A spatially extended stochastic model of the bacterial chemotaxis signalling pathway

Lattice ultrasensitivity produces large gain in E. coli chemosensing

E. coli use a regular lattice of receptors and attached kinases to detect and amplify faint chemical signals. Kinase output is characterized by precise adaptation to a wide range of background ligand levels and large gain in response to small relative changes in ligand concentration. These characteristics are well described by models which achieve their gain through equilibrium cooperativity. But these models are challenged by two experimental results. First, neither adaptation nor large gain are present in receptor binding assays. Second, in cells lacking adaptation machinery, fluctuations can sometimes be enormous, with essentially all kinases transitioning together. Here we introduce a far-from equilibrium model in which receptors gate the spread of activity between neighboring kinases. This model achieves large gain through a mechanism we term lattice ultrasensitivity (LU). In our LU model, kinase and receptor states are separate degrees of freedom, and kinase kinetics are dominated by chemical rates far-from-equilibrium rather than by equilibrium allostery. The model recapitulates the successes of past models, but also matches the challenging experimental findings. Importantly, unlike past lattice critical models, our LU model does not require parameters to be fine tuned for function.

Noise properties of adaptation-conferring biochemical control modules

A key goal of synthetic biology is to develop functional biochemical modules with network-independent properties. Antithetic integral feedback (AIF) is a recently developed control module in which two control species perfectly annihilate each other's biological activity. The AIF module confers robust perfect adaptation to the steady-state average level of a controlled intracellular component when subjected to sustained perturbations. Recent work has suggested that such robustness comes at the unavoidable price of increased stochastic fluctuations around average levels. We present theoretical results that support and quantify this trade-off for the commonly analyzed AIF variant in the idealized limit with perfect annihilation. However, we also show that this trade-off is a singular limit of the control module: Even minute deviations from perfect adaptation allow systems to achieve effective noise suppression as long as cells can pay the corresponding energetic cost. We further show that a variant of the AIF control module can achieve significant noise suppression even in the idealized limit with perfect adaptation. This atypical configuration may thus be preferable in synthetic biology applications.

The Chemoreceptor Sensory Adaptation System Produces Coordinated Reversals of the Flagellar Motors on an Escherichia coli Cell

In isotropic environments, an Escherichia coli cell exhibits coordinated rotational switching of its flagellar motors, produced by fluctuations in the intracellular concentration of phosphorylated CheY (CheY-P) emanating from chemoreceptor signaling arrays. In this study, we show that these CheY-P fluctuations arise through modifications of chemoreceptors by two sensory adaptation enzymes: the methyltransferase CheR and the methylesterase CheB. A cell containing CheR, CheB, and the serine chemoreceptor Tsr exhibited motor synchrony, whereas a cell lacking CheR and CheB or containing enzymatically inactive forms did not. Tsr variants with different combinations of methylation-mimicking Q residues at the adaptation sites also failed to show coordinated motor switching in cells lacking CheR and CheB. Cells containing CheR, CheB, and Tsr [NDND], a variant in which the adaptation site residues are not substrates for CheR or CheB modifications, also lacked motor synchrony. TsrΔNWETF, which lacks a C-terminal pentapeptide-binding site for CheR and CheB, and the ribose-galactose receptor Trg, which natively lacks this motif, failed to produce coordinated motor switching, despite the presence of CheR and CheB. However, addition of the NWETF sequence to Trg enabled Trg-NWETF to produce motor synchrony, as the sole receptor type in cells containing CheR and CheB. Finally, CheBc, the catalytic domain of CheB, supported motor coordination in combination with CheR and Tsr. These results indicate that the coordination of motor switching requires CheR/CheB-mediated changes in receptor modification state. We conclude that the opposing receptor substrate-site preferences of CheR and CheB produce spontaneous blinking of the chemoreceptor array's output activity. IMPORTANCE Under steady-state conditions with no external stimuli, an Escherichia coli cell coordinately switches the rotational direction of its flagellar motors. Here, we demonstrate that the CheR and CheB enzymes of the chemoreceptor sensory adaptation system mediate this coordination. Stochastic fluctuations in receptor adaptation states trigger changes in signal output from the receptor array, and this array blinking generates fluctuations in CheY-P concentration that coordinate directional switching of the flagellar motors. Thus, in the absence of chemoeffector gradients, the sensory adaptation system controls run-tumble swimming of the cell, its optimal foraging strategy.

Fluctuations in Intracellular CheY-P Concentration Coordinate Reversals of Flagellar Motors in E. coli

Signal transduction utilizing membrane-spanning receptors and cytoplasmic regulator proteins is a fundamental process for all living organisms, but quantitative studies of the behavior of signaling proteins, such as their diffusion within a cell, are limited. In this study, we show that fluctuations in the concentration of the signaling molecule, phosphorylated CheY, constitute the basis of chemotaxis signaling. To analyze the propagation of the CheY-P signal quantitatively, we measured the coordination of directional switching between flagellar motors on the same cell. We analyzed the time lags of the switching of two motors in both CCW-to-CW and CW-to-CCW switching (∆t_CCW-CW and ∆t_CW-CCW). In wild-type cells, both time lags increased as a function of the relative distance of two motors from the polar receptor array. The apparent diffusion coefficient estimated for ∆t values was ~9 µm²/s. The distance-dependency of ∆t_CW-CCW disappeared upon loss of polar localization of the CheY-P phosphatase, CheZ. The distance-dependency of the response time for an instantaneously applied serine attractant signal also disappeared with the loss of polar localization of CheZ. These results were modeled by calculating the diffusion of CheY and CheY-P in cells in which phosphorylation and dephosphorylation occur in different subcellular regions. We conclude that diffusion of signaling molecules and their production and destruction through spontaneous activity of the receptor array generates fluctuations in CheY-P concentration over timescales of several hundred milliseconds. Signal fluctuation coordinates rotation among flagella and regulates steady-state run-and-tumble swimming of cells to facilitate efficient responses to environmental chemical signals.

Regulation of the chemotaxis histidine kinase CheA: A structural perspective

Bacteria sense and respond to their environment through a highly conserved assembly of transmembrane chemoreceptors (MCPs), the histidine kinase CheA, and the coupling protein CheW, hereafter termed "the chemosensory array". In recent years, great strides have been made in understanding the architecture of the chemosensory array and how this assembly engenders sensitive and cooperative responses. Nonetheless, a central outstanding question surrounds how receptors modulate the activity of the CheA kinase, the enzymatic output of the sensory system. With a focus on recent advances, we summarize the current understanding of array structure and function to comment on the molecular mechanism by which CheA, receptors and CheW generate the high sensitivity, gain and dynamic range emblematic of bacterial chemotaxis. The complexity of the chemosensory arrays has motivated investigation with many different approaches. In particular, structural methods, genetics, cellular activity assays, nanodisc technology and cryo-electron tomography have provided advances that bridge length scales and connect molecular mechanism to cellular function. Given the high degree of component integration in the chemosensory arrays, we ultimately aim to understand how such networked molecular interactions generate a whole that is truly greater than the sum of its parts. This article is part of a Special Issue entitled: Molecular biophysics of membranes and membrane proteins.

