Near-Perfect Adaptation in E. coli Chemotaxis Signal Transduction Network

Yang Yang & Sima Setayeshgar

Jan, 2007

E. coli

•1-3 microns long

•1 micron in diameter

•4-6 flagella

• Small genome (4288 genes)

static.howstuffworks.com/gif/cell-ecoli.gif

www.hatetank.dk

Bacterial Chemotaxis

http://www.rowland.harvard.edu/labs/bacteria/index_movies.html

Increasing attractants or Decreasing repellents

Run Tumble

Chemotaxis signal transduction network in E. coli

well-characterized model system

CheW: Coupling CheA to MCPs CheB: CheBp demethylate MCPs CheR: Methylate MCPs CheY: flagella motor regulator protein CheZ: Dephosphoryte CheYp; CheA: Histidine kinase

ligand binding

Methylation

Phosphorylation

)()( )(7/7~5/5

)( CheRLTCheRTL pnkmkkmk

pn

ppnmkmk

ppn

pnckck

pn

CheBTLCheBTL

CheRTLCheRTL

)(14~1

)(

)(14~1

)(

)()(

)()(

PCheBCheB

PCheYCheZCheZCheY

CheBCheRTCheBCheRTL

CheYCheRTCheYCheRTL

ADPCheRTLATPCheRTL

kmbp

kmyp

pnkb

np

nky

np

npkk

n

)()()(

)()()(

)()()()( 9~7

T3 T4T2

T4pT2p T3p

LT3 LT4

LT4p

LT2

LT3pLT2p

phosphorylation

methylation

Lig

and

bind

ing

Full realistic model

Chemical reactions

Perfect adaptation

Steven M., et al. Journal of bacteriology. 1983

This property allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli

Robustness

U. Alon et al. Nature,1999

inputReaction rates

proteins outpu

t

Tau-Mu Yi* et al. Biophysics,2000

Can be achieved

by

Motivation

shedding light on biochemical steps and feedback mechanisms underlying robustness

shed light on values of unknown or partially known paramters

SOLVE: we develop a novel method for elucidating regions in parameter space allowing perfect adaptation.

QUESITON: basis of robustness of perfect adaptation?

START with a fine-tuned model of chemotaxis network that:

reproduces key features of experiments

is NOT robust

AUGMENT the model explicitly with the requirements that:

steady state value of CheYp

values of reaction rate constants, are independent of the external stimulus, s, thereby

achieving robustness of perfect adaptation.

s

k

F

u

skuFdt

ud

0);;(

: state variables

: reaction kinetics

: reaction constants

: external stimulus

Algorithm

Decretizing s

into H points

0

||

0);;(

ds

kdds

du

skuFdt

ud

N

02

|2

|

0);;(

)1(

11

11

s

kks

uu

skuFdt

ud

sjss

jm

jm

j

jN

jN

jjj

jlowj

Augmented system

The steady state concentration of proteins in the network must satisfy:

The steady state concentration of CheYp must satisfy:

At the same time, the reaction rate constants must be independent of stimulus:

: allows for near-perfect adaptation

= CheYp

0ds

kd

0);;( skuFdt

ud

N

N

u

ds

du

||

Implementation

Newton-Raphson, to solve for the steady state of augmented system:

multidimensional root finding method Efficient way of converging to a root with a sufficiently good initial guess.

x

f(x) 1

2 f(x)1

x

2

Works well unfortunate case fortunate case

x

f(x) 1

32

0

||

0);;(

ds

kdds

du

skuFdt

ud

N

Dsode (stiff ODE solver), to verify Time dependent behavior of proteins for different ranges of external stimulus by solving: 0);;( skuF

dt

ud

Implementation

Working progress

Exploring the parameter spaces of E. coli chemotaxis signaling transduction network

Exploring the unknown parameter ranges of chemotaxis signaling transduction network of species with multiple CheYs

E .coli

Relative change of CheYp:• less than 5%• less than 3%• less than 1%• pairwise trajectory

Pairwise result: 3D surface result:

parameter spaces of E. coli

E. coli

Relative change of CheYp:• less than 5%• less than 3%• less than 1%• pairwise trajectory

Pairwise result: 3D surface result:

parameter spaces of E. coli

Pairwise result: 3D surface result:

Relative change of CheYp:• less than 5%• less than 3%• less than 1%• pairwise

trajectory

E .coli

parameter spaces of E. coli

1%

k1c : 0.17 s-1 1 s-1

k8 : 15 s-1 12.7 s-1

Violating and restoring perfect adaptation 9%

k1c : 0.17 s-1 1 s-1(1,15)

(1,12.7)

At 250s, giving step stimulus from 0 to 1e-6M

Consistency with recent work done by Bernardo A. mello and Yuhai Tu

They list a series of conditions which allow near-perfect adaptations

They are a active-ependent model which the receptors are either in active or inactive state

Our parameter space remarkable shows the same consistency with their predictions about the relationships of the parameter values although we are using a active-independent model.

