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The machinery for transduction of chemotactic stimuli in the bacterium
Most organisms have developed signal detection systems that extract information from their environment to enable them to find food and mates, initiate developmental changes, avoid harmful environments or execute any of the multitude of actions and behaviors in their repertoire. Since most organisms maintain a clear distinction between inside and outside, many primary environmental signals do not penetrate the organism very far, and therefore mechanisms for transducing an external signal into an internal signal, and where appropriate, an internal response are needed. For example, at the cellular level extracellular hydrophilic
The response at the individual level to changes in the signal involves changes in the bias of the flagellar motor, and this can also lead to a response in the form of spatial pattern formation at the population level.
Signal transduction systems often filter the signal as well, since not all features of a signal are equally important. Often the important information in a signal is the shortterm change in amplitude, rather than the absolute amplitude itself, and many systems have evolved to ignore constant background signals, yet remain responsive to changes in the signal. In such systems a step change in the signal from one constant level to another may elicit a transient change in one or more components of the internal state and some behavior of the organism, followed by a return to a basal level of that component or behavior. The process that leads to termination of the response in the face of a constant stimulus is called desensitization, habituation, or adaptation, depending on the context, but here we use adaptation when the stimulus does not provoke any gross rearrangements or alterations in the signalprocessing machinery, whereas desensitization may involve structural changes such as the degradation of receptors. The visual system and mechanoreceptors in the dermis of mammals provide examples of adaptation to certain stimuli, but this capability is very common in sensory systems. In general adaptation also involves maintenance of sensitivity to further changes in the signal, and here we define an adapting sensory system as one that responds transiently to a transient change in the signal, returns to a basal activity level in the presence of a prolonged constant stimulus, and retains sensitivity to further changes in the stimulus. These characteristics are shown schematically for another cellular model system in
At the cellular level and higher, the response to environmental signals frequently involves
A chemoeffector alters the probabilities that the flagella will rotate in a given direction, thereby changing the frequencies of runs and tumbles, and these probabilities change in response to
Two essential properties of the
Chemoreceptors are the transmembrane methylaccepting chemotaxis proteins (MCP) that bacteria use to detect chemicals, light, or temperature. Among the five classes, Tsr and Tar are the majortype receptors with a few thousand copies per cell; Tap, Trg, and Aer are the minor types with a few hundred copies per cell. The functional form of chemoreceptors is a helical, intertwined homodimer. Each monomer consists of a variable periplasmic ligandbinding domain, a transmembrane domain, and a conserved cytoplasmic signaling domain. The ligandbinding domain contains a four
The cytoplasmic domain extends from the transmembrane domain and bends back via a “U” turn (
In addition to the chemoreceptors, the excitation phase involves a twocomponent signal transduction system to controlmotor behavior, based on CheA, a histidine protein kinase (HPK), and CheY, a response regulator. HPK is linked to a sensory unit that detects changes in the environmental condition and when activated by the unit, the kinase catalyzes phosphotransfer from ATP to its own histidine residue. The response regulator, when phosphorylated by HPK, acts directly to modify the bias of the motor, and thereby leads to a change in cellular behavior. In
The adaptation phase involves CheR and CheB, proteins involved in changes of the methylation level of chemoreceptors. CheR methylates glutamate (
The high signaling sensitivity and wide response range in
With the preceding description of the structure of the chemoreceptor clusters, which involve multiple levels of organization, at hand, we discuss the structurefunction relationship (summarized in
Finally, we discuss the conformational aspect of the signaling mechanism of receptor clusters. A homodimer is usually treated as a twostate (active or inactive) switch. Ligand binding, methylation/demethylation, and interaction with neighboring receptors can shift the equilibriumbetween the two signaling states in a dimer. In more detail, attractant binding initiates a pistonlike sliding of the transmembrane signaling helix (TM2 in
The chemotaxis signal transduction system in
This raises the question as to how adaptation can be ensured in a system of chemical reactions. To illustrate that this is not easy to answer in general, we observe that for any network of chemical reactions, adaptation of a given component does not necessarily ensure adaptation of a particular species located further “downstream” in the kinetic pathway. This is demonstrated with several schematic counterexamples in
In addition to the problem inherent in specifying an upstream adapting quantity
In view of the fact that