Cell polarity: having and making sense of direction-on the evolutionary significance of the primary cilium/centrosome organ in Metazoa

Cell-autonomous polarity in Metazoans is evolutionarily conserved. I assume that permanent polarity in unicellular eukaryotes is required for cell motion and sensory reception, integration of these two activities being an evolutionarily constrained function. Metazoans are unique in making cohesive multicellular organisms through complete cell divisions. They evolved a primary cilium/centrosome (PC/C) organ, ensuring similar functions to the basal body/flagellum of unicellular eukaryotes, but in different cells, or in the same cell at different moments. The possibility that this innovation contributed to the evolution of individuality, in being instrumental in the early specification of the germ line during development, is further discussed. Then, using the example of highly regenerative organisms like planarians, which have lost PC/C organ in dividing cells, I discuss the possibility that part of the remodelling necessary to reach a new higher-level unit of selection in multi-cellular organisms has been triggered by conflicts among individual cell polarities to reach an organismic polarity. Finally, I briefly consider organisms with a sensorimotor organ like the brain that requires exceedingly elongated polarized cells for its activity. I conclude that beyond critical consequences for embryo development, the conservation of cell-autonomous polarity in Metazoans had far-reaching implications for the evolution of individuality.

Propagation of regulatory fluctuations induces coordinated switching of flagellar motors in chemotaxis signaling pathway of single bacteria

The random motion of E. coli is driven by multiple flagella motors. When all motors rotate in the counter clockwise direction, the bacteria swims smoothly. A recent experimental report by Terasawa et al. [Biophys J,100,2193,(2011)] demonstrated that a coordination of the motors can occur through signaling pathways, and perturbation of a regulatory molecule disrupted the coordination. Here, we develop a mathematical model to show that a large temporal fluctuation in the regulator concentration can induce a correlated switching of the multiple motors. Such a large fluctuation is generated by a chemotaxis receptor cluster in unilateral cell pole, which then exhibits a spatial propagation through the cytoplasm from the receptor position to the motor around cell periphery. Our numerical simulation successfully reproduces synchronized switching and the lag time in the motions of two distant motors, which has been observed experimentally. We further show that the large fluctuation in the regulator concentration at the motor positions can expand the dynamic range that the motor can respond, which confers robustness to the signaling system.

Multiple sources of slow activity fluctuations in a bacterial chemosensory network

Cellular networks are intrinsically subject to stochastic fluctuations, but analysis of the resulting noise remained largely limited to gene expression. The pathway controlling chemotaxis of Escherichia coli provides one example where posttranslational signaling noise has been deduced from cellular behavior. This noise was proposed to result from stochasticity in chemoreceptor methylation, and it is believed to enhance environment exploration by bacteria. Here we combined single-cell FRET measurements with analysis based on the fluctuation-dissipation theorem (FDT) to characterize origins of activity fluctuations within the chemotaxis pathway. We observed surprisingly large methylation-independent thermal fluctuations of receptor activity, which contribute to noise comparably to the energy-consuming methylation dynamics. Interactions between clustered receptors involved in amplification of chemotactic signals are also necessary to produce the observed large activity fluctuations. Our work thus shows that the high response sensitivity of this cellular pathway also increases its susceptibility to noise, from thermal and out-of-equilibrium processes.

Phenotypic diversity and temporal variability in a bacterial signaling network revealed by single-cell FRET

We present in vivo single-cell FRET measurements in the Escherichia coli chemotaxis system that reveal pervasive signaling variability, both across cells in isogenic populations and within individual cells over time. We quantify cell-to-cell variability of adaptation, ligand response, as well as steady-state output level, and analyze the role of network design in shaping this diversity from gene expression noise. In the absence of changes in gene expression, we find that single cells demonstrate strong temporal fluctuations. We provide evidence that such signaling noise can arise from at least two sources: (i) stochastic activities of adaptation enzymes, and (ii) receptor-kinase dynamics in the absence of adaptation. We demonstrate that under certain conditions, (ii) can generate giant fluctuations that drive signaling activity of the entire cell into a stochastic two-state switching regime. Our findings underscore the importance of molecular noise, arising not only in gene expression but also in protein networks.

Emergent properties of bacterial chemotaxis pathway

The chemotaxis pathway of Escherichia coli is the most studied sensory system in prokaryotes. The highly conserved general architecture of this pathway consists of two modules which mediate signal transduction and adaptation. The signal transduction module detects and amplifies changes in environmental conditions and rapidly transmits these signals to control bacterial swimming behavior. The adaptation module gradually resets the activity and sensitivity of the first module after initial stimulation and thereby enables the temporal comparisons necessary for bacterial chemotaxis. Recent experimental and theoretical work has unraveled multiple quantitative features emerging from the interplay between these two modules. This has laid the groundwork for rationalization of these emerging properties in the context of the evolutionary optimization of the chemotactic behavior.

Inferring Models of Bacterial Dynamics toward Point Sources

Experiments have shown that bacteria can be sensitive to small variations in chemoattractant (CA) concentrations. Motivated by these findings, our focus here is on a regime rarely studied in experiments: bacteria tracking point CA sources (such as food patches or even prey). In tracking point sources, the CA detected by bacteria may show very large spatiotemporal fluctuations which vary with distance from the source. We present a general statistical model to describe how bacteria locate point sources of food on the basis of stochastic event detection, rather than CA gradient information. We show how all model parameters can be directly inferred from single cell tracking data even in the limit of high detection noise. Once parameterized, our model recapitulates bacterial behavior around point sources such as the “volcano effect”. In addition, while the search by bacteria for point sources such as prey may appear random, our model identifies key statistical signatures of a targeted search for a point source given any arbitrary source configuration.

The role of membrane-mediated interactions in the assembly and architecture of chemoreceptor lattices

In vivo fluorescence microscopy and electron cryo-tomography have revealed that chemoreceptors self-assemble into extended honeycomb lattices of chemoreceptor trimers with a well-defined relative orientation of trimers. The signaling response of the observed chemoreceptor lattices is remarkable for its extreme sensitivity, which relies crucially on cooperative interactions among chemoreceptor trimers. In common with other membrane proteins, chemoreceptor trimers are expected to deform the surrounding lipid bilayer, inducing membrane-mediated anisotropic interactions between neighboring trimers. Here we introduce a biophysical model of bilayer-chemoreceptor interactions, which allows us to quantify the role of membrane-mediated interactions in the assembly and architecture of chemoreceptor lattices. We find that, even in the absence of direct protein-protein interactions, membrane-mediated interactions can yield assembly of chemoreceptor lattices at very dilute trimer concentrations. The model correctly predicts the observed honeycomb architecture of chemoreceptor lattices as well as the observed relative orientation of chemoreceptor trimers, suggests a series of “gateway” states for chemoreceptor lattice assembly, and provides a simple mechanism for the localization of large chemoreceptor lattices to the cell poles. Our model of bilayer-chemoreceptor interactions also helps to explain the observed dependence of chemotactic signaling on lipid bilayer properties. Finally, we consider the possibility that membrane-mediated interactions might contribute to cooperativity among neighboring chemoreceptor trimers.

