a model of excitation and adaptation in bacterial chemo tax is

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Proc. Natl. Acad. Sci. USAVol. 94, pp. 72637268, July 1997Biochemistry

A model of excitation and adaptation in bacterial chemotaxisPETER A. SPIRO *, JOHN S. P ARKINSON , AND H ANS G. O THMER *

Departments of *Mathematics and Biology, University of Utah, Salt Lake City, UT 84112

Communicated by Charles S. Peskin, New York University, Hartsdale, NY, April 24, 1997 (received for review August 26, 1996)

ABSTRACT Bacterial chemotaxis is widely studied becauseof its accessibility and because it incorporates processes that areimportant in a number of sensory systems: signal transduction,excitation, adaptation, and a change in behavior, all in responseto stimuli. Quantitative data on the change in behavior areavailable for this system, and the major biochemical steps in thesignal transduction processing pathway have been identified. We have incorporated recent biochemical data into a mathemat-ical model that can reproduce many of the major features of theintracellular response, including the change in the level of chemotactic proteins to step andramp stimuli suchas those usedin experimental protocols. The interaction of the chemotacticproteins with the motor is not modeled, but we can estimate the

degree of cooperativity needed to produce the observed gainunder the assumption that the chemotactic proteins interactdirectly with the motor proteins.

Chemotaxis (or more accurately, chemokinesis), the process by which a cell alters its speed or frequency of turning in responseto an extracellular chemical signal, has been most thoroughlystudied in the peritrichous bacterium Escherichia coli. E. coliexhibits sophisticated responses to many beneficial or harmfulchemicals. In isotropic environments the cell swims about in arandom walk produced by alternating episodes of counterclockwise (CCW) and clockwise (CW) flagellar rotation. CCW rota-tion pushes thecell forward in a fairlystraightrun, CW rotationtriggers a random tumble that reorients the cell. The durationsof both runs and tumbles are exponentially distributed, with

means of 1.0 s and 0.1 s, respectively (1). In a chemoeffectorgradient, the cell carries out chemotactic migration by extendingruns that happen to carry it in favorabledirections.Using specificchemoreceptors to monitor its chemical environment, E. coliperceives spatial gradients as temporal changes in attractant orrepellent concentration. The cell in effect compares its environ-ment duringthe past second with the previous 34 s and respondsaccordingly. Attractant increases and repellent decreases tran-siently raise the probability of CCW rotation, or bias, and thena sensory adaptationprocessreturns thebias to baseline,enablingthe cell to detect and respond to further concentration changes.The response to a small step change in chemoeffector concen-tration in a spatially uniform environment occurs over a 2- to 4-stime span (2).Saturating changes in chemoeffector concentrationcan increase the response time to several minutes (3).

Many bacterial chemoreceptors belong to a family of trans-membrane methyl-accepting chemotaxis proteins (MCPs) (re- viewed in ref. 4). Among the best-studied MCPs is Tar, the E. coli receptor for the attractant aspartate. Tar has a periplasmicbinding domain and a cytoplasmic signaling domain thatcommunicates with the flagellar motors via a phosphorelaysequence involving the CheA, CheY, and CheZ proteins (seeFig. 1). CheA, a histidine kinase, first autophosphorylates andthen transfers its phosphoryl group to CheY.

PhosphoCheY interacts with switch proteins in the flagellarmotors to augment CW rotation, CCW being the default state inthe absence of PhosphoCheY. CheZ assists in dissipating CWsignals by enhancing the dephosphorylation of CheY. Tar con-trols the flux of phosphate through this circuit by forming a stableternary complex (Tar CheW CheA) that modulates CheA au-tophosphorylation in response to changes in ligand occupancy ormethylation state. CheA phosphorylates more slowly when thereceptor is occupied than when it is not.

Changes in MCP methylation state are responsible for sensoryadaptation. Tar has four residues that are reversibly methylatedby a methyltransferase, CheR, and demethylated by a methyles-terase, CheB. CheR activity is unregulated, whereas CheB, likeCheY, is activated by phosphorylation via CheA. Thus, receptormethylation level is regulated by feedback signals from thesignaling complex, which can probably shift between two confor-mational states having different rates of CheA autophosphory-lation. Attractantbindingand demethylation shift theequilibriumtoward a low CheA activity state; attractant release and methyl-ation shift the equilibrium toward a high CheA activity state. Areceptor complex that is bound to attractant but not highlymethylated can be thought of as a sequestered state (5) whichis unable to autophosphorylate at a significant rate.

E. coli can sense and adapt to ligand concentrations thatrange over five orders of magnitude (5). In addition, themachinery for detection and transduction is exquisitely sensi-tive to chemical stimuli. The cell can respond to exponentialincreases (ramps) in attractant levels that correspond torates of change in fractional occupancy of only 0.1% persecond. Consequently, the cell can respond even if only a smallfraction of its receptors have changed occupancy state duringa typical sampling period.

These experimental observations raise several questions. First,howdoes thecell achieve theextreme sensitivity that is observed?Second, why are there multiple methylation states of the recep-tor? In this paper we describe a mathematical model based onknown kinetic properties of Tar and the phosphorelay signalingcomponents that accounts for this exquisite sensitivity. In themodel, excitation results from the reduction in the autophos-phorylation rate of CheA when Tar is bound to a ligand, andadaptation arises from methylation of the receptor. The disparityin the time scales of these processes produces a derivative- ortemporal-sensing mechanism with respect to the ligand con-centration. The model makes essential use of the multiple meth-

ylation states to achieve adaptation, and can be used to derivequantitative relations between certain key rates and the behav-ioral responses to experimental protocols, and to predict thecooperativity needed to achieve the observed gain.

A Qualitative Description of the Responseto Different Stimuli

We assume that Tar is the only receptor type, that the TarCheACheW complex does not dissociate, and that Tar, CheA,and CheW are found only in this complex. We also assume thatmethylation of the multiple sites occurs in a specified order (6, 7).The publication costs of this article were defrayed in part by page charge

payment. This article must therefore be hereby marked advertisement inaccordance with 18 U.S.C. 1734 solely to indicate this fact. 1997 by The National Academy of Sciences 0027-8424 97 947263-6$2.00 0PNAS is available online at http: www.pnas.org.

Abbreviations: CCW, counterclockwise; CW, clockwise; MCP, methyl-accepting chemotaxis protein.To whom reprint requests should be addressed.

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Our primary objective is to model the response to attractant, which probably involves onlyincreases in theaverage methylationlevelabove theunstimulatedlevel ofabout 1.52methylesterspertransducer(8).For this reasonandforsimplicity,we consideronlythe three highest methylation states of Tar. In the model wepostulate that only the phosphorylated form of CheB (CheB p)has demethylation activity. We assume that the phosphorylationstate of CheA does not affect the ligand-binding reactions of thereceptor with which it forms a complex, and does not affect theactivities of CheR or CheB p. We further assume that the forma-tion of a complex between Tar and CheR does not affect theligand-binding properties of Tar, and that it does not affect theautophosphorylation rate of the attached CheA until the transi-tion to the more highly methylated state (and dissociation of thecomplex) occurs. We assume that CheZ activity is unmodulated,although it has been suggested (9) that an as-yet-unidentifiedmechanism for modulating CheZ activity could account for theobserved gain. Later we consider the case of modulated CheZactivity. Finally, we assume that the rates of the phosphotransferreactions between CheA and CheY or CheB are not affected bythe occupancy or methylation state of the receptor.

