Page 1

Chemotaxis in Escherichia coli: A Molecular

Model for Robust Precise Adaptation

Clinton H. Hansen1, Robert G. Endres2¤, Ned S. Wingreen2*

1 Department of Physics, Princeton University, Princeton, New Jersey, United States of America, 2 Department of Molecular Biology, Princeton University, Princeton, New

Jersey, United States of America

The chemotaxis system in the bacterium Escherichia coli is remarkably sensitive to small relative changes in the

concentrations of multiple chemical signals over a broad range of ambient concentrations. Interactions among

receptors are crucial to this sensitivity as is precise adaptation, the return of chemoreceptor activity to prestimulus

levels in a constant chemoeffector environment. Precise adaptation relies on methylation and demethylation of

chemoreceptors by the enzymes CheR and CheB, respectively. Experiments indicate that when transiently bound to

one receptor, these enzymes act on small assistance neighborhoods (AN) of five to seven receptor homodimers. In this

paper, we model a strongly coupled complex of receptors including dynamic CheR and CheB acting on ANs. The model

yields sensitive response and precise adaptation over several orders of magnitude of attractant concentrations and

accounts for different responses to aspartate and serine. Within the model, we explore how the precision of adaptation

is limited by small AN size as well as by CheR and CheB kinetics (including dwell times, saturation, and kinetic

differences among modification sites) and how these kinetics contribute to noise in complex activity. The robustness of

our dynamic model for precise adaptation is demonstrated by randomly varying biochemical parameters.

Citation: Hansen CH, Endres RG, Wingreen NS (2008) Chemotaxis in Escherichia coli: A molecular model for robust precise adaptation. PLoS Comput Biol 4(1): e1. doi:10.1371/

journal.pcbi.0040001

Introduction

Through the process of chemotaxis, the bacterium Escher-

ichia coli swims up the concentration gradients of attractants

(nutrients) and down the concentration gradients of repel-

lents. E. coli moves via the rotation of multiple flagella. When

the flagella rotate counterclockwise, they bundle and propel

the bacterium forward. Rotation in a clockwise direction

causes the flagella to fly apart, and the organism tumbles to

change direction. Swimming up a gradient of attractant

causes a decrease in the probability of tumbling, whereas

swimming up a gradient of chemorepellent causes an increase

in the probability of tumbling. The result is that E. coli

performs a biased random walk toward chemoattractants and

away from chemorepellents [1].

The signaling pathway that governs E. coli chemotaxis is

well characterized [2–4]. Out of five different membrane-

bound chemotaxis receptors, Tar and Tsr are expressed at

high levels, whereas Tap, Trg, and Aer are expressed at lower

levels. The receptors form homodimers that can each bind

one molecule of ligand [5]. The homodimers in turn form

trimers of dimers, and associate with CheW and CheA. CheW

is a linker protein, and CheA is a histidine kinase [6,7].

Receptor signaling activates CheA autophosphorylation, and

the phosphoryl group is transferred to the response

regulator, CheY. Phosphorylated CheY diffuses and binds to

the flagellar motors, favoring clockwise rotation and tum-

bling. CheY is dephosphorylated by the phosphatase CheZ.

E. coli are able to react to small relative changes in

concentration over a range of several orders of magnitude.

In experiments done by Mao et al. [8], bacteria responded to

changes in concentration from 10 mM to as low as 3.2 nM of

the attractant aspartate. Two properties of the network that

underlie the broad range of responsiveness are interactions

among receptors and precise adaptation [9]. In vivo fluo-

rescence resonance energy transfer (FRET) measurements

[10,11] suggest that signaling is mediated by strongly coupled

complexes of 10–20 receptor homodimers that are all active

or inactive together. FRET also reveals that levels of

phosphorylated CheY adapt precisely following a transient

response to steps of chemoeffector concentration. Precise

adaptation occurs though the methylation by CheR and

demethylation by CheB of eight sites on each homodimer

receptor [12,13]. Methylation at each site increases the

activity level of receptors, whereas demethylation decreases

activity. Each Tar or Tsr receptor has a tether at its C

terminus, with a pentapeptide site that can bind one CheR or

CheB [14]. Experiments indicate that when transiently bound

to one receptor, each CheR or CheB can act on five to seven

adjacent receptor homodimers, defining an ‘‘assistance

neighborhood’’ (AN) [15].

The dynamics of receptor modification in complexes is not

well understood. A two-state single-receptor model was

proposed by Barkai and Leibler in which the modification

Editor: Philip E. Bourne, University of California San Diego, United States of

America

Received June 4, 2007; Accepted November 19, 2007; Published January 4, 2008

A previous version of this article appeared as an Early Online Release on November

20, 2007 (doi:10.1371/journal.pcbi.0040001.eor).

Copyright: ? 2008 Hansen et al. This is an open-access article distributed under

the terms of the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original author

and source are credited.

Abbreviations: AN, assistance neighborhood; BL, Barkai–Leibler; FRET, fluores-

cence resonance energy transfer; MeAsp, alpha-methyl aspartate; MWC, Monod–

Wyman–Changeux

* To whom correspondence should be addressed. E-mail: wingreen@princeton.edu

¤ Current address: Division of Molecular Biosciences and CISBIC, Imperial College,

London, United Kingdom

PLoS Computational Biology | www.ploscompbiol.org January 2008 | Volume 4 | Issue 1 | e10014

Page 2

activities of CheR and CheB depend only on receptor activity,

not ligand concentration or methylation level [16]. This

simple model naturally leads to precise adaptation, but does

not include interactions among receptors. More recent

approaches have incorporated receptor interactions using a

Monod–Wyman–Changeux (MWC) model [17] in which a

complex of receptors is either active (on) or inactive (off) as a

whole [11,18–20]. The free-energy difference between the on

and off states of a complex dictates the probability of its

being in the active state. To preserve precise adaptation

within the MWC model, the Barka–Leibler (BL) model was

extended [18,19] to include the action of CheR/CheB on ANs

of receptors. Static nonoverlapping ANs of size 6 were

utilized. Here, we build on this earlier model by incorporat-

ing the binding and unbinding of CheR and CheB, creating

dynamic ANs. This extension allows us to consider limits to

precise adaptation from small AN size as well as from CheR

and CheB kinetics, including dwell times, saturation, and

kinetic differences among modification sites.

Models for the E. coli chemotaxis network are complex and

depend on numerous parameters, bringing into question how

well the essential property of precise adaptation is preserved

when parameters are altered. Barkai and Leibler showed that

their simple two-state single-receptor model of the chemo-

taxis network was robust to parameter variation [16]. We find

a similar robustness of our dynamic AN model.

Model

Complex Activity

In this paper, we explore a MWC model of mixed and

strongly coupled Tar and Tsr chemoreceptor homodimers.

Within the MWC model, a complex of receptors is either on

or off as a whole (Figure 1). The average complex activity is

the probability that the complex is in the on state and is

determined by the free-energy difference between the on and

off states [19]. Here, the MWC model is used to calculate the

thermal equilibrium complex activity from the instantaneous

attractant concentration and receptor methylation state.

We assume that each receptor homodimer is a two-state

system, being either on or off. Each receptor homodimer can

bind a ligand molecule in either state, albeit with different

affinities. Therefore, the four possible configurations for each

homodimer and their free energies are (1) on with no ligand

bound, Eon

(3) off with no ligand bound, Eoff

bound, Eoff

binding constants in the on and off states for a specific type

of receptor r, and m is the methylation level (m¼0,...,8). Based

on experimental data, these binding constants are assumed to

be independent of ligand concentration or methylation level

[21–23]. For the two on states, the sum of the equilibrium

Boltzmann factors is

?

rðmÞ, (2) on with ligand bound, Eon

rðmÞ? logð½L?=Kon

and Koff

r

rÞ,

rðmÞ, and (4) off with ligand

rÞ. Here Kon

rðmÞ? logð½L?=Koff

r

are the

exp½? Eon

rðmÞ?þexp ? Eon

rðmÞþlog½L?

Kon

r

?

¼ exp ? Eon

rðmÞþlog 1þ½L?

Kon

r

????

;

therefore, the combined free energy of the two on states is

fon

Kon

r

energy of the two off states is foff

All energy units are expressed in units of the thermal energy,

kBT. Since binding of attractant favors the off state, Kon.

Koff. The opposite applies to repellents: Kon, Koff.