The chemotaxis system allows bacteria to respond to minute changes in chemical concentration, and serves as a paradigm for biological signal processing and the self-assembly of large protein lattices in living cells. The sensitivity of the chemotaxis system relies crucially on cooperative interactions among chemoreceptor trimers, which are organized into intricate honeycomb lattices. Chemoreceptors are membrane proteins and, hence, are expected to deform the surrounding lipid bilayer, leading to membrane-mediated interactions between chemoreceptor trimers. Using a biophysical model of bilayer-chemoreceptor interactions we show that the membrane-mediated interactions induced by chemoreceptor trimers provide a mechanism for the observed self-assembly of chemoreceptor lattices. We find that the directionality of membrane-mediated interactions between trimers complements protein-protein interactions in the stabilization of the observed honeycomb architecture of chemoreceptor lattices. Our results suggest that the symmetry of membrane protein complexes such as chemoreceptor trimers is reflected in the anisotropy of membrane-mediated interactions, yielding a general mechanism for the self-assembly of ordered protein lattices in cell membranes.

Multi-state modeling of biomolecules

Multi-state modeling of biomolecules refers to a series of techniques used to represent and compute the behavior of biological molecules or complexes that can adopt a large number of possible functional states. Biological signaling systems often rely on complexes of biological macromolecules that can undergo several functionally significant modifications that are mutually compatible. Thus, they can exist in a very large number of functionally different states. Modeling such multi-state systems poses two problems: the problem of how to describe and specify a multi-state system (the “specification problem”) and the problem of how to use a computer to simulate the progress of the system over time (the “computation problem”). To address the specification problem, modelers have in recent years moved away from explicit specification of all possible states and towards rule-based formalisms that allow for implicit model specification, including the κ-calculus [1], BioNetGen [2]–[5], the Allosteric Network Compiler [6], and others [7], [8]. To tackle the computation problem, they have turned to particle-based methods that have in many cases proved more computationally efficient than population-based methods based on ordinary differential equations, partial differential equations, or the Gillespie stochastic simulation algorithm[9], [10]. Given current computing technology, particle-based methods are sometimes the only possible option. Particle-based simulators fall into two further categories: nonspatial simulators, such as StochSim [11], DYNSTOC [12], RuleMonkey [9], [13], and the Network-Free Stochastic Simulator (NFSim) [14], and spatial simulators, including Meredys [15], SRSim [16], [17], and MCell [18]–[20]. Modelers can thus choose from a variety of tools, the best choice depending on the particular problem. Development of faster and more powerful methods is ongoing, promising the ability to simulate ever more complex signaling processes in the future.

Fold-change detection in a whole-pathway model of Escherichia coli chemotaxis

There has been recent interest in sensory systems that are able to display a response which is proportional to a fold change in stimulus concentration, a feature referred to as fold-change detection (FCD). Here, we demonstrate FCD in a recent whole-pathway mathematical model of Escherichia coli chemotaxis. FCD is shown to hold for each protein in the signalling cascade and to be robust to kinetic rate and protein concentration variation. Using a sensitivity analysis, we find that only variations in the number of receptors within a signalling team lead to the model not exhibiting FCD. We also discuss the ability of a cell with multiple receptor types to display FCD and explain how a particular receptor configuration may be used to elucidate the two experimentally determined regimes of FCD behaviour. All findings are discussed in respect of the experimental literature.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

Prolonged stimuli alter the bacterial chemosensory clusters

The clustering of membrane-bound receptors plays an essential role in various biological systems. A notable model system for studying this phenomenon is the bacterial chemosensory cluster that allows motile bacteria to navigate along chemical gradients in their environment. While the basic structure of these chemosensory clusters is becoming clear, their dynamic nature and operation are not yet understood. By measuring the fluorescence polarization of tagged receptor clusters in live Escherichia coli cells, we provide evidence for stimulus-induced dynamics in these sensory clusters. We find that when a stimulus is applied, the packing of the receptors slowly decreases and that the process reverses when the stimulus is removed. Consistent with these physical changes we find that the effective cooperativity of the kinase response slowly evolves in the presence of a stimulus. Time-lapse fluorescence imaging indicates that, despite these changes, the receptor clusters do not generally dissociate upon ligand binding. These data reveal stimulus-dependent plasticity in chemoreceptor clusters.

A "trimer of dimers"-based model for the chemotactic signal transduction network in bacterial chemotaxis

The network that controls chemotaxis in Escherichia coli is one of the most completely characterized signal transduction systems to date. Receptor clustering accounts for characteristics such as high sensitivity, precise adaptation over a wide dynamic range of ligand concentrations, and robustness to variations in the amounts of intracellular proteins. To gain insights into the structure-function relationship of receptor clusters and understand the mechanism behind the high-performance signaling, we develop and analyze a model for a single trimer of dimers. This new model extends an earlier model (Spiro et al. in Proc. Natl. Acad. Sci. 94:7263-7268, 1997) to incorporate the recent experimental findings that the core structure of receptor clusters is the trimer of receptor dimers. We show that the model can reproduce most of the experimentally-observed behaviors, including excitation, adaptation, high sensitivity, and robustness to parameter variations. In addition, the model makes a number of new predictions as to how the adaptation time varies with the expression level of various proteins involved in signal transduction. Our results provide a more mechanistically-based description of the structure-function relationship for the signaling system, and show the key role of the interaction among dimer members of the trimer in the chemotactic response of cells.

Smoldyn on graphics processing units: massively parallel Brownian dynamics simulations

Space is a very important aspect in the simulation of biochemical systems; recently, the need for simulation algorithms able to cope with space is becoming more and more compelling. Complex and detailed models of biochemical systems need to deal with the movement of single molecules and particles, taking into consideration localized fluctuations, transportation phenomena, and diffusion. A common drawback of spatial models lies in their complexity: models can become very large, and their simulation could be time consuming, especially if we want to capture the systems behavior in a reliable way using stochastic methods in conjunction with a high spatial resolution. In order to deliver the promise done by systems biology to be able to understand a system as whole, we need to scale up the size of models we are able to simulate, moving from sequential to parallel simulation algorithms. In this paper, we analyze Smoldyn, a widely diffused algorithm for stochastic simulation of chemical reactions with spatial resolution and single molecule detail, and we propose an alternative, innovative implementation that exploits the parallelism of Graphics Processing Units (GPUs). The implementation executes the most computational demanding steps (computation of diffusion, unimolecular, and bimolecular reaction, as well as the most common cases of molecule-surface interaction) on the GPU, computing them in parallel on each molecule of the system. The implementation offers good speed-ups and real time, high quality graphics output

Radial and spiral stream formation in Proteus mirabilis colonies

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form a variety of spatial patterns in colonies. Colony formation involves two distinct cell types: swarmer cells that dominate near the surface and the leading edge, and swimmer cells that prefer a less viscous medium, but the mechanisms underlying pattern formation are not understood. New experimental investigations reported here show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to previously-reported concentric rings. These new observations suggest that swimmers are motile and that indirect interactions between them are essential in the pattern formation. To explain these observations we develop a hybrid model comprising cell-based and continuum components that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams results from a swimming bias of the cells near the surface of the substrate. The spatial patterns generated from the model are in qualitative agreement with the experimental observations.

