An Allosteric Model for Transmembrane Signaling in...

doi:10.1016/j.jmb.2004.08.046 J. Mol. Biol. (2004) xx, 1–13

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An Allosteric Model for Transmembrane Signaling inBacterial Chemotaxis

Christopher V. Rao1,2*, Michael Frenklach2,3 and Adam P. Arkin1,2,4

1Department of BioengineeringUniversity of CaliforniaBerkeley, CA 94720, USA

2Lawrence Berkeley NationalLaboratory, Berkeley, CA 94720USA

3Department of MechanicalEngineering, University ofCalifornia, Berkeley, CA 94720USA

4Howard Hughes MedicalInstitute, Berkeley, CA 94720USA

0022-2836/$ - see front matter q 2004 E

Abbreviation used: TD, trimer ofE-mail address of the correspond

[email protected]

Bacteria are able to sense chemical gradients over a wide range ofconcentrations. However, calculations based on the known number ofreceptors do not predict such a range unless receptors interact with oneanother in a cooperative manner. A number of recent experiments supportthe notion that this remarkable sensitivity in chemotaxis is mediated bylocalized interactions or crosstalk between neighboring receptors. Anumber of simple, elegant models have proposed mechanisms for signalintegration within receptor clusters. What is a lacking is a model, based onknown molecular mechanisms and our accumulated knowledge ofchemotaxis, that integrates data from multiple, heterogeneous sources. Toaddress this question, we propose an allosteric mechanism for transmem-brane signaling in bacterial chemotaxis based on the “trimer of dimers”model, where three receptor dimers form a stable complex with CheW andCheA. The mechanism is used to integrate a diverse set of experimentaldata in a consistent framework. The main predictions are: (1) trimers ofreceptor dimers form the building blocks for the signaling complexes; (2)receptor methylation increases the stability of the active state and retardsthe inhibition arising from ligand-bound receptors within the signalingcomplex; (3) trimer of dimer receptor complexes aggregate into clustersthrough their mutual interactions with CheA and CheW; (4) cooperativityarises from neighboring interaction within these clusters; and (5) clustersize is determined by the concentration of receptors, CheA, and CheW. Themodel is able to explain a number of seemingly contradictory experimentsin a consistent manner and, in the process, explain how bacteria are able tosense chemical gradients over a wide range of concentrations bydemonstrating how signals are integrated within the signaling complex.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: chemotaxis; allostery; receptor clustering; signal transduction;Monte Carlo
*Corresponding author
Introduction

Chemotaxis is the process by which cells sensechanges in their chemical environment and move tomore favorable conditions.1 In enteric bacteria suchas Escherichia coli, cells bias their motion in chemicalgradients by transitioning between straight runsand re-orientating tumbles through the rotation ofthe flagella that dot their surface. The primarysignal transduction module for the chemotaxispathway involves a stable, ternary signaling com-plex comprising transmembrane receptors, CheW

lsevier Ltd. All rights reserve

receptor dimers.ing author:

adaptor proteins, and CheA histidine kinases,2

where both the receptors and the CheA kinaseform homodimers. E. coli cells modulate the fre-quency of runs and tumbles by regulating the rateof CheA autophosphorylation.3 The phosphoryl-group on CheA is transferred to a soluble, cyto-plasmic response regulator (CheY) that interactswith the flagellar motor and increases the likelihoodof re-orienting tumbles.4 A dedicated phosphataseCheZ enhances the rate of CheY dephosphoryla-tion. E. coli cells respond only to temporal changesin the concentration of chemoeffectors and perfectlyadapt their sensory response in reference to thebackground stimuli. In particular, their stimulatedresponse always returns to pre-stimulus valuesdespite the sustained presence of attractants orrepellents. Methylation of the receptors increases

d.

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the rate of CheA autophosphorylation anddecreases the sensitivity of receptors to attractants.The antagonizing action of the receptor methyl-transferase CheR and methylesterase CheB adaptsthe sensory response by adding and removingmethyl groups on the chemoreceptors.5,6

While the basic mechanism for adaptation inenteric bacteria has been elucidated and numerousmathematical models have been proposed,7–12 oneunresolved question is how bacteria are able tosense chemical gradients over a wide range ofconcentrations.13,47 The signaling gain, measured asthe fractional change in signaling response perfractional change in ligand concentration, isroughly constant over a range of concentrationsspanning five orders of magnitude in vivo.14,15

However, in vitro data suggest only a limitedrange.5,16 The models based on these in vitro datapredict that the gain is constant over a range ofconcentrations spanning one to two orders ofmagnitude. This difference between the modelsand experiments, both in vitro and in vivo, demon-strates that the current understanding for chemo-taxis is incomplete and additional mechanisms arenecessary to explain the signaling gain.