As we will see later, the signal transduction system in a single bacteriuum can be described by a finite number of state variables and an evolution equation that determines how the state changes under prescribed inputs or stimuli. We denote the state vector by
where
More generally,
Evidently this definition allows for the trivial case when
A widelyused model system that illustrates some of the essential features of an adaptive system is given as follows. Suppose that there are two internal state variables
In these equations the function
Since this system is linear, the solution can be obtained by quadrature once the stimulus is specified. For the special case in which
Thus the response occurs on two time scales, the scale of excitation, which is characterized by
This is just the pseudosteadystate value of
We note from
This simple model illustrates some of the basic features necessary in an adapting system, but there is no explicit biochemical basis for it. However the excitation variable
We first focus on theoretical studies of the excitation and adaptation characteristics of the signal transduction pathway and the underlying mechanism for system robustness. The early modeling studies were directed toward understanding the observed adaptation in bacterial chemotaxis [
A longstanding question is how biological systems maintain the stability of their functions in the face of perturbations of parameters or state variables such as reaction rates or molecular concentrations. Barkai and Leibler [
A comparison of the finetuned and the robust adaptation models shows that the significant difference lies in the treatment of methylation by CheR and demethylation by CheBp especially the latter. In the finetuned systems [
Next we review a large number of models that employ a receptor clusteringbased explanation for the high signaling sensitivity, large transduction gain, and wide dynamic range of the signaling system. Bray
The models in this category are based on the hypothesis that receptors exist in an extended, weaklycoupled lattice network and use the Isingtype framework. As an implementation of Bray’s idea [
These models hypothesize that receptors exist in several strongly coupled clusters and use the MonodWymanChangeux (MWC)type framework. The FRET experiments in Sourjik and Berg [
The models addressed above, either of Isingtype or MWCtype, treat the receptor homodimer as the basic functional unit and thus the receptor interaction is at the dimerdimer level. However, it is now established that the trimer of homodimers serves as the core unit in signaling, especially in kinase control [
Thus far the MWCtype models reviewed all have a prescribed stoichiometry and a fixed size of receptor clusters. In cells, the number of the receptor complexes involved in chemotactic responses probably varies with the stimulus magnitude. Recently, theoreticians have begun to develop models for dynamic signaling receptor clusters with a variable size. Endres
Next we discuss several models analyzing other features of the system. Lipkow
Lastly, we introduce a recent, trimer of dimersbased model [
The single trimer model cannot reproduce the higher receptor cooperativity,
where
Similarly, we obtain the freeenergy levels for the puretype trimer of dimers, shown in
Using Boltzmann’s law, the probability that the puretype trimer of dimers is active is as below.
Now, we consider a receptor cluster with
Finally, we consider a cluster of mixedtype receptors. Here, we work on the case of two types, Tar and Tsr, as an example. In this case, a trimer could contain three Tar homodimers, three Tsr homodimers, or a combination of Tar and Tsr homodimers. For simplicity, we use the approximation wherein only trimers made of the same homodimers exist in a mixedtype receptor cluster. With this assumption, a TarTsr cluster only consists of the pure Tar trimers and the pure Tsr trimers. We only consider the response of the mixedtype cluster to a single type of ligands. The probability of the mixedtype receptor cluster with
We use this model to explain the observed ultrahigh cooperativity. We simulate the responses to methylaspartate (MeAsp) and serine in the
At the population level, cellcellsignaling and chemotaxis provide a mechanism for longrange cellcell communication and formation of multicellular spatial patterns [
Chemotaxis of
The enteric bacterium
The formation of these bacterial patterns involves a complex interplay between different processes, including consumption of nutrients, production of chemoattractants, tactic movement towards the attractant, and hydrodynamic interaction with the environment. Moreover, formation of these spatial patterns can involve millions of cells. Mathematical models of the patterns include continuum models that incorporate these processess in a phenomenological way, and hybrid cellbased models that allow detailed description of the microscopic behavior. The continuum models are easier to implement numerically and more amenable to mathematical analysis, but justification of these models has to be addressed. Hybrid cellbased models can be used to integrate better descriptions of the experimental picture, but can be computationally expensive.
Continuum approaches to bacterial pattern formation
A variety of PatlakKellerSegel type systems
have been developed and applied to model
Traveling wave solutions of the system (
Numerous extensions of the PKS model at system 14 have been proposed to model the
Hybrid cellbased models for bacterial pattern formation have also been developed [
