APS-DFD

November, 2006

Measurements and Models of

Cytoskeletal Rheology

Roger D. Kamm

MITDepartment of Mechanical Engineering

Department of Biological Engineering

Acknowledgements: TaeYoon Kim, Hyunsuk Lee, Belinda Yap,Jeff Hsu, Jorge Ferrer, Seok Chung

APS-DFD

November, 2006

The Neutrophil -- naturally-occurring

deformations

(micrograph courtesy of C. Dongand R. Skalak)

- spherical- mean diameter = 6.8 µm

SEM

Cross-section

- multi-lobed nucleus- granules/cytoskeleton- actin-rich cortical region

(micrograph courtesy of C.Dong and R. Skalak)

Bathe et al., 2002

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November, 2006

Cells have a

multi-component

fibrous, gel-like

matrix.

Cytoskeletal composition/structure

Ingber, Scientific

American

Hartwick,http://expmed.bwh.harvard.edu

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November, 2006

Actin -- One of the primary structural

components of cells

Biochemistry, Wiley 1995

• 40kDa (375 AA residues)

• G-actin to F-actin : Nucleation (lag phase), elongation (growth phase),

steady state (equilibrium phase)

• Semi-flexible filament (neither flexible nor stiff) lp lc

• Viscoelastic network

• Monomer size << mesh size lp

• Self assembly with polarity

• Binding to a variety of ABPs (Actin Binding Proteins)

• Actin polymerization plays a key role in fundamental cellular processes

APS-DFD

November, 2006

G*= G + i G =

˜ T (t)˜ (t)

is a geometry-dependent prefactor determined

from finite element analysis

G*= G0

0

x 1

1 + i ( ) (2 x)cos2

(x 1)

+ i µ

is the Gamma-function; G0, 0 and x are

adjustable parameters

Consistent with models for soft, glassy materials

Magnetic Twisting Cytometry (MTC)

Cells exhibit a power-law rheology,

reminiscent of a soft , glassy

material.

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November, 2006

Different behaviors for different types of

measurement

Hoffman, et al., PNAS, 2006

But values for G range from 10’s of Pa to several

kPa, depending on the method of measurement.

Data appear to fall

along two characteris-

tic slopes, one for

methods that

emphasize cortical

structure, and another

for internal, cytoskele-

tal structure.

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November, 2006

Reconstituted actin gel rheology

Hypotheses for breakdown of

the cytoskeleton due to

mechanical deformation

1) Sudden

depolymerization of

F-actin

2) Rupture of actin

cross-links between

F-actin filaments

Incremental shear moduli exhibit strain hardening up to a

point, followed by a catastrophic drop in stiffness.

Tseng et. al. (2004) J Biol Chem; 279: 1819-1826

APS-DFD

November, 2006

a b c d

e f g h

Cytoskeletal shear modulus

computed from the Brownian

fluctuations of granules

As a function of time after enteringchannel, until protrusion

In adherent, migratingneutrophils

In non-adherent,neutrophils

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November, 2006

Entrance Time -- Passive Behavior

0.01

0.1

1

10

100

0.1 1 10 100

Pressure (mm H2O)

En

tra

nce

tim

e (

s)

DL, 37deg

DS, 37deg

DL, 23deg

Entrance time : interval

between leading edge

touching channel

entrance and trailing

edge clearing channel

entrance after

deformation

Entrance time appears

primarily influenced by

the magnitude of

mechanical stimulation;

less so by biological

activity

Yap & Kamm, APS, 2005

To what extent can the

cytoskeleton be treated as

a cross-linked actin gel?

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November, 2006

Viscoelastic moduli of neutrophils following deformation

0

50

100

150

200

250

300

0 10 15

(N=16)

20

(N=37)

25

(N=12)

30

(N=20)

Time (s)

G' (

dyn

/cm

2 )

DS,10mmH2O,37deg (n=4)

Control

0

50

100

150

200

250

300

0 10 15

(N=9)

20

(N=27)

25 30

Time (s)

G' (

dyn

/cm

2)

DL,1mmH2O,37deg (n=5)

Control

*

**

**

** ** **

†††

0

100

200

300

400

500

600

0 10 15

(N=9)

20

(N=27)

25 30

Time (s)

G"

(dyn/c

m2)

DL,1mmH2O,37deg (n=5)