As the ternary signaling complexes localize at thepoles of the cell,17–19 the most popular explanationfor sensitivity is that the receptor complexesaggregate to form higher-order structures. Onemodel, advocated by Bray and colleagues, proposesthat the chemoreceptors form a cooperative lat-tice.20–23 In their model, a single ligand-boundreceptor interacts with neighboring receptors, pro-pagating the inhibiting signal to adjacent receptorsin the lattice. While this model is appealing, as it canexplain how subtle changes in the concentration ofligand elicit large responses, there is little evidenceto suggest the existence of a globally cooperativelattice beyond polar localization. Furthermore, invivo and in vitro experiments paint a mixed pictureregarding cooperativity, where some experimentsmeasure little and others measure a lot.24

On the other hand, there is substantial experi-mental evidence to suggest localized interactionsamong the individual receptor dimers. The firstevidence is that the cytoplasmic domains of theserine receptor crystallize to form trimers ofreceptor dimers.25 Using this crystal structurealong with the atomic structures for CheW andCheA, Shimizu and colleagues proposed a struc-tural model for the ternary signaling complexwhere three receptor dimers form a complex withthree CheW monomers and three CheA dimers.23

The second line of evidence is that mixed receptorsinteract with one another. There are five differentkinds of chemoreceptors in E. coli, each specializedto particular chemical signals. Using genetic andcrosslinking experiments, Parkinson and colleaguesdemonstrated that the serine (Tsr) and aspartate(Tar) receptors interact with one another in com-plexes consistent with the “trimer of dimers”model.26,27 In DcheBR cells, the sensitivity toaspartate is amplified when Tsr is deleted.15,28

Using the data reported by Sourjik & Berg,15

Mello & Tu29 argued that the interactions betweendifferent kinds of receptors enable cells to sensegradients over a wide range of concentrations.Further evidence is provided from the analysis ofthe low-abundance receptor Trg; multivalentligands directed towards the Trg receptor amplifythe response of the Tsr receptor to serine.30

In a recent series of experiments,31 Sourjik & Bergmeasured significant cooperativity in vivo. Remark-ably, they were able to modulate the cooperativityby independently varying the expression of recep-tors, CheW, and CheA. Their results demonstratethat receptors potentially interact with each other inlarge complexes, though the structure of thesecomplexes is still unknown. Their results alsopotentially explain the discrepancy between pre-vious cooperativity measurements, if we attributethe differences to disparities in expression. Finally,Sourjik & Berg were able to establish that differentkinds of receptors interact with one another in adose-dependent fashion, clearly establishing theexistence of cooperative structures.

A number of simple, elegant models haveproposed mechanisms for signal integration withinreceptor clusters.20,29 What is lacking is a model,based on known molecular mechanisms and ouraccumulated knowledge of chemotaxis, that inte-grates data from multiple, heterogeneous sources.To address this question, we propose a two-stateallosteric mechanism based on the trimer of dimersstructural model to explain how bacteria sensechemical gradients over a wide range of concen-trations. We explore this allosteric mechanism usingnumerical simulations and demonstrate that theassociated model is able to explain a number of invitro and in vivo experiments. The main predictionsare: (1) trimers of receptor dimers form the buildingblocks for the signaling complexes; (2) receptormethylation increases the stability of the active stateand retards the inhibition arising from ligand-bound receptors within the signaling complex; (3)trimer of dimer receptor complexes aggregate intoclusters through their mutual interactions withCheA and CheW; (4) cooperativity arises fromneighboring interaction within these clusters; and(5) cluster size is determined by the concentration ofreceptors, CheA, and CheW. Numerous other modelshave advanced alternative hypotheses,7,20,21,29,32,33

and the model proposed here does not necessarilyargue against any of these mechanisms. Rather, ourgoal was to explore a molecular mechanism involv-ing a minimal number of assumptions that isconsistent with what is known about chemotaxis,able to explain both the in vitro and in vivo data, anddoes not introduce additional mechanisms such aslong-range signaling20 or mechanisms involvingdynamics clustering or feedback.7,22,32

Theory: Assumptions and Models

We propose a two-state allosteric model for

Figure 1. The trimer of receptor rimers (TD) allostericmodel. The TD model assumes that three receptor dimersassociate with CheW and CheA to form a stable complex.The model assumes that the complex exists either in anactive (circles) or inactive conformation (squares). Eachdimer subunit can bind one ligand (shaded circle orsquare). The arrows denote the transitions associatedwith ligand binding. The transitions between active andinactive conformations are omitted from the Figure foraesthetics.

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receptor signaling akin to the Monod, Wyman, andChangeux (MWC) allosteric model.34 The allostericmechanism invokes the molecular model proposedby Shimizu and colleagues,23 where three receptordimers form a stable complex with three CheWproteins and three CheA dimers. Using the meta-phors of allostery, the model assumes that the entirecomplex exists in either a tense or relaxed form,where the tense form corresponds with the activestate and the relaxed form corresponds with theinactive state. Implicit in the allosteric model is theassumption that the associated CheA kinaseswithin the complex act in unison. The rate ofCheA autophosphorylation is proportional to theprobability that the receptor complex exists in anactive state. The model assumes that receptormethylation increases the stability of the active (ortense) state, whereas ligand destabilizes the activestate and increases the probability that the complexwill adopt an inactive (or relaxed) state. Methyl-ation increases the stability of the receptor complexby retarding the destabilizing effect of ligand. A keyassumption is that the sensitivity to ligand isprimarily determined by the cooperative inter-actions of the receptors within the complex.