Here the superscript
Since particles are conserved in the turning process
The foregoing individualbased model for cell movement can be coupled with continuum reactiondiffusion equations to describe the evolution of the extracellular nutrient and attractant concentrations. The combined system is solved with a hybrid scheme in which the movement of each cell is simulated by a Monte Carlo method while the reactiondiffusion equations are solved with an alternating direction method. The details of this algorithm can be found in [
To model the experiments done by Budrene and Berg, the hybrid cellbased model was coupled with the following equations for the attractant and nutrient concentrations in [
where
To model the traveling band formation observed by Adler, the hybrid model was coupled with the following equation for
where
The hybrid cellbased model has also been applied to model radial and spiral stream formation of
Hybrid cellbased models based on simplified descriptions of cell signaling have also been used to study
The hybrid cellbased models described above can be used to integrate details on cell signaling and movement faithfully. However, when used to simulate population behavior, it becomes computationally intensive, especially when some parameters of the model are unknown and parameter exploration is needed. The continuum models such as those based on PKS equations are computationally managable and analytically amenable. However, justification of these models under different scenarios of signals, and the relationship between macroscopic parameters in these models with parameters known in the signal transduction steps, are not rigorously established in the original PKS equation.
To fill in this gap, significant effort has been put in deriving continuum models from cellbased models for chemotaxis of
and involves asymptotic approximations of the resulting master equation of the velocity jump process,
Here
where
This PKS equation, in which the chemotactic sensitivity is defined in terms of cell parameters, provides a very good approximation to the spatialtemporal dynamics observed in the cellbased models under a variety of signal regimes, as long as the signal gradient times the cell is small compared to the reciprocal of the adaptation time [
However when the signal changes rapidly along the cell’s trajectory, the underlying assumption of the derivation is not satisfied and
The signal transduction pathway that governs bacterial chemotaxis in
At the receptor homodimer level, there are questions concerning conformational changes of the cytoplasmic domain for signaling, especially in the HAMP region and the signaling region. At the trimer of dimers level, further studies to determine the stoichiometry of the ternary MCPCheACheWsignaling complexes, and how the signalinginduced conformational changes of one homodimer affect other dimer members within a trimer, are needed. Major questions remain at the receptor cluster level, where a cluster of trimers of dimers is probably used. One hypothesis that could be tested in this context is that for shortrange interaction among three dimers of the same trimer, the protein interaction is primarily due to the direct coupling of dimers in the cytoplasmic domain, while for longerrange interaction among trimers (dimers of different trimers), the protein interaction is due to indirect coupling through the interconnected CheA and CheW network, or possibly membranemediated elastic interaction.
At the population level open questions include: how to derive a quantitative macroscopic description of bacterial population chemotaxis when the external signal changes rapidly, or whether such an approach is even possible. Currently continuum models are only derived from coarsegrained or abstract models of signal transduction that can perform excitation and adaptation. Derivation of continuum models from cellbased models that take into account detailed descriptions of cell signaling is currently under investigation and will be published elsewhere.
This research is supported in part by NIH grant # GM 29123 and by NSF grant DMS 0517884. CX would like to thank the Mathematical Biosciences Institute at the Ohio State University for support as a longterm visitor through NSFDMS 0931642.
Two examples of the response of an adapting system to changes in the stimulus level. We show the predicted cyclic AMP (cAMP) relay response, as measured by the secreted cAMP, to extracellular cAMP stimuli in the cellular slime mold
A schematic of the signal transduction pathway in
The structure of chemoreceptors. The schematic view of a chemoreceptor monomer (left) demonstrates the primary architecture consisting of ligandbinding domain (
Examples of various adapting and nonadapting systems. (
The response to single (
Signal transduction network. The basic unit of the network is the signaling complex, denoted by T. The three indices used to denote the properties of the complex are shown in the upper left corner. In the reaction network, vertical transitions are ligand binding and release, horizontal transitions are methylation and demethylation, and fronttorear and reverse transitions are kinase activation, deactivation, phosphorylation and dephosphorylation. The details of the phosphotransfer transitions are depicted at the left. Adopted with permission from [
Responses of receptor Tsr
Simulated
Spiral streams in a growing
Comparison of solutions of the derived PDE (
Structurefunction relationship of chemoreceptor clusters in
Dimer  Trimer of dimers  Cluster of trimers  