Bacteria frequently colonize surfaces and grow as biofilm communities embedded in a gel-like polysaccharide matrix, and when this occurs on catheters, heart valves and other medical implants, it can lead to serious, hard-to-treat infections. Proteus mirabilis is an enteric bacterium that forms biofilms on urinary catheters, but in laboratory experiments it can swarm over hard surfaces and form a variety of spatial patterns. Understanding these patterns is a first step toward understanding biofilm formation, and here we describe new experimental results and mathematical models of pattern formation in Proteus. The experiments show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to concentric rings. To explain these observations we develop a model that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams can be predicted by incorporating a swimming bias of the cells near the surface of the substrate.

Adapt locally and act globally: strategy to maintain high chemoreceptor sensitivity in complex environments

In bacterial chemotaxis, several types of ligand-specific receptors form mixed clusters, wherein receptor-receptor interactions lead to signal amplification and integration. However, it remains unclear how a mixed receptor cluster adapts to individual stimuli and whether it can differentiate between different types of ligands. Here, we combine theoretical modeling with experiments to reveal the adaptation dynamics of the mixed chemoreceptor cluster in Escherichia coli. We show that adaptation occurs locally and is ligand-specific: only the receptor that binds the external ligand changes its methylation level when the system adapts, whereas other types of receptors change methylation levels transiently. Permanent methylation crosstalk occurs when the system fails to adapt accurately. This local adaptation mechanism enables cells to differentiate individual stimuli by encoding them into the methylation levels of corresponding types of chemoreceptors. It tunes each receptor to its most responsive state to maintain high sensitivity in complex environments and prevents saturation of the cluster by one signal.

Lateral density of receptor arrays in the membrane plane influences sensitivity of the E. coli chemotaxis response

In chemotactic bacteria, transmembrane chemoreceptors, CheA and CheW form the core signalling complex of the chemotaxis sensory apparatus. These complexes are organized in extended arrays in the cytoplasmic membrane that allow bacteria to respond to changes in concentration of extracellular ligands via a cooperative, allosteric response that leads to substantial amplification of the signal induced by ligand binding. Here, we have combined cryo-electron tomographic studies of the 3D spatial architecture of chemoreceptor arrays in intact E. coli cells with computational modelling to develop a predictive model for the cooperativity and sensitivity of the chemotaxis response. The predictions were tested experimentally using fluorescence resonance energy transfer (FRET) microscopy. Our results demonstrate that changes in lateral packing densities of the partially ordered, spatially extended chemoreceptor arrays can modulate the bacterial chemotaxis response, and that information about the molecular organization of the arrays derived by cryo-electron tomography of intact cells can be translated into testable, predictive computational models of the chemotaxis response.

A modular gradient-sensing network for chemotaxis in Escherichia coli revealed by responses to time-varying stimuli

Combining in vivo FRET with time-varying stimuli, such as steps, ramps, and sinusoids allowed deduction of the molecular mechanisms underlying cellular signal processing.

The bacterial chemotaxis pathway can be described as a two-module feedback circuit, the transfer functions of which we have characterized quantitatively by experiment. Model-driven experimental design allowed the use of a single FRET pair for measurements of both transfer functions of the pathway.

The adaptation module's transfer function revealed that feedback near steady state is weak, consistent with high sensitivity to shallow gradients, but also strong steady-state fluctuations in pathway output.

The measured response to oscillatory stimuli defines the frequency band over which the chemotaxis system can compute time derivatives.

In searching for better environments, bacteria sample their surroundings by random motility, and make temporal comparisons of experienced sensory cues to bias their movement toward favorable directions (Berg and Brown, 1972). Thus, the problem of sensing spatial gradients is reduced to time-derivative computations, carried out by a signaling pathway that is well characterized at the molecular level in Escherichia coli. Here, we study the physiology of this signal processing system in vivo by fluorescence resonance energy transfer (FRET) experiments in which live cells are stimulated by time-varying chemoeffector signals. By measuring FRET between the active response regulator of the pathway CheY-P and its phosphatase CheZ, each labeled with GFP variants, we obtain a readout that is directly proportional to pathway activity (Sourjik et al, 2007). We analyze the measured response functions in terms of mechanistic models of signaling, and discuss functional consequences of the observed quantitative characteristics.

Experiments are guided by a coarse-grained modular model (Tu et al, 2008) of the sensory network (Figure 1), in which we identify two important ‘transfer functions': one corresponding to the receptor–kinase complex, which responds to changes in input ligand concentration on a fast time scale, and another corresponding to the adaptation system, which provides negative feedback, opposing the effect of ligand on a slower time scale. For the receptor module, we calibrate an allosteric MWC-type model of the receptor–kinase complex by FRET measurements of the ‘open-loop' transfer function G([L],m) using step stimuli. This calibration provides a basis for using the same FRET readout (between CheY-P and CheZ) to further study properties of the adaptation module.

It is well known that adaptation in E. coli's chemotaxis system uses integral feedback, which guarantees exact restoration of the baseline activity after transient responses to step stimuli (Barkai and Leibler, 1997; Yi et al, 2000). However, the output of time-derivative computations during smoothly varying stimuli depends not only on the presence of integral feedback, but also on what is being integrated. As this integrand can in general be any function of the output, we represent it by a black-box function F(a) in our model, and set out to determine its shape by experiments with time-varying stimuli.

We first apply exponential ramp stimuli—waveforms in which the logarithm of the stimulus level varies linearly with time, at a fixed rate r. It was shown many years ago that during such a stimulus, the kinase output of the pathway changes to a new constant value, a_c that is dependent on the applied ramp rate, r (Block et al, 1983). A plot of a_c versus r (Figure 5A) can thus be considered as an output of time-derivative computations by the network, and could also be used to study the ‘gradient sensitivity' of bacteria traveling at constant speeds.

To obtain the feedback transfer function, F(a), we apply a simple coordinate transformation, identified using our model, to the same ramp-response data (Figure 5B). This function reveals how the temporal rate of change of the feedback signal m depends on the current output signal a. The shape of this function is analyzed using a biochemical reaction scheme, from which in vivo kinetic parameters of the feedback enzymes, CheR and CheB, are extracted. The fitted Michaelis constants for these enzymatic reactions are small compared with the steady-state abundance of their substrates, thus indicating that these enzymes operate close to saturation in vivo. The slope of the function near steady state can be used to assess the strength of feedback, and to compute the relaxation time of the system, τ_m. Relaxation is found to be slow (i.e. large τ_m), consistent with large fluctuations about the steady-state activity caused by the near-saturation kinetics of the feedback enzymes (Emonet and Cluzel, 2008).

Finally, exponential sine-wave stimuli are used to map out the system's frequency response (Figure 5C). The measured data points for both the amplitude and phase of the response are found to be in excellent agreement with model predictions based on parameters from the independently measured step and ramp responses. No curve fitting was required to obtain this agreement. Although the amplitude response as a function of frequency resembles a first-order high-pass filter with a well-defined cutoff frequency, ν_m, we point out that the chemotaxis pathway is actually a low-pass filter if the time derivative of the input is viewed as the input signal. In this latter perspective, ν_m defines an upper bound for the frequency band over which time-derivative computations can be carried out.