Control

0

100

200

300

400

500

600

0 10 15

(N=16)

20

(N=37)

25

(N=12)

30

(N=20)

Time (s)

G"

(dyn/c

m2)

DS,10mmH2O,37deg (n=4)

Control

**

**

*

†

††

G G”

Moduli fall immediately upon deformation, and recover on a time

scale of about one minute

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November, 2006

0

50

100

150

200

250

300

0 10 15

(N=16)

20

(N=37)

25

(N=12)

30

(N=20)

Time (s)

G' (

dyn

/cm

2 )

DS,10mmH2O,37deg (n=4)

Control

Storage modulus drops

immediately, then recovers on a

time scale of 30-60s.

A similar time-course is

observed for F-actin content

following passage through

pores.

Dynamic shearmodulus and F-actinfollow similar patterns

Post-deformation

Resting

neutrophils

Actin depolymerization appears to

explain part, but perhaps not all, of

the drop in modulus

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November, 2006

Similar behavior -- abrupt fall, then

recovery of stiffness -- has been

observed in other cell types

Cytoskeletal fluidization is a common

attribute of cells

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November, 2006

Cytoskeletal remodeling

Forces are transmitted via theCSK and actin cross-links

Bonds may rupture andreform, leading to remodeling

Signaling pathways mightinitiate depolymerization

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November, 2006

Some outstanding questions

What is the physical basis for soft, glassy rheology?

How can strain stiffening in reconstituted actin gels bereconciled with fluidization in cells under abrupt strains?

Are the differences in behavior seen with differentmeasurement methods real, and how can they bereconciled?

Are the different power law behaviors really due to cortex vs.cytoskeleton? If so, why?

APS-DFD

November, 2006

• Estimate viscoelasticity of the actin cytoskeleton using

microbead tracking rheology, both active and passive.

• Investigate the effect of ACPs on viscoelasticity

Cytoskeleton Growth

Microbead Rheology

• Investigate the effects and roles of the properties of actin

cross-linking proteins (ACPs) on cytoskeletal structure

Modeling Objectives

Simulate actin cytoskeletal growth, rheology, and force-induced changes in biochemical activity using moleculardynamics, Brownian dynamics, and continuum models.

Relevant recent works: Storm, et al., Nature (London), 2005

Hoffman, et al., http://arxiv.org/pdf/physics/0504051

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November, 2006

Cytoskeletal Modeling

2

2( )i j

i j

d r drm f d t

dt dt= +

Equation of motion: Langevin equation

** * * * * *

*( ) ( )i j i j

i j i j

drf d t dr f d t dt

dt= + = +

Integration Method: Euler forward (dropping inertia)

Drag computed (initially)ignoring hydrodynamicinteractions

Interaction forces: Scalings:

ULJ

(r) =4

r

12

r

6

+ r 21/6

0 r > 21/6

U

Bend=

1

2 Bend 0( )2

U

Spring=

1

2k

s(r r

0)

U

Torsion=

1

2 Torsion 0( )2

Truncated Lennard-

Jones

Bending stiffness

Extensional

stiffness

Torsional stiffness

t*=

t2 / k

BT

Length scaled with

actin radius, energy

with kBT

2

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November, 2006

Cross-linking

Bundling cross-linkers (e.g., fimbrin,fascin, scruin, -actinin)

Network cross-linkers (e.g., filamin)

r

ACP= 0.75,

1= ,

2=

2,

3= 0

r

ACP= 1.5,

1= 1.158 (rad),

2=

2,

3=

2

Unbinding obeys Bellʼsequation and is force-dependent.

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November, 2006

Monomers : 8,000, ACPs : 1,000

Vol. % = 0.82% (+ACP ~0.1%), R=0.125

Conc = 150 M, Domain = 560 nm

Comparison of cross-linked and bundled structures(varied by changing the torsional stiffness and orientation of actin binding sites)