In Fig. 2 we illustrate the receptor states and the network of transitions between them that are used in the model. Detailsof the phosphotransfer reactions are shown in Fig. 3, together with the dephosphorylation reactions for CheBand CheY. Thedetails of the reactions in these figures are given in Table 1.Before we introduce the equations we present a qualitativedescription of how the system works.

First consider the response of the network to a step increasein attractant. The ligand-binding reactions are the fastest, sothe first component of the response is a shift in the distributionof states toward the ligand-bound states (i.e., from states at thetop to those at the bottom in Fig. 2). This increases the fractionof receptors in the sequestered states ( LT 2, LT 2 p, and possibly LT 3, LT 3 p), and because these states have the same phospho-transfer rates but much lower autophosphorylation rates thanthe corresponding ligand-free states, the distribution subse-

quently shifts toward states containing unphosphorylatedCheA. This in turn leads to a reduction in CheY p, a decreasedrate of tumbling, and an increase in the run length. Thisconstitutes the excitation response of the system.

Next methylation and demethylation, which are the slowestreactions, begin to exert an effect.CheR methylates ligand-boundreceptors more rapidly than unbound receptors, and thedecreasein CheA p that results fromexcitationcausesa decrease in thelevelof CheBp, therebyreducing the rate of demethylation. As a result,thedistribution shifts toward higher methylation states.However,autophosphorylation of CheA is faster, the higher the methyl-ation state of theTarCheACheWcomplex, andthereforein thelast phase of the response there is a shift toward the statescontaining CheA p via transitions along the right face of thenetwork. Consequently, the net effect of an increase in attractantis to shift the distribution of receptor states toward those whichare ligand-bound and more highly methylated, but the total levelof receptor complex containing CheA p (the sum of the states atthe rear face of the network) returns to baseline. As a result, thetotal phosphotransfer rate from CheA p species to CheY returnsto the prestimulus level, which means that CheY p returns to itsprestimulus level. Under the assumption that only CheY p andCheY interact with the motor complex, this in turn implies thatthe bias returns to its prestimulus level. Thus the cell can respondto an increase in ligand by a transient decrease in tumbling, butit adapts to a constant background level of ligand and retainssensitivity to further changes in the ligand concentration. Of course this qualitative description must be supplemented bynumerical results which demonstrate that the model can alsoproduce quantitatively correct results using experimentally basedrate coefficients. This is done in the following section.

FIG . 1. Signaling components and pathways for E. coli chemotaxis.Chemoreceptors (MCPs) span the cytoplasmic membrane (hatchedlines), with a ligand-binding domain deployed on the periplasmic sideand a signaling domain on the cytoplasmic side. The cytoplasmic Chesignaling proteins are identified by single letterse.g., A CheA.MCPs form stable ternary complexes with the CheA and CheWproteins to generate signals that control the direction of rotation of theflagellar motors. The signaling currency is in the form of phosphorylgroups ( P), made available to the CheY and CheB effector proteinsthrough autophosphorylation of CheA. CheY p initiates flagellar re-sponses by interacting with the motor to enhance the probability of clockwise rotation. CheB p is part of a sensory adaptation circuit thatterminates motor responses. MCP complexes have two alternative

signaling states. In the attractant-bound form, the receptor inhibitsCheA autokinase activity; in the unliganded form, the receptorstimulates CheA activity. The overall flux of phosphoryl groups toCheB and CheY reflects the proportion of signaling complexes in theinhibited and stimulated states. Changes in attractant concentrationshift this distribution, triggering a flagellar response. The ensuingchanges in CheB phosphorylation state alter its methylesterase activ-ity, producing a net change in MCP methylation state that cancels thestimulus signal (see ref. 4 for a review).

FIG. 2. The ligand-binding, phosphorylation, and methylation reac-tions of the TarCheACheW complex, denoted by T . LT indicates aligand-bound complex. Vertical transitions involve ligand binding andrelease, horizontal transitions involve methylation and demethylation,and front-to-rear transitions involve phosphorylation and the reverseinvolve dephosphorylation. The details of the phosphotransfer steps aredepicted in Fig. 3. Numerical subscripts indicate the number of methyl-ated sites onTar, anda subscriptp indicates that CheA is phosphorylated.The rates for the labeled reaction pairs are given in Table 3.

FIG. 3. Detail of the phosphotransfer reactions corresponding tothe labels 813 in Fig. 2.

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In ref. 10 we examine the response of a simplified model thatincorporates just two methylation states to step changes and toa slow exponential ramp. The results of this analysis imposedifferent conditions on the methylation and demethylationrates, because a significant ramp response results only whenthese rates are considerably slower than those which reproduceobserved adaptation times to small steps. To capture both of these responses a model must contain at least three methyl-ation states, and the transitions between the lowest andintermediate methylation states must be considerably fasterthan the transitions between the intermediate and higheststates. In such a scheme the adaptation time to small steps iscontrolled by the transitions between the doubly and triplymethylated states in Fig. 2, and the ramp response is controlledby the slower transitions between the triply and fully methyl-ated states. The three-methylation-state model described inthe following section also captures the correct time scale forthe adaptation response to saturating levels of attractant.

A Three-State Model Captures the Ramp, Step, and

Saturation Responses A minimal model must have three methylation states, and weassume fastermethylation and demethylation rates for transitionsbetween the two lowest methylation states, and slower transitionratesbetweenthetwo highestmethylationstates.We useMichae-lisMentenmethylation kinetics, since CheRactivityis thoughttobe saturated under normal conditions (11). This precludes ananalytical derivation of the relationships between kinetic param-eters which guarantee perfect adaptation, by which we mean thatthe bias returns precisely to baseline in the face of any constantattractant level below saturation, (10). Instead we have tuned therate constants of the full system by trial and error so that it adapts well over a large range of ligand concentrations.

The mathematical description of the model is based on massaction kinetics for all the steps except methylation. Because theequations that govern the evolution of the amounts in the variousstates of the receptor complex are similar and easy to derive, weonlydisplay oneof them,andwe indicate theprocesscontributingto the rate of change beneath the corresponding rate expression.

dT 2 dt

Lk5T 2 k 5 LT 2Ligand binding release

k8T 2

Phosphorylation B p k 1T 3

Demethylation

k yT 2 p Y 0 Y p k bT 2 p B0 B p

Phosphotransfer

V maxT 2

K R T 2methylation [1]

We write the equations for CheY p and CheB p as

dY p dt

k y P Y 0 Y p k y ZY p, [2]

dB p dt

k b P B0 B p k b B p, [3]

where P T 2 p LT 2 p T 3 p LT 3 p T 4 p LT 4 p is the totalamount of phosphorylated receptor complex. In addition, thetotal amounts of T , Y , B, and R are conserved. These fourconditions can be used to eliminate four variables, which wasdone for Y and B in Eqs. 2 and 3, and the resulting system can be

integrated numerically, using the parameter values given in thetables. As we remarked earlier, we assume that the motorinteracts only with CheY p and possibly CheY, and in view of thesmall numberofmotor complexesin a cell, wefurther assumethatthe motor does not affect the budgets of these species. It thenfollows from the conservation of Y that perfect adaptation of thebias only requires perfect adaptation of CheY p.