Adaptation to attractant occurs through methylation,

which favors the on state. Therefore, the offset energy

erðmÞ¼ Eon

difference between the on and off states of a single receptor is

rðmÞ¼ Eon

rðmÞ? log 1 þ

½L?

??

. Similarly, the combined free

rðmÞ¼ Eoff

rðmÞ? log 1 þ

½L?

Koff

r

??

.

rðmÞ? Eoff

rðmÞdecreases as m increases. The free-energy

frðmÞ¼ fon

rðmÞ? foff

rðmÞ¼ erðmÞþ log

1 þ

1 þ

½L?

Koff

r

½L?

Kon

r

0

@

1

A:

ð1Þ

The free-energy difference F of a complex of receptors is

the summation of the individual fr(m)of all of the receptors in

the complex,

X

The average activity A of the complex of receptors is its

probability of being in the on state and is given according to

equilibrium statistical mechanics by

F ¼

n

i¼1

friðmiÞ:

ð2Þ

A ¼

1

1 þ eF:

ð3Þ

Receptor Modification by CheR and CheB

Within our model, receptors dynamically bind and unbind

the adaptation enzymes CheR and CheB (Figure 1A). We

assume that a receptor-bound CheR only methylates recep-

tors when the complex is off, whereas a receptor-bound CheB

only demethylates receptors when the complex is on (Figure

1B). For precise adaptation to occur, the rates of methylation

and demethylation by CheR and CheB must depend only on

the activity of the complex A. We assumed each bound CheR

adds methyl groups at a rate kR(1?A), and each bound CheB

protein removes methyl groups at a rate kBA. In most

simulations, we assumed saturated kinetics for CheR and

CheB. Specifically, we assumed each receptor-bound CheR or

CheB acts on available sites in the AN with equal probability,

independent of the number of available modification sites.

However, we also explored the effects of nonsaturated

kinetics by introducing a factor N/(N þ Msat) into the

methylation/demethylation rates, where N is the total number

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e10015

Author Summary

Bacteria swim in relatively straight lines and change directions

through tumbling. In the process of chemotaxis, a network of

receptors and other proteins controls the tumbling frequency to

direct an otherwise random walk toward nutrients and away from

repellents. Receptor clustering and adaptation to persistent stimuli

through covalent modification allow chemotaxis to be sensitive over

a large range of ambient concentrations. The individual components

of the chemotaxis network are well characterized, and signaling

measurements by fluorescence microscopy quantify the network’s

response, making the system well suited for modeling and analysis.

In this paper, we expand upon a previous model based on

experiments indicating that the covalent modifications required

for adaptation occur through the action of enzymes on groups of

neighboring receptors, referred to as assistance neighborhoods.

Simulations show that our proposed molecular model of a strongly

coupled complex of receptors produces accurate responses to

different stimuli and is robust to parameter variation. Within this

model, the correct adaptation response is limited by small

assistance-neighborhood size as well as enzyme kinetics. We also

explore how these kinetics contribute to noise in the chemotactic

response.

Model for Robust Precise Adaptation

Page 3

of available sites for methylation/demethylation and Msatis a

constant. Smaller values of Msatmean more nearly saturated

kinetics, with full saturation corresponding to Msat¼0. For all

of our simulations, the methylation/demethylation rate is zero

if there are no available modification sites (N ¼ 0).

Unlike the previous AN model [18,19], we include

dynamical CheR/CheB binding and unbinding to receptors.

Free receptors bind CheR/CheB molecules at a rate cbind

receptor-bound CheR/CheB molecules unbind at a rate

cunbind

R=B

. In addition, each receptor can bind at most one

CheR or one CheB. At steady state, this gives for the average

proportion of receptors bound by CheR (with a similar

expression for CheB):

cbind

R

1 þ cbind

The CheR/CheB binding rates cbind

linearly with the concentration of free CheR/CheB, whereas

the CheR/CheB unbinding rates cunbind

independent of concentration. As the concentration of CheR,

and therefore cbind

R

, rises, the proportion of CheR-bound

receptors hCheRi also rises. Since each receptor can bind at

most one CheR or one CheB, the proportion of CheR-bound

receptors hCheRi decreases with an increase in the CheB

binding rate cbind

B

. The CheR/CheB unbinding rates also

define an average dwell time each CheR or CheB is bound to a

receptor to be 1=cunbind

R=B

.

A fixed hexagonal arrangement of 19-receptor homo-

dimers (Figure 1A) was used for every simulation. For

simplicity, from this point, we refer to each homodimer as

a receptor. The ANs consist of a receptor and its nearest

neighbors. This creates 19 possible ANs, each centered on one

receptor: seven ANs of size 7, six ANs of size 5, and six ANs of

size 4. The average AN size is 5.4 receptors. Six receptors were

Tar, and 13 receptors were Tsr, consistent with the wild-type

ratio [24]. We also explored the effect of AN size through the

use of size-one AN complexes and half AN complexes. For

R=B, and

hCheRi ¼

=cunbind

R

þ cbind

R=Bare assumed to increase

R

=cunbind

R

B

=cunbind

B

:

ð4Þ

R=B

are assumed to be

size-one AN clusters, each CheR or CheB can only modify the

bound receptor. For half AN complexes, we randomly chose

half of each receptor’s nearest neighbors to be in the AN and

used the same configuration for all simulations. For the six

receptors with three nearest neighbors, three have half ANs

including two adjacent neighbors, and the other three have

half ANs including only one adjacent neighbor.

Mean Field Theory

By considering the average net methylation rate of a

complex, we derived a simple mean field theory for complex

activity. The average methylation rate of a complex is the rate

of methylation by a single CheR times the number of bound

CheR with available methylation sites. Similarly, the average

demethylation rate of the complex is the rate of demethyla-

tion by a single CheB times the number of bound CheB with

available demethylation sites. Therefore, the average net

methylation rate for a single receptor is

dhmi

dt

¼ kRð1 ? AÞhCheRið1 ? Pmax

ANÞ ? kBAhCheBið1 ? P0

ANÞ;

ð5Þ

where Pmax

methylated and fully demethylated ANs, respectively. The

factors 1 ? Pmax

cannot methylate an already fully methylated AN, and CheB

cannot demethylate an already fully demethylated AN. The

condition that the average net methylation rate is zero

(dhmi=dt ¼ 0) determines the average steady-state activity,

A ¼ 1 þkBhCheBið1 ? P0

kRhCheRið1 ? Pmax

As long as Pmax

precisely to

?

AN

and P0

ANare the average proportions of fully

ANand 1 ? P0

ANaccount for the fact that CheR

ANÞ

ANÞ

???1

:

ð6Þ

AN¼ P0

AN¼ 0, the activity will always adapt

A?¼ 1 þkBhCheBi

kRhCheRi

??1

;

ð7Þ

Figure 1. Two-State Receptor Complex and Precise-Adaptation Model

(A) Top view of the hexagonal arrangement of the 19-receptor homodimer used in our simulations. Each receptor can bind to either CheR or CheB. Each

bound CheR or CheB can then act on an ‘‘assistance neighborhood’’ of adjacent receptors (dashed line) through methylation (CheR) or demethylation

(CheB).

(B) Side view of receptor complex in the on state (top) and off state (bottom) with, respectively, active and inactive receptor-bound kinases CheA. Active

CheA kinases autophosphorylate to CheA-P. In our adaptation model, CheB only demethylates receptors when the complex is in the on state (top), and

CheR only methylates receptors when the complex is in the off state (bottom).

doi:10.1371/journal.pcbi.0040001.g001

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e1 0016

Model for Robust Precise Adaptation

Page 4

which is independent of ligand concentration or methylation

level, since hCheRi and hCheBi depend only on the concen-

trations and binding rates of CheR and CheB.

Fluctuations in methylation and demethylation will lead to

finite values of Pmax

the probability that a neighborhood will become fully

methylated or fully demethylated by chance will be very

small (as long as the average receptor methylation level is not

close to hmi ¼ 8 or hmi ¼ 0), so Pmax

approximation. Therefore, we expect the activity in large-AN

models to adapt to A*over a broad range of ligand

concentrations.

As the average receptor-methylation level reaches hmi ¼ 8,

the methylation level cannot increase to compensate for the

increased free-energy difference due to attractant binding.

Therefore, activity drops to zero beyond the limiting ligand

concentration at which receptors become fully methylated.