Cross-linked Bundled Mixed

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November, 2006

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November, 2006

Computed 3D images

Computed Network Structures

“Confocal” images

1.12 µm x 1.12 µm

90° cross-

links

“Bundling”

cross-links200nm

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November, 2006

Analysis of the Polymerized Structure

Monomers : 8000, ACPs : 173

Vol. % = 0.82% = 150 M

Nucleation rate = 1ⅹ10-8 s-1, Domain = 560 nm

Filament length = total contour length

Connectivity = number of filaments attached

via cross-links

Avg = 328, S.D. = 112

APS-DFD

November, 2006

Analysis of the Polymerized Structure

Monomers : 8000, ACPs : 173

Vol. % = 0.82% = 150 M

Nucleation rate = 1ⅹ10-8 s-1, Domain = 560 nm

Mesh size = distance between cross-links

Pore size = max diameter of a sphere that can fit

inside a pore

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November, 2006

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November, 2006

F-actin networks formed by -actinin

R = 0.01 R = 0.1 R = 0.2 R = 0.5

10 m

Stress fibers form of increasing diameter as the amountof -actinin is increased

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November, 2006

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November, 2006

F-actin networks formed by filamin

Mean ± SD

0.93 ± 0.80 m

10 m

Rf = 10-1 Rf = 10-2 Rf = 10-3 Rf = 10-4

Rf = 10-1 Rf = 10-2 Rf = 10-4

Mean ± SD

0.98 ±

0.56 m

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November, 2006

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November, 2006

Passive microrheology

Monitor thermal fluctuations of

embedded microspheres to

estimate the frequency

dependent complex modulus

High spatial and temporal

resolution using quadrant

photo detector

Fluctuation-dissipation theoremFluctuation-dissipation theorem

Kramers-Kronig Kramers-Kronig relationrelation

Generalized Stokes-EinsteinGeneralized Stokes-Einstein

relationrelation

Thermally driven dynamicsThermally driven dynamics

of the of the microspheremicrosphere

TrappingTrapping

laserlaser

MicrosphereMicrosphere

F-F-actinactin matrix matrix

[ ]*

2

*

*

2

1/

( )(1/ ) 1 ( )

'( ) ( ) cos( ( ) / 2)

''( ) ( ) sin( ( ) / 2)

d ln ( )( )

d ln

B

t

k TG

a r

G G

G G

r t

t=

+

=

=

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November, 2006

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November, 2006

Apply a driving force to an embedded microsphere and

monitor its trajectory

• Sinusoidal and stepwise

Bulk rheometer

• Parallel plate, AR2000, TA Instruments

• Gap: 120 m – 40mm flat plate

• Frequency sweep

Active microrheology

timetime

xxtr

ap

trap(( tt

))

timetime

xxtr

ap

trap(( tt

))

tinput

tinput

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November, 2006

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November, 2006

Comparison at different filamin concentrations

– Active sinusoidal forcing

• One order magnitude difference of moduli as Rf increases10 fold

• G´a ~ Rf0.2 for 10 M actin, Rf = 0.01 and 0.1

• G´a of Rf = 0.1 is comparable to cellular rheology

● Rf = 0.1

● Rf = 0.01○ Rf = 0.1

○ Rf = 0.01

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November, 2006

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November, 2006

Comparison at different filamin concentrations –

Passive measurement

• Longer plateau region in G´p of Rf = 0.01

• Similar values of G´p in low frequency region

● Rf = 0.1

● Rf = 0.01

○ Rf = 0.1

○ Rf = 0.01

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November, 2006

Width : 924 nm, 1331 Monomers

Bead : 224 nm, Conc ~ O(10 M)

ComputationG’

G’’

Los

s sh

ear

Mod

ulus

, G’’

[Pa]

Sample : 20 x 5 x 0.2 mm

Bead dia. : 1 m, Conc. = 10 M

Experiment

Comparison of complex moduli

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November, 2006

Summary and remaining questions

• Simulated networks exhibit structures and rheology

“similar” to those found in experiments in cross-linked

gels.

• Can a computational model be created that replicates

cellular measurements?

• What is the fundamental basis for cytoskeletal fluidization,

correct power-law scaling?

• What is the role of measurement method (1- vs. 2-particle

methods, active vs. passive, bead coating, particle size

vs. mesh size) in cell rheological measurements?

"If we knew what it was we were doing, it would notbe called research, would it?” - Albert Einstein

APS-DFD

November, 2006

AcknowledgementsStudents:

TaeYoon Kim

Hyungsuk Lee

Jorge Ferrer

Belinda Yap

Jeff Hsu

Postdocs:

Seok Chung

Faculty collaborators:

Matt Lang

Peter So

Douglas Lauffenburger

Subra Suresh

Funding agencies:

NHLBI

NIGMS