The responses obtained from this three-methylation-statemodel are shown in Fig. 4AC and the concentrations andrates used are listed and compared with measured values inTables 2 and 3. The system exhibits both a significant responseto a slow ramp and an appropriate adaptation time for a smallstep. It also exhibits the correct adaptation time to a saturatingstep in ligand concentration.

Analysis of the GainIn the preceding figures we have chosen the step size so thatthe maximum deviations of the CheY p concentration frombaseline are equal in Fig. 4A and B, and thus the maximumchange in bias in response to the corresponding ramp (0.015s 1) and step (11% change in receptor occupancy) will be thesame. Experimentally it is found that the maximum change inbias in response to this ramp is about 0.3 (figure 4A in ref. 16),and so our model, coupled with a scheme for the interactionof CheY p with the motor, would produce a maximum gain of about 0.3 0.11 2.73 in response to the step used in Fig. 4 B.This is consistent with the finding (17) that the maximum gainis about 6, though not with the much higher gain of 55 reported(18). The model thus demonstrates that ramp experiment

results are consistent with the lower estimates regarding thegain for small steps. However the source of the gain remainsundetermined, because even though the deviations of CheY pconcentration from baseline are significant ( 9%), they aresmall compared with the reported change (0.3) in bias for thisramp stimulus (16). Thus one or more mechanisms to amplifythe internal CheY p signal must be involved, and the majorpossibilities are cooperative effects in the interaction of CheYspecies with the motor and modulation of CheZ activity.

To explore these possibilities, we define the gain as

g db

d ln p, [4]

where b is the bias, or probability that a flagellum will be

rotating CCW, and p k y P is the pseudo-first order rateconstant for phosphorylation of CheY. We have chosen thisdefinition of gain for mathematical convenience, but it can beshown (10) that for small steps this definition of g is consistent with the definition of gain given in ref. 17, namely, the changein bias per percent change in receptor occupancy.

When the phosphorylation reactions are at pseudo-equilibrium, one can show that

y p

p z , [5]

where y Y p Y 0 [0,1] is the dimensionless amount of CheYin phosphorylated form, and p and z k-y Z are the productionand loss coefficients, respectively, of y. We note that

Table 1. Details of reactions depicted in Figs. 2 and 3

DescriptionReaction

labels Kinetics and rate labels

Methylation (first orderkinetics)

14 T 2 R 3 k1

T 3 R

Methylation (MichaelisMenten kinetics)

14 T 2 R k1 b

k1 aT 2 R 3

k1 cT 3 R

Demethylation 14 T 3 B p 3 k 1

T 2 B p

Ligand binding 57 T 2 L k5

k 5 LT 2

Autophosphorylation 813 T 2 3 k8

T 2 pPhosphotransfer 8 13 T 2 p B 3

k bT 2 B p

T 2 p Y 3 k y

T 2 Y pDephosphorylation B p 3

k b B

Y p Z 3 k y

Y Z

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ln yln p

z p z

1. [6]

Thus without modulation of CheZ activity, the fractional changein CheYp levels must be less than the change in receptoroccupancy, so that no signal amplification is possible upstream of the interaction of CheY and CheY p with the motor.

To examine theeffect on thegain of cooperative interactionsof CheY species at the motor and modulation of CheZ activity, we expand Eq. 4, using Eq. 5, to obtain

g db dy

y 1 y 1 d ln z d ln p v 1 w . [7]

Here v y(1 y) db dy is the gain due only to cooperativeinteraction of CheY with the motor, and w dln z dln p is,for small steps at least, the fractional change in CheZ activity

per fractional change in receptor occupancy (10). We haveassumed here that CheZ does not interact directly with themotor (19), and acts only by dephosphorylating CheY. Thisexpression makes it clear that the two possible sources of cooperativity act multiplicatively.

To determine the gain achievable, consider first the factor v. We assume that CheY species interact with the motor bybinding to the f lagellar switch, and we allow for the possibilitythat both CheY p and unphosphorylated CheY may bind (19).Interaction with the motor may involve cooperativity, either

via cooperative binding, or possibly via some cooperativeinteraction among subunits of the switch which bind individ-ually to CheY molecules. Here we consider the first possibility(the second is treated in ref. 10), and we assume that a motorunit binds either Che Y or Che Y p according to

jY p M M Y p j, [8]

kY M M Y k. [9]

Here j and k are the number of molecules of CheY p and CheY,respectively, which bind to the motor, M represents motorunbound to either CheY species, M (Y p) j represents motor incomplex with Y p , and M (Y ) k represents motor in complex withY . We assume that the binding reactions Eqs. 8 and 9 equili-brate rapidly, and study only the steady-state quantities

m M

M 0

1

1 A j yj

A k 1 y k , [10]

m j M Y p j

M 0

A j y j

1 A j y j A k 1 y k, [11]

m k M Y k

M 0

A k 1 y k

1 A j y j A k 1 y k. [12]

Here M 0 is the total concentration of motor, and A j and A k arethe ratios of the association and dissociation constants forCheY p and unphosphorylated CheY, respectively.

We assume that the unbound motor M exists in CCW mode with probability one (19), and that the M j and M k states haveprobabilities f j and f k, respectively, of being in CCW mode.Then the bias is given by

Table 2. Conserved quantities used in the model (see ref. 3)

Species T R B Y Z

Concentration, M 8 0.3 1.7 20 40

FIG. 4. Response of the system depicted in Fig. 2 to various chemoattractant (aspartate) stimulus protocols: a ramp of rate 0.015 s 1 ( Left), astep at t 10 s from zero concentration to a concentration (0.11 M) that is 11% of the K d for ligand binding (Center ), and a step at t 100 sfrom zero concentration to a concentration (1 mM) that is 1,000 times the K d for ligand binding ( Right). Time series in the top row of figures arefor Che Y p (solid line), CheB p (short dashes), and aspartate (long dashes). The concentrations and rates used are listed and compared to measured values in Tables 2 and 3. The maximum change in [Che Yp] from baseline for both the ramp and small step responses is 9%. Time series in thebottom row are the corresponding bias responses, using a Hill coefficient of 11 for binding of both Che Y p and the competitive inhibitor Che Yto the motor. (Compare the left trace to figure 4 A in ref. 12; the center trace to figure 4 in ref. 2, figure 2 in ref. 14, and figure 8 in ref. 13; andthe right trace to figure 8 in ref.3.