The model is supported by the following exper-iments. Numerous lines of experimental evidenceindicate that transmembrane signaling is accom-plished by a subtle conformational shift betweenthe helices of a receptor dimer35 and the receptordimers within the signaling complex.36 Usingmolecular models for the serine receptor andcrystallography data, it was shown that inactivereceptors are more dynamic than active receptors interms of the average temperature factor.22 In theproposed mechanism, the dynamic (inactive) statecorresponds with the relaxed state. The confor-mational shift drives the complex towards therelaxed state by weakening the interactions withinthe complex. In vitro studies with purified com-plexes involving identical receptors demonstratethat methylation does not significantly change thesensitivity to ligand.5,28,37 Instead, receptor methyl-ation only increases the activity of the CheAkinase.38 With in vivo studies involving multiplekinds of receptors, methylation changes the sensi-tivity by several orders of magnitude.15,28 Theseresults suggest that the sensitivity is modulatedthrough the interactions among different kinds ofreceptors rather than by changing the affinity forligand.

Sourjik & Berg also used an allosteric modelbased on the MWC formulation to analyze theirdata.31 However, they only considered a genericMWC model and did not investigate how differentkinds of receptors within the complex alter themodel. They also did not attempt to adapt theirmodel to the particulars of signaling in bacterialchemotaxis.

To analyze the data, we first considered a modelwhere three receptor dimers associate with CheWand CheA to form stable, independent signalingcomplexes (Figure 1). The trimer of dimers (TD)

model assumes that the signaling complex adoptseither an active (A) or an inactive (I) conformation.Each receptor dimer within the signaling complex iseither bound with ligand (1) or free (0). Asindividual receptor dimers exhibit half-site satur-ation, the model assumes that each dimer subunitbinds only one ligand.39 For example, the notationA010 designates an active receptor complex withthe first and third subunits free and the secondbound with ligand. Thus, the model has a total ofeight state variables (Figure 1). The model includesfour possible transitions among the states. The firstis the transition from an inactive conformation (I) toan active conformation (A) and the second is thereverse transition from an active conformation (A)to an inactive complex (I). As these transitions areisomerization reactions, they were modeled as first-order reactions. For example, the probability of atransition from an inactive to an active confor-mation is proportional to the probability that thecomplex adopts an inactive conformation. The thirdtransition occurs when ligand binds the receptor,and the fourth transition occurs when the receptorreleases the ligand. We assume that there is excessligand and modeled the ligand binding reactionsalso as first-order processes. If we invoke detailedbalancing, then there are a total of 11 freeparameters in the model that characterize thesteady-state behavior. If we include dynamics,then we double the number of free parameters aswe need to account for both the forward and reverserates rather than just their ratios at steady-state.Consequently, we focused on the steady-statebehavior as there are no dynamic data to test themodel against nor do dynamics significantly changethe predictions (results not shown).

For each ligation state in the model, we cancharacterize the equilibrium constant between activeand inactive complexes in terms of free energy:

KðAxyz# IxyzÞ ¼ expðKDGmxyz=RTÞ (1)

Figure 2. (a) Isolated trimer of dimers signalingcomplex. The dark gray circles denote receptor dimers,the white squares denote CheW, and the light gray circlesdenote CheA dimers. (b) Clustered trimer of dimerssignaling complex. The dark gray circles denote receptordimers, the white squares denote CheW, and the lightgray circles denote CheA dimers. Note that some CheAdimers interact with multiple receptor TD complexesdepending on the structure of the cluster. Cooperativity isassumed to increase with the number of interacting TDreceptor complexes.


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where the subscript xyz denotes an arbitrary ligationstate and the superscript m is the average number ofresidues methylated (the effect of differential meth-ylation is ignored for simplicity). In terms of thephysical picture, the free energy characterizes thestability of the active (or tense) state for each ligationand methylation state. Receptor methylationincreases the change in free energy (e.g.DGi

xyz%DGiC1xyz ). As the number of ligand-bound

subunits increases, the free energy decreases. Forexample:

DGm000RDGm

100RDGm110RDGm

111 (2)

For complexes with the same type of receptor, themodel assumes that the subunits are identical:DGm

100ZDGm010ZDGm

001 and DGm110ZDGm

101ZDGm011,

and thus reduces the number of free parametersto five. For notational convenience, we makethe definitions: DGm

0 bDGm000, DGm

1 bDGm100,

DGm2 bDGm

110, and DGm3 bDGm

111. In addition tothe change in free energy, the model distinguishesbetween the subunits in terms of ligand binding.The binding coefficients, for example:

KðA000#A100Þ (3)

and:

KðA000#A010Þ (4)

are not necessarily equal. They are equal only whenthe receptors are identical (e.g. Tar only) anddifferent when the receptors are not (e.g. Tar andTsr).

