . . . .. . . . . . . . . . . . . . . . . . 85

5.2.4 Structure of parameter space and model reductions . . .. . . . . . . . . . . . . . . . . . 87

5.3 Proposed experimental protocol . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 88

5.4 Simulated experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 92

5.4.1 Simulated FRAP titrations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 92

5.4.2 Fitting simulated data . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 93

5.5 Experimental outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 94

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 97

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99

6 Effects of intracellular calcium and actin cytoskeleton on TCR mobility . . . . . . . . . . . . . . . 103

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 103

6.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 104

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 105

6.3.1 TCR mobility is similar in naive and activated human T cells . . . . . . . . . . . . . . . . 105

6.3.2[

Ca2+]

iincrease affects TCR mobility by an actin cytoskeleton dependent mechanism . . 106

6.3.3[

Ca2+]

iincrease induces polymerization of the actin cytoskeleton. . . . . . . . . . . . . 107

6.3.4 Modeling reveals effective TCR binding parameters tothe actin cytoskeleton . . . . . . . 107

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 109

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Contents vi

7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.1 Serial engagement in T cell activation . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 115

7.1.1 Serial engagements in TCR clusters . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 115

7.1.2 Localized serial engagements as a mechanism of pMHC discrimination . . . . . . . . . . 117

7.2 Parameter estimation using FRAP . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 117

7.2.1 Estimating diffusion coefficients using FRAP . . . . . . .. . . . . . . . . . . . . . . . . 117

7.2.2 Estimating binding rates between cell membrane proteins using FRAP . . . . . . . . . . . 118

7.2.3 Estimating effective binding rates between cell membrane proteins and the actin cytoskele-

ton using FRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 120

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122

A Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) . . . . . . . . . . . 125

B Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) . . . . . . . . . . . . . . . . 130

B.1 Model of pMHC and TCR cluster dynamics . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 130

B.1.1 Simulation Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 132

B.2 Determiningψ andχ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.2.1 PDE governing pMHC dynamics in a TCR cluster . . . . . . . . .. . . . . . . . . . . . 133

B.2.2 Laplace transform solution of the PDE system . . . . . . . .. . . . . . . . . . . . . . . . 135

B.2.3 Rapid evaluation ofψ andχ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.2.4 Definitions ofΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

B.3 Connection betweenP (t), p(t, θ), and MFPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141

C Analysis of errors due to geometric approximations (Chapter 4) . . . . . . . . . . . . . . . . . . . 142

D Asymptotic reductions of the full model (Chapter 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 144

D.1 Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 144

D.2 Dimensionless form of full model . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 145

D.3 Pure diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 145

D.4 Weighted diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 146

D.5 Independent diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 147

D.6 Reaction dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 148

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150

Contents vii

E Details of model fitting to simulated data (Chapter 5) . . . . . . . . . . . . . . . . . . . . . . . . . 151

F Materials and methods (Chapter 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

F.1 Cell isolation and culture . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 154

F.2 Fluorescence recovery after photobleaching . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 154

F.3 Fitting procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 155

F.4 Quantification of F-actin by phalloidin staining and flowcytometry . . . . . . . . . . . . . . . . . 155

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156

viii

List of Tables

2.1 Parameter Definitions and Estimations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 26

2.2 Estimates of TCR hits and pMHC transport by a mobile TCR cluster . . . . . . . . . . . . . . . . 29

2.3 CD8 augments low affinity TCR/pMHC interactions . . . . . . .. . . . . . . . . . . . . . . . . 32

2.4 TCR hits and pMHC transport by multiple mobile TCR clusters . . . . . . . . . . . . . . . . . . 40

4.1 Fitting the 1D and 2D models to the sphere simulations (Figure 4.4, solid lines) . . . . . . . . . . 74

5.1 Parameters used for numerically simulating FRAP recovery curves shown in Figure 5.3. . . . . . 92

5.2 Parameter values used in numerical simulations, and theestimates from fitting the simulated FRAP

titration data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 95

6.1 Diffusion coefficient and mobile fraction under variouspharmacological treatments . . . . . . . . 105

6.2 Model fitting to FRAP recovery curves . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 109

B.1 The state of pMHC determines the possible reactions . . . .. . . . . . . . . . . . . . . . . . . . 133

D.1 Summary of Asymptotic Limits . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 149

E.1 Model Fitting to Simulated Experiment A . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 151

E.2 Model Fitting to Simulated Experiment B . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 152

E.3 Model Fitting to Simulated Experiment C . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 153

ix

List of Figures

1.1 Interactions of TCR and pMHC at the T cell-APC contact junction . . . . . . . . . . . . . . . . . 2

1.2 Early signaling events in T cell activation . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3

1.3 Three models of T cell activation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5

1.4 Early studies of the immune synapse reveal protein segregation . . . . . . . . . . . . . . . . . . . 8

1.5 Recent studies of the immune synapse reveal the existence of many TCR clusters . . . . . . . . . 9

1.6 Schematic representation of Fluorescence Recovery After Photobleaching (FRAP) experiments . . 11

2.1 The most general reaction scheme we consider . . . . . . . . . .. . . . . . . . . . . . . . . . . 23

2.2 Reactions schemes we consider . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 27

2.3 Residence time and total TCR hits in a TCR cluster as a function of the initial position . . . . . . 28

2.4 Maximum dissociation constant (KCD) of coreceptors required to achieve transport of agonist

pMHC to cSMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 33

2.5 Number of pMHC engagements to clustered TCR is almost independent of cluster size . . . . . . 38

2.6 Probability of cSMAC arrival by pMHC depends onkon andkoff . . . . . . . . . . . . . . . . . . . 42

3.1 Models of TCR-pMHC reactions with and without coreceptor-MHC interactions . . . . . . . . . 53

3.2 Comparisons of stochastic simulation and PDE computations . . . . . . . . . . . . . . . . . . . . 54

3.3 TCR concentration impacts the accuracy in the estimation of kon and engagement localization . . 57

3.4 Total time required to achieve 10 binding events . . . . . . .. . . . . . . . . . . . . . . . . . . . 57

3.5 Clustering of coreceptors with TCR improves ligand detection . . . . . . . . . . . . . . . . . . . 58

3.6 Confidence in the estimation ofkoff is independent ofkon and TCR concentration . . . . . . . . . 59

4.1 Experimental geometry for protocol 1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 65

4.2 Experimental geometry for protocol 2 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 67

4.3 Accuracy of the 1D model and the closed form circularly symmetric solution of Soumpasis . . . . 72

4.4 FRAP simulations based on the geometry of protocol 2 . . . .. . . . . . . . . . . . . . . . . . . 73

4.5 Statistical signifiance of the 2D model . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 75

5.1 The geometry of a typical confocal FRAP experiment . . . . .. . . . . . . . . . . . . . . . . . . 83

List of Figures x

5.2 Quantifying full model limits . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 88

5.3 Numerically simulated FRAP titration experiments demonstrate the effect of increasing ligand

density on FRAP recovery dynamics . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 93

6.1 TCR mobility does not depend on the activation state of T cells . . . . . . . . . . . . . . . . . . . 106

6.2 TCR mobility is modulated by intracellular calcium via the actin cytoskeleton . . . . . . . . . . . 107

6.3[

Ca2+]

iincrease induces actin polymerization in PBL . . . . . . . . . . . .. . . . . . . . . . . 108

6.4 Fitting the diffusion and binding model to FRAP recoverycurves . . . . . . . . . . . . . . . . . . 109

7.1 Possible relationship between 3D and 2D reaction rates .. . . . . . . . . . . . . . . . . . . . . . 119

B.1 Illustration of model geometry and the interaction between diffusing pMHC and advecting TCR

clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 131

B.2 Flow chart illustrating the main loop in the simulation .. . . . . . . . . . . . . . . . . . . . . . . 133

C.1 Potential source of error in using the 2D model . . . . . . . . .. . . . . . . . . . . . . . . . . . 142

xi

List of Abbreviations

APC Antigen presenting cell

AIC Akaike’s information criterion

BCR B cell receptor

CAC Cortical actin cytoskeleton

CD4/CD8 T cell coreceptors

CLSM Confocal laser scanning microscope

cSMAC Central supramolecular activation cluster

FCS Fluorescence correlation spectroscopy

FRAP Fluorescence recovery after photobleaching

FRET fluorescence resonance energy transfer

ICAM1 Inter-cellular adhesion molecule 1

IL-2 Interleukin 2

IS Immunological synapse

ITAMs Immunoreceptor tyrosine-based activation motifs

LCK Lymphocyte-specific protein tyrosine kinase

LFA1 Leukocyte function-associated antigen 1

MHC Major-histocompatibility-complex

PDE Partial differential equation

pMHC Peptide-major-histocompatibility-complex

pSMAC Peripheral supramolecular activation cluster

SPR Surface Plasmon Resonance

TCR T cell receptor

TIRF Total internal reflection fluorescence (microscopy)

xii

Glossary

Antigen A substance that can cause an immune response. In the contextof the present work, an antigen is

usually displayed as a pMHC on an APC.

Antigen presenting cell (APC) Cell displaying antigen, in the form of pMHC, to immune cells.

CD4/CD8 T cell coreceptors that bind the MHC molecule directly and may help stabilize the TCR-pMHC bond.

Coreceptors are also important for signaling.

Supramolecular activation cluster (SMAC) A region on the membrane enriched with signaling molecules

and receptors. In the immune synapse, a central-SMAC (cSMAC) and a peripheral-SMAC (pSMAC) have been

characterized in detail.

Fluorescence recovery after photobleaching (FRAP) An experimental technique used to quantify the mobil-

ities of proteins (e.g. by determining a diffusion coefficient).

Inter-cellular adhesion molecule 1 (ICAM1) See LFA1.

Immunological synapse A term used to describe the interface between a T cell and an antigen presenting cell

(APC).

Interleukin 2 (IL-2) A molecule that is secreted by activated T cells to mobilize an immune response.

Leukocyte function-associated antigen 1 (LFA1) An adhesion molecule found on T cells and other immune

cells that binds to ICAM1 on APCs. The LFA1/ICAM1 interaction facilitates cell-cell adhesion and the formation

of the immunological synapse.

Lymphocyte-specific protein tyrosine kinase (Lck) Lck iniates a signaling cascade by phosphorylating the

signaling modules of TCR that are bound to pMHC.

Peptide-major-histocompatibility-complex (pMHC) Complexes derived from self and foreign proteins dis-

played to T cells on the surface of antigen presenting cells (APCs, see above).

Glossary xiii

Surface plasmon resonance (SPR)A technique used to measure binding parameters between proteins. In

typical experiments, one protein is confined to a two-dimensional chip while a second protein is in solution.

T cell receptor (TCR) The primary receptor used by T cells to scan the surfaces of antigen presenting cells

(APCs, see above) for foreign material in the form of pMHC (see above).

Total internal reflection fluorescence (TIRF) microscopy A technique used to obtain high resolution images

of thin surfaces, like the plasma membrane. It is often used to study the T cell membrane during the formation of

the immune synapse.

xiv

Acknowledgements

The work presented here is the culmination of my PhD studies at the University of British Columbia (UBC). Many

people and institutions, in various capacities, have contributed to my work. I wish to thank the Natural Sciences

and Engineering Research Council, Mathematics of Information Technology and Complex Systems, UBC, the

International Graduate Training Centre in Mathematical Biology, and Green College for providing financial assis-

tance which allowed me to dedicate the majority of my time to research. I thank the Department of Mathematics

and the Institute of Applied Mathematics for providing excellent facilities and I thank my engaging colleagues

for all our discussions. I would also like to thank the administrative staff in the Mathematics Department for their

assistance and extreme efficiency. The Mathematical Biology group has been of tremendous support and provided

invaluable feedback throughout my time at UBC. I hope to find myself working in such an atmosphere in the

future.

I would also like to thank several people who went out of theirway to assist me with my research. I thank

Michael Ward and Eric Cytrynbaum for sitting on my proposal committee, assisting with mathematical analysis,

and for providing feedback on my modeling endeavors. I thankAlan Lindsay, Yoichiro Miro, Anmar Khadra, and

Sasha Jilkine for mathematical advice. I thank James Baileyfor helpful discussions about theoretical immunology.

I thank Talia Cohen for helping me illustrate figures. I thankDavid Depoil, Sebastien Soubies, Sabina Mueller,

and Salvatore Valitutti for introducing me to the experimental side of immunology. I thank Michael Gold and

Leah Keshet for being on my proposal committee and my thesis committee. In addition, I thank Leah Keshet for

her encouragement ever since I got to UBC. I would like to thank Raibatak (Dodo) Das for exhaustively sharing

his knowledge of biochemistry and immunology during our many discussions. Daniel Coombs, my PhD advisor,

has been a tremendous force over the past few years and I am indebted to him for taking the time to mentor me.

Finally, I wish to thank my parental units, Yoram and Ilana Dushek, and my sibling unit, Hana Dushek, for their

continual support in the form of love and rugelach.

xv

Dedication

To my parental units, Yoram and Ilana Dushek.

xvi

Co-Authorship Statement

My PhD. supervisor, Daniel Coombs, has been involved in all projects presented in this thesis. He has pro-

vided feedback, assisted in solving analytical and computational difficulties, analyzed results, and has edited all

manuscripts. All computations and simulations were performed by me using Matlab or C. I produced all plots

using Matlab. Schematics were designed by me and illustrated by me.

Chapter 2 Daniel Coombs and I identified and designed the research on serial engagement in the context of

TCR clusters. I constructed, parameterized, and analyzed the mathematical model and subsequently wrote a draft

manuscript. Daniel Coombs edited the manuscript.

Chapter 3 I proposed that serial engagement can explain T cell specificity by allowing for multiple samples

of the reaction parameters of a given pMHC. Together with Daniel Coombs, I designed a set of numerical exper-

iments to support the hypothesis. I constructed and analyzed the necessary simulations and subsequently wrote a

draft manuscript. Daniel Coombs edited the manuscript.

Chapter 4 Daniel Coombs and I felt that FRAP analysis of protein mobility on cell membranes could be

improved and together identified an improved protocol. I performed the necessary calculations and simulations

and subsequently wrote a draft manuscript. Daniel Coombs rigorously edited the manuscript.

Chapter 5 I proposed that FRAP can be used to determine binding reactions between membrane proteins.

Through discussions with Raibatak Das and Daniel Coombs we designed the research. I constructed the math-

ematical model, performed model analysis, and fit simulateddata. Raibatak Das and I wrote a draft manuscript

which was edited by Daniel Coombs.

Chapter 6 Salvatore Valitutti, David Depoil, and I designed the research to probe the interplay between

intracellular calcium, the actin cytoskeleton, and TCR mobility. David Depoil and I designed the experimental

protocol. All experimental data was obtained in Toulouse (France) by Sabina Mueller, Sebastien Soubies, and

David Depoil. I performed all data fitting. All authors on this work analyzed the data. Salvatore Valitutti,

Sebastien Soubies, and Daniel Coombs wrote the draft manuscript which I edited.

1

Chapter 1

Introduction

Thymus derived lymphocytes, or T cells, play a central role in cellular immunity (1). Continually circulating be-

tween peripheral lymphoid tissues (e.g. lymph nodes), naive T cells scan the surfaces of dendritic cells for specific

antigens. Antigens are proteins that are recognized by the adaptive immune system. Upon encountering a specific

antigen, the T cell becomes activated and begins a program ofproliferation and differentiation that culminates in

the generation of many mature effector T cells that can recognize the specific antigen. Effector T cells can acti-

vate specific B cells to make antibody and may leave the lymphoid tissue to perform targeted killings of infected

cells and activate infected macrophages. These effector functions also prevent viruses from efficiently replicating

within cells and spreading between them. Therefore, T cellsare critical for the elimination of viral infections but,

in addition, they play an important role in clearing many other pathogens. Like the initial activation of naive T

cells, activation of effector functions is only carried outupon recognition of specific antigen on the surface of

the relevant cell. Consequently, how a T cell recognizes a specific antigen is a central question in contemporary

immunology (2; 3).

T cells use their antigen receptor, the T cell receptor (TCR), to scan the surfaces of dendritic cells, macrophages,

and B cells, collectively known as antigen presenting cells(APC), for foreign antigen, see Figure 1.1. A given T

cell has thousands of identical TCR on its cell surface and ingeneral, each T cell in the body has a different TCR

that is able to detect a different set of antigens. In other words, all TCR on a single T cell have the same specificity

for antigen. Foreign antigen is displayed on the surface of APCs in the form of peptide fragments, derived from

foreign proteins, attached to major-histocompatibility-complex (MHC). If the foreign protein is produced within

the cell, fragments of it will be displayed on class I MHC along with peptides derived from all other proteins

produced within the cell. If the foreign protein is endocytosed, fragments of it will be displayed on class II MHC

along with all other peptides endocytosed by the cell. In certain APCs, displayed peptides on a particular class

of MHC can arise by both routes, a phenomena known as cross-presentation (4; 5). In addition to the APCs

mentioned above that express both MHC class I and II, all cells in the body express MHC class I. In each case a

different subset of T cells detect the peptide-MHC (pMHC) complex: CD8 T cells detect peptides bound to MHC

class I while CD4 T cells detect peptides bound to MHC class II. The labels CD8 and CD4 refer to cell surface

proteins that, respectively, bind to MHC class I and MHC class II directly. In both cases, it is expected that the

Chapter 1. Introduction 2

vast majority of displayed pMHC will not be antigenic but will represent pMHC derived from self-proteins (i.e.

proteins used by cells in the body for normal functions). Indeed, experiments have shown that T cells can detect

as few as 1-10 antigenic pMHC among tens of thousands of self-pMHC (6; 7; 8; 9; 10), in part by direct binding

of TCR to antigenic pMHC.

T CELL APC

TCR Agonist pMHC Endogenous pMHC

Figure 1.1: Interactions of TCR and pMHC at the T cell-APC contact junction. Binding events between TCRand pMHC take place at the contact junction between a T cell and APC. The two proteins are mobile on theirrespective cell membranes. Each T cell has∼40 000 identical copies of TCR while each APC contains roughly10 000 different pMHC, although relatively few pMHC are recognized by any specific T cell in an individual.

The early signaling events following the engagement of TCR by pMHC have been extensively studied (11).

A simplified signaling diagram is shown in Figure 1.2, which focuses on four signaling molecules: Lck, ZAP70,

LAT, and SLP76. The engagement of TCR by pMHC results in the phosphorylation (phosphate addition) of

specific tyrosine residues within specialized signaling motifs (ITAMs) on the cytoplasmic domains of the TCR-

associated signaling modules by leukocyte-specific protein tyrosine kinase (Lck). In Figure 1.2 the signaling mod-

ules are shown as CD3 andζ-chain on which specific ITAMs reside. Once phosphorylated,ζ-chain-associated

protein kinase 70 (ZAP70) can bind to the ITAMs. The next stepin the signaling cascade is the phosphorylation

of ZAP70 by Lck. Phosphorylated ZAP70 then activates linkerof activated T cells (LAT) and SH2-domain-

containing leukocyte protein of 76 kDa (SLP76). LAT and SLP76 form a large signaling platform which recruits

many additional signaling molecules and subsequently activates multiple downstream responses, such as actin

polymerization, integrin activation, TCR internalization, calcium release, and ultimately full effector function.


The process of signaling is augmented by the action of the CD4and CD8 coreceptors which bind the MHC

molecule directly. Lck is constitutively associated with the cytoplasmic tails of coreceptors and therefore core-

ceptors help localize Lck to the TCR-pMHC complex (12; 13; 14; 15; 16; 17). However, the spatial and temporal

orchestration of signaling in T cells is complex and does notproceed in a simple linear cascade. Rather, various

feedback loops have been described (11). Although the interactions between many signaling molecules have been

characterized, the quantitative rates have generally not been measured.

Lck

CD3

TCRCD4

ZAP70

67PLS

TA

L

P

P P

FURTHER DOWNSTREAM SIGNALING

APC

T CELL

CD8/

pMHC

ζ - Chain

P

P

Figure 1.2: Early signaling events in T cell activation. Upon engagement of TCR by pMHC, lymphocyte-specificprotein tyrosine kinase (Lck) phosphorylates specific immunoreceptor tyrosine-based activation motifs (ITAMs)located on the CD3 andζ chains of the TCR. Lck is constitutively associated with thecytoplasmic domain ofthe T cell coreceptors CD4 and CD8. The next step in the signaling cascade involves the binding of ZAP70 tophosphorylated ITAMs. Lck then phosphorylates ZAP70 whichproceeds to phosphorylate SLP76 and LAT, twocritical signaling molecules that mediate many downstreamresponses.

The biophysical parameters governing mobility and interactions for TCR and pMHC have been extensively

studied. Reactions between TCR and pMHC have been shown to obey the law of mass-action (18) and have been

measured to be weak, with solution dissociation constants,KD, in the range of 0.1-100µM and half-lives in the


range of 1-50s (19; 20; 21; 22). For comparison, typical specific interactions between membrane proteins have

KD = 0.1 µM (23). TCR and pMHC have been shown to undergo effective diffusion on their respective cell

membranes withD ≈ 0.05 µm2/s (24; 25; 26; 27), which is comparable to the diffusion coefficients of other cell

surface proteins (28).

The concept that T cell activation in response to antigenic pMHC can follow the productive interaction with a

single APC has existed since the 1980s (29) and detailed studies of the T cell-APC contact region first appeared

in 1998 (30) and 1999 (21). These latter studies revealed a large and homogenous enrichment of TCR and pMHC

in the center of the contact region. Given the weak reactionsbetween TCR and pMHC, this central accumulation

was thought to mediate T cell activation by increasing the probability of TCR/pMHC engagement. As we shall

discuss below, subsequent studies utilized higher resolution microscopy to paint a much more dynamic picture

of the contact region (31; 32; 33; 34; 35; 36). The term “immunological synapse” (or immune synapse) is now

in common use to describe the T cell-APC contact region because it has many structural and functional similar-

ities to the neurological synapse (37). Structurally, bothsynapses are composed of closely opposing membranes

with many receptor-ligand interactions. Functionally, both synapses utilize their close membrane opposition for

directed secretion (37; 38).

In summary, CD4 and CD8 T cells are thought to form immunological synapses with APCs in order to facil-

itate the interaction between TCR and pMHC. Antigenic pMHC is extremely diluted by thousands of irrelevant

pMHC on a given APC yet the T cell is able to recognize and become activated by few antigenic pMHC. These

observations form one of the central questions in modern immunology: how can a few pMHC having low affinity

(i.e. weak reaction kinetics) to the TCR drive the activation of a T cell? In what follows, I will discuss previous

experimental and theoretical work aimed at answering this question. I will then discuss recent experimental work

characterizing the spatial and temporal dynamics of TCR andpMHC in the immune synapse. My own work on

the subject, presented in Chapters 2 and 3, focuses in part onhow these recent experimental observations impact

the previous theoretical work aimed at understanding T cellactivation.

1.1 Models of T cell activation

1.1.1 The serial engagement hypothesis

A partial solution to how T cells become activated by few antigenic pMHC was first proposed by Valitutti et al.

(39) in 1995. In experiments where a T cell is in contact with an APC bearing as few as 100 pMHC, Valitutti


observed the internalization of thousands of TCRs from the Tcell surface. Assuming each internalized TCR has

contacted a pMHC, Valitutti estimated that a single pMHC serially engages roughly 200 TCRs in the immune

synapse. Mathematically, the rate of serial engagement by asingle pMHC or the “hitting rate”, is simply the

reciprocal of the total time for a single TCR-pMHC binding cycle,

hitting rate=1

1/ (konT ) + 1/koff=

koffkonT

koff + konT(1.1)

whereT is the concentration of free TCR (in units ofµm−2), kon is the two-dimensional bimolecular reaction

constant (in units ofµm2/s), andkoff is the reaction off-rate (in units of s−1). The hitting rate is plotted in Figure

1.3a as a function ofkoff . In order to determine the total number of TCR hits from the hitting rate, an estimate

of the time that a pMHC is able to interact with TCR is required. Wofsy et al. (40) used a TCR/pMHC reaction-

diffusion framework to estimate the mean escape time, defined as the mean time a single pMHC is expected to

remain in the immune synapse (and interact with TCRs). During the time of their research (1999-2000), available

data indicated that the TCR and pMHC were roughly homogenously distributed in the immune synapse (21; 30).

Using this assumption they computed the mean escape time andmultiplied it by the hitting rate to find that a

typical pMHC can engage≈ 5−35 TCRs in the immune synapse before diffusing out. Although smaller than the

estimates of Valitutti, they found that substantial serialengagement of TCR is plausible. Taken together, these ex-

perimental and theoretical results suggest that a single pMHC can initiate multiple signaling cascades by serially

binding multiple TCRs. The summation of TCR signaling from few serially binding pMHC could therefore be

sufficient to activate the T cell. A further prediction from the serial engagement hypothesis is that the stimulatory

potency of pMHC improves with increasingkon andkoff .

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

log10

(koff

) (s−1

)

Hittin

g R

ate

(s

−1)

Serial Engagement Hypothesis

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

log10

(koff

) (s−1

)

Pro

b.

of

Pro

du

ctive

En

ga

ge

me

nt

The Kinetic Proofreading Model

−3 −2 −1 0 1 2 30

0.01

0.02

0.03

0.04

log10

(koff

) (s−1

)

Pro

du

ctive

Hittin

g R

ate

(s−

1)

Serial Engagement and Kinetic ProofreadingA) B) C)

Figure 1.3: Three models of T cell activation. The serial engagement model (A) predicts that the hitting rate, thenumber of TCR engaged by a single pMHC per unit time, is an important determinant of T cell activation. Incontrast, the kinetic proofreading model (B) predicts thatthe quality of each hit is important (i.e. the duration ofbinding). Combining the two models (C) reveals a peak in T cell stimulation as a function ofkoff . In contrast,the equilibrium model predicts that T cell stimulation willbe a strictly decreasing function withkoff (not shown).Parameters:kon = 0.01µm2/s,T = 100µm−2, kp = 0.5s−1,N = 5.


1.1.2 The kinetic proofreading model

In the same year that Valitutti proposed the serial engagement hypothesis, McKeithan (41) proposed a kinetic

proofreading mechanism of pMHC discrimination based on thework of Hopfield (42). Kinetic proofreading,

as applied to the T cell, postulates that upon pMHC binding tothe TCR, a series of biochemical modifications

begin to accumulate at the TCR (e.g. phosphorylation of ITAMs, binding of ZAP70, etc.). However, if the

pMHC dissociates before a critical modification has occurred, the TCR-pMHC engagement does not result in a

downstream signal and the TCR reverts to a resting unmodifiedstate. Assuming there areN modification steps

and that the rate of each modification is equal tokp, an ordinary differential equation (ODE) formulation can be

used to show that the probability of a productive engagement, fkp, is (43),

fkp =

(

kp

kp + koff

)N

. (1.2)

In Figure 1.3b we plotfkp as a function ofkoff . We see from the Figure and from equation 1.2 that whenkoff kp,

the probability of a productive engagement approaches unity. The kinetic proofreading mechanism, in contrast to

the serial engagement hypothesis, predicts that the stimulatory potency of pMHC will improve with decreasing

koff . Weak evidence for this model can be found in studies showingthat pMHC having smallerKD (= koff/kon)

are more stimulatory than pMHC with largerKD (18; 20; 21; 22).

1.1.3 Combining kinetic proofreading and serial engagement

The kinetic proofreading and serial engagement models werecombined in 2002 by Coombs et al. (43). Their

reaction-diffusion partial differential equation (PDE) model of the immune synapse captured serial engagement,

kinetic proofreading, and TCR internalization. One simpleresult from their study is that theproductive hitting

rate is,

productive hitting rate=koffkonT

koff + konT

(

kp

kp + koff

)N

(1.3)

which is simply the hitting rate, equation 1.1, multiplied by the probability of a productive engagement, Equation

1.2. The balance between the hitting rate and productive engagement reveals that the stimulatory potency of a

pMHC is maximized whenkoff ≈ kp/(N − 1), see Figure 1.3c. This observation was also made in a Monte Carlo

study of TCR on a lattice that incorporated both serial engagement and kinetic proofreading (44). The existence

of a peak in T cell stimulation as a function ofkoff has been observed experimentally (43; 45).


1.1.4 The equilibrium model

The combined serial engagement and kinetic proofreading model has been challenged over the years (20; 22; 46;

47; 48). In particular, the equilibrium model of T cell activation argues that T cells discriminate pMHC based on

KD, which at equilibrium determines the number of TCR-pMHC bonds. Experimental evidence for the model

comes from studies showing a significant negative correlation betweenKD and T cell stimulation (20; 22; 46; 47).

For example, Holler and Kranz (20) allowed specific T cells tointeract with APCs bearing pMHC known to stim-

ulate that particular T cell. After several hours of incubation they measured the release of interleukin 2 (IL-2),

a cytokine secreted by T cells upon activation. In separate experiments they used Surface Plasmon Resonance

(SPR) to measure the solution binding parameters between the TCR and pMHC and found a good correlation

betweenKD and T cell stimulation. Similar results were obtained by Andersen et al. (46) and Tian et al. (22).

The combined kinetic proofreading and serial engagement model and the equilibrium model both assume a

standard mass action reaction and diffusion model for TCR and pMHC in the immune synapse. These assumptions

were considered reasonable based on the early studies of theimmune synapse that showed diffusive enrichment

of TCR and pMHC to the contact region (21; 30). However, our understanding of the spatial-temporal dynamics

of TCR and pMHC at the immune synapse has dramatically improved over the past ten years and this new under-

standing can give clues into the T cell activation process.

1.2 The immune synapse is the site of T cell activation

Given that TCR/pMHC interactions take place within the immune synapse there has been considerable interest in

characterizing the spatial and temporal dynamics of TCR andpMHC within the immune synapse in the hope that

they can be used to uncover important mechanisms of T cell activation.

Using fluorescent markers and digital three-dimensional microscopy, Monks et al. (30) were able to image

mature T cell-APC junctions on fixed cells. Fixed cells are obtained from live cells by treatment with a fixative

agent (e.g. formaldehyde) that preserves the morphological structure at the time of fixing. In these junctions they

found two distinct domains that collectively formed a bullseye, see Figure 1.4. In the central domain, which

they called the central supramolecular activation cluster(cSMAC), they observed a large aggregation of TCRs. In

the second outer domain, a region known as the peripheral SMAC (pSMAC), they observed a ring of leukocyte

function-associated antigen 1 (LFA1) surrounding the cSMAC. LFA1 is a cell surface receptor involved in cell

adhesion. Grakoui et al. (21) confirmed these results by showing that fluorescently labeled pMHC and ICAM1,


the complementary receptors to TCR and LFA1, aggregated in the c and p SMAC, respectively. In summary,

these studies showed that cell surface receptors and intracellular proteins localize to either the p or cSMAC at the

T cell-APC junction.

EN FACE VIEW

T CELL APC

cSMAC

pSMAC

TCR Agonist pMHC Endogenous pMHC

LFA1/ICAM1

TCR/pMHC

Figure 1.4: Early studies of the immune synapse reveal protein segregation. In the center of the contact region(cSMAC) aggregation of TCR and pMHC is observed while in the periphery of the contact region (pSMAC)aggregation of the adhesion molecules LFA1 and ICAM1 is observed.

Grakoui et al. (21) made several observations that allowed them to surmise a purpose for the immune synapse.

First, they observed that the extracellular length of TCR are small (∼15nm (49)) compared to many other cell

surface receptors (LFA1∼42nm, CD45∼80nm (49)). Synapse formation would then allow for close membrane

opposition in the cSMAC allowing TCRs to easily interact with pMHC molecules. Second, the low affinity of the

TCR-pMHC complex would be enhanced by confinement within such a small space. Further, aggregation of these

molecules in the cSMAC increases their concentrations leading to further increases in signaling. Thus, synapse

formation seemed essential for “sustained TCR engagement and signaling” (21).

I briefly mention that soon after these experimental studiesappeared, mathematical modeling of synapse for-

mation showed that purely passive processes, including diffusion, reactions, and membrane tension and bending

rigidity, could lead to observed molecular patterns duringsynapse formation through a spontaneous self-assembly

process (50; 51; 52; 53; 54). The basic idea behind these models is that minimization of membrane bending and

tension will faciliate protein sorting/aggregation according to their extracellular bond-length.

Two years following the publication of the studies by Monks et al. (30) and Grakoui et al. (21), two studies

showed that the signaling important for T cell activation isconfined to the pSMAC and that minimal signaling


occurs in the cSMAC, despite the fact that it contains a largeconcentration of TCR and pMHC. Krummel et al.

(55) showed that the coreceptor CD4 colocalizes with TCR in the early stages of immune synapse formation but

is largely absent from the cSMAC, suggesting that Lck cannotefficiently localize with TCR in the cSMAC. Lee

et al. (56) stained for global tyrosine phosphorylation, phosphorylated Lck, and phosphorylated ZAP70 during

immune synapse formation. They found that all phosphorylation takes place in the pSMAC and concluded that the

cSMAC is not the site of T cell activation. With these observations in hand, studies began using higher resolution

microscopy to resolve the dynamics of TCR and pMHC at the siteof T cell activation: the pSMAC.

1.3 T cell receptor clusters in T cell activation

The concept that TCR clustering can facilitate T cell activation (57) existed for many years before T cell receptor

clusters were first observed (36). Over the past three years,experiments utilizing high resolution total internal

reflection fluorescence (TIRF) microscopy of the T cell membrane resting on a supported planar bilayer have

shown that TCR aggregate into sub-micron clusters that formin the pSMAC and are actively transported to the

cSMAC (31; 32; 33; 34; 35), see Figure 1.5.

EN FACE VIEW

T CELL

cSMAC

pSMAC

t = 0 - 60s t > 60s

TCR Agonist pMHC Signaling TCR Cluster Non-Signaling TCR Cluster

PLANAR BILAYER

Figure 1.5: Recent studies of the immune synapse reveal the existence of many TCR clusters. Interactions betweenTCR and pMHC are confined to the contact interface between theT cell and APC. Studies of the contact interfaceuse a surrogate APC, in the form of a planar bilayer, because the contact interface remains flat over long timescales.Within 60 s of contact the interface begins to accumulate with newly formed TCR clusters. After∼ 60s the clustersbegin to aggregate in the center of the interface (forming the cSMAC) by traveling along radially oriented actinfilaments. Generation of new signaling TCR clusters in the pSMAC and their subsequent translocation to thecSMAC continues for hours.

The emerging picture of TCR cluster dynamics is that they form very rapidly (∼ 5s) after T cell stimulation

with antigenic pMHC. After a brief initial period (≈ 60s), they begin to translocate along radially aligned actin

filaments towards the synapse center where they coalesce to form the cSMAC, see Figure 1.5. Generation of


new TCR clusters in the pSMAC and their subsequent translocation to the cSMAC continues for hours. Effective

TCR cluster velocities have been measured to be≈ 0.05µm/s (31; 34). Each TCR cluster contains≈ 50 − 150

TCRs (31; 32) and covers an area≈ 0.35 − 0.50µm2 (32). Therefore the concentration of TCR in a cluster is

= 100/0.35 ≈ 285µm−2 which is almost three times greater than typical concentration of TCR on resting T cells.

The spatial and temporal dynamics of the early signaling molecules shown in Figure 1.2 have also been in-

vestigated. Rapid localization of Lck, ZAP70, SLP76, and LAT to newly formed pSMAC TCR clusters has been

observed (31; 32). As with earlier studies, none of these signaling molecules were found in the cSMAC which is

presently believed to be the site of TCR internalization (33). In summary, TCRs form clusters in the periphery of

the immune synapse (pSMAC) in response to agonist pMHC. Signaling relevant to T cell activation takes place

within the peripheral TCR clusters during their journey to the synapse center (cSMAC), where signaling is extin-

guished and TCRs are internalized.

Chapter 2 examines the degree of serial engagement in the context of TCR clustering. I find that TCR/pMHC

binding alone does not allow for substantial serial engagement of TCR and that the pMHC molecules are usually

not transported to the center of the contact region by a single TCR cluster. I show that the presence of TCR core-

ceptors such as CD4 and CD8, or pMHC dimerization on the APC, can substantially increase serial engagement

and directed transport of pMHC by a single TCR cluster. Finally, I analyze effects of multiple TCR clusters.

Chapter 3 takes an alternative view of serial engagement. Byconsidering the stochastic nature of TCR/pMHC

interactions, I suggest that serial engagement is used by the T cell to sample the reaction parameters of pMHC.

Integrating multiple binding events allows the T cell to accurately probe the reaction parameters of a pMHC and

hence discriminate between pMHC. I further show that TCR clustering is essential to this process, in part, by

allowing for rapid and localized binding events.

1.4 Modeling relies on parameter estimates

Mathematical modeling of biological processes is heavily dependent on accurate estimates of various parameters.

Typical parameters used in biological modeling are diffusion coefficients and reaction rates. Specific to T cell

biology, models of T cell activation and immune synapse formation typically rely on estimates of diffusion coef-

ficients for TCR and pMHC, reaction rates between TCR and pMHC, and effective reaction rates between TCRs

and the actin cytoskeleton.

1.4.1 FRAP experiments reveal protein diffusion coefficients

A common tool used to investigate protein mobility on the cell membrane is Fluorescence Recovery After Pho-

tobleaching (FRAP), see Figure 1.6. In this experimental technique, a fluorescent marker is attached to a protein

of interest. When viewed under a confocal microscope, a homogenous distribution of fluorescence is observed at

the cell membrane (t < 0). High laser intensity in a confined region bleaches (denatures) the fluorescent marker,

reducing the fluorescence to background levels (t = 0). Over time, the fluorescence intensity in the bleached

region recovers due to motion of unbleached proteins (t > 0). Using image analysis software one can integrate

the total fluorescence from the bleached region over time to obtain the iconographic FRAP recovery curve. To

obtain an estimate of the diffusion coefficient, a mathematical solution of the diffusion equation that captures the

experimental protocol (e.g. the shape and dimensions of thebleach region) is fit to the FRAP recovery curve.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

Time (s)N

orm

aliz

ed

Flu

ore

sce

nce

In

ten

sity

D=0.005 µm2/s

D=0.05 µm2/s

D=0.5 µm2/s

t < 0 t = 0 t > 0 Large t

Figure 1.6: Schematic representation of Fluorescence Recovery After Photobleaching (FRAP) experiments. Be-fore the experiment (t < 0), a uniform distribution of fluorescence is observed indicating that the fluorescentlylabeled protein is homogenously distributed on the cell membrane. Att = 0 high laser intensity is applied toa small localized region of the cell membrane, termed the bleached region (white rectangle), bringing the fluo-rescence intensity to background levels. Over time,t > 0, the fluorescence intensity recovers and returns to itsoriginal level after a long time (larget). The FRAP recovery curve is obtained by integrating the fluorescenceintensity in the bleached region over time. The speed of recovery depends, in part, on the diffusion coefficient.

Chapter 4 shows that common mathematical models and experimental protocols used for estimating diffusion

coefficients for cell surface proteins give rise to appreciable errors. I propose an improved protocol that utilizes

a small bleach region on the cell membrane and provide the relevant solution of the two-dimensional diffusion

equation required for fitting. This FRAP protocol is used in the theoretical study presented in Chapter 5 and in

experiments probing TCR mobility in Chapter 6.

1.4.2 FRAP can determine binding constants

In recent years the use of FRAP has been extended to study the binding properties of proteins in vivo (58; 59; 60;

61). For example, Sprague et al. (58) used FRAP to study the binding kinetics of diffusing glucocorticoid receptor

(in the nucleus) binding to an immobile substrate. The basictenet of these experiments is that FRAP recovery will


be slower in the presence of binding reactions. Having an independent estimate of the free diffusion coefficient,

one can determine whether the FRAP recovery curve for the protein of interest is consistent with free diffusion or

if binding reactions need to be invoked to explain slower recovery compared to free diffusion.

Inspired, in part, by the work of Sprague et al. (58), in Chapter 4 we show how FRAP can be used to determine

the reaction parameters governing the TCR/pMHC bond. Briefly, we propose a set of FRAP experiments in the

immune synapse formed in response to various concentrationof pMHC. We show how fitting the resulting set of

FRAP experiments to a mathematical model can uncover the reaction off-rate and the two-dimensional bimolec-

ular reaction on-rate. The importance of this protocol is underlined by the fact that the majority of experiments

characterize TCR/pMHC reaction rates in solution and not inthe context of the plasma membrane (2; 62).

Over the past decade there has been mounting work suggestingthat cell surface proteins can be regulated by

the actin cytoskeleton (63; 64; 65; 66; 67). In the T cell system, several groups have shown that TCR can undergo

directed motion by the cytoskeleton (32; 33; 34; 68). The study of Kaizuka et al. (34) provided indirect evidence

that TCRs undergo dynamic reactions with the cytoskeleton by comparing the directed velocity of TCR towards

the cSMAC and the rate of actin treadmilling. They showed that the actin treadmilling rate is twice as fast as the

TCR velocity, a result implying that TCRs are not constantlybound to actin filaments but cycle between bound

and unbound states. Mathematical models attempting to explain immune synapse formation have assumed that

TCR can undergo mass action reactions with the cytoskeleton(52; 54). However, effective reaction rates between

the TCR and the cytoskeleton have not been reported.

In Chapter 6 we establish a role for intracellular calcium inregulating the interaction between surface TCR

and the cytoskeleton. Using the FRAP protocol developed in Chapter 4, in addition to other experimental tools,

we show that TCR mobility decreases upon T cell activation bythe cortical actin cytoskeleton. Using the binding

and fitting procedures developed in Chapters 4 and 5, we are able to extract the first estimates of the effective

reaction rates between TCR and the cytoskeleton.