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b m f j m j f k m k1 f j A j y j f k A k 1 y k

1 A j y j A k 1 y k.[13]

One can show that the optimal values of f j and f k are 0 and 1,respectively (10), and using these v is given by

v y 14 j 1 y

ky a k

1 a k. [14]

where a k A k(1 y)k and y is the baseline CheY p concen-tration. Because this expression is monotonically increasing in a k, v( y ) is maximized with respect to a k when a k is as large aspossible, indicating that both CheY species should have strongbinding affinities for the flagellar switch. In the limit as a k 3

, Eq. 14 becomes

v y 14

j 1 y ky . [15]

Finally, maximizing Eq. 15 with respect to y (subject to y [0, 1])shows that the maximum possible gain at the flagellar switch is

v y 14 n , n max j, k . [16]

Thus, a gain of 2.7 requires a Hill coefficient of at least 11 if theonly cooperativity is in theinteraction of CheY species withthe switch. In Fig. 4 DF we show the bias resulting from theCheY p response in Fig. 4 AC using a Hill coefficient of 11 forboth CheY species ( j k 11).

The full expression for the gain now becomes

g14 n 1 w , [17]

and so the fractional change in CheZ activity relative to thefractional change in receptor occupancy is

w 4 g n

1. [18]

Thus, without cooperative effects at the switch, we must have w 10, and for moderate cooperativity at the switch, forexample n 6 (20), we must have w 0.8. A gain of 6 (17)requires w 23 and w 3 in these respective cases.

Finally, we note that an additional cooperative step probablyoccurs in the interactions between f lagella, since the bias of anindividual flagellum in the absence of stimulation [ 0.64 (2)]is less than that of a swimming cell [ 0.9, assuming mean runand tumble durations of 1.1 s and 0.14 s, respectively (21)]. Wecan estimate what the threshold number of flagella might befor this cooperative interaction if we adopt the voting hy-pothesis (21, 22), whereby the biases of the individual flagellaare identical and independent, and the probability that theflagella will form a bundle is one when the number of flagellaturning CCW equals or exceeds a threshold and zerootherwise. Then the bias B of the cell is given by

B j

N N j b

j 1 b N j, [19]

(not B j N b j(1 b) N j, as claimed by Weis and Koshland

(22)], where N is the total number of flagella and b is the biasof an individual flagellum. For b 0.64 and either n 6 or n 8 (23), we find that B 0.9 when N 2, and thus asimple majority rules.

Discussion

In our model of aspartate signal transduction via Tar, excitationis the result of the reduced autophosphorylation rate of theligand-bound state of the receptor, which reduces the level of CheA p and CheY p, thereby reducing the tumbling rate. Adapta-tion results fromtheenhanced rate of methylation ofbound statesand the fact that methylation increases the rate of autophosphorylation, which returns CheAp and CheY p to their prestimulus

Table 3. Rates used in the model, and corresponding values from the literature

Reaction Rate constant Value Literature value Ref.

T 2 R 3 T 3 R k1 c 0.17 s 1 0.17 s 1 12T 3 R 3 T 4 R k2 c 0.1 k1 c 0.02 k1 c* 13 LT 2 R 3 LT 3 R k3 c 30 k1 c 15 k1 c 30 k1 c 13 LT 3 R 3 LT 4 R k4 c 30 k2 c 15 k2 c 30 k2 c 13T n R T n R k1 b k1 a , . . . , k4 b k4 a 1.7 M 1.7 M 12T 3 B p 3 T 2 B p k 1 4 10 5 M 1 s 1 3 104 M 1 s 1 11

T 4 B p 3 T 3 B p k 2 3 104

M1

s1

3.0 k 1 13 LT 3 B p 3 LT 2 B p k 3 k 1 k 1 13 LT 4 B p 3 LT 3 B p k 4 k 2 k 2 13 L T 3 LT k5, k6, k7 7 10 7 M 1 s 1 7 107 M 1 s 1 3 LT 3 L T k 5, k 6, k 7 70 s 1 70 s 1 3T 2 3 T 2 p k8 15 s 1 17 s 1 14T 3 3 T 3 p k9 3 k8T 4 3 T 4 p k10 3.2 k8 LT 2 3 LT 2 p k11 0 LT 3 3 LT 3 p k12 1.1 k8 LT 4 3 LT 4 p k13 0.72 k10 k10 15 B T np 3 B p T n k b 8 10 5 M 1 s 1 8 105 M 1 s 1 11Y T np 3 Y p T n k y 3 10 7 M 1 s 1 3 107 M 1 s 1 14 B p 3 B k b 0.35 s 1 0.35 s 1 14Y p Z 3 Y Z k y 5 10 5 M 1 s 1 5 105 M 1 s 1 14

*Methylation rates of different methylation sites vary by a factor of up to 50.Ligand binding increases methylation rates of different methylation sites by a factor of 1530.Demethylation rates of different methylation sites vary by a factor of up to 3.Ligand binding has little effect on demethylation rate.Estimated from figure 10 in ref. 3.Estimated from figure 3 in ref. 11.

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levels. The results we present demonstrate that the model canreproduce the experimentally observed responses to both stepincreases andslow ramps using experimentallydeterminedvaluesfor most of the parameters. The disparity between the time scaleof excitation, which is fast, and that of adaptation, which is slow,implies that the transduction system can function as a derivativesensor with respect to the ligand concentration: the DC com-ponent of a signal is ultimately ignored if it is not too large. Thisprovidesa bacteriumwitha temporal sensing mechanism withoutthe need for any type of memory beyond that embodied in thedisparity in time scales between excitation and adaptation.

In ref. 10 we show that, as is seen experimentally (16), themagnitude of the response to a slow ramp is an increasingfunction of ramp rate, and so we may understand the rampresponse as the result of a difference between the rate at whichreceptors enter the sequestered state via increases in receptoroccupancy,and the rate at which they exit via transitionsbetweenmethylation states. Steeper ramps result in larger differencesbetween the entrance and exit rates, and thus in larger responses.The threshold ramp response (16) occurs when these rates areequal. In response to a ramp, the deviation from baseline of thelevel of sequestered receptor is an approximately linear functionof ramp rate. If thedownstreamsteps in thetransduction pathwayoperate in a linear range, then the bias response will in turn be

approximately linear in the ramp rate, consistent with the findingof Block et al. (16).Three methylation states were necessary to accurately repro-

duce the responses to both step and ramp stimuli, because theseresponses place competing restrictions on the effective methyl-ation and demethylation rates.For step stimuli given at a baselineattractantconcentration of zero (17,18), adaptation is dominatedby the fast transitions between the two lowest methylation states, which provide a sufficiently fast adaptation response. At thehigher attractant concentrations of ramp experiments (16, 18),receptors are on average more highly methylated, and so theslower transitions between the two highest methylation states aremore prominent, providing a significant ramp response.