At full methylation of the complex,

Fm¼8¼ nsfsð8Þþ nafað8Þ

0

ANand P0

AN. However, for large enough ANs,

AN¼ P0

AN¼ 0 will be a good

¼ ns esð8Þþ log

1 þ

1 þ

½L?

Koff

s

½L?

Kon

s

@

1

A

2

4

3

5þ na esð8Þþ log

1 þ

1 þ

½L?

Koff

a

½L?

Kon

a

0

@

1

A

ð8Þ

2

4

3

5;

where nsis the number of Tsr receptors in the complex and na

is the number of Tar receptors. Therefore, failure of precise

adaptation begins near the ligand concentration [L] for which

Fm¼8¼ F*, i.e., the value of F at precise adaptation. Further

increase in attractant concentration causes a rapid decay in

activity:

2

A ¼

1 þ exp½nsesð8Þþ naeað8Þ?

1 þ

1 þ½L?

½L?

Koff

s

Kon

s

0

@

1

A

ns

1 þ

1 þ½L?

½L?

Koff

a

Kon

a

0

@

1

A

na

4

3

5

?1

:

Analytical Expression for Noise

Here, we derive an analytical expression for the fluctua-

tions of complex activity due to discrete methylation and

demethylation events by receptor-bound CheR and CheB.

First, we calculate the variance of the total complex

methylation level within the Langevin approximation [25].

If the free-energy difference between the active and inactive

state of a receptor depends linearly on the methylation level

m, then for a single receptor

frðmÞ¼ e0? mde þ log

1 þ

1 þ

½L?

Koff

r

½L?

Kon

r

0

@

1

A;

ð9Þ

and for a complex of nsTsr receptors and naTar receptors

(with total methylation level M),

0

F ¼ ðnsþ naÞe0? Mde þ nslog

1 þ

1 þ

½L?

Koff

s

½L?

Kon

s

@

1

Aþ nalog

1 þ

1 þ

½L?

Koff

a

½L?

Kon

a

0

@

1

A:

ð10Þ

Since CheR/CheB methylation/demethylation rates depend

on complex activity, fluctuations in the free-energy differ-

ence F translate into fluctuations in complex activity A.

Linearization of Equation 5 with Pmax

d_ M,

AN¼ P0

AN¼ 0 yields for

d_ M ¼@_ M

@AdA þ noise ¼ ?@A

@MðNRkRþ NBkBÞdM

þ

X

NR

i¼1

gRðiÞþ

X

NB

i¼1

gBðiÞ;

ð11Þ

where NR/NBare the number of bound CheR/CheB enzymes,

gR(i)/B(i)are independent Langevin noise terms for each bound

CheR/CheB, and

@F

transform and integration of the power spectrum, we obtain

(with h~ g2

h~ g2

hdM2i ¼

@A

@M¼@A

@F

@M¼ Að1 ? AÞde. After a Fourier

Ri ¼ single CheR methylation rate ¼ kRð1 ? hAiÞ and

Bi ¼ single CheB demethylation rate¼kBhAi)

NRh~ g2

Ri þ NBh~ g2

Bi

2dehAið1 ? hAiÞ½NRkRþ NBkB?¼1

The variance in methylation level depends only (inversely)

on de, the step in free-energy difference per added methyl

group. The variance in activity is therefore

?

de:

ð12Þ

hdA2i ¼

@A

@M

?2

hdM2i ¼ de½hAið1 ? hAiÞ?2:

ð13Þ

Results

Figure 2 shows simulated response curves of complexes of

19 chemoreceptor dimers to step increases in concentration

of alpha-methyl aspartate (MeAsp), an attractant. The results

shown include the dynamics of CheR and CheB (see Model)

and are similar to those obtained with static ANs [18]. Precise

adaptation occurs over four orders of magnitude of MeAsp

concentration, with methylation levels increasing to com-

pensate for drops in activity due to increases in attractant

concentration.

In Figure 2, the Tar-only complexes exemplify two differ-

ent limits of precise adaptation at high attractant concen-

trations, as in the static AN model [18]. For the Tar-only

complex with higher Koff

a

¼ 0:06 mM (dot-dashed curve), the

activity continues to adapt precisely, but the activity stops

responding to increases of MeAsp. In this case, the receptors

become saturated, and further increases in MeAsp do not

produce changes in the free-energy difference between the

on and off states of the complex. The average methylation of

the complex reaches a constant value, below full methylation

(Figure 2B). In contrast, for the complex with lower

Koff

a

¼ 0:02 mM (dashed curve), the activity approaches zero

at high concentrations. In this case, full methylation occurs

before saturation of the receptors with MeAsp. When MeAsp

concentrations increase further, the resulting increase in the

free-energy difference between the on and off state of the

complex cannot be compensated by additional methylation,

so the activity drops without recovering.

Compared to the Tar-only complexes, the heterogeneous

receptor complex with six Tar and 13 Tsr receptors (solid

curve) continues to respond to MeAsp increases and to adapt

precisely over an extended range. The Tar receptors in this

complex have Koff

a

¼ 0:02 mM, as in the second case

considered above (dashed curve); these six Tar receptors

allow for a sensitive response at low concentrations of MeAsp.

In contrast, the 13 Tsr receptors in the complex have low

affinity for MeAsp. Therefore, at low MeAsp concentrations

the Tsr receptors act as extra methylation sites, increasing the

range of precise adaptation. As MeAsp concentrations

PLoS Computational Biology | www.ploscompbiol.org January 2008 | Volume 4 | Issue 1 | e10017

Model for Robust Precise Adaptation

Page 5

increase, the Tar receptors become fully saturated, but the

Tsr receptors begin to bind MeAsp. This increases the upper

limit of response to well over 100 mM MeAsp.

For homogeneous complexes, the limit of adaptation at

high attractant concentration depends on which occurs first,

saturation of receptors by attractant or full methylation of

receptors. Which of these occurs first depends on the ratio

Kon

adaptation can be estimated in mean field theory (Equation

2). At high ligand concentrations:

F ’nfrðhmiÞ¼ nerðhmiÞþ nlogKon

r=Koff

r. The crossover ratio between the two limits of

r

Koff

r

;

ð14Þ

where n is the number of receptors in the complex. Loss of

activity occurs if the offset energy at full methylation er(8)

cannot compensate for the free-energy difference per

receptor due to saturating attractant, logðKon

fore, for F*denoting the precisely adapted free-energy

difference, if

F?,nerð8Þþ nlogKon

r=Koff

rÞ. There-

r

Koff

r

;

ð15Þ

or equivalently, if Kon

will occur at high concentrations of attractant (dashed curve

in Figure 2A). In contrast, if Kon

of response will occur at high concentrations of attractant

(dot-dashed curve in Figure 2A). For fixed er(m), F*, and

complex size n, the limit of adaptation depends only on the

r=Koff

r.expðF?=n ? erð8ÞÞ, loss of activity

r=Koff

r

,expðF?=n ? erð8ÞÞ, loss

ratio Kon

Koff

¼ log2 ¼ 0.693. Therefore, the expected crossover ratio is

exp(F*/n ? er(8)) ¼ 20.8. For Tar-only complexes with

Koff

a

¼ 0:02 mM, Kon

loss of activity as observed in Figure 2A (dashed curve). For

Tar-only complexes with Koff

a

¼ 0:06 mM, Kon

adaptation fails through loss of response, as also observed in

Figure 2A (dot-dashed curve).

Experiments indicate that the adapted tumbling rate, and

therefore, also the adapted receptor activity, increases with

the concentration of CheR [26]. At low levels of CheR, the

binding rate cbind

R

is proportional to the concentration of

CheR. In Figure 3, we show the adapted activity as a function

of the CheR binding rate cbind

R

. Adapted activity is the average

activity calculated according to Equation 3, after allowing the

complex to reach equilibrium (see Methods). As the binding

rate of CheR increases, the proportion of CheR-bound

receptors hCheRi also increases (Figure 3, inset). The increase

in hCheRi causes the rate of methylation for the whole

complex to rise, therefore increasing activity. The complex

with full ANs (including all nearest neighbors) closely follows

the expected mean-field-theory result (Equation 7), whereas

the complex with ANs of size one deviates to higher activity

over a wide range of CheR binding rates. In these simulations,

no attractant is present, and therefore, the average methyl-

ation level of the receptors is low. Consequently, complete

demethylation of individual receptors is likely to occur in the

r=Koff

r, not on the individual magnitudes of Kon

r. For our simulation, er(8)¼?30, n¼19, and A*¼1/3, so F*

r

and

a=Koff

a

¼ 25, so adaptation fails through

a=Koff

a

¼ 8:3, and

Figure 2. Averaged Response Curves for Step Increases in Attractant Concentration (0–100 mM)

A mixed receptor complex of six Tar and 13 Tsr receptors (solid curve) is compared to complexes of 19 Tar receptors with Koff

curve) and Koff

a¼ 0.02 mM (dashed curve). Other parameters are given in the Model section. Simulations were averaged over 500 runs.