1.5 Brief chapter summary

My thesis is composed of two major themes: serial engagementin T cell activation (Chapters 2 and 3) and pa-

rameter estimation using FRAP (Chapters 4, 5, and 6). A tableof abbreviations and a glossary of terms used

frequently throughout my thesis can be found in the prefactory pages (page xi).


Serial engagement in T cell activation

Chapter 2 Several models of T cell activation rely on serial engagement as a mechanism of signal ampli-

fication. Given recent experimental observations showing that signaling is confined to mobile TCR clusters, I

investigate the degree of serial engagement within TCR clusters.

Chapter 3 In this chapter I propose that serial engagements may explain the specificity of T cells. I propose

that multiple TCR engagements allow the T cell to sample the biochemical parameters of a given pMHC and

hence discriminate between pMHC. I show that TCR clusteringis required, in part, to achieve rapid and localized

binding events.

Parameter estimation using FRAP

Chapter 4 In this chapter I investigate the use of FRAP to determine diffusion coefficients for cell surface

proteins. I find that established methods give rise to appreciable errors. I propose an alternative protocol and

provide the equation required for FRAP fitting which minimizes errors for cell surface FRAP experiments.

Chapter 5 Motivated by the lack of experimental methods to determine reaction parameters between mem-

brane proteins, I investigated how FRAP experiments can be used to uncover these parameters. I propose an

experimental protocol and provide the mathematical analysis required to fit experimental data. Using simulated

data I confirm the utility of the method.

Chapter 6 In collaboration with experimental laboratory of Salvatore Valitutti (INSERM U563, Toulouse,

France), we use a set of experiments to show that TCR mobilityis modulated by the actin cytoskeleton. Using the

FRAP protocol developed in Chapter 4, I fit experimental FRAPrecovery data for TCR when T cells are treated

with various pharmacological agents. Using methods similar to those described in Chapter 5, I am able to estimate

effective reaction parameters between the TCR and the actincytoskeleton.

14

References

[1] Janeway, C. A. 2005. Immunobiology: the immune system inhealth and disease, 6th Ed. Garland Science

Publishing.

[2] Davis, S. J., S. Ikemizu, E. J. Evans, L. Fugger, T. R. Bakker, and P. A. van der Merwe. 2003. The nature of

molecular recognition by T cells.Nat Immunol. 4:217–224.

[3] Choudhuri, K., A. Kearney, T. R. Bakker, and P. A. van der Merwe. 2005. Immunology: how do T cells

recognize antigen?Curr Biol. 15:R382–R385.

[4] Heath, W. R., and F. R. Carbone. 2001. Cross-presentation, dendritic cells, tolerance and immunity.Annu

Rev Immunol. 19:47–64.

[5] Ackerman, A. L., and P. Cresswell. 2004. Cellular mechanisms governing cross-presentation of exogenous

antigens.Nat Immunol. 5:678–684.

[6] Irvine, D. J., M. A. Purbhoo, M. Krogsgaard, and M. M. Davis. 2002. Direct observation of ligand recogni-

tion by T cells.Nature. 419:845–849.

[7] Demotz, S., H. M. Grey, and A. Sette. 1990. The minimal number of class II MHC-antigen complexes

needed for T cell activation.Science. 249:1028–1030.

[8] Harding, C. V., and E. R. Unanue. 1990. Quantitation of antigen-presenting cell MHC class II/peptide

complexes necessary for T-cell stimulation.Nature. 346:574–576.

[9] Sykulev, Y., M. Joo, I. Vturina, T. J. Tsomides, and H. N. Eisen. 1996. Evidence that a single peptide-MHC

complex on a target cell can elicit a cytolytic T cell response. Immunity. 4:565–571.

[10] Krogsgaard, M., and M. M. Davis. 2005. How t cells ’see’ antigen.Nat Immunol. 6:239–245.

[11] Lin, J., and A. Weiss. 2001. T cell receptor signalling.J Cell Sci. 114:243–244.

[12] Wyer, J., B. Willcox, G. Gao, U. Gerth, S. Davis, J. Bell,P. van der Merwe, and B. Jakobsen. 1999. T

cell receptor and coreceptor CD8 alphaalpha bind peptide-MHC independently and with distinct kinetics.

Immunity. 10:219–25.


[13] Xiong, Y., P. Kern, H. Chang, and E. Reinherz. 2001. T Cell Receptor Binding to a pMHCII Ligand Is

Kinetically Distinct from and Independent of CD4.J Biol Chem. 276:5659–67.

[14] Holdorf, A., K. Lee, W. Burack, P. Allen, and A. Shaw. 2002. Regulation of Lck activity by CD4 and CD28

in the immunological synapse.Nat Immunol. 3:259–64.

[15] Garcia, K., C. Scott, A. Brunmark, F. Carbone, P. Peterson, I. Wilson, and L. Teyton. 1996. CD8 enhances

formation of stable T-cell receptor/MHC class I molecule complexes.Nature. 384:577–81.

[16] Li, Q., A. Dinner, S. Qi, D. Irvine, J. Huppa, M. Davis, and A. Chakraborty. 2004. CD4 enhances T cell

sensitivity to antigen by coordinating Lck accumulation atthe immunological synapse.Nat Immunol. 5:791–

9.

[17] Huse, M., L. Klein, A. Girvin, J. Faraj, Q. Li, M. Kuhns, and M. Davis. 2007. Spatial and temporal dynamics

of T cell receptor signaling with a photoactivatable agonist. Immunity. 27:76–88.

[18] Sykulev, Y., R. J. Cohen, and H. N. Eisen. 1995. The law ofmass action governs antigen-stimulated cytolytic

activity of CD8+ cytotoxic T lymphocytes.Proc Natl Acad Sci U S A. 92:11990–11992.

[19] Krogsgaard, M., N. Prado, E. J. Adams, X. lin He, D.-C. Chow, D. B. Wilson, K. C. Garcia, and M. M.

Davis. 2003. Evidence that structural rearrangements and/or flexibility during TCR binding can contribute

to T cell activation.Mol Cell. 12:1367–1378.

[20] Holler, P. D., and D. M. Kranz. 2003. Quantitative analysis of the contribution of TCR/pepMHC affinity and

CD8 to T cell activation.Immunity. 18:255–264.

[21] Grakoui, A., S. K. Bromley, C. Sumen, M. M. Davis, A. S. Shaw, P. M. Allen, and M. L. Dustin. 1999. The

immunological synapse: a molecular machine controlling T cell activation.Science. 285:221–227.

[22] Tian, S., R. Maile, E. J. Collins, and J. A. Frelinger. 2007. CD8+ T cell activation is governed by TCR-

peptide/MHC affinity, not dissociation rate.J Immunol. 179:2952–2960.

[23] Davis, M., M. Krogsgaard, J. Huppa, C. Sumen, M. Purbhoo, D. Irvine, L. Wu, and L. Ehrlich. 2003.

Dynamics of cell surface molecules during T cell recognition. Annu Rev Biochem. 72:717–42.

[24] Wade, W. F., J. H. Freed, and M. Edidin. 1989. Translational diffusion of class II major histocompatibility

complex molecules is constrained by their cytoplasmic domains. J Cell Biol. 109:3325–3331.

[25] Edidin, M., S. C. Kuo, and M. P. Sheetz. 1991. Lateral movements of membrane glycoproteins restricted by

dynamic cytoplasmic barriers.Science. 254:1379–1382.


[26] Munnelly, H. M., C. J. Brady, G. M. Hagen, W. F. Wade, D. A.Roess, and B. G. Barisas. 2000. Rotational

and lateral dynamics of i-a(k) molecules expressing cytoplasmic truncations.Int Immunol. 12:1319–1328.

[27] Favier, B., N. J. Burroughs, L. Wedderburn, and S. Valitutti. 2001. TCR dynamics on the surface of living T

cells. Int Immunol. 13:1525–1532.

[28] Saxton, M., and K. Jacobson. 1997. Single-particle tracking: applications to membrane dynamics.Annu

Rev Biophys Biomol Struct. 26:373–99.

[29] Norcross, M. A. 1984. A synaptic basis for T-lymphocyteactivation.Ann Immunol (Paris). 135D:113–134.

[30] Monks, C. R., B. A. Freiberg, H. Kupfer, N. Sciaky, and A.Kupfer. 1998. Three-dimensional segregation of

supramolecular activation clusters in T cells.Nature. 395:82–86.

[31] Yokosuka, T., K. Sakata-Sogawa, W. Kobayashi, M. Hiroshima, A. Hashimoto-Tane, M. Tokunaga, M. L.

Dustin, and T. Saito. 2005. Newly generated T cell receptor microclusters initiate and sustain T cell activation

by recruitment of Zap70 and SLP-76.Nat Immunol. 6:1253–1262.

[32] Campi, G., R. Varma, and M. L. Dustin. 2005. Actin and agonist MHC-peptide complex-dependent T cell

receptor microclusters as scaffolds for signaling.J Exp Med. 202:1031–1036.

[33] Varma, R., G. Campi, T. Yokosuka, T. Saito, and M. L. Dustin. 2006. T cell receptor-proximal signals

are sustained in peripheral microclusters and terminated in the central supramolecular activation cluster.

Immunity. 25:117–127.

[34] Kaizuka, Y., A. D. Douglass, R. Varma, M. L. Dustin, and R. D. Vale. 2007. Mechanisms for segregating T

cell receptor and adhesion molecules during immunologicalsynapse formation in jurkat T cells.Proc Natl

Acad Sci U S A. 104:20296–20301.

[35] DeMond, A. L., K. D. Mossman, T. Starr, M. L. Dustin, and J. T. Groves. 2008. T cell receptor microcluster

transport through molecular mazes reveals mechanism of translocation.Biophys J. 94:3286–3292.

[36] Bunnell, S., D. Hong, J. Kardon, T. Yamazaki, C. McGlade, V. Barr, and L. Samelson. 2002. T cell receptor

ligation induces the formation of dynamically regulated signaling assemblies.J Cell Biol. 158:1263–75.

[37] Dustin, M. L., and D. R. Colman. 2002. Neural and immunological synaptic relations.Science. 298:785–

789.

[38] Coombs, D., and B. Goldstein. 2004. Effects of the geometry of the immunological synapse on the delivery

of effector molecules.Biophys J. 87:2215–20.


[39] Valitutti, S., S. Mller, M. Cella, E. Padovan, and A. Lanzavecchia. 1995. Serial triggering of many T-cell

receptors by a few peptide-MHC complexes.Nature. 375:148–151.

[40] Wofsy, C., D. Coombs, and B. Goldstein. 2001. Calculations show substantial serial engagement of T cell

receptors.Biophys J. 80:606–612.

[41] McKeithan, T. 1995. Kinetic proofreading in T-cell receptor signal transduction.Proc Natl Acad Sci U S A.

92:5042–6.

[42] Hopfield, J. 1974. Kinetic proofreading: a new mechanism for reducing errors in biosynthetic processes

requiring high specificity.Proc Natl Acad Sci U S A. 71:4135–9.

[43] Coombs, D., A. M. Kalergis, S. G. Nathenson, C. Wofsy, and B. Goldstein. 2002. Activated TCRs remain

marked for internalization after dissociation from pMHC.Nat Immunol. 3:926–931.

[44] Chan, C., A. J. George, and J. Stark. 2001. Cooperative enhancement of specificity in a lattice of T cell

receptors.Proc Natl Acad Sci U S A. 98:5758–5763.

[45] Kalergis, A., N. Boucheron, M. Doucey, E. Palmieri, E. Goyarts, Z. Vegh, I. Luescher, and S. Nathenson.

2001. Efficient T cell activation requires an optimal dwell-time of interaction between the TCR and the

pMHC complex.Nat Immunol. 2:229–34.

[46] Andersen, P., C. Geisler, S. Buus, R. Mariuzza, and K. Karjalainen. 2001. Role of the T cell receptor ligand

affinity in T cell activation by bacterial superantigens.J Biol Chem. 276:33452–7.

[47] McMahan, R., J. McWilliams, K. Jordan, S. Dow, D. Wilson, and J. Slansky. 2006. Relating TCR-peptide-

MHC affinity to immunogenicity for the design of tumor vaccines.J Clin Invest. 116:2543–51.

[48] Boulter, J., N. Schmitz, A. Sewell, A. Godkin, M. Bachmann, and A. Gallimore. 2007. Potent T cell agonism

mediated by a very rapid TCR/pMHC interaction.Eur J Immunol. 37:798–806.

[49] Dustin, M., and A. Shaw. 1999. Costimulation: buildingan immunological synapse.Science. 283:649–50.

[50] Qi, S. Y., J. T. Groves, and A. K. Chakraborty. 2001. Synaptic pattern formation during cellular recognition.

Proc Natl Acad Sci U S A. 98:6548–6553.

[51] Coombs, D., M. Dembo, C. Wofsy, and B. Goldstein. 2004. Equilibrium thermodynamics of cell-cell adhe-

sion mediated by multiple ligand-receptor pairs.Biophys J. 86:1408–1423.

[52] Burroughs, N. J., and C. Wlfing. 2002. Differential segregation in a cell-cell contact interface: the dynamics

of the immunological synapse.Biophys J. 83:1784–1796.


[53] Weikl, T., and R. Lipowsky. 2004. Pattern formation during T-cell adhesion.Biophys J. 87:3665–78.

[54] Figge, M., and M. Meyer-Hermann. 2006. Geometrically repatterned immunological synapses uncover

formation mechanisms.PLoS Comput Biol. 2:e171.

[55] Krummel, M. F., M. D. Sjaastad, C. Wlfing, and M. M. Davis.2000. Differential clustering of CD4 and

CD3zeta during T cell recognition.Science. 289:1349–1352.

[56] Lee, K.-H., A. D. Holdorf, M. L. Dustin, A. C. Chan, P. M. Allen, and A. S. Shaw. 2002. T cell receptor

signaling precedes immunological synapse formation.Science. 295:1539–1542.

[57] Germain, R. 1997. T-cell signaling: the importance of receptor clustering.Curr Biol. 7:R640–4.

[58] Sprague, B. L., R. L. Pego, D. A. Stavreva, and J. G. McNally. 2004. Analysis of binding reactions by

fluorescence recovery after photobleaching.Biophys J. 86:3473–3495.

[59] Sprague, B. L., F. Mller, R. L. Pego, P. M. Bungay, D. A. Stavreva, and J. G. McNally. 2006. Analysis of

binding at a single spatially localized cluster of binding sites by fluorescence recovery after photobleaching.

Biophys J. 91:1169–1191.

[60] Braga, J., J. G. McNally, and M. Carmo-Fonseca. 2007. A reaction-diffusion model to study RNA motion

by quantitative fluorescence recovery after photobleaching. Biophys J. 92:2694–2703.

[61] Mueller, F., P. Wach, and J. G. McNally. 2008. Evidence for a common mode of transcription factor in-

teraction with chromatin as revealed by improved quantitative fluorescence recovery after photobleaching.

Biophys J. 94:3323–3339.

[62] Dustin, M., S. Bromley, M. Davis, and C. Zhu. 2001. Identification of self through two-dimensional chem-

istry and synapses.Annu Rev Cell Dev Biol. 17:133–57.

[63] Wulfing, C., and M. Davis. 1998. A receptor/cytoskeletal movement triggered by costimulation during T

cell activation.Science. 282:2266–9.

[64] Kusumi, A., K. Suzuki, and K. Koyasako. 1999. Mobility and cytoskeletal interactions of cell adhesion

receptors.Curr Opin Cell Biol. 11:582–90.

[65] Dustin, M., and J. Cooper. 2000. The immunological synapse and the actin cytoskeleton: molecular hardware

for T cell signaling.Nat Immunol. 1:23–9.

[66] Liu, S., D. Calderwood, and M. Ginsberg. 2000. Integrincytoplasmic domain-binding proteins.J Cell Sci.

113 ( Pt 20):3563–71.


[67] Cairo, C., R. Mirchev, and D. Golan. 2006. Cytoskeletalregulation couples LFA-1 conformational changes

to receptor lateral mobility and clustering.Immunity. 25:297–308.

[68] Moss, W., D. Irvine, M. Davis, and M. Krummel. 2002. Quantifying signaling-induced reorientation of T

cell receptors during immunological synapse formation.Proc Natl Acad Sci U S A. 99:15024–9.

20

Chapter 2

Analysis of serial engagement and

peptide-MHC transport in T cell receptor

microclusters1

2.1 Introduction

T cells play a central role in adaptive immunity by regulating immune responses and performing targeted killing

of infected cells. In order for T cells to carry out these functions they must be stimulated by antigen-presenting

cells (APC) bearing cognate antigen. Stimulation is mediated by interactions between T cell receptors (TCR) and

specific antigen presented in the form of peptide-major-histocompatibility complexes (pMHC), in a tight adhesion

region between the T cell and APC (69; 70). In this region, known as the immunological synapse (IS), fewer than

10 agonist pMHC molecules have been shown to transduce sufficient intracellular signaling to cause measurable

stimulation of T cells (71; 72; 73). Further, the TCR-pMHC bond is weak, with solutionKD usually in theµM

range (74).

A partial explanation of the sensitivity of T cells to such weak stimuli was proposed by Valitutti et al. (75).

TCR downregulation was measured over the course of several hours of T cell interaction with APC carrying a

known number of antigenic pMHC. The ratio of downregulated TCR to the number of pMHC was found to be

as high as 200. Assuming that every internalized TCR has previously bound pMHC, this suggests that pMHC

sequentially bind hundreds of TCR in the IS. This is known as the “serial engagement” hypothesis. Similar re-

sults were found by Itoh et al. (76) but the notion of serial engagement was weakened by findings that TCR

that have never bound antigenic pMHC can be internalized in apMHC-dependent manner (77). Further inves-

tigation revealed that when few pMHC are present on the APC, TCR are downregulated so rapidly that a strict

serial engagement model could not fit the data (78; 79). Rather, a model allowing for downregulation of nearby,

1A version of this chapter has been published. Dushek O, Coombs D (2008) Analysis of serial engagement and peptide-MHC transport inT cell receptor microclusters.Biophys J. 94:3447-3460

Chapter 2. pMHC dynamics in TCR microclusters 21

unstimulated TCR, in a TCR-density-dependent manner, provided a more plausible explanation (79). The impor-

tance of serial engagement of TCR for T cell activation has been further challenged by data of Holler and Kranz

(80) which did not show a decline in T cell activation by high affinity pMHC. We re-analyze parts of this data here.

We see that the rate of engagements (the “hitting rate”) between TCR and pMHC is of great importance. Using

a simple mathematical model, Wofsy et al. (81) calculated the hitting rate for a single pMHC moving diffusively

on the APC surface. Using measured parameters for the TCR-pMHC bond, it was found, for instance, that a single

agonist pMHC can engage 5-35 TCR during one sojourn in the IS.The highest rates of serial engagement were

achieved by those pMHC that bind TCR very transiently. On theother hand, fast-dissociating pMHC would lead

to only weak signaling of individual TCR. These considerations led to a model of overall pMHC signaling efficacy

based on the lifetime of the TCR-pMHC bond, where the optimalpMHC has a lifetime that is high enough for

reliable signaling of individual TCR, but low enough for serial engagement to proceed efficiently (75; 82; 83).

These predictions were confirmed experimentally and theoretically in a series of experimental and theoretical pa-

pers (84; 85; 86).

In the calculations of Wofsy et al. (81) it was assumed that TCR are distributed homogeneously within the

immune synapse. It has since been shown that TCR aggregate into half-micron sized clusters, often referred to

as TCR microclusters, in T cell synapses formed with both APC(87) and suspended planar bilayers (87; 88; 89)

bearing agonist pMHC. These TCR clusters form in the periphery of the immune synapse, a region known as the

peripheral supramolecular activation cluster (pSMAC), and migrate towards the center of the immune synapse,

(the central-SMAC (cSMAC)) (87). Various molecules in the signaling cascade that begins with a pMHC inter-

acting with a TCR have been shown to localize in pSMAC clusters and not in the cSMAC (87; 88). TCR clusters

are observed to form when bilayer pMHC densities are as low as0.2µm−2 (89).

Weak TCR-pMHC interactions can also be stabilized by more complex binding reactions. For instance,

the cell-surface coreceptors CD4 and CD8 (90) are thought tobind the nonpolymorphic regions of the pMHC

molecule (91), strengthening the TCR-pMHC interaction. The cytoplasmic tails of CD8 and CD4 are also associ-

ated with Lck, a molecule known to interact with the cytoplasmic tail of engaged TCR, perhaps further stabilizing

the complex (92). In a similar vein, pMHC are thought to form multimeric complexes that may allow for stabi-

lization of bonds formed by the individual pMHC (further discussion below).

In this paper we revisit the calculations of Wofsy et al. (81)in the context of moving TCR clusters and stabi-

lization of the TCR-pMHC bond by coreceptors and pMHC dimerization. We show that serial engagement, in the


context of mobile clusters, is negligible unless the TCR-pMHC bond is stabilized by coreceptors and/or pMHC

dimerization, in which case substantial serial engagementof TCR by a single pMHC molecule will occur. We

also show that stabilization is necessary for pMHC transport to the cSMAC within a single cluster and that pMHC

transport to the cSMAC is well correlated with T cell stimulation. Finally, we investigate effects of multiple TCR

clusters. In the case of immobile TCR clusters we are able to obtain an analytical solution while in the case of

mobile TCR clusters we perform a full Monte Carlo simulationof an immune synapse containing multiple mobile

TCR clusters.

2.2 Mathematical model

We derive a model for the escape time of a pMHC molecule diffusing within the immunological synapse and

potentially binding and unbinding from TCR. We do not resolve individual TCR in this model, nor do we con-

sider TCR signaling following pMHC engagement of TCR. Principally, we are concerned with understanding

how physical parameters of the TCR-pMHC interaction are related to pMHC motion within the immune synapse

and serial engagement of TCR by pMHC. TCR signaling and more complex effects within this model could be

addressed using an agent-based approach (93).

2.2.1 Four state escape time formulation from a mobile TCR cluster

We want to calculate the length of time that a pMHC is expectedto remain within a TCR cluster before escaping.

We model a particle (i.e. a pMHC molecule) advecting and diffusing in a 2D membrane. The particle transitions

between four states (i=A, B, C and D) by a Markov process with first order reactions between the states, as shown

in Figure 2.1. We define the escape time from some domain, for aparticle in state i at position(x, y), to beti(x, y).

This is calculated for a diffusing particle (without reactions) by solving the Poisson equation on the region of

interest using Dirichlet (zero) boundary conditions (see Appendix A of (94) for a heuristic derivation based on a

two-dimensional random walk). Following the procedure in (81; 94), we derive a system of equations governing


A B

D C

λ-1

λ+1

λ+3

λ-3

λ+2

λ-2

λ-4

λ+4

Figure 2.1: The most general reaction scheme we consider. Transitions between the four states occur with firstorder reactions, with transition rates given byλ±i. Detailed balance (microscopic reversibility) requires thatλ1λ2λ3λ4 = λ−1λ−2λ−3λ−4. Specific reaction schemes are illustrated in Figure 2.2.

theti(x, y):

DA∇2tA − ~VA · ∇tA + λ+1 (tB − tA) + λ−4 (tD − tA) = −1

DB∇2tB − ~VB · ∇tB + λ+2 (tC − tB) + λ−1 (tA − tB) = −1

DC∇2tC − ~VC · ∇tC + λ+3 (tD − tC) + λ−2 (tB − tC) = −1

DD∇2tD − ~VD · ∇tD + λ+4 (tA − tD) + λ−3 (tC − tD) = −1. (2.1)

Di are the diffusion coefficients of the particle in each state,~Vi are the advection velocity vectors,λi are the

transition rates, and we setti = 0 on the domain boundary.

Although we will apply the general reaction scheme illustrated in Figure 2.1 to several scenarios, we will

always identify statei = A with a free pMHC on the APC or supported planar bilayer and statesi = B,C,D

with pMHC bound to molecule(s) on the T cell (e.g. TCR, CD4, CD8), see Figure 2.2. Since the pMHC molecule

undergoes directed motion towards the cSMAC only when boundto molecules within a cluster, we can make the

equations simpler by working in the cluster reference frame(i.e. ~VA = −~V and~VB = ~VC = ~VD = 0) and picking

the cluster velocity vector along a coordinate axis~V = (V, 0). We further assume that, when bound, the pMHC

has a negligible rate of diffusion (i.e.DB = DC = DD = 0), to find

DP∇2tA + V ∂tA/∂x+ λ+1 (tB − tA) + λ−4 (tD − tA) = −1

λ+2 (tC − tB) + λ−1 (tA − tB) = −1

λ+3 (tD − tC) + λ−2 (tB − tC) = −1

λ+4 (tA − tD) + λ−3 (tC − tD) = −1 (2.2)

where we setDA = DP , the free pMHC diffusion coefficient. We use the last three equations to solve fortB and

tD in terms oftA and so obtain a single equation fortA :

DP∇2tA + V∂tA∂x

= −β. (2.3)

β = 1 + Λ1 + Λ1Λ2 + Λ1Λ2Λ3, whereΛi = λi/λ−i are the transition affinities. We eliminatedΛ4 from β using

the principle of detailed balance (microscopic reversibility) which requires thatΛ1Λ2Λ3Λ4 = 1. For simplicity

we will assume that the cluster is a square of sideb.

We formulate the solution to Equation 2.3 astA(x, y) = βΦ(x, y), with

Φ(x, y) =∞∑

n=1

qn(x)sin(nπy

b

)

. (2.4)

After substituting in, we obtain

qn(x) =2b2 ((−1)n − 1)

DP (nπ)3

[

exp

(−x (V + wn)

2DP

)

− 1

+exp

(

V (b− x)

2DP

)(

1 − exp

(−b(V + wn)

2DP

))

sinh

(

xwn

2DP

)

/sinh

(

bwn

2DP

)]

(2.5)

where wn =√

V 2 + 4D2P (nπ/b)2. We note that the escape time is only dependent on transitionaffinities,

through the parameterβ, and not on the individual forward and backward reaction rates. We obtain the mean

escape time by averaging over all possible starting positions of the pMHC,

< t >= β < Φ(x, y;V,DP , b) >=β

b2

∫ b

0

∫ b

0

Φ(x, y;V,DP , b)dxdy. (2.6)

This averaging gives us a simple measure of the time a pMHC will spend in a single cluster, allowing us to easily

understand the effects of changing the parameters of the TCR-pMHC interaction. Finally, the mean distance a

pMHC molecule is transported by the cluster is then< L >= V < t >.

2.2.2 Mean escape time and reaction rates determine the number of hits

We now derive expressions for the number of transitions between each state in Figure 2.1 during a sojourn in

a TCR cluster. We begin by calculating the probability,Pi, of finding the particle in statei by considering the

transition matrix associated with Figure 2.1 augmented by the conservation equationPA + PB + PC + PD = 1,

−λ1 − λ−4 λ−1 0 λ4

λ1 −λ2 − λ−1 λ−2 0

0 λ2 −λ3 − λ−2 λ−3

λ−4 0 λ3 −λ4 − λ−3

1 1 1 1

PA

PB

PC

PD

=

0

0

0

0

1

The solution of this system is(PA , PB, PC, PD) = (1/β)(1,Λ1,Λ1Λ2,Λ1Λ2Λ3).

We can now use thePi to approximate the mean number of transitions between each state. As an example,

consider the mean number of transitions between statei = A and statei = B. The total amount of time the particle

spends in state A is< t > PA . The particle transitions out of state A into state B or D witha total rateλ1 + λ−4

and therefore the number of transitions is< t > PA(λ1 + λ−4). We can calculate how many of these transitions

were strictly to state B by noting that the probability of transitioning, given that a transition occurred, from A to B

isλ1/(λ1 +λ−4). Therefore, the number of transitions from A to B is[< t > PA(λ1 + λ−4)] [λ1/(λ1 + λ−4)] =

λ1 < t > PA . A similar calculation shows that hits(B → A)=hits(A → B). In this way we can estimate the

mean number of transitions (hits) along any arrow in the reaction network, while the particle (pMHC) stays in the

cluster:

hits(A B) = λ1 < t > PA , (2.7a)

hits(B C) = λ2 < t > PB, (2.7b)

hits(C D) = λ3 < t > PC, (2.7c)

hits(D A) = λ4 < t > PD. (2.7d)

2.3 Results

2.3.1 TCR/pMHC binding events in a mobile TCR cluster

We begin our study of the TCR/pMHC dynamics by focusing on a single TCR cluster that forms in the pSMAC

and travels towards the cSMAC. We first calculate an upper bound on the number of TCR a pMHC can hit (bind

to) during one visit to that cluster. Yokosuka et al. (87) experimentally measured the mean velocity of 81 TCR

clusters as a function of their initial formation location.They found a mean velocity ofV = 0.0249µm/s for pS-

MAC clusters traveling towards the cSMAC and found that these clusters migrated a maximum ofLmax = 4.5µm

towards the cSMAC (Figure S1 in their work). Therefore the maximum journey time istmax ≈ 181 s. If during

this time a pMHC molecule can continually engage TCR (i.e. the pMHC molecule cannot escape the cluster) and

the mean time for a free pMHC to bind a TCR is negligible (i.e. the reaction on-rate is very large) then the number

of TCR engagements, or ‘hits’, can be approximated by dividing tmax by the mean TCR-pMHC bond lifetime,

1/koff. Typical off-rates for agonist pMHC are 0.03-0.3 s−1 (69) and therefore the maximum number of possible

hits if the pMHC remains in the cluster for 181 s is in the rangeof ∼5-54. Parameters are summarized in Table 2.1.

This rough calculation suggests that, in principle, a single pMHC molecule can engage a substantial number

of TCR in a cluster before arriving at the cSMAC. However, thenumber of engagements can be significantly

reduced by considering the effects of pMHC diffusion, TCR cluster composition and mobility, and the finite re-

action on-rate. In the next section we will use our escape time formulation to quantify the decrease in the number

of hits when these effects are considered.

Table 2.1: Parameter Definitions and EstimationsParameter Description Reported parameter rangesAmc TCR microcluster area 0.35-0.5µm2 (88)b TCR microcluster dimension Amc=b2, b = 0.59 − 0.71µm or

Amc=πb2, b = 0.33 − 0.40µmR Synapse radius 5-6µm (69)NT TCR number in a microcluster 40-150 (87; 88)Tmc TCR concentration in microclusters 80-430µm−2

T0 TCR concentration outside microclusters 50µm−2 (87)Cmc Coreceptor concentration in microclusters Not knownKD TCR/pMHC dissociation constant 0.005-1500µM (see Table 2.2)KC

D Coreceptor/pMHC dissociation constant ∼ 200µM (95; 96)V TCR microcluster velocity 0.0249µm/s (87)DP pMHC diffusion coefficient 0.03µm2/s (81; 97)

Two state escape time formulation from a mobile TCR cluster

To investigate the behavior of a single pMHC within a mobile TCR cluster, we consider the pMHC to be in one

of two states; unbound (i=A) or bound (i=B) to a TCR; see Figure 2.2 (basic model). The reaction scheme is

A B, with a forward rateλ1, backward rateλ-1, and transition affinityΛ1 = λ1/λ−1.

We define the number of “hits” for this simple model to be the number of times (on average) the pMHC binds

a TCR during its time in the cluster. We compute the mean escape time, the pMHC transport distance, and the

number of hits by reducing the four state model described earlier to a two state model. This reduction is achieved

by settingΛ2 = Λ3 = Λ4 = 0 in Equations 2.6 and 2.7a. The number of hits in this two statemodel is then

hits(A B) =< t >λ1

1 + Λ1(2.8)

A B C D

B) Coreceptor Heterodimerization Model

A B

A) Basic Model

TCR pMHC Coreceptor

C) Dimer Model

DCBA

Agonist-Agonist pMHC /

Endogenous-Agonist pMHC

Figure 2.2: Reactions schemes we consider. (a) two state reaction scheme between single pMHC and TCR.Definition of states: A - pMHC is unbound from TCR, B - pMHC is bound to TCR. (b) inclusion of coreceptorsrequires a four state reaction scheme. Definition of states:A - pMHC is unbound from TCR and coreceptor, B- pMHC is bound only to TCR, C - pMHC is bound to both TCR and coreceptor, and D - pMHC is bound onlyto coreceptor. (c) inclusion of pMHC dimers also requires a four state reaction scheme. Definition of states: A -Both pMHC are unbound from TCR, B - Agonist is bound to TCR, C - Both pMHC are bound to TCR, and D -Endogenous pMHC (or the other Agonist pMHC, depending on thedimer) is bound to TCR.

We plot the mean escape time,tA , as a function of the initial position of the pMHC in Figure 2.3 for different

values ofV . We find that microcluster velocities< 0.1µm/s do not substantially affect the escape time.

Estimating the pMHC transport distance and total TCR engagements

Before we can determine the number of TCR engagements by a pMHC molecule in a cluster we need to estimate

parameters. Several recent studies have characterized TCRcluster composition and mobility (87; 88; 89). By

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

0

0.3

0.6

0.8

1.1

1.4

Hits

x (µm)

t A (

s)

V=0 µm/sV=0.05 µm/sV=0.1 µm/sV=0.5 µm/sV=1 µm/s

Figure 2.3: Residence time and total TCR hits in a TCR clusteras a function of the initial position. We plottA(x, y) as a function ofx for y = b/2 for several values of the microcluster velocityV . We use reactionparameters for the MCC88-103 peptide (Table 2):b = 0.59µm, Tmc = 286µm−2, DP = 0.03µm2/s, kon =0.0057µm2/s andkoff = 0.057s−1

comparing background and cluster anti-TCR-fab fluorescence intensity, Campi et al. (88) determined that 140

TCR are contained in a single cluster. Consistent with thesefindings, Yokosuka et al. (87) reported 40-150 CD3ζs

per cluster. Cluster area was observed to be 0.35-0.5µm2 (88). The concentration range of TCR in a cluster is

thenTmc = 80 − 430µm−2. We also take the diffusion coefficient of pMHC to beDP = 0.03µm2/s (81). These

parameters are summarized in Table 2.1.

We summarize experimentally determined reaction rates between various TCR and pMHC in Table 2.2. The

transition rates in our model are related to these using the relations,

λ1 = konTmc , λ−1 = koff (2.9)

wherekon is the 2D on-rate which can be related to the experimentally measured 3D on-rate using a confinement

length (see Table 2.2) (98; 99). The two-dimensional forward rate constantkon for the peptide MCC88-103 was

calculated from itskoff value and the two-dimensional dissociation constant determined in (69). For the other

peptides,kon was calculated by assuming that the proportionality constant was the same as for MCC99-103. In

relating the transition rates to the reaction rates in this way (equation 2.9) we have assumed that there is no com-

petition among pMHC for TCR. This assumption is reasonable because pMHC concentration is low (< 10µm−2)

compared to the cluster TCR concentration (> 100µm−2).

Table 2.2: Estimates of TCR hits and pMHC transport by a mobile TCR cluster< t >* < L >*

KD kon kon koff (s) (µm) Hits*

TCR pMHC (µM) (M−1s−1 ) (µm2s−1) (s−1) CR− CR+ CR− CR+ CR− CR+

Data from Grakoui et al. (69)2B4 MCC88-103 60.2 900 0.0057 0.057 12 121 0.30 3.0 0.67 6.72B4 T102S 238 1500 0.0095 0.36 3.5 35 0.087 0.87 1.1 112B4 T102G 1520 3400 0.022 5.0 0.91 9.2 0.023 0.23 2.5 253.L2 Hb64-76 12.1 5557 0.035 0.064 65 180 1.6 4.5 4.1 123.L2 N72T 9.90 15374 0.097 0.136 84 181 2.1 4.5 11 253.L2 N72I 14.9 16600 0.11 0.248 50 181 1.2 4.5 12 45Data from Krogsgaard et al. (100)2B4 102S 90 2240 0.014 0.20 8.7 87 0.22 2.2 1.7 172B4 PCC 32 1080 0.0068 0.035 23 181 0.58 4.5 0.80 6.22B4 MCC95-103 8.7 2200 0.014 0.019 86 181 2.1 4.5 1.6 3.42B4 K2 8.7 6670 0.042 0.058 86 181 2.1 4.5 4.9 102B4 K3 33 2120 0.013 0.071 23 181 0.56 4.5 1.6 12.62B4 K5 2.9 4900 0.031 0.014 181 181 4.5 4.5 2.5 2.5Data from Garcia et al. (101)172.10 MBP1-11 5.9 37200 0.24 0.219 126 181 3.1 4.5 28 391934.4 MBP1-11 31 5130 0.032 0.160 24 181 0.60 4.5 3.8 28Data from Willcox et al. (102)JM22z HLFA-A2 17 69000 0.44 1.2 43 181 1.1 4.5 51 215* We calculate the mean time the pMHC is expected to remain within the TCR cluster (< t >), the totaldistance the cluster transports the pMHC (< L >), and the number of TCR serially bound by pMHC (hits)as described in the main text in the absence (CR−) and prescence (CR+) of coreceptors. The 3D molarvalues are converted to 2D values using a confinement length of 0.262µm (81). Parameters:b = 0.59µm,Tmc = 100TCR/(0.59µm)2 = 286µm−2, Cmc = Tmc, KC

D = 200µM, V = 0.0249µm/s, andDP =0.03µm2/s.

We can now compute the mean escape time< t > (Equation 2.6,β = 1 + Λ1), pMHC transport distance

< L >, and total hits (using Equation 2.8). We summarize these three results for specific TCR and pMHC in the

the CR− columns of Table 2.2. We can draw two main conclusions: (1) pMHC interactions with mobile TCR in

a cluster are insufficient to consistently transport pMHC tothe cSMAC in a single journey (< L > < 4.5µm);

(2) most pMHC molecules engage less than 5 TCR in the cluster.However, exceptions to these conclusions exist.

For example, clusters comprised of 172.10 TCR transport MBP1-11 to the cSMAC and MBP1-11 can engage 40

TCR. There are also examples of substantial TCR engagement even though the pMHC is not transported to the

cSMAC (e.g. 172.10/MBP1-11, JM22z/HLFA-A2).

In constructing Table 2.2 we have imposed the total journey time, tmax = 181s, as an upper limit to< t >,

which subsequently imposes an upper bound on< L > and hits. We impose this upper bound because once in

the cSMAC, TCR begin to be internalized (89), an effect not accounted for in our model. We also focus on the

cluster journey because experiments have shown that pMHC dependent signaling through the TCR occurs during

the journey to the cSMAC and not in the cSMAC itself (87; 88; 89; 103).

We can also determine the required TCR/pMHC affinity to achieve pMHC transport to the cSMAC by a single

mobile cluster from Equation 2.6. Using the parameters given above, we calculate< Φ >= 0.41s (equation 2.4).

Setting< t >= tmax, we find that on average the pMHC remains in the microcluster for tmax providedβ > 442.2.

For a TCR concentration ofTmc = 286µm−2 this corresponds toKD < 0.65µm−2. The maximum 3D dis-

sociation constant that permits transport to the cSMAC, based on the TCR-pMHC interaction alone is therefore

K∗D = 4.1µM.

We note that estimates of TCR numbers in clusters and estimates of cluster size, discussed above, rely on opti-

cal fluorescence microscopy (87; 88). Since clusters cannotbe resolved by optical microscopy, the latter estimate

is probably an upper bound. We find that variations in clustersize do not have a significant effect on< t > and

hits, provided we fix the number of TCR per cluster, because a decrease (increase) in cluster size is proportionally

balanced by a larger (smaller) reaction on-rate.

2.3.2 Coreceptors augment TCR/pMHC interactions

There is a growing body of evidence suggesting that T cell activation by pMHC molecules is dependent upon

coreceptors CD4 (69; 73; 104; 105) or CD8 (90; 95; 106; 107). In cases where the pMHC concentration is low

(69; 73; 108) or the pMHC exhibits small affinity to TCR (107),T cells lacking coreceptors have been observed to

be less likely to form an immune synapse (69; 73), flux calcium(73; 108), proliferate (69), or secrete IL-2 (107).

How coreceptors facilitate the activation of T cells remains largely unknown, in part because the experimentally

determined affinity between CD4 or CD8 and MHC is very weak (KD ∼ 200µM) (95; 96) (reviewed in (109)).

CD4 and CD8 are known to associate with TCR via the signaling molecules Lck and ZAP-70 (110; 111), possibly

providing additional stabilization to the TCR-pMHC-coreceptor complex.

In this section we will show that coreceptors, although having weak binding to MHC, sufficiently augment the

TCR/pMHC interactions such that substantial TCR engagement and pMHC transport to the cSMAC are achieved.

Four state escape time formulation from a mobile TCR cluster

To determine the degree to which coreceptors augment the TCR/pMHC interaction we will need to consider the

full four state escape time problem. In this reaction scheme, the pMHC molecule can exist in four states: unbound

from both TCR and coreceptor (i = A), bound to TCR (i = B), bound to TCR and coreceptor (i = C), and

bound to a coreceptor (i = D), see Figure 2.2 (coreceptor model). There are two fundamental reactions in this


scheme: TCR/pMHC (transition rates:λ1,λ−1) and coreceptor/pMHC (transition rates:λ2,λ−2). We assume the

reactions occur independently and therefore identify the transitions between states C and D with those of A and

B (i.e. λ−3 = λ1, λ3 = λ−1) and transitions between states D and A with those of B and C (i.e. λ−4 = λ2,

λ4 = λ−2).