The sensitivity, or gain, of the signal transduction system isfound experimentally to be quite high, but the source of thissensitivity is unknown. We have shown that the sensitivity ob-served in response to ramp stimuli (16, 18) is consistent with themoderate estimate of 6 for the maximum gain of the system (17)obtained fromstep experiments,but that cooperativity equivalentto a Hill coefficient on the order of 11 is necessary to produce thedesired gain. There are several potential sources of cooperativity.Binding of CheYp to the flagellar switch is thought to becooperative (20), and we have shown that competitive inhibitionby unphosphorylated CheY can enhance the sensitivity. Becausethe motor contains 26 or 27 subunits of switch protein FliF (19),it is conceivable that high cooperativity occurs either in bindingto the switch or else in interactions among switch subunits boundindividually to CheYp. CheZ phosphatase activity may also bemodulated in a manner that exhibits cooperativity. The systemgain could be significant if CheZ activity were positively corre-lated with the level of sequestered CheA, becausesmall fractionalchanges in receptor occupancy can correspond to large fractionalchanges in sequestered CheA. As an example, the cells possess ashort form of CheA (CheA s) which may bind several moleculesof CheZ (24). If it does so when the associated transducer isunbound to attractant or highly methylated, and releases thesemolecules into the cytoplasm upon attractant binding, the effec-tive concentration of CheZ will be raised following the latterevent. Alternatively, the CheA sCheZ complex may amplifyCheZ phosphatase activity [Wang (1996) cited in ref. 25], whichcould produce high gain if the complex were to form uponattractant binding. Polymerization of CheZ in the presence of CheY p is another potential mechanism for signal amplification,though it is more likely that this reaction instead enhances theadaptationresponse (25). A furthersource of gain might be foundin cooperative interactions among receptors in close proximity to

one another, a possibility raised by the existence of a nose spotof elevated receptor density (26).

We have shown that the effects of cooperativity at differentlocations in the signal transduction pathway are likely to bemultiplicative, and thus the total system gain may be the resultof moderate cooperativity occurring at two or more of theabove-mentioned locations. For example, a Hill coefficient of 6 at the flagellar switch (20) and a moderate degree of CheZmodulation are sufficient to produce the desired gain.

Although the model analyzed herein is specific to signaltransduction in bacterial chemotaxis, the structure of thenetwork in Fig. 2 is very similar to those of other signaltransduction processes, such as those modeled in refs. 2729.This suggests that the type of analysis done here will haveapplicability to other systems. A more complete discussion of aspects of adaptation not treated here, including an evaluationof general models of adaptation, is given in ref. 10.

This work was supported in part by National Institutes of HealthGrant GM29123 to H.G.O. and National Institutes of Health GrantGM19559 to J.S.P.

1. Berg, H. C. (1990) Cold Spring Harbor Symp. Quant. Biol. 55,539545.

2. Block, S. M., Segall, J. E. & Berg, H. C. (1982)Cell 31, 215226.3. Stock, J. B. (1994) in Regulation of Cellular Signal Transduction

Pathways by Desensitization and Amplification, eds. Sibley, D. R.& Houslay, M. D. (Wiley, New York), pp. 324.

4. Stock, J. B. & Surette, M. G. (1996) Escherichia coli and Salmo- nella: Cellular and Molecular Biology (Am. Soc. Microbiol.,Washington, DC).

5. Bourret, R. B., Borkovich, K. A. & Simon, M. I. (1991) Annu. Rev. Biochem. 60, 401441.

6. Engstrom, P. & Hazelbauer, G. L. (1980) Cell 20, 165171.7. Springer, M. S., Zanolari, B. & Pierzchala, P. A. (1982) J. Biol.

Chem. 257, 68616866.8. Boyd, A. & Simon, M. I. (1980) J. Bacteriol. 143, 809815.9. Bray, D., Bourret, R. B. & Simon, M. I. (1993) Mol. Biol. Cell 4,

469482.10. Spiro, P. A. (1997) Ph.D. dissertation (Univ. of Utah, Salt Lake

City).11. Lupas, A. & Stock, J. B. (1989) J. Biol. Chem. 264, 1733717342.12. Simms, S. A., Stock, A. M. & Stock, J. B. (1987)J. Biol. Chem.

262, 85378543.13. Terwilliger, T. C., Wang, J. Y. & Koshland, D. E. (1986) J. Biol.

Chem. 261, 1081410820.14. Bray, D. & Bourret, R. B. (1995) Mol. Biol. Cell 6, 13671380.15. Borkovich, K. A., Alex, L. A. & Simon, M. I. (1992)Proc. Natl.

Acad. Sci. USA 89, 67566760.16. Block, S. M., Segall, J. E. & Berg, H. C. (1983)J. Bacteriol. 154,

312323.17. Khan,S., Castellano, F.,Spudich, J.,McCray, J., Goody,R., Reid,

G. & Trentham, D. (1993) Biophys. J. 65, 23682382.18. Segall, J. E., Block, S. M. & Berg, H. C. (1986)Proc. Natl. Acad.

Sci. USA 83, 89878991.19. Macnab, R. M. (1995) in Two-Component Signal Transduction,

eds. Hoch, J. A. & Silhavy, T. J. (Am. Soc. Microbiol., Washing-ton, DC), pp. 181199.

20. Kuo, S. C. & Koshland, D. E. (1989) J. Bacteriol. 171, 62796287.21. Ishihara, A., Segall, J. E., Block, S. M. & Berg, H. C. (1983) J. Bacteriol. 155, 228237.

22. Weis, R. M. & Koshland, D. E. (1990) J. Bacteriol. 172, 10991105.

23. Stewart, R. C. & Dahlquist, F. W. (1987) Chem. Rev. 87, 9971025.

24. Amsler, C. D. & Matsumura, P. (1995) in Two-Component SignalTransduction, eds. Hoch, J. A. & Silhavy, T. J. (Am. Soc. Micro-biol., Washington, DC), pp. 89 103.

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7/11Nature Macmillan Publishers Ltd 1997

Robustness in simplebiochemical networksN. Barkai & S. LeiblerDepartments of Physics and Molecular Biology, Princeton University, Princeton, New Jersey 08544, USA. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . .

Cells use complex networks of interacting molecular componentsto transfer and process information. These computationaldevices of living cells1 are responsible for many importantcellular processes, including cell-cycle regulation and signaltransduction. Here we address the issue of the sensitivity of thenetworks to variations in their biochemical parameters. Wepropose a mechanism for robust adaptation in simple signaltransduction networks. We show that this mechanism applies inparticular to bacterial chemotaxis 27 . This is demonstrated withina quantitative model which explains, in a unied way, many aspects of chemotaxis, including proper responses to chemicalgradients 812. The adaptation property 10,1316 is a consequence of the networks connectivity and does not require the ne-tuning

of parameters. We argue that the key properties of biochemicalnetworks should be robust in order to ensure their properfunctioning.