The two types of measurements yield complementary information regarding time-derivative computations by E. coli. The ramp-responses characterize the asymptotically constant output when a temporal gradient is held fixed over extended periods. Interestingly, the ramp responses do not depend on receptor cooperativity, but only on properties of the adaptation system, and thus can be used to reveal the in vivo adaptation kinetics, even outside the linear regime of the kinase response. The frequency response is highly relevant in considering spatial searches in the real world, in which experienced gradients are not held fixed in time. The characteristic cutoff frequency ν_m is found by working within the linear regime of the kinase response, and depends on parameters from both modules (it increases with both cooperativity in the receptor module, and the strength of feedback in the adaptation module).

Both ramp responses and sine-wave responses were measured at two different temperatures (22 and 32°C), and found to differ significantly. Both the slope of F(a) near steady state, from ramp experiments, and the characteristic cutoff frequency, from sine-wave experiments, were higher by a factor of ∼3 at 32°C. Fits of the enzymatic model to F(a) suggest that temperature affects the maximal velocity (V_max) more strongly than the Michaelis constants (K_m) for CheR and CheB.

Successful application of inter-molecular FRET in live cells using GFP variants always requires some degree of serendipity. Genetic fusions to these bulky fluorophores can impair the function of the original proteins, and even when fusions are functional, efficient FRET still requires the fused fluorophores to come within the small (<10 nm) Förster radius on interactions between the labeled proteins. Thus, when a successful FRET pair is identified, it is desirable to make the most of it. We have shown here that combined with careful temporal control of input stimuli, and appropriately calibrated models, a single FRET pair can be used to study the structure of multiple transfer functions within a signaling network.

The Escherichia coli chemotaxis-signaling pathway computes time derivatives of chemoeffector concentrations. This network features modules for signal reception/amplification and robust adaptation, with sensing of chemoeffector gradients determined by the way in which these modules are coupled in vivo. We characterized these modules and their coupling by using fluorescence resonance energy transfer to measure intracellular responses to time-varying stimuli. Receptor sensitivity was characterized by step stimuli, the gradient sensitivity by exponential ramp stimuli, and the frequency response by exponential sine-wave stimuli. Analysis of these data revealed the structure of the feedback transfer function linking the amplification and adaptation modules. Feedback near steady state was found to be weak, consistent with strong fluctuations and slow recovery from small perturbations. Gradient sensitivity and frequency response both depended strongly on temperature. We found that time derivatives can be computed by the chemotaxis system for input frequencies below 0.006 Hz at 22°C and below 0.018 Hz at 32°C. Our results show how dynamic input–output measurements, time honored in physiology, can serve as powerful tools in deciphering cell-signaling mechanisms.

A dynamic-signaling-team model for chemotaxis receptors in Escherichia coli

The chemotaxis system of Escherichia coli is sensitive to small relative changes in ambient chemoattractant concentrations over a broad range. Interactions among receptors are crucial to this sensitivity, as is precise adaptation, the return of chemoreceptor activity to prestimulus levels in a constant chemoeffector environment through methylation and demethylation of receptors. Signal integration and cooperativity have been attributed to strongly coupled, mixed teams of receptors, but receptors become individually methylated according to their ligand occupancy states. Here, we present a model of dynamic signaling teams that reconciles strong coupling among receptors with receptor-specific methylation. Receptor trimers of dimers couple to form a honeycomb lattice, consistent with cryo-electron microscopy (cryoEM) tomography, within which the boundaries of signaling teams change rapidly. Our model helps explain the inferred increase in signaling team size with receptor modification, and indicates that active trimers couple more strongly than inactive trimers.

Quantitative understanding of cell signaling: the importance of membrane organization

Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.

An agent-based model of signal transduction in bacterial chemotaxis

We report the application of agent-based modeling to examine the signal transduction network and receptor arrays for chemotaxis in Escherichia coli, which are responsible for regulating swimming behavior in response to environmental stimuli. Agent-based modeling is a stochastic and bottom-up approach, where individual components of the modeled system are explicitly represented, and bulk properties emerge from their movement and interactions. We present the Chemoscape model: a collection of agents representing both fixed membrane-embedded and mobile cytoplasmic proteins, each governed by a set of rules representing knowledge or hypotheses about their function. When the agents were placed in a simulated cellular space and then allowed to move and interact stochastically, the model exhibited many properties similar to the biological system including adaptation, high signal gain, and wide dynamic range. We found the agent based modeling approach to be both powerful and intuitive for testing hypotheses about biological properties such as self-assembly, the non-linear dynamics that occur through cooperative protein interactions, and non-uniform distributions of proteins in the cell. We applied the model to explore the role of receptor type, geometry and cooperativity in the signal gain and dynamic range of the chemotactic response to environmental stimuli. The model provided substantial qualitative evidence that the dynamic range of chemotactic response can be traced to both the heterogeneity of receptor types present, and the modulation of their cooperativity by their methylation state.

The chemoreceptor dimer is the unit of conformational coupling and transmembrane signaling

Transmembrane chemoreceptors are central components in bacterial chemotaxis. Receptors couple ligand binding and adaptational modification to receptor conformation in processes that create transmembrane signaling. Homodimers, the fundamental receptor structural units, associate in trimers and localize in patches of thousands. To what degree do conformational coupling and transmembrane signaling require higher-order interactions among dimers? To what degree are they altered by such interactions? To what degree are they inherent features of homodimers? We addressed these questions using nanodiscs to create membrane environments in which receptor dimers had few or no potential interaction partners. Receptors with many, few, or no interaction partners were tested for conformational changes and transmembrane signaling in response to ligand occupancy and adaptational modification. Conformation was assayed by measuring initial rates of receptor methylation, a parameter independent of receptor-receptor interactions. Coupling of ligand occupancy and adaptational modification to receptor conformation and thus to transmembrane signaling occurred with essentially the same sensitivity and magnitude in isolated dimers as for dimers with many neighbors. Thus, we conclude that the chemoreceptor dimer is the fundamental unit of conformational coupling and transmembrane signaling. This implies that in signaling complexes, coupling and transmembrane signaling occur through individual dimers and that changes between dimers in a receptor trimer or among trimer-based signaling complexes are subsequent steps in signaling.

Cellular automata modelling of biomolecular networks dynamics

The modelling of biological systems dynamics is traditionally performed by ordinary differential equations (ODEs). When dealing with intracellular networks of genes, proteins and metabolites, however, this approach is hindered by network complexity and the lack of experimental kinetic parameters. This opened the field for other modelling techniques, such as cellular automata (CA) and agent-based modelling (ABM). This article reviews this emerging field of studies on network dynamics in molecular biology. The basics of the CA technique are discussed along with an extensive list of related software and websites. The application of CA to networks of biochemical reactions is exemplified in detail by the case studies of the mitogen-activated protein kinase (MAPK) signalling pathway, the FAS-ligand (FASL)-induced and Bcl-2-related apoptosis. The potential of the CA method to model basic pathways patterns, to identify ways to control pathway dynamics and to help in generating strategies to fight with cancer is demonstrated. The different line of CA applications presented includes the search for the best-performing network motifs, an analysis of importance for effective intracellular signalling and pathway cross-talk.