As before, the mean escape time is given by equation 2.6, withβ = (1 + Λ1)(1 + Λ2) in this case. We also

keep track of the total number of times the pMHC binds TCR or “hits” (in this case determined by summing

equations 2.7a and 2.7c. ThePi in equations 2.7 can be simplified to the form

PA

PB

PC

PD

=1

β

1

Λ1

Λ1Λ2

Λ2

. (2.10)

Coreceptors increase pMHC transport distance and total hits

The only additional parameter we introduce by including coreceptors is their transition affinity for the MHC

molecule,Λ2. This parameter is related to the equilibrium dissociationconstant as follows,

Λ2 = Cmc/KCD (2.11)

whereCmc is the coreceptor concentration in the cluster andKCD is the 2D dissociation constant between pMHC

and coreceptors. The 3D dissociation constant has been reported to be∼ 200µM between class I MHC and

CD8αα (95) and∼ 199µM between class II MHC and CD4 (96). As before, we convert 3D values to 2D values

using a confinement length (see Table 2.2).

The results when coreceptors are incorporated into the TCR/pMHC model are summarized in the CR+

columns of Table 2.2. We find that in most cases, the addition of coreceptors maximizes the mean escape time,

which results in the transport of pMHC to the cSMAC. Furthermore, pMHC molecules that exhibit few TCR

engagements in the absence of coreceptors are able to engagea substantial number of TCR in the presence of

coreceptors. These increases are observed despite the large dissociation constant for the coreceptor-MHC interac-

tion.

2.3.3 T cell stimulation is correlated to pMHC transport to cSMAC

The importance of coreceptors to T cell stimulation was highlighted in a study by Holler and Kranz (107). Briefly,

they measured IL-2 release following interactions betweenAPC bearing a particular pMHC and CD8− or CD8+

T cells. In separate experiments, they determined the binding constants for some of the TCR-pMHC pairs in

their experiments. They found that coreceptors were necessary for T cell stimulation only when the dissociation

constant for the TCR/pMHC bond roughly exceeded5µM. This is comparable to our previous estimate for the

minimal affinity required to retain pMHC within a TCR clusterfor travel to the cSMAC (described above) of

4.1µM.

In Table 2.3 we summarize the reaction rates for TCR/pMHC combinations used in their study and indicate

the particular experiments where coreceptors were required for T cell stimulation. In this table we also compute

< t >, < L >, and total hits in the absence (CR−) and presence (CR+) of coreceptors. We find a striking corre-

lation between the experimental determination of CD8 dependence and our theoretical determination of whether

the pMHC can be expected to travel to the cSMAC. We also find that, if T cell stimulation is independent of CD8,

the predicted number of hits is also independent of the presence of coreceptors, but the opposite is true when CD8

is required for T cell stimulation.

Table 2.3: CD8 augments low affinity TCR/pMHC interactions< t > < L >

KD kon kon koff (s) (µm) HitsTCR pMHC (µM) (M−1s−1) (µm2s−1) (s−1) CR− CR+ CR− CR+ CR− CR+

T cell stimulation is CD8 independent2C QL9/L 3.9 6300 0.040 0.025 181 181 4.5 4.5 4.5 4.5m6α QL9/L 0.0055 155000 0.98 0.0008 181 181 4.5 4.5 0.15 0.15m6α Y5/L 0.0051 115000 0.73 0.0006 181 181 4.5 4.5 0.11 0.11m6α M5/L 0.034 147000 0.93 0.005 181 181 4.5 4.5 0.90 0.90m6α H5/L 0.0778 NA - NA 181 181 4.5 4.5 - -m6α Q5/L 0.167 NA - NA 181 181 4.5 4.5 - -m67α SIYR/K 0.0159 277000 1.75 0.44 181 181 4.5 4.5 79 79T cell stimulation is CD8 dependent2C dEV8/K 84.1 2200 0.014 0.185 9.2 92 0.24 2.3 1.6 16m33α dEV8/K 38 NA - NA 20 181 0.49 4.5 - -m67α dEV8/K 6.57 86000 0.54 0.567 113 181 2.8 4.5 64 1022C SIYR/K 31.9 2300 0.015 0.075 23 181 0.58 4.5 1.7 13Data taken from Holler and Kranz (107). Calculations and parameters are described in Table 2.2.

We can write a simple formula to estimate the maximum pMHC-TCR dissociation constant,K∗D, that gives

pMHC transport to the cSMAC. Rearrangingβ = (1 + Λ1)(1 + Λ2) and substituting the physical parameters for


Λ1 andΛ2 we obtain,

K∗D =

Tmc

(

KCD + Cmc

)

βKCD −

(

KCD + Cmc

) (2.12)

We obtainK∗D = 6.65µm−2 (using parametersβ = 442.2,K

CD = 31.6µm−2, andCmc = Tmc = 286µm−2). The

3D value is thenK∗D = 42.1µM, an order of magnitude larger than in the case without coreceptors. Examining

Tables 2.2 and 2.3 we can see that many TCR/pMHC have a lowerKD than this, permitting pMHC transport and

maximal hits when coreceptors are present.

In our analysis of the effects of coreceptors on pMHC transport and TCR hits we have assumed equal concen-

trations of coreceptors and TCR. Lower concentration of coreceptors would decreaseK∗D. In Figure 2.4 we plot

KD versusKCD using Equation 2.12 forCmc = [Tmc, Tmc/2, Tmc/5, Tmc/10]. We find that at lower coreceptor con-

centration, a dissociation constant of∼ 200µM is too large to achieve pMHC transport to cSMAC. We conclude

that if coreceptors facilitate pMHC transport to the cSMAC,they must be present in clusters at concentrations

comparable to that of TCR.

20 40 60 80 100

200

400

600

800

1000

1200

KD

(µM)

KDC

(µ

M)

1:1

1:2

1:5

1:10

Figure 2.4: Maximum dissociation constant (KCD) of coreceptors required to achieve transport of agonist pMHC

to cSMAC. The dissociation constant between the agonist pMHC and TCR is given on the x-axis (KD). Resultsare shown for different ratios of coreceptor:TCR in the microcluster.

2.3.4 Potential effects of pMHC dimers on the APC

Biochemical assays have provided evidence that MHC class IImolecules form dimers (112; 113) and it is rea-

sonable to suppose that they may form dimers in experiments using supported bilayers or APC. pMHC dimers

have also been shown to be the minimal unit required for T cellactivation in an assay where soluble multimeric

pMHC complexes were used to stimulate T cells (114), although it is not clear to what extent this result informs

the physiological situation where binding occurs at a cell-cell interface. Furthermore the coreceptor and the TCR

that it associates with may bind different pMHC complexes (105; 115). This forms a “pseudodimer” model of

TCR triggering, proposed in part to explain the observationthat a single agonist pMHC complex can lead to TCR

triggering. It was suggested that, when a TCR with an associated coreceptor binds to an agonist-pMHC, the core-

ceptor binds a distinct self/null-pMHC complex. A TCR pseudodimer is formed when a second TCR binds to this

self-pMHC complex.

We will show that pMHC dimers can boost the effective affinityof their constituent pMHC for the cluster and

thus allow enhanced pMHC transport to the cSMAC. Therefore,higher-order complexes such as pseudo-dimers

will also allow enhanced transport.

To study the effects of pMHC dimers we use the full four state escape time formulation, see Figure 2.2 (dimer

model). This is exactly analogous to the coreceptor theory,with Λ2 in this case related to the second pMHC

molecule in the pMHC dimer,

Λ2 = Tmc/K(2)

D (2.13)

whereK(2)D is the dissociation constant for the TCR-second pMHC bond. In the case of homodimers (identical

presented peptide and MHC molecule), we ignore any cooperative effects and setK(2)D = KD. In the case of

heterodimers,K(2)D will be different fromKD.

As in the previous cases, we find a simple relationship to determine the maximum dissociation constant,K∗D,

required for pMHC transport to the cSMAC. Using the definition of β we find

K∗D =

Tmc

(

K(2)

D + Tmc

)

βK(2)

D −(

K(2)

D + Tmc

) . (2.14)

This equation is equivalent to Equation 2.12 whenCmc = Tmc and is shown in Figure 2.4 (1:1 case). In the case

of pMHC homodimers, we find that forKD < 80µM the dimer will be transported to the cSMAC. Consequently,

transport to the cSMAC is expected for almost all homodimersof pMHC listed in Tables 2.2 and 2.3.

The observation that endogenous peptides accumulate in theIS (108) and that heterodimers of agonist/endogenous

pMHC stimulate T cells (115) suggest that endogenous peptides may play a role in pMHC transport to the cS-

MAC and in serial triggering. Endogenous peptides generally have an undetectable affinity for TCR and therefore

have a dissociation constant> 200µM. Equation 2.14 (shown in Figure 2.4, 1:1 case)) indicates that endoge-


nous peptides are able to transport the heterodimer only when the dissociation constant of the agonist peptide is

stronger than∼ 40µM. The physical reason for such transport is that the endogenous pMHC is able to anchors

the heterodimer to clustered TCR and therefore increase thetransport distance. In cases when the agonist pMHC

dissociation constant is> 40µM, we predict that the addition of endogenous peptides alonewill not sufficiently

augment the interaction to result in pMHC transport to the cSMAC and substantial serial engagement of TCR.

2.3.5 TCR/pMHC binding events with multiple TCR clusters

In the previous sections we have shown that agonist pMHC can be expected to escape from TCR clusters during

the journey to the cSMAC unless the reaction is augmented by coreceptors or pMHC dimers. Substantial TCR

engagements and transport to the cSMAC could be possible in the absence of these factors if a single pMHC

molecule visits multiple TCR clusters whose collective motion sieve the pMHC molecules into the cSMAC.

2.3.6 Effects of multiple stationary clusters

We leave a complete examination of pMHC dynamics in a field of moving clusters for the next section, focusing

here on the question of how many engagements (hits) a pMHC would experience during a single sojourn in an IS

that holds a number of immobile clusters enriched in TCR as well as a background concentration of non-clustered

TCR. Varma et al. (89) generated such synapses by treating T cells with latrunculin A, a drug that prevents actin

polymerization, shortly after synapse formation. The result, shown in Figures 6F-6J of their work, is an immobile

and stable field of TCR clusters. In Figures 3C and 3D of their work they also illustrate the rapid (∼ 60s) reduction

in calcium signaling upon administering latrunculin A to T cells forming an IS. In order to form a clear picture of

the potential effects of multiple TCR clusters, in what follows we will not include coreceptors.

We reduce system 2.1 by settingλ±i = 0, for all i 6= 1, and removing the advective field by setting~Vi = 0,

for all i. We take the synapse to be a disc of radiusR containingN disc-shaped clusters centred at the points

r1, r2, ...rN and having a radius ofb. Each cluster containsNT TCR. These simplifications allow us to write a

single equation governing the escape timetsyn,

DP∇2tsyn = −1 − λ1

λ−1I(r) − λ∗1

λ−1(1 − I(r)) (2.15)

I(r ; r1, r2, ...rN ) =

1 | r − r j |≤ b

0 | r − r j |> b(2.16)

where the superscript, syn, indicates the escape time is from the entire synapse. The boundary condition istsyn = 0

when|r | = R. To account for the difference between background and clustered TCR concentrations we have split

the reaction term into two parts using the indicator function I, which is zero everywhere except within clusters

where it is equal to one. Therefore the term withλ1/λ−1, captures reactions within clusters having a forward

transition rateλ1. The term withλ∗1/λ−1 captures reactions outside of clusters with a forward rateλ∗1. The tran-

sition rates are related to the physical parameters by equation 2.9 and the relationλ∗1 = konT0 whereT0 is the

TCR concentration outside clusters, which we take to be50µm−2 (87). The TCR concentration within clusters is

NT /(πb2).

As discussed earlier, there is uncertainty in the amount of area covered by TCR in clusters. Consequently, in

the analysis that follows we fix the number of TCR per cluster,NT, and vary the cluster size,b. We can decompose

the solution to Equation 2.15 into three parts,tsyn = t1 + t2 + t3, satisfying

DP∇2t1 = −1, (2.17a)

DP∇2t2 = − λ1

λ−1I(r), (2.17b)

DP∇2t3 = − λ∗1λ−1

(1 − I(r)). (2.17c)

All the ti = 0 on the synapse boundary (i.e.ti(r = R) = 0). t1(r) = (R2 − r2)/(4D) is the escape time

from a synapse without any TCR, leading to the average time ofescape< t1 >= R2/8DP (averaged over all

possible starting positions).t2 is the time spent bound to clustered TCR, andt3 is the time spent bound to non-

clustered TCR. By linearity,t3 = (λ∗1/λ−1) t1−(λ∗1/λ1) t2. Substituting in for the physical parameters we obtain

t3 = (T0kon/koff)t1+(T0πb2/NT )t2. This shows thatt3 ≈ (T0kon/koff)t1 asb→ 0, in agreement with the theory

for a uniform distribution of TCR presented in (81). We can obtain the mean number of transitions in this case

by dividing the total time spent bound to TCR,t2 + t3, by the mean time per binding event,1/λ−1 (i.e. hits =

λ−1 < t2 + t3 >).

Weak dependence of escape time on cluster size

If b/R 1 then we can use matched asymptotics to obtaint2 as a power series in the small parameterε = b/R,

see Appendix A for detailed calculations. Averaging over all starting positions of the pMHC, we find, to first

order inε,

< t2 > =konNT

4πkoffDP

N∑

k=1

(

1 −∣

∣

∣

r k

R

∣

∣

∣

2)

(2.18)

< tsyn> =

(

1 +T0kon

koff

)

R2

8DP+

(

1 − T0πb2

NT

)

konNT

4πkoffDP

N∑

k=1

(

1 −∣

∣

∣

r k

R

∣

∣

∣

2)

(2.19)

We see that, to first order inε, < tsyn > depends only on physical parameters and the locations of theclusters

relative to the synapse boundary, and that centrally located clusters have the biggest impact on the mean time a

pMHC spends in the synapse.

We can also consider the case where we assume that the pMHC starts within a cluster. In this case, we average

over all possible starting positions within clusters to obtain the mean time spent bound to clustered TCR:

< tin2 > =NT kon

πkoffDP

1

8− log(b/R)

2+

1

N

N∑

j=1

Aj,0

(2.20)

where theAj,0 are constants that depend only on the cluster positions (seeAppendix A). This formulation reveals

the weak (logarithmic) dependence of escape time on TCR cluster size,b.

pMHC engagement of clustered TCR is nearly independent of cluster size

We begin by randomly placing TCR clusters within a synapse ofradius 5.5µm. In Figure 2.5a we show the

locations,r j , of N=5 (black), N=25 (black + dark grey), and N=50 (all discs) TCR clusters. In Figure 2.5b we plot

the total hits to clustered TCR (solid line) and to non-clustered TCR (dashed line) as a function ofb for the three

values of N. We obtain< t2 >, and hence the total hits, by using a central difference scheme for the Laplacian

in equation 2.17 whenb > 0.07µm. Whenb < 0.1µm we use the asymptotic solution, Equation 2.18, which

agrees with the numerical solution on the overlapping rangeof b. We take reaction rates for a typical TCR-pMHC

interaction:kon = 0.05µm2/s,koff = 0.05s−1.

We find that the total hits to clustered TCR increases withN but is independent of the cluster size. Indeed,

the asymptotic solution, Equation 2.18, is independent ofb. As b → 0, for all values ofN , the total hits to TCR

external to clusters asymptotes to the value it would have ifthere were no clusters, as it should. For larger values

of b, the total hits to non-clustered TCR decreases asN increases. The small changes in the number of hits at

largeb ' 1µm occur because, in the simulation, parts of the microclusters end up outside the idealized synapse

region.

(a) (b)

0 2 4 6 8 10 120

2

4

6

8

10

12

x (µm)

y (µ

m)

−2 −1.5 −1 −0.5 00

100

200

300

400

500

600

log10

(b) (µm)

<hits

>

N=5

N=5

N=25N=25

N=50

N=50

hits to MC TCRhits to non−MC TCR

(c) (d)

0 0.05 0.1 0.15 0.2 0.25200

250

300

350

400

450

500

550

600

total microcluster area / synapse area

<hits

>

hits to MC TCRhits to non−MC TCR

Figure 2.5: Number of pMHC engagements to clustered TCR is almost independent of cluster size. We computethe number of binding events (hits) in two scenarios. In (a) we use an idealized disc synapse of radius 5.5µmcontaining N randomly distributed TCR microclusters each of radiusb and containingNT = 100 TCR. Thepositions of N=5 (black), N=25 (black + dark grey), and N=50 (all discs) TCR clusters is shown. In (b) we usethe TCR microcluster distribution from a) to compute the total hits to clustered (solid line) and non-clustered(dashed line) TCR for the three values of N as a function of themicrocluster size,b. In (c) we use experimentaldata (Figure 6H in (89)) to obtain a physiological microcluster distribution, see main text for details. Also shownin (c) are three steps along the multiple erosion operationsperformed on the microcluster stencil. Using thesemicrocluster distributions we compute the total hits (d) asa function of the total microcluster area. Parameters:kon = 0.05µm2/s, koff = 0.05s−1,DP = 0.03µm2/s.

As discussed earlier, Varma et al. (89) created synapses containing immobile TCR clusters. In order to obtain

a physiological TCR microcluster distribution, we performed simple image analysis on figure 6H of their work.

We acquired a high resolution version of figure 6H and performed thresholding to convert the grayscale TIRF

image into a binary (black/white) image. Morphological open and close operations were performed to remove

isolated pixels. The resulting TCR microcluster stencil, shown in Figure 2.5c, was used as the indicator function

for a numerical solution of Equation 2.17. The synapse boundary, whereti = 0, was obtained by alternate thresh-

olding and multiple morphological open operations on the TIRF image (see black outline in Figure 2.5c).

To examine the effect of cluster size we use a morphological operation that removes pixels that are not sur-

rounded. Applying this morphological operation multiple times on the TCR cluster stencil progressively decreases


the total TCR cluster area. We show the TCR cluster stencil atthree instances in Figure 2.5c. We keep the to-

tal number of TCR in clusters fixed at 5000, equivalent to 100 TCR distributed across 50 clusters (N=50 in the

idealized synapse). We takekon, koff , andDP as in the idealized synapse calculation. In Figure 2.5d we plot the

total hits as a function of the microcluster to synapse area ratio (synapse area is fixed at85µm2). The total hits to

clustered TCR remains almost unchanged.

The main trend observed in Figure 2.5, that the number of hitsto clustered TCR is only weakly dependent on

cluster size, is a result of keeping the number of TCR per cluster constant. Decreases in cluster area are balanced

by increases in the reaction on-rate within clusters leaving the number of hits unchanged. The number of hits that

a single pMHC makes with clustered and non-clustered TCR during a single sojourn in the IS is substantial, even

without the aid of bond stabilization by coreceptors or dimerization.

2.3.7 Effects of multiple mobile clusters

The analysis above focused on a stationary (steady-state) distribution of signaling TCR clusters generated by treat-

ing T cells with a cytoskeleton poison. However, in healthy synapses new TCR clusters are continually formed

in the periphery of the synapse and signal as they migrate towards the cSMAC, where signaling is abolished

(87; 88; 89). The time-dependent positions of TCR clusters precludes the use of an escape time formulation to

study effects of multiple mobile TCR clusters. Instead, we use a Monte Carlo method to simulate the formation

of TCR clusters in the pSMAC, at a rate ofkmc, and their mobility towards the cSMAC. The simulation begins

with the pMHC unbound in a random location within a TCR cluster located 4.5µm away from the cSMAC. As

before, the pMHC undergoes free diffusion unless it binds tomolecules within a TCR cluster in which case it

undergoes directed motion. In contrast to the single TCR cluster calculations, once the pMHC escapes the cluster

it continues to diffuse and may subsequently bind to anotherTCR cluster. The simulation is terminated once the

pMHC has escaped the synapse or has entered the cSMAC. The simulation efficiently and accurately captures the

multiple time and space scales involved in the process. For clarity, the full description of the simulation is given

in Appendix B.

The only additional parameter introduced by incorporatingmultiple TCR clusters is their formation rate,kmc.

Assuming there is roughly a constant number of TCR clusters in the pSMAC (∼ 50) we expect the rate of cluster

formation to equal the rate of cluster arrival in the cSMAC. For an individual cluster, the latter quantity is simply

V/L = 0.0249µm/s/4.5µm= 0.0055s−1. Therefore,kmc ≈ (50)(0.0055) = 0.28s−1. Parameters related to the

geometry of the synapse (e.g. cSMAC radius, pSMAC radius, etc), which have been experimentally measured


(69; 87; 89), are discussed in Appendix B.

Few TCR clusters are required for pMHC transport to cSMAC and substantial TCR engagements

In Table 2.4 we summarize the results of the Monte Carlo simulations for all pMHC found in Table 2.2. In

contrast to the escape time calculations that focused on a single mobile TCR cluster, we find that pMHC have a

large probability,> 0.7, of entering the cSMAC when we consider multiple TCR clusters. For comparison, the

probability of cSMAC arrival in the absence of TCR clusters is ∼ 0.2, which can be computed by solving the

diffusion equation on an annulus with zero Dirichlet boundary conditions (not shown). We further find that an

individual pMHC may escape and subsequently enter> 5 TCR clusters. However, we find that re-entry into the

same TCR cluster occurs frequently. In fact, many pMHC listed in Table 2.4 have a large probability of cSMAC

arrival,> 0.9, but only utilize∼ 2 unique TCR clusters to achieve this large probability.

Table 2.4: TCR hits and pMHC transport by multiple mobile TCRclusterskon koff Prob. of Cluster Unique Total Hits per

TCR pMHC (µm2s−1) (s−1) Transporta Visitsb Visitsb Hits Clusterc

Data from Grakoui et al. (69)2B4 MCC88-103 0.0057 0.057 0.77 5.4 2.5 14.0 2.62B4 T102S 0.0095 0.36 0.59 17.5 4.6 49.4 2.82B4 T102G 0.022 5.0 0.23 33.5 5.1 109.5 3.33.L2 Hb64-76 0.035 0.064 0.93 4.1 1.8 20.4 5.03.L2 N72T 0.097 0.136 0.94 4.8 1.7 42.7 8.93.L2 N72I 0.11 0.248 0.93 7.9 2.1 65.9 8.3Data from Krogsgaard et al. (100)2B4 102S 0.014 0.20 0.74 11.8 3.4 39.2 3.32B4 PCC 0.0068 0.035 0.84 3.7 2.0 10.1 2.72B4 MCC95-103 0.014 0.019 0.93 2.2 1.4 8.2 3.72B4 K2 0.042 0.058 0.94 3.6 1.6 20.0 5.62B4 K3 0.013 0.071 0.85 5.6 2.3 18.3 3.32B4 K5 0.031 0.014 0.95 1.8 1.3 9.6 5.3Data from Garcia et al. (101)172.10 MBP1-11 0.24 0.219 0.96 5.2 1.8 81.1 15.61934.4 MBP1-11 0.032 0.160 0.88 8.1 2.4 37.2 4.6Data from Willcox et al. (102)JM22z HLFA-A2 0.44 1.2 0.91 14.6 2.4 275.9 18.9a The probability of transport refers to the fraction of the simulations that terminated with thepMHC transported to the cSMAC.b These values correspond to the number of clusters a pMHCmolecule visits before the simulation is terminated by pMHCarrival in the cSMAC or pMHCexiting the synapse. Unique visits refers to the number ofdifferent clusters the pMHC visits.c

Obtained by dividing the Total Hits by Cluster Visits. Reported values are averages over 1000simulations. Calculations and parameters are described inTable 2.2 and in the main text. SeeAppendix B for simulation details

We find that by visiting multiple TCR clusters, a single pMHC can engage a substantial number of TCRs,

see Total Hits in Table 2.4. However, we find that the average number of TCR hits per cluster visit is small. A


greater number of TCR hits per cluster visit can be achieved by the use of coreceptors, see Table 2.2. Therefore a

trade-off exists between the number of localized TCR engagements and the total number of TCR engagements.

pMHC discrimination by cSMAC arrival

A recent study found that B cell receptor (BCR) clustering and mobility can discriminate antigens in B cells. Fleire

et al. (116) observed the time-course of B cell synapse formation and found that antigens with certain reaction

rates to the BCR efficiently accumulated in the B cell cSMAC. Using Monte Carlo simulations that capture the

essential features of the experiments, they showed that antigen collection to the B cell cSMAC by BCR clusters is

dependent on the BCR-antigen affinity and proposed that thismechanism allows the B cell to discriminate antigen.

Motivated by this observation, we investigated the possibility that T cells can discriminate pMHC using mobile

TCR clusters. We used our Monte Carlo simulation, as described in the previous section, with the exception that

the initial position of pMHC was not confined to a TCR cluster.

In Figure 2.6a we plot the probability that the pMHC enters the cSMAC. For a given parameter set, we per-

formed the simulation 500 times and record the fraction of the simulations that terminated by pMHC arrival in

the cSMAC. As expected, we find that the probability of cSMAC arrival depends on the reaction rates. For high

affinity interactions we find that the probability saturatesto ∼ 0.9, reflecting the probability of finding a TCR

cluster. Indeed, in these simulations we find that a single TCR cluster visit is sufficient for cSMAC transport (data

not shown). For low affinity interactions we see that the probability of cSMAC arrival approaches∼ 0.2, which

reflects the random, diffusive, probability of entering thecSMAC. In these simulations we find that pMHC rarely

bind to TCR clusters (data not shown). Including the effectsof coreceptors, see Figure 2.6b, we find that the

probability of cSMAC arrival is nearly independent of the pMHC reaction rates because the pMHC remains in a

single TCR cluster (in agreement with the escape time calculations above). Therefore, pMHC discrimination by

the probability of cSMAC arrival is only possible in the absence of coreceptors.

We note that the results presented for the Monte Carlo model involving coreceptors apply equally well to

the model of agonist pMHC dimerization with an endogenous pMHC (Figure 2.2, dimer model) if we make the

assumptions that 1) the agonist pMHC diffuses as a dimer and 2) that the reaction rates between the endogenous

pMHC/TCR are similar to the rates between coreceptors and MHC.


(a) (b)

−4 −3 −2 −1 0 10

0.2

0.4

0.6

0.8

1

log10

(kon

) (µm2/s)

Pro

b. o

f cS

MA

C A

rriv

al

−4 −3 −2 −1 0 10

0.2

0.4

0.6

0.8

1

log10

(kon

) (µm2/s)

Pro

b. o

f cS

MA

C A

rriv

al

koff

=0.01 s−1

koff

=0.1 s−1

koff

=1 s−1

koff

=10 s−1

Figure 2.6: Probability of cSMAC arrival by pMHC depends onkon andkoff. Results are shown in the absence (a)and presence (b) of coreceptors. Legend for panel (a) is shown in panel (b). The probability of cSMAC arrival inthe absence of TCR clusters is computed to be 0.19 (dotted black line) using an analytical solution of the diffusionequation (not shown). Error bars indicated standard error in the mean. Simulation performed as described in themain text.

2.4 Discussion

Experimental observations show pMHC-dependent signalingin TCR clusters and the accumulation of pMHC in

the cSMAC. Using a series of mathematical models we have analyzed serial engagement of TCR and transport of

pMHC by mobile TCR clusters. We have shown that the TCR-pMHC interaction alone does not support substan-

tial serial engagement of TCR or pMHC transport to the cSMAC by a single cluster but that if the TCR-pMHC

bond can be stabilized (for instance, by coreceptor molecules such as CD4/8 or by dimerized pMHC, or by com-

bined effects thereof), transport to the cSMAC and serial engagement within a single cluster can be expected.

We have calculated minimum affinities of the TCR-pMHC bond that allow pMHC transport to the cSMAC to

proceed efficiently in each scenario. We found evidence thatcoreceptors CD4/8 must be present in concentrations

comparable to that of clustered TCR for pMHC transport in a coreceptor-dependent manner. Using experimental

data (107) we were able to correlate predicted pMHC transport to the cSMAC with T cell stimulation as measured

by IL-2 production. We also analyzed the role of multiple clusters in trapping pMHC in the synapse and boosting

serial engagement. Our conclusions are based on a number of modeling assumptions and suggest future directions

of experimental and theoretical enquiry. We discuss these in turn.

Parameter estimation Our results underline the importance of measuring the kinetic parameters for TCR-

pMHC bonds. The parameters which have the largest uncertainty in our model are probably the two-dimensional

dissociation constant,KD, and two-dimensional on-rate,kon. We obtained 2D values by converting their respec-

tive experimentally measured 3D values using a constant factor (confinement length) determined for a specific

TCR/pMHC interaction. In a review, Davis et al. (74) discussthe importance of directly determining 2D rates,

as they can be substantially different from their respective 3D values. Recently, we have proposed a method to

directly determine 2D affinities and on-rates using live cells based on fluorescence recovery after photobleaching

(FRAP) (O. Dushek, R. Das and D. Coombs, see Chapter 5).

In this paper we used the macroscopic or long range diffusioncoefficient for pMHC which can be substan-

tially smaller than the microscopic diffusion coefficient (117). Increases inDP would decrease our predictions

of < t >, pMHC transport distance and hits. We also remark that we have assumed that pMHC binding to TCR

creates TCR-pMHC complexes that cannot diffuse. Diffusionof these complexes within the cluster would also

decrease< t >. In the absence of other uncertainties concerning the binding parameters, these considerations

would mean that the values of< t > in Tables 2.2, 2.3, and 2.4 would be upper bounds. FRAP experiments could

indicate the extent to which diffusion constants change following TCR-pMHC binding.

Role of clusters in TCR signaling Our calculations of numbers of TCR-pMHC engagement (“hits”) in single

clusters are smaller than those calculated for a homogeneous TCR distribution (Wofsy et al. (81)) and a clus-

tered TCR distribution (this work) across the whole synapse. However, despite the relatively small numbers of

engagements, the spatial and temporal proximity of TCR binding events probably has a major impact on sig-

nal transduction, perhaps due to spatiotemporal localization of signaling proteins in the cluster. In this work we

looked only at physical stabilization of the TCR-pMHC bond by coreceptors and null pMHC. The presence of

the coreceptor (and associated Lck) at the TCR during pMHC binding has been shown, experimentally and the-

oretically, to boost TCR signaling (105). Furthermore, it has been shown that stronger agonists are able to take

better advantage of the pool of null or endogenous pMHC present on the APC (115). Further experimental and

theoretical analysis will refine our understanding of the costs and benefits of TCR clustering on antigen detection

and effector function.

Multiple cluster effects We have calculated the degree of TCR engagement by a diffusing pMHC in a field of

immobile TCR clusters and found that substantial engagement of clustered and non-clustered TCR is expected.

However, experiments show no change in Zap-70 recruitment to non-clustered TCR once the TCR-pMHC in-

teraction is blocked (89) but do show large accumulation of Zap-70, Lck, SLP-76, and LAT to clustered TCR

(87; 88; 89), suggesting that only clustered TCR signal. Using a full Monte Carlo simulation of mobile TCR

clusters we find that pMHC achieve the maximum number of TCR engagements in the absence of coreceptors or

pMHC dimerization by visiting multiple TCR clusters. However, the average number of TCR engagements per

cluster is smaller without coreceptors or pMHC dimerization (compare TCR hits in Table 2.2 to hits per cluster

in Table 2.4). We therefore report a trade-off between the total number of TCR hits and the number of localized


TCR hits.

Using the full Monte Carlo simulation we have also shown thatpMHC transport to the cSMAC is possible

without the aid of coreceptors or dimerization. In this case, we find that the probability that a given pMHC will

arrive in the cSMAC (after many simulations) depends on the TCR/pMHC reaction rates and may serve to discrim-

inate pMHC. However, discrimination will only be possible when pMHC are presented in large concentrations, as

is the case for presented antigen to B cells. When discrimination is required based on 1-10 presented pMHC, as is

the case for T cells, the probability of cSMAC arrivial, which is an ensemble quantity, will not be well correlated

to the reaction parameters. Although a plausible mechanismfor antigen discrimination in B cells, we suggest that

such large scale (across the whole immune synapse) discrimination is unlikely to be taking place in T cells.

Future particle tracking studies of labeled pMHC molecules(on a cell or supported bilayer) during T cell

contact will inform on the duration that they spend within clusters and will resolve whether pMHC move between

clusters. Such studies will reveal whether maximizing the total number of TCR hits or the number of TCR hits

per cluster visit is important. Our work has highlighted therole of coreceptors in pMHC trapping in a single

TCR cluster and suggest that blocking the action of coreceptors (e.g. by antibody blocking or mutations in the

coreceptor binding domain to MHC) will lead to dramatic changes in the spatial-temporal dynamics of pMHC in

the IS.

Importance of pMHC-TCR bond affinity Fundamentally, T cell detection of pMHC rests on the TCR-pMHC

interaction and is parameterized by just a few measured constants -kon, koff and/orKD = koff/kon. The im-

portance ofkoff has been emphasized in some studies (for instance, (84; 85; 86; 118)) where an intermediate

“optimal” lifetime is proposed to balance serial engagement with effective TCR signaling, suggesting that theKD

of the interaction is less important. On the other hand, (107) presents striking evidence for the importance ofKD.

Here, we have found maximum values forKD that allow effective pMHC transport, independent of the kinetic rate

constants, and a potentially important role for factors strengthening the TCR-pMHC bond. The data presented

in tables 2 and 3 show no correlation betweenkoff and hits in the absence of coreceptors, but good correlation

(for data from (115) and (107)) in the presence of coreceptors. More work is definitely needed to elucidate the

importance ofkoff vsKD in T cell signaling.

Importance of serial engagement of TCR In this paper, we have focused on calculating levels of serial en-

gagement of TCR by pMHC. The importance of serial engagementto T cell activation, however, remains unclear.


It has been shown that very few agonist pMHC are able to stimulate signaling downstream from the TCR within

5-10 s of contact (73; 119). Furthermore, cytotoxic T cell killing can occur extremely rapidly, before the forma-

tion of the full immune synapse (120). On the other hand, sustained TCR signaling during a prolonged interaction

(many hours) has been shown to be important for cytokine production and cellular proliferation (121). We suggest

that there are different levels of TCR signaling, leading tovarious cellular responses, and that serial engagement

of TCR by pMHC may be important for sustained signaling on thetimescale of minutes to hours. Furthermore, the

density of agonist pMHC for a particular T cell is probably lower in vivo than in the usual experimental situations

and the TCR-pMHC bond may be weaker.

Understanding signal amplification by various mechanisms (coreceptor action, pMHC dimers and pseudodimers,

serial engagement of TCR by pMHC, and serial encounters of T cells with APC) is a major theme in T cell acti-

vation research. Hopefully, our knowledge of the roles of these mechanisms will improve with new experimental

and theoretical insights.

46

References


immunological synapse: A molecular machine controlling T cell activation.Science. 285:221–227.

[70] Cemerski, S., and A. Shaw. 2006. Immune synapses in T-cell activation.Curr. Opin. Immunol. 18:298–304.

[71] Harding, C. V., and E. R. Unanue. 1990. Quantitation of antigen-presenting cell MHC class-II peptide

complexes necessary for T-cell stimulation.Nature. 346:574–576.

[72] Demotz, S., H. M. Grey, and A. Sette. 1990. The minimal number of class-II MHC antigen complexes

needed for T-cell activation.Science. 249:1028–1030.

[73] Irvine, D. J., M. A. Purbhoo, M. Krogsgaard, and M. M. Davis. 2002. Direct observation of ligand recogni-

tion by T cells.Nature. 419:845–849.

[74] Davis, M. M., M. Krogsgaard, J. B. Huppa, C. Sumen, M. A. Purbhoo, D. J. Irvine, L. C. Wu, and L. Ehrlich.

2003. Dynamics of cell surface molecules during T cell recognition. Ann. Rev. Biochem. 72:717–742.

[75] Valitutti, S., S. Muller, M. Cella, E. Padovan, and A. Lanzavecchia. 1995. Serial triggering of many T-cell


[76] Itoh, Y., B. Hemmer, R. Martin, and R. N. Germain. 1999. Serial TCR engagement and down-modulation by

peptide:MHC molecule ligands: relationship to the qualityof individual TCR signaling events.J. Immunol.

162:2073–2080.

[77] Jose, E. S., A. Borroto, F. Niedergang, A. Alcover, andB. Alarcon. 2000. Triggering the TCR complex cause

the downregulation of nonengaged receptors by a signal transduction-dependent mechanism.Immunity.

12:161–170.

[78] Sousa, J., and J. Carneiro. 2000. A mathematical analysis of TCR serial triggering and down-regulation.

Eur. J. Immunol. 30:3219–3227.

[79] Utzny, C., D. Coombs, S. Muller, and S. Valitutti. 2006.Analysis of peptide/MHC-induced TCR downreg-

ulation. Cell Biochem. Biophys. 46:101–111.


[80] Holler, P. D., and D. M. Kranz. 2003. Quantitative analysis of the contribution of TCR/pepMHC affinity and

CD8 to T cell activation.Immunity. 18:255–264.


receptors.Biophys. J. 80:606–612.

[82] McKeithan, K. 1995. Kinetic proofreading in T-cell receptor signal transduction.Proc. Natl. Acad. Sci.

USA. 92:5042–5046.

[83] Coombs, D., and B. Goldstein. 2005. T cell activation: Kinetic proofreading, serial engagement and cell

adhesion.J. Comp. Appl. Math. 184:121–139.

[84] Kalergis, A. M., N. Boucheron, M. A. Doucey, E. Palmieri, E. C. Goyarts, Z. Vegh, I. F. Luescher, and S. G.

Nathenson. 2001. Efficient cell activation requires an optimal dwell-time of interaction between the TCR

and the pMHC complex.Nature Immunol. 2:229–234.


marked for internalization after dissociation from peptide-MHC. Nature Immunol. 3:926–931.

[86] Gonzalez, P. A., L. J. Carreno, D. Coombs, J. E. Mora, E. Palmieri, B. Goldstein, S. G. Nathenson, and

A. M. Kalergis. 2005. T cell receptor binding kinetics required for T cell activation depend on the density of

cognate ligand on the antigen-presenting cell.Proc. Natl. Acad. Sci. USA. 102:4824–4829.

[87] Yokosuka, T., K. Sakata-Sogawa, W. Kobayashi, M. Hiroshima, A. Hashimoto-Tane, M. Tokunaga, and

M. L. Dustin. 2005. Newly generated T cell receptor microclusters initiate and sustain T cell activation by

recruitment of Zap70 and SLP-76.Nature Immunol. 6:1253–1262.


receptor microclusters as scaffolds for signaling.J. Exp. Med. 202:1031–1036.



Immunity. 25:117–127.

[90] Garcia, K. C., C. A. Scott, A. Brunmark, F. R. Carbone, P.A. Peterson, I. A. Wilson, and L. Teyton. 1996.

CD8 enhances formation of stable T-cell receptor MHC class Imolecule complexes.Nature. 384:577–581.

[91] Gao, G., J. Tormo, U. Gerth, J. Wyer, A. McMichael, D. Stuart, J. Bell, E. Jones, and B. Jakobsen. 1997.

Crystal structure of the human CD8aa and HLA-A2.Nature. 387:630–634.

[92] Janeway, C. A. 1992. The T cell receptor as a multicomponent signalling machine: CD4/CD8 coreceptors

and CD45 in t cell activation.Annu. Rev. Immunol. 10:645–674.


[93] Lee, K. H., A. R. Dinner, C. Tu, G. Campi, S. Raychaudhuri, R. Varma, T. N. Sims, W. R. Burack, H. Wu,

J. Wang, O. Kanagawa, M. Markiewicz, P. M. Allen, M. L. Dustin, A. K. Chakraborty, and A. S. Shaw. 2003.

The immunological synapse balances T cell receptor signaling and degradation.Science. 302:1218–1222.

[94] Goldstein, B., R. Griego, and C. Wofsy. 1984. Diffusion-limited forward rate constants in two dimensions:

application to the trapping of cell surface receptors by coated pits.Biophys. J. 46:573–583.

[95] Wyer, J. R., B. E. Willcox, G. F. Gao, U. C. Gerth, S. J. Davis, J. I. Bell, P. A. van der Merwe, and B. K.

Jakobsen. 1999. T cell receptor and coreceptor CD8αα bind peptide-MHC independently and with distinct

kinetics. Immunity. 10:219–225.

[96] Xiong, Y., P. Kern, H. Chang, and E. Reinherz. 2001. T cell receptor binding to a pMHCII ligand is kineti-

cally distinct from and independent of CD4.J. Biol. Chem. 276:5659–5667.

[97] Saxton, M. J., and K. Jacobson. 1997. Single -particle tracking: applications to membrane dynamics.Ann.

Rev. Biophys. Biomol. Struct. 26:373–399.

[98] Dustin, M. L., L. M. Ferguson, P. Y. Chan, T. A. Springer,and D. E. Golan. 1996. Visualization of CD2 in-

teraction with LFA-3 and determination of the two-dimensional dissociation constant for adhesion receptors

in a contact area.J. Cell. Biol. 132:465–474.

[99] Shaw, A. S., and M. L. Dustin. 1997. Making the T cell receptor go the distance: a topological view of T

cell activation.Immunity. 6:361–369.

[100] Krogsgaard, M., N. Prado, E. J. Adams, X. L. He, D. C. Chow, D. B. Wilson, K. C. Garcia, and M. M.


to T cell activation.Mol. Cell. 12:1367–1378.

[101] Garcia, K. C., C. G. Radu, J. Ho, R. J. Ober, and E. S. Ward. 2001. Kinetics and thermodynamics of T cell

receptor-autoantigen interactions in murine experimental autoimmune encephalomyelitis.Proc. Natl. Acad.

Sci. USA. 98:6818–6823.

[102] Willcox, B. E., G. F. Gao, J. R. Wyer, J. E. Ladbury, J. I.Bell, B. K. Jakobsen, and P. A. van der Merwe.

