Approach START with a fine-tuned model of chemotaxis network that: reproduces key features of…

PCheBCheB

PCheYCheZCheZCheY

CheBCheRTCheBCheRTL

CheYCheRTCheYCheRTL

ADPCheRTLATPCheRTL

CheBTLCheBTL

CheRTLCheRTL

CheRLTCheRTL

b

y

b

y

cc

kp

kp

pnk

np

nk

np

npkk

n

ppnkk

ppn

pnkk

pn

pnkk

kkpn

)()()(

)()()(

)()()()(

)()(

)()(

)()(

97

41

41

75

75

~)(1

~)(

)(1~

)(

)(~

~)(

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

reproduces key features of experiments (adaptation times to small and large ramps, perfect adaptation of the steady state value of CheYp) is NOT robustAUGMENT 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.

: state variables: reaction kinetics: reaction constants: external stimulus

Time (s)

Con

cent

ratio

n (µ

M)

Verify steady state NR solutions dynamically using DSODE for different stimulus ramps: k3c= 5 s-1

k10 = 101 s-1 k-2 = 6.3e+4 M-1s-1

Validation

Violating and Restoring Perfect Adaptation

1% k1c : 0.17 s-1 1 s-1

k8 : 15 s-1 12.7 s-1

Step stimulus from 0 to 1e-6M at t=250s

(1,15)

(1,12.7)

T2 Methylation rate (k1c)

T 2 a

utop

hosp

hory

latio

n ra

te

(k8)

Demethylation rate/mmethylation rate is proportional to autophosphorylation rate:

T3 autophosphorylation rate T 3 d

emet

hyla

tion

rate

/ T2

met

hyla

tion

rate

[3]

T4 autophosphorylation rate

T 4 d

emet

hyla

tion

rate

/ T3

met

hyla

tion

rate

[3]

LT3 autophosphorylation rate

T 3 d

emet

hyla

tion

rate

/ T2

met

hyla

tion

rate

[3]

LT4 autophosphorylation rate

LT4 d

emet

hyla

tion

rate

/ LT 3

met

hyla

tion

rate

[3]

Che

B p

hosp

hory

latio

n ra

te (k

b) /

liter

atur

e va

lue[

3]

Che

Y ph

osph

oryl

atio

n ra

te

(ky)

/ li

tera

ture

val

ue[3

]

(L)Tn autophosphorylation rate / literature value

(L)Tn autophosphorylation rate / literature value

● T2● T3● T4● LT3● LT4

● T2● T3● T4● LT3● LT4

Che

B p

hosp

hory

latio

n ra

te

LT2 autophosphorylation rate

Che

Y ph

osph

oryl

atio

n ra

te

LT2 autophosphorylation rate

Conditions for Perfect Adaptation(con’d)

CheB, CheY phosphorylation rate is proportional to autophosphorylation rate:

Diversity of Chemotaxis Systems

Eg., Rhodobacter sphaeroides, Caulobacter crescentus and several rhizobacteria possess multiple CheYs while lacking of CheZ homologue.

In different bacteria, additional protein components as well as multiple copies of certain chemotaxis proteins are present.

Response regulatorCheY1

CheY2 Phosphate “sink”

Requiring:• Faster phosphorylation/autodephosphorylation rates of CheY than CheY1

• Faster phosphorylation rate of CheB

Che

Y1p (

µM)

Time(s)

Exact adaptation in modified chemotaxis network with CheY1, CheY2 and no CheZ:

Near-Perfect Adaptation in Bacterial ChemotaxisYang Yang & Sima Setyeshgar

Department of Physics, Indiana University, Bloomington, Indiana 47405

Run TumbleA network of interacting proteins converts an external stimulus (attractant/repellent) into an internal signal - the phosphorylated form of the Y chemotaxis protein - which in turn interacts with the flagellar motor to bias the cell’s motion between runs and tumbles.The chemotaxis signal transduction network is a well-characterized model system for studying the properties of the two-component superfamily of receptor-regulated phosphorylation pathways in general.

Chemotaxis signal transduction network in E. coliChemotaxis in E. coli - motion toward desirable chemicals and away from harmful ones - is an important behavioral response also shared by many other prokaryotic and eukaryotic cells. It consists of a series of modulated runs and tumbles, leading to a biased random walk in the desired direction.

Fast responseSlow adaptation

[1] V. Sourjik et al., (2002), PNAS, 99, 123 [2] U. Alon et al., (1999), Nature, 397, 168

FRET signal [CheY-P]

CheR fold expression

Adap

tatio

n Pr

ecis

on

Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation)Precision of adaptation insensitive to changes in network parameters (robustness) [2]

It is an important property of the chemotaxis network: rapid response - in the form of change in concentration of intracellular response regulator and corresponding change in running versus tumbling bias - to a step change in external signal, followed by exact adaptation back to the pre-stimulus value.Recent work has highlighted the fact that the underlying design of the chemosensory pathway is such that exact adaptation is "robust" or insensitive to changes in network parameters such as total protein concentrations and reaction rates.