When the receptors are identical, the modelassumes that methylation does not change theaffinity for ligand. However, we needed to relaxthis assumption somewhat when applying themodel to experiments involving mixed complexesof receptors. We did this by introducing a parameterc less than or equal to 1 such that:

cKðA0yz#A1yzÞ ¼ KðAx0z#Ax1zÞ

¼ KðAxy0#Axy1Þ (5)

or:

cKðA0yz#A1yzÞ ¼ cKðAx0z#Ax1zÞ

¼ KðAxy0#Axy1Þ (6)

The parameter c measures the change in ligandaffinity when the receptor complexes are mixed. Itsuffices to consider only the high-affinity receptorfor a particular ligand in a mixed complex. If theparameter c is less than 1, then the ligand affinity isdecreased in mixed receptor complexes. As with theconformational changes, we parameterize the par-ameter c in terms of free energy:

cZ expðKDGmc =RTÞ (7)

By introducing the parameter c, we are effectivelyassuming that receptor methylation changes theaffinity for ligand.

The TD model is able to explain a number of

experiments. However, in those experiments wherethe measured cooperativity is greater than 3, the TDmodel no longer works. To include these exper-iments, we needed to assume that the receptorsassemble into large clusters. That said, we did notwish to abandon the TD model as there issubstantial evidence in favor of it. To reconcilethese apparent differences, we invoked a math-ematical model based on the structural modelproposed by Shimizu and colleagues. In theShimizu model,23 TD receptor complexes clustertogether through their shared interactions withCheW and CheA. We extended their model andassumed that each CheA dimer is able to interactwith three, rather than two, TD receptor complexeswith CheWas the glue in between (Figure 2(a)). Thisextension was necessary for the model to attain Hillcoefficients greater than 6. A key element of thismodel is that the TD receptor complex is thesmallest functional unit of receptors in a signalingcomplex.

There are two components to the clustered TDmodel. The first component dealt with the size andstructure of the cluster (Figure 2(b)). To modelcluster formation, we extended the lattice modelproposed by Bray and colleagues40 and simulatedcluster formation using the Metropolis algorithm.In this model, TD receptor complexes randomlydiffuse and rotate on a hexagonal lattice. CheW can


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bind to the TD receptor complexes with interactionenergy DGRW and CheA dimers can bind to CheWwith interaction energy DGAW. Clusters form whenmultiple TD receptor complexes bind to the sameCheA dimer through their shared interactions withCheW. We were interested particularly in theaverage number of TD receptor complexes boundto a given CheA dimer, as it provides a measure ofcooperativity.

The second component of the model is theallosteric mechanism governing signal transductionin the cluster. We assumed that the state of the CheAkinase, being either active or inactive, is determinedby the ligation and methylation states of the TDreceptor complexes coupled to it. If a single TDcomplex is clustered to a given CheA dimer, thenthe clustered TD model is identical with theallosteric mechanism discussed previously forisolated TD receptor complexes. If two or threeTD complexes are clustered to a given CheA dimer,then the model extends the allosteric mechanism byincreasing the number of receptor subunits fromthree to six or nine (depending on the clusterconfiguration) with appropriate corrections for thetransitions among the different states. To limit theexplosion in the number of free parameters, wemade a few simplifying assumptions discussed inResults. As a result of this clustering formulation,

Figure 3. Parametric sensitivity of homo-TD model. The n0 kcal/mol, DG1Z0 kcal/mol, DG2Zmax (DG1, 0 kcal/mremaining parameters were determined by the thermodynamgenerated by varying one parameter with respect to the nochanges when a specific parameter is increased.

signaling is localized and does not propagatethrough the entire cluster as with the spin-glassmodels proposed by Bray and colleagues.20,21 Long-range signaling is a legitimate possibility and weconsidered it, but did not find it necessary tointegrate the data.

The clustered model is supported by a number ofexperiments, in particular the experimental workby Studdert & Parkinson27 and Sourjik & Berg.31

Using a directed crosslinking approach based onthe structural model for the receptors,25 Studdert &Parkinson argued that trimers of receptor dimersform the basic building blocks for the signalingcomplexes, as their structural integrity does notdepend on CheW, CheA, methylation, or theligation state. The connections with the Sourjik &Berg experiments are discussed in Results.

Results

Parametric sensitivity of homo-TD model

If we fix the ligand affinity, then four parameterscharacterize the TD model when the subunits areidentical, which we designate the homo-TD model.To determine the parametric behavior of the homo-TD model, we plotted the fraction of active

ominal parameters are: KðA000#A100Þ ¼ 100 mM, DG0Zol), DG3ZK2.7 kcal/mol, and DGcZ0 kcal/mol. The

ic linkage conditions. The ligand-response curves wereminal parameters. The arrows indicate how the model