Context-dependent interaction leads to emergent search behavior in social aggregates

Locating the source of an advected chemical signal is a common challenge facing many living organisms. When the advecting medium is characterized by either high Reynolds number or high Peclet number, the task becomes highly nontrivial due to the generation of heterogeneous, dynamically changing filamental concentrations that do not decrease monotonically with distance to the source. Defining search strategies that are effective in these environments has important implications for the understanding of animal behavior and for the design of biologically inspired technology. Here we present a strategy that is able to solve this task without the higher intelligence required to assess spatial gradient direction, measure the diffusive properties of the flow field, or perform complex calculations. Instead, our method is based on the collective behavior of autonomous individuals following simple social interaction rules which are modified according to the local conditions they are experiencing. Through these context-dependent interactions, the group is able to locate the source of a chemical signal and in doing so displays an awareness of the environment not present at the individual level. This behavior illustrates an alternative pathway to the evolution of higher cognitive capacity via the emergent, group-level intelligence that can result from local interactions.

Chemotaxis: how bacteria use memory

Bacterial chemotaxis represents one of the simplest and best studied examples of unicellular behavior. Chemotaxis allows swimming bacterial cells to follow chemical gradients in the environment by performing temporal comparisons of ligand concentrations. The process of chemotaxis in the model bacteriumEscherichia colihas been studied in great molecular detail over the past 40 years, using a large range of experimental tools to investigate physiology, genetics and biochemistry of the system. The abundance of quantitative experimental data enabled detailed computational modeling of the pathway and theoretical analyses of such properties as robustness and signal amplification. Because of the temporal mode of gradient sensing in bacterial chemotaxis, molecular memory is an essential component of the chemotaxis pathway. Recent studies suggest that the memory time scale has been evolutionary optimized to perform optimal comparisons of stimuli while swimming in the gradient. Moreover, noise in the adaptation system, which results from variations of the adaptation rate both over time and among cells, might be beneficial for the overall chemotactic performance of the population.

E. coli superdiffusion and chemotaxis-search strategy, precision, and motility

Escherichia coli motion is characterized by a sequence of consecutive tumble-and-swim events. In the absence of chemical gradients, the length of individual swims is commonly believed to be distributed exponentially. However, recently there has been experimental indication that the swim-length distribution has the form of a power-law, suggesting that bacteria might perform superdiffusive Lévy-walk motion. In E. coli, the power-law behavior can be induced through stochastic fluctuations in the level of CheR, one of the key enzymes in the chemotaxis signal transmission pathway. We use a mathematical model of the chemotaxis signaling pathway to study the influence of these fluctuations on the E. coli behavior in the absence and presence of chemical gradients. We find that the population with fluctuating CheR performs Lévy-walks in the absence of chemoattractants, and therefore might have an advantage in environments where nutrients are sparse. The more efficient search strategy in sparse environments is accompanied by a generally larger motility, also in the presence of chemoattractants. The tradeoff of this strategy is a reduced precision in sensing and following gradients, as well as a slower adaptation to absolute chemoattractant levels.

Designing and encoding models for synthetic biology

A key component of any synthetic biology effort is the use of quantitative models. These models and their corresponding simulations allow optimization of a system design, as well as guiding their subsequent analysis. Once a domain mostly reserved for experts, dynamical modelling of gene regulatory and reaction networks has been an area of growth over the last decade. There has been a concomitant increase in the number of software tools and standards, thereby facilitating model exchange and reuse. We give here an overview of the model creation and analysis processes as well as some software tools in common use. Using markup language to encode the model and associated annotation, we describe the mining of components, their integration in relational models, formularization and parametrization. Evaluation of simulation results and validation of the model close the systems biology 'loop'.

The core signaling proteins of bacterial chemotaxis assemble to form an ultrastable complex

The chemosensory pathway of bacterial chemotaxis forms a polar signaling cluster in which the fundamental signaling units, the ternary complexes, are arrayed in a highly cooperative, repeating lattice. The repeating ternary units are composed of transmembrane receptors, histidine-kinase CheA, and coupling protein CheW, but it is unknown how these three core proteins are interwoven in the assembled ultrasensitive lattice. Here, to further probe the nature of the lattice, we investigate its stability. The findings reveal that once the signaling cluster is assembled, CheA remains associated and active for days in vitro. All three core components are required for this ultrastable CheA binding and for receptor-controlled kinase activity. The stability is disrupted by low ionic strength or high pH, providing strong evidence that electrostatic repulsion between the highly acidic core components can lead to disassembly. We propose that ultrastability arises from the assembled lattice structure that establishes multiple linkages between the core components, thereby conferring thermodynamic or kinetic ultrastability to the bound state. An important, known function of the lattice structure is to facilitate receptor cooperativity, which in turn enhances pathway sensitivity. In the cell, however, the ultrastability of the lattice could lead to uncontrolled growth of the signaling complex until it fills the inner membrane. We hypothesize that such uncontrolled growth is prevented by an unidentified intracellular disassembly system that is lost when complexes are isolated from cells, thereby unmasking the intrinsic complex ultrastability. Possible biological functions of ultrastability are discussed.

Signal amplification in a lattice of coupled protein kinases

The bacterium Escherichia coli detects chemical attractants and repellents by means of a cluster of transmembrane receptors and associated molecules. Experiments have shown that this cluster amplifies the signal about 35-fold and current models attribute this amplification to cooperative interactions between neighbouring receptors. However, when applied to the mixed population of receptors of wild-type E. coli, these models lead to indiscriminate methylation of all receptor types rather than the selective methylation observed experimentally. In this paper, we propose that cooperative interactions occur not between receptors but in the underlying lattice of CheA molecules. In our model, each CheA molecule is stimulated by its neighbours via their flexible P1 domains and modulated by the ligand binding and methylation states of associated receptors. We test this idea with detailed, molecular-based stochastic simulations and show that it gives an accurate reproduction of signalling in this system, including ligand-specific adaptation.

Introducing simulated cellular architecture to the quantitative analysis of fluorescent microscopy

Biological cells are complex and highly dynamic: many macromolecules are organized in loose assemblies, clusters or highly structured complexes, others exist most of the time as freely diffusing monomers. They move between regions and compartments through diffusion and enzyme-mediated transport, within a heavily crowded cytoplasm. To make sense of this complexity, computational models, and, in turn, quantitative in vivo data are needed. An array of fluorescent microscopy methods is available, but due to the inherent noise and complexity inside the cell, they are often hard to interpret. Using the example of fluorescence recovery after photobleaching (FRAP) and the bacterial chemotaxis system, we are here introducing detailed spatial simulations as a new approach in analysing such data.