Robust Perfect Adaptation

T 4 a

utop

hosp

hory

latio

n ra

te

(k10

)

T 4 a

utop

hosp

hory

latio

n ra

te

(k10

)

● 3%<<5% ● 1%<<3●0%<<1%

Parameter Surfaces

LT2 methylation rate (k3c)

T4 demethylation rate (k-2)

• Successful implementation of a novel method for elucidating regions in parameter space allowing precise adaptation

• Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of• conditions required for perfect adaptation, consistent with and extending

previous works [5-7]]

• numerical ranges for unknown or partially known kinetic parameters• Extension to modified chemotaxis networks, for example with no CheZ homologue

and multiple CheYs [5] N. Barkai et al., (1997), Nature , 387,855 [6] T. M. Yi et al., (2000), PNAS ,97,4649. [7] B. A. Mello et al., (2003), Biophys. J., 84, 2943

Conclusion

Work in progressExtension to other signaling networks:• vertebrate phototransduction • mammalian circadian clock

allowing determination of• parameter dependences underlying robustness• plausible numerical values for unknown network parameters

• Ligand binding

We begin with a detailed model of the chemotaxis pathway in E. coli, including ligand binding, methylation/demethylation and phosphorylation steps. This model is not assume the two-state active/inactive description of the receptor complex: instead receptor activity is allowed to be graded through the variable autophosphorylation rate of the histidine kinase, CheA.Although capturing the main features of the chemotactic response, this model is "broken" in that the values of reaction rates and protein concentrations are fine-tuned to achieve perfect adaptation of the response.

E.Coli Chemotaxis Signaling Network

• Phosphorylation

• Methylation

• New computational scheme for determining conditions and numerical ranges for parameters allowing robust (near-)perfect adaptation in the E. coli chemotaxis network

• Comparison of results with previous works

• Extension to other modified chemotaxis networks, with additional protein components

• Conclusions and future work

This work: outline

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 There are n system variables, m system parameters and 1 small variable to allow near perfect adaptation, giving a total of (n+m+1)H equations and (n+m+1)H variables.

Discretizing s

into H points

0dskd

Augmented system

||ds

duN

Nu

0);;( skuFdtud

02

|2

|

0);;(

)1(

11

11

skksuu

skuFdtud

sjss

jm

jm

j

jN

jN

jjj

jlowj

Physical Interpretation of Parameter ε Measurement of c = [CheY-P] by flagellar motor constrained by diffusive noise Relative accuracy[3],

Signaling pathway required to adapt “nearly” perfectly, to within this lower bound

[3] H. C. Berg et al., (1977) , Biophys. Journal. 20, 193 .

: diffusion constant (~ 3 µM): linear dimension of motor C-ring (~ 45 nm): CheY-P concentration (at steady state ~ 3 µM): measurement time (run duration ~ 1 second)

%101~

cDac

c

caD

Use Newton-Raphson (root finding algorithm with back-tracking), to solve for the steady state of augmented system,

Use Dsode (stiff ODE solver), to verify time- dependent behavior for different ranges of external stimulus by solving:

Implementation

},,{

0

||

0);(

kuy

dskddsdy

syFN

);;( skuFdtud

T 3 demethylation

rate (k -

1)

T3 autophosphorylation rate (k9)

T4 autophosphorylation rate (k10)

T 4 demethylation

rate (k -

2)

LT3 autophosphorylation rate (k12)

LT3 demethylation rate

(k-3)

LT4 autophosphorylation rate (k13)

LT4 demethylation rate

(k-4)

Demethylation rate is proportional to autophosphorylation rate2

:

Conditions for Perfect Adaptation

T2 autophosphorylation rate (k8)

T 2 M

ethy

latio

n ra

te

(k1c

)

T3 autophosphorylation rate (k9)

T 3 M

ethy

latio

n ra

te (k

2c)Methylation rate is proportional to autophosphorylation rate:

LT2 autophosphorylation rate (k12)

LT2 M

ethy

latio

n ra

te

(k3c

)

LT3 autophosphorylation rate (k13)

LT3 M

ethy

latio

n ra

te

(k4c

)

)(|)()(|

beforeYbeforeYafterY

p

pp

[1]

[1]

skFu

skuFdtud

0);;(

Literature value[3]

k 10 (

s-1)

k-2 (M-1s-1)k3c (s-1)

k-2 (M-1s-1)

k 10 (

s-1)

k3c (s-1)k-2 (M-1s-1)

Literature value[3]

[3] P. A. Spiro et al., (1997), Proc. Natl. Acad. Sci. USA , 94, 7263

[3] P. A. Spiro et al., (1997), Proc. Natl. Acad. Sci. USA , 94, 7263

[3] P. A. Spiro et al., (1997), Proc. Natl. Acad. Sci. USA , 94, 7263