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complexes as a function of ligand concentration fordifferent values of the four parameters (Figure 3).The free energy for the unbound receptor complexDG0 increases the basal activity of the complex, as itincreases the probability that the unbound complexadopts an active conformation. The free energy DG0,however, does not change the apparent Km (definedas the half-maximal activity [L]0.5) or the coopera-tivity between subunits, because it only character-izes the behavior of the unbound complex. On theother hand, the free energy of the complex with oneligand-bound subunit DG1 or two ligand-boundsubunits DG2 simultaneously increases the appar-ent Km and the cooperativity, though it does notchange the basal activity. Note that the affinity forligand is not the same as the apparent Km in themodel, because the affinity is fixed separately fromthe free energies. As the free energies DG1 and DG2

increase, one or two ligand-bound subunits do notdeactivate the complex; only when two or moresubunits are bound with ligand is the complexdeactivated. The cooperativity increases with DG1

and DG2, because multiple ligand-bound subunitsare necessary to deactivate the complex, hence theincrease in the apparent Km and cooperativity. Thefree energy of the complex when all three subunitsare ligand-bound, DG3, increases the apparent Km

and the activity under saturating ligand concen-trations. The increase in the apparent Km is due todetailed balancing; increasing the free energy DG3

decreases the affinity for ligand in the inactive(relaxed) state.

Figure 4. (a) Homo-TD model applied to the aspartatereceptor data. The circle markers denote the model andthe continuous lines denote the data for the aspartatereceptor reported by Bornhorst & Falke.37 We averagedtheir data and presented their data as the solution of theequation A0K

n=ðKnCLnÞ, where for mZ1, A0Z0.06, KZ16 mM, and nZ1.6; for mZ2, A0Z0.16, KZ20 mM, and nZ2.2; for mZ3, A0Z0.35, KZ63 mM, and nZ2.4; and formZ4, A0Z0.6, KZ97 mM, and nZ2.2. (b) Parameters forhomo-TD model. For all methylation states, the affinityfor ligand is KðA000#A100Þ ¼ 1 mM.

Analysis of the aspartate receptor

We first compared the TD model to the datapublished by Bornhorst & Falke.16,37 Other studieshave also experimentally investigated the aspartatereceptor and reported similar findings,5,28 thoughthe Bornhorst & Falke study appears to be the mostcomprehensive. In their in vitro experiments, theymeasured the effect of methyl-aspartate on CheAkinase activity using modified receptors with aminoacid substitutions that mimic different methylationstates. There are at least three major trends in theirdata. The first is that the kinase activity increaseswith the number of modifications (akin to thenumber of residues methylated), the second is thatthe apparent Km also increases with the number ofmodifications, and the third is that there is limitedcooperativity that changes only moderately with themodification state. All three trends correlate nicelywith our previous analysis of the homo-TD model.

The TD model fits their data well (Figure 4(a)),where the estimated parameters (fitted by hand) areplotted in Figure 4(b). We see a clear trend in theparameters as a result of receptor methylation. Thefree energies DG0, DG1, DG2, and DG3 all increase asthe average number of modifications increases. Thistrend is expected, as methylation increases both thekinase activity and apparent Km. Furthermore,changes in free energy (DG0KDG1) and (DG1KDG2)decrease as the average number of modifications

increases. This trend explains how methylationincreases the apparent Km. When only a fewmethylation sites are modified, one or two ligand-bound subunits can destabilize the complex. As thenumber of modified sites increases, two or threeligand-bound subunits are needed to destabilize thecomplex. These results suggest, at least under theconditions measured by Bornhorst and Falke, thatthe signaling complex involves at least threereceptor dimer subunits. Obviously, larger com-plexes are possible, though there are no features inthe Bornhorst & Falke data to suggest morecomplex models.

Analysis of the serine receptor

We next compared the model to the data

Figure 5. (a) Clustered TD model applied to the serinereceptor data. In this model formulation, each CheAdimer associates with two TD receptor complexes. Thecircle markers denote the model and the continuous linesdenote the serine receptor data reported by Li & Weis.41

Their data are presented as the solution of the equationA0K

n=ðKnCLnÞ, where for mZ0, A0Z0.028, KZ0.2 mM,and nZ1; for mZ2, A0Z0.062, KZ5.2 mM, and nZ2.5;and for mZ4, A0Z0.11, KZ1 mM, and nZ5.3.(b) Parameters for the clustered TD model. The parameterDGn denotes the free energy for the active state when atotal of n subunits within the cluster are bound withligand. For all methylation states, the affinity for ligandfor the active state is 128 mM.


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published by Li & Weis.41 The Li and Weisexperiments are similar to the Bornhorst & Falkeexperiments, except that they investigated theserine receptor instead of the aspartate receptor.However, their results are significantly different.Unlike the measurements made for the aspartatereceptor,5,16,28 Li & Weis found that receptormodifications strongly influence the apparent Km

(a factor of 4600, 0.2 mM to 1 mM versus a factor of 6,16 mM to 97 mM). They also measured significantcooperativity, with a Hill coefficient of 5.3 for thefully modified receptor. As the Hill coefficient is sohigh, we were unable to fit the TD model to theirdata unless we assumed that the signaling com-plexes are clustered such that each CheA dimer wasbound to at least two TD receptor complexes.

