Changing cellular location of CheZ predicted by molecular simulations

A Mathematical Description of Bacterial Chemotaxis in Response to Two Stimuli

Bacteria are often exposed to multiple stimuli in complex environments, and their efficient chemotactic decisions are critical to survive and grow in their native environments. Bacterial responses to the environmental stimuli depend on the ratio of their corresponding chemoreceptors. By incorporating the signaling machinery of individual cells, we analyze the collective motion of a population of Escherichia coli bacteria in response to two stimuli, mainly serine and methyl-aspartate (MeAsp), in a one-dimensional and a two-dimensional environment, which is inspired by experimental results in Y. Kalinin et al., J. Bacteriol. 192(7):1796-1800, 2010. Under suitable conditions, we show that if the ratio of the main chemoreceptors of individual cells, namely Tar/Tsr, is less than a specific threshold, the bacteria move to the gradient of serine, and if the ratio is greater than the threshold, the group of bacteria moves toward the gradient of MeAsp. Finally, we examine the theory with Monte Carlo agent-based simulations and verify that our results qualitatively agree well with the experimental results in Y. Kalinin et al. (2010).

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.

Allosteric Priming of E. coli CheY by the Flagellar Motor Protein FliM

Phosphorylation of Escherichia coli CheY protein transduces chemoreceptor stimulation to a highly cooperative flagellar motor response. CheY binds to the N-terminal peptide of the FliM motor protein (FliM_N). Constitutively active D13K-Y106W CheY has been an important tool for motor physiology. The crystal structures of CheY and CheY ⋅ FliM_N with and without D13K-Y106W have shown FliM_N-bound CheY contains features of both active and inactive states. We used molecular dynamics (MD) simulations to characterize the CheY conformational landscape accessed by FliM_N and D13K-Y106W. Mutual information measures identified the central features of the long-range CheY allosteric network between D57 phosphorylation site and the FliM_N interface, namely the closure of the α4-β4 hinge and inward rotation of Y- or W106 with W58. We used hydroxy-radical foot printing with mass spectroscopy (XFMS) to track the solvent accessibility of these and other side chains. The solution XFMS oxidation rate correlated with the solvent-accessible area of the crystal structures. The protection of allosteric relay side chains reported by XFMS confirmed the intermediate conformation of the native CheY ⋅ FliM_N complex, the inactive state of free D13K-Y106W CheY, and the MD-based network architecture. We extended the MD analysis to determine temporal coupling and energetics during activation. Coupled aromatic residue rotation was a graded rather than a binary switch, with Y- or W106 side-chain burial correlated with increased FliM_N affinity. Activation entrained CheY fold stabilization to FliM_N affinity. The CheY network could be partitioned into four dynamically coordinated sectors. Residue substitutions mapped to sectors around D57 or the FliM_N interface according to phenotype. FliM_N increased sector size and interactions. These sectors fused between the substituted K13-W106 residues to organize a tightly packed core and novel surfaces that may bind additional sites to explain the cooperative motor response. The community maps provide a more complete description of CheY priming than proposed thus far.

Probing chemotaxis activity in Escherichia coli using fluorescent protein fusions

Bacterial chemotaxis signaling may be interesting for the development of rapid biosensor assays, but is difficult to quantify. Here we explore two potential fluorescent readouts of chemotactically active Escherichia coli cells. In the first, we probed interactions between the chemotaxis signaling proteins CheY and CheZ by fusing them individually with non-fluorescent parts of stable or unstable ‘split’-Green Fluorescent Protein. Wild-type chemotactic cells but not mutants lacking the CheA kinase produced distinguishable fluorescence foci, two-thirds of which localize at the cell poles with the chemoreceptors and one-third at motor complexes. Fluorescent foci based on stable split-eGFP displayed small fluctuations in cells exposed to attractant or repellent, but those based on an unstable ASV-tagged eGFP showed a higher dynamic behaviour both in the foci intensity changes and the number of foci per cell. For the second readout, we expressed the pH-sensitive fluorophore pHluorin in the cyto- and periplasm of chemotactically active E. coli. Calibrations of pHluorin fluorescence as a function of pH demonstrated that cells accumulating near a chemo-attractant temporally increase cytoplasmic pH while decreasing periplasmic pH. Both readouts thus show promise for biosensor assays based on bacterial chemotaxis activity.