Cellular biochemical networks are highly interconnected: a per-turbation in reaction rates or molecular concentrations may affectnumerous cellular processes. The complexity of biochemical net-works raises the question of the stability of their functioning. Onepossibility is that to achieve an appropriate function, the reactionrate constants and the enzymatic concentrations of a network needto be chosen in a very precise manner, and any deviation from thene-tuned values will ruin the networks performance. Another

possibility is that the key properties of biochemical networks arerobust; that is, they are relatively insensitive to the precise values of biochemical parameters. Here we explore the issue of robustness of one of the simplest and best-known signal transduction networks: abiochemical network responsible for bacterial chemotaxis. Bacteriasuch as Escherichia coliare able to sense (temporal) gradients of chemical ligands in their vicinity 2. The movement of a swimmingbacterium is composed of a series of smooth runs, interrupted by events of tumbling, in which a new direction for the next run ischosen randomly. By modifying the tumbling frequency, a bac-terium is able to direct its motion either towards attractants or away from repellents.A well established feature of chemoxisis itsproperty of adaptation 10,1316: the steady-state tumbling frequency in ahomogeneous ligand environment is insensitive to the value of ligand concentration. This property allows bacteria to maintaintheir sensitivity to chemical gradientsover a wide range of attractantor repellent concentrations.

The different proteins that are involved in chemotactic responsehave been characterized in great detail, and much is known aboutthe interactions between them (Fig. 1a). In particular, the receptorsthat sense chemotactic ligands are reversibly methylated. Biochem-ical data indicate that methylation is responsible for the adaptationproperty: changes in methylation of the receptor can compensate

for the effect of ligand on tumbling frequency. Theoretical modelsproposed in the past assumed that the biochemical parameters arene-tuned to preserve the same steady-state behaviour at differentligand concentrations 17,18. We present an alternative picture inwhich adaptation is a robust property of the chemotaxis networkand does not rely on the ne-tuning of parameters.

We have analysed a simple two-state model of the chemotaxisnetwork closely related to the one proposed previously 2,19. The two-state model assumes that the receptor complex has two functionalstates: active and inactive. The active receptor complex shows akinase activity: it phosphorylates the response regulator molecules,

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Figure 1 a , The chemotaxis network. Chemotactic ligands bind to specializedreceptors (MCP) which form stable complexes (E), with the proteins CheA andCheW. CheA is a kinase that phosphorylates the response regulator, CheY,whose phosphorylated form (CheY p) binds to the agellar motor and generatestumbling.Bindingof the ligand to the receptormodies thetumblingfrequencybychanging the kinase activity of CheA. The receptor can also be reversiblymethylated. Methylation enhances the kinases activity and mediates adaptationto changes in ligand concentration. Two proteins are involved in the adaptationprocess: CheR methylates the receptor, CheB demethylates it. A feedbackmechanism is achieved through the CheA-mediated phosphorylation of CheB,whichenhances its demethylationactivity. b , Mechanismfor robust adaptation. Eis transformed to a modied form, E m , by the enzyme R; enzyme B catalyses thereverse modication reaction. E m is active with a probability of a m ( l), whichdepends on the input level l. Robust adaptation is achieved when R works atsaturation and B acts only on the active form of E m . Note that the rate of reversemodication is determined by the systems output and does not depend directlyon the concentration of E m (vertical bar at the end of the arrow).

Figure 2 Chemotactic response and adaptation. The system activity, A, of amodel system (thereference system described in Methods)which wassubject toa series of step-like changes in the attractant concentration, is plotted as afunction of time. Attractant was repeatedly added to the system and removedafter 20 min,with successiveconcentrationsteps of l of 1, 3, 5 and 7 M. Note theasymmetry to addition compared with removal of ligand, both in the responsemagnitude and the adaptation time. The chemotactic drift velocity of this systemis presentedin theinset. Inset: thedifferent curves correspond to gradients V l 0,0.01, 0.025 and 0.05 M/ m. An average change in receptor occupancy of lessthan 1% per second is sufcient to induce a mean drift velocity of the order ofmicrons per second.



which then bind to the motors and induce tumbling. The receptorcomplexes can be either in the active or in the inactive state,although with probabilities that depend on both their methylationlevel and ligand occupancy. The average complex activity can beconsidered as the output of the network, whereas its input is theconcentration of the ligand. A quantitative description of the modelconsists of a set of coupled differential equations describing inter-actions between protein components (Box 1).

The two-state model correctly reproduces the main features of bacterial chemotaxis. When a typical model system is subject to astep-like change in attractant concentration, l (Fig. 2), it is able torespond and to adapt to the imposed change. The adaptation isnearly perfect for all ligand concentrations. The addition (removal)of attractant causes a transientdecrease (increase) in system activity,and thus of tumbling frequency. We observe a strong asymmetry in

the response to the addition compared with the removal of ligand.This asymmetry has been observed experimentally 14. The chemo-tactic response of the system has been measured by the average driftvelocity in the presence of a linear gradient of attractant (Fig. 2,inset). Thesystem is very sensitive: an average changein thereceptoroccupancy of 1% per second is enough to induce a drift velocity of

1 micron per second.Figure 3a illustrates the most striking result of the model: we have

found that the system shows almost perfect adaptation for a widerange of values of the networks biochemical parameters. Typically,one can change simultaneously each of the rate constants several-fold and still obtain, on average, only a few per cent deviation fromperfect adaptation. For instance, over 80 per cent of model systems,obtained from a perfectly adaptive one by randomly changing all of its biochemical parameters by a factor of two, still show 15%

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Box 1 Two-state model of the bacterial chemotactic networkThe main component of the two-state model 2,19 is the receptor complex,MCP CheA CheW (Fig. 1a), considered here as a single entity, E. Thecomplex is assumed to have two functional statesactive and inactive. A

receptor complex in the active state shows a kinase activity of CheA; byphosphorylating the response regulators, CheY, it sends a tumbling signal tothemotors.The output of thenetworkis thusthe averagenumberof receptorsin the active state, the system activity A. It is assumed that this quantitydetermines the tumbling frequency of the bacteria. The transformationbetween A and the tumbling frequency depends on the kinetics of CheYphosphorylation and dephosphorylation, as well as on the interaction ofCheY with the motors, which are not considered explicitly in the presentmodel.

The receptor complexes are assumed to exist in different forms. Considera complex methylatedon m sites ( m 1; M). Such a complex caneitherbeoccupied or unoccupied by the ligand. We denote the concentration of thesecomplexesby E0 m and Eu m , respectively. Eachform of thereceptorcomplex canbe in theactive state witha probability dependingon bothits methylationlevel

andits ligand occupancy. Weassumethat an occupied receptorcomplexhasthe probability 0 m of being in the activestate; for an unoccupied receptor, thisprobabilityis m. If l is theligandconcentration, B(R)the concentrationof CheB(CheR), and { Eu m B} the concentration of the Eu mCheB complex and so on, themodel reactions can then be illustrated schematically (see gure).