To fit the Li & Weis data to the model, weassumed that each CheA dimer was bound to twoTD receptor complexes. To limit the number ofparameters, we also assumed that each receptordimer subunit within the hexamer is identical. Inaddition, Li & Weis measured residual kinaseactivity when the receptors are saturated withligand. The clustered TD model was unable tocapture this residual activity, because we had toassume that the saturated complexes were com-pletely inactive in order to fit the measured range ofapparent Km values. If we ignore the residualactivity, then the clustered TD model fits the datatrends well (Figure 5(a)). The parameters areplotted in Figure 5(b), where we again see a cleartrend in the parameters. For a single modificationsite (e.g. one residue methylated), CheA is deacti-vated by only one ligand-bound subunit in thecluster, whereas the fully modified receptorrequires that all six subunits are bound with ligand.The large change in free energy is needed to accountfor the broad range of apparent Km values. Also, forthe same reasons, the change in free energy undersaturating ligand conditions does not increase asthe number of modification sites increases.

Clearly, the serine data suggest a different modelthan the aspartate receptor. In a second study,38

Levit & Stock present data for the serine receptorwhere receptor methylation has less effect on theapparent Km (a factor of 23, 10 mM for unmodifiedreceptors to 230 mM for fully modified receptors,versus a factor of 4600 for the Li & Weis data). Theirdata also indicate that there is limited cooperativity(w1.7). We did not attempt to fit the model to theLevit & Stock data, as their results are normalized,though the trends are similar to the aspartatereceptor (clearly the unclustered TD model can fitthese measurements). In a third study, Sourjik &Berg31 measured significant cooperativity in vivo(nZ9) for DcheBR cells expressing only Tsr recep-tors, which is consistent with the Li & Weisexperiments. However, they did not measure howmethylation changes the effective Km, so we did notfit their data directly. It is not definitively clear whythese experimental results are significantly differ-ent, but the model predicts that the differences are

due to the amounts of protein expressed. Weelaborate in the subsection Clustering.

Mixed TD model: aspartate and serine receptors

We next compared the TD model to the datapublished by Sourjik & Berg.15 Unlike the previousexperiments, they were able to measure the effect ofmethyl-aspartate on CheA kinase activity withmodified receptors in vivo. They performed theirexperiments with cells expressing both aspartate(tar) and serine (tsr) receptors. Unlike the in vitroexperiments involving solely the aspartate receptor,Sourjik and Berg measured a large change in the


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apparent Km (a factor of 3000, 2.6 mM to 75 mM)with respect to changes in receptor methylation.They also measured two apparent Km values inDcheBR cells, where the first is due to the aspartatereceptor binding methyl-aspartate at low concen-trations and the second is due to the serine receptorbinding methyl-aspartate at high concentrations.Mello & Tu29 used these data to propose a modelthat predicts that the large changes in sensitivity aredue to interactions between the aspartate and serinereceptor. The model described next is consistentwith their hypothesis and builds on their results byproviding a molecular mechanism for signaling.

To analyze the Sourjik & Berg data, we needed toformulate the TD model with non-identical sub-units, which we refer to here as the mixed-TDmodel. Because three is an odd number, weconsidered a mixed-TD model with two Tarreceptors and one Tsr receptor and another withone Tar receptor and two Tsr receptors. Bothconfigurations fit the Sourjik & Berg data well(Figure 6(a)). Unlike the homo-TD model, themixed-TD model assumes that:

KðA000#A100ÞsKðA000#A001Þ (8)

Figure 6. (a) The TD model applied to aspartate and serinereceptor complexes with two Tar receptors and one Tsr receptcomplexes with one Tar receptor and two Tsr receptors. Thedenote the data for the aspartate and serine receptors reportedof the equation: bðA0K1=ðK1CLÞÞC ð1KbÞðA0K2=ðK2CLÞÞ, whbZ1; for cheR, A0Z0.03, K1Z3.3 mM, K2Z3.3 mM, and bZ1; fomZ1, A0Z0.75, K1Z80 mM, K2Z77 mM, and bZ0.46; for mmZ3, A0Z0.9, K1Z440 mM, K2Z110 mM, and bZ0.27; and(b) Parameters for the TD model. The left Figure denotes the reand one Tsr receptor and the right Figure denotes the resultsTsr receptors. For all methylation states, the affinity for ligan

and DGcs0. The estimated parameters are plottedin Figure 6(b). Again, we see a clear trend in the freeenergies as the number of modification sites on theaspartate receptor increases for both configurations(the serine receptors are fixed in their experimentswith two sites amidated). In agreement with Mello& Tu, the model predicts that unstimulated wild-type receptor complexes are minimally methylated,corresponding to at most one residue methylated oneither the Tar or Tsr receptor. Likewise, the DcheBmutant data match the predictions when all of theresidues on both receptors are modified andthe DcheR mutant data match the prediction whenthe receptors are unmethylated. These predictionswere subsequently verified in a second study bySourjik & Berg.31.

