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:

Robust Perfect Adaptation in Bacterial Chemotaxis Yang Yang & Sima Setayeshgar Department of Physics, Indiana University, Bloomington

MotivationThe biochemical basis of robustness of perfect adaptation is not as yet fully understood. In this work, we develop a novel method for elucidating regions in parameter space of which the E. coli chemotaxis network adapts perfectly:

Broader impactThis method should have applicability to other cellular signal transduction networks and engineered systems that exhibit robust homeostasis.

The shapes of resulting manifolds determine relationships between reaction parameters (for example, methylation and phosphorylation rates) that must be satisfied in order for the network to exhibit perfect adaptation, thereby shedding light on biochemical steps and feedback mechanisms underlying robustness.

Given lack of complete data on values of in vivo reaction rates, the numerical ranges of the resulting manifolds will shed light on values of unknown or partially known parameters.

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

Work in progress Extension to other signaling networks:

• vertebrate phototransduction • mammalian circadian clock

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

0|2

|

|2

|

0);;(

)1(

11

11

s

kk

s

uu

skuFdt

ud

siss

iN

iN

i

iN

iN

iii

ilowi

AlgorithmSTART 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.

s

k

F

u

skuFdt

ud

0);;(

: state variables

: reaction kinetics

: reaction constants

: external stimulus

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: 0

ds

kd

0);;( skuFdt

ud

N

N

u

ds

du

||

: allows for near-perfect adaptation

= CheYp

E. coli exhibits an important behavioral response known as chemotaxis - motion toward desirable chemicals (usually nutrients) and away from harmful ones - which is also shared by various other prokaryotic and eukaryotic cells. The cell’s motion consists of series of “runs” punctuated by “ tumbles”.

E .coli

It is considered to be an ideal model organism for understanding the behavior of cells at the molecular level from the perspectives of several scientific disciplines-anatomy, genetics, chemistry and physics since :• Ease of experimentation, through microscopy and genetic analysis• Small genome (4288 genes), most of which encode proteins

2

Importantance of perfect adaptation

Chemotaxis signal transduction network in E. coli

Importance of robustnessThe E. coli chemotaxis signal transduction network exhibits robust perfect adaptation, where the concentration of CheYp returns to its prestimulus value despite large changes in the values of many of the biochemical reaction rate constants. These rate constants depend on concentrations of enzymes, which are often present in small copy numbers, making fluctuations in their numbers significant.

Chemotaxis in E. coli involves temporal measurement of the change in concentration of an external stimulus. This is achieved through the existence of fast and slow reaction time scales, in the chemotaxis signal transduction network: fast measurement of the current external concentration is compared with the cell’s “memory” of the concentration some time ago to determine whether to extend a run in a given direction or to tumble, thereby randomly selecting a new direction.

E. coli is a single-celled organism that lives primary in our intestines. It is approximately 1-2 microns long and 1 micron in diameter, and weights 1 picogram. Each cell has 4-6 flagella, approximately 10-20 microns long, driven by an intracellular rotary motor operated by the protonmotive force.

The chemotaxis signal transduction pathway in E. coli – a network of ~50 interacting proteins – converts an external stimulus (change in concentration of chemoattractant / repellent) into an internal stimulus (change in concentration of intracellular response regulator, CheYp) which in turn interacts with the flagella motor to bias the cell’s motion.

It is used as a well-characterized model system for the study of properties of (two-component) cellular signaling networks in general.

Perfect adaptation is an important and generic property of signaling systems, where the response (e.g. running bias in chemotaxis) returns precisely to the pre-stimulus level while the stimulus persists. This property allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli.Thus, E. coli is able to respond to changes in chemoattractant concentrations spanning 5 orders of magnitude! Similarly, the vertebrate visual system responds to changes in light intensity spanning 10 orders of magnitude during the night-day cycle.

Ref: H. C. Berg, “Motile behavior of bacterial”, Physics Today, January 2000

Ref: P. A. Spiro, J. S. Parkinson, and H. G. Othmer, “A model of excitation and adaptation in bacterial chemotaxis”, Proc. Natl. Acad. Sci. USA 94, 7263(1997)

Ref: N.Barkai & S. Leibler, “Robustness in simple biochemical network”, Nature 387, 913(1997)

2

Chemical reactions: Ligand binding

Methylation

Phosphorylation

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

)( CheRLTCheRTL pnkmkkmk

pn

ppnmkmk

ppn

pnckck

pn

CheBTLCheBTL

CheRTLCheRTL

)(14~1

)(

)(14~1

)(

)()(

)()(

PCheBCheB

PCheYCheZCheZCheY

BCheYCheRTBCheYCheRTL

ADPCheRTLATPCheRTL

kmbp

kmyp

pnbky

np

npkk

n

)()()()()(

)()()()()(

9~7

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

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

0

||

0);(

ds

kdds

dysyF

N

},,{ kuy

);;( skuFdt

ud

T4

au

top

hosp

hory

lati

on

rate

(k

10)