1999. TCR binding to peptide-MHC stabilizes a flexible recognition interface.Immunity. 10:357–365.

[103] Lee, K. H., A. D. Holdorf, M. L. Dustin, A. C. Chan, P. M. Allen, and A. S. Shaw. 2002. T cell receptor

signaling precedes immunological synapse formation.Science. 295:1539–1542.

[104] Wulfing, C., C. Sumen, M. D. Sjaastad, L. C. Wu, M. L. Dustin, and M. M. Davis. 2002. Costimulation

and endogenous MHC ligands contribute to T cell recognition. Nature Immunol. 3:42–47.


[105] Li, Q. J., A. R. Dinner, S. Y. Qi, D. J. Irvine, J. B. Huppa, M. M. Davis, and A. K. Chakraborty. 2004.

CD4 enhances T cell sensitivity to antigen by coordinating Lck accumulation at the immunological synapse.

Nature Immunol. 5:791–799.

[106] Cho, B. K., K. C. Lian, P. Lee, A. Brunmark, C. McKinley,J. Z. Chen, D. M. Kranz, and H. N. Eisen. 2001.

Differences in antigen recognitionand cytolytic activityof CD8+ and CD8− T cells that express the same

antigen-specific receptor.Proc. Natl. Acad. Sci. USA. 98:1723–1727.

[107] Holler, P., and D. Kranz. 2003. Quantitative analysisof the contribution of TCR/pepMHC affinity and CD8

to T cell activation.Immunity. 18:255–264.

[108] Wulfing, C., I. Tskvitaria-Fuller, N. Burroughs, M. D.Sjaastad, J. Klem, and J. D. Schatzle. 2002. Interface

accumulation of receptor/ligand couyples in lymphocyte activation: methods, mechanisms, and significance.

Immunological Reviews. 189:64–83.

[109] van der Merwe, P. A., and S. J. Davis. 2003. Molecular interactions mediating T cell antigen recognition.

Annu. Rev. Immunol. 21:659–684.

[110] Thome, M., P. Duplay, M. Guttinger, and O. Acuto. 1995.Syk and ZAP-70 mediate recruitment of

p56lck/CD4 to the activated T cell receptor/CD3/zeta complex. J. Exp. Med. 181:1997–2006.

[111] Thome, M., V. Germain, J. P. DiSanto, and O. Acuto. 1996. The p56lck SH2 domain mediates recruitment

of CD8/p56lck to the activated T cell receptor/CD3/zeta complex. Eur. J. Immunol. 26:2093–2100.

[112] Schafer, P., and S. Pierce. 1994. Evidence for dimers of MHC class II molecules in B lymphocytes and

their role in low affinity T cell responses.Immunity. 1:699–707.

[113] Lindstedt, R., N. Monk, G. Lombardi, and R. Lechler. 2001. Amino acid substitutions in the putative MHC

class II “dimer of dimers” interface inhibit CD4(+) T cell activation. J. Immunol. 166:800–808.

[114] Cochran, J. R., T. O. Cameron, and L. J. Stern. 2000. Therelationship of MHC-peptide binding and T cell

activation probed using chemically defined MHC class II oligomers.Immunity. 12:241–250.

[115] Krogsgaard, M., Q. J. Li, C. Sumen, J. B. Huppa, M. Huse,and M. M. Davis. 2005. Agonist/endogenous

peptide-MHC heterodimers drive T cell activation and sensitivity. Nature. 434:238–243.

[116] Fleire, S., J. Goldman, Y. Carrasco, M. Weber, D. Bray,and F. Batista. 2006. B cell ligand discrimination

through a spreading and contraction response.Science. 312:738–41.

[117] Vrljic, M., S. Y. Nishinura, S. Brasselet, W. E. Moerner, and H. M. McConnell. 2002. Translational

diffusion of individual class II MHC membrane proteins in cells. Biophys. J. 83:2681–2692.


[118] Lee, S.-J. E., Y. Hori, J. T. Groves, M. L. Dustin, and A.K. Chakraborty. 2002. Correlation of a dynamic

model for immunological synapse formation with effector functions: two pathways to synapse formation.

Trends. Immunol. 23:492–499.

[119] Li, Q. J., J. Chau, P. J. Ebert, G. Sylvester, H. Min, G. Liu, R. Braich, M. Manoharan, J. Soutschek, P. Skare,

L. O. Klein, M. M. Davis, and C. Z. Chen. 2007. miR-181a is an intrinsic modulator of T cell sensitivity and

selection.Cell. 129:147–161.

[120] Purbhoo, M. A., D. J. Irvine, J. B. Huppa, and M. M. Davis. 2004. T cell killing does not require the

formation of a stable mature immunological synapse.Nat. Immunol. 5:524–530.

[121] Huppa, J. B., and M. M. Davis. 2003. T-cell-antigen recognition and the immunological synapse.Nat. Rev.

Immunol. 3:973–983.

51

Chapter 3

T cell receptor clustering is essential for

accurate ligand discrimination1

3.1 Introduction

T cells are responsive to small numbers of particular pMHC complexes presented on APC. However, the bonds

formed between TCR and agonist pMHC are weak and transient (122; 123; 124), with solution dissociation con-

stants,KD, in the range of 0.1-100µM and half-lives in the range of 1-100 s. Experiments using a variety of

TCR, pMHC, altered peptide ligands (122; 123; 124; 125; 126)and point mutations in the CDR3 loops of the

TCR (127) demonstrate clear correlations between T cell stimulation, as measured by cytokine production and

cytotoxicity, and the solution reaction parameters governing the TCR-pMHC interaction. For example, Holler

and Kranz (123) measured IL-2 production from T cells incubated with APC pulsed with a particular peptide. In

separate experiments they measured the solution reaction parameters between the TCR and pMHC using surface

plasmon resonance (SPR), finding a good correlation betweenIL-2 production andKD. Other work has shown

that modest variations (factors of∼ 1.5) in the measured binding parameters lead to measurable changes in the

cellular response (122; 126).

TCR aggregation into sub-micron scale clusters has been shown to occur in T cells upon stimulation by pMHC

presented on a suspended planar bilayer (128; 129; 130) and on an APC (129). Many signaling molecules impor-

tant to T cell activation have been shown to localize to theseTCR clusters (128; 129; 130) and there is evidence

that the coreceptors CD4 and CD8 are present with Lck, an important signaling molecule associated with the

cytoplasmic tail of these coreceptors, in TCR clusters (128; 131). The function of TCR clustering is discussed

controversially.

Normally, experiments are undertaken to correlate the binding parameters of a given TCR-pMHC interaction

with the cellular response. Here, we take an alternative approach by asking how well a T cell can measure the

1A version of this chapter has been submitted for publication. Dushek O, Coombs D (2008) T cell receptor clustering is essential foraccurate ligand discrimination.

Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 52

binding parameters of the TCR-pMHC interaction. With this idea in mind, we imagine that the T cell seeks sim-

ply to measure the binding parameters, and the decision to act (making the pMHC ade facto agonist) follows this

measurement. As we shall see, since TCR-pMHC bonds form and break rapidly and stochastically, pMHC must

be repeatedly sampled (by binding TCR) before the reaction parameters can be confidently estimated. We will

show that TCR clustering is required, in part, to ensure thatthe multiple binding events are localized, allowing for

rapid and accurate detection of presented pMHC.

3.2 Methods

3.2.1 Simulation method

The modeling approach we have used in this study is the next reaction Monte Carlo simulation, originally pro-

posed by Gillespie (132). The primary purpose of our simulations is to record the duration of binding events and

the time interval between binding events as a function of several parameters. We chose the next reaction method

because, unlike other Monte Carlo simulations that rely on iteration numbers or fixed time steps, this method

advances time in a natural way that is in good agreement with an underlying continuum model (132). A Monte

Carlo approach is used, instead of a PDE, because we wish to investigate how stochastic binding events (duration

between binding events and duration of individual binding events) may be used by the T cell to determine kinetic

parameters.

Our spatial reaction-diffusion simulation is based on the work of Isaacson and Peskin (133). We simulate a

single pMHC diffusing on the APC membrane and binding to TCR on the T cell membrane, see Figure 3.1 (basic

model). We use a lattice simulation which allows for reactions when the pMHC is in the same lattice site as a

TCR. The binding rate in this case iskmicro = kon/h2, wherekon is the experimentally measured (e.g. by SPR)

bimolecular reaction constant andh is the lattice constant. The unbinding rate iskoff and diffusion is captured by

a first order reaction, of rateD/h2, to nearest neighbours lattice sites. We takeh to be roughly the size of a TCR,

h = 10 nm, in our simulations making it possible to capture both reaction-limited and diffusion-limited regimes.

Depending on the state of the pMHC only certain reactions arepossible and the quantitya is the combined rate.

In the basic model, Figure 3.1, three states are possible: pMHC is unbound and not above a TCR (a = 4D/h2),

pMHC is unbound above TCR (a = 4D/h2 + kon/h2), or pMHC is bound (a = koff). Based on the overall rate,

a, we compute the time for the next reaction,

τnext =1

aln

(

1

r1

)


wherer1 is a uniformly distributed random number between 0 and 1. Knowing τnext, we use a second random

number,r2, to determine which reaction actually took place. Finally,we simulate the reaction, advance time by

τnext, recomputea, and repeat.

A B C D

TCR pMHC Coreceptor

Coreceptor Heterodimerization Model

A B

Basic Model

Figure 3.1: Models of TCR-pMHC reactions with and without coreceptor-MHC interactions.

In each simulation we record the location in the membrane of each binding event, the time between binding

events (τon), and the duration of binding at each event (τoff). The pMHC is initially bound and the simulation

is stopped after 10 binding events. We compute the percent error in estimatingkon from τon asEon = |1 −

(1/〈τon〉)/(kon/h2)|×100 and the percent error in estimatingkoff from τoff asEoff = |1− (1/〈τoff〉)/(koff)|×100.

Angled brackets indicate the mean after 10 binding events. We collect statistics for each set of parameters by

repeating the simulation 500 times. Confidence estimates are defined as the fraction of simulations that result in

Eon (orEoff) less than50%.

We tested the accuracy of our stochastic simulations by comparing them to a partial differential equation

(PDE) model. We performed simulations, as described above,but terminated the simulation when the pMHC

reached a distanceR. This minor change converted our stochastic simulation to astochastic first passage time

simulation. We performed 1000 simulations and in Figure 3.2we show the binned (and normalized) first passage

time (grey line). The PDE describing this first passage time process is,

∂fA

∂t= −konTfA + kofffB +D∇2fA ,

∂fB

∂t= konTfA − kofffB (3.1)

with fA = 0 on the boundary (atr = R), fA(t = 0) = δ(r)/r, andfB(t = 0) = 0. To obtain the probability

of a first passage we numerically solve the above PDE and compute the total flux through the boundary,p, as

a function of time. We plotp in Figure 3.2 (dotted black line) and find that it is in good agreement with the


stochastic simulation in both the reaction- and diffusion-limited regimes. Good agreement is also observed over a

wide range of parameters.

(a) (b)

0 50 100 150 200 250 3000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

time (s)

prob

abili

ty o

f firs

t pas

sage

, p(t

)

Stochastic simulationPDE computation

0 0.5 1 1.5 2

x 105

0

1

2

3

4

x 10−5

time (s)

prob

abili

ty o

f firs

t pas

sage

, p(t

)

Stochastic simulationPDE computation

Figure 3.2: Comparisons of stochastic simulation and PDE computations. In a) we simulated a reaction-limitedprocess (kon = 0.005µm2/s) and in b) we simulate a diffusion limited process (kon = 5µm2/s). Parameters:R = 1µm,DP = 0.05µm2/s,koff = 1s−1.

We investigated an alternate stochastic simulation that allowed pMHC to bind TCR when it was in one of eight

nearest neighbour lattice sites, in addition to being centered on the TCR. In contrast to the hard-shell simulations,

the overall binding rate in these soft-shell simulations issmeared over nine lattice sites. The rate of binding in

the neighbouring lattice sites decreased according to a Gaussian with a full-width-half-maximum ofh, the grid

spacing. We normalize the binding rates such that the sum of the rates in all nine sites is equal tokon. We obtained

good agreement with the PDE and repeating all simulations using this model we find no substantial variation in

the results.

3.2.2 Simulation parameters

The simulations we have used rely on several parameters, many of which have been experimentally determined.

The solution reaction parameters for many TCR/pMHC have been measured using SPR (122; 123; 124; 126).

Solution on-rates have been reported in the range of 500-300,000 M−1s−1 and off-rates 0.0001-5 s−1. SPR mea-

surements have revealed similar equilibrium binding constants between CD4 or CD8 and MHC (134; 135). Wyer

et al. (134) report the solution on-rate to be> 100, 000 M−1s−1 and the off-rate to be> 18 s−1. In our simula-

tions we have used the solution off rates directly. We note that the actual value ofkoff for TCR-pMHC only alters

the total time in Figure 3.4 and does not affect results in other figures.

We have simulated reactions between membrane proteins and hence the bimolecular reaction parameter,kon,

is a 2D quantity. However, reaction measurements between TCR-pMHC and CD4/CD8-MHC using SPR provide


solution or 3D on-rates. To obtain estimates of 2D quantities we multiplied the 3D on-rate (in units of M−1s−1)

by a factor of1015/NA (whereNA is Avagadro’s number) to obtain units ofµm3/s. We next divide this 3D quan-

tity by a confinement length to obtain the 2D on-rate in units of µm2/s. We use a confinement length of 0.262

nm (136). This conversion indicates a range of∼ 0.001 − 2µm2/s for TCR-pMHC on-rates and∼ 0.1µm2/s for

CD4/CD8-MHC on-rates. The accuracy of this method is unknown and we therefore explore a larger range in the

TCR-pMHC on-rate (see x-axis in figures).

We have taken the diffusion coefficient of pMHC to be0.05µm2/s (137). Simulations withD = 0.1µm2/s or

D = 0.01µm2/s give similar results. The diffusion of TCR also warrants some discussion. Our goal was to create

a simple stochastic simulation to characterizeτon, the time between binding events, as a function ofD, T , kon, and

koff . We have assumed that upon binding to a TCR, the diffusion coefficient of the TCR-pMHC complex is zero.

We have also assumed a diffusion coefficient of zero for TCR. These assumptions, although quantitatively altering

our results, do not alter our conclusions. When initially unbound, a pMHC molecule diffuses around looking for

another TCR. The probability of encountering another TCR depends on the TCR concentration and simulating

TCR mobility will not alter the time it takes to find another TCR, on average. We neglect TCR diffusion for com-

putational efficiency. On a grid of 5 by 5µm at a TCR concentration of 5000µm−2 it would be very expensive to

simulate diffusion of 125 000 TCR.

3.3 Results

3.3.1 T cells cannot make a direct and reliable estimate ofKD

Given the observed correlation betweenKD and T cell stimulation (123; 126), we first consider whether the T cell

can determineKD directly. KD is an equilibrium quantity determining the concentration of pMHC-TCR bonds,

according toKD = PT/B whereB is the concentration of bound TCR,T of free TCR, andP of free pMHC

in the contact region. There are substantial obstacles to making an accurate measurement ofKD within a reason-

able timeframe. First, agonist pMHC may be present at low numbers (138), and may enter and leave the contact

area during the sample time, so there will be large fluctuations in the number of TCR that are bound. Second,

non-specific (null) pMHC are likely present in large numbers, weakly binding with TCR and interfering with

agonist binding (as well as possibly generating weak signals that must be ignored). In summary, TCR occupancy

alone cannot be used to estimateKD and so pMHC quality must be assessed by some means that uses the kinetic

parameters of the reactions (kon andkoff), possibly allowing subsequent estimation ofKD.


3.3.2 Multiple binding events are required to determine theTCR/pMHC reaction rates

We now imagine an APC bearing a single pMHC that binds and unbinds from TCR on the T cell surface. How

many times must a pMHC bind to TCR in order that the true unbinding ratekoff can be estimated with a given

accuracy? Since unbinding is an exponential stochastic process, the maximum likelihood estimate ofkoff is the re-

ciprocal of the mean binding time. Even with perfect measurement of pMHC-TCR bond durations, the number of

samples needed to have 85% confidence that the estimate ofkoff is within50% of the actual value can be calculated

as 10 (139). In order to ensure that these multiple binding events arise from the same pMHC and for effective

integration of signals by the intracellular signalling machinery, the binding events presumably must be local-

ized, for instance within TCR microclusters (128; 129; 130). pMHC are expected to bind multiple TCR within a

single cluster only if the TCR-pMHC bond is stabilized, for instance by the TCR coreceptors CD4 and CD8 (136).

Continuing with this line of reasoning, we ask how well the binding rate of the pMHC can be estimated from

the time between binding events. This rate depends on the two-dimensional chemical forward rate constant (kon),

the diffusion coefficient of the pMHC (D) and the TCR concentration (T ). The forward rate constantkon is

measured in units ofµm2/s, and is related to the rate of formation of bonds between a single pMHC-TCR pair,

kmicro, by kmicro = kon/h2 whereh ∼ 10 nm is the reaction radius. Reported values ofkon for agonist pMHC

range from0.001 − 2µm2/s (see methods). We performed spatial Monte Carlo simulations of a pMHC diffus-

ing (D = 0.05µm2/s (137)) through the contact area and interacting with a homogeneous distribution of TCR

(132; 133) (Figure 3.1: basic model) and took the estimate ofkmicro to be the reciprocal of the mean time between

binding events. Simulations begin with the pMHC in the boundstate and are stopped after 10 binding events.

In Figure 3.3a we report the statistical confidence that the estimate ofkon is within 50% of the actual value, for

varying TCR concentrations. We observe that, at physiological TCR concentrations (T ∼ 100µm−2 for resting

T cells and∼ 500µm−2 within TCR clusters (128; 129; 136)), the confidence in estimating kon is low unless

kon ≥ 1µm2/s.The estimates ofkon for slowly-binding pMHC at physiological TCR concentrations are poor be-

cause they spend a lot of time diffusing between binding events. In fact, ifkon is smaller thanD then the system is

reaction-rate-limited and direct estimates ofkon might be made using information about the TCR concentration.

However, in such cases, pMHC binding events are infrequent and widely distributed, making it hard to imagine

how the TCR signaling machinery might integrate them successfully or how it is possible to distinguish the ago-

nist signals from those due to null pMHC.

(a) (b)

−4 −3 −2 −1 0 10

20

40

60

80

100

log(kon

) (µm2/s)

% C

on

fidence in E

stim

ate

of

kon

[TCR]=100 µm

−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

−4 −3 −2 −1 0 10

2

4

6

8

log(kon

) (µm2/s)

Mea

n D

ista

nce

(µm

)

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

Figure 3.3: TCR concentration impacts the accuracy in the estimation ofkon and engagement localization. (a)statistical confidence that the estimate ofkon from 10 binding events is within 50% of the true value. (b) meandistance between 10 TCR engagements. Parameters arekoff = 0.1 s−1,D = 0.05µm2/s (simulation methods andmodel parameters are discussed in the supplementary information)

3.3.3 TCR enrichment impacts pMHC detection

The mean distance and time between binding events scale as(D/konT )1/2 and1/konT respectively, so increases

in TCR concentration andkon improve spatial and temporal localization of signals (Figure 3.3b and 3.4a). How-

ever, even at clustered TCR concentrations of 500µm−2, the mean distance and time between events exceed the

cluster size (∼0.5µm) and cluster lifetime (100s) ifkon < 10−2µm2/s (129). Summarizing, we have shown that

membrane organization, in the form of TCR enrichment, impacts the accuracy, localization, and speed of pMHC

detection and discrimination. However, TCR clustering as observedin vitro does not allow accurate discrimina-

tion of kon during a 10-sample period or confinement of 10 binding eventswithin a single cluster.

(a) (b)

−4 −3 −2 −1 0 1

200

400

600

800

1000

1200

1400

log(kon

) (µm2/s)

Sim

ulat

ion

Tim

e (s

)

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

−4 −3 −2 −1 0 190

100

110

120

130

140

150

log(kon

) (µm2/s)

Sim

ulat

ion

Tim

e (s

)

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

Figure 3.4: Total time required to achieve 10 binding events. The simulation is performed as described in themain text in the absence (a) and prescence (b) of coreceptors. As expected, for largekon the total simulation timeis dependent only onkoff and plateaus to 10*(1/koff)≈100 s. In the absence of coreceptors and TCR enrichmentwe see that the detection time of pMHC with smallkon is substantial. Parameters:D = 0.05µm2/s, koff = 0.1s−1.

3.3.4 TCR enrichment with coreceptors improves pMHC detection for weakly binding

pMHC

We now investigate whether bond stabilization by the TCR coreceptors CD4 and CD8 can improve pMHC detec-

tion and discrimination. To this end, we extend our previoussimulation to allow for coreceptor binding to pMHC

(Figure 3.1). We assume a 1:1 TCR:coreceptor stoichiometryand use experimentally determined reaction param-

eters (134; 135) (see methods). Repeating the simulations we obtain Figures 3.5a-b and Figure 3.4b. We find a

dramatic improvement in the accuracy ofkon (Figure 3.5a) when the TCR concentration is∼ 500µm−2 or larger.

Under these conditions, we also find that pMHC binding eventsare well localized (< 0.5µm) and rapid (10 events

within 100 s). Coreceptors are not required to achieve accuracy, localization, and rapid detection of pMHC having

largekon. We note that other mechanisms of bond stabilization (for instance, dimerization of agonist/self-pMHC

(140)) could create more stable complexes and achieve similar results. However, our results provide a simple

explanation for the observations of Holler and Kranz (123),who found that TCR coreceptors must be present for

pMHC detection when the TCR-pMHC bond has largeKD.

(a) (b)

−4 −3 −2 −1 0 10

20

40

60

80

100

log(kon

) (µm2/s)

% C

onfid

ence

in E

stim

ate

of k

on

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

−4 −3 −2 −1 0 10

0.5

1

1.5

2

log(kon

) (µm2/s)

Mea

n D

ista

nce

(µm

)

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

Figure 3.5: Clustering of coreceptors with TCR improves ligand detection. Identical to Figure 3.3 expect that thecoreceptor model is used, see Figure 3.1. Reaction rate between coreceptors and MHC was taken to be0.1µm2/s(on-rate) and 50 s−1 (off-rate)

TCR clustering and the presence of coreceptors effectivelyreduce the mobility of pMHC enabling the T cell

to accurately, locally, and rapidly, probe the kinetic parameters of a given pMHC using TCR. Consistent with

these observations we find that a very small pMHC diffusion coefficient (D < 0.001µm2/s in contrast to mea-

surements of∼ 0.05µm2/s (137)) would allow for pMHC detection in the absence of coreceptors and substantial

TCR enrichment (results not shown). Similarily, increasing the on-rate for CD4/CD8-MHC by a factor of102

allowed coreceptors alone, without the aid of TCR enrichment, to immobilize pMHC (results not shown). Lastly,

the confidence in estimatingkoff in our simulations is independent ofkon, the TCR concentration, and the presence

of coreceptors, see Figure 3.6.


(a) (b)

−4 −3 −2 −1 0 120

40

60

80

100

log(kon

) (µm2/s)

% C

onfid

ence

in E

stim

ate

of k

off

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

−4 −3 −2 −1 0 120

40

60

80

100

log(kon

) (µm2/s)

% C

onfid

ence

in E

stim

ate

of k

off

[TCR]=100 µm−2

[TCR]=500 µm−2

[TCR]=1000 µm−2

[TCR]=5000 µm−2

Figure 3.6: Confidence in the estimation ofkoff is independent ofkon and TCR concentration. Statistical confi-dence that the estimate ofkoff, from the recipercol of the mean waiting time after 10 engagements, is within50%of the true value in the absence (a) and presence (b) of coreceptors. The confidence is≈ 85% as expected (seemain text). Parameters:D = 0.05µm2/s,koff = 0.1s−1.

In applying our results to TCR clusters we have ignored the directed motion of TCR clusters to the center of

the contact region (129). We have previously shown that the diffusion coefficient is the primary determinant of

pMHC residence time within a cluster and that the cluster velocity only determines the transport distance (136).

Consequently, the time between binding events and hence ourresults will not be altered by TCR cluster mobility.

3.4 Discussion

The present work suggests a model where pMHC binding to TCR (141) can trigger the formation of a TCR cluster.

Using a simple model where TCR diffuse and are trapped in a localized region (142), thus forming a microcluster,

we estimate that a TCR cluster can form in∼ 1s, which is smaller than typical TCR-pMHC bond half-lives. After

the cluster is formed, the reaction rates of the pMHC are rapidly and efficiently measured via a series of spatially

localized binding events. Self-pMHC that may subsequentlyenter the small cluster can be shown to transiently

bind few TCR (during the lifetime of the cluster). Discrimination and signaling within a cluster may then be

explained by existing nanoscale models of TCR signaling (143; 144; 145). Having outlined the necessity of TCR

clustering to detect the TCR-pMHC biochemical parameters,it remains to be determined how this information is

translated, via TCR signaling or otherwise, into differential release of cytokines (122; 123; 124; 126).

The serial engagement hypothesis (146) proposes that multiple TCR binding by a single pMHC is important

for signal amplification. Here, we have found that serial engagement is essential for cells to determine biochemi-

cal rate constants rapidly and robustly. In contrast to analyses based on serial engagement (147; 148), which have

focused largely on the role of the off-rate of the TCR-pMHC bond, however, we have found a role for a large

kon in allowing resampling of the pMHC within a single TCR cluster. These effects will be important in future


studies of T cell signaling that analyze how the spatial distribution of ligand binding affects downstream signaling.

Receptor clustering has recently been observed in B cells responding to membrane bound antigen (149). Prior

to affinity maturation, the reaction parameters between B cell receptors (BCR) and their ligands are comparable

to those of TCR-pMHC (150). Although detailed characterization of BCR clusters is not available yet, it appears

that they are similar to TCR clusters and therefore we suspect that they play a similar role in determining antigen

quality via multiple localized sampling.

61

References

[122] Krogsgaard, M., N. Prado, E. J. Adams, X. L. He, D. C. Chow, D. B. Wilson, K. C. Garcia, and M. M.


to T cell activation.Mol. Cell. 12:1367–1378.

[123] Holler, P. D., and D. M. Kranz. 2003. Quantitative analysis of the contribution of TCR/pepMHC affinity

and CD8 to T cell activation.Immunity. 18:255–264.



[125] Evavold, B., and P. Allen. 1991. Separation of IL-4 production from Th cell proliferation by an altered T

cell receptor ligand.Science. 252:1308–1310.

[126] Tian, S., R. Maile, E. Collins, and J. Frelinger. 2007.CD8(+) T cell activation is governed by TCR-

peptide/MHC affinity, not dissociation rate.J. of Immunol. 179:2952–2960.

[127] Kalergis, A. M., N. Boucheron, M. A. Doucey, E. Palmieri, E. C. Goyarts, Z. Vegh, I. F. Luescher, and

S. G. Nathenson. 2001. Efficient cell activation requires anoptimal dwell-time of interaction between the

TCR and the pMHC complex.Nature Immunol. 2:229–234.



[129] Yokosuka, T., K. Sakata-Sogawa, W. Kobayashi, M. Hiroshima, A. Hashimoto-Tane, M. Tokunaga,

M. Dustin, and T. Saito. 2005. Newly generated T cell receptor microclusters initiate and sustain T cell

activation by recruitment of Zap70 and SLP-76.Nat Immunol. 6:1253–62.



Immunity. 25:117–127.

[131] Krummel, M. F., M. D. Sjaastad, C. Wulfing, and M. M. Davis. 2000. Differential clustering of CD4 and

CD3ζ during T cell recognition.Science. 289:1349.


[132] Gillespie, D. 1977. Exact stochastic simulations of coupled chemical reactions.J. Phys. Chem. 81:2340–

2361.

[133] Isaacson, S. A., and C. S. Peskin. 2006. Incorporatingdiffusion in complex geometries into stochastic

chemical kinetics simulations.SIAM J. Sci. Comp. 28:47–74.

[134] Wyer, J. R., B. E. Willcox, G. F. Gao, U. C. Gerth, S. J. Davis, J. I. Bell, P. A. van der Merwe, and B. K.

Jakobsen. 1999. T cell receptor and coreceptor CD8αα bind peptide-MHC independently and with distinct

kinetics. Immunity. 10:219–225.

[135] Xiong, Y., P. Kern, H. Chang, and E. Reinherz. 2001. T cell receptor binding to a pMHCII ligand is

kinetically distinct from and independent of CD4.J. Biol. Chem. 276:5659–5667.

[136] Dushek, O., and D. Coombs. 2008. Analysis of serial engagement and peptide-MHC transport in T cell

receptor microclusters.Biophys J. 94:3447–60.

[137] Saxton, M. J., and K. Jacobson. 1997. Single -particletracking: applications to membrane dynamics.Ann.

Rev. Biophys. Biomol. Struct. 26:373–399.

[138] Irvine, D. J., M. A. Purbhoo, M. Krogsgaard, and M. M. Davis. 2002. Direct observation of ligand recog-

nition by T cells.Nature. 419:845–849.

[139] Trivedi, K. S. 1982. Probability and statistics with reliability, queuing and computer science applications,

2nd Ed. Prentice-Hall.

[140] Krogsgaard, M., Q. J. Li, C. Sumen, J. B. Huppa, M. Huse,and M. M. Davis. 2005. Agonist/endogenous

peptide-MHC heterodimers drive T cell activation and sensitivity. Nature. 434:238–243.

[141] Choudhuri, K., and P. van der Merwe. 2007. Molecular mechanisms involved in t cell receptor triggering.

Seminars in Immunology. 19:255–261.

[142] Weaver, D. L. 1983. Diffusion-mediated localizationon membrane surfaces.Biophys. J. 41:81–86.

[143] Chan, C., A. J. T. George, and J. Stark. 2001. Cooperative enhancement of specificity in a lattice of T cell

receptors.Proc. Natl. Acad. Sci. USA. 98:5758–5763.

[144] Burroughs, N. J., Z. Lazic, and P. A. van der Merwe. 2006. Ligand detection and discrimination by spatial

relocalization: A kinase-phosphatase segregation model of TCR activation.Biophysical Journal. 91:1619–

1629.

[145] Germain, R. 1997. T-cell signaling: The importance ofreceptor clustering.Curr Biol. 7:R640–R644.


[146] Valitutti, S., S. Muller, M. Cella, E. Padovan, and A. Lanzavecchia. 1995. Serial triggering of many T-cell



receptors.Biophys. J. 80:606–612.


marked for internalization after dissociation from peptide-MHC. Nature Immunol. 3:926–931.

[149] Depoil, D., S. Fleire, B. Treanor, M. Weber, N. Harwood, K. Marchbank, V. Tybulewicz, and F. Batista.

2008. CD19 is essential for B cell activation by promoting B cell receptor-antigen microcluster formation in

response to membrane-bound ligand.Nat Immunol. 9:63–72.

[150] Paus, D., T. Phan, T. Chan, S. Gardam, A. Basten, and R. Brink. 2006. Antigen recognition strength

regulates the choice between extrafollicular plasma cell and germinal center B cell differentiation.J Exp

Med. 203:1081–91.

64

Chapter 4

Improving parameter estimation for cell

surface FRAP data1

4.1 Introduction

Fluorescence Recovery After Photobleaching (FRAP) using the confocal laser scanning microscope (CLSM) is

routinely used to determine the diffusion coefficient and mobile fraction of cell surface proteins (e.g. (151; 152;

153)). The method relies on the ability to conjugate a fluorescent probe, known as a fluorophore, to the protein of

interest. In a specific region of the cell surface, high laserintensity denatures/bleaches the fluorophores reducing

the fluorescence in that region to background levels. Motion(thermal or otherwise) of unbleached proteins from

outside the bleached region causes fluorescence in the bleached region to recover over time. The exact form of

the recovery is governed by the mobility properties of the labeled protein itself, modulated by the details of the

experimental protocol (i.e. size and shape of the bleached region, amount of bleaching, etc.). Excellent reviews

of the technique are given in (154; 155; 156; 157). In particular, the book chapter by Rabut and Ellenberg (157)

provides a complete general description of the technique, including numerous experimental practicalities.

As described in the review of Goodwin and Kenworthy (156), a common method to obtain the diffusion coef-

ficient (D) of the labeled protein and the fraction of these proteins that are mobile (Mf) is to fit an equation to the

fluorescence recovery in the bleached region. The equation is derived from a model that captures the experimental

protocol and the underlying physical processes. Commonly used models are described by Ellenberg et al. (158)

and Axelrod et al. (159). We will show here that for the space and time-scales of FRAP experiments for cell sur-

face proteins, certain key assumptions of the common modelsare violated resulting in significant errors inD and

Mf. We will then present a refined experimental protocol and fitting procedure for use in obtaining the diffusion

coefficient and mobile fraction of cell surface molecules.

1A version of this chapter has been published. Dushek O, Coombs D (2008) Improving parameter estimation for cell surface FRAP data.J. Biochem. Biophys. Methods. 70:1224-1231

Chapter 4. Improving parameter estimation for cell surfaceFRAP data 65

4.2 Methods

4.2.1 Experimental protocols for cell surface FRAP

In what follows, we assume, as is usually the case for a CLSM, that the bleaching beam and monitoring beam are

parallel to each other and to thez-axis.

There are two common protocols for performing confocal FRAPfor cell surface molecules using a CLSM.

Protocol 1, the simpler of the two, is more suitable if the cell is able to spread considerably on a substrate. In this

case, the focal plane of the microscope (depicted as thexy plane in Figure 4.1) is adjusted to obtain fluorescence

from membrane proteins that are directly above the substrate. A region of the cell membrane, generally a rectan-

gle measuringLxb by Lyb, is selected for bleaching and in a subset of this region, which we call the monitoring

region, the fluorescence is integrated to obtain the iconographic FRAP recovery curve. The dimensions of the

monitoring region areLxm by Lym. In this case, it is quite easy to visualize both the bleachedand monitoring

regions. Studies where this protocol is employed are (156; 160; 161; 162) and (163). This protocol cannot always

be used because many cells do not spread sufficiently and further, light scattering can cause problems when the

microscope is focused close to the substrate.

Figure 4.1: Experimental geometry for protocol 1 (not to scale). Shown here is a schematic of a cell that hasspread on a substrate. The focal plane of the CSLM is adjustedto obtain fluorescence from membrane anchoredproteins proximal to the substrate. The bleached region is arectangle measuringLxb by Lyb and the monitoringregion is a subset of this region measuringLxm byLym.

The geometry is more complicated in protocol 2 because the focal plane of the microscope is set to the equa-

torial plane of the cell. We assume the cell is spherical and centered at the origin of a Cartesian coordinate system.

A ring of fluorescence, represented as a circle in thexy plane, is observed from the equatorial plane of the cell

(Figure 4.2a). The thickness of this ring is dependent on theconfocal pinhole, which controls the depth,Wz,


about the focal plane where fluorescence is collected as shown in Figure 4.2b. The box (Wx by Wy) in Figure

4.2a represents a typical region selected for bleaching. Unlike protocol 1, here one cannot directly observe the

bleached region on the cell surface. However, since we do know that the bleaching beam penetrates the entire

depth of the cell (157), we can safely assume that the bleached region is a projection of the box of Figure 4.2a

onto the cell surface. Therefore the bleached region is a stripe, shown in Figure 4.2c, measuringLxb byLyb. The

monitoring region in this case is the intersection of the focal plane and the bleached stripe. Its dimensions areLxm

by Lym as indicated in Figure 4.2c. The fluorescence in the monitoring region is integrated to obtain the FRAP

recovery curve. Examples where this protocol is employed are (151; 152) and (153). Note that, in describing

protocol 2, we label experimentally adjustable parameterswith aW and the quantities describing the actual shape

of the bleached and monitoring regions byL.

The most common equation used to fit FRAP recovery curves generated by protocol 1 or 2 is the empirical

equation presented by Ellenberg et al. (158). We restate their 1D approximation as

F (t)

Fp=

(

1 − Fo

Fp

)

Mf

(

1 −(

4πDt

(Lyb)2+ 1

)− 1

2

)

+Fo

Fp(4.1)

whereF (t) is the time-dependent fluorescence intensity during the recovery phase,Fp the fluorescence prior to

bleaching,Fo the fluorescence immediately after bleaching,D the diffusion coefficient, andLyb the width of the

stripe. The mobile fraction,Mf, is defined as

Mf =F∞ − Fo

Fp − Fo(4.2)

whereF∞ is the asymptotic fluorescence (reached after a long recovery period). The exact analog of the empirical

formula given in Equation 4.1 can be derived by modeling the cell surface as an infinite (flat) plane of diffusing

labeled proteins. The initial fluorescence distribution models the experimental geometry, i.e. a stripe of widthLyb

and infinite length (Lxb = ∞) with zero intensity and a constant intensity everywhere else. This initial condition

reduces the problem from 2D to 1D. The solution obtained fromthe diffusion equation is then integrated over the

width of the stripe and is normalized. This solution agrees well with the empirical Equation 4.1 (158).

Henceforth we will refer to Equation 4.1 as the 1D model. To achieve an effective 1D diffusion on a finite

cell, an effectively infinite stripe must be bleached on the cell surface. This can be achieved by bleaching a stripe

that wraps around the cell. In protocol 1 this is accomplished by extending the bleached rectangle, shown in

Figure 4.1, through the entire length of the cell and in protocol 2 it is accomplished by pickingWx = 2R. In both

protocols, a symmetric 1D recovery is ensured. However, as we discuss in the results, several sources of error


(a) (b)

(c)

Figure 4.2: Experimental geometry for protocol 2 (not to scale). The focal plane of the CSLM is adjusted tocapture an equatorial section of the cell, depicted as thexy plane in (a). The thickness of the focal plane, labeledWz in (b), is set by the pinhole. In the focal plane a rectangularbox (Wx byWy) is selected for bleaching. Sincethe bleaching beam penetrates the entire depth of the cell, the bleached rectangle translates into a bleached stripe(Lyb by Lxb) on the surface of the cell, as shown in c). The fluorescence recovery is monitored from a subset ofthis bleached region: the intersection of the focal plane and the bleached region. The monitoring box measuresLxm by Lym. We note that the bleaching and monitoring lasers are both parallel to the z-axis. (a) and (b) depictplanar slices of the cell. Experimentally adjustable quantities are labeled W and the quantities L describe theactual dimensions of the bleached and monitoring regions onthe curved cell surface.


exist when a large stripe is bleached.

4.2.2 An improved protocol for cell surface FRAP

As we will show in the results section, an improvement to the common experimental protocols for cell surface

FRAP can be made by bleaching a sufficiently small region and using a 2D model of diffusion. Below we derive

the necessary equations and describe the protocol. We beginby deriving the 2D equation to use when a finite

rectangular region is bleached, as shown in Figures 4.1 and 4.2.

We begin by solving the 2D infinite plane diffusion equation,

∂f(x, y, t)

∂t= D

(

∂2f(x, y, t)

∂x2+∂2f(x, y, t)

∂y2

)

(4.3)

We assume that the initial intensity is unity everywhere except in the bleached rectangle, of dimensionsLxb by

Lyb, where the intensity is zero. Equation 4.3 can be solved by using Fourier transforms. The solution, after

applying the initial conditions and performing an inverse Fourier transform is

f(x, y, t) = 1 − 1

4

[

erf

(

Lxb/2 − x√4Dt

)

+ erf

(

Lxb/2 + x√4Dt

)]

×[

erf

(

Lyb/2 − y√4Dt

)

+ erf

(

Lyb/2 + y√4Dt

)]

(4.4)

erf is the usual error function. This equation gives the spatial time course of the recovery. To obtain the theoretical

FRAP recovery curve this equation must be integrated and normalized, a procedure analogous to averaging in this

case. The solution is integrated over the monitoring region, a rectangle measuringLxm by Lym centered at the

origin, and then normalized by the total intensity in the monitoring region prior to the bleaching event. The FRAP

recovery curve, when bleaching is complete and the mobile fraction is 1, is

G(t,D) = 1 − 18πLxmLym

√πLxm

[

erf(

Lxb−Lxm

4√

Dt

)

+ erf(

Lxb+Lxm

4√

Dt

)]

+√πLxb

[

erf(

Lxb+Lxm

4√

Dt

)

− erf(

Lxb−Lxm

4√

Dt

)]

+4√Dt[

exp(

− (Lxm+Lxb)2

16Dt

)

− exp(

− (−Lxm+Lxb)2

16Dt

)]

×√

πLym

[

erf(

Lyb−Lym

4√

Dt

)

+ erf(

Lyb+Lym√16Dt

)]

+√πLyb

[

erf(

Lyb+Lym

4√

Dt

)

− erf(

Lyb−Lym

4√

Dt

)]

+4√Dt[

exp(

− (Lym+Lyb)2

16Dt

)

− exp(

− (−Lym+Lyb)2

16Dt

)]

(4.5)


Allowing for the possibility of incomplete bleaching and animmobile fraction, we have

F (t)

Fp=

(

1 − Fo

Fp

)

Mf G(t,D) +Fo

Fp(4.6)

This equation, withG(t,D) as defined in Equation 4.5 and all other quantities as defined earlier, can easily be fit

directly to FRAP data when rectangular bleach and monitoring regions are used.

The use of Equation 4.6 to analyze data obtained using the geometry of protocol 1 is straightforward. A rect-

angle (Lxb by Lyb) is chosen for bleaching, as shown in Figure 4.1. In every subsequent frame, the intensity in

the rectangular monitoring region (Lxm by Lym) is integrated to obtain the experimental FRAP recovery curve.

This integration can be accomplished with standard imagingsoftware. Equation 4.6 can then be fit directly to

experimental FRAP recovery curves to obtain the diffusion coefficient and mobility fraction.