The differential equations describing our model can be written in astandard way from the gure. For instance, the kinetic equation for Eu m is

dE u m dt

k l lEu m k l E

0 m

1 d m ;0 1

ab a m Eu m B db E

u m B

8 9

kr Eu m 1 R

8 9 2 3

1 d mM 1

a ra m Eu m R a r 1 a m

1

Eu m R db Eu m R

8 9

kb Eu m 1 B

8 9

h i

m 1; M 1

The presence of a m in the equation is due the fact that CheB demethylatesonlythe active receptors; we havealso includedtwo different association rateconstantsof CheR tothe active( a r) andtheinactive ( a r) receptors(see below).d jk is the Kroneckers delta ( d jk 1, when j k, and is zero otherwise). Similarequations can be written to describe kinetics of { Eu m B}, { Eu m R}, E0 m , { E0 m B} and{ E0 m R}, with additional parameters a

0 m dening the probabilities of E0 m to be in

the active state. For xed a m and a 0 m the biochemical parameters of thissystem include nine different rate constants ( k l; k l; a r; a r ; d r; k r; a b ; d b ; k b )and three enzyme concentrations (total concentrations of CheR, CheB andreceptor complexes).

The present model is by no means the only two-state model of thechemotactic network that exhibits robust adaptation and proper chemotacticresponse. Rather, it is one of the simplest variants that is consistent with theexperimental data on the response and adaptation of wild-type E. coli . Themain assumptions underlying this model are as follows.X The input to the system is the ligand concentration; rapid binding (andunbinding) of the ligand to the receptor induces an immediate change in the

activity of the complex. For simplicity, the binding afnity is assumed to beindependent of the receptors activity and its degree of methylation. Thisassumption can be relaxed without affecting the main conclusions of ourmodel.X The methylation and demethylationreactionoccur on slower timescales.A central assumption is that CheB can only demethylate active receptors. Inaddition, the demethylation rate constants do not depend explicitly either onligand occupancy or on the methylation of the receptor, so that all activereceptors are demethylated at the same rate. In the variant of the modeldiscussed here, the phosphorylation of CheB is not considered explicitly; inmolecular terms, we assume that the phosphorylated form of CheB, CheB p ,does not move freely in the cell. Rather,. a CheB p molecule can onlydemethylate thesame receptorthat hasphosphorylated it.We note,however,that this assumption can also be readily relaxed (N.B. et al. , manuscript in

preparation). Robust adaptation is maintained as long as both CheB andCheB p demethylate only the active receptors.X The methylating enzyme, CheR, acts both on active and inactivereceptors. Here we assume that the association rate constant for thisreaction depends only on the activity of the receptor ( a r for inactive, a r foractive), whereas the dissociation rate constant d r and the catalytic rateconstant kr are the same for all forms of the receptor. This assumption canagain be relaxed in various ways. In particular, the accumulated biochemicaldata indicate that CheR works at saturation and operates at its maximalvelocity. In thiscase, theconclusions of ourmodel arenot altered,even if dr, a rand a r depend on the ligand occupancy and on the methylation level 21.



deviation from perfect adaptation (Fig. 3a, lower panel). Whenvaried separately, most of the rate constants may be changed by several orders of magnitude without inducing a signicant devia-tion from perfect adaptation.

In our model we have assumed MichaelisMenten kinetics forsimplicity. However, we have found that cooperative effects in theenzymatic reactionscan be addedwithout destroying the robustnessof adaptation. Similarly, robust adaptation is obtained for systemswith different numbers of methylation sites. Multiple methylationsites are thus not required for robust adaptation, butpossibly areforallowing strong initial responses for a wide range of attractant andrepellent stimuli (N.B. et al., manuscript in preparation).

The adaptation itself, as measured by its precision (Fig. 3a), isthus a robust property of the chemotactic network. This does notmean, however, that all the properties are equally insensitive tovariations in the network parameters. For instance, Fig. 3b showsthat the adaptation time, t , which characterizes the dynamics of relaxation to the steady-state activity, displays substantial variationsin the altered systems. Robustness is thus a characteristic of specicnetwork properties andnot of thenetwork as a whole:whereas someproperties are robust, others can show sensitivity to changes in thenetwork parameters.

Plots similar to the ones depicted in Fig. 3 can be obtained in

quantitative experiments. A large collection of chemotacticmutants can be analysed for variations in the biochemical rateconstants of the chemotactic network components. Alternatively,the rate constants of the enzymes could be systematically modiedor their expression varied. At the same time, their various physio-logical characteristics can be measured, such as steady-statetumbling frequency, precision of adaptation, adaption time, andso on. In this way, thepredictions of themodel can be quantitatively checked.

What features of the chemotactic network make the adaptationproperty so robust? We propose here a general and simple mechan-ism for robust adaptation. Let us introduce this mechanism for oneof the simplest networks (Fig. 1b), which can be viewed either as anadaptation module, or, as a simplifying reduction of a more

complex adaptive network, such as the one presented for bacterialchemotaxis. Consideran enzyme, E, which is sensitiveto an externalsignal l , such as a ligand. Each enzyme molecule is at equilibriumbetween two functional states: an active state, in which it catalyses areaction,and an inactive state, inwhich it doesnot. The signal level l affects the equilibrium between two functional states of the enzyme:we suppose that a change in l causes a rapid response of the systemby shifting this equilibrium. Thus, l is the input of this signaltransduction system and the concentration of active enzymes (thatis, the system activity, A) can be considered as its output. Theenzyme E can be reversibly modied, for example by addition of methyl or phosphate groups. The modication of E affects theprobabilities of the active and inactive states, and hence cancompensate for the effect of the ligand. In general, then, Al a l E a ml Em , where Em and E are the concentrationsof the modied and unmodied enzyme, respectively, and a m(l )and a (l ) are the probabilities that the modied and unmodiedenzyme is active. After an initial rapid response of the system to achange in the input level, l , slower changes in the system activity proceed according to the kinetics of enzyme modication.

The system is adaptive when its steady-state activity, Ast, isindependent of l . A mechanism for adaptation can be readily obtained by assuming a ne-tuned dependence of the biochemicalparameters on the signal level, l . This kind of mechanism has beenproposed for an equivalent receptor system 17,18. A mechanism forrobust adaptation, on the other hand, can be obtained when therates of the modication and the reverse-modication reactionsdepend solely on the system activity, A, and not explicitly on theconcentrations Em and E. This system can be viewed as a feed-backsystem, in which the output A determines the rates of modication

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Figure 3 Robustness of adaptation. a , The precision of adaptation, P, and b ,adaptation time, t , to a step-like addition of saturating amount of attractant areplotted as a function of the total parameter variation k, for an ensemble of modelsystems (see Methods).The time evolution of the system activity A is depicted inthe inset for the reference system (solid curve) and for an altered model system,obtained by randomly increasing or decreasing by a factor of two allbiochemicalparametersof the reference system (dashed curve). Each point in the top graphsin a and b corresponds to a different altered system, out of the total number of6,157. The reference system is denoted by a black diamond; the particular alteredsystem from the inset is denoted by an open square. Bottom graphs: a , theprobability that P is larger than 0.95; b , the probability that t deviates from theadaptation time of the reference system ( 10min) by less than 5% (solid curve)and by a factor 5 (dashed curve). c , Individuality in the chemotaxis model. Theinverse steady-state activity A 1 is plotted as a function of the adaptation time, t .Each point represents an altered system, obtained from the reference system(arrow) by varying the concentration of CheR (between 100 to 300 molecules percell).