1. The timescale for ligand binding is much shorter than the methylation and phosphorylation timescale. This condition allows us to neglect ligand-binding/unbinding kinetics.

2. The association rates between the receptor and the methylation/demethylation enzymes, CheR and CheB-P, are linearly related to the activity of the receptor and are zero for n = 4 and n = 0, respectively: and : The dissociation rates of the enzyme receptor bound states are independent of λ.

3. The receptor activities of the nonmethylated and the maximally methylated receptors are independent of l: P0v = P0o, P4v = P4o.

4. The ratios between the CheR catalytic rate kR n and the CheB-P catalytic rate of the next methylation level kB n+1 are the same for all methylation states n: kB n+1 / kR n = const:

5. The phosphate transfer rates from CheA to CheB or CheY are proportional to CheA autophosphorylation rate:

6. The explicit dependence on [TFn] distribution can be

removed from the expression

this condition can only be strictly satisfied when

nRn PPK 4 0PPK n

Bn

nPYnn

PBn PkPk ;

4

0

2 ][)][][

(n

FnnB

F

R

F

TPK

B

K

R

B

F

R

F

KB

KR ][][

List of conditions:

Condition 2

nancnRn kkPPK 4

ckbkakPPK nanamnnBn 2

0 )(

Condition 3 P0v = P0o, P4v =

P4o

Condition 4 kB n+1 / kR n =

constan

nc

nm kkk

)1()1(

Condition 5 n

PYnn

PBn PkPk ;

kb /k8-13 and ky/k8-13 are linearly related

*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997) while the inset is not since the literature value is zero for k11.

Condition 6 B

F

R

F

KB

KR ][][

kb /k8-13 and ky/k8-13 are linearly related

*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997) while the inset is not since the literature value is zero for k11.

Two CheY system

•Rhodobacter sphaeroides, Caulobacter crescentus have multiple CheYs while lack of CheZ protein.•Similar chemotaxis behaviors.

Two CheY system

Our work:Reproduce the key feature of chemotaxis behavior in two CheY system by replacing CheZ with CheY2.

Parameter spaces of two CheY system

Introducing [CheY2] and CheY2 (de-)phosphorylation rates.Exploring the parameter values which can give perfect adaptation.

Two CheY

Relative change of CheYp:• less than 5%• less than 3%• less than 1%

Other parameter value were set as the literature value except Kb= 1e+6 M-1s-1 instead of 8e+5 M-1s-1.

parameter spaces comparison of two and single CheY case

The parameter space for single CheY case seems more restrict than the two CheY case

Conclusions

Successful implementation of the augmented model of the chemotaxis signal transduction network in E. coli that explicitly takes into account robust perfect adaption.

Preliminary results on projections of robustness manifolds in parameter space of E. coli and two CheY system

Complete construction of manifolds in parameter space, allowing insight into parameter dependence giving rise to robustness

Work in progress

Future workApplying the method to other cellular signal

transduction networks exhibiting robust homeostasis, such as phototransduction

Signal flow in visual transduction, Leon Lagnado and Denis Baylor,Neuron,1992

Thanks and comments!

Physics limitation in signal sensing

25 years ago Berg and Purcell had showed that the physics limitation of the single celled organism.The derivation is mainly assumed a perfect measurement device and they determined the relative measurement accuracy is :

But for multiple and noninterating receptors shaped as a ring, the formula is derived by Willam and Sima recently as:

With know parameter value, we can get the actual physics limit to measurements of CheYp concentration corresponds to

cDac

C 1 : diffusion constant

: device size: average concentration: sampling time

210 )2

1(

1

a

g

mbcDc

C

C

a

D

0g

b

m : receptor numbers: single receptor size: geometric factor of order unity

c

M

c

C 035.3

*Computational models of chemotaxis signal transduction network

Activity dependent model

Barkai & Leibler (1997)

The concept of robustness in biochemical networks introduced, showing how it may arise in bacterial chemotaxis through activity-dependent kinetics. The chemoreceptor is either in active state or inactive state.Simulations show that precision of adaptation is a robust property, while adaptation time is not, and that adaptation time is inversely proportional to receptor-complex activity.It did not show how the parameter space will change which is very important for understanding the robustness mechanism.

Activity independent model

Spiro et al. (1997)

Simplified three-methylation-state model, fine-tuned by trial and error, simulates ramp, step and saturation responses to aspartate.Although it can not achieve the robust perfect adaptation, but it is a more realistic model without assuming any of the activity dependent parameter values.And our work is start from implementing this fine-tuned model.

k1c/km1, k2c/km2, k3c/km3, k4c/km4 are linearly related:

*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997).