The use of Equation 4.6 with the geometry of protocol 2 is morecomplicated. Here the bleached and moni-

toring regions are not directly observable and therefore the quantitiesLxb, Lyb, Lxm, andLym in Equation 4.5 are

unknown. Protocol 2 is generally used when cells are unable to spread and maintain a roughly spherical shape,

for example unstimulated T cells (152). By approximating the cell as a sphere (at least near the bleached and

monitoring regions) the unknown quantities can be related to the known quantities by simple geometry,

Lxb = 2R arccos(1 −Wx/R) Lyb = 2R arcsin(Wy/2R)

Lxm = 2R arcsin(Wz/2R) Lym = 2R arcsin(Wy/2R) (4.7)

whereWx, Wy, andWz are chosen (as shown in Figure 4.2) andR is the cell radius. Knowing the quantities

describing the dimensions of the bleach and monitoring regions, Equation 4.6 can be fit directly to the integrated

fluorescence from the monitoring region for FRAP experiments utilizing protocol 2. There are two potential

sources of error in this method which we explore in detail in Appendix C. We ultimately find that they are negli-

gible for relevant experimental parameter regimes.

4.3 Results

4.3.1 Error estimates for common FRAP protocols

For most models used in fitting FRAP recovery curves, there are three requirements that must be satisfied:

1. Total time of bleaching event compared to characteristicdiffusion time must be small.


2. Size of bleached region compared to total surface area should be small.

3. Effects due to the finite area of the cell surface are not important.

When bleaching a large region, as in the case of a large stripe, it is difficult to satisfy all these requirements.

The first requirement that most models assume is that the bleaching event is instantaneous. It has been

recommended that the bleaching time be at least 10 times smaller than the characteristic diffusion time (157;

164). The characteristic diffusion time when bleaching a large stripe is∼ 150 s (calculated asLxbLyb/4D =

(30µm)(2µm)/4(0.1µm2/s)). Typical times required to bleach a large stripe extending through the entire cell are

tens of seconds, depending on the particular FRAP protocol and apparatus being used. In that case, we would

expect appreciable recovery before the bleaching event is complete, leading to errors inD. The correction for the

finite bleach time has been investigated in detail in (165).

The second requirement addresses the fact that most models assume that, in the absence of an immobile

fraction, recovery should be100%. However, bleaching a large region can remove upwards of20% of the total

fluorescence of the cell (LxbLyb/4πR2 = (30µm)(2µm)/4π(5µm)2) preventing recovery to pre-bleach levels. If

not corrected for, an error of20% is expected inMf when fitting Equation 4.1 to the recovery curve. Of course,

Mf andD are fit simultaneously to the data and so the error in the estimate ofMf may be reduced, at the expense

of introducing additional errors in the estimate ofD.

Lastly, the third requirement (previously considered by Carrero et al. (155) for cytosolic FRAP and Wey et

al. (166) for disk membranes ) addresses the fact that most models assume an infinite plane of fluorescence.

This assumption is valid provided the characteristic time for molecules to diffuse from the unbleached poles

(x = 0, y = ±R, z = 0 in Figure 4.2a for protocol 2 and at maximum y-values in Figure 4.1 for protocol 1)

into the bleached stripe is longer than the time-scale of theFRAP experiment. That is, the derivation of the equa-

tion assumes that boundary effects are minor during the time-scale of the experiment. The distance,S, from the

unbleached poles to the bleached stripe is roughly7 − 15µm for cells of radius5 − 10 µm. The characteristic

diffusion time is thenS2/4D ≈ 150 − 1000 s forD = 0.05 − 0.1 µm2/s (typical of cell surface molecules).

This range is well within the time-scale of typical FRAP experiments and so the infinite plane of fluorescence

assumption is violated for both protocols. In fact, we estimate a factor of 3 error inD under certain conditions

(see Table 4.1 and discussion in a subsequent section).

The three requirements are not satisfied, for either protocol, because a large stripe that extends through the

entire length of the cell is bleached (Figure 4.2c withLxb ≈ 2πR). One obvious way to satisfy these requirements


is to bleach a sufficiently small region by decreasingLxb. In this case, the bleaching event will be shorter, the ratio

of bleach area to total cell surface area will be smaller, andthe infinite plane assumption will be valid for a longer

period of time since the effective boundaries will be further away. We provide quantitative recommendations in a

subsequent section.

Many authors bleach a smaller region using both protocols and analyze their results with either the 1D model,

Equation 4.1 (as in (152; 160; 161; 162; 163)), or the circularly symmetric model outlined in Axelrod et al. (159)

(used by Fabra et al. (153)). WhenLxb is less than the cell circumference the fluorescence recovery into the

monitoring region is 2D. Continuing to use the 1D model in this case leads to appreciable errors inD. In Figure

4.3 we explore the percent difference inD (|(1 −Dfit/D) × 100%|) by fitting the 1D model to the solution of

the 2D infinite-plane diffusion equation, for various aspect ratios (Lxb/Lyb) of the bleached stripe. All fitting

procedures in this paper were carried out using the Matlab function nlinfit. We label quantities arising from the

fiting procedure with subscript fit (i.e.Dfit andMf-fit ). As an example, consider an aspect ratio of≈ 5 (Lxb ≈10

µm andLyb ≈2µm) typical of a small bleach region. The 1D model gives rise toa percent difference of130% (or

Dfit/D = 2.3) in either protocol 1 or 2. For aspect ratios in the range of 5-12 the 1D model gives rise to percent

differences inD of 50 − 130%. In making these calculations, we have ignored potential errors introduced by the

finite extent of the cell surface. These error contributionsare discussed in the next section but ultimately do not

affect the main conclusion drawn from Figure 4.3, which is that the 1D model is an inaccurate model for these

experimental protocols.

In the case of protocol 2, it is problematic to use the closed form variant of Axelrod’s formula presented by

Soumpasis (167) because it is impossible, in this geometry,to bleach a circle on the surface of the cell and to

monitor the fluorescence solely from this circle. The closest shape that can be achieved, in theory, is a square,

with Lxb = Lyb = Lxm = Lym, which gives an error inD of 40% when the square side length is 1µm. In practice,

the signal to noise ratio is too low when such a small region isbleached. The dotted line in Figure 4.3 describes

the error inD when the Soumpasis solution is used, as a function of the aspect ratio. Since it is more natural for

most CSLM to bleach a rectangular region, the use of a circularly symmetric model for protocol 1 has not been

widely used.

We see that substantial errors are introduced in estimates of D by the use of Equation 4.1.


Figure 4.3: Accuracy of the 1D model and the closed form circularly symmetric solution of Soumpasis (167).We solve the 2D diffusion equation on an infinite plane using as initial condition unit intensity everywhere excepton a rectangle (Lxb by Lyb) where the intensity is set to zero, simulating the bleachedregion. We integratethe solution over a subset of this rectangle (Lxm by Lym), i.e. the monitoring region. We then fit the 1D andSoumpasis solutions to the generated recovery curve for various values ofLxb. Each recovery curved is generatedusing 200 equally spaced time points reaching a recovery of99%. The graph shows the percent difference inD(|(1 −Dfit/D) × 100%|) for both models. AsLxb increases, the percent difference decreases when using the1Dmodel and increases when using the solution of Soumpasis, asexpected. When using a small bleach region withprotocol 1 or 2, a typical aspect ratio of≈ 5 (Lxb ≈ 10 µm andLyb ≈2 µm for example) is used. The 1D modelgives rise to a percent difference of130% (orDfit/D = 2.3) and the Soumpasis solution gives rise to a percentdifference of80% (orDfit/D = 0.2). Here we have ignored the finite extent of the cell surface. We note that thisgraph is independent of the actual value ofD.

4.3.2 Utility of the 2D model

In the previous section we stated that bleaching a sufficiently small region satisfies the three requirements impor-

tant for FRAP experiments but that continuing to use the 1D model, even though 2D diffusion occurs, resulted in

significant errors in the estimate ofD. In section 2.2 we described a 2D model of diffusion that can be used with

a small bleach region.

To confirm the utility of the 2D model we simulated FRAP experiments for cell surface proteins according

to protocol 2. We numerically solved the diffusion equationon the surface of a sphere using an explicit finite

difference scheme with 13000 grid points. We replicate the initial conditions of a bleached stripe and integrate the

solution over the monitoring region, both as shown in Figure4.2. All simulations were carried out in Matlab. We

usedD = 0.1 µm2/s, Mf = 1, R =5 µm,Wy =2 µm,Wz =2 µm, and variedWx. Figure 4.4 shows the results

of the simulations along with the predicted FRAP recovery curves from the 2D and 1D models. Figures 4.4 a),

b), and c) reveal that the 1D model is inaccurate for allWx and that the 2D model is the most accurate whenWx

is small (Figure 4.4c).

We fit the 2D and 1D models directly to the simulated FRAP recovery curves (Figure 4.4, solid lines). We


(a) (b)

(c)

Figure 4.4: FRAP simulations based on the geometry of protocol 2. We solve the diffusion equation on the surfaceof a sphere using an explicit scheme with 13000 grid points. We replicate the initial conditions of a bleachedstripe and integrate the solution over the monitoring region, both as shown in Figure 4.2. We useD = 0.1µm2/s,Mf = 1, R = 5µm, Wy = 2µm, andWz = 2µm. In (a), (b), and (c) the solid line shows the simulatedFRAP recovery while the dotted lines show the predictions ofthe 2D and 1D models with the same simulationparameters except that the mobile fraction is chosen to beMf − LxbLyb/4πR

2 (corrected for loss of fluorescenceduring bleach). In (a) we setWx = 10µm, bleaching a stripe that wraps around the cell. In (b),Wx = 4.5µm,and in (c)Wx = 1µm. In all cases we see that the 1D model is a poor predictor of the recovery. The 2D modelis accurate whenWx is small because the three requirements important for FRAP for cell surface proteins aresatisfied.


fit for the diffusion coefficientDfit and the mobile fractionMf-fit ; results shown in Table 4.1. We find percent

differences inD of ∼ 200% for both models whenWx = 2R = 10µm (Figure 4.4a). DecreasingWx substantially

reduces the error in the estimate ofD to 1% (Dfit/D = 1.01) but only for the 2D model. We would expect this

error reduction because decreasing the size of the bleach region addresses all three requirements in the previous

section.

Table 4.1: Fitting the 1D and 2D models to the sphere simulations (Figure 4.4, solid lines)2D Model 1D Model

Wx Dfit % err Mf r2 Dfit % err Mf r2

(µm) (µm2/s) (µm2/s)10.0 (fig. 4.4a) 0.36 260 1.0 0.93 0.27 173 1.0 0.944.5 (fig. 4.4b) 0.14 35 1.0 0.99 0.12 20 1.1 0.981.0 (fig. 4.4c) 0.10 1 1.0 1.00 0.15 48 1.1 0.96

a. As described in Figure 4.4, the sphere simulations were carried out with D=0.1µm2/s andMf = 1.

b. For each fit, we correct the fitted value ofMf to reflect the loss of fluorescence during the bleaching

process (Mf = Mf-fit + LxbLyb/4πR2).

c. Degree-of-freedom adjusted coefficient of determination,r2, is a goodness-of-fit measure.

We note that in all the cases in Table 4.1 the fitted diffusion coefficient,Dfit , was overestimated. This effect

is counterintuitive as one would expectDfit to be underestimated because boundary effects (due to the finite cell

size) slow diffusion into the monitoring region. However, we are simultaneously fitting forDfit andMf-fit and

this introduces a problem. In the simulations,Mf = 1. However, because a finite fraction of the cell surface

is bleached, the fitting procedure will necessarily estimate Mf-fit as less thanMf. Given this fact,Dfit will be

estimated as greater thanD to match the initial slope of the FRAP curve. We refer to this effect as the mobile

fraction effect.

For the 2D model, the mobile fraction effect decreases as theratio of bleached area to total surface area de-

creases. The decrease in this effect is observed in Table 4.1sinceDfit converges to its true value and we can

always correctMf-fit by the fraction of lost fluorescence during bleaching (giving the correct valueMf = 1 for all

cases with this model). We use this correction fraction because the bleaching event effectively removes a portion

of the labeled protein. In the 1D model, as the mobile fraction effect decreases, errors introduced by 2D diffusion

prevent accurate determinations of bothDfit andMfit . Although the 1D model gives a reasonable estimate forD

whenWx = 4.5µm, it overestimatesMf by 10%.

We note that in all cases both models provided excellent fits with r2 (degree-of-freedomadjusted coefficient of


determination) statistic close to one. Therefore, goodness of fit with an inappropriate model should not be relied

upon in fitting FRAP data on cell surfaces.

We next investigated the effects of instrumental noise. Since some amount of experimental noise is expected

in all FRAP experiments, we wanted to ensure that the 1D and 2Dmodels gave results that were statistically

different. To this end, we added15% normally distributed noise to FRAP curves generated using Equation 4.5

for variousD values. For eachD value we generated 10 noisy FRAP curves, simulating 10 FRAP experiments.

We then fit each of the 10 curves for a givenD value using the 1D model 4.1 and 2D model 4.6. In Figure 4.5,

grey up-triangles show the results for the 1D model and blackdown-triangles show the results for the 2D models.

Dotted lines show the mean value ofDfit . For all D values the 1D and 2D models give statistically different mean

value ofDfit (using a paired t-test at the 5% level). We found that when the means were different by a factor of

∼ 2 and a large amount of noise (15%) was present, the mean diffusion coefficients predicted by each model were

statistically different with 10 experiments. Performing only 5 experiments, we occasionally found that values

were not statistically different. However, most authors make≥ 10 measurements. We also note that the spread in

Dfit/D for a givenD value is large for the 1D model and much smaller for the 2D model.

Figure 4.5: Statistical signifiance of the 2D model.15% normally distributed noise was added to FRAP curvesgenerated using Equation 4.5 for variousD (x-axis). For eachD value, 10 noisy FRAP curves were fit using the1D model, Equation 4.1, and 2D model, Equation 4.6. Grey up-triangles show the ratio ofDfit to the actualD forthe 1D model and black down-triangles show the ratio for the 2D model. Dotted lines show the means. For all Dvalues the 1D and 2D models give statistically different mean value ofDfit (using a paired t-test at the 5% level).

We ultimately conclude that it is not advisable to use the 1D model in situations where the three requirements


discussed in the previous section are violated. The 2D modelcombined with a small bleach region provides a

more accurate protocol.

4.3.3 Practical recommendations for cell surface FRAP experiments

In order to satisfy the three requirements important for cell surface FRAP experiments Equation 4.6 with protocols

1 or 2 should be used. We recommend bleaching the smallest region possible while maintaining a good signal to

noise ratio. As rough guidelines, we recommend that the bleaching event take less than 3 seconds (characteristic

diffusion times for an aspect ratio of≈ 5 are> 45 s) and that the ratio of bleached region area to total cell surface

area be less than5%. These rough guidelines can be made more precise for a particular FRAP experiment by

examining the arguments presented above. We also recommendthat the bleach region should be in the center of

the cell for protocol 1 and that protocol 2 be used for cells most resembling spheres. We hope that the use of the

2D model, Equation 4.6, in conjunction with protocols 1 or 2 will provide more consistent results for the diffusion

coefficient and mobile fraction within and between laboratories.

4.4 Discussion

We have used theoretical arguments to show that common analytic methods used to obtain the diffusion coeffi-

cient and mobile fraction are generally not ideal for cell surface FRAP. When the 1D model is used appropriately

by bleaching a large stripe wrapping around the cell, it is difficult to satisfy the three requirements important for

FRAP experiments listed earlier. Ignoring possible errorsarising from a finite bleach time, the diffusion coeffi-

cient in this case will be overestimated by a factor of∼ 3 due to the mobile fraction effect. Bleaching a smaller

region will reduce this effect but new errors (factor of∼ 1.5) will be introduced if the 1D model is used because

2D diffusion into the monitoring region now takes place. The1D model with a large stripe can be used for cell

surface proteins provided the three requirements important for FRAP are satisfied, as may be the case for very

large cells.

We have presented a more accurate equation to use in the case of a small bleach region where 2D diffusion

occurs and we have provided guidelines for the use of this equation when using protocol 1 and 2 for cell surface

proteins. We note that this equation can be used for rectangular geometries within vitro andin vivo studies of

protein mobilities, analogous to the circularly symmetricsolution of Soumpasis (167) based on the theory of Ax-

elrod et al. (159).


Our method for fitting FRAP data relies on Equation 4.5, whichis derived from an approximate model. In

deriving the equation, we replaced the curved, finite, cell surface with an infinite flat plane. An alternative to

Equation 4.5 could be found by analytically solving the FRAPproblem on a finite, spherical surface, yielding a

solution in the form of an infinite series of special functions. This solution would be difficult to work with in

fitting data, and in any case would provide only a very slight improvement in the estimate ofD in situations where

the three FRAP requirements are met. As we have shown, Equation 4.5 represents a substantial improvement over

existing formulae for fitting FRAP data without adding much computational effort in fitting.

Improvements in the absolute determination ofD, as reported here, are useful in comparing different experi-

mental systems and in physical modeling. Ultimately, the use of the 2D equation along with a small bleach region

should lead to more consistent reports of diffusion coefficients.

78

References

[151] Sloan-Lancaster, J., J. Presley, J. Ellenberg, T. Yamazaki, J. Lippincott-Schwartz, and L. E. Samelson.

1998. ZAP-70 association with T cell receptorζ (TCRζ): Fluorescence imaging of dynamic changes upon

cellular stimulation.J. Cell. Biol. 143:613–624.

[152] Favier, B., N. J. Burroughs, L. Wedderburn, and S. Valitutti. 2001. TCR dynamics on the surface of living

T cells. Int. Immunol. 13:1525–1532.

[153] Fabra, S., V. Lang, J. Harriague, A. Jobart, T. G. Unterman, A. Trautmann, and G. Bismuth. 2005. Stable

activation of phosphatidylinositol 3-kinase in the T cell immunological synapse stimulates Akt signaling to

FoxO1 nuclear exclusion and cell growth control.J. Immunol. 174:4161–4171.

[154] Klonis, N., M. Rug, I. Harper, M. Wickham, A. Cowman, and L. Tilley. 2002. Fluorescence photobleaching

analysis for the study of cellular dynamics.Eur. Biophys. J. 31:36–51.

[155] Carrero, G., D. McDonald, E. Crawford, G. Vries, and M.Hendzel. 2003. Using FRAP and mathematical

modeling to determine the in vivo kinetics of nuclear proteins. Methods. 29:14–28.

[156] Goodwin, J. S., and A. Kenworthy. 2005. Photobleaching approaches to investigate diffusional mobility

and trafficking of Ras in living cells.Methods. 37:154–164.

[157] 2004. Live Cell Imaging: A Laboratory Manual, chapterPhotobleaching techniques to study mobility and

molecular dynamics of proteins in live cells: FRAP, iFRAP and FLIP. Cold Spring Harbor Laboratory Press.

[158] Ellenberg, J., E. D. Siggia, J. E. Moreira, C. L. Smith,J. F. Presley, H. J. Worman, and J. Lippincott-

Schwartz. 1997. Nuclear membrane dynamics and reassembly in living cells: Targeting of an inner nuclear

membrane protein in interphase and mitosis.J. Cell Biol. 138:1193–1206.

[159] Axelrod, D., D. E. Koppel, J. Schlessinger, E. Elson, and W. W. Webb. 1976. Mobility measurement by

analysis of fluorescence photobleaching recovery kinetics. Biophys. J. 16:1055–1069.

[160] Cha, B., A. Kenworthy, R. Murtazina, and M. Donowitz. 2004. The lateral mobility of NHE3 on the apical

membrane of renal epithelial OK cells is limited by the PDZ domain proteins NHERF1/2, but is dependent

on an intact actin cytoskeleton as determined by FRAP.J. Cell. Sci. 117:3353–3365.


[161] Matsumoto, G., A. Stojanovic, C. I. Holmberg, S. Kim, and R. I. Morimoto. 2005. Structural properties

and neuronal toxicity of amyotrophic lateral sclerosis-associated Cu/Zn superoxide dismutase 1 aggregates.

J. Cell. Biol. 141:75–85.

[162] Kim, S., E. A. A. Nollen, K. Kitagawa, V. P. Bindokas, and R. I. Morimoto. 2002. Polyglutamine protein

aggregates are dynamic.Nature. Cell. Biol. 4:826–831.

[163] Cho, H., Y. A. Kim, J. Yoon, D. Lee, J. H. Kim, S. H. Lee, and W. Ho. 2005. Low mobility of phos-

phatidylinositol 4,5-bisphosphate underlies receptor specificity of Gq-mediated ion channel regulation in

atrial myocytes.Proc. Natl. Acad. Sci. USA. 42:15241–15246.

[164] Meyvis, T. K. L., S. C. D. Smedt, P. V. Oostveldt, and J. Demeester. 1999. Fluorescence recovery after

photobleaching: A versatile tool for mobility and interaction measurements in pharmaceutical research.

Pharm. Res. 16:1153–1162.

[165] Braga, J., J. M. P. Desterro, and M. Carmo-Fonseca. 2004. Intracellular macromolecular mobility measured

by fluorescence recovery after photobleaching with confocal laser scanning microscopes.Mol. Biol. Cell.

15:4749–4760.

[166] Wey, C. W., R. A. Cone, and M. A. Edidin. 1981. Lateral diffusion of rhodopsin in photoreceptor cells

measured by fluorescence photobleaching and recovery.Biophys. J. 33:225–232.

[167] Soumpasis, D. M. 1983. Theoretical analysis of fluorescence photobleaching recovery experiments.Bio-

phys. J. 41:95–97.

80

Chapter 5

Analysis of membrane-localized binding

kinetics with FRAP1

5.1 Introduction

Many interfacial biological phenomena consist of interactions between molecules that are confined to two appos-

ing membranes, often at the site of a cell-cell contact. A familiar example is the formation of an immune synapse

between a T cell and an antigen presenting cell (APC), drivenby interactions between T cell receptor (TCR) and

peptide-MHC (pMHC) complexes, and a host of coreceptors andadhesion molecules. Immune synapse forma-

tion can also be induced by substrate immobilized lipid bilayers containing a cognate pMHC and the adhesion

molecule ICAM-1 (168). Substrate-immobilized lipid bilayers containing a cognate ligand have also been used

to study stimulated activation of mast cells through the IgE-receptor complex (169). In order to properly under-

stand these phenomena, in immunology and elsewhere, we require good experimental estimates of kinetic binding

parameters for the ligand-receptor interaction. For instance, T cell activation has been shown to depend on TCR-

pMHC binding kinetics (reviewed in (170)).

Measurement of kinetic parameters for ligand-receptor binding can, for example, be performed using a Bia-

core optical sensor (Biacore, AB, Uppsala, Sweden). The receptor of interest is immobilized on the surface of a

chip, and ligands flow through a cell above the chip leading tochanges in its optical properties (171; 172). It is

then possible to estimate the on- and off-rates of the ligand-receptor interaction. However, a limitation of Biacore,

and indeed of most conventional techniques for estimating binding kinetics, is that one or both binding partners

are freely diffusing in a three dimensional volume, whereasin a cell-cell or a cell-bilayer localized interaction

they are restricted to diffuse on two dimensional surfaces.Thus, Biacore measurements give three dimensional

(3D) reaction rates, that depend on the the molar concentration in solution, rather than 2 dimensional (2D) rates

that depend on the surface densities of the the binding partners. Moreover, the receptor is immobilized on the chip

surface and therefore, the potentially important influenceof the plasma membrane is neglected.

1A version of this chapter has been published. Dushek O, Das R,Coombs D (2008) Analysis of membrane-localized binding kinetics withFRAP.Eur Biophys J. 37(5):627-638

Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 81

A few alternative approaches have been used, with varying degrees of success, to estimate 2D binding affini-

ties and reaction rates. Using quantitative fluorescence microscopy, the 2D affinity of the T cell surface adhesion

molecules, CD2 and CD28 to their ligands has been determined(173; 174; 175) as well as the 2D affinity of

certain TCR/pMHC interactions (168). Fluorescence resonance energy transfer (FRET) is another fluorescence

microscopy technique that can be used to observe the association between two fluorophores (176). Because of

its exquisite sensitivity to the distance between two fluorophores, FRET is ideal for measuring nanometer scale

changes in spatial separation. FRET measurements have beenprimarily used for assaying qualitative changes in

the association between interacting proteins (177; 178). Another possibility for quantifying binding interactions

is to use dual color fluorescence correlation spectroscopy (FCS) which measures correlated fluctuations in the

signals from two different fluorophores to infer a binding interaction between them (179; 180). Both FRET and

FCS measurements are, at the present time, technically challenging to implement. Moreover, these techniques

require distinct fluorescent tags on both the binding partners of an interacting pair. These difficulties may explain

why there are very few measurements of kinetic parameters for membrane-bound protein interactions.

In this paper we focus on the technique of fluorescence recovery after photobleaching (FRAP), most com-

monly used to measure diffusion coefficients of biomolecules (181; 182), reviewed in (183). FRAP has recently

been used to measure kinetic parameters of 3D binding interactions between a fluorescently labeled ligand and

its unlabeled binding partner (184; 185; 186; 187; 188), comprehensively reviewed in (189). These studies show

that FRAP can yield 3D kinetic parameters in a variety of settings, but subject to certain assumptions that will

not apply in the case of membrane-restricted binding interactions. For instance, (184; 185; 187) assumed simple

geometries and immobile binding sites for the labeled receptor. More recently, Braga et al. extended this analysis

to the case when the binding sites are also mobile and appliedtheir analysis to examine the diffusion of mRNA

in the the nucleus (188). A potential limitation of all thesestudies, discussed further in (190), is that they yield

compound parameters such as the true equilibrium constant multiplied by the free receptor concentration. Be-

cause the receptor is commonly assumed to be in excess in these studies, some care is required in interpreting the

compound parameters obtained.

Here, we present an experimental protocol that can be utilized to measure true (not compound) kinetic parame-

ters for a binding interaction between two mobile proteins diffusing on nearby membranes. We allow for diffusion

of both the labeled receptor and the unlabeled ligand, and perform our analysis using the rectangular geometry

that is commonly used in cell-surface experiments with a laser confocal scanning microscope. We systematically

identify regions of parameter space in which the kinetic parameters can (theoretically) be measured, as well as

those in which they cannot. Finally, we fit a set of simulated FRAP recovery curves using the outlined protocol,

and we provide guidelines for fitting experimental FRAP recovery curves to infer kinetic binding parameters.

5.2 Modeling

5.2.1 Full model

We begin our analysis with a reaction-diffusion system thatgoverns the dynamics of two diffusing species on a

2-dimensional surface which bind with a 2-dimensional association ratekon, and unbind at a ratekoff . One of the

two species, the receptor, is labeled with a fluorophore, andit binds to a complementary ligand that is unlabeled.

After photobleaching there is a subpopulation of unlabeledreceptors and unlabeled receptor-ligand complexes.

We use the following notation to describe the surface densities of all the relevant species:

f = [labeled receptor], f ′ = [unlabeled (photobleached) receptor],

c = [labeled complex], c′ = [unlabeled (photobleached) complex],

s = [unlabeled ligand],

(5.1)

These quantities evolve according to the PDE system

∂f/∂t = Df∇2f − konfs+ koffc, (5.2a)

∂c/∂t = Dc∇2c+ konfs− koffc, (5.2b)

∂f ′/∂t = Df∇2f ′ − konf′s+ koffc

′, (5.2c)

∂c′/∂t = Dc∇2c′ + konf′s− koffc

′, (5.2d)

∂s/∂t = Ds∇2s− konf′s+ koffc

′ − konfs+ koffc, (5.2e)

whereDf, Ds, andDc are diffusion coefficients of the free receptors, free ligands and bound complexes, re-

spectively. We assume that the system is at chemical equilibrium prior to the FRAP experiment and that the

fluorophores in the bleaching region are instantaneously photobleached at timet = 0. Thus, fort < 0, we have

f(t) = Feq, f ′(t) = 0, c(t) = Ceq, c′(t) = 0, ands(t) = Seq where the subscript eq indicates the equilibrium

density of the three species. The photobleaching event att = 0 creates a new subpopulation of unlabeled receptors

and unlabeled complexes in a localized region, i.e.f ′(t) > 0 andc′(t) > 0 for t > 0. Moreover, because photo-

bleaching only removes the label and does not disturb the chemical equilibrium, we can assume thats(t) = Seq


for all t. Therefore, we define a pseudo on-ratek∗on = konSeq, and rewrite Equations 5.2a and 5.2b as:

∂f/∂t = Df∇2f − k∗onf + koffc, (5.3)

∂c/∂t = Dc∇2c+ k∗onf − koffc. (5.4)

To obtain theoretical FRAP recovery curves we must solve Equations 5.3 and 5.4 with appropriate initial and

boundary conditions reflecting the experimental geometry,and integrate the total fluorescencef + c over the

recovery region. The resulting quantity,Gfm(t), gives the idealized FRAP recovery curve and is a function of

the bleaching geometry, diffusion coefficients, and reaction rates. In a subsequent section we describe how to

efficiently evaluateGfm.

Note that becauses(t) is invariant in time, the dynamics of the labeled species aredecoupled from those of

the unlabeled species, resulting in a simpler system of equations to analyze. With these assumptions, Equations

5.3 and 5.4 govern the dynamics of the two labeled species andby solving them we obtain expressions for the ob-

served fluorescence recovery (see below). As illustrated inFigure 5.1, we assume a rectangular bleaching region

of dimensionsXb andYb along thex andy directions, respectively, and a monitoring region of dimensionsXm

by Ym that is a smaller subset of this bleaching region.

Xb

Xm

Yb

Ym

x

y

monitoring

bleached

region

region

Figure 5.1: The geometry of a typical confocal FRAP experiment.

5.2.2 Reduced models

The full model defined by Equations 5.3 and 5.4 can be well-approximated by four simpler models in certain

situations. These model reductions are briefly described below, and formally derived in Appendix D. In each of


these reduced models (1-4 below), the spatio-temporal dynamics are governed by simple diffusion equations of

the form

∂u/∂t = D∇2u (5.5)

whereD is a relevant diffusion coefficient for that model. The problem of fitting FRAP data to estimateD can

then be solved as usual for the experimental geometry that isused (181; 191; 192; 193). The solution for the

geometry shown in Figure 5.1 is described below.

1. Weighted diffusion: One limit arises whenk∗on andkoff are not well-separated, but the lifetimes of the free and

bound states are much smaller than their characteristic diffusive timescales. In this regime, the labeled receptor

switches rapidly between its unbound form, with diffusion coefficientDf, and its bound form with diffusion

coefficientDc. The FRAP recovery curveGwd(t) = U(t,Dwd) (Equation 5.9) then depends only on the effective

(or weighted) diffusion coefficient:

Dwd =1

1 +K∗Df +K∗

1 +K∗Dc (5.6)

whereK∗ = k∗on/koff. In this regime, onlyDwd can be determined from the data.

2. Independent diffusion: When the diffusive timescales are much shorter than the lifetimes of the free and

bound states, a labeled receptor remains in its (un)bound state on the timescale of the FRAP experiment. In this

case, there is no exchange between the pools of free and boundfluorophores, and FRAP recovery is a result of

“independent diffusion” of these two pools with diffusion coefficientDf andDc, respectively. In this case, the

FRAP recovery curveGid(t) is simple a weighted average of two independent FRAP recovery curves, (Gpf(t) for

free receptors andGpc(t) for bound receptors) :

1

1 +K∗Gpf(t) +K∗

1 +K∗Gpc(t). (5.7)

3. Receptor diffusion dominant: Whenkoff k∗on, the bound complex rapidly dissociates, and the labeled

receptor is almost always in the free state. The FRAP recovery curveGpf(t) is then a function ofDf alone. This

regime is a subset of both the weighted and independent diffusion regimes.

4. Complex diffusion dominant: Whenk∗on koff , the labeled receptor is almost always in the bound state.

The FRAP recovery curveGpc(t) is then a function ofDc alone. Like the receptor diffusion dominant regime,


this regime is a subset of both the weighted and independent diffusion regimes.

The crucial difference between the full model and the reduced models is that, for the full model, but not any

of the reduced models, all four parameters,Df,Dc, k∗on andkoff are required to completely describe the FRAP re-

covery curves. In contrast, in parameter regimes where the full model is reduced to one of these effective models,

FRAP recovery curves are well-described by a smaller subsetof parameters. Thus, in the weighted and indepen-

dent diffusion regimes, the necessary parameters areDf, Dc andK∗, and in the two diffusion dominant regimes,

onlyDf orDc are sufficient to fully describe the FRAP recovery curves. Therefore in parameter regimes where

one of these reduced models is applicable, one cannot, with any statistical confidence, estimate all four model pa-

rameters. Specifically, in the two diffusion dominant regimes no binding parameters can be estimated, and in the

weighted and independent diffusion regimes, only the binding affinityK∗ can be estimated. This categorization

of the FRAP recovery curves is the basis of our proposed protocol described below.

5.2.3 Evaluation of theoretical FRAP recovery curves

Simple FRAP analysis In typical surface photobleaching experiments, the photobleached area is much smaller

than the total area of the plasma membrane. Therefore, we canassume that the domain is effectively infinite in

extent without introducing appreciable error. For the parameter regimes that reduce to simple diffusion (cases

1-3), the FRAP recovery curve is determined by Equation 5.5 with initial conditions

u(t = 0) = 1 − [H(x+Xb/2) −H(x−Xb/2)] [H(y + Yb/2)−H(y − Yb/2)] (5.8)

whereH(x) is the Heaviside step-function. This is easily solved by Fourier transform and the desired FRAP

recovery curve is then obtained by integratingu over the monitoring region (−Xm/2 ≤ x ≤ Xm/2, −Ym/2 ≤

y ≤ Ym/2), to obtain the following expression for the theoretical FRAP recovery curve (193):

U(t,D) = 1 −√Dt

2πXmYm

√π [X+ erf(X+) −X− erf(X−)] +

[

exp(−X2+) − exp(−X2

−)]

×√

π [Y+ erf(Y+) − Y− erf(Y−)] +[

exp(−Y 2+) − exp(−Y 2

−)]

(5.9)

where we define

X± =Xb ±Xm

4√Dt

and Y± =Yb ± Ym

4√Dt

. (5.10)


This expression can be fit to data to determine the parameterD.

FRAP with binding kinetics We are now concerned with solving Equations 5.3 and 5.4. Taking the Fourier

transform (denoted with hats), we obtain the linear system

∂f/∂t = Df(−4π2q2x − 4π2q2y)f − k∗onf + koff c, (5.11a)

∂c/∂t = Dc(−4π2q2x − 4π2q2y)c+ k∗onf − koff c. (5.11b)

We defineqf = 4π2Df(q2x + q2y) + k∗on andqc = 4π2Dc(q

2x + q2y) + koff to obtain

∂f

∂t= −qf f + koff c, (5.12a)

∂c

∂t= k∗onf − qcc, (5.12b)

whose solutions are

f =1

v

[

(1/2)f(t = 0) (exp(D1t)(qc − qf + v) + exp(D2t)(−qc + qp + v))

+ koff c(t = 0)(exp(D1t) − exp(D2t))] , (5.13a)

c =1

v[(1/2)c(t = 0) (exp(D1t)(−qc + qf + v) + exp(D2t)(qc − qf + v))

+ k∗onf(t = 0)(exp(D1t) − exp(D2t))]

(5.13b)

wherev = [(qc − qf )2 + 4k∗onkoff ]1/2,D1 = (−qc − qf + v)/2,D2 = (−qc − qf − v)/2. f(t = 0) andc(t = 0)

are Fourier transforms of the initial conditions,

f(t = 0) = −Feqsinc(πXbqx) sinc(πYbqy) + Feqδ(qx)δ(qy) (5.14a)

c(t = 0) = −Ceqsinc(πXbqx) sinc(πYbqy) + Ceqδ(qx)δ(qy). (5.14b)

whereFeq = koff/(k∗on + koff) andFeq = k∗on/(k

∗on + koff). We obtain these relationships by assuming that the sys-

tem is in equilibrium prior to the FRAP experiment (k∗onFeq = koffCeq) and the FRAP recovery can be normalized

to unity (Feq + Ceq = 1), see (185).

It is not possible to obtain an analytical inverse Fourier transform of these equations, and therefore theoretical

FRAP recovery curves must be numerically computed. These are obtained by integrating the sumf + c over the

monitoring region, and dividing by the area of the monitoring region to obtain the mean fluorescence intensity

over this area. Thus, for a rectangular monitoring region ofdimensionsXm(≤ Xb) by Ym(≤ Yb), the recovery

curve is given by

Gfm(t) =

(

∫ Xm/2

−Xm/2

dx

∫ Ym/2

−Ym/2

dy [f(t, x, y) + c(t, x, y)]

)

/(XmYm) , (5.15)

and, by definition of the inverse Fourier transform,

Gfm(t) =1

4XmYm

∫ Xm

−Xm

dx

∫ Ym

−Ym

dy

∫ ∞

−∞dqx

∫ ∞

−∞dqy

[

f(t, qx, qy) + c(t, qx, qy)]

× exp(2πıqxx) exp(2πıqyy)

=

∫ ∞

−∞dqx

∫ ∞

−∞dqy

[


sinc(πqxXm) sinc(πqyYm)

= 4

∫ ∞

0

dqx

∫ ∞

0

dqy

[


sinc(πqxXm) sinc(πqyYm), (5.16)

which we compute by numerical integration. We found integrating to 15-20 inqx andqy yields sufficient accu-

racy for our purposes. Using the Matlab functionquadl, the above integral can be evaluated 100 times (typical

number of images in FRAP experiments) in approximately 3 seconds, allowing parameter fits in a reasonable

timeframe using a standard fitting procedure. All fits presented in this work were carried out using the Matlab

functionlsqcurvefit.

5.2.4 Structure of parameter space and model reductions

In the model reductions described above, we have made no assumptions about the relative magnitudes ofDf and

Dc. If Df = Dc then we can see by adding Equations 5.3 and 5.4 that the full system reduces to a single diffusion

equation and no binding information can be obtained. More generally, ifDf andDc are close to each other, the

region in parameter space where the full model applies is small. On the other hand, if the two diffusion coefficients

are well separated then the analysis of Sprague et al (185) applies (see Appendix D).

We illustrate how the separation betweenDf andDc impacts the regions of parameter space where the model

reductions are relevant in Figure 5.2. We evaluate the sum ofthe squared residuals (SSR) between the full model

and either the weighted or the independent diffusion reductions. In Figure 5.2a-c we plot SSR contours as a

function ofk∗on andkoff for biologically relevant values ofDf andDc (194; 195). Following (185), we consider

the model reduction a good estimate to the FRAP recovery if SSR< 1. We note that the closer the two diffusion

coefficients are to each other, the smaller is the region where only the two-parameter full model provides a better

fit to the data (compare Figure. 5.2, a and b). When the diffusion coefficients differ only by a factor of two, the


region where only the two-parameter full model disappears (Figure. 5.2 c). The structure of parameter space is

also illustrated in Figure 5.3d where all four regimes are shown (discussed below). As discussed in the following

section, we utilize the existence of these different regions in thek∗on − koff parameter-space to quantify binding

kinetics of the receptor-ligand interaction.

(a) (b)

0.1

0.1

0.1

0.1

0.1

0.1

0.1

1

1

1

1

1

0.1

0.10.1

0.1

0.1

0.1

0.1

0.1

1

1

1

1

1

1

log(koff

)

log(

kon*

)

Df=0.50 µm2/s

Dc=0.05 µm2/s

−4 −3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3Weighted DiffusionIndependent Diffusion

0.1

0.1

0.10.1

0.1

0.1

1

1

1

1

1

0.1

0.1

0.1

0.1

0.1

0.1

1

1

1

1

1

log(koff

)

log(

kon*

)

Df=0.50 µm2/s

Dc=0.10 µm2/s

−4 −3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2


(c)

0.1

0.1

0.1

0.1

0.1

0.1

0.5

0.50.5

0.5

0.1

0.1

0.1

0.1

0.1

0.5

0.5 0.5

0.5

log(koff

)

log(

kon*

)

Df=0.50 µm2/s

Dc=0.25 µm2/s

−4 −3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2


Figure 5.2: Quantifying full model limits. We compute the sum of the squared residuals (SSR) between the fullmodel, Equation 5.16, and cases 1 and 2 for 200 equally spacedtime points reaching 99% recovery. We plot SSRcontours as a function ofk∗on andkoff . In (a-c) we see that the region where the full model is valid progressivelydecreases asDc approachesDf. Parameters:Xb = Yb = Xm = Ym = 2µm.

5.3 Proposed experimental protocol

In the previous section we identified four parameter regimes, in which the combined reaction-diffusion system

(Equations 5.3 and 5.4) is effectively reduced to a single diffusion equation (Equation 5.5). As discussed above,

in those parameter regimes, it is not possible to estimate the binding parametersk∗on andkoff independently. To do

so, we must first identify regions of parameter space where the full model fits the data, i.e. outside regimes 1-4.

In this section, we propose an experimental protocol that, under suitable conditions, can be used to obtain these

binding parameters by fitting a series of FRAP recovery curves. We consider an experimental system consisting

of a cell bearing fluorescently labeled receptors on its surface, that is in contact with a suspended planar bilayer

bearing unlabeled ligands at concentrationST. We assume that the system is at chemical equilibrium beforeeach

experiment (achieved by waiting a sufficiently long time after the initial contact between the cell and the bilayer).

A. Measuring diffusion coefficients The first step is to establish the diffusion coefficientsDf andDc, if they

are not already known. This can be done by performing a straightforward FRAP experiment in the absence of

the ligand (i.e.ST = 0), and fitting the recovery curve to Equation 5.5 to establishDf. The FRAP experiment

can then be repeated with a saturating density of the ligand (such that all the receptors are engaged) to establish

Dc. These two FRAP experiments correspond to the two pure-diffusion limits of the full model (cases 3 and 4),

respectively. IfDf is found to be close toDc then successful measurement of the binding kinetic parameters is

not expected in part B.