(A) Averaged response of complex activity A (Equation 3). Upper curves are each displaced vertically by 0.4. Inset: response curve for one simulation of a

single mixed complex. The average activity is superimposed in gray.

(B) Averaged methylation m of receptor homodimers. Insets: distribution of methylation levels at 0 mM, 1 mM, and 100 mM.

doi:10.1371/journal.pcbi.0040001.g002

a¼0.06 mM (dot-dashed

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e10018

Model for Robust Precise Adaptation

Page 6

AN ¼ 1 model (P0

attempts, and therefore, to an increase in adapted activity

according to Equation 6. In effect, for the AN ¼ 1 model, the

demethylation rate is lower than it ‘‘should be’’ because by

chance, some individual receptors are already fully demethy-

lated, and therefore, CheB fails to act sometimes when it

‘‘should.’’

In Figure 4, we explore precision of adaptation over a

broad range of MeAsp concentrations for several variants of

our model. In general, deviations from precise adaptation

occur if and only if the rates of methylation or demethylation

cease to depend exclusively on complex activity (Equation 6).

We find that large AN sizes, saturated kinetics of CheR/CheB,

and short CheR/CheB dwell times favor precise adaptation. In

all cases, we consider the same complex of 19 receptors

(Figure 1A) composed of six Tar receptors and 13 Tsr

receptors. For comparison, we also show the mean field

theory result (see Model).

In Figure 4A, we show the effect of AN size on precise

adaptation. Within each AN, there is a ‘‘ladder’’ of possible

methylation levels. Fluctuations cause the methylation level

to move up and down the ladder, deviating from the average.

For small ANs, the ladder is shorter, and fluctuations are

more likely to produce fully methylated or fully demethylated

neighborhoods. At low levels of MeAsp and low average

methylation, fluctuations are likely to produce fully demethy-

lated neighborhoods, lowering the rate of demethylation and

increasing activity according to Equation 6. Similarly, at high

levels of MeAsp and high average methylation, neighbor-

hoods may become fully methylated, lowering the rate of

methylation by CheR and decreasing activity.

As shown in Figure 4A, complexes with ANs of size one

have a drastically reduced precision of adaptation, but half

AN¼1. 0), leading to missed demethylation

neighborhood complexes have a precision of adaptation close

to that of full AN complexes. Beyond a certain AN size, the

methylation ladder is already long enough to effectively

prevent fluctuations from causing full methylation or full

demethylation of neighborhoods. Therefore, increasing AN

size improves precision of adaptation only up to a point,

beyond which AN size only affects activity near the concen-

tration at which all receptors become fully methylated. For

our parameters, receptors do not become fully demethylated

even at zero attractant concentration, but full demethylation

could be induced by addition of repellent.

We also performed simulations with varying degrees of

saturation of CheR and CheB (Figure 4B). Specifically, we

introduced a factor of N/(NþMsat) into the rates of CheR and

CheB action, where N is the total number of available sites for

methylation/demethylation and Msatis a constant (see Model).

For all other simulations, CheR and CheB were assumed to

work at saturation, independent of methylation level (Msat¼

0). Increasing Msatmakes the rate of action of CheR and CheB

more dependent on the number of available modification

sites. For finite Msat, in low concentrations of MeAsp, and

therefore, low average methylation levels, the rate of

demethylation by CheB is significantly lower than the

saturated (maximal) rate. Conversely, there are many avail-

able sites for methylation, so the rate of methylation is near

maximal. Therefore NB/(NBþ Msat) , NR/(NRþ Msat) ’ 1. A

relative decrease in the rate of demethylation by CheB

compared to the rate of methylation by CheR causes an

increase in the activity of the complex as seen below 0.1 mM

MeAsp in Figure 4B. As the average methylation level

increases with increasing MeAsp concentration, the inequal-

ity is reversed so that NR/(NRþ Msat) , NB/(NBþ Msat) ’ 1

results in a relative decrease in the rate of methylation by

Figure 3. Adapted Activity as a Function of CheR Binding Rate cbind

ResultsareshownforANsofsizeone,ANsofalladjacentreceptors,andthemeanfieldtheory.Theotherenzymebindingratesarecunbind

and cbind

B

¼ 0:01s?1. Inset: hCheRi (proportion of receptors bound to CheR) as a function of CheR binding rate.

doi:10.1371/journal.pcbi.0040001.g003

R

R

¼ cunbind

B

¼ 0:1s?1

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e10019

Model for Robust Precise Adaptation

Page 7

CheR compared to the rate of demethylation by CheB.

Therefore, at high MeAsp concentration, above 0.1 mM, the

adapted activity decreases below the expected value for

precise adaptation.

Within mean field theory for Msat. 0, we can approximate

the crossover concentration, i.e., the concentration of

attractant at which the activity of complexes is equal to the

expected precisely adapted activity. The crossover occurs

when the saturation factors of CheR and CheB are equal, NB/

(NBþ Msat) ¼ NR/(NRþ Msat). This occurs when the average

methylation level hmi is 4, which occurs at 0.36 mM MeAsp.

This is close to the crossover concentration observed in our

simulations (Figure 4B).

Simulations were also performed in which the average

dwell time of CheR and CheB was varied (Figure 4C). The

average dwell time is equal to 1=cunbind

number of enzymes bound to the complex depends on the

ratios cbind

R=B

(Equation 4). Therefore, in order to

change the average dwell time while conserving the average

number of CheR and CheB enzymes bound to the complex,

we altered both cbind

R=B

model, when a CheR or CheB is bound for a long time, the

enzyme catalyzes the same reaction numerous times before

unbinding. The methylation level in the neighborhood will

therefore move along the ladder in one direction, possibly

reaching the end, i.e., full methylation or full demethylation.

As for the AN¼1 model in Figure 4A, the result in Figure 4C

for long CheR and CheB dwell times is higher activity at low

MeAsp concentrations (where full demethylation is more

R=B

, whereas the average

R=B=cunbind

R=Band cunbind

by the same factor. In the

Figure 4. Precision of Adaptation

The adapted activity is shown as a function of MeAsp concentration for variants of the model.

(A) Various AN sizes (and mean field theory). For ‘‘Half AN,’’ the assistance neighbors of each receptor were chosen at random (see Model).

(B) Different saturation factors Msat, for both CheR and CheB. The rate of methylation/demethylation varies as N/(N þ Msat), where N is the number of

available modification sites.

(C)Differentbinding/unbindingratesfor2CheRandCheB.Theratiocunbind

R=B

=cbind

1 mM MeAsp for cunbind

R=B

¼ 1:0s?1; 0:01s?1, and 10?4s?110?4s?1.

doi:10.1371/journal.pcbi.0040001.g004

R=Bwaskeptconstantat10.Inset:distributionofreceptormethylationlevelsat

PLoS Computational Biology | www.ploscompbiol.org January 2008 | Volume 4 | Issue 1 | e10020

Model for Robust Precise Adaptation

Page 8

likely) and lower activity at high MeAsp (where full

methylation is more likely).

Deviations from mean field theory occur if the average

dwell time of CheR or CheB is long enough to allow full

methylation or demethylation of neighborhoods. Below 0.001

mM MeAsp, the average adapted methylation level per

receptor homodimer is ’ 2.2. Since there are on average

5.4 receptors per AN, the average distance to the bottom of

the methylation ladder is 2.2 3 5.4 ’ 12. Therefore, precise

adaptation is expected to fail when the demethylation rate is

’12 times the CheB unbinding rate (kBA?’12cunbind

our parameters, A*¼ 1/3 and kB¼ 0.2 s?1, we expect precise

adaptation to fail for cunbind

B

[ 0:006 s?1. Consistent with this

calculation, our simulations show that deviations from

precise adaptation begin to occur at low MeAsp concen-

trations for cunbind

R=B

The fact that most receptors are either fully methylated or

fully demethylated for long dwell times of CheR and CheB is

clearly shown by the distribution of methylation levels for

different average dwell times (Figure 4C, inset). As dwell time

increases, the single-peaked methylation distribution flattens

out and becomes bimodal, i.e., most receptors become fully

methylated or fully demethylated. Addition of ligand causes a

shift in the amplitudes of the two peaks, but the peak

positions, at m ¼ 0 and m ¼ 8, do not change. We can exploit

this fact along with mean field theory to estimate the

crossover attractant concentration where the activity of the

complex crosses A*. The average methylation (demethylation)

rate has a correction factor equal to the proportion of not

fully methylated (not fully demethylated) ANs (Equation 6).