Differential affinity and catalytic activity of CheZ in E. coli chemotaxis

Push–pull networks, in which two antagonistic enzymes control the activity of a messenger protein, are ubiquitous in signal transduction pathways. A classical example is the chemotaxis system of the bacterium Escherichia coli, in which the kinase CheA and the phosphatase CheZ regulate the phosphorylation level of the messenger protein CheY. Recent experiments suggest that both the kinase and the phosphatase are localized at the receptor cluster, and Vaknin and Berg recently demonstrated that the spatial distribution of the phosphatase can markedly affect the dose–response curves. We argue, using mathematical modeling, that the canonical model of the chemotaxis network cannot explain the experimental observations of Vaknin and Berg. We present a new model, in which a small fraction of the phosphatase is localized at the receptor cluster, while the remainder freely diffuses in the cytoplasm; moreover, the phosphatase at the cluster has a higher binding affinity for the messenger protein and a higher catalytic activity than the phosphatase in the cytoplasm. This model is consistent with a large body of experimental data and can explain many of the experimental observations of Vaknin and Berg. More generally, the combination of differential affinity and catalytic activity provides a generic mechanism for amplifying signals that could be exploited in other two-component signaling systems. If this model is correct, then a number of recent modeling studies, which aim to explain the chemotactic gain in terms of the activity of the receptor cluster, should be reconsidered.

In both prokaryotes and eukaryotes, extra- and intracellular signals are often processed by biochemical networks in which two enzymes together control the activity of a messenger protein via opposite modification reactions. A well-known example is the chemotaxis network of Escherichia coli that controls the swimming behavior of the bacterium in response to chemical stimuli. Recent experiments suggest that the two counteracting enzymes in this network are colocalized at the receptor cluster, while experiments by Vaknin and Berg indicate that the spatial distribution of the enzymes by itself can markedly affect the response of the network. We argue using mathematical modeling that the most widely used model of the chemotaxis network is inconsistent with these experimental observations. We then present an alternative model in which part of one enzyme is colocalized with the other enzyme at the receptor cluster, while the remainder freely diffuses in the cytoplasm; moreover, the fraction at the cluster both binds more strongly to the messenger protein and modifies it faster. This model is consistent with a large number of experimental observations and provides a generic mechanism for amplifying signals.

Stochastic modelling for quantitative description of heterogeneous biological systems

Two related developments are currently changing traditional approaches to computational systems biology modelling. First, stochastic models are being used increasingly in preference to deterministic models to describe biochemical network dynamics at the single-cell level. Second, sophisticated statistical methods and algorithms are being used to fit both deterministic and stochastic models to time course and other experimental data. Both frameworks are needed to adequately describe observed noise, variability and heterogeneity of biological systems over a range of scales of biological organization.

MULTISCALE MODELS OF TAXIS-DRIVEN PATTERNING IN BACTERIAL POPULATIONS

Spatially-distributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants. In the past decade a great deal has been learned about signal transduction, metabolism and movement in E. coli and other bacteria, but translating the individual-level behavior into population-level dynamics is still a challenging problem. However, this is a necessary step because it is computationally impractical to use a strictly cell-based model to understand patterning in growing populations, since the total number of cells may reach 10(12) - 10(14) in some experiments. In the past phenomenological equations such as the Patlak-Keller-Segel equations have been used in modeling the cell movement that is involved in the formation of such patterns, but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation. Significant progress has been made for bacterial species that employ a "run-and-tumble" strategy of movement, in that macroscopic equations based on simplified schemes for signal transduction and turning behavior have been derived [14, 15]. Here we extend previous work in a number of directions: (i) we allow for time-dependent signals, which extends the applicability of the equations to natural environments, (ii) we use a more general turning rate function that better describes the biological behavior, and (iii) we incorporate the effect of hydrodynamic forces that arise when cells swim in close proximity to a surface. We also develop a new approach to solving the moment equations derived from the transport equation that does not involve closure assumptions. Numerical examples show that the solution of the lowest-order macroscopic equation agrees well with the solution obtained from a Monte Carlo simulation of cell movement under a variety of temporal protocols for the signal. We also apply the method to derive equations of chemotactic movement that are governed by multiple chemotactic signals.

The Intersection of Theory and Application in Elucidating Pattern Formation in Developmental Biology

We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and vice-versa. The chosen systems - Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns - illustrate the fundamental physical processes of signaling, growth and cell division, and cell movement involved in pattern formation and development. These systems exemplify the current state of theoretical and experimental understanding of how these processes produce the observed patterns, and illustrate how theoretical and experimental approaches can interact to lead to a better understanding of development. As John Bonner said long ago'We have arrived at the stage where models are useful to suggest experiments, and the facts of the experiments in turn lead to new and improved models that suggest new experiments. By this rocking back and forth between the reality of experimental facts and the dream world of hypotheses, we can move slowly toward a satisfactory solution of the major problems of developmental biology.'

Dependence of bacterial chemotaxis on gradient shape and adaptation rate

Simulation of cellular behavior on multiple scales requires models that are sufficiently detailed to capture central intracellular processes but at the same time enable the simulation of entire cell populations in a computationally cheap way. In this paper we present RapidCell, a hybrid model of chemotactic Escherichia coli that combines the Monod-Wyman-Changeux signal processing by mixed chemoreceptor clusters, the adaptation dynamics described by ordinary differential equations, and a detailed model of cell tumbling. Our model dramatically reduces computational costs and allows the highly efficient simulation of E. coli chemotaxis. We use the model to investigate chemotaxis in different gradients, and suggest a new, constant-activity type of gradient to systematically study chemotactic behavior of virtual bacteria. Using the unique properties of this gradient, we show that optimal chemotaxis is observed in a narrow range of CheA kinase activity, where concentration of the response regulator CheY-P falls into the operating range of flagellar motors. Our simulations also confirm that the CheB phosphorylation feedback improves chemotactic efficiency by shifting the average CheY-P concentration to fit the motor operating range. Our results suggest that in liquid media the variability in adaptation times among cells may be evolutionary favorable to ensure coexistence of subpopulations that will be optimally tactic in different gradients. However, in a porous medium (agar) such variability appears to be less important, because agar structure poses mainly negative selection against subpopulations with low levels of adaptation enzymes. RapidCell is available from the authors upon request.

Monte Carlo simulation and linear stability analysis of Turing pattern formation in reaction-subdiffusion systems

Subdiffusion is an important physical phenomenon observed in many systems. However, numerical techniques to study it, especially when coupled to reactions, are lacking. In this paper, we develop an efficient Monte Carlo algorithm based on the Gillespie algorithm and the continuous-time random walk to simulate reaction-subdiffusion systems. Using this algorithm, we investigate Turing pattern formation in the Schnakenberg model with subdiffusion. First, we show that, as the system becomes more subdiffusive, the homogeneous state becomes more difficult to destablize and Turing patterns form less easily. Second, we show that, as the number of particles in the system decreases, the magnitude of fluctuations increases and again the Turing patterns form less easily. Third, we show that, as the system becomes more subdiffusive, the ratio between the two diffusive constants must be higher in order to observe Turing patterns. Finally, we also carry out linear stability analysis to validate the results obtained from our algorithm.