Similar experimental trends were reported byDunten & Koshland,28 where they showed that therange of apparent Km values for aspartate receptormodifications in DcheBR cells in response toD-aspartate increased from a range of 5–90 mM inthe absence of the serine receptor to a range of 15–3000 mM in the presence of the serine receptor. Wedid not attempt to fit the model to their data as theyonly reported the Km. They also reported only a

receptors data. The left Figure denotes the results for TDor and the right Figure denotes the results for TD receptorcircle markers denote the model and the continuous linesby Sourjik & Berg Their data are presented as the solutionere for wild-type, A0Z0.5, K1Z2.6 mM, K2Z2.6 mM, andr mZ0, A0Z0.65, K1Z38 mM, K2Z83 mM, and bZ0.65; forZ2, A0Z0.8, K1Z150 mM, K2Z105 mM, and bZ0.36; forfor cheB, A0Z0.95, K1Z75 mM, K2Z75 mM, and bZ0.

sults for the TD receptor complexes with two Tar receptorsfor TD receptor complexes with one Tar receptor and twod is KðA00#A10Þ ¼ 200 mM and KðA00#A01Þ ¼ 10 M.


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single Km, though the difference may be due to thesensitivity of their assay (Dunten & Koshlandrecorded the behavior of swimming cells, whereasSourjik & Berg measured changes in phosphoryl-ated CheY) or differences between D-aspartate andmethyl-aspartate. Despite the differences, bothstudies clearly demonstrate that different kinds ofreceptors interact with one another by altering thestability of the signaling complex.

The trend in the estimated parameters for themixed TD model is the same as the homo-TD model(Figures 4(b) and 6(b)). The difference is that withthe mixed-TD model the subunits do not have thesame affinity for ligand. This difference means thatthe sensitivity to ligand for the transition charac-terized by DG1 is far greater than the sensitivity toligand for the transition characterized by DG3. If thefree energy change is large for either (DG0KDG1) or(DG1KDG2), then the receptor complex is destabi-lized at very low concentrations of ligand. If, on theother hand, the free energy changes for (DG0KDG1)and (DG1KDG2) are negligible, then the receptorcomplex is destabilized only at very high concen-trations of ligand. When the changes in free energyfor both (DG0KDG1) and (DG1KDG2) are neitherlarge nor small, then there are two apparent Km

values. In terms of the Sourjik & Berg data, therelative values of the free energy DG1 and, to a lesserextent, DGc are the dominant parameters (Figure

Figure 7. Parametric sensitivity of the mixed-TD model. ThKðA00#A01Þ ¼ 10 M, DG0Z0.5 kcal/mol, DG1Zmax(K0.2 kK2.9 kcal/mol, and DGcZK0.25 kcal/mol. The remaining paconditions. The ligand-response curves were generated byparameters. The arrows indicate how the model changes wh

6(b)). These parameters are characterized both bythe modification state of the aspartate and the serinereceptors. In this regard, the model parameters arecharacterized by the collective behavior of thesubunits within the complex.

These results still do not explain how bacteria areable to sense gradients equally well over such awide range of concentrations. To explore the mixed-TD model further, we plotted the fraction of activecomplexes as a function of ligand at different valuesof the four parameters DG0, DG1, DG3, and DGc for aconfiguration with two Tar receptors and one Tsrreceptor (Figure 7). Of the four parameters, it isclear that DG1 has the greatest effect on thesensitivity, as the other two parameters controlsensitivity only at higher concentrations of ligand.Based on these results, we hypothesized that DG1 isthe dominant parameter controlling sensitivity. Inother words, the model predicts that the mostsensitive parameter is the stability of the active statewhen one subunit is bound with ligand.

To test this hypothesis, we generated a series ofligand-response curves for different values of DG1.At each concentration, we picked the value of DG1

for which the sensitivity is the greatest: the valuethat causes the greatest change in activity for a 10%increase in ligand concentration. These resultspredict that cells are able to sense gradients over arange of concentrations spanning almost six orders

e nominal model parameters are KðA00#A10Þ ¼ 200 mM,cal/mol, DG0), DG2Zmax(K0.4 kcal/mol, DG1), DG3Zrameters were determined by the thermodynamic linkage

varying one parameter with respect to the nominalen a specific parameter is increased.


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of magnitude by just increasing DG1 (Figure 8). Themodel also predicts that the sensitivity has twomaxima at concentrations of 25 mM and 25 mM,respectively. By examining the free energy DG1 forwhich the sensitivity is maximal, it is evident thatthe first peak is associated when a single kind ofreceptor (Tar) can destabilize a mixed complex andthe second when both kinds of receptors (Tar andTsr) are necessary to destabilize a mixed complex.Similar trends were reported in the experimentalstudy by Sourjik & Berg,15 though they reportmaximal sensitivity at concentrations of 100 mM and10 mM. The differences are expected, as we variedonly a single parameter, and receptor methylationlikely changes all four parameters. Based on theseresults, the model predicts that bacteria modulatetheir sensitivity by increasing the stability of thereceptor complex, a process controlled by receptormethylation.