The crossover attractant concentration will occur where

these two correction factors are equal, namely where

Pmax

adapted activity of A*¼1/3, and requiring hmi ¼ 4, we obtain

a crossover concentration of 30 mM MeAsp, consistent with

the simulation results shown in Figure 4C.

In all our simulations, the methylation levels of receptors

fluctuate, translating into fluctuations in complex activity.

Figure 5 shows the distribution of activities due to fluctuating

methylation levels at 0 mM, 1 mM, and 100 mM of MeAsp.

Within the MWC model, complex activity is strictly either

zero or one. However, we assume that switching between

these two states is rapid, so we consider the distribution of

thermally averaged complex activities given by Equation 3.

Even for the full AN model, for which adaptation is precise,

there is a broad range of complex activities. Note though that

for the observed variation in activity of ’50% for a single

complex and assuming ’500 independent receptor com-

plexes per cell, the resulting variation in total activity would

be only ’2.5%. As shown in Figure 5, for size-one ANs at 0

mM and 100 mM MeAsp, the activity distributions are shifted

relative to the activity distributions for full ANs because

adaptation is not precise when CheR and CheB act only on

single bound receptors (cf. Figure 4A). Also shown in Figure 5,

long dwell times of CheR and CheB cause a bimodal

distribution of complex activities, corresponding to the

bimodal distribution of receptor methylation levels (cf.

Figure 4C, inset).

Within our model, noise is caused by fluctuations in both

binding/unbinding of CheR and CheB and methylation/

demethylation by CheR/CheB. For short average dwell times,

fluctuations in the number of bound CheR and CheB enzymes

B

). For

around 0.01 s?1.

AN¼ P0

AN, which implies hmi ¼ 4. For our mean field–

are rapidly averaged out, and the dominant source of noise is

the discrete methylation/demethylation events by receptor-

bound CheR/CheB. We have estimated the resulting variance

in complex methylation hdM2i and activity hdA2i with the

linear noise approximation (see Model and Figure 6). In this

limit, the only factor that affects the variance hdM2i is the

free-energy difference de per methyl group, with hdM2i ¼

1=de. In the opposite limit of long average dwell times,

fluctuations in the number of CheR and CheB enzymes bound

to the complex add to the variance in methylation levels and

thus activity. As seen in Figure 6A and 6C, low binding and

unbinding rates cbind=unbind

R=B

cause an increase in noise over the

calculated theoretical noise limit due to the discreteness of

methylation and demethylation events. Increasing complex

size can decrease the noise due to CheR and CheB binding/

unbinding, but not the noise due to CheR/CheB methylation/

demethylation. Therefore, increasing complex size only

decreases noise for long average dwell times of CheR and

CheB, but has no effect in the case of short dwell times, where

noise is near the theoretical limit (Figure 6B and 6D).

It was observed experimentally by Chalah and Weis [27]

that CheR and CheB methylate/demethylate the four differ-

ent methyl-attachment sites on each receptor monomer at

different rates. These observations suggest two possible

scenarios: either CheR and CheB have different rates of

action on different modification sites, or CheR and CheB

divide their time unequally among the sites (or some

combination of these two). To test the first scenario, we

extended our model to include variation in the rates of action

of CheR and CheB, with the results shown in Figure 7.

Specifically, we assumed that when a CheR or CheB is

tethered to a receptor, it divides its time equally among all

available modification sites in the AN. The total rate of action

by a bound CheR or CheB is therefore the average over the

rates for all available modification sites in the AN. The

catalytic rates for a methylation/demethylation reaction were

assumed to vary in the ratio 1:2:4:8 for the four different sites

[27]. We studied two cases. In the first case, the ratios of

methylation and demethylation matched for each site (i.e., for

sites 1–4, the ratios for both kBand kRwere 1:2:4:8). In the

second case, the ratios for methylation and demethylation

were inverted relative to each other (i.e., for sites 1–4, the

ratios for kBare 1:2:4:8 and for kR8:4:2:1).

As shown in Figure 7, when the ratios of methylation and

demethylation match for each site, precise adaptation is

preserved. In this case, since every site has the same ratio of

kB/kRas every other site, the average methylation levels of all

sites remain the same, as shown in the inset. The average

methylation and demethylation rates over sites is therefore

constant, independent of ligand concentration, preserving

precise adaptation. In contrast, inverted ratios of methylation

and demethylation rates among the sites fail to produce

precise adaptation. In this case, the ratio kB/kRvaries among

the four methylation sites, causing varying equilibrium

methylation levels (Figure 7, inset). The sites with a low kB/

kRratio are the first to become methylated at low concen-

trations of MeAsp, leading to low average rates of demethy-

lation compared to methylation, and therefore to high

adapted activity. As average methylation levels rise with

increasing MeAsp, these low kB/kRsites ‘‘fill up,’’ leading to

high average rates of demethylation compared to methyl-

ation, and therefore to low adapted activity.

PLoS Computational Biology | www.ploscompbiol.org January 2008 | Volume 4 | Issue 1 | e10021

Model for Robust Precise Adaptation

Page 9

The second scenario suggested by the Chalah and Weis data

[27], namely different dwell times for CheR and CheB among

the modification sites, leads more robustly to precise

adaptation. As long as CheR and CheB work near saturation,

differences in dwell times between sites will not affect total

rates of methylation and demethylation, and precise adapta-

tion will be preserved, according to Equation 6. Indeed, as

shown in Figure 7, even if the relative dwell times for each site

are inverted for CheR and CheB, precise adaptation is

preserved.

Experiments by Berg and Brown [9] on wild-type E. coli

indicate that whereas adaptation to aspartate is precise over a

large concentration range, precise adaptation to serine fails

at relatively low concentrations. In Figure 8, we compare our

model to these experiments. In both cases, adaptation to

aspartate (or MeAsp) is precise over four orders of

magnitude. However, adaptation to serine fails at approx-

imately 0.1 mM. Within our model, this difference with

respect to attractants reflects the presence of more Tsr

receptors (13) in the complex than Tar receptors (six). More

Tsr receptors amplify the change in complex free energy due

to serine, which results in an increased sensitivity at low

concentrations, but also results in full methylation of the

complex and loss of activity beginning at 0.1 mM serine.

We tested robustness of our theoretical model by randomly

varying parameters as described in the Model section. The

results shown in Figure 9 demonstrate that precise adaptation

is a robust property of our model. Almost ideal adaptation

occurs for all parameter sets up to a total parameter variation

of K ’ 1018. For larger parameter variations, in the range of K

¼104–105, 77% of the altered models still have a precision of

adaptation within 10%. These results are similar to those

obtained from the simple single-receptor model of Barkai

and Leibler [16]. However, in one regard, our MWC model

with ANs is more robust than the single-receptor model. In

the single-receptor model, precise adaptation requires that

the activity of the receptor is zero at full demethylation and

one at full methylation. Our model has the property of

precise adaptation without this assumption.

Discussion

The chemotaxis system in the bacterium E. coli is

remarkably sensitive to small relative changes in the concen-

trations of multiple chemical signals over a broad range of

ambient concentrations. We have presented a model of

complexes of strongly coupled chemoreceptors to account

for precise adaptation, as well as other properties of the

chemotaxis network. Similarly to the BBL model of precise

adaptation for a single two-state receptor, CheR only

Figure 5. Distribution of Adapted Complex Activities (Reflecting Distribution of Complex Methylation Levels) at Different MeAsp Concentrations

The MeAsp concentrations are as follows: (A) 0 mM, (B) 1 mM, and (C) 100 mM. Distributions are shown for AN¼1 and full AN at cunbind

full AN at cunbind

R=B

¼ 10?4s?1. The ratio cunbind

doi:10.1371/journal.pcbi.0040001.g005

R=B

¼ 0:1s?1, and for

R=B

=cbind

R=Bwas kept constant at 10.