Yes 
Yes  Yes  
Yes 
Yes  Yes  
Yes 
Yes  Yes  
No  Yes 
Yes  
Low 
Moderate 
High 
Mathematical models of bacterial chemotaxis (1982–2012).
Excitation, adaptation, and robustness  

Model  Methods  Assumptions and Outcomes 
Goldbeter and Koshland Jr [ 
ODE  Includes ligand binding and onesite methylation; Uses twostate assumption (methylated and demethylated); Demonstrates that perfect adaptation could be achieved via methylation whose reaction rates depend on receptor occupancy. 
Block 
ODE  Uses twostate assumption (CW and CCW); Includes adaptation; Demonstrates that transition between the run and tumble states depends on adaptation to the sensory input. 
Asakura and Honda [ 
ODE  Includes ligand binding and multiplesite methylation; Uses twostate assumption (methylated and demethylated); Shows adaptation to attractants and repellents at both low and high background concentrations via multiple methylation. 
Segel 
ODE  Similar with Goldbeter and Koshland Jr [ 
Bray 
ODE  Includes ligand binding, phosphorylation cascade, and motor control; Reproduces the motor bias response to step changes in attractants and repellents ; Does not include methylation/demethylation and model for adaptation. 
Bray and Bourret [ 
ODE  Models the ternary MCP/CheA/CheWsignaling complex formation and adds it into Bray 
Hauri and Ross [ 
ODE  Models the complete signal transduction pathway and reproduces the excitation and adaptation phases of bacterial chemotaxis in the experimentally agreed timescales; Assumes that CheA autophosphorylation rate dependent on the methylation level of receptors. 
Spiro 
ODE  Models the complete signal transduction pathway with reduced three methylation states and reproduces excitation and adaptation in the experimentally agreed timescales. Assumes the autophosphorylation rate increases with the methylation level, the methylation rate is greater for attractantbound than attractantfree receptors, and the demethylation rate is independent of ligand binding of receptors. 
Barkai and Leibler [ 
ODE  Includes ligand binding and methylation/demethylation for a threecomponent system (MCP, CheR and CheB); Uses twostate assumption (active or inactive for receptors); Assumes that CheR works at saturation in a constant rate and CheB acts only on active receptors in a rate independent of ligand binding; Shows perfect adaptation of receptor activity and robustness of the ratio of adapted steadystate receptor activity over prestimulus activity for a wide range of parameter values. 
Levin 
ODE  Investigates the effect of changes in chemotactic protein expression levels on the concentration of CheYp, and compares the finetuned and the robust adaptation models in this aspect. 
MortonFirth and Bray [ 
Freeenergybased stochastic simulation  Includes phosphorylation cascade; Simulates the temporal fluctuation of CheYp. 
MortonFirth 
Freeenergybased stochastic simulation  Includes phosphorylation cascade (based on [ 
Yi 
ODE  Analyzes the Barkai and Leibler’s model and shows an integral feedback control imbedded in the system that leads to robust perfect adaptation. 
Almogy 
ODE  Proposes an alternative adaptation mechanism that is through dephosphorylation of CheYp by both CheZ and the CheAs–CheZ complex rather than methylation/demethylation of receptors. 
Mello and Tu [ 
ODE  Studies the robust adaptation problem analytically and proposes six conditions for achieving perfect adaptation, confirming those key assumptions that Barkai and Leibler use [ 
Arocena and Acerenza [ 
ODE  Studies the response range of bacterial chemotaxis, and shows the wider range when receptor modification is through methylation and phosphorylation than through attractant binding. 
Kollmann 
ODE  Uses a simplified signaling network only including a single methylation site; Shows the robustness to the intercellular variation in chemotactic protein concentrations arising from gene expression, and the variation of CheYp is much smaller than that of other proteins. 
Tu 
ODE, meanfield theory  Simulates chemotactic responses to timevarying exponential ramp, sine wave, and impulsive signals. 
Bray 
probability analysis  Conceptual model; Shows that receptor clustering and conformational spread among neighboring receptors can explain high sensitivity. 
Shi and Duke [ 
statistical mechanics, Ising model  Isingtype model and meanfield theory applied; Shows that receptor coupling strength affects response more than attractant binding. 
Duke and Bray [ 
Monte Carlo methods  Monte Carlo simulation of [ 
Shi [ 
statistical mechanics, Ising model  Adaptive Isingtype model with CheR, CheBp, and their negative feedback effect on receptor activity included; More robust than [ 
Shi [ 
Ising model  Compares simulations of the models [ 
Shi [ 
Ising model, Monte Carlo methods  Considers the receptor movement and allows them to float; Shows strong correlation for neighboring receptors and exponential decay with increasing receptorreceptor distance. 
