March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 1

Near-Perfect Adaptation in Bacterial Chemotaxis

Yang Yang and Sima Setayeshgar

Department of Physics

Indiana University, Bloomington, IN

Chemotaxis Signal Transduction Network in E. coli

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO

2

Histidine kinase Methylesterase

Couples CheA to MCPs Response regulator

Methyltransferase Dephosphorylates CheY-P

CheB

CheW

CheZ

CheR

CheY

Signal Transduction

Pathway

Motor Response

[CheY-P]

Stimulus

Flagellar Bundling

Motion

Run Tumble

Robust Perfect Adaptation

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO

Fast response Slow adaptation

From Sourjik et al., PNAS (2002).

FRET signal [CheY-P]

From Alon et al., Nature (1999).

CheR fold expressionAd

apta

tio

n

Pre

ciso

n

Steady state [CheY-P] / running bias independent of value constant external stimulus (adaptation)

Precision of adaptation insensitive to changes in network parameters (robustness)

This Work: Outline

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 4

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

E. coli Chemotaxis Signaling Network

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 5

Ligand binding

Methylation

Phosphorylation

)()( )(7~5

7~5)( CheRLTCheRTL pn

kmkm

kkpn

ppnmkmk

ppn

pnckck

pn

CheBTLCheBTL

CheRTLCheRTL

)(14~1

)(

)(14~1

)(

)()(

)()(

PCheBCheB

PCheYCheZCheZCheY

CheBCheRTCheBCheRTL

CheYCheRTCheYCheRTL

ADPCheRTLATPCheRTL

kmbp

kmyp

pnkb

np

nky

np

npkk

n

)()()(

)()()(

)()()()( 9~7T3 T4T2

T2p T4pT3p

LT3 LT4

LT4p

LT2

LT3pLT2p

phosphorylation

methylation

Lig

an

d b

ind

ing

Approach …

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 6

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 CheY-P

values of reaction rate constants,

are independent of the external stimulus, s, thereby explicitly incorporating perfect adaptation.

s

k

F

u

skuFdt

ud

0);;(

: state variables

: reaction kinetics

: reaction rates

: external stimulus

The steady state concentration of proteins in the network satisfy:

The steady state concentration of = [CheY-P] must be independent of stimulus, s:

where parameter allows for “near-perfect” adaptation.

Reaction rates are constant and must also be independent of stimulus, s:

Augmented System

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 7

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

0ds

kd

0);;( skuFdt

ud

||ds

duN

Nu

Discretize s in

range {slow, shigh}

Physical Interpretation of Parameter, : Near-Perfect Adaptation

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 8

Measurement of c = [CheY-P] by flagellar motor constrained by diffusive noise Relative accuracy*,

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

(*) Berg & Purcell, Biophys. J. (1977).

%101

~

cDac

c

: 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)c

a

D

},,{ kuy

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);(

ds

kdds

dysyF

N

);;( skuFdt

ud

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 9

T4 demethylation rate (km2)

T4

aut

oph

osp

hory

latio

n ra

te

(k10

)

LT2 methylation rate (k3c)

LT

4 a

uto

ph

osp

ho

ryla

tion

ra

te

(k10

)

Parameter Surfaces

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 10

● 3%<<5% ● 1%<<3% ● 0%<<1%

Surface: 2D projections:

)(

|)()(|

beforeY

beforeYafterY

p

pp

Validation

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 11

Time (s)

Co

nce

ntr

atio

n (

µM

)Verify steady state NR solutions dynamically using DSODE for different stimulus ramps:

{k3c= 5 s-1, k10 = 101 s-1, km2 = 6.3e+4 M-1s-1}

1%

k1c : 0.17 s-1 1 s-1

k8 : 15 s-1 12.7 s-1

Violating and Restoring Perfect Adaptation

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 12

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

(1,15)

(1,12.7)

T2 Methylation rate (k1c)

T2

auto

phos

phor

ylat

ion

rate

(k

8)

Conditions for Perfect Adaptation

Methylation Rate Autophosphorylation Rate

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 14

T2 autophosphorylation rate (k8)

T2 M

eth

yla

tion

ra

te (

k 1c)

T3 autophosphorylation rate (k9)

T3 M

eth

yla

tion

ra

te (

k 2c)

LT2 autophosphorylation rate (k12)

LT

2 M

eth

yla

tion

ra

te (

k 3c)

LT3 autophosphorylation rate (k13)

LT

3 M

eth

yla

tion

ra

te (

k 4c)

Demethylation Rate Autophosphorylation Rate2

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 15

T3 autophosphorylation rate (k9)

T3 d

em

eth

yla

tion

ra

te (

k m1)

T4 autophosphorylation rate (k10)

T4 d

em

eth

yla

tion

ra

te (

k m2)

LT3 autophosphorylation rate (k12)

LT

3 d

em

eth

yla

tion

ra

te (

k m3)

LT4 autophosphorylation rate (k13)

LT

4 d

em

eth

yla

tion

ra

te (

k m4)

Demethylation Rate/Methylation Rate Autophosphorylation Rate

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 16

T3 autophosphorylation rate

T3

dem

ethy

latio

n ra

te/ T

2 m

ethy

latio

n ra

te

T4 autophosphorylation rate

T4

dem

ethy

latio

n ra

te/ T

3 m

ethy

latio

n ra

te

LT3 autophosphorylation rate

T3

dem

ethy

latio

n ra

te/ T

2 m

ethy

latio

n ra

te

LT4 autophosphorylation rate

LT4

dem

ethy

latio

n ra

te/ L

T3

met

hyla

tion

rate

CheB, CheY Phosphorylation Rate Autophosphorylation Rate

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 17

Che

B p

hosp

hory

latio

n ra

te (

k b)

/ lit

erat

ure

val

ue

Che

Y p

hosp

hory

latio

n ra

te (

k y)

/ lit

erat

ure

val

ue

(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 p

hosp

hory

latio

n ra

te

LT2 autophosphorylation rate

Diversity of Chemotaxis Systems

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 18

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 regulator

Phosphate “sink”

CheY1CheY2

Two CheY System

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 19

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

Ch

eY1

p (µ

M)

Ch

eY1

p (µ

M)

Time(s) Time(s)

Requiring: Faster phosphorylation/autodephosphorylation rates of CheY2 than CheY1

Faster phosphorylation rate of CheB

Conclusions

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 20

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

II. Numerical results for (near-) perfect adaptation manifolds in parameter space for the E. coli chemotaxis network, allowing determination of

i. conditions required for perfect adaptation, consistent with and extending previous works [1-3]

ii. numerical ranges for unknown or partially known kinetic parameters

I. Extension to modified chemotaxis networks, for example with no CheZ homologue and multiple CheYs

[1] Barkai & Leibler, Nature (1997). [2] Yi et al., PNAS (2000). [3] Tu & Mello, Biophys. J. (2003).

Future Work

March 8, 2007 Yang Yang, March APS Meeting, Denver, CO 21

Extension to other signaling networks

vertebrate phototransduction mammalian circadian clock

allowing determination of

a) parameter dependences underlying robustness

b) plausible numerical values for unknown network parameters