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e1 0022

Model for Robust Precise Adaptation

Page 10

methylates inactive receptors, and CheB only demethylates

active receptors [16]. A previous MWC model [18,19]

extended the BL model to fit observations of in vivo receptor

clustering [10,11] and of the action by CheR and CheB on ANs

of five to seven adjacent receptor homodimers [15]. Our

model builds upon this earlier MWC model, but includes

dynamic ANs, created by the transient binding of CheR and

CheB to tethering sequences at the C termini of receptors

[14]. The importance of tethering of enzymes has recently

attracted considerable theoretical interest [28–31]. Transient

CheR and CheB binding is of particular relevance because the

experimentally observed ratio of receptor homodimers to the

enzymes CheR and CheB is approximately 50:1 and 30:1,

respectively [4]. Therefore, receptors are not likely to be

continuously in the AN of an CheR or CheB enzyme. Here, we

have shown that a model with dynamical CheR/CheB binding

and unbinding to receptors can reproduce precise adaptation

as in the previous AN model [18]. However, CheR/CheB

dynamics can both limit precise adaptation and increase

noise in complex activity. In addition, we have expanded our

results to show robust adaptation, to explain the experimen-

tally observed difference between the responses to aspartate

and serine, and to account for the persistence of precise

adaptation despite the experimentally observed kinetic

variation among methylation sites.

Although both MWC models [11,18–20,32] and Ising-type

lattice models [33–35] have been used to represent inter-

actions among receptors, analysis of FRET data provides

evidence for strongly coupled MWC complexes [36]. Mello

and Tu [20,32] successfully fit the Sourjik and Berg FRET data

[10] using an identical MWC model to ours [18,19], but did

not include CheR/CheB kinetics. Although Mello and Tu

considered methylation-dependent ligand-binding constants

Kon/off, fitting results do not require variable Kon/off. CheR and

CheB dynamics have been explored in a mixed-receptor

Ising-type lattice model by assuming Michaelis-Menten

kinetics, with each CheR or CheB only able to methylate or

demethylate the bound receptor once before detaching

[34,35]. In principle, combining catalysis with unbinding

increases precision of adaptation by decreasing the likelihood

of fully methylating or fully demethylating a receptor, but

enzyme tethering suggests each bound enzyme may catalyze

multiple methyl transfers before unbinding.

Within our model, deviations from precise adaptation

occur only if CheR/CheB methylation/demethylation rates

become dependent on the receptor-methylation level. Small

AN size and long dwell times of CheR and CheB cause full

methylation or demethylation of neighborhoods, resulting in

methylation-dependent rates and failure of precise adapta-

tion. Although the average dwell time of CheR or CheB has

not been experimentally determined, the diffusion-limited

association rate of protein–protein interactions is on the

order of 105–107M?1s?1[28,37]. Multiplying by the exper-

imentally determined dissociation constant of 11 lM for

CheR [38] yields unbinding rates in the range of 1–100 s?1,

well above the CheR unbinding rate cunbind

precise adaptation (Figure 4). Precise adaptation also requires

saturated enzyme kinetics, meaning that each bound CheR or

CheB acts at a rate independent of the number of available

modification sites. (Unsaturated kinetics would imply de-

creased demethylation rates at low average methylation

levels, and decreased methylation rates at high average

methylation levels.) In our model, precise adaptation is

robust to differences in dwell times of CheR or CheB on

R

required for

Figure 6. Variance in Complex Methylation and Activity Levels

Curves are shown for CheR/CheB binding/unbinding rates cbind=unbind

noise approximation (see Model).

(A,C) Variance in complex methylation level as a function of the free-energy step de per methyl group (A) and complex size (C) (see Methods).

(B, D) Variance in complex activity level as a function of de (C) and complex size (D).

doi:10.1371/journal.pcbi.0040001.g006

R=B

¼ 1:0s?1;0:1s?1, and 0.01 s?1, as well as for the theoretical limit from the linear

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e10023

Model for Robust Precise Adaptation

Page 11

different modification sites, but not, in general, to different

rates of CheR or CheB action on these sites, pointing to

different dwell times as the explanation for the site-depend-

ent methylation/demethylation rates observed by Chalah and

Weis [27].

As with the BL model for a single receptor, our model for a

receptor complex robustly yields precise adaptation over a

wide range of parameters (Figure 9). One improvement is that

our model based on ANs does not require fully methylated

receptors to be fully active, or fully demethylated receptors to

be fully inactive. Experiments indicate that adaptation time

and adapted activity level vary even among genetically

identical cells [39]. Consistent with observations by Alon et.

al. [26], our model predicts that varying CheR and/or CheB

concentrations will lead to different adapted activities

(Figure 3) while preserving precise adaptation. The robust-

ness of the essential properties of the network (e.g., sensitivity

and precise adaptation) presumably also allows for genetic

polymorphisms in the binding and reaction rates of network

proteins, making the network robust to evolutionary change.

Within our model, we assume that the rates of modification

by CheR and CheB depend directly on complex activity. In

fact for precise adaptation, only one enzyme needs to

respond directly to complex activity. This is consistent with

experiment as CheR rates are affected directly by activity [40],

whereas CheB is phosphorylated to an active form by the

receptor-regulated kinase CheA [41,42], implying a global

feedback mechanism. If, hypothetically, all feedback were

global, there would be no direct ‘‘return force’’ on the activity

of individual complexes, only an indirect return force on the

average complex activity. As a result, sensitivity would be lost

as most complexes would drift to nonresponsive methylation

levels, becoming either fully active or fully inactive. However,

within our model, precise adaptation still occurs if the CheB

feedback mechanism is disabled without destroying CheB’s

demethylating ability as long as direct feedback from

complex activity to CheR is maintained. Indeed, experiments

mutating the phosphorylation site of CheB demonstrate that

CheB phosphorylation is not required for precise adaptation

[26], but is important to keep adapted CheY-P levels in the

range of motor sensitivity [43].

Our model helps explain the advantage of multiple

methylation sites per receptor. First, the number of receptors

that a tethered CheR or CheB can modify is constrained by

the physical length of the tether. Therefore, to provide

enough steps in the ladder of methylation levels to prevent

full demethylation or full methylation of neighborhoods (and

therefore loss of precise adaptation), the number of

modification sites per receptor must be sufficiently large.

Second, if the number of methylation sites per receptor were

small, then to allow precise adaptation over a large concen-

tration range would require a large change in free energy per

methyl group. However, large free-energy steps per methyl

group increase the noise in activity (Figure 6), and prevent

complexes from operating in the regime of maximal

sensitivity.

One longstanding puzzle has been the observed difference

in E. coli’s chemotactic response to serine and aspartate [9].

Our model explains both the observed broad range of precise

adaptation to aspartate/MeAsp and the failure of adaptation

at relatively low serine concentrations (Figure 8). Based on

receptor in vivo expression levels, complexes contain more

Tsr receptors then Tar receptors, so the Tsr receptors act as

extra methylation sites and increase the range of precise

Figure 7. Precision of Adaptation for Receptors with Site-Dependent and Site-Independent Methylation/Demethylation Rates

Shown are site-dependent methylation/demethylation rates with matching ratios (kR¼ 0.0125,0.025,0.05,0.1 s?1; kB¼ 0.025,0.05,0.1,0.2 s?1), with

inverted ratios (kR¼0.0125,0.25,0.05,0.1 s?1; kB¼0.2,0.1,0.05, 0.025 s?1), for receptors with site-independent methylation/demethylation rates (kR¼0.1

s?1; kB¼0.2 s?1), and for receptors with site-dependent CheR dwell times in the ratio 1:2:4:8 and inverted site-dependent CheB dwell times in the ratio

8:4:2:1. Inset: average site methylation at 1 mM MeAsp for receptors with site-dependent methylation/demethylation rates for both the matching-ratio

case (filled bars) and the inverted-ratio case (open bars), and for receptors with inverted site-dependent CheR/CheB dwell times (gray bars).

doi:10.1371/journal.pcbi.0040001.g007

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e1 0024

Model for Robust Precise Adaptation

Page 12

adaptation to aspartate/MeAsp. As the Tar receptors become

fully saturated, the Tsr receptors bind aspartate/MeAsp,

thereby also broadening the range of response. In contrast,

the high proportion of Tsr receptors amplifies the complex

free-energy change due to serine and leads to full methylation

of receptors and, therefore, loss of activity, beginning at 0.1

mM serine. The prevalence of Tsr receptors suggests that

chemotaxis to low concentrations of serine is biologically

important.