B. Ligand titration experiment Having established the diffusion coefficients of the free and bound forms, the

key experiment is to ‘titrate’ the ligand density, by performing FRAP experiments with a range of intermediate

ligand densities. As the total ligand density,ST , increases, the equilibrium free ligand concentration,Seq, also

increases according to the following equation (derived from a conservation equation):

Seq =K(ST − FT ) − 1 +

√

[1 +K(ST + FT )]2 − 4STFT

2K, (5.17)

whereFT is the total receptor concentration andK = kon/koff is the 2D equilibrium constant. As a result, the

pseudo-on rate,k∗on = konSeq, also increases, and the system traverses thek∗on − koff parameter-space parallel to

thek∗on axis (Figure 5.3d). As shown in Figure 5.3d, for differentk∗on values, a different model will best describe

the FRAP recovery curve. In other words, as the total ligand concentration,ST , is varied in a controlled fashion,

each resulting FRAP recovery curve is best fitted by a particular model, depending on the location of the system

on thek∗on − koff phase plane.

In Figure 5.3a-c the recovery dynamics are progressively slower as the ligand density increases because

receptor-ligand binding effectively slows down the diffusion of the receptors,Dc < Df . The idea is to perform a

series of FRAP experiments traversing thek∗on−koff phase plane (Figure 5.3d) and, for each experiment, establish

which model best describes the FRAP recovery curve. For example, for the recovery curves shown in Figure 5.3b,

with koff = 0.01s−1, the system traverses from case 3 (receptor diffusion dominant), via the full model regime, to

case 4 (complex diffusion dominant).

C. Data fitting The goal is now to classify each of the recovery curves for intermediate ligand densities into case

1, case 2 or the full model. Each recovery curve is initially fit using all three different models: the two-parameter

full model (Equation 5.16), a one-parameter diffusion model (case 1;Equations 5.6,5.9), and a one-parameter

independent diffusion model (case 2; Equations 5.7,5.9). For each fit, we determine the best-fit value of the

parameter(s) for that model. The relative quality of the three fits are then compared on the basis of the sum-of-

squared-residuals (SSR), where a lower SSR indicates a better fit. However, to compensate for the difference in

number of parameters, we use Akaike’s Information criterion (AIC) as a metric for comparing the quality of fits

(196; 197). AIC, defined as:

AIC = n ln(SSR/n) + 2P (5.18)

wheren is the number of data points, andP is the number of parameters, balances the gain due to a lowered SSR

with the penalty for an increase in the number of parameters.The model with the lowest AIC value is considered

to be the best descriptor of the experimental data.

We use the quality of fits, as quantified by AIC (Equation 5.18), to select the appropriate model(s) for param-

eter estimation. For each recovery curve, we identify the fitwith the lowest AIC value. The other two fits are

quantified by the difference in their AIC value from this fit. The difference,∆AIC, is a measure of the probability

of the other model being an equally likely descriptor for theFRAP recovery curves. The probability of each model

is given by

p =exp(−∆AIC/2)

1 + exp(−∆AIC/2). (5.19)

In practice, a fit with∆AIC ≥ 10 indicates that the model under consideration is extremely unlikely compared to

the best model, and a fit with∆AIC ≤ 2 indicates that the model is nearly as likely as the best model(197).

We use the relative quality-of-fit for the three models to determine (for each one of the the FRAP experiments)

whether or notk∗on, koff andK∗ can be found. Three scenarios emerge:

Scenario 1. ∆AIC < 10 for both one-parameter models. The system is in a pure diffusion limit and no estimates

of binding parameters can be obtained.

Scenario 2. ∆AIC < 10 for exactly one of the one-parameter models. The system is inthe regime best described

by that model, and the best-fit value of the pseudo-affinityK∗ is obtained.

Scenario 3. ∆AIC ≥ 10 for both the one-parameter models. The system is in the full model regime and best-fit

values ofk∗on andkoff are obtained.

This procedure rejects any fits with∆AIC ≥ 10 as being too unreliable for parameter estimation. Categoriz-

ing the fits into these three scenarios is motivated by the existence of different model regimes in thek∗on − koff

parameter space. As shown in Figure 5.3d, the points corresponding to the recovery curves for a titration will

transition from one diffusion-dominant regime to another diffusion-dominant regime as the ligand concentration

is increased. In the diffusion-dominant regimes, the recovery dynamics are well-described by either of the two

one-parameter models (scenario 1). For intermediate concentrations, however, the system is in either the full

model regime, or in a one-parameter model regime. When the system is in a weighted-diffusion or independent-

diffusion regime, the recovery dynamics are best fitted exclusively by the corresponding one-parameter model,

and not the other one (scenario 2). Finally, in scenario 3, the system is in the full model regime, and neither of the

two one-parameter reductions are appropriate descriptorsof recovery dynamics.

D. Parameter estimation In scenario 1, the system is in a regime where no binding information can be obtained.

In scenario 2, we determine the pseudo-affinityK∗ using the one-parameter model with∆AIC < 10. If this is

the independent diffusion model, then the parameterK∗ is directly determined from the fit. If this is the weighted

diffusion model, then the fitted parameter isDwd, andK∗ is obtained by rearranging Equation 5.6:

K∗ =Df −Dwd

Dwd −Df(5.20)

Additionally, in scenario 2, it is also possible that∆AIC < 10 for the full model, and we have another estimate

for K∗ from the full model fit, given by the ratiok∗on/koff. We combine the parameter estimates from the two fits

by using the probability,p (Equation 5.19), of each model as a weight, and calculate theweighted mean of the

best-fitK∗. In scenario 3 fitting the full model provides estimates ofk∗on andkoff, as well asK∗ = k∗on/koff.

Estimates of the pseudo-affinity,K∗ (scenario 2) and the pseudo-association rate,k∗on (scenario 3), can be con-

verted to the true 2D affinity,K = K∗/Seq and association rate,kon = k∗on/Seq using the free ligand concentration

Seq, given by:

Seq = ST − K∗

1 +K∗FT , (5.21)

whereST andFT are the known total ligand and total receptor concentrations, respectively, and we use the best-fit

estimate ofK∗ from the fits. However, the estimate of the true parameters isunreliable whenSeq ≈ 0 because

of the small value of the denominator in the expressions forkon andK. In practice, we find that when the ratio


Seq/ST becomes small, that is, at equilibrium only a negligible fraction of the total ligand is unbound, the esti-

mates of the true parameters can deviate significantly from their actual value.

5.4 Simulated experiments

5.4.1 Simulated FRAP titrations

To test our protocol, we simulated three sets of numerical experiments, and applied the above data-fitting and

parameter-estimation procedure to the simulated datasets. These simulations model the interaction between a

receptor-ligand pair, with a fixedkon, and three differentkoff values, representing, for example, a range of bind-

ing affinities of pMHC for a particular TCR. Table 5.1 lists the parameter values used to generate the simulated

data. We use biologically relevant reaction rates between cell surface proteins (168; 198), typical diffusion coeffi-

cients for membrane proteins (194; 195), and commonly used sizes for the bleach and monitoring regions (195).

The Einstein-Stokes relation predicts that if two particles with individual diffusion coefficientsDf andDs bind

together, the complex will have diffusion coefficientDfDs/(Df +Ds) (assuming their drag coefficients are addi-

tive). SupposingDf ' Ds for similar surface proteins, this estimate yields that thecomplex will be roughly half

as diffusive. However, proteins diffusing in a lipid membrane may not behave as ideal Brownian particles (199),

and upon binding the resulting complex, that may undergo a conformational change or begin signaling, may have

a drag coefficient different from the sum of the individual drag coefficients. We therefore take this estimate as

only an upper bound onDc, and assumeDc = 0.1Df in our simulations.

Table 5.1: Parameters used for numerically simulating FRAPrecovery curves shown in Figure 5.3.

Parameter ValueDf 0.5 (µm)2 s−1

Dc 0.05 (µm)2 s−1

FT 100 (µm)−2

Xb 6 µmYb 3 µmXm 2 µmYm 3 µmkon 0.01 (µm)2s−1

10−4 s−1 (Experiment A, Figure 5.3a)koff 10−2 s−1 (Experiment B, Figure 5.3b)

1 s−1 (Experiment C, Figure 5.3c)


For each experiment, we chose a set of increasingST values to simulate the effect of titrating in an increasing

density of pMHC into the lipid bilayer. The corresponding recovery curves are shown in Figures 5.3a-c. Individ-

ual recovery curves were generated using the full model for the given choice of parameter values. 10% Gaussian

noise was added to the solution, to model typical experimental errors in FRAP measurements. Each FRAP recov-

ery curve shown in Figures 5.3a-c is the average of ten such individual recovery curves. The location of points on

thek∗on − koff parameter-space are indicated by corresponding colored dots in Figure 5.3d. The SSR contours in

Figure 5.3d indicate that for differentST values (i.e. differentk∗on values through Equation 5.17) a different model

will best describe the FRAP recovery.

(a) (b)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Time (s)

Nor

mal

ized

Inte

nsity

, G(t

)

koff

=0.0001 s−1k

on=0.01µm2s−1

ST=0 µm−2

ST=1µm−2

ST=40µm−2

ST=80µm−2

ST=200µm−2

ST=600µm−2

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Time (s)

Nor

mal

ized

Inte

nsity

, G(t

)

koff

=0.01 s−1k

on=0.01µm2s−1

ST=0 µm−2

ST=1µm−2

ST=40µm−2

ST=80µm−2

ST=200µm−2

ST=600µm−2

(c) (d)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Time (s)

Nor

mal

ized

Inte

nsity

, G(t

)

koff

=1 s−1k

on=0.01µm2s−1

ST=0 µm−2

ST=1µm−2

ST=50µm−2

ST=200µm−2

ST=1000µm−2

ST=5000µm−2

log(koff

)

log(

kon*

)

Df=0.5 µm2/s

Dc=0.05 µm2/s

Diffusion Dominant

WeightedDiffusion

IndependentDiffusion

Diffusion Dominant

FullModel

−6 −4 −2 0 2 4−6

−4

−2

0

2

4

Figure 5.3: Numerically simulated FRAP titration experiments demonstrate the effect of increasing ligand densityon FRAP recovery dynamics. FRAP curves (a-c) were numerically simulated as described in the text, withparameter values listed in Table 5.1, and the total ligand densityST indicated in the legend. The correspondingpoints on thek∗on− koff parameter space are shown in (d), along with the contours indicating the applicable modelregimes. Only contours of SSR=1 are shown (grey - pure diffusion (free), dotted grey - pure diffusion (bound),black - weighted diffusion, dotted black - independent diffusion). The contour lines were generated as describedin Figure 5.2.

5.4.2 Fitting simulated data

We applied the fitting procedure described above to the simulated FRAP titrations. For each experiment, we first

fit the recovery curves withST = 0 to Equation 5.5, and determined the best-fit values ofDf = [0.50, 0.50, 0.49]


(µm)2s−1, for experiments A-C (Figure 5.3, a-c), respectively. Similarly, for each experiment, we also fit the

titrations with the highestST value and obtained the best-fit values ofDc = [0.050, 0.051, 0.059] (µm)2s−1, re-

spectively.

Having established the diffusion coefficient of the free receptor and bound complex, we next fit the interme-

diate FRAP recovery curves in each experiment to all three models, i.e. weighted diffusion (case 1), independent

diffusion (case 2), and the full model. We used the best-fit values ofDf andDc, as determined from fitting the

two limiting FRAP curves, as known constants in these equations. Thus, for each intermediate FRAP curve in a

titration we obtained the best-fit value of the the model parametersDwd,K∗, k∗on andkoff . These best-fit parameter

values, the SSR from the corresponding fits, and the resulting AIC statistics for the intermediate FRAP recovery

curves from all three titrations are shown in Appendix E.

Based on the∆AIC values of the three fits for each curve, we identify the particular scenario that is applicable

for parameter estimation. As described in detail above, fora fit in scenario 2 we can determine the pseudo-affinity,

K∗, and for a fit in scenario 3 we can also determine the pseudo-onratek∗on and the off-ratekoff . In Table 5.2

we list the identified fitting scenario for each FRAP curve, and parameter estimates for fits in scenarios 2 and

3. Comparing the fitted parameter values with those used for simulating the data, we note that for all fits under

scenarios 2 and 3, the fitted parameter values well match their actual values.

We also list the ratioSeq/ST in Table 5.2. This is determined using the knownST , andFT values, and the

fittedK∗ values. When this ratioSeq/ST > 0.01, we are additionally able to estimate the true 2D on ratekon

(Experiment B), and the true 2D affinityK∗ (Experiments, B and C) for the ligand-receptor pair. The ratio Seq/ST

measures the fraction of ligands that are in an unbound stateat equilibrium. Thus, this fraction is small for high

affinity ligands, and also at low ligand densities. Thus, under these conditions, we are unlikely to be able to re-

liably determine the true 2D kinetic parameters. In the numerical simulations above, the receptor-ligand affinity

is the greatest for experiment A, for which we observe the smallest values of this ratio. For the lower affinities

in experiments B and C,Seq/ST is large enough for progressively higher ligand concentrations, allowing reliable

estimates of the true 2D parameters.

5.5 Experimental outlook

In the previous sections we proposed a confocal FRAP-based protocol to measure true 2D binding parameters be-

tween a membrane-associated receptor-ligand pair, and tested its applicability with numerically simulated FRAP


Table 5.2: Parameter values used in numerical simulations,and the estimates from fitting the simulated FRAPtitration data.

Simulation parameterskoff K ST K∗ k∗on Fit Scenarios−1 (µm)−2 (µm)−2 s−1

1 0.0101 1.01×10−6 10.0001 100 40 0.666 6.66×10−5 2(Experiment A) 80 3.99 3.99×10−4 2

200 1.00×10−4 1.00 1

1 0.0100 1.00×10−4 10.01 1 40 0.649 6.49×10−3 3(Experiment B) 80 3.29 0.0329 3

200 1.01×10−2 1.01 1

1 5.01×10−3 5.01×10−3 11 0.01 50 0.281 0.281 2

(Experiment C) 200 1.41 1.41 21000 9.10 9.10 2

Fitted ParametersExpt. ST K∗ k∗on koff Seq/ST K kon

(µm)−2 s−1 s−1 (µm)−2 (µm)2 s−1

A 140 0.667 1.25 × 10−4

80 3.88 0.00631200

B 140 0.692 7.67×10−3 0.0111 -0.022280 3.31 0.0309 0.00933 0.0401 1.03 0.00963200

C 140 0.269 0.576 0.0093480 1.46 0.703 0.0104200 11.082 0.908 0.0122

recovery curves. Here we point out some experimental challenges that need to be overcome for a successful ap-

plication of this protocol.

A key requirement of our protocol is the prior knowledge of the diffusion coefficients of free and bound

forms of the receptor. We suggest conducting FRAP experiments in the absence of any ligand to establish the

former, and in the presence of a saturating concentration ofthe ligand, to establish the latter. A limitation of this

approach may be the low surface density of free receptors resulting in a low signal to noise ratio in the FRAP ex-

periments. Nonetheless, translational diffusion coefficients of a variety of cell-surface proteins, including a GFP

conjugated TCR, have been successfully measured using FRAP(194; 195), and we expect that for an experimen-

tal system of interest such measurements will be feasible. Alternatively, overexpressing the receptor of interest

could further enhance the signal to noise ratio for receptors with low surface density. When the receptors are

engaged, possible complications in FRAP measurement may arise from clustering of receptors into microclusters,

observed for TCR-pMHC interactions (200) or a substantial immobile fraction (194). However, these effects are

cytoskeleton-dependent, and are abrogated in the presenceof pharmacological inhibitors of actin polymerization

such as latranculin-A.

The more challenging requirement of our protocol is the accurate and reproducible measurement of the sur-

face densities of both binding partners. In cells expressing a genetically encoded fluorescently-labeled protein,

one often observes a heterogenous population with varying expression levels, that may be difficult to quantify,

without a detailed knowledge of the photophysical properties of the fluorophore in an intracellular environment.

Fluorescent calibration beads have been successfully usedto quantify the surface density of TCR on cells in simi-

lar experiments (200). Further, it is possible to maintain arelatively uniform expression level by isolating a single

clone and proliferating it, or alternatively by sorting cells on the basis of fluorescence intensity and retaining a

population with a relatively homogenous expression level.Likewise, by loading liposomes with varying concen-

trations of the specific ligand during bilayer preparation,it is possible to control the ligand surface density on a

lipid bilayer (168; 201).

Lastly, as with all FRAP experiments care must be taken to minimize errors due to the finite bleach time

(202; 203), the shape of the bleaching beam (202), the finite domain (193). Incomplete FRAP recovery, in the

absence of binding, could arise from an immobile fraction (203). By definingMf to be the fraction of proteins

that are mobile, we can correct for situation whenMf < 1 by simplying multiplying the FRAP recovery by this

fraction (see (193; 203)). Additionally, our method and indeed most common FRAP analyses are not applicable

when there is appreciable membrane curvature. For a detailed discussion of the applicability of FRAP to curved


membranes see (204).

5.6 Conclusions

We have outlined an experimental protocol for determination of kinetic parameters for membrane-restricted bind-

ing of a ligand and receptor, based on titrating the density of the unlabeled substrate (ligand) and measuring

FRAP recovery dynamics of the labeled receptor. As the ligand density increases, the system transitions from

a diffusion-dominant regime where the protein is predominantly unbound, to another diffusion-dominant regime

where the protein is predominantly bound. The passage between these two diffusion dominant regimes is through

an intermediate regime where either the full model, or one ofeither the weighted-diffusion model, or the indepen-

dent diffusion model are applicable.

As described above, and illustrated in Figure 5.2, the extent of this intermediate regime in parameter space

is determined by the separation between the diffusion coefficients of the free and the bound forms of the fluo-

rophore. Thus, when the diffusion coefficients of the bound and free forms of a protein are sufficiently different,

this method allows for an estimation of the true two-dimensional on and off rates for the binding or the true two-

dimensional affinity. Alternatively, if the diffusion coefficients are not sufficiently different, this technique cannot

discern any kinetic information. Due to this constraint, itmay be necessary to slow down the diffusion of the

ligand (and therefore, presumably of the bound complex), for instance via adhesion to a large, or slowly-diffusing

particle.

A distinct advantage of our technique is the need to fluorescently label only one of the binding partners. Fur-

ther, the feasibility of our approach is at least partially evident from a number of recent papers where FRAP has

been used to measure 3D binding constants in a variety of settings (e. g. (184; 185; 187; 188)). The study of

Sprague et al. assumed that the unlabeled binding partner was immobile, whereas we analyze a more general

model without this constraint. Further, for sufficiently different diffusion coefficients, our method allows us to

fit for the 2D rate constantkon rather than the compound ratek∗on = konSeq. We additionally specialize to the

commonly used geometry of confocal FRAP on cell surfaces. Our method also represents an alternative to that of

(188), where FRAP measurements were used to elucidate the binding-mediated diffusion of mRNA fragments in

the nucleus of HeLa cells. In that study, a relatively complex method was used to fit the data, involving repeated

numerical solution of the partial differential equations for the diffusing species. Our fitting technique is concep-

tually simpler, involving fitting only a relatively simple expression to the data.


We also point out some experimental challenges for a successful implementation of our protocol. One of the

key requirement of our method is an accurate estimation of the surface density of the two binding partners, and we

foresee this to be a key experimental challenge toward a successful implementation. Our analysis has also ignored

other sources of experimental error such as background photobleaching of the fluorophore, fluorescence recovery

during the finite bleaching time, geometric effects stemming from the beam shape, etc. discussed elsewhere (183).

Notwithstanding these difficulties, our proposed experimental technique offers a conceptually simple alternative

to existing methods for quantifying receptor-ligand binding interactions on membranes.

99

References



[169] Wu, M., D. Holowka, H. G. Craighead, and B. Baird. 2004.Visualization of plasma membrane compart-

mentalization with patterned lipid bilayers.Proc Natl Acad Sci U S A. 101:13798–803.

[170] Coombs, D., and B. Goldstein. 2005. T cell activation:Kinetic proofreading, serial engagement and cell

adhesion.J. Comp. Appl. Math. 184:121–139.

[171] Garland, P. B. 1996. Optical evanescent wave methods for the study of biomolecular interactions.Q. Rev.

Biophys. 29:91–117.

[172] Rich, R. I., and D. G. Myszka. 2000. Advances in surfaceplasmon resonance biosensor analysis.Curr.

Opin. Biotech. 11:54–61.

[173] Dustin, M. L., L. M. Ferguson, P. Y. Chan, T. A. Springer, and D. E. Golan. 1996. Visualization of

CD2 interaction with LFA-3 and determination of the two-dimensional dissociation constant for adhesion

receptors in a contact area.J Cell Biol. 132:465–74.

[174] Dustin, M. L., D. E. Golan, D. M. Zhu, J. M. Miller, W. Meier, E. A. Davies, and P. A. van der Merwe.

1997. Low affinity interaction of human or rat T cell adhesionmolecule CD2 with its ligand aligns adhering

membranes to achieve high physiological affinity.J Biol Chem. 272:30889–98.

[175] Bromley, S. K., A. Iaboni, S. J. Davis, A. Whitty, J. M. Green, A. S. Shaw, A. Weiss, and M. L. Dustin.

2001. The immunological synapse and CD28-CD80 interactions. Nat Immunol. 2:1159–66.

[176] Selvin, P. R. 2000. The renaissance of fluorescence resonance energy transfer.Nat. Struct. Biol. 7:730.

[177] Zal, T., and N. R. Gascoigne. 2004. Using live FRET imaging to reveal early protein-protein interactions

during T cell activation.Curr. Opin. Immunol. 16:418.

[178] Hoffmann, C., G. Gaietta, M. Bunemann, S. R. Adams, S.Oberdorff-Maass, B. Behr, J. P. Vilardaga, R. Y.

Tsien, M. H. Ellisman, and M. J. Lohse. 2005. A FlAsH-based FRET approach to determine G protein-

coupled receptor activation in living cells.Nature Methods. 2:171.


[179] Kim, S. A., and P. Schwille. 2003. Intracellular applications of fluorescence correlation spectroscopy:

prospects for neuroscience.Curr Opin Neurobiol. 13:583–90.

[180] Kohl, T., E. Haustein, and P. Schwille. 2005. Determining protease activity in vivo by fluorescence cross-

correlation analysis.Biophys J. 89:2770–82.

[181] Axelrod, D., D. E. Koppel, J. Schlessinger, E. Elson, and W. W. Webb. 1976. Mobility measurement by

analysis of fluorescence photobleaching recovery kinetics. Biophys. J. 16:1055–1069.

[182] Schlessinger, J., D. E. Koppel, D. Axelrod, K. Jacobson, W. W. Webb, and E. L. Elson. 1976. Lateral

transport on cell membranes: mobility of concanavalin A receptors on myoblasts.Proc Natl Acad Sci U S

A. 73:2409–13.

[183] Rabut, G., and J. Ellenberg. 2004. Photobleaching techniques to study mobility and molecular dynamics of

proteins in live cells: FRAP, iFRAP and FLIP.In Live Cell Imaging: A Laboratory Manual. R. D. Goldman,

and D. L. Spector, editors, chapter 6. Cold Spring Harbor Laboratory Press. 101–127.

[184] Carrero, G., D. McDonald, E. Crawford, G. de Vries, andM. J. Hendzel. 2003. Using FRAP and mathe-

matical modeling to determine the in vivo kinetics of nuclear proteins.Methods. 29:14–28.



[186] Beaudouin, J., F. Mora-Bermudez, T. Klee, N. Daigle, and J. Ellenberg. 2006. Dissecting the contribution

of diffusion and interactions to the mobility of nuclear proteins.Biophys J. 90:1878–94.

[187] Hinow, P., C. Rogers, C. Barbieri, J. A. Pietenpol, A. Kenworthy, and E. DiBenedetto. 2006. The DNA

binding activity of p53 displays reaction-diffusion kinetics. Biophys. J. 91:330–342.

[188] Braga, J., J. G. McNally, and M. Carmo-Fonseca. 2007. Areaction-diffusion model to study RNA motion

by quantitative fluorescence recovery after photobleaching. Biophys J. 92:2694–703.

[189] Sprague, B. L., and J. G. McNally. 2005. FRAP analysis of binding: proper and fitting.Trends Cell. Biol.

15:84–91.

[190] Pando, B., S. P. Dawson, D.-O. D. Mak, and J. E. Pearson.2006. Messages diffuse faster than messengers.

Proc. Natl. Acad. Sci. USA. 14:5338–5342.

[191] Ellenberg, J., E. D. Siggia, J. E. Moreira, C. L. Smith,J. F. Presley, H. J. Worman, and J. Lippincott-

Schwartz. 1997. Nuclear membrane dynamics and reassembly in living cells: Targeting of an inner nuclear

membrane protein in interphase and mitosis.J. Cell Biol. 138:1193–1206.




[193] Dushek, O., and D. Coombs. 2008. Improving parameter estimation for cell surface frap data.J Biochem

Biophys Methods. 70:1224–1231.

[194] Favier, B., N. J. Burroughs, L. Wedderburn, and S. Valitutti. 2001. TCR dynamics on the surface of living

T cells. Int Immunol. 13:1525–32.

[195] Kenworthy, A. K., B. J. Nichols, C. L. Remmert, G. M. Hendrix, M. Kumar, J. Zimmerberg, and

J. Lippincott-Schwartz. 2004. Dynamics of putative raft-associated proteins at the cell surface.J Cell

Biol. 165:735–46.

[196] Akaike, H. 1973. Information theory as an extension ofthe maximum likelihood principle.In Second Inter-

national Symposium on Information Theory. B. Petrov, and F.Csaki, editors. Akademiai Kiado, Budapest.

267–281.

[197] Motulsky, H., and A. Christopoulos. 2004. Fitting Models to Biological Data Using Linear and Nonlinear

Regression: A Practical Guide to Curve Fitting,, chapter 22. Oxford University Press, 138–142. Available

online at http://www.graphpad.com/index.cfm?cmd=library.page&pageID=7&categoryID=1.

[198] Davis, M. M., M. Krogsgaard, J. B. Huppa, C. Sumen, M. A.Purbhoo, D. J. Irvine, L. C. Wu, and

L. Ehrlich. 2003. Dynamics of cell surface molecules duringT cell recognition.Ann. Rev. Biochem. 72:717–

742.

[199] Saffman, P. G., and M. Delbruck. 1975. Brownian motionin biological membranes.Proc Natl Acad Sci U

S A. 72:3111–3113.





Immunity. 25:117–127.

[202] Braga, J., J. M. Desterro, and M. Carmo-Fonseca. 2004.Intracellular macromolecular mobility measured

by fluorescence recovery after photobleaching with confocal laser scanning microscopes.Mol Biol Cell.

15:4749–60.




[204] Sbalzarini, I., A. Hayer, A. Helenius, and P. Koumoutsakos. 2006. Simulations of (an)isotropic diffusion

on curved biological surfaces.Biophys. J. 90:878–885.

103

Chapter 6

Effects of intracellular calcium and actin

cytoskeleton on TCR mobility measured

by fluorescence recovery1

6.1 Introduction

The activation of T lymphocytes by antigenic ligands displayed on the surface of antigen presenting cells (APC) is

the central event in developing an adaptive immune response. During the T cell/APC interaction, T cell recpetors

(TCR) bind to peptide-major-histocompatibility-complexes (pMHC) present on the surface of the APC (205). T

cells are observed to respond sensitively and specifically to antigenic stimulation (206; 207).

During the interaction between T cells and APC, TCR and accessory molecules travel towards the T cell/APC

contact site, forming a signaling area at the interface thathas been named the immunological synapse (208; 209).

TCR movement towards the immune synapse has been thoroughlydocumented and shown to reflect the intensity

of antigenic stimulation (210; 211) but we do not have a complete picture of how surface receptor signaling mod-

ulates TCR motion.

TCR motion on live cells has been studied using fluorescence recovery after photo-bleaching (FRAP) and

fluorescence loss in photo-bleaching (FLIP) techniques (212). In a primary study, Sloan-Lancaster et al. showed

that GFP-tagged CD25ζ-chain (i.e. fluorescently labeled TCR) chimeras expressedon the surface of HeLa cells

exhibit very low mobility (212). Following this observation, we employed a TCRβ-chain deficient Jurkat T cell

line transfected with a GFP/fusion TCRβ-chain to measure lateral mobility of fully assembled TCR/CD3 com-

plexes using FRAP. We showed that TCR are mobile on the surface of Jurkat T cells (213). Using a different

approach, M. Krummel and colleagues also showed that TCR aremobile on the surface of murine T cells and that

1Dushek O∗, Mueller S∗, Soubies S, Depoil D, Caramalho I, Coombs D∗, Valitutti S∗ (2008) Effects of intracellular calcium and actincytoskeleton on TCR mobility measured by fluorescence recovery. PLoS One. in press.∗ indicates equal contribution

Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 104

they undergo directed transport towards the immune synapseduring antigen recognition (214). More recently, it

has been shown that TCR dynamically form micrometer-scale clusters that move in a directed fashion towards

the center of the immune synapse during antigen recognition(215; 216; 217). TCR recruitment to the center

of the immune synapse has been associated with productive signaling (209; 218) and with TCR internalization

(217; 219).

The directed motion of TCR requires an energy-consuming mechanism and a natural candidate for this mech-

anism is an interaction between TCR and the cortical actin cytoskeleton (CAC). T cells need a functional actin

cytoskeleton to form productive conjugates with APC (217; 220; 221) and cortical actin has further been impli-

cated in the transport of immune synapse components to the T cell/APC contact (222; 223; 224). However, an

explicit link between the actin cytoskeleton and TCR dynamics has not been established at the molecular level. In

particular it is not clear how activation signals triggeredby TCR engagement might modulate TCR mobility via

CAC-dependent mechanisms.

Here, we report on efforts to characterize TCR mobility in human T lymphocytes using fluorescence recovery

techniques. We stimulated T cells using ionomycin, a drug known to increase the concentration of intracellular

calcium,[

Ca2+]

i. We show that

[

Ca2+]

iincrease markedly reduces TCR mobility on the T cell surfacevia an

actin cytoskeleton-dependent mechanism. We also show thatactin polymerization increases following[

Ca2+]

i

increase, suggesting a direct link between calcium inducedCAC polymerization and constraints on TCR mobility

on the T cell surface.

6.2 Materials and methods

A full discussion of the materials and methods can be found inAppendix F. Briefly, CD4 T cells were isolated from

the blood of healthy donors. The TCR of these cells was fluorescently labeled and visualized using a confocal

laser scanning microscope. Cells were either untreated or treated with various pharmacological agents before

performing FRAP experiments. The protocol used for FRAP experiments is described in (225) (Chapter 4) and

Matlab was used for model fitting. The amount of actin in a filamentous form was quantified using flow cytometry

by fixing cells and staining them with phalloidin.


6.3 Results

6.3.1 TCR mobility is similar in naive and activated human T cells

We initially focused on studying TCR mobility on unstimulated T cells. We used three CD4+ T cell popula-

tions: naive T lymphocytes freshly isolated from cord blood(CBTL); T cells from a line obtained by culturing

for two weeks in vitro naive T cells in the presence of magnetic beads coated with anti-CD3 plus anti-CD28

mAb (activated CBTL); freshly isolated T lymphocytes from peripheral blood of adult healthy donors (PBL). We

stained cells from each population using anti-CD3 Fab antibodies labeled with Cy5 (211; 226) and performed

FRAP experiments. We applied the method described in Dushekat al. (225) to analyze FRAP recovery curves

and extracted the mobility parameters of the TCR. Two parameters were calculated: the diffusion coefficient (D,

generally expressed inµm2/s) and the mobile fraction (Mf ), indicating the fraction of TCR that are mobile on

the timescale of the experiment. The results of this analysis are presented in Figure 6.1 and Table 6.1. TCR had

a diffusion coefficient (D, expressed as mean± SEM)) of 0.048± 0.008µm2/s in CBTL, 0.035± 0.006µm2/s

in activated CBTL and 0.061± 0.008µm2/s in PBL. The values of D were not found to be significantly different

(p>0.05). The mobile fraction (Mf ) of TCR was very high in all cell populations, supporting thenotion that TCR

are constitutively mobile on living T cells (Table 6.1).

These results show that the basic parameters of TCR dynamicson the surface of T cells are not significantly

different in T cells at different stages of activation, indicating that TCR mobility is not involved in allowing acti-

vated T cells to be more responsive to antigenic stimulation.

Table 6.1: Diffusion coefficient and mobile fraction under various pharmacological treatmentsDiffusion Coefficient Mobile Fraction P-Value of Diffusion

Condition n (µm2/s) (Mf ) Relative to PBL∗

PBL 61 0.061±0.008 0.96±0.04 1.00CBTL 20 0.048±0.008 1.00±0.05 0.40Activated CBTL 20 0.035±0.006 1.09±0.04 0.08PBL + Cyto D 20 0.068±0.014 0.93±0.05 0.66PBL + Lat B 10 0.082±0.016 0.95±0.06 0.34PBL + Iono 37 0.017±0.002 1.00±0.04 0.0001PBL + Iono + Cyto D 25 0.043±0.005 0.92±0.04 0.17PBL + Iono + Lat B 10 0.045±0.009 0.99±0.05 0.45∗ P-values were obtained by comparing diffusion coefficientsfrom each experiment to theuntreated PBL data using a two-sample T-test. With the exception of PBL + Iono, no signifi-cant differences in the diffusion coefficient were observed(P-values> 0.05). No significantdifferences in the mobile fraction were observed (all P-values> 0.05).


0 10 20 30 40 50 60 70 80 900.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

No

rma

lize

d I

nte

nsity

PBL

CBTL

Activated CBTL

Figure 6.1: TCR mobility does not depend on the activation state of T cells. Mean FRAP recovery curves areshown for peripheral blood T Lymphocytes (PBL, blue curve n=61), Cord Blood T Lymphocytes either freshlyisolated (CBTL, green curve n=20) or activated in culture (activated CBTL, red curve n=20) were submitted toFRAP experiments. Data are from 2 (for CBTL) and 4 (for PBL) independent experiments.

6.3.2[

Ca2+]

iincrease affects TCR mobility by an actin cytoskeleton dependent

mechanism

We next investigated whether calcium signaling in T cells could affect TCR mobility. To this end, T cells were

loaded with the calcium probe FLUO-4 (a dye that increases its light emission when bound to Ca2+) before

staining with anti-CD3/Cy5 Fab antibodies (the fluorescenttag that binds TCR). This allowed us to detect sig-

nal transduction in parallel with TCR dynamics (211; 227). We stimulated T cells with ionomycin to mimic

the[

Ca2+]

iincrease that follows TCR productive engagement (228). Ourresults show that

[

Ca2+]

iincrease

results in a substantial decrease in TCR lateral diffusion in PBL, as measured by a reduction of D, see Figure

6.2 and Table 6.1. Pre-treatment of T cells with 10µM cytochalasin D (a drug that inhibits actin cytoskeleton

function) abolished the effect of ionomycin on TCR mobility, Figure 6.2a. The values of D were 0.017± 0.002

µm2/s in PBL treated with ionomycin, 0.068± 0.014µm2/s in PBL treated with cytochalasin D only and 0.043

± 0.004µm2/s in PBL treated with both cytochalasin D and ionomycin. Similar results were obtained when T

cells were treated with 50 nM latrunculin B instead of cytochalasin D, Figure 6.2b. Interestingly, cytochalasin

D/latranculin B treatements did not significantly affect basal lateral mobility of TCR indicating that this process

is not dependent on actin cytoskeleton function. Ionomycintreatment, although affecting TCR lateral diffusion

(D values) did not affect the mobile fraction of TCR that was again close to 1 in the different samples (Table 6.1).

Our results indicate that the activation of the calcium signaling pathway reduces TCR mobility in T cell by an

actin cytoskeleton-dependent mechanism. They also show that[

Ca2+]

iincrease, although reducing the effective

mobility of individual TCR, does not reduce the number of mobile molecules.


(a) (b)

0 50 100 1500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

No

rma

lize

d I

nte

nsity

PBL

PBL+CytoD

PBL+Iono

PBL+CytoD+Iono

0 50 100 1500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

No

rma

lize

d I

nte

nsity

PBL

PBL+Lat B

PBL+Iono

PBL+Lat B+Iono

Figure 6.2: TCR mobility is modulated by intracellular calcium via the actin cytoskeleton. a. Mean FRAP recov-ery curves are shown for Peripheral blood T Lymphocytes (PBL, blue curve n=61), PBL treated with cytochalasinD (PBL + CytoD, green curve n=20), PBL treated with ionomycin(PBL + iono, orange curve n=37), and PBLtreated with ionomycin and cytochalasin D (PBL + Iono + CytoD, light blue curve n=25). b. Mean FRAP recov-ery curves are shown for PBL (blue line, n=61) PBL treated with latrunculin B (PBL + Lat B, green curve n=10),PBL treated with ionomycin (PBL + iono, orange curve n= 37), and PBL treated with ionomycin and latrunculinB (PBL + Iono + LatB, light blue curve n=10).

6.3.3[

Ca2+]

iincrease induces polymerization of the actin cytoskeleton

The above results suggested that, following[

Ca2+]

iincrease, the actin cytoskeleton becomes a constraint for

TCR mobility. To define a link between[

Ca2+]

iincrease and actin cytoskeleton function, we used FACS anal-

ysis to measure F-actin (polymerized actin). PBL were either untreated or treated with 2.5µg/ml ionomycin

or increasing concentrations of latrunculin B (as a controlof actin cytoskeleton de-polymerization) for 15 min-

utes at37 C. Cells were fixed, permeablized, and stained with Alexa-488labelled phalloidin to selectively tag

F-actin. As shown in Figure 6.3, treatment with latrunculinB decreased the amount of F-actin in T cells in a

dose-dependent fashion, supporting the notion that FACS analysis is suitable for quantification of F-actin (229).

Treatment of T cells with ionomycin induced a moderate increase in F-actin. These results indicate that[

Ca2+]

i

increase augments the amount of F-actin in T cells and suggest that the increase in actin polymerization could be

responsible for the observed effect of[

Ca2+]

iincrease on TCR mobility.

6.3.4 Modeling reveals effective TCR binding parameters tothe actin cytoskeleton

We have shown that the observed decrease in TCR mobility following ionomycin treatment is mediated by the

cortical actin cytoskeleton. We can extend the pure diffusion model, used to obtain the results in Table 6.1, to


Figure 6.3:[

Ca2+]

iincrease induces actin polymerization in PBL. PBL were treated for 15 minutes with lantran-

culin A (50 nM, 100 nM or 500 nM) or with ionomycin (2.5µg/ml). F-actin cellular content was measured byFACS analysis. Data are from one representative experimentout of three.

include binding interactions between TCR and the actin cytoskeleton as follows,

∂T/∂t = D∇2T − k∗onT + koffC,

∂C/∂t = k∗onT − koffC. (6.1)

where T is the concentration of diffusing TCR, C is the concentration of TCR bound to the cytoskeleton, D is the

diffusion coefficient of TCR,koff is the unbinding rate, andk∗on is the effective binding rate. The effective binding

rate can be written ask∗on = konS, wherekon is the intrinsic binding rate of TCR directly to the cytoskeleton or

binding through a putative mediator that links the TCR to thecytoskeleton.S is the concentration of binding

sites available to diffusing TCR which are assumed to be homogenously distributed. This binding model has been

previously used in FRAP analysis to extract binding parameters (230).

In Table 6.2 we summarize the results of fitting the pure diffusion model and the binding model to the PBL

data and the PBL + ionomycin data. When fitting the binding model we only fit for the on- and off-rates and

set D=0.0609µm2/s, the free diffusion coefficient. The fitting procedure we used is described in Dushek et al.

(231). We find that both the pure diffusion and binding modelsaccurately fit both data sets, Figure 6.4. Using

a statistical test to compare the explanatory power of each model (Akaike’s Information Criterion, AIC) we find

that both models explain the PBL data equally well. However,we find that the binding model has significant

additional power in explaining the PBL + ionomycin data. This suggests that TCR binding is quite possibly re-

sponsible for the slow FRAP recovery after ionomycin treatment as opposed to explanations involving a smaller

mobile fraction or a change in the underlying TCR diffusivity. In this case, the binding model reveals rapid bind-

ing kinetics between the TCR and the putative cytoskeletal binding partner withk∗on = 0.22 s−1 andkoff = 0.05

s−1. Put another way, a TCR might be expected to bind the cytoskeleton on average every 5 s and unbind after 20 s.


Table 6.2: Model fitting to FRAP recovery curvesPBL Data PBL + Ionomycin Data

SSR AIC 4AIC Prob. SSR AIC 4AIC Prob.Pure Diffusion Model 0.0449 -680.06 0.799 0.40 0.0281 -722.27 38.71 0.00Diffusion + Binding Model 0.0445 -680.86 0.000 0.60 0.0183 -760.97 0.00 1.00SSR - Sum squared residuals, AIC - Akaike Information Criterion, Prob - Probability that the givenmodel is the best descriptor of the FRAP recovery based on4AIC.

0 50 100 1500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s)

No

rma

lize

d I

nte

nsity

PBL

PBL+Ionomycin

Pure Diffusion Model

Binding + Diffusion Model

Figure 6.4: Fitting the diffusion and binding model to FRAP recovery curves. We show the fit of the purediffusion model (red) and the binding model (black) to the PBL data and to the ionomycin-treated PBL. Bothmodels provide a good fit to the recovery curves but a statistical test reveals that the binding model is significantlybetter at explaining the FRAP recovery when the PBL are treated with ionomycin (Table 6.2).