Modeling the chemotactic response of Escherichia coli to time-varying stimuli

In their natural environment, cells need to extract useful information from complex temporal signals that vary over a wide range of intensities and time scales. Here, we study how such signals are processed by Escherichia coli during chemotaxis by developing a general theoretical model based on receptor adaptation and receptor-receptor cooperativity. Measured responses to various monotonic, oscillatory, and impulsive stimuli are all explained consistently by the underlying adaptation kinetics within this model. For exponential ramp signals, an analytical solution is discovered that reveals a remarkable connection between the dependence of kinase activity on the exponential ramp rate and the receptor methylation rate function. For exponentiated sine-wave signals, spectral analysis shows that the chemotaxis pathway acts as a lowpass filter for the derivative of the signal with the cutoff frequency determined by an intrinsic adaptation time scale. For large step stimuli, we find that the recovery time is determined by the constant maximum methylation rate, which provides a natural explanation for the observed recovery time additivity. Our model provides a quantitative system-level description of the chemotaxis signaling pathway and can be used to predict E. coli chemotaxis responses to arbitrary temporal signals. This model of the receptor system reveals the molecular origin of Weber's law in bacterial chemotaxis. We further identify additional constraints required to account for the related observation that the output of this pathway is constant under exponential ramp stimuli, a feature that we call "logarithmic tracking."

Looking inside the box: bacterial transistor arrays

One often compares cells to computers, and signalling proteins to transistors. Location and wiring of those molecular transistors is paramount in defining the function of the subcellular chips. The bacterial chemotactic sensing apparatus is a large, stable assembly consisting of thousands of receptors, signal transducing kinases and linking proteins, and is responsible for the motile response of the bacterium to environmental signals, whether chemical, mechanical, or thermal. Because of its rich functional repertoire despite its relative simplicity, this chemosome has attracted much attention from both experimentalists and theoreticians, and the bacterial chemotaxis response becoming a benchmark in Systems Biology. Structural and functional models of the chemotactic device have been developed, often based on particular assumptions regarding the topology of the receptor lattice. In this issue of Molecular Microbiology, Briegel et al. provide a detailed view of the receptor arrangement, unravelling the wiring of the molecular signal processors.

Protein exchange dynamics at chemoreceptor clusters in Escherichia coli

Signal processing in bacterial chemotaxis relies on large sensory complexes consisting of thousands of protein molecules. These clusters create a scaffold that increases the efficiency of pathway reactions and amplifies and integrates chemotactic signals. The cluster core in Escherichia coli comprises a ternary complex composed of receptors, kinase CheA, and adaptor protein CheW. All other chemotaxis proteins localize to clusters by binding either directly to receptors or to CheA. Here, we used fluorescence recovery after photobleaching (FRAP) to investigate the turnover of chemotaxis proteins at the cluster and their mobility in the cytoplasm. We found that cluster exchange kinetics were protein-specific and took place on several characteristic time scales that correspond to excitation, adaptation, and cell division, respectively. We further applied analytical and numerical data fitting to analyze intracellular protein diffusion and to estimate the rate constants of cluster equilibration in vivo. Our results indicate that the rates of protein turnover at the cluster have evolved to ensure optimal performance of the chemotaxis pathway.

Relationship between cellular response and behavioral variability in bacterial chemotaxis

Over the last decades, bacterial chemotaxis in Escherichia coli has emerged as a canonical system for the study of signal transduction. A remarkable feature of this system is the coexistence of a robust adaptive behavior observed at the population level with a large fluctuating behavior in single cells [Korobkova E, Emonet T, Vilar JMG, Shimizu TS, Cluzel P (2004) Nature 428:574-578]. Using a unified stochastic model, we demonstrate that this coexistence is not fortuitous but a direct consequence of the architecture of this adaptive system. The methylation and demethylation cycles that regulate the activity of receptor-kinase complexes are ultrasensitive because they operate outside the region of first-order kinetics. As a result, the receptor-kinase that governs cellular behavior exhibits a sigmoidal activation curve. We propose that the steepness of this kinase activation curve simultaneously controls the behavioral variability in nonstimulated individual bacteria and the duration of the adaptive response to small stimuli. We predict that the fluctuating behavior and the chemotactic response of individual cells both peak within the transition region of this sigmoidal curve. Large-scale simulations of digital bacteria suggest that the chemotaxis network is tuned to simultaneously maximize both the random spread of cells in the absence of nutrients and the cellular response to gradients of attractant. This study highlights a fundamental relation from which the behavioral variability of nonstimulated cells is used to infer the timing of the cellular response to small stimuli.

Modelling reaction kinetics inside cells

In the past decade, advances in molecular biology such as the development of non-invasive single molecule imaging techniques have given us a window into the intricate biochemical activities that occur inside cells. In this chapter we review four distinct theoretical and simulation frameworks: (i) non-spatial and deterministic, (ii) spatial and deterministic, (iii) non-spatial and stochastic and (iv) spatial and stochastic. Each framework can be suited to modelling and interpreting intracellular reaction kinetics. By estimating the fundamental length scales, one can roughly determine which models are best suited for the particular reaction pathway under study. We discuss differences in prediction between the four modelling methodologies. In particular we show that taking into account noise and space does not simply add quantitative predictive accuracy but may also lead to qualitatively different physiological predictions, unaccounted for by classical deterministic models.

Chemotaxis in Escherichia coli: a molecular model for robust precise adaptation

The chemotaxis system in the bacterium Escherichia coli is remarkably sensitive to small relative changes in the concentrations of multiple chemical signals over a broad range of ambient concentrations. Interactions among receptors are crucial to this sensitivity as is precise adaptation, the return of chemoreceptor activity to prestimulus levels in a constant chemoeffector environment. Precise adaptation relies on methylation and demethylation of chemoreceptors by the enzymes CheR and CheB, respectively. Experiments indicate that when transiently bound to one receptor, these enzymes act on small assistance neighborhoods (AN) of five to seven receptor homodimers. In this paper, we model a strongly coupled complex of receptors including dynamic CheR and CheB acting on ANs. The model yields sensitive response and precise adaptation over several orders of magnitude of attractant concentrations and accounts for different responses to aspartate and serine. Within the model, we explore how the precision of adaptation is limited by small AN size as well as by CheR and CheB kinetics (including dwell times, saturation, and kinetic differences among modification sites) and how these kinetics contribute to noise in complex activity. The robustness of our dynamic model for precise adaptation is demonstrated by randomly varying biochemical parameters.

Bacteria swim in relatively straight lines and change directions through tumbling. In the process of chemotaxis, a network of receptors and other proteins controls the tumbling frequency to direct an otherwise random walk toward nutrients and away from repellents. Receptor clustering and adaptation to persistent stimuli through covalent modification allow chemotaxis to be sensitive over a large range of ambient concentrations. The individual components of the chemotaxis network are well characterized, and signaling measurements by fluorescence microscopy quantify the network's response, making the system well suited for modeling and analysis. In this paper, we expand upon a previous model based on experiments indicating that the covalent modifications required for adaptation occur through the action of enzymes on groups of neighboring receptors, referred to as assistance neighborhoods. Simulations show that our proposed molecular model of a strongly coupled complex of receptors produces accurate responses to different stimuli and is robust to parameter variation. Within this model, the correct adaptation response is limited by small assistance-neighborhood size as well as enzyme kinetics. We also explore how these kinetics contribute to noise in the chemotactic response.