LT4 a

uto

ph

osp

hory

lati

on

rate

(k

10)

● 3%<<5% ● 1%<<3% ● 0%<<1%

Parameter Surfaces

Time (s)

Con

cen

trati

on

(µ

M)

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

•{k3c= 5 s-1, k10 = 36 s-1, km2 = 3e+4 M-1s-1}

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

Validation

(1,12.7)

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)

T2 a

uto

ph

osp

hory

lati

on

rate

(k

8)

Conditions for Perfect Adaptation

T3 autophosphorylation rate

T3 d

em

eth

yla

tion r

ate

/ T2

meth

yla

tion

rate

T4 autophosphorylation rate

T4 d

em

eth

yla

tion r

ate

/ T3

meth

yla

tion

rate

LT3 autophosphorylation rate

T3 d

em

eth

yla

tion r

ate

/ T2

meth

yla

tion

rate

LT4 autophosphorylation rate

LT4 d

em

eth

yla

tion r

ate

/ LT

3

meth

yla

tion

rate

CheB

phosp

hory

lati

on

rate

(k

b)

/ lit

era

ture

valu

e

CheY p

hosp

hory

lati

on

rate

(k

y)

/ lit

era

ture

valu

e

(L)Tn autophosphorylation rate / literature value

(L)Tn autophosphorylation rate / literature value

● T2● T3● T4● LT3● LT4

● T2● T3● T4● LT3● LT4

Ch

eB

ph

osp

hory

lati

on

rate

LT2 autophosphorylation rate

Ch

eY p

hosp

hory

lati

on

ra

te

LT2 autophosphorylation rate

T3

dem

eth

yla

tion r

ate

(k

m1)

T3 autophosphorylation rate (k9)

T4 autophosphorylation rate (k10)

T4

dem

eth

yla

tion r

ate

(k

m2)

LT3 autophosphorylation rate (k12)

LT3

dem

eth

yla

tion r

ate

(k

m3)

LT4 autophosphorylation rate (k13)

LT4

dem

eth

yla

tion r

ate

(k

m4)

Demethylation Rate is proportional to Autophosphorylation Rate2

T2 autophosphorylation rate (k8)

T2

Meth

yla

tion r

ate

(k

1c)

T3 autophosphorylation rate (k9)

T3

Meth

yla

tion r

ate

(k 2

c)Methylation Rate is proportional to Autophosphorylation Rate

LT2 autophosphorylation rate (k12)

LT2

Meth

yla

tion r

ate

(k

3c)

LT3 autophosphorylation rate (k13)

LT3

Meth

yla

tion r

ate

(k

4c)

Demethylation Rate/Methylation Rate is proportional to Autophosphorylation Rate

CheB, CheY Phosphorylation Rate is proportional to Autophosphorylation Rate

Condition for Robust Perfect Adaptation

By varying 3 parameters(Ttot, k11, k3c) in the code to find a region where Ttot can vary a lot while the

others remain constant.

L=0(solid)L=1µM(dashed)L=1mM(dashed dot)

Spiro model.

Barkai–Leibler model

L=0(solid)L=1µM(dashed)L=1mM(dashed dot)

Ref: P. A. Spiro, J. S. Parkinson, and H. G. Othmer, “A model of excitation and adaptation in bacterial chemotaxis”, Proc. Natl. Acad. Sci. USA 94, 7263(1997)T.M.Yi, Y. Huang, M. I. Simon, and J. Doyle, “Robust perfect adaptation in bacterial chemotaxisthrough integral feedback control”,PNAS, 97, 4649 (2000)

Verifying the conditions for perfect adaptation of two-state model

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

Ch

eY

1p

(µM

)

Time(s)

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

Inactive active

1. Fast ligand (un-)binding reaction

2. Only acitive receptor can bind to CheB

3. Only inactive receptor can bind to CheR

4. Autophosphorylation rates of receptors

are proportional to the activity

Slope=0.18

Slope=0.15

T2

dem

eth

yla

tion c

ata

lyti

c ra

te

T1 methylation catalytic rate

T1

dem

eth

yla

tion c

ata

lyti

c ra

te

T0 methylation catalytic rate

T3

dem

eth

yla

tion c

ata

lyti

c ra

te

T2 methylation catalytic rate

T4

dem

eth

yla

tion c

ata

lyti

c ra

te

T3 methylation catalytic rate

5. Phosphorylation transferrate form CheA to CheY and CheB are porportional to

the activity

6. The ratios between the CheR catalytic rate and CheB-p catalytic rate of the next

methylation level are the same for all methylation states.Verifying condition 6: reference value kb/kr=0.155/0.819=0.19

Ref: B. Mello,  Y. Tu,”Perfect and near-perfect adaptation in a model of bacterial chemotaxis”, Biophysical Journal, 2003

Slope=0.15 Slope=0.18 Slope=0.15 Slope=0.19