6.4 Discussion

In the present work we have shown, in agreement with previousstudies (213; 214), that TCR are constitutively

mobile on the surface of T cells. This lateral mobility (witha diffusion coefficient in the region of 0.05-0.1µm2/s)

is in principle sufficient to allow individual TCR to patrol the entire T cell surface within about 30 minutes. The

diffusivity of TCR is similar in T cells at different stages of activation, indicating that this parameter is not in-

volved in allowing activated T cells to be more responsive toantigenic stimulation.

Fast lateral mobility and rapid cell surface patrolling by TCR may be instrumental in resting conditions to al-

low T cells to rapidly sense antigenic stimuli. However it seems likely that, upon antigenic stimulation, regulated

TCR motion would be needed to permit spatially localized signaling. Accordingly we find that[

Ca2+]

iincrease

(a central and early signal in T cell activation) induces a significant decrease of TCR mobility. Our results also

show that this reduction in TCR mobility requires a functional actin cytoskeleton. This suggests that following[

Ca2+]

iincrease, TCR become associated with actin cytoskeleton orwith actin binding proteins. Fitting a model

for TCR diffusion and binding to an immobile partner to our data suggests that this may indeed be the case.

Our observations, by showing that TCR may be bound or trappedby F-actin, raise the question of how they


could move towards the immune synapse. It is well established that F-actin is enriched at the immune synapse

(232). Moreover, work from M.M. Davis and colleagues showedthat cortical F-actin moves towards the T

cell/APC contact site in antigen stimulated T cells (223). Our present work points towards an important role

of actin cytoskeleton in control of TCR motion although we did not focus in this study on TCR mobility during T

cell/APC cognate interaction.

We also showed that increase[

Ca2+]

iby a pharmacological agent results in increased levels of F-actin in

T cells. The concentration of ionomycin at which a clear shift in F-actin cellular content was observed was 2.5

µg/ml. In the FRAP experiments, 0.5µg/ml ionomycin (a concentration previously used to activate T cells (220))

was sufficient to induce a reduction of TCR mobility. The reason for this discrepancy is presently elusive. It is

possible that the effect on TCR mobility precedes significant actin polymerization. Alternatively, differences in

sensitivity between the two experimental methods could be the reason for the observed discrepancy. This result

suggests that upon initial conjugation of a T cell with a cognate APC the productive and localized engagement

of a small number of TCR may lead rapidly, via[

Ca2+]

iincrease and actin polymerization, to global control of

TCR motion.

Mathematical models of immune synapse formation commonly include TCR transport towards the center of

the immune synapse, driven by the actin cytoskeleton (218; 233; 234; 235; 236). Important parameters in these

models are effective reaction rates between TCR and the actin cytoskeleton, which to our knowledge have not

been previously reported. For example, in the study of Burroughs et al. (234), it was assumed that the effective

on- and off-rates were identical and equal to 0.1 s−1. It would be interesting to investigate the effects of a larger

on-rate and smaller off-rate, as reported in the present work, on the results of this and other models.

Several lines of evidence established a link between CAC polymerization and TCR signaling (237). Treatment

of T cells with drugs affecting actin cytoskeleton such as cytochalasin D, aborts antigen induced[

Ca2+]

iincrease

indicating that actin polymerization is required to sustain signaling in T cells (220). Moreover, deficiency or mu-

tation in several molecules regulating the actin cytoskeleton such as Vav-1, Cdc42 and Wiskott-Aldrich syndrome

protein (WASP) affect IS formation (238). Some of these proteins such Nck and WASP are recruited to the TCR

signaling area together with polymerized actin (239). Interestingly, while actin cytoskeleton polymerization is

usually considered to be required for the activation of calcium pathway (220; 237), here a reverse mechanism is

proposed:[

Ca2+]

iincrease was shown to enhance actin cytoskeleton polymerization and in turn modulate TCR

mobility.


We speculate that the binding or trapping of TCR by the actin cytoskeleton network could also be instrumen-

tal in favoring the assembly of the TCR associated signalingcascade. The notion that TCR move very rapidly is

difficult to reconcile with the necessity of assembling complexes of adaptor molecules and signaling components

at the immune synapse. The observation that TCR mobility is affected by global[

Ca2+]

iincrease is compatible

with the formation of localized signaling scaffolds favoring complex molecular interactions. This hypothesis is in

agreement with reported data by M. Dustin and colleagues showing that a functional actin cytoskeleton is required

to allow TCR micro-clusters to signal and that high concentration of polymerized actin is confined to TCR clusters

(217). It is tempting to speculate that global[

Ca2+]

iincrease induced by the engagement of a few TCR at the T

cell/APC contact site (via polimerization of CAC) might lead to the rapid formation of a much greater number of

TCR clusters in the nascent immune synapse, If so, this wouldequip T cells with pre-formed signaling units ideal

for sensitive detection of a limited number of antigenic ligands.

Our work outlines a previously unexpected link between the calcium pathway, the actin cytoskeleton and TCR

mobility. Although we have not examined the detailed mechanisms of this triangular interaction our results high-

light the ability of TCR population to control their own mobility via calcium signaling.

112

References

[205] Davis, M., M. Krogsgaard, M. Huse, J. Huppa, B. Lillemeier, and Q. Li. 2007. T cells as a self-referential,

sensory organ.Annu Rev Immunol. 25:681–95.

[206] Harding, C., and E. Unanue. 1990. Quantitation of antigen-presenting cell MHC class II/peptide complexes

necessary for T-cell stimulation.Nature. 346:574–6.

[207] Irvine, D., M. Purbhoo, M. Krogsgaard, and M. Davis. 2002. Direct observation of ligand recognition by

T cells. Nature. 419:845–9.

[208] Monks, C., B. Freiberg, H. Kupfer, N. Sciaky, and A. Kupfer. 1998. Three-dimensional segregation of

supramolecular activation clusters in T cells.Nature. 395:82–6.

[209] Grakoui, A., S. Bromley, C. Sumen, M. Davis, A. Shaw, P.Allen, and M. Dustin. 1999. The immunological

synapse: a molecular machine controlling T cell activation. Science. 285:221–7.

[210] Purtic, B., L. Pitcher, N. van Oers, and C. Wulfing. 2005. T cell receptor (TCR) clustering in the immuno-

logical synapse integrates TCR and costimulatory signaling in selected T cells.Proc Natl Acad Sci U S A.

102:2904–9.

[211] Depoil, D., R. Zaru, M. Guiraud, A. Chauveau, J. Harriague, G. Bismuth, C. Utzny, S. Muller, and S. Vali-

tutti. 2005. Immunological synapses are versatile structures enabling selective T cell polarization.Immunity.

22:185–94.

[212] Sloan-Lancaster, J., J. Presley, J. Ellenberg, T. Yamazaki, J. Lippincott-Schwartz, and L. Samelson. 1998.

ZAP-70 association with T cell receptor zeta (TCRzeta): fluorescence imaging of dynamic changes upon

cellular stimulation.J Cell Biol. 143:613–24.

[213] Favier, B., N. Burroughs, L. Wedderburn, and S. Valitutti. 2001. TCR dynamics on the surface of living T

cells. Int Immunol. 13:1525–32.




[215] Campi, G., R. Varma, and M. Dustin. 2005. Actin and agonist MHC-peptide complex-dependent T cell


[216] Yokosuka, T., K. Sakata-Sogawa, W. Kobayashi, M. Hiroshima, A. Hashimoto-Tane, M. Tokunaga,

M. Dustin, and T. Saito. 2005. Newly generated T cell receptor microclusters initiate and sustain T cell

activation by recruitment of Zap70 and SLP-76.Nat Immunol. 6:1253–62.

[217] Varma, R., G. Campi, T. Yokosuka, T. Saito, and M. Dustin. 2006. T cell receptor-proximal signals are sus-

tained in peripheral microclusters and terminated in the central supramolecular activation cluster.Immunity.

25:117–27.

[218] Cemerski, S., J. Das, J. Locasale, P. Arnold, E. Giurisato, M. Markiewicz, D. Fremont, P. Allen,

A. Chakraborty, and A. Shaw. 2007. The stimulatory potency of T cell antigens is influenced by the forma-

tion of the immunological synapse.Immunity. 26:345–55.

[219] Lee, K., A. Dinner, C. Tu, G. Campi, S. Raychaudhuri, R.Varma, T. Sims, W. Burack, H. Wu, J. Wang,

O. Kanagawa, M. Markiewicz, P. Allen, M. Dustin, A. Chakraborty, and A. Shaw. 2003. The immunological

synapse balances T cell receptor signaling and degradation. Science. 302:1218–22.

[220] Valitutti, S., M. Dessing, K. Aktories, H. Gallati, and A. Lanzavecchia. 1995. Sustained signaling leading

to T cell activation results from prolonged T cell receptor occupancy. Role of T cell actin cytoskeleton.J

Exp Med. 181:577–84.

[221] Negulescu, P., T. Krasieva, A. Khan, H. Kerschbaum, and M. Cahalan. 1996. Polarity of T cell shape,

motility, and sensitivity to antigen.Immunity. 4:421–30.

[222] Kaizuka, Y., A. Douglass, R. Varma, M. Dustin, and R. Vale. 2007. Mechanisms for segregating T cell

receptor and adhesion molecules during immunological synapse formation in Jurkat T cells.Proc Natl Acad

Sci U S A. 104:20296–301.

[223] Wulfing, C., and M. Davis. 1998. A receptor/cytoskeletal movement triggered by costimulation during T

cell activation.Science. 282:2266–9.

[224] Dustin, M. 2008. T-cell activation through immunological synapses and kinapses.Immunol Rev. 221:77–

89.

[225] Dushek, O., and D. Coombs. 2008. Improving parameter estimation for cell surface FRAP data.J Biochem



[226] Wiedemann, A., D. Depoil, M. Faroudi, and S. Valitutti. 2006. Cytotoxic T lymphocytes kill multiple

targets simultaneously via spatiotemporal uncoupling of lytic and stimulatory synapses.Proc Natl Acad Sci

U S A. 103:10985–90.

[227] Faroudi, M., C. Utzny, M. Salio, V. Cerundolo, M. Guiraud, S. Muller, and S. Valitutti. 2003. Lytic versus

stimulatory synapse in cytotoxic T lymphocyte/target cellinteraction: manifestation of a dual activation

threshold.Proc Natl Acad Sci U S A. 100:14145–50.

[228] Lewis, R. 2001. Calcium signaling mechanisms in T lymphocytes.Annu Rev Immunol. 19:497–521.

[229] Valitutti, S., M. Dessing, and A. Lanzavecchia. 1993.Role of cAMP in regulating cytotoxic T lymphocyte

adhesion and motility.Eur J Immunol. 23:790–5.

[230] Sprague, B., R. Pego, D. Stavreva, and J. McNally. 2004. Analysis of binding reactions by fluorescence

recovery after photobleaching.Biophys J. 86:3473–95.

[231] Dushek, O., R. Das, and D. Coombs. 2008. Analysis of membrane-localized binding kinetics with FRAP.

Eur Biophys J. 37:627–38.

[232] Roumier, A., J. Olivo-Marin, M. Arpin, F. Michel, M. Martin, P. Mangeat, O. Acuto, A. Dautry-Varsat, and

A. Alcover. 2001. The membrane-microfilament linker ezrin is involved in the formation of the immunolog-

ical synapse and in T cell activation.Immunity. 15:715–28.



[234] Burroughs, N. J., and C. Wlfing. 2002. Differential segregation in a cell-cell contact interface: the dynamics

of the immunological synapse.Biophys J. 83:1784–1796.

[235] Weikl, T. R., and R. Lipowsky. 2004. Pattern formationduring t-cell adhesion.Biophys J. 87:3665–3678.

[236] Figge, M. T., and M. Meyer-Hermann. 2006. Geometrically repatterned immunological synapses uncover

formation mechanisms.PLoS Comput Biol. 2:e171.

[237] Billadeau, D., J. Nolz, and T. Gomez. 2007. Regulationof T-cell activation by the cytoskeleton.Nat Rev

Immunol. 7:131–43.

[238] Tooley, A., J. Jacobelli, M. Moldovan, A. Douglas, andM. Krummel. 2005. T cell synapse assembly:

proteins, motors and the underlying cell biology.Semin Immunol. 17:65–75.

[239] Barda-Saad, M., A. Braiman, R. Titerence, S. Bunnell,V. Barr, and L. Samelson. 2005. Dynamic molecular

interactions linking the T cell antigen receptor to the actin cytoskeleton.Nat Immunol. 6:80–9.

115

Chapter 7

Discussion

In my thesis I investigated two major themes: serial engagement in T cell activation (Chapters 2 and 3) and

parameter estimation using FRAP (Chapters 4, 5, and 6). These will be discussed in turn.

7.1 Serial engagement in T cell activation

The serial engagement hypothesis has received much attention over the years (240; 241; 242; 243; 244; 245; 246).

The fact that serial engagements are inevitable in systems where mobile proteins undergo reactions, as in the case

for TCR/pMHC, will ensure that it remains a central component in models of T cell activation. Serial engage-

ments are probably necessary but not sufficient for T cell activation, as the serial engagement hypothesis, in its

standard form, does not provide a mechanism for TCR triggering or accurate pMHC discrimination (246).

The serial engagement hypothesis predicts that the total number of TCRs engaged by a single pMHC will

determine the stimulatory potency of that pMHC. Experimentally, it is difficult to quantify the number of binding

events between pMHC and TCR because fluorescence microscopy, the most commonly used tool to probe the

immune synapse (IS), provides information only on the spatial dynamics of proteins. Therefore, mathematical

modeling is necessary to estimate the number of serial engagements.

7.1.1 Serial engagements in TCR clusters

In Chapter 2, I used Monte Carlo simulations and escape time formulations (or mean first passage time cal-

culations) to explore serial engagement in the context of TCR clusters. Using a simple two-state escape time

formulation I showed that 1) pMHC are expected to escape froma mobile TCR cluster before the cluster reaches

the cSMAC and 2) pMHC engage few TCRs before exiting the TCR cluster. I was able to capture the effects of

CD4 and CD8 coreceptors or pMHC dimerization by extending the model to include 4 states. Including corecep-

tors or pMHC dimerization allowed the pMHC to remain in the TCR cluster until it arrives in the cSMAC and in

addition, allowed for substantially more TCR engagements.

Chapter 7. Discussion 116

The role of coreceptors in T cell activation has remained controversial (246; 247). Here, I have shown that

coreceptors are predicted to increase the trapping time of pMHC in TCR clusters. These predictions can be exper-

imentally tested. Using single particle tracking (SPT) of pMHC molecules in the immune synapse will reveal if

pMHC move between clusters or remain in a single cluster. If the latter is true, my work suggests that coreceptors

and/or pMHC dimerization play an important role in such confinement. Antibody blocking of coreceptors when

presented pMHC are monomeric should allow the pMHC to easilyescape the TCR cluster. In addition to trapping

pMHC, I have shown that coreceptors allow pMHC to make multiple localized engagements. The association

of Lck with coreceptors (246; 248) suggest that coreceptor facilitated TCR engagements are important in signal

amplification.

By examining a single TCR cluster I have shown that, in the absence of coreceptors and dimerization, the

pMHC will escape from a TCR cluster well before the cluster arrives in the cSMAC. I therefore examined the

effects of multiple TCR clusters. First I examined an IS thatcontains multiple immobile TCR clusters. I was

able to obtain an asymptotic solution in the limit that the TCR clusters are small relative to the IS and used a

numerical solution when the TCR clusters are large. I found that substantial serial engagements are possible but

that engagements are no longer spatially localized becausethe pMHC visits multiple TCR clusters before leaving

the IS. I then examined a more relevant model that accounted for the mobility of TCR clusters. By using a fully

stochastic Monte Carlo simulation I showed that pMHC have a high probability of entering the cSMAC and of

achieving substantial TCR engagements by visiting multiple TCR clusters.

Mathematically, there are several possible extensions of the TCR cluster models I have investigated. I have

assumed that interactions between TCR and pMHC in TCR clusters obey the law of mass-action. However,

molecular crowding within the clusters and/or steric hindrances by molecules surrounding the TCR cluster may

trap pMHC within a TCR cluster without the aid of pMHC dimerization or coreceptors. These effects can be

incorporated into a nano-scale Monte Carlo simulation of dynamics within a single TCR cluster, similar to the

model presented in Chapter 3.

Future work is also possible on the models of multiple TCR clusters that I have presented. First, I was only

able to obtain asymptotic solutions in the case of immobile TCR clusters. It may be possible to obtain asymptotic

solutions for a PDE model of multiple mobile TCR clusters by using matched asymptotic analysis in the reference

frame of the TCR clusters. As the number of TCR clusters increases the asymptotic solution will break down

and full PDE computations will be required. In the limit of many TCR clusters, a mean field approach may be

possible. This analysis will be mathematically interesting and studying how the number of TCR clusters alters the


spatial-temporal dynamics of diffusing pMHC is immunologically relevant.

7.1.2 Localized serial engagements as a mechanism of pMHC discrimination

In Chapter 3 I investigated an alternative role for serial engagement of TCRs by a single pMHC. Given that down-

stream responses of T cells are highly sensitive to the dissociation constant,KD, governing the TCR/pMHC bond

(249; 250), I investigated how the T cell can determineKD. I found that a direct estimate ofKD, for example

by the fraction of TCR bound to pMHC at equilibrium, is unlikely and that estimates of the kinetic parameters

governing the TCR/pMHC bond are required to allow the T cell to estimateKD. I then showed that the stochastic

nature of the TCR/pMHC interaction requires multiple samples of the pMHC reaction rates. Multiple samples are

achieved by serial engagements of TCR by a single pMHC. I showed that large TCR concentrations, similar to

the concentration of TCR in TCR clusters, and coreceptors (for weakly binding pMHC) are required for localized

engagements. I suggest that localized samples are important to ensure that binding events arise from the same

pMHC and for effective integration of signals that arise from each binding event.

By viewing serial engagements as a sampling process, I proposed that membrane organization, in the form of

TCR clusters, is required for accurate pMHC discrimination. This view of serial engagement poses several chal-

lenges regarding the molecular mechanisms that store and integrate signals from multiple binding events. Analysis

of the experimentally determined time course of activity for early signaling molecules like Lck, ZAP-70, LAT,

and SLP-76 in response to various pMHC having different reaction parameters may elucidate the mechanism of

pMHC discrimination. The feasibility of this analysis is atleast partially evident by recent work that characterized

the phosphorylation time course of LAT (251). A long term, and a more ambitious goal is to understand how the

complex signaling network in T cells (252) faithfully translates information about the reaction rates between TCR

and pMHC into downstream responses, such as cytokine release.

7.2 Parameter estimation using FRAP

7.2.1 Estimating diffusion coefficients using FRAP

Fluorescence Recovery After Photobleaching (FRAP) is a widely used technique to study the mobility of proteins

in the cytoplasm and the cell membrane (253; 254; 255). In Chapter 4, I investigated the application of FRAP

to cell surface proteins. I found that two commonly used protocols to estimate the diffusion coefficient of cell

surface proteins give rise to appreciable errors. The first utilitizes a large bleach region but does not account for


the finite extent of the cell membrane. The second uses a smallbleach region but the equation used to fit the FRAP

recovery curve does not capture the experimental geometry.I proposed a protocol that is simple to implement and

provided an analytical equation to fit to the FRAP curve. Thisprotocol provides improved parameter estimation

for diffusion coefficients and was successfully implemented in the study of TCR mobility in Chapter 6.

There have been several other attempts to improve the absolute accuracy in estimating the diffusion coefficient

from FRAP experiments. One study improved parameter estimation by capturing the diffusion of highly mobile

proteins during the bleaching event (256). Other studies focused on the inhomogeneous environment of diffus-

ing proteins and implemented a numerical solution of the diffusion equation to improve parameter estimation

(257; 258). Although these studies provide improved parameter estimates, few FRAP studies have implemented

these improvements, probably because implementation is nontrivial. The protocol I proposed to improve parame-

ter estimates is simple, primarily because I provide an analytical formula that can be fit directly to the experiment

data.

Reports of diffusion coefficients for membrane proteins span roughly two orders of magnitude,D = 0.005 −

0.5µm2/s (259). Even for the same protein, estimates in the diffusion coefficient range over an order of magni-

tude. The reason for these large differences is probably dueto different experimental apparatus and, in addition,

because of appreciable errors in the fitting procedure. Minimizing the errors in the fitting procedure, by the pro-

tocol I proposed or otherwise, will allow for more consistent reports of diffusion coefficients between laboratories.

7.2.2 Estimating binding rates between cell membrane proteins using FRAP

Communication between nearby cells in the immune system, and elsewhere, relies on interactions between pro-

teins on opposing membranes. The strength of these interactions often determines the subsequent activity of

each cell. In the case of T cells interacting with APCs bearing antigenic pMHC, the reaction rates between

TCR and pMHC have been shown to govern T cell activation and the amount of cytokine released by the T cell

(249; 250; 260; 261). Hence, measuring reaction rates between membrane proteins is critical to the understanding

of cellular activity following cell-cell (and hence protein-protein) interactions.

In Chapter 5 I outlined an experimental protocol for determining reaction rates between receptors and their

ligands on opposing membranes, based on titrating the density of the ligand and performing a series of FRAP

experiments in the junction between a cell and a planar bilayer. Using asymptotic analysis and model fitting, I

showed that as the ligand density increases the system transitions between a diffusion-dominant regimes where


the receptor is primarily unbound to another diffusion-dominant regime where the receptor is primarily bound to

the ligand. In both of these diffusion-dominant regimes no binding information can be obtained. However, in the

regime that exists between the transition from these diffusion-dominant regimes it is possible to obtain the 2D

bond affinity (independent diffusion or weighted diffusionregimes) or the 2D reaction on-rate and off-rate (full

model regime). I confirmed these theoretical predictions using simulated data.

A key advantage of this FRAP-based method to determine reaction rates, compared to other existing meth-

ods, is that the reaction rates are determinedin vivo. The most widely used method for determining reaction

rates between proteins is Surface Plasmon Resonance (SPR) (262; 263; 264). However, this method requires

that the proteins of interest be removed from their native environment (i.e. the plasma membrane) and placed

in solution. Therefore, SPR provides solution or 3D reaction rates whereas the actual rates are 2D because the

proteins are confined to the plasma membrane, which is a 2D surface. The relationship between experimentally

measured 3D reaction rates and the physiologically relevant 2D rates is presently unknown (265). Most studies

assume a linear relationship but non-linear relationshipsmay arise, see figure 7.1. For example, taking into ac-

count conformational changes, Qi et al. (266) have shown that the relationship between 2D and 3D off rates is

not a simple linear function. By using the FRAP-based methodto directly determine 2D reaction rates for pro-

teins whose 3D rates have already been measured, the effective relationship between these rates can be uncovered.

3D (Solution) Reaction Rate

2D

(M

em

bra

ne)

Reaction R

ate

Linear

Quadratic

Square Root

Figure 7.1: Possible relationship between 3D and 2D reaction rates. The simplest relationship is that 3D and2D rates are related by a constant factor (linear model). However, non-linear effects may be introduced intothe conversion (quadratic and square root model) by considering effects of the plasma membrane, molecularcrowding, and/or cooperative effects. The actual relationship between 3D and 2D reaction rates is presentlyunknown. An arbitrary scale has been used.

I briefly mention that a FRAP-based method for determining 2Dreaction parameters has recently been pub-

lished (267; 268). In this work, the authors perform FRAP experiments in the contact junction between a cell

and planar bilayer and report, for the first time, direct measurements of the 2D reaction on- and off-rates between

membrane proteins. Interestingly, they found that the reaction off-rate was 100 times smaller than the reaction


off-rate measured by SPR.

The analysis and concepts of using FRAP to determine bindingkinetics developed in Chapter 5 can be ap-

plied to many scenarios. For example, the asymptotic analysis was performed without imposing any geometry

and therefore can be directly applied to studies that use circular bleaching profiles instead of the rectangular pro-

files often used for cell membrane FRAP. Although applied to the interaction between a receptor and ligand on

opposing membranes, the procedure can readily be applied tointeracting proteins on the same membrane or to

interacting proteins in the cytosol. The procedure can alsobe used to determine effective reaction parameters

between membrane proteins and lipid rafts (269).

7.2.3 Estimating effective binding rates between cell membrane proteins and the actin

cytoskeleton using FRAP

It is now well established that integrins regulate the actincytoskeleton and that the cytoskeleton, in turn, regu-

lates the mobility of integrins (270; 271). However, quantitative estimates for the effective binding reactions have

not been measured. In Chapter 6 I presented work, performed in collaboration with the experimental laboratory

of Salvatore Valitutti, aimed at characterizing the relationship between TCR mobility, the cytoskeleton, and in-

tracellular calcium. Using the protocol developed in Chapter 4 we performed a series of FRAP experiments to

determine the mobility of TCR when T cells were treated with various pharmacological agents. In particular, we

showed that treatment with ionomycin decreases TCR mobility in an actin cytoskeleton dependent manner. We fit

a diffusion-binding model, using the tools developed in Chapter 5, and extract the first estimates of the effective

binding parameters governing the TCR and cytoskeleton interaction. We find an effective on-rate of 0.22 s−1 and

an effective off-rate of 0.05s−1.

These estimates for the effective on- and off-rates are relevant to studies of immune synapse formation. The

study of Qi et al. (272) argued that TCR enrichment in the cSMAC can come about by passive processes alone

while the study of Burroughs et al. (273) argued that the actin cytoskeleton is required to transport TCR to the

cSMAC. In the latter study, it was assumed that TCR undergo reversible first order reactions with the actin cy-

toskeleton and that when bound to the cytoskeleton, TCR are actively transported to the cSMAC. The reaction

rates used by the authors (273) are similar to the experimentally determined values in our study.

The FRAP experiments we have performed provided ensemble estimates of the reaction rates between TCR

and the actin cytoskeleton. In the future, it will be important to confirm the interaction between TCR and the


cytoskeleton using single particle tracking (SPT) (259). Trajectories of individual TCR using SPT will reveal

if they are undergoing free diffusion or in addition, binding to the actin cytoskeleton. Our results suggests that

whichever nanoscale model governs the mobility of TCR, on the micron scale TCR are effectively diffusing and

binding to the actin cytoskeleton.

In the future, it will be important to confirm these results ina more physiological setting by activating T cells

with an APC bearing antigenic pMHC. These experiments pose achallenge for FRAP analysis. Unlike treatment

with soluble pharmacological agents, stimulation with an APC will likely polarize the actin cytoskeleton. The

FRAP recovery curve will no longer represent binding of TCR to an immobile actin cytoskeleton but to mobile

and dynamic strucutre. A previous study (274) was able to investigate the mobility of TCR towards the contact

junction between a T cell and an APC during immune synapse formation and a similar analysis may be performed

in our case.

The mechanism by which TCRs aggregate to form TCR clusters onthe cell membrane remains controversial.

There is evidence that protein-protein interactions are required (275) while other evidence suggest that the cy-

toskeleton is required (276). Having shown that TCRs can be immobilized by the cytoskeleton in Chapter 5 and

knowing that signaling through a TCR can initiate local actin polymerization (271), our work suggests a model

whereby productive engagement of TCR by pMHC leads to local actin polymerization. The resulting increase

in localized actin increases the probability of TCR bindingand hence promotes the formation of a TCR cluster.

Indeed, high concentration of polymerized actin has been shown to be confined to TCR clusters (276).

122

References

[240] Valitutti, S., S. Mller, M. Cella, E. Padovan, and A. Lanzavecchia. 1995. Serial triggering of many T-cell


[241] Itoh, Y., B. Hemmer, R. Martin, and R. N. Germain. 1999.Serial TCR engagement and down-modulation

by peptide:MHC molecule ligands: relationship to the quality of individual TCR signaling events.J Im-

munol. 162:2073–2080.

[242] Chan, C., A. J. George, and J. Stark. 2001. Cooperativeenhancement of specificity in a lattice of T cell

receptors.Proc Natl Acad Sci U S A. 98:5758–5763.


marked for internalization after dissociation from pMHC.Nat Immunol. 3:926–931.

[244] Berg, H. A. V. D., N. J. Burroughs, and D. A. Rand. 2002. Quantifying the strength of ligand antagonism

in TCR triggering.Bull Math Biol. 64:781–808.

[245] van den Berg, H. A., and D. A. Rand. 2007. Quantitative theories of T-cell responsiveness.Immunol Rev.

216:81–92.

[246] Choudhuri, K., and P. A. van der Merwe. 2007. Molecularmechanisms involved in T cell receptor trigger-

ing. Semin Immunol. 19:255–261.

[247] Davis, S., and P. van der Merwe. 2003. TCR triggering: co-receptor-dependent or -independent?Trends

Immunol. 24:624–6; author reply 626–7.

[248] Li, Q., A. Dinner, S. Qi, D. Irvine, J. Huppa, M. Davis, and A. Chakraborty. 2004. CD4 enhances T

cell sensitivity to antigen by coordinating Lck accumulation at the immunological synapse.Nat Immunol.

5:791–9.

[249] Holler, P. D., and D. M. Kranz. 2003. Quantitative analysis of the contribution of TCR/pepMHC affinity

and CD8 to T cell activation.Immunity. 18:255–264.

[250] Tian, S., R. Maile, E. J. Collins, and J. A. Frelinger. 2007. CD8+ T cell activation is governed by TCR-

peptide/MHC affinity, not dissociation rate.J Immunol. 179:2952–2960.


[251] Huse, M., L. Klein, A. Girvin, J. Faraj, Q. Li, M. Kuhns,and M. Davis. 2007. Spatial and temporal

dynamics of T cell receptor signaling with a photoactivatable agonist.Immunity. 27:76–88.

[252] Lin, J., and A. Weiss. 2001. T cell receptor signalling. J Cell Sci. 114:243–244.





[255] 2004. Live Cell Imaging: A Laboratory Manual, chapterPhotobleaching techniques to study mobility and

molecular dynamics of proteins in live cells: FRAP, iFRAP and FLIP. Cold Spring Harbor Laboratory Press.

[256] Braga, J., J. M. P. Desterro, and M. Carmo-Fonseca. 2004. Intracellular macromolecular mobility measured

by fluorescence recovery after photobleaching with confocal laser scanning microscopes.Mol. Biol. Cell.

15:4749–4760.

[257] Siggia, E. D., J. Lippincott-Schwartz, and S. Bekiranov. 2000. Diffusion in inhomogeneous media: theory

and simulations applied to whole cell photobleach recovery. Biophys J. 79:1761–1770.

[258] Sniekers, Y. H., and C. C. van Donkelaar. 2005. Determining diffusion coefficients in inhomogeneous

tissues using fluorescence recovery after photobleaching.Biophys J. 89:1302–1307.

[259] Saxton, M., and K. Jacobson. 1997. Single-particle tracking: applications to membrane dynamics.Annu

Rev Biophys Biomol Struct. 26:373–99.


immunological synapse: a molecular machine controlling T cell activation.Science. 285:221–227.

[261] Krogsgaard, M., N. Prado, E. J. Adams, X. lin He, D.-C. Chow, D. B. Wilson, K. C. Garcia, and M. M.


to T cell activation.Mol Cell. 12:1367–1378.

[262] Margulies, D. H., D. Plaksin, S. N. Khilko, and M. T. Jelonek. 1996. Studying interactions involving the

T-cell antigen receptor by surface plasmon resonance.Curr Opin Immunol. 8:262–270.

[263] Schuck, P. 1997. Reliable determination of binding affinity and kinetics using surface plasmon resonance

biosensors.Curr Opin Biotechnol. 8:498–502.

[264] Schuck, P. 1997. Use of surface plasmon resonance to probe the equilibrium and dynamic aspects of

interactions between biological macromolecules.Annu Rev Biophys Biomol Struct. 26:541–566.


[265] Davis, S. J., S. Ikemizu, E. J. Evans, L. Fugger, T. R. Bakker, and P. A. van der Merwe. 2003. The nature

of molecular recognition by T cells.Nat Immunol. 4:217–224.

[266] Qi, S., M. Krogsgaard, M. M. Davis, and A. K. Chakraborty. 2006. Molecular flexibility can influence the

stimulatory ability of receptor-ligand interactions at cell-cell junctions.Proc Natl Acad Sci U S A. 103:4416–

4421.

[267] Wu, J., Y. Fang, V. I. Zarnitsyna, T. P. Tolentino, M. L.Dustin, and C. Zhu. 2008. A coupled diffusion-

kinetics model for analysis of contact-area FRAP experiment. Biophys J. 95:910–919.

[268] Tolentino, T. P., J. Wu, V. I. Zarnitsyna, Y. Fang, M. L.Dustin, and C. Zhu. 2008. Measuring diffusion and

binding kinetics by contact area FRAP.Biophys J. 95:920–930.

[269] Kenworthy, A. K., B. J. Nichols, C. L. Remmert, G. M. Hendrix, M. Kumar, J. Zimmerberg, and

J. Lippincott-Schwartz. 2004. Dynamics of putative raft-associated proteins at the cell surface.J Cell

Biol. 165:735–746.

[270] DeMali, K. A., K. Wennerberg, and K. Burridge. 2003. Integrin signaling to the actin cytoskeleton.Curr

Opin Cell Biol. 15:572–582.

[271] Billadeau, D. D., J. C. Nolz, and T. S. Gomez. 2007. Regulation of T-cell activation by the cytoskeleton.

Nat Rev Immunol. 7:131–143.



[273] Burroughs, N. J., and C. Wulfing. 2002. Differential segregation in a cell-cell contact interface: the dynam-

ics of the immunological synapse.Biophys J. 83:1784–1796.



[275] Douglass, A. D., and R. D. Vale. 2005. Single-moleculemicroscopy reveals plasma membrane mi-

crodomains created by protein-protein networks that exclude or trap signaling molecules in T cells.Cell.

121:937–950.



Immunity. 25:117–127.

125

Appendix A

Matched asymptotic solution for multiple

stationary TCR clusters (Chapter 2)

In this section we use matched asymptotic analysis to solve for t2 (equation 2.17b) in the limit thatb → 0. We

begin by rescaling the equation fort2 usingu = t2/t∗ andx = r/Rs. After some rearranging we obtain,

πkoffDP t∗

konNT

∇2u = −R2s

b2I (A.1)

where we have used the fact thatTmc = NT /πb2. We identify the dimensionless quantityb/Rs with a small

parameterε = b/Rs and definet∗ = konNT /πkoffDP . We obtain the following equation that we wish to solve in

the limit ε→ 0:

∇2u = − 1

ε2Iε(x; x1...xN) (A.2)

with u(|x| = 1) = 0. The indicator function (equation 2.16) becomes,

Iε(x; x1, ...xN ) =

1 | x − xj |≤ ε

0 | x − xj |> ε.

Inner solution In the region near the the jth TCR microcluster we introduce the local space variabley =

(x − xj)/ε. Consequently we writeuin(y) = u(xj + εy) to obtain the governing equation near thej’th cluster,

∇2yu

inj =

−1 | y |≤ 1

0 | y |> 1. (A.3)

We expand the inner solution in powers ofv = −1/log(ε),

uinj = uj,-1v

−1 + uj,0 +∑

i=1

viuj,i (A.4)

Appendix A. Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) 126

where thei subscript indicates the order of the solution. Substituting eqn. A.4 into eqn. A.3 and matching orders

of v we obtain,

v−1; ∇2yuj,-1 = 0

v0; ∇2yuj,0 =

−1 | y |≤ 1

0 | y |> 1

vi; ∇2yuj,i = 0, i ≥ 1 (A.5)

The solution to thei = 0 equation is (up to an additive constant),

uj,0 =

− 12 log | y | | y |> 1

14 (1 − y2) | y |≤ 1

The solutions to all other orders areuj,i = c1log | y | +Aj,i (i 6= 0). We setc1 = 0 to avoid blow up. In summary,

the solution to the inner problem around the jth TCR microcluster is

uinj = Aj,-1v

−1 + uj,0 +Aj,0 +∑

i=1

viAj,i (A.6)

As | y |→ ∞ the inner solution becomes

uinj = Aj,-1v

−1 − 1

2log | x − xj | +

1

2log(ε) +Aj,0 +

∑

i=1

viAj,i . (A.7)

The unknown constantsAj,i will be determined by matching this exact inner solution with the outer solution.

Outer solution The outer problem is

∇2uout = 0 , x ∈ Ω\

∑

i=1

| x − xj |≤ ε

(A.8)

with uout(| x |= 1) = 0 anduout → −(1/2)log(| x − xj |) asx → xj . The related Green’s function problem is,

∇2G = δ(x − xk), G(| x |= 1) = 0 (A.9)

Using the method of images the solution can be written as

G(x, xk) = G(x, xk) +R(x, xk), where (A.10)

G(x, xk) =1

2πlog | x − xk | and R(x, xk) = − 1

2πlog

[ | x | xk |2 −xk || xk |

]

.

R is known as the regular part of the Green’s function. We see that we obtain the required log dependence and

consequently reformulate the outer problem as

∇2uout = −π∑

k=1

δ(x − xk) , x ∈ Ω (A.11)

with uout(| x |= 1) = 0. We write the solution in terms of the Green’s function

uout = −π∑

k=1

G(x, xk). (A.12)

As x → xj the solution becomes

uout = −1

2log | x − xj | −πR(xj , xj) − π

∑

k=1k 6=j

G(xj , xk). (A.13)

Matching condition We can now match the inner expansion (equation A.7) with the outer expansion (equation

A.13) to all orders ofv. We obtainAj,-1 = 1/2, Aj,0 = −π R(xj , xj)− π∑

k=1,k 6=jG(xj , xk) andAj,i = 0 for

i ≥ 1. Therefore the infinite expansion in the inner region reduces to just two terms.

Mean escape time We compute the mean ofu by averaging over the whole domain.

 =1

|Ω|

∫∫

Ω

(1 − Iε)uout + Iε

N∑

j=1

uinj

dx

where the outer solution, equation A.12, is valid outside clusters, and the inner solution, equation A.6, is valid

within clusters. Rearranging and computing the integral weobtain

 =1

|Ω|

∫∫

Ω

uout +1

|Ω|

∫∫

Ω

Iε

N∑

j=1

uinj − uout

dx

=1

4

N∑

k=1

(

1 − |xk|2)

+ ϑ(ε2) (A.14)

We show how to explicitly compute the first integral below. The second integral is of orderε2 because of the

appearance of the indicator function,Iε. Finally, we multiply by t∗ to obtain the unscaled quantity< t2 >.

Assuming the initial position of the particle is within one of theN microclusters we may compute:

< uin > =1

Nπε2

∫∫

Ω

IεN∑

j=1

uinj

After computing the integral and unscaling the equation we obtain equation 2.20.

Mean escape time integral In this section we evaluate the first integral in equation A.14. After substituting the

outer solution, equation A.12, we obtain

 = − π

|Ω|

N∑

k=1

∫∫

Ω

G (x, xk) = − π

|Ω|

N∑

k=1

[

L1k + L2

k

]

(A.15)

where

L1k =

1

2π

∫∫

Ω

log|x − xk|dx and L2k = − 1

2π

∫∫

Ω

log

∣

∣

∣

∣

|xk|x − xk

|xk|

∣

∣

∣

∣

dx. (A.16)

We first calculateL2k . Without loss of generality we assume thatxk lies on the positive x-axis and therefore we let

x = reiθ r andxk = ρkr . With these we find,

L2k = − 1

2π

∫ 2π

0

dθ

∫ 1

0

drlog|ρkreiθ − 1| (A.17)

Next we letz = eiθ to obtain a computable contour integral,

L2k = − 1

2π

∫ 1

0

rdr

∫

|z|=1

dz−ilog|ρkrz − 1|

z

= − 1

2π

∫ 1

0

rdr (−i2πiRes(z = 0)) = − 1

2π

∫ 1

0

rdr (2πlog| − 1|) = 0

The only singularity for|z| ≤ 1 occured wherez = 0 and consequently we findL2k = 0. To evaluateL1

k , we make

the same substitutions forx andxk to obtain

L1k =

1

2π

∫ 2π

0

dθ

∫ 1

0

drlog|reiθ − ρk|.

Here we cannot make the complex substitution directly sincethe log term will have a singularity within the contour

integral. Instead, we rearrange the integrand to obtain

log|reiθ − ρk| =1

2log(r2 + ρ2

k) +1

2log(1 − µ cos θ) (A.18)

whereµ = 2rρk/(r2 + ρ2

k). Substituting and performing the integration overθ we obtain

L1k =

1

2π

∫ 1

0

rdr

[

πlog(r2 + ρ2k) + πlog

(

1 +√

1 − µ2

2

)]

where we have used the identity

∫ 2π

0

log(1 − µ cos θ) dθ = 2πlog

(

1 +√

1 − µ2

2

)

.

Next we note that

1 +√

1 − µ2

2=

r2

r2+ρ2

kr > ρk

ρ2

kr2+ρ2

kr < ρk

so we can establish

L1k =

1

2π

[

πlog(ρ2k)

∫ ρk

0

rdr + π

∫ 1

ρk

rlog(r2)dr

]

=1

4(ρ2

k − 1). (A.19)

Finally, we combine the results forL1k andL2

k to obtain

 = −1

4

N∑

k=1

[

|xk|2 − 1]

(A.20)

where we have used|Ω| = π andρk = |xk|.