Levin 
Monte Carlo methods  Studies effect of binding and diffusion of CheR through receptor clusters with the model [ 
Shimizu 
Ising model, freeenergybased stochastic simulation  Ising model incorporated into [ 
Mello and Tu [ 
Ising model  Deterministic version of Isingtype model; Includes receptor interactions between Tar and Tsr; Includes methylation/demethylation (same assumptions as [ 
Mello 
Ising model, meanfield theory, Monte Carlo methods  Meanfield theory applied to and Monte Carlo simulation of [ 
Goldman 
Lattice gas model, Monte Carlo methods  Applies 2D lattice gas model of protein association to chemoreceptor clusters. 
Sourjik and Berg [ 
MWC model  Applies MWC model to explain their FERT data. 
Albert 
ODE  Model for dynamic formation of trimer of dimers; Assumes the time scale of association and dissociation of trimer of dimers comparable to that of ligand binding and kinase activity, which was disproved later by experiments [ 
Rao 
MWC model  Model of static trimer of dimers; Reproduces 
Mello and Tu [ 
MWC model  Generalizes MWC model for allosteric interaction and multiple signal integration in heterogeneous receptor clusters; Reproduces measured responses for 14 mutant strains with varied expression levels of Tar and/or Tsr [ 
Keymer 
MWC model  Proposes two regimes for a twostate receptor: regime I is characterized by low to moderate kinase activity and a low, constant inhibition number for halfmaximal activity 
Endres and Wingreen [ 
MWC model  Adaptation model based on ‘assistantneighborhoods’ [ 
[ 
MWC model, Ising model  Compares activity response of receptor clusters generated by onedimensional Isingtype model, twodimensional Isingtype model, and tworegime MWCtype model; Shows that the outputs of Isingtype models are not consistent with the FRET data on activity responses to steps of attractants for wildtype and 
Mello and Tu [ 
MWC model  Studies the mechanism how the cells maintain high sensitivity over a wide range of backgrounds based on a simplified version of [ 
Endres 
statistical mechanics, MWC model  Model of static trimer of dimers; Reproduces 
Park 
sensitivity analysis  Performs sensitivity analysis for trimer of dimers and shows enhanced signaling sensitivity compared with dimers. 
Hansen 
MWC model  Robust adaptation model extended from [ 
Endres 
MWC model, statistical method  Determines the sizes of signaling clusters through best fitting 
Hansen 
statistical mechanics, MWC model  Model of dynamic signaling clusters of trimers of dimers, the boundaries of which are variable in simulation; Shows stronger coupling of active trimers of dimers than inactive. 
Meir 
MWC model, ODE  Analyzes the characteristics of precise adaptation and finds the asymmetries ( 
Clausznitzer 
MWC model, ODE  Studies the dynamics (time courses) of adaptation and evaluate the existing adaptation models. 
Khursigara 
MWC model  Study with experiments and simulations combined; A cutoff distance used to determine the range of interacting receptors and the size of signaling receptor clusters variable; Shows that the size of receptor arrays is relatively stable, noncorrelated with the protein expression level, and the packing density is slightly varied in difficult growth media. 
Xin and Othmer [ 
ODE  Model of trimer of dimers; Simulates dynamics for the overall pathway; Explains a line of 
Rao 
ODE  Compares signaling pathways between 
Lipkow 
spatiotemporal stochastic simulation  3D stochastic simulation of CheY phosphorylation, CheY/CheYp diffusion, CheYp binding to FliM and dephosphorylation; Studies effects of CheZ localization, motor position, and macromolecular crowding on spatial concentration of CheYp; Shows a constant concentration of CheYp throughout the cytoplasm when CheZ is restricted to anterior ends and an exponential gradient across the length of the cell when CheZ diffuses freely. 
Lipkow [ 
spatiotemporal stochastic simulation  Studies the effect of CheZ localization; Suggests that clustering of CheZ–CheAs–CheYp at the cell poles, introducing a negative feedback to the CheYp level, serves a secondary adaptation mechanism and explains the overshoot of CheYp in 
Endres [ 
statistical mechanics  Free energybased model for formation of clusters of trimer of dimers; Studies the determining factors of the size of polar receptor clusters. 
Roberts 
ODE  Develops a control engineering method and applies it to elucidating the signaling pathways of 
Tindall 
ODE  Studies the signal integration mechanism in 
Hamadeh 
control theory  Studies the feedback configuration of 
Freeenergy levels for a puretype trimer of dimers.
State  Freeenergy Level (unit: 

On with 0 ligand bound 

On with 1 ligands bound 

On with 2 ligands bound 

On with 3 ligands bound 

 
Off with 0 ligand bound  
Off with 1 ligands bound 

Off with 2 ligands bound 

Off with 3 ligands bound 