reactions, which in turn determinethe slow changes in A. With suchactivity-dependent kinetics, the value of the steady-state activity, Ast, is independent of the ligand level, therefore the system isadaptive. Activity-dependent kinetics can be achieved in a variety of ways. As a simple example, consider a system for which only themodied enzyme can be active ( a 0); the enzyme R, whichcatalyses the modication reaction E Em , works at saturation,and the enzyme B, whichcatalyses the reverse-modication reactionEm E, can only bind to active enzymes. In this case, the modica-tion rate is constant at all times, whereas the reverse modicationrate is a simple function of the activity

dEmdt

V Rmax V Bmax

AK b A

1

where V Bmax and V Bmax are the maximal velocities of the modicationand the reverse-modication reactions, respectively, and K b isthe Michaelis constant for the reverse modication reaction; wehave assumed V Rmax V Bmax . For simplicity, we have assumedthat the enzymes follow MichaelisMenten (quasi-steady-state)kinetics. The functioning of the feedback can now be analysed: thesystem activity is continuously compared to a reference stead-statevalue

Ast K bV Rmax

V Bmax V Rmax:

For A Ast, the amount of modication increases, leading to anincrease in A; for A Ast, the modication decreases, leading to adecrease in A. In this way, the system always returns to its steady-state value of activity, exhibiting adaptation. Moreover, with theseactivity-dependent kinetics, the adaptation properties is insensitiveto thevalues of systemparameters (such as enzymeconcentrations),so adaptation is robust.

Note, however, that the steady-state activity itself, which is not arobust property of the network, depends on the enzyme concentra-tions. Thus, the mechanism presented here still provides a way to

control the system activity on long timescales, for example by changing the expression level of the modifying enzymes whilepreserving adaptation itself on shorter timescales.

A quantitative analysis demonstrates that, on methylation time-scales, the kinetics of the two-state model of chemotaxis can, for awide range of parameters, be mathematically reduced to thesimpleactivity-dependent kinetics shown in equation (1) (N.B. et al.,manuscript in preparation). Robust adaptation thus follows natu-rally as consequenceof the simple mechanism described above. Thedeviations from perfect adaptation (Fig. 3) are in fact connected todepartures from the assumptions underlying this mechanism (suchas V Rmax V Bmax). This simple mechanism suggests that the variousdetailed assumptions about the systems biochemistry can be easily altered, provided that the activity-dependent kinetics of receptormodications is preserved. Allvariantsof themodelobtained in thisway still exhibit robust adaptation (N.B. et al., manuscript inpreparation).

Two main observations argue in favour of a robust, rather than ane-tuned, adaptation mechanism for chemotaxis. First, the adap-tation property is observed in a large variety of chemotacticbacterial populations. It is easier to imagine how a robust mechan-ism allows bacteria to tolerate genetic polymorphism, which may change the networks biochemical parameters. In addition, ingenetically identical bacteria some features of the chemotacticresponse, such as the values of adaptation time and of steady-state tumbling frequency, vary signicantly from one bacterium toanother, while the adaptation property itself is preserved 20. Thisindividuality can be readily explained in the framework of thepresent model. The concentrations of some cellular proteins, forexample the methylating enzyme CheR, are very low 2, and thus may

be subject to considerable stochastic variations. In consequence,both adaptation time and steady-state tumbling frequency, whichare not robust properties of the network, should vary signicantly.Moreover, the present model predicts that both these quantitiesshould show a strong correlation in their variation (Fig. 3c), whichhas been observed experimentally 20.

How general are the results presented here? In addition toexplaining response and adaptation in chemotaxis, the presentmodel accounts, in a unifying way, for other taxis behaviour of bacteria mediated by the same network. Indeed, as the networksdynamics is solely determined by the system activity, the system willrespond and adapt to any environmental change that affects thisactivity. Mechanisms of robust adaptation similar to the oneintroduced above could apply to a widerclass of signal transductionnetworks. Robustness may be a common feature of many key cellular properties and could be crucial for the reliable performanceof many biochemical networks. Robust properties of a network willbe preserved even if its components are modied through randommutations, or are produced in modied quantities. Systems whosekey properties are robust could have an important advantage inhaving a larger parameter space in which to evolve and to adjust toenvironmental changes.

The degree of robustness in many biochemical networks can be

quantitatively investigated. This can be achieved by characterizing abehavioural, a physical or biochemical property while varyingsystematically the expression level and the rate constants of thenetworks components.

The complexity of biological systems introduce several concep-tual and practical difculties, however. Among the most importantis the difculty of isolating smaller subsystems that could beanalysed separately. For instance, in the present analysis, we haveneglected the existence of different types of receptors and any crosstalk between them. We have also disregarded the interactionsbetween the chemotaxis network and other components of the cell.In addition, the complexity and stochastic variability of biologicalnetworks may preclude their complete molecular description. Rateconstants and concentrations of many enzymes can only be mea-

sured outside their natural cellular environment and many othernetwork parameters remain unknown. Robustness may provide away out of both these quandaries: robust properties do not dependon the exact values of the networks biochemical parameters andshould be relatively insensitive to the inuence of the othersubsystems. It should then be possible to extract some of theprinciples underlying cell function without a full knowledge of the molecular detail.. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . .

Methods

Numerical integration of the kinetics equations dening the two-state model(see Box 1) was used to investigate its properties. Computer programs in C language were executed on an SGI (R4000) workstation using a standardroutine ( lsodefrom LLNL). Typical CPU time for nding a numerical solutionof a model system is of the order of 1 min. A particular model system wasobtained by assigning values to the rate constants and the total enzymeconcentrations. Most of our results were obtained for a reference systemdened by the following biochemical parameters: the equilibrium bindingconstant of ligand to receptor is 1 M and the time constant for the reaction is1ms (k1 1 ms 1 M 1 , k 1 1 ms 1). CheR methylates both active andinactive receptors at the same rate, with a Michaelis constant of 1.25 M, and atime constant of 10s ( ar a r 80 s 1 M 1, dr 100s 1, kr 0:1 1),CheB (CheBp) demethylates only active receptors with a Michaelis constantof 1.25 M and a time constant of 10 s ( ab 800s 1 M 1 , db 1;000s 1,kb 0:1 s 1). The number of enzyme molecules per cell are: 10,000 receptorcomplexes, 2,000 CheB and 200 CheR (cell volume of 1:4 10 15l). Theprobabilities that a receptor with m 1; 4 methylated sites is in its activestate are: 1 0:1, 2 0:5, 3 0:75, 4 1 if it is unoccupied, and 01 0,

02 0:1, 03 0:5, 04 1 if it is occupied.

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a model of excitation and adaptation in bacterial chemo tax is

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