Mathematical Analysis of the Escherichia coli Chemotaxis Signalling Pathway

We undertake a detailed mathematical analysis of a recent nonlinear ordinary differential equation (ODE) model describing the chemotactic signalling cascade within an Escherichia coli cell. The model includes a detailed description of the cell signalling cascade and an average approximation of the receptor activity. A steady-state stability analysis reveals the system exhibits one positive real steady state which is shown to be asymptotically stable. Given the occurrence of a negative feedback between phosphorylated CheB (CheB-P) and the receptor state, we ask under what conditions the system may exhibit oscillatory-type behaviour. A detailed analysis of parameter space reveals that whilst variation in kinetic rate parameters within known biological limits is unlikely to lead to such behaviour, changes in the total concentration of the signalling proteins do. We postulate that experimentally observed overshoot behaviour can actually be described by damped oscillatory dynamics and consider the relationship between overshoot amplitude, total cell protein concentration and the magnitude of the external ligand stimulus. Model reductions in the full ODE model allow us to understand the link between phosphorylation events and the negative feedback between CheB-P and receptor methylation, as well as elucidate why some mathematical models exhibit overshoot and others do not. Our paper closes by discussing intercell variability of total protein concentration as a means of ensuring the overall survival of a population as cells are subjected to different environments.

Lighting up my life: a LOV-based fluorescent reporter for Campylobacter jejuni

In this study, a LOV-based fluorescent reporter (light, oxygen, or voltage-sensing domains of phototropin), termed iLOV, was adapted for Campylobacter jejuni and used to investigate promoter activity via monitoring fluorescence intensity and to study the localisation of two chemotaxis proteins. The pC46 complementation vector contains coding sequence from cj0046, a C. jejuni NCTC11168 pseudo-gene and is used to integrate cloned genes onto the C. jejuni chromosome. The pC46 vector was used to construct plasmids containing iLOV, driven by three different C. jejuni constitutive promoters and plasmids containing transcriptional fusions of the iLOV reporter and two chemoreceptors, tlp5 and tlp8. Expression from the porA promoter, pporA, produced the highest fluorescence signals compared to pfdxA (intermediate level) and pmetK (lowest level). The cellular localisation pattern of transducer-like protein (Tlp) clusters, containing Tlp5 and Tlp8, was predominately polar, with Tlp5 positioned only at one and Tlp8 at both poles. Here, we demonstrate that a iLOV fluorescent reporter can be used as a promoter probe or as a gene fusion reporter in Campylobacter spp. This is a new system uniquely placed for studying Campylobacter spp., as it combines resistance to photobleaching and functionality under microaerobic conditions.

Ultrasensitivity and fluctuations in the Barkai-Leibler model of chemotaxis receptors in Escherichia coli

A stochastic version of the Barkai-Leibler model of chemotaxis receptors in Escherichia coli is studied here with the goal of elucidating the effects of intrinsic network noise in their conformational dynamics. The model was originally proposed to explain the robust and near-perfect adaptation of E. coli observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. In the model, a receptor is either active or inactive and can stochastically switch between the two states. The enzyme CheR methylates inactive receptors while CheB demethylates active receptors and the probability for a receptor to be active depends on its level of methylation and ligand occupation. In a simple version of the model with two methylation sites per receptor (M = 2), we show rigorously, under a quasi-steady state approximation, that the mean active fraction of receptors is an ultrasensitive function of [CheR]/[CheB] in the limit of saturating receptor concentration. Hence the model shows zero-order ultrasensitivity (ZOU), similar to the classical two-state model of covalent modification studied by Goldbeter and Koshland (GK). We also find that in the limits of extremely small and extremely large ligand concentrations, the system reduces to two different two-state GK modules. A quantitative measure of the spontaneous fluctuations in activity is provided by the variance σa2 in the active fraction, which is estimated mathematically under linear noise approximation (LNA). It is found that σa2 peaks near the ZOU transition. The variance is a non-monotonic, but weak function of ligand concentration and a decreasing function of receptor concentration. Gillespie simulations are also performed in models with M = 2, 3 and 4. For M = 2, simulations show excellent agreement with analytical results obtained under LNA. Numerical results for M = 3 and M = 4 are qualitatively similar to our mathematical results in M = 2; while all the models show ZOU in mean activity, the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to affect the motility of the organism, and may have a beneficial effect on it. The response of mean receptor activity to small time-dependent changes in the external ligand concentration is computed within linear response theory, and found to have a bilobe form, in agreement with earlier experimental observations.