For stimulation of low-abundance (minor) receptors, our

model predicts a limited range of response. With approx-

imately one minor receptor of each type per complex, there

is no amplification of free-energy change, so sensitivity is

limited to the off-state ligand affinity Koff

response is then constrained by the on-state ligand affinity

Kon

binding of ligand to other receptors.

The effect of AN size may be testable experimentally

through shortening the flexible tether at the C terminus of

receptors while preserving the pentapeptide binding site for

CheR and CheB. Decreasing neighborhood size should

produce deviations from precise adaptation as the ends of

the methylation ladders for ANs are reached (Figure 4A). In

addition, the consequence of nonsaturated kinetics may be

testable through mutations in CheR/CheB that reduce their

affinities for the methyl-modification sites on receptors. Our

model predicts global failure of precise adaptation for large

deviations from fully saturated kinetics, but even small

deviations from full saturation have noticeable consequences

near full methylation (Figure 4B).

Experiments demonstrate that precise adaptation is a

robust property of the E. coli chemotaxis network [26]. The

elegant BL model exhibits robust adaptation through integral

feedback control [44], but does not include interactions

among receptors. Our model provides a molecular mecha-

nism illustrating how integral feedback control is imple-

mented in the presence of receptor clustering, and highlights

the importance of ANs to effectively increase the ladder of

methylation levels.

r. The range of

r, unless the range of response is extended though weak

Methods

Model parameters. In calculating complex activity, we used the

same offset energies for both Tar and Tsr receptors: er(0)¼ 1.0; er(1)¼

Figure 8. Adapted Complex Activity versus Concentration of MeAsp (Filled Circles) or Serine (Open Circles)

The saturation factor is Msat¼1, and other parameters are given in the Model section. Setting Msat¼0 would sharpen and shift the drop in activity to

higher serine concentration [18]. Inset: experimental measurement of the fractional change in run length versus concentration of aspartate (open

circles) or serine (closed circles) [9].

doi:10.1371/journal.pcbi.0040001.g008

Figure 9. Robustness of Assistance-Neighborhood Model for Adaptation

Ratio of adapted activity at 1 mM MeAsp to adapted activity at 0 mM

MeAsp plotted as a function of total parameter variation logK ¼Pjlogkj

parameter sets.

doi:10.1371/journal.pcbi.0040001.g009

(see Methods). The scatter plot shows results for 3,000 different

PLoS Computational Biology | www.ploscompbiol.org January 2008 | Volume 4 | Issue 1 | e10025

Model for Robust Precise Adaptation

Page 13

0.5; er(2)¼0.0; er(3)¼?0.3; er(4)¼?0.6; er(5)¼?0.85; er(6)¼?1.1; er(7)¼?2.0;

and er(8)¼?3.0. Both MeAsp and serine were considered as attractants.

For MeAsp (Tar ¼ a and Tsr ¼ s): Koff

Koff

s

¼ 100 mM,

Kon

s

¼ 0:0025 mM, Kon

chosen for Tar-MeAsp binding and Tsr-serine binding are approx-

imately consistent with experimental data [10,45]. The high values of

Koff

a

and Kon

a

for serine indicate that Tar does not bind serine at the

concentrations considered (?2 M). On the other hand, Tsr binds

MeAspat lower concentrations sinceKoff

serine are attractants, so Kon. Koff. For the demethylation rate, we

used the (rounded-off) observed value kB¼0.2 s?1[46,47], and for the

methylation rate we set kR¼0.1 s?1. Since kB¼2kR, this sets an adapted

activity of 1/3, assuming that the bound levels of CheR and CheB are

the same. Unless otherwise specified, we used the same rates of

binding/unbinding for CheR and CheB: cbind

cunbind

R=B

¼ 0:1 s?1, yielding hCheRi ¼ hCheBi ¼ 0:083.

Simulation algorithm. To simulate the dynamics of an MWC

complex of receptors, we used an exact stochastic Gillespie algorithm

[48]. We assumed that the rates of ligand binding/unbinding and on/

off switching of complexes are much faster than the rates of receptor

modification and the rates of CheR/CheB binding and unbinding.

Therefore, methylation/demethylation and CheR/CheB dynamics

were modeled explicitly, whereas the average activity of the complex

was calculated using Equation 3. The Gillespie algorithm involves

three different steps for the generation of each data point. First, the

reaction that occurs is picked randomly, with weighting directly

proportional to the individual rates of each event. The possible

events are methylation, demethylation, binding of CheR or CheB, and

unbinding of CheR or CheB. A receptor cannot have both CheR and

CheB bound at the same time. Next, the site of the event is randomly

chosen. The time is then incremented by s ¼?(log r)/C, where r is a

random variable picked from a uniform distribution over [0,1], and C

is the sum of the rates of all possible events.

For each attractant concentration, simulations to determine

adapted activity and distributions (Figures 3–8) were averaged over

200 runs of 10,000 Gillespie steps, each following 10,000 steps to allow

time for equilibration.

Robustness. In order to test the robustness of our dynamical

model, we randomly varied the parameters and tested the precision

of adaptation. Altered systems were obtained by modifying eight

parameters (kR, kB, Kon

R

factors of kn¼1,...,8. Total parameter variation is expressed by

logK ¼P8

References

1.Berg HC (2000) Motile behavior of bacteria. Phys Today 53: 24–29.

2.Sourjik V (2004) Receptor clustering and signal processing in E. coli

chemotaxis. Trends Microbiol 12: 569–576.

3. Bourret RB, Stock AM (2002) Molecular information processing: lessons

from bacterial chemotaxis. J Biol Chem 277: 9625–9628.

4.Li M, Hazelbauer GL (2004) Cellular stoichiometry of the components of

the chemotaxis signaling complex. J Bacteriol 186: 3687–3694.

5. Kim KK, Yokota H, Kim SH (1999) Four-helical-bundle structure of the

cytoplasmic domain of a serine chemotaxis receptor. Nature 400: 787–

792.

6.Ames P, Studdert CA, Reiser RH, Parkinson JS (2002) Collaborative

signaling by mixed chemoreceptor teams in E. coli. Proc Natl Acad Sci U S A

99: 7060–7065.

7.Maddock JR, Shapiro L (1993) Polar location of the chemoreceptor

complex in the E. coli cell. Science 259: 1717–1723.

8.Mao H, Cremer PS, Manson MD (2003) A sensitive versatile microfluidic

assay for bacterial chemotaxis. Proc Natl Acad Sci U S A 100: 5449–5454.

9. Berg HC, Brown DA (1972) Chemotaxis in E. coli analysed by three-

dimensional tracking. Nature 239: 500–504.

10. Sourjik V, Berg HC (2002) Receptor sensitivity in bacterial chemotaxis.

Proc Natl Acad Sci U S A 99: 123–127.

11. Sourjik V, Berg HC (2004) Functional interactions between receptors in

bacterial chemotaxis. Nature 428: 437–441.

12. Anand GS, Goudreau PN, Lewis JK, Stock AM (2000) Evidence for

phosphorylation-dependent conformational changes in methylesterase

CheB. Protein Sci 9: 898–906.

13. Djordjevic S, Goudreau PN, Xu Q, Stock AM, West AH (1998) Structural

basis for methylesterase CheB regulation by a phosphorylation-activated

domain. Proc Natl Acad Sci U S A 95: 1381–1386.

14. Barnakov AN, Barnakova LA, Hazelbauer GL (1999) Efficient adaptational

demethylation of chemoreceptors requires the same enzyme-docking site

as efficient methylation. Proc Natl Acad Sci U S A 96: 10667–10672.

15. Li M, Hazelbauer GL (2005) Adaptational assistance in clusters of bacterial

chemoreceptors. Mol Microbiol 56: 1617–1626.

a

¼ 0:02 mM, Kon

serine:

¼ 1:0mM. The constants

a¼ 0:5 mM,

Koff

a

¼ 105mM,

Kon

s

¼ 106mM.For

a¼ 106mM, Koff

s

s

¼ 100 mM.Both MeAsp and

R=B¼ 0:01 s?1and

a, Koff

a, cbind

, cunbind

R

, cbind

B

, and cunbind

B

) by

n¼1jlogknj. Each K was randomly chosen as K ¼ 105r,

where r is a random variable picked from a uniform distribution

over [0,1]. Values of the jlog knj were randomly chosen over [0,1], and

were then normalized to yield the correct sum for log K. The sign of

each log knwas then chosen with equal probability to be negative or

positive. These systems were then subject to a concentration change

from 0 mM of ligand to 1 mM. Precision of adaptation was calculated

by dividing the adapted activity at 1 mM by the adapted activity at 0

mM.