Clustering

Sourjik & Berg,31 in a second study, measured the

Figure 8. (a) Sensitivity to ligand for the mixed-TDmodel as a function of DG1 for a 10% increase in aspartateconcentration. Sensitivity is defined as:ðAðLCDLÞKAðLÞÞ=DL, where DLZ0.1L. (b) Value of DG1

when the sensitivity is maximized for a 10% increaserelative to background concentration of aspartate.

effect of methyl-aspartate and serine on CheAactivity in vivo by varying the expression of theaspartate and serine receptors, CheA, and CheW.Significant cooperativity was observed when eitherthe concentration of receptors or CheW in the cellwas increased. However, cooperativity decreasedwith increasing concentrations of CheA. Becausethe range of cooperativity is so broad (from 2 to 11in the case of the aspartate receptor), their datasuggest a range of clusters that depend on theconcentrations of receptors, CheA, and CheW. Toexplore this hypothesis, we used a Monte-Carlomodel where TD receptor complexes aggregatewith CheA and CheW on a triangular lattice.

The results of the Monte-Carlo simulations aresummarized in Figure 9, where the degree ofclustering is the average number of TD receptorcomplexes bound to each CheA dimer. Expressionof CheW modulates the degree of clusteringbiphasically, consistent with previous experimentalmeasurements.2 At relatively low concentrations ofCheW, the degree of clustering is small becausethere is insufficient CheW to couple all of the TDreceptor complexes to CheA. At relatively highconcentrations, the degree of clustering is also smallbecause CheW inhibits the formation of clusters byinstead forming receptor-CheW and CheW-CheAintermediates. Maximal clustering is achievedwhen CheW is expressed at intermediate concen-trations. Expression of the receptors increases thedegree of clustering monotonically, as there aremore TD receptor complexes to bind CheA.Expression of CheA, on the other hand, decreasesthe degree of clustering by effectively shifting thecurves to the left (results not shown).

Collectively, these results explain many of theexperimental observations made by Sourjik &Berg.31 As the degree of clustering increases from1 to 3, the maximum cooperativity for the associatedallosteric model increases from 3 to 9 (results notshown). These numbers approach the measure-ments made by Sourjik & Berg (11 for the aspartatereceptor and 10 for the serine receptor when over-expressed). They also explain the different coopera-tivity measurements made for the same kind ofreceptor by different groups. As noted by Bornhorst& Falke,16 the receptor densities in some of these invitro assays are greater than wild-type levels.Perhaps, the measurements made by Li & Weis41

and Levit & Stock38 are different because ofexpression. By increasing the number of receptors,the degree of clustering between adjacent TDreceptor complexes increases and, as a result, thecooperativity also increases. Such a hypothesis isconsistent with the model and measurements madeby Sourjik & Berg.31

Discussion

Using an allosteric model based on the TD model,we were able to integrate a diverse set of data usingknown signaling mechanisms. The overall

Figure 9. Clustering and the roleof protein expression. Simulationswere performed on a 100!100triangular lattice with periodicboundary conditions, theparameters DGRWZDGAWZK0.8 kcal/mol, and 250 CheA mol-ecules. Two million Monte-Carlosteps were used to generate eachdata point. The number of receptorswere 100 (circles), 600 (!-marks),1100 (squares), 1700 (diamonds),and 2100 (stars).


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conclusion is that the TD model is consistent withboth the in vitro and in vivo experiments if we allowthe TD receptor complexes to aggregate into largercooperative structures. The conclusions are appeal-ing as the experimental data are arguably conver-ging towards this molecular model. In fact, we arenot the first to propose this molecular model23–25

and sought instead to show that an allostericmechanism based on this model is in fact consistentwith the data.

Enteric bacteria are the paradigm for chemotaxis,though the paradigm does not translate to allspecies of bacteria. Despite the differences, allknown motile species of bacteria have homologuesto the E. coli transmembrane receptors, CheW, andCheA.42 The question then is whether the samesignaling mechanism is conserved in divergentspecies of bacteria. In the case of Bacillus subtilis,different kinds of receptors can compensate formethylation defects in the asparagine (McpB)receptor.43 These results suggest that mixed recep-tors may interact with one another in B. subtilisusing a mechanism similar to E. coli. Whether thereceptors form trimers of dimers or utilize alternatemechanisms to increase sensitivity is stillunknown.44

Two aspects that we did not address with themodel are adaptation and localization. It haspreviously been suggested that CheB phosphoryl-ation plays an integral role in sensitivity.7 We arecurrently attempting to integrate the proposedallosteric mechanism into a model for the fullchemotaxis pathway. The challenge is that weneed to integrate parameter estimates from mul-tiple, often heterogeneous, sources of data. Finally,the model provides minimal insight regardingclustering and polar localization. Obviously, thelattice model is consistent with localization, but itdoes not explain why the receptors localize to the

cell poles, as the model often, depending on theparameters, predicts numerous small clusters uni-formly distributed on the surface of the cell. Onehypothesis orthogonal to the formation of coopera-tive structures is that localization increases theconcentration of kinase and, as a result, the rate ofmessenger (CheY) phosphorylation.45 It also couldbe that localization has nothing to do withsensitivity and, perhaps, is involved in methylationand adaptation.24,46

Acknowledgements

This work was generously supported by HHMI,DOE, and DARPA. We thank the anonymousreferees for their constructive criticism andsuggestions.

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Edited by I. B. Holland

(Received 16 March 2004; received in revised form 9 August 2004; accepted 11 August 2004)

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