Single-cell E. coli response to an instantaneously applied chemotactic signal

In response to an attractant or repellant, an Escherichia coli cell controls the rotational direction of its flagellar motor by a chemotaxis system. When an E. coli cell senses an attractant, a reduction in the intracellular concentration of a chemotaxis protein, phosphorylated CheY (CheY-P), induces counterclockwise (CCW) rotation of the flagellar motor, and this cellular response is thought to occur in several hundred milliseconds. Here, to measure the signaling process occurring inside a single E. coli cell, including the recognition of an attractant by a receptor cluster, the inactivation of histidine kinase CheA, and the diffusion of CheY and CheY-P molecules, we applied a serine stimulus by instantaneous photorelease from a caged compound and examined the cellular response at a temporal resolution of several hundred microseconds. We quantified the clockwise (CW) and CCW durations immediately after the photorelease of serine as the response time and the duration of the response, respectively. The results showed that the response time depended on the distance between the receptor and motor, indicating that the decreased CheY-P concentration induced by serine propagates through the cytoplasm from the receptor-kinase cluster toward the motor with a timing that is explained by the diffusion of CheY and CheY-P molecules. The response time included 240 ms for enzymatic reactions in addition to the time required for diffusion of the signaling molecule. The measured response time and duration of the response also revealed that the E. coli cell senses a similar serine concentration regardless of whether the serine concentration is increasing or decreasing. These detailed quantitative findings increase our understanding of the signal transduction process that occurs inside cells during bacterial chemotaxis.

Helicobacter pylori CheZ(HP) and ChePep form a novel chemotaxis-regulatory complex distinct from the core chemotaxis signaling proteins and the flagellar motor

Chemotaxis is important for Helicobacter pylori to colonize the stomach. Like other bacteria, H. pylori uses chemoreceptors and conserved chemotaxis proteins to phosphorylate the flagellar rotational response regulator, CheY, and modulate the flagellar rotational direction. Phosphorylated CheY is returned to its non-phosphorylated state by phosphatases such as CheZ. In previously studied cases, chemotaxis phosphatases localize to the cellular poles by interactions with either the CheA chemotaxis kinase or flagellar motor proteins. We report here that the H. pylori CheZ, CheZ(HP), localizes to the poles independently of the flagellar motor, CheA, and all typical chemotaxis proteins. Instead, CheZ(HP) localization depends on the chemotaxis regulatory protein ChePep, and reciprocally, ChePep requires CheZ(HP) for its polar localization. We furthermore show that these proteins interact directly. Functional domain mapping of CheZ(HP) determined the polar localization motif lies within the central domain of the protein and that the protein has regions outside of the active site that participate in chemotaxis. Our results suggest that CheZ(HP) and ChePep form a distinct complex. These results therefore suggest the intriguing idea that some phosphatases localize independently of the other chemotaxis and motility proteins, possibly to confer unique regulation on these proteins' activities.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

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.

The HtrA protease of Borrelia burgdorferi degrades outer membrane protein BmpD and chemotaxis phosphatase CheX

Borrelia burgdorferi, the spirochaetal agent of Lyme disease, codes for a single HtrA protein, HtrABb (BB0104) that is homologous to DegP of Escherichia coli (41% amino acid identity). HtrABb shows physical and biochemical similarities to DegP in that it has the trimer as its fundamental unit and can degrade casein via its catalytic serine. Recombinant HtrABb exhibits proteolytic activity in vitro, while a mutant (HtrABbS198A) does not. However, HtrABb and DegP have some important differences as well. Native HtrABb occurs in both membrane-bound and soluble forms. Despite its homology to DegP, HtrABb could not complement an E. coli DegP deletion mutant. Late stage Lyme disease patients, as well as infected mice and rabbits developed a robust antibody response to HtrABb, indicating that it is a B-cell antigen. In co-immunoprecipitation studies, a number of potential binding partners for HtrABb were identified, as well as two specific proteolytic substrates, basic membrane protein D (BmpD/BB0385) and chemotaxis signal transduction phosphatase CheX (BB0671). HtrABb may function in regulating outer membrane lipoproteins and in modulating the chemotactic response of B. burgdorferi.

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

Adaptation at the output of the chemotaxis signalling pathway

In the bacterial chemotaxis network, receptor clusters process input^1–3, and flagellar motors generate output⁴. Receptor and motor complexes are coupled by the diffusible protein CheY-P. Receptor output (the steady-state concentration of CheY-P) varies from cell to cell⁵. However, the motor is ultrasensitive, with a narrow [CheY-P] operating range⁶. How might the match between receptor output and motor input be optimized? Here we show that the motor can shift its operating range by changing its composition. The number of FliM subunits in the C-ring increases in response to a decrement in the concentration of CheY-P, increasing motor sensitivity. This shift in sensitivity explains the slow partial adaptation observed in mutants that lack the receptor methyltransferase and methylesterase^7–8 and why motors exhibit signal-dependent FliM turnover⁹. Adaptive remodelling is likely to be a common feature in the operation of many molecular machines.