130

Appendix B

Monte Carlo simulation of pMHC

dynamics with multiple mobile TCR

clusters (Chapter 2)

In this appendix we develop an agent-based Monte Carlo simulation that models the formation and mobility of

multiple TCR clusters, the diffusion of pMHC, and the interaction between pMHC and clustered TCR in the

immune synapse (IS). Although we do not explicitly simulateindividual binding events between pMHC and

TCR/coreceptors in clusters, the model is fully stochastic(details below).

B.1 Model of pMHC and TCR cluster dynamics

Diffusion module We model the diffusion of a single pMHC on a lattice where the rate of hopping between the

four nearest neighbour sites isD/h2 (see Isaacson and Peskin (277)), whereD is the diffusion coefficient andh

is the lattice spacing. We takeh = 0.01µm, approximately the size of pMHC or TCR in the membrane, in order

to ensure that we capture both reaction limited and diffusion limited regimes. Our domain is the immune synapse

which is a disc of radiusRs = 6µm, see figure B.1. However, diffusing pMHC on the APC or a supported planar

bilayer can diffuse beyondRs and subsequently re-enter the synapse. To allow for synapsere-entry, we terminate

the simulation once the pMHC has reached a radius ofRe = 7µm. The accumulation of pMHC in the cSMAC

suggests that pMHC become trapped in the cSMAC upon entry (278; 279; 280). We therefore also terminate

the simulation if the pMHC has reached the cSMAC, which has a radius ofRc = 1µm (278). In summary, we

simulate the diffusion of pMHC in a large annulus and terminate the simulation once the pMHC has diffused out

of the annulus atRc orRe, see figure B.1.

TCR cluster module We model multiple TCR clusters (of radiusb) that are stochastically advecting along actin

filaments towards the cSMAC. In the immune synapse, actin filaments have been shown to be radially organized

and to be undergoing treadmilling, presumably moving TCR clusters to the cSMAC (281). The stochasticity in

Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 131

V

b

1.

2.

3.

4.5.

θ

y

xRc Rp Rs Re

Figure B.1: Illustration of model geometry and the interaction between diffusing pMHC and advecting TCRclusters (not to scale). TCR clusters are continually generated in a random location betweenRp andRs andadvect with velocityV towards the cSMAC center (the origin). 1. Simulations beginwith the pMHC at a radiusof 5µm. 2. Diffusion of pMHC and/or advection of TCR clusters may result in the pMHC entering a cluster. Onceinside a cluster, the pMHC may bind to TCR/coreceptors. Uponbinding, the pMHC remains in the cluster for aperiod of time determined byψ during which it is carried by the cluster towards the cSMAC. 3. Upon unbinindg,the pMHC exits the cluster at an angle determined fromχ and resumes free diffusion. 4. It may subsequentlyenter and bind to TCR/coreceptors in a different cluster. 5.The simulation is terminated once the pMHC hasentered the cSMAC.

TCR cluster velocity in our model implicitly captures the stochasticity in actin treadmilling and binding events

between a TCR cluster and actin. The rate of advection in our model is given byV/h, whereV is the effective

TCR cluster velocity andh is the lattice spacing. TCR clusters exit our simulation once they fully enter the

cSMAC. The formation of new TCR clusters is observed in the periphery of the immune synapse (279; 280; 282)

and we therefore simulate the formation of new TCR clusters,with a ratekmc, in a random location between the

synapse boundaryRs = 6µm andRp = 5µm. We denote the total number of TCR clusters in the synapse at

any given time asN . In summary, this module generates new TCR clusters in an annulus in the periphery of the

immune synapse (betweenRp andRs) that undergo stochastic directed motion towards the cSMACwhere they

exit from our simulation.

Full simulation Combining both the pMHC diffusion module and the TCR clustermodule we can now consider

reactions whenever the pMHC diffuses into a TCR cluster. Contained within the TCR cluster is some number of

TCR and coreceptors that are able to bind pMHC. Since these reactions have very fast kinetics we do not resolve

individual binding events in order to create efficient simulations. Instead, once the pMHC has bound a molecule in

the TCR cluster, we solve the relevant partial differentialequation (PDE) to determine the cumulative probability

distribution of escape times,ψ(t) (i.e. how long the pMHC has spent in the cluster) and escape angles,χ(t, θ)

(i.e. where does it pop out of the cluster). These distributions depend on the initial state of the pMHC (bound to

TCR or coreceptor), the initial position within the cluster, the concentration of TCR (Tmc) and coreceptors (Cmc),

the TCR cluster radius (b), and the reaction rates between pMHC and TCR/coreceptors.Once initially bound

in a TCR cluster, we generate two random numbers which determine the escape time,τesc, and escape angle,

θesc, based on the distributionsψ andχ. In the sections below we show to evaluateψ andχ. Summarizing, the

diffusing pMHC may enter and bind a molecule in a TCR cluster.Once bound the pMHC advects with the cluster

towards the cSMAC for a certain period of time, stochastically determined by the distributionψ. In figure B.1 we

illustrate a sequence events that a diffusing pMHC may undergo. By allowing the pMHC to interact with multiple

TCR clusters, the purely diffusive motion of the pMHC is biased towards cSMAC arrival by the motion of TCR

clusters.

B.1.1 Simulation Engine

The main engine of our simulation is the next reaction methodfirst proposed by Gillespie (283). We begin the

main loop by first examining the state of the pMHC to determineall possible reactions, see table B.1, from which

we determine the overall ratea. Based ona we determine the time for the next reaction as,

τnext =1

aln

(

1

r1

)

wherer1 is a uniformly distributed random number between 0 and 1. If the pMHC is unbound, we proceed to use

a second random number,r2, to determine which of the possible reactions took place. Wesimulate the reaction,

advance the simulation time,tc, byτnext, and return to the start of the loop. A flow chart of the simulation is shown

in figure B.2. If a binding event takes place we computeτesc usingψ and store the time at which the pMHC is

expected to exit the cluster,te = tc + τesc. After each computation ofτnext we check if the pMHC has escaped the

TCR cluster (i.e.tc + τnext > te). If pMHC escape has occured, we simply settc = te and return to the start of

the loop.

In each realization of the simulation we collect statisticson the number of clusters the pMHC has visited, the

number of TCR engagements, and whether the pMHC has arrived in the cSMAC. The number of TCR engage-

ments is calculated using equations 2.7, where< t > is replaced by the stochastically determined escape time,

τesc. In order to collect sufficient statistics, we run the simulation for a given parameter set at least 500 times. For

a discussion of the parameters see main text.

Table B.1: The state of pMHC determines the possible reactionspMHC state unbound unbound within a cluster bound within a clusterPossible reactions diffusion diffusion cluster advection

cluster advection cluster advection cluster formationcluster formation cluster formation unbinding

bindingOverall rate a = 4D/h2

+ NV/h a = 4D/h2+ NV/h a = NV/h + kmc

+kmc +kmc + λ1 + λ−4

pMHC boundpMHC unbound

tc + τnext < te tc + τnext > te

simulate reaction

tc = tc + τnexttc = te

Determine pMHC State

(compute τnext)

Determine pMHC State

(compute τnext)

Figure B.2: Flow chart illustrating the main loop in the simulation

B.2 Determiningψ and χ

B.2.1 PDE governing pMHC dynamics in a TCR cluster

In order to carry out our simulations we need to compute the cumulative distributions for pMHC escape time,

ψ(t), and escape angle,χ(t, θ). The PDE governing pMHC dynamics within a TCR cluster is,

∂fA

∂t= λ4fD − λ−4fA − λ1fA + λ−1fB +D∇2fA,

∂fB

∂t= λ1fA − λ−1fB − λ2fB + λ−2fC,

∂fC

∂t= λ2fB − λ−2fC − λ3fC + λ−3fD,

∂fD

∂t= λ3fC − λ−3fD − λ4fD + λ−4fA , (B.1)

where thefi are the probability density functions for each state,λi are the transition rates, andD is the pMHC

diffusion coefficient. We assume a disc domain of radiusr = a and take the boundary condition to befA(r =


b) = 0. The initial conditions for thefi are,

fA(t = 0) = g(r, θ)ΓA

fB(t = 0) = g(r, θ)ΓB

fC(t = 0) = g(r, θ)ΓC

fD(t = 0) = g(r, θ)ΓD (B.2)

where,

g(r, θ) =δ(r − r0)δ(θ)

r(B.3)

We take the initial position of the particle (i.e. pMHC) to be(r, θ) = (r0, 0) without loss of generality. The

values ofΓi indicate the initial state of the particle. For example, ifΓ = (ΓA ,ΓB,ΓC,ΓD) = (0, 1, 0, 0) then

the particle begins bound to TCR in statei = B. The probability that the particle has escaped the cluser is

ψ(t) = 1 − P (t), whereP (t) is the probability that the particle is in the domain. This latter quantity is obtained

by simply integrating the solution of equations B.2 over thewhole domain,

P (t) =

∫ a

0

rdr

∫ 2π

0

dθ (fA + fB + fC + fD) (B.4)

In the last section of this appendix we show how to efficientlyevaluateP (t).

Given an escape time, we now wish to determine the probability distribution for escape angles,χ(t, θ). We

define the escape angle,θ, relative to the initial angle which we take to be 0, see step 3in figure B.1. We define

p(t, θ) to be the flux through the boundary,

p(t, θ) = −D (∇fA · dr) |r=b (B.5)

Note that onlyfA appears in this equation because in all other states the particle is bound and hence immobile.

The probability that the pMHC has escaped at an angleθ at timet is then given by,

χ(t, θ) =

∫ θ

0p(t, θ

′

)dθ′

∫ 2π

0p(t, θ′)dθ′

(B.6)

In the last section of this appendix we show how to evaluate this quantity.

Once the pMHC has bound a molecule in a TCR cluster we computeψ(t) using information about the binding

event (e.g. location within the cluster, reaction rates of the simulation, etc). We generate a random number,r3,


and solve the equationr3 = ψ(τesc) for τesc. Next we generate another random number,r4, and solve the equation

r4 = χ(τesc, θesc) for θesc. In the sections that follow we show to to solve equations B.1and how this solution is

used to efficiently evaluateψ andχ. Using this evaluation method we use the Matlab function fzero to solve for

τescandθesc in the above equations.

B.2.2 Laplace transform solution of the PDE system

We begin by finding a solution to equations B.1 with initial conditions B.2 and boundary conditionfA(r = b) = 0.

An explicit solution will not be possible and instead we willfind a solution in the transformed space which we can

numerically invert. We begin by taking the Laplace transform of equations B.1 and denoting the Laplace variable

ass and the transformed concentrations asFi,

sFA − fA(t = 0) = λ4FD − λ−4FA − λ1FA + λ−1FB +D∇2FA ,

sFB − fB(t = 0) = λ1FA − λ−1FB − λ2FB + λ−2FC,

sFC − fC(t = 0) = λ2FB − λ−2FC − λ3FC + λ−3FD,

sFD − fD(t = 0) = λ3FC − λ−3FD − λ4FD + λ−4FA , (B.7)

We can use the last three equations to solve forFB,FC, andFD in terms ofFA . We obtain,

FB = Λ1BFA + Λ2

Bg(r, θ),

FC = Λ1CFC + Λ2

Cg(r, θ),

FD = Λ1DFD + Λ2

Dg(r, θ), (B.8)

whereΛ variables are rational functions of the Laplace variables. They are explicitly given in the end of this

appendix. We can now subsituteFB andFD (from equations B.8) into the equation forFA (first equation in B.7)

to obtain (after some rearrangements),

∇2FA − µ2FA = −σδ(r − r0)δ(θ)

r(B.9)

where,

σ = (ΓA + λ4Λ2D + λ−1Λ

2B)/D (B.10)

µ2 = (s+ λ1 + λ−4 − λ4Λ1D − λ−1Λ

1B)/D (B.11)


We write the Laplacian for polar coordinates and use the variable subsitution z= µr to obtain,

z2 ∂2FA

∂z2+ z

∂FA

∂z+∂2FA

∂θ2− z2FA = −σzδ(z− z0)δ(θ) (B.12)

where z0 = µr0. We use the method of eigenfunction expansion to write down aplausible solution forFA ,

FA =∑

n

[Vn(z) sin(nθ) + Un(z) cos(nθ)] (B.13)

Subsituting this equation into equation B.12 and using the orthogonality property of the eigenfunctions we obtain

two equations,

z2V ′′n + zV ′

n − (z2 + n2)Vn = 0 (B.14)

z2U ′′n + zU ′

n − (z2 + n2)Un = −σzδ(z− z0)

αn(B.15)

where,

αn =

2π n = 0

π n > 0.

The two conditions on equations B.14 and B.15 are (1) solutions remain finite on the domain and (2) solutions

are zero on the domain boundary (r = b), za = µb. The solutions to equation B.14 are modified Bessel functions

of the first and second kind. After applying the two conditions we find thatVn = 0 for all n. The homogeneous

solutions of equation B.15 are also modified Bessel functions,CnIn(z) + DnKn(z). To obtain the particular

solution of equation B.15 we use the method of variation of parameters,

Un(z) = Xn(z)In(z) + Yn(z)Kn(z) (B.16)

where Xn(z) and Yn(z) are determined by substituting the above expression into equation B.15. The ingredients

required for this substitution are,

U ′n = XnI′n + YnK′

n

U ′′n = X′

nI′n + XnI′′n + Y′nK′

n + YnK′′n

and as usual we have set,

X′nIn + Y′

nKn = 0 (B.17)


Carrying out the substitution followed by some rearrangements we find,

X′nI′n + Y′

nK′n = −σδ(z− z0)

zαn(B.18)

Together, equations B.17 and B.18 form a system of equationsfor the derivatives of our unknown functions. We

integrate to find,

Xn(z) = −σKn(z0)

αnH(z− z0) (B.19)

Yn(z) =σIn(z0)

αnH(z− z0) (B.20)

whereH is the Heaviside step function. We can now write the full solution to equation B.15,

Un(z) = CnIn(z) + DnKn(z) + Xn(z)In(z) + Yn(z)Kn(z)

We set Dn = 0 for all n to obtain a finite solution. Applying the condition thatUn(za) = 0 we find,

Cn =σ [In(za)Kn(z0) − In(z0)Kn(za)]

αnIn(za)(B.21)

Finally, we have all necessary quantities to write the solution forFA ,

FA =∑

n

(CnIn(z) + Xn(z)In(z) + Yn(z)Kn(z)) cos(nθ) (B.22)

We obtainFB, FC, andFD using equation B.8.

B.2.3 Rapid evaluation ofψ and χ

Recall thatψ(t) = 1 − P (t) and we therefore begin by evaluatingP (t) as defined in equation B.4. Substituting

in the transformed quantities and using the change of variables z= µr we obtain,

P (t) =

∫ a

0

rdr

∫ 2π

0

dθ (fA + fB + fC + fD)

= L−1

(1/µ2)

∫ za

0

zdz∫ 2π

0

dθ (FA + FB + FC + FD)

whereL−1 is the inverse Laplace transform. Using equations B.8 we substitute all transformed varialbes in terms

of FA and subsequently use the solution forFA (equation B.22) to simplify the expression. This simplification


requires some tedious algebra along with the use of the following identities,

zI1 =

∫

zI0dz

−zK1 =

∫

zK0dz

InKn+1 + In+1Kn = 1/z

We finally arrive at the following simple expression,

P (t) = L−1

(

Λ2B + Λ2

C + Λ2D

)

+σ

µ2

(

1 + Λ1B + Λ1

C + Λ1D

)

(

1 − I0(z0)

I0(za)

)

(B.23)

Using a numerical inverse Laplace transform in Matlab we caninvert this quantity in under 1 millisecond (AMD

Athlon) and findψ(t) = 1 − P (t).

We now wish to evaluateχ(t, θ). We begin by evaluatingp(t, θ) using equation B.5. Substituting in the

transformed quantities and using the change of variables z= µr we obtain,

p(t, θ) = −D (∇fA · dr) |r=b

= L−1

−D ∂FA

∂r

∣

∣

∣

∣

r=b

= L−1

−Dµ ∂FA

∂z

∣

∣

∣

∣

z=za

.

(B.24)

Taking the derivative of equation B.22, using the Bessel function identities quoted above, along with the following

ones,

I′n = In+1 +n

zIn

K′n = −Kn+1 +

n

zKn ,

(B.25)

we can obtain, after some tedious algebra, the following simple expression

p(t, θ) = L−1

Dσµ

za

∑

n

In(z0)

αnIn(za)cos(nθ)

. (B.26)


Subsituting this expression into equation B.6 we obtain,

χ(t, θ) =

[

∑

n

(

L−1

Dσµ

a

In(z0)

In(za)

)(

sin(nθ)

nαn

)

]

[(

L−1

Dσµ

b

I0(z0)

I0(za)

)(

2π

α0

)]

(B.27)

Once again, using a numerical inverse Laplace transform routine we can invert this expression in less than 1 ms.

B.2.4 Definitions ofΛ

Λ1B =

[

λ1s2 + (λ4λ1 + λ1λ−3 + λ3λ1 + λ−2λ1)s+ λ−3λ−2λ−4 + λ3λ1λ4 + λ1λ−2λ4 + λ1λ−2λ−3

]

/Λ

Λ2B =

[

ΓBs2 + (λ3ΓB + ΓBλ−3 + λ−2ΓC + λ−2ΓB + ΓBλ4)s+ λ3ΓBλ4 + ΓBλ−2λ−3 + λ−3λ−2ΓD+

λ−2λ−3ΓC + λ−2λ4ΓC + ΓBλ−2λ4] /Λ

Λ1C = [(λ−3λ−4 + λ1λ2)s+ λ−3λ−1λ−4 + λ−3λ1λ2 + λ−3λ2λ−4 + λ4λ1λ2] /Λ

Λ2C =

[

ΓCs2 + (λ−3ΓD + ΓCλ−1 + λ−3ΓC + ΓCλ2 + λ4ΓC + ΓBλ2)s+ λ4ΓBλ2 + λ−3ΓBλ2 + λ−3λ−1ΓD+

λ4ΓCλ−1 + λ4ΓCλ2 + λ−3ΓCλ2 + λ−3ΓCλ−1 + λ−3λ2ΓD] /Λ

Λ1D =

[

λ−4s2 + (λ−4λ−1 + λ−4λ3 + λ−2λ−4 + λ−4λ2)s+ λ3λ−1λ−4 + λ3λ2λ−4 + λ−2λ−1λ−4+

λ1λ3λ2] /Λ

Λ2D =

[

ΓDs2 + (λ2ΓD + λ−2ΓD + ΓDλ−1 + λ3ΓD + ΓCλ3)s+ λ3λ2ΓD + ΓCλ3λ2 + ΓCλ3λ−1+

λ−2λ−1ΓD + ΓBλ3λ2 + λ3λ−1ΓD] /Λ

where,

Λ = s3 + (λ3 + λ−2 + λ−3 + λ4 + λ2 + λ−1)s2 + (λ−2λ−3 + λ−2λ4 + λ4λ3 + λ2λ−3+

λ4λ2 + λ4λ−1 + λ−3λ−1 + λ3λ2 + λ3λ−1 + λ−2λ−1)s+ λ3λ−1λ4 + λ−2λ−1λ−3+

λ−2λ−1λ4 + λ3λ2λ4

(B.28)

B.3 Connection betweenP (t), p(t, θ), and MFPT

Given that earlier sections in Chapter 2 were concerned withcalculating the mean escape time from a TCR cluster

it is worth pointing out the connection between the quantities derived above and the mean escape time. Consider a

particle released at a specific location within our domain and that we measure how long it takes for the particle to

first reach the boundary. Repeating this exact experiment many times will produce a distribution,p(t), of escape


times (or first passage times) due to stochasticity in the diffusion process. The mean escape time (or mean first

passage time (MFPT)) can be obtained fromp(t) as follows,

τ =

∫ ∞

0

tp(t)dt (B.29)

whereτ is the mean escape time and is equal totA , as defined in the main text, when there are no reactions. As

defined above,P (t), is the probability that the particle has not escaped the domain at timet. The distribution of

escape times,p(t), is related toP (t) as follows,

p(t) = −∂P (t)

∂t(B.30)

By multiplying equation B.30 byt and integrating from0 to ∞ we find,

τ =

∫ ∞

0

P (t)dt

where we have used the fact thatp(0) = 0 (i.e. probability of escape is zero att = 0) andP (0) = 1 (i.e att = 0

the particle is in the domain). Lastly, we can obtainp by integratingp(t, θ) overθ from 0 to2π.

141

References

[277] Isaacson, S. A., and C. S. Peskin. 2006. Incorporatingdiffusion in complex geometries into stochastic

chemical kinetics simulations.SIAM J. Sci. Comp. 28:47–74.


immunological synapse: a molecular machine controlling t cell activation.Science. 285:221–227.

[279] Yokosuka, T., K. Sakata-Sogawa, W. Kobayashi, M. Hiroshima, A. Hashimoto-Tane, M. Tokunaga, M. L.

Dustin, and T. Saito. 2005. Newly generated t cell receptor microclusters initiate and sustain t cell activation

by recruitment of zap70 and slp-76.Nat Immunol. 6:1253–1262.

[280] Varma, R., G. Campi, T. Yokosuka, T. Saito, and M. L. Dustin. 2006. T cell receptor-proximal signals are

sustained in peripheral microclusters and terminated in the central supramolecular activation cluster.Immu-

nity. 25:117–127.

[281] Kaizuka, Y., A. D. Douglass, R. Varma, M. L. Dustin, andR. D. Vale. 2007. Mechanisms for segregating

t cell receptor and adhesion molecules during immunological synapse formation in jurkat t cells.Proc Natl

Acad Sci U S A. 104:20296–20301.

[282] Campi, G., R. Varma, and M. L. Dustin. 2005. Actin and agonist mhc-peptide complex-dependent t cell


[283] Gillespie, D. 1977. Exact stochastic simulations of coupled chemical reactions.J. Phys. Chem. 81:2340–

2361.

142

Appendix C

Analysis of errors due to geometric

approximations (Chapter 4)

There are two subtleties with the application of equation 4.6 to the geometry of protocol 2. First, the cell radius

must be determined. This is a trivial task since, as shown in figure 4.1, a ring of fluorescence from the labeled

surface molecule is observed at the equatorial plane of the cell. The cell diameter can be determined from this ring

using most image editing software. Since equations 4.7 varyslowly withR, we find that a10% error in estimating

R gives an error of3% in the diffusion coefficient when using equation 4.6, see figure C.1a. Therefore negligible

error is expected to arise inD from inaccuracies in determiningR.

Figure C.1: Potential source of error in using the 2D model. Percent differences (defined in figure 4.3) inD as aresult of estimating the cell radius are shown as contours. Data was generated using equation 5.9 at various cellradii (ActualR) and were fit using a different cell radius (EstimatedR). The bleached and monitoring regionparameters,L, were calculated in each case using equations 4.7 withWx = 2µm,Wy = 2µm, andWz = 2µm.

The second subtlety has due to with approximating the bleached and monitoring regions as perfect rectangles

on the surface of the cell. Protocol 2 assumes that a perfect rectangle (of areaAr = LxbLyb) is bleached on the

surface of the cell. However, althoughLxb represents the curved length at the middle of the stripe, this distance

changes throughout the stripe width,Lyb. It maximally differs fromLxb on the edges of the bleached stripe.

Appendix C. Analysis of errors due to geometric approximations (Chapter 4) 143

Taking into account this curvature, we calculate the area ofthe bleached stripe as,

Ac =

∫ π2+arcsin

(

Wy2R

)

π2−arcsin

(

Wy2R

)

2R sin(φ) arccos

(

1 − Wx

R sin(φ)

)

Rdφ (C.1)

The total area of the bleached stripe when we assume a perfectrectangular stripe (neglecting curvature effects) on

the surface of the cell is,

Ar = LxbLyb = 4R2 arccos(1 −Wx/R) arcsin(Wy/2R) (C.2)

We compare these values for a large range ofWx,Wy, andR = 5µm. We find that the difference betweenAr and

Ac is negligible (< 0.1%) for all experimental conditions with a large cushion and that increasingR reduces this

difference.

144

Appendix D

Asymptotic reductions of the full model

(Chapter 5)

D.1 Full model

The full model, described by equations 5.3 and 5.4, has four relevant timescales; two diffusive timescales (τf =

L2/Df, τc = L2/Dc) and two reaction timescales (1/k∗on, 1/koff) . Using formal asymptotic analysis, we demon-

strate below, that when there exists a separation of timescales the full model can be reduced to a simpler model

consisting of only simple diffusion equations. We non-dimensionalize the full model with the transformations:

f = f/Feq, c = c/Ceq, x = x/L, t = t/τ , resulting in the following nondimensional form of the fullmodel:

∂f/∂t =τDf

L2∇2f + τk∗on(c− f), (D.1a)

∂c/∂t =τDc

L2∇2c+ τkoff(f − c). (D.1b)

In this form, all timescales are explicit and all terms are oforder one. The length scaleL is determined from the

experimental setup; for example, an appropriate choice is the dimension of the bleaching region. Intuitively, at

equilibrium, we expect the labeled ligand to be in a free state (corresponding to the diffusion coefficientDf) for

a fractionkoff/(k∗on + koff) of the time, and in the bound complex (with diffusion coefficientDc) for a fraction

k∗on/(k∗on + koff). Therefore, an appropriate choice of timescale for the fullsystem is:

τ = L2/

(

koff

k∗on + k∗onDf +

k∗on

k∗on + koffDc

)

.

= L2/

(

1

1 +K∗Df +K∗

1 +K∗Dc

)

(D.2)

whereK∗ = k∗on/koff is a pseudo-affinity constant that will be the parameter of interest in the model reductions

described below. Note that, by definition,Feq = 1/(1 +K∗), andCeq = K∗/(1 +K∗)

Appendix D. Asymptotic reductions of the full model (Chapter 5) 145

D.2 Dimensionless form of full model

We now examine the dimensionless system (equation D.1) under different limiting conditions. To do so, we define

a small parameter,ε as the ratio of two timescales, and expand the solutions in powers of this parameter,

f ∼ f0 + εf1 + . . . (D.3a)

c ∼ c0 + εc1 + . . . (D.3b)

to obtain leading order equations forf0 and c0. As described below, each reduction leads to a simple diffusion

equation of the form

∂u

∂t= D∇2u (D.4)

with a different definition ofu, and the parameterD for each limiting case. This diffusion eqution can be solved

using standard techniques, and for each limit, we sum the dimensional forms of the leading order solutions to

obtain an expression for fluorescence intensity of the form:

g(x, t,D) = f0(x, t) + c0(x, t)

=koff

k∗on + kofff0(x/L, t/τ) +

k∗on

k∗on + koffc0(x/L, t/τ) (D.5)

Integrating (and normalizing)g over the the monitoring region gives rise to a functionG that can be fit directly to

FRAP data.

D.3 Pure diffusion

There are two parameter regimes where we obtain “pure diffusion” limits. The first is when the free pool is large,

τfk∗on = ε, τfkoff = 1/ε (D.6)

In this case the timescale of the FRAP experiment reduces toτ = τf and equation D.1 reduces to,

∂(f0 + εf1)

∂t= ∇2(f0 + εf1) + ε(c0 + εc1 − (f0 + εf1)) (D.7a)

∂(c0 + εc1)

∂t= τfτc∇2(c0 + εc1) + (1/ε)(f0 + εf1 − (c0 + εc1)), (D.7b)


whose first order terms are:

ε0 :∂f0

∂t= ∇2f0 (D.8a)

f0 = c0. (D.8b)

Transforming these equations back to their dimensional form we find,

∂f0∂t

= Df∇2f0 (D.9)

and, therefore, in this case,g = gpf = u(x, t,Df).

The second “pure diffusion” regime occurs when the bound pool is large,

τfk∗on = 1/ε, τfkoff = ε (D.10)

and by analogy with the previous case,g = gpc = u(x, t,Dc).

D.4 Weighted diffusion

When the reaction timescales are fast compared to the diffusive timescales we obtain another reduction which we

label as “weighted diffusion”. The appropriate definition for ε in this case is,

τfk∗on = 1/ε (D.11)

We note that there is no separation of scale between the on- and off-rate (i.e.k∗on/koff = constant) and the two

diffusion timescales (τf/τc = constant). The timescaleτ does not simplify in this case. We rearrange the terms in

the full system D.1 to obtain,

∂f

∂t=

1 +K∗

1 +K∗(Dc/Df)∇2f + k∗onτf

1 +K∗

1 +K∗(Dc/Df)(c− f) (D.12a)

∂c

∂t=

1 +K∗

K∗ + (Df/Dc)∇2c+ k∗onτf

1 +K∗

K∗ + (Df/Dc)(f − c) (D.12b)

where we can clearly identify the small parameter in the second term of each equation. We expandf and c to


obtain,

∂(f0 + εf1)

∂t=

1 +K∗

1 +K∗(Dc/Df)∇2(f0 + εf1) + k∗onτf

1 +K∗

1 +K∗(Dc/Df)(c0 + εc1 − (f0 + εf1)) (D.13a)

∂(c0 + εc1)

∂t=

1 +K∗

K∗ + (Df/Dc)∇2(c0 + εc1) + k∗onτf

1 +K∗

K∗ + (Df/Dc)(f0 + εf1 − (c0 + εc1)) (D.13b)

Equating terms with like powers ofε we obtain the following equations,

ε0 : f0 = c0 (D.14a)

ε1 :∂f0

∂t=

1 +K∗

1 +K∗(Dc/Df)

(

∇2f0 + c1 − f1

)

(D.14b)

∂c0

∂t=

1 +K∗

K∗ + (Df/Dc)∇2c0 +

1 +K∗

K∗ + (Df/Dc)(f1 − c1) (D.14c)

Eliminatingf1 − c1 and rearranging terms we obtain,

∂f0

∂t= ∇2f0 (D.15)

recast in the dimensional form to,

∂f0∂t

=

(

1

1 +K∗Df +K∗

1 +K∗Dc

)

∇2f0 = Dwd∇2f0 (D.16)

where we have defined an effective diffusion coefficientDwd =(

11+K∗

Df + K∗

1+K∗Dc

)

, that is a weighted av-

erage of the individual diffusion coefficients. Sincef0 = c0 we determine thatg = gwd = u(x, t,Dwd). Thus,

in this parameter regime we ultimately obtain a simple diffusion equation with the effective diffusion coefficient,

Dwd. This regime is equivalent to the “pseudo-effective diffusion” regime of Braga et al. (? ).

D.5 Independent diffusion

Finally, when the reaction timescales are slow compared to the diffusive timescales we obtain another reduction

which we label as “independent diffusion”. The appropriatedefinition forε in this case is,

τfk∗on = ε (D.17)

As in the weighted diffusion reduction, we do not assume any scale separation within the reaction or diffusion

timescales and no simplification toτ is possible. We identify the smaller parameter in equation D.12 and subsitute


equation D.3 to obtain,

∂(f0 + εf1)

∂t=

1 +K∗

1 +K∗(Dc/Df)∇2(f0 + εf1) + ε

1 +K∗

1 +K∗(Dc/Df)(c0 + εc1 − (f0 + εf1)) (D.18a)

∂(c0 + εc1)

∂t=

1 +K∗

K∗ + (Df/Dc)∇2(c0 + εc1) + ε

1 +K∗

K∗ + (Df/Dc)(f0 + εf1 − (c0 + εc1)) (D.18b)

The leading order equations are simply,

ε0 :∂f0

∂t= τDf∇2f0 (D.19a)

∂c0

∂t= τDc∇2c0 (D.19b)

rewritten in their dimensional form as the following systemof uncoupled diffusion equations,

∂f0∂t

= Df∇2f0 (D.20a)

∂c0∂t

= Dc∇2c0 (D.20b)

and the functiong is the weighted sum of the solutions,g = gid = (1/(1 + K∗))u(x, t,Df) + (K∗/(1 +

K∗))u(x, t,Dc). In other words, in this parameter regime there is effectively no exchange between the free and

bound pool on the timescale of the FRAP experiment, and therefore the two pools diffuse independently.

D.6 Reaction dominant

In the above asymptotic reductions we have not assumed any scale separation betweenDf andDc. We can recover

the results of Sprague et al. (284), who analyzed the reaction-diffusion system in equation D.1 withDc = 0, when

we assume a large scale separation betweenDf andDc. Definingε as,

τfk∗on = ε, τf/τc = ε2 (D.21)

and taking the timescale to be,τ = 1/koff, we can reduce equation D.1 to,

∂f/∂t = 1koffτf

∇2f + k∗on/koff(c− f) (D.22)

∂c/∂t = 1koffτc

∇2c+ (f − c) (D.23)


Identifying the small parameter, we arrive at the followingequations,

∂(f0 + εf1)

∂t= 1

ε∇2(f0 + εf1) + k∗on/koff(c− (f0 + εf1)) (D.24)

∂(c0 + εc1)

∂t= ε∇2(c0 + εc1) + ((f0 + εf1) − (c0 + εc1)) (D.25)

The leading order equations are simply,

ε0 : ∇2f0 = 0 (D.26a)

∂c0

∂t= f0 − c0 (D.26b)

Solving these equations we recover the reaction dominant solution presented in (284),

GRD(t) = 1 − K∗

1 +K∗ e−kofft (D.27)

Note: This suggests that for the approximationDc = 0 to be valid,Dc/Df must beO(ε2).

Table D.1 below summarizes the small parameters for each model reduction.

Table D.1: Summary of Asymptotic LimitsReduction Small Parameter DefinitionPure Diffusion (free pool) τfk

∗on = ε, τfkoff = 1/ε

Pure Diffusion (bound pool) τfk∗on = 1/ε, τfkoff = ε

Weighted Diffusion τfk∗on = 1/ε

Independent Diffusion τfk∗on = ε

Reaction Dominant τfk∗on = ε, τf/τc = ε2

150

References



151

Appendix E

Details of model fitting to simulated data

(Chapter 5)

Table E.1: Model Fitting to Simulated Experiment AModel SSR AIC ∆AIC p Fitted ParametersST = 1µm−2

WD 0.02469 -828.64 2.93 0.24Dwd = 0.4795µm2/sID 0.02398 -831.57 0.00 0.63K∗ = 0.0160FM 0.02450 -827.41 4.15 0.14k∗on = 3.162× 10−5s−1

koff = 1.774× 10−3s−1

ST = 40µm−2


koff = 1.045× 10−3s−1

ST = 80µm−2


koff = 8.806× 10−5s−1

ST = 200µm−2

WD 0.02123 -843.77 0.84 0.37Dwd = 0.0503µm2/sID 0.02105 -844.61 0.00 0.46K∗ = 198.0FM 0.02127 -841.56 3.05 0.17k∗on = 75.92s−1

koff = 4.856× 10−2s−1

WD - Weighted Diffusion, ID - Independent Diffusion, FM - FullModel, p - Probability (equation 5.19 in main text)

Appendix E. Details of model fitting to simulated data (Chapter 5) 152

Table E.2: Model Fitting to Simulated Experiment BModel SSR AIC ∆AIC p Fitted ParametersST = 1µm−2


koff = 5.965× 10−1s−1

ST = 40µm−2


koff = 1.109× 10−2s−1

ST = 80µm−2


koff = 9.335× 10−3s−1

ST = 200µm−2


koff = 5.331× 10−4s−1


Appendix E. Details of model fitting to simulated data (Chapter 5) 153

Table E.3: Model Fitting to Simulated Experiment CModel SSR AIC ∆AIC p Fitted ParametersST = 1µm−2

WD 0.02221 -839.26 0.00 0.80Dwd = 0.5083µm2/sID 0.02334 -834.29 4.97 0.12K∗ = 4.978× 106

FM 0.02310 -833.33 5.93 0.08k∗on = 3.162× 10−5s−1

koff = 1.000× 103s−1

ST = 50µm−2


koff = 1.401s−1

ST = 200µm−2


koff = 6.030× 10−1s−1

ST = 1000µm−2

WD 0.02090 -845.33 0.00 0.58Dwd = 0.0940µm2/sID 0.07804 -713.57 131.76 0.00K∗ = 2.542FM 0.02072 -844.17 1.17 0.42k∗on = 3.024s−1

koff = 2.812× 10−1s−1


154

Appendix F

Materials and methods (Chapter 6)

F.1 Cell isolation and culture

Purification of CD4+ human PBL T cells were isolated from the whole blood of healthy donors (Centre de Trans-

fusion Sanguine, CHU Purpan, Toulouse) as described previously (285). Briefly, PBMC were purified from blood

by centrifugation through Ficoll-Hypaque (Pharmacia Biotech, Sweden). CD4+ T lymphocytes were purified

using RosetteSep Kit (StemCell Technologies, Vancouver, Canada). Cell purity was assessed by FACS analysis

(Facscan, Becton Dickinson) using FITC-labelled anti-CD4mAb (clone RPA-T4, BD Pharmingen). CD4+ frac-

tions were routinely∼ 90% pure. Before use, purified T cells were cultured in complete RPMI 1640 (Gibco,

Paisley, Scotland) containing5% human serum. In parallel experiments, Cord Blood T Lymphocytes (CBTL)

were purified from whole cord blood (CHU Purpan, Toulouse) asdescribed above. In some cases CBTL were cul-

tured and expanded for 14 days in RPMI 1640,5% human serum, IL-2 (150 IU/ml) using Dynabeads CD3/CD28

T Cell Expander (Dynalbiotech, Oslo, Norway). Bead to cell ratio was 1:1.

F.2 Fluorescence recovery after photobleaching

Anti-CD3ε mAbs (TR66, IgG1, (286)) were digested by using the IgG1 Fab and F(ab)’2 kit (Pierce biotech-

nologyTM, Rockford). Fab fragments (5 mg/ml in 0.1 M sodium carbonate, pH= 9.3) were labelled with Cy5

Monoreactive dye pack (Amersham BioscienceTM, Piscataway). The labelled Fab fragments were separated from

non-conjugated dye by gel chromatography by the use of Nap-10 Columns (Amersham Bioscience). This step

was followed by an overnight dialysis in 1×PBS by using a Lyser Dialysis cassettes (Pierce biotechnology) (287).

PBMC derived CD4+ T cells were washed and re-suspended in RPMI, 5% FCS, 10 mM Hepes. Cells were loaded

with 2µM Fluo4 AM (Molecular Probes, Leiden, The Netherlands) for 30 minutes at37 C. After washing, cells

were stained with 30µg/mL CD3/Cy5 Fab in RPMI,1% FCS at4 C for 30 minutes. Cells were washed and kept

in ice until used for FRAP experiments. FRAP was performed inLab-Tek chambers (Nalgene Nunc, Rochester,

NY) in pre-warmed5% FCS/10/mM HEPES at37 C, 5% CO2 on a confocal microscope (LSM 510 ; Carl Zeiss,

Jena, Germany) using a Plan-Apochromate 63x objective and the 633 nm laser. A rectangular region (2µm by

Appendix F. Materials and methods (Chapter 6) 155

1.4µm) was defined on the surface of T cells. This region was irradiated using the 633 nm, 543 nm, and 488 nm

lasers with 100% intensity for 3 s. The aperture of the pinhole was adjusted toobtain optical slices of 2µm depth.

Before and after bleaching, the whole field was visualized byirradiating with 488 nm laser with4% intensity and

633 nm laser with17% intensity. Images were taken at 1 s intervals. In some experiments, cells were treated

either with ionomycin (0.5µg/ml) at the time of the time-lapse recording. In some experiments cells were treated

with cytochalasin D (10µM) or latrunculin B (50 nM) before the beginning of the assay.

F.3 Fitting procedures

We used the method for FRAP measurements of surface diffusion described in (288). Briefly, a small region of

the cell membrane was bleached and the fluorescence recoverycurve was fit using a two-dimensional recovery

equation to estimate the TCR diffusion constant and the mobile fraction of TCR. The Matlab function lsqcurvefit

was used for fitting. We also fit a simple exponential to every FRAP experiment. Using the time-constant of the

recovery as an indicator of TCR mobility gave the same conclusions as using the diffusion coefficient (analysis

not shown).

F.4 Quantification of F-actin by phalloidin staining and flow cytometry

A method described by Downey et al (289) was used with some modifications (290). Briefly, PBL were fixed with

3% PFA, permeabilized with0.1% saponin and stained with 160 nM Alexa 488-labelled phalloidin. Cells were

analyzed on a FACSCalibur (Becton Dickinson). Gating was done to exclude by forward and side scatter criteria

cell debris and cell clumps. In some samples T cells were treated with 2.5µg/ml ionomycin or with 50 to 500 nM

latrunculin B for 15 minutes at37 C, 5%, before fixation.

156

References

[285] Zaru, R., T. Cameron, L. Stern, S. Muller, and S. Valitutti. 2002. Cutting edge: TCR engagement and

triggering in the absence of large-scale molecular segregation at the T cell-APC contact site.J Immunol.

168:4287–91.

[286] Lanzavecchia, A., and D. Scheidegger. 1987. The use ofhybrid hybridomas to target human cytotoxic T

lymphocytes.Eur J Immunol. 17:105–11.

[287] Depoil, D., R. Zaru, M. Guiraud, A. Chauveau, J. Harriague, G. Bismuth, C. Utzny, S. Muller, and S. Vali-

tutti. 2005. Immunological synapses are versatile structures enabling selective T cell polarization.Immunity.

22:185–94.

[288] Dushek, O., and D. Coombs. 2008. Improving parameter estimation for cell surface FRAP data.J Biochem


[289] Downey, G., C. Chan, S. Trudel, and S. Grinstein. 1990.Actin assembly in electropermeabilized neu-

trophils: role of intracellular calcium.J Cell Biol. 110:1975–82.

[290] Valitutti, S., M. Dessing, and A. Lanzavecchia. 1993.Role of cAMP in regulating cytotoxic T lymphocyte

adhesion and motility.Eur J Immunol. 23:790–5.

mathematical modeling in cellular immunology: t cell ... · mathematical modeling in cellular...

Documents