Noise. Simulations to test the effect of the free-energy difference

de per methyl group on the variances in methylation and activity

levels (Figure 6) were performed at 10 mM of MeAsp, with the

constant free-energy offset e0set to yield an average adapted receptor

methylation level of hmi¼4 (i.e., e0¼?1.0 þ4de). de ¼0.5 was chosen

to approximate the free-energy difference per methyl group used in

all other simulations. For increased complex sizes, we used two or

three strongly coupled 19-receptor complexes (i.e., all receptors on or

off together) to produce complexes of size 38 or 57, respectively,

preserving the original AN pattern.

Supporting Information

Accession Numbers

The primary protein accession numbers (in parentheses) from the

Swiss-Prot databank (http://www.ebi.ac.uk/swissprot) for the proteins

mentioned in the text are as follows: CheA E. coli CHEA_ECOLI

(P07363), CheB E. coli CHEB_ECOLI (P07330), CheR E. coli

CHER_ECOLI (P07364), CheW E. coli O157 CHEW_ECO57

(P0A966), CheY E. coli O157 CHEY_ECOLI (P0AE67), Tar E. coli

MCP2_ECOLI (P07017), and Tsr E. coli MCP1_ECOLI (P02942).

Acknowledgments

The authors thank Victor Sourjik for critical reading of the

manuscript.

Author contributions. CHH, RGE, and NSW analyzed data and the

literature, RGE and NSW conceived and designed the model. CHH

performed the model calculations. CHH, RGE, and NSW wrote the

paper.

Funding. RGE and NSW acknowledge funding from the Human

Frontier Science Program (HFSP).

Competing interests. The authors have declared that no competing

interests exist.

16. Barkai N, Leibler S (1997) Robustness in simple biochemical networks.

Nature 387: 913–917.

17. Monod J, Wyman J, Changeux JP (1965) On the nature of allosteric

transitions: a plausible model. J Mol Biol 12: 88–118.

18. Endres RG, Wingreen NS (2006) Precise adaptation in bacterial chemotaxis

through ‘assistance neighborhoods.’ Proc Natl Acad Sci U S A 103: 13040–

13044.

19. Keymer JE, Endres RG, Skoge M, Meir Y, Wingreen NS (2006) Chemo-

sensing in Escherichia coli: two regimes of two-state receptors. Proc Natl

Acad Sci U S A 103: 1786–1791.

20. Mello BA, Tu Y (2005) An allosteric model for heterogeneous receptor

complexes: understanding bacterial chemotaxis responses to multiple

stimuli. Proc Natl Acad Sci U S A 102: 17354–17359.

21. Borkovich KA, Alex LA, Simon MI (1992) Attenuation of sensory receptor

signaling by covalent modification. Proc Natl Acad Sci U S A 89: 6756–6760.

22. Bornhorst JA, Falke JJ (2001) Evidence that both ligand binding and

covalent adaptation drive a two-state equilibrium in the aspartate receptor

signaling complex. J Gen Physiol 118: 693–710.

23. Dunten P, Koshland DE (1991) Tuning the responsiveness of a sensory

receptor via covalent modification. J Biol Chem 266: 1491–1496.

24. Clarke S, Koshland DE (1979) Membrane receptors for aspartate and serine

in bacterial chemotaxis. J Biol Chem 254: 9695–9702.

25. Van Kampen NG (1981) Stochastic processes in physics and chemistry.

Amsterdam: North-Holland. 419 p.

26. Alon U, Surette MG, Barkai N, Leibler S (1999) Robustness in bacterial

chemotaxis. Nature 397: 168–171.

27. Chalah A, Weis RW (2005) Site-specific and synergistic stimulation of

methylation on the bacterial chemotaxis receptor tsr by serine and CheW.

BMC Microbiol 5: 12.

28. Levin MD, Shimizu TS, Bray D (2002) Binding and diffusion of CheR

molecules within clusters of membrane receptors. Biophys J 82: 1809–1817.

29. Camacho CJ, Kimura SR, DeLisi C, Vajda S (2000) Kinetics of desolvation-

mediated protein-protein binding. Biophys J 78: 1094–1105.

30. Windisch B, Bray D, Duke T (2006) Balls and chains—a mesoscopic

approach to tethered protein dynamics. Biophys J 91: 2383–2392.

PLoS Computational Biology | www.ploscompbiol.orgJanuary 2008 | Volume 4 | Issue 1 | e10026

Model for Robust Precise Adaptation

Page 14

31. Segall DE, Nelson PC, Phillips R (2006) Excluded-volume effects in

tethered-particle experiments: bead size matters. Phys Rev Lett 96:

088306.

32. Mello BA, Tu Y (2007) Effects of adaptation in maintaining high sensitivity

over a wide range of backgrounds for Escherichia coli chemotaxis. Biophys J

92: 2329–2337.

33. Shimizu TS, Aksenov SV, Bray D (2003) A spatially extended stochastic

model of the bacterial chemotaxis signalling pathway. J Mol Biol 329: 291–

309.

34. Mello BA, Tu Y (2003) Perfect and near-perfect adaptation in a model of

bacterial chemotaxis. Biophys J 84: 2943–2956.

35. Mello BA, Shaw L, Tu Y (2004) Effects of receptor interaction in bacterial

chemotaxis. Biophys J 87: 1578–1595.

36. Skoge ML, Endres RG, Wingreen NS (2006) Receptor-receptor coupling in

bacterial chemotaxis: evidence for strongly coupled clusters. Biophys J 90:

4317–4326.

37. Northrup SH, Erickson HP (1992) Kinetics of protein-protein association

explained by brownian dynamics computer simulation. Proc Natl Acad Sci

U S A 89: 3338–3343.

38. Yi X, Weis RM (2002) The receptor docking segment and s-adenosyl-l-

homocysteine bind independently to the methyltransferase of bacterial

chemotaxis. Biochim Biophys Acta 1596: 28–35.

39. Spudich JL, Koshland DE (1976) Non-genetic individuality: chance in the

single cell. Nature 262: 467–471.

40. Stock JB, Koshland DE (1981) Changing reactivity of receptor carboxyl

groups during bacterial sensing. J Biol Chem 256: 10826–10833.

41. Lupas A, Stock JB (1989) Phosphorylation of an N-terminus regulatory

domain activates the CheB methylesterase in bacterial chemotaxis. J Biol

Chem 264: 17337–17342.

42. Stewart RC, Russell CB, Roth AF, Dahlquist FW (1988) Interaction of CheB

with chemotaxis signal transduction components in Escherichia coli:

modulation of the methylesterase activity and effects on cell swimming

behavior. Cold Spring Harb Symp Quant Biol 53 (Part 1): 27–40.

43. Kollmann M, Lvdok L, Bartholom K, Timmer J, Sourjik V (2005) Design

principles of a bacterial signalling network. Nature 438: 504–507.

44. Yi T-M, Huang Y, Simon MI, Doyle J (2000) Robust perfect adaptation in

bacterial chemotaxis through integral feedback control. Proc Natl Acad Sci

U S A 97: 4649–4653.

45. Levit MN, Stock JB (2002) Receptor methylation controls the magnitude of

stimulus-response coupling in bacterial chemotaxis. J Biol Chem 277:

36760–36765.

46. Barnakov AN, Barnakova LA, Hazelbauer GL (2002) Allosteric enhance-

ment of adaptational demethylation by a carboxyl-terminal sequence on

chemoreceptors. J Biol Chem 277: 42151–42156.

47. Anand GS, Stock AM (2002) The kinetic basis for the stimulatory effect of

phosphorylation on the methylesterase activity of CheB. Biochemistry 41:

6752–6760.

48. Gillespie DT (1977) Exact stochastic simulation of coupled chemical

reactions. J Phys Chem 81: 2340–2361.

PLoS Computational Biology | www.ploscompbiol.org January 2008 | Volume 4 | Issue 1 | e10027

Model for Robust Precise Adaptation