Theoretical insights into bacterial chemotaxis

Research into understanding bacterial chemotactic systems has become a paradigm for Systems Biology. Experimental and theoretical researchers have worked hand-in-hand for over 40 years to understand the intricate behavior driving bacterial species, in particular how such small creatures, usually not more than 5 µm in length, detect and respond to small changes in their extracellular environment. In this review we highlight the importance that theoretical modeling has played in providing new insight and understanding into bacterial chemotaxis. We begin with an overview of the bacterial chemotaxis sensory response, before reviewing the role of theoretical modeling in understanding elements of the system on the single cell scale and features underpinning multiscale extensions to population models.

Coordinated regulation of multiple flagellar motors by the Escherichia coli chemotaxis system

Escherichia coli cells swim toward a favorable environment by chemotaxis. The chemotaxis system regulates the swimming behavior of the bacteria by controlling the rotational direction of their flagellar motors. Extracellular stimuli sensed by chemoreceptors are transduced to an intracellular signal molecule, phosphorylated CheY (CheY-P), that switches the rotational direction of the flagellar motors from counterclockwise (CCW) to clockwise (CW) or from CW to CCW. Many studies have focused on identifying the proteins involved in the chemotaxis system, and findings on the structures and intracellular localizations of these proteins have largely elucidated the molecular pathway. On the other hand, quantitative evaluations of the chemotaxis system, including the process of intracellular signaling by the propagation of CheY-P and the rotational switching of flagellar motor by binding of CheY-P molecules, are still uncertain. For instance, scientific consensus has held that the flagellar motors of an E. coli cell switch rotational direction asynchronously. However, recent work shows that the rotational switching of any two different motors on a single E. coli cell is highly coordinated; a sub-second switching delay between motors is clearly correlated with the relative distance of each motor from the chemoreceptor patch located at one pole of the cell. In this review of previous studies and our recent findings, we discuss the regulatory mechanism of the multiple flagellar motors on an individual E. coli cell and the intracellular signaling process that can be inferred from this coordinated switching.

Exploring the spatial and temporal organization of a cell's proteome

To increase our current understanding of cellular processes, such as cell signaling and division, knowledge is needed about the spatial and temporal organization of the proteome at different organizational levels. These levels cover a wide range of length and time scales: from the atomic structures of macromolecules for inferring their molecular function, to the quantitative description of their abundance, and spatial distribution in the cell. Emerging new experimental technologies are greatly increasing the availability of such spatial information on the molecular organization in living cells. This review addresses three fields that have significantly contributed to our understanding of the proteome's spatial and temporal organization: first, methods for the structure determination of individual macromolecular assemblies, specifically the fitting of atomic structures into density maps generated from electron microscopy techniques; second, research that visualizes the spatial distributions of these complexes within the cellular context using cryo electron tomography techniques combined with computational image processing; and third, methods for the spatial modeling of the dynamic organization of the proteome, specifically those methods for simulating reaction and diffusion of proteins and complexes in crowded intracellular fluids. The long-term goal is to integrate the varied data about a proteome's organization into a spatially explicit, predictive model of cellular processes.

Adaptation and control circuits in bacterial chemotaxis

Bacteria are capable of sensing and responding to changes in their environment. One of the ways they do this is via chemotaxis, regulating swimming behaviour. The chemotaxis pathway senses chemoattractant gradients and uses a feedback loop to change the bacterial swimming pattern; this feedback loop differs in detail between species. In the present article, we summarize the current understanding of the regulatory mechanisms in three species and how these pathways can be viewed and analysed through the ideas of feedback control systems engineering.

Origins and diversification of a complex signal transduction system in prokaryotes

The molecular machinery that controls chemotaxis in bacteria is substantially more complex than any other signal transduction system in prokaryotes, and its origins and variability among living species are unknown. We found that this multiprotein "chemotaxis system" is present in most prokaryotic species and evolved from simpler two-component regulatory systems that control prokaryotic transcription. We discovered, through genomic analysis, signaling systems intermediate between two-component systems and chemotaxis systems. Evolutionary genomics established central and auxiliary components of the chemotaxis system. While tracing its evolutionary history, we also developed a classification scheme that revealed more than a dozen distinct classes of chemotaxis systems, enabling future predictive modeling of chemotactic behavior in unstudied species.

Detailed simulations of cell biology with Smoldyn 2.1

Most cellular processes depend on intracellular locations and random collisions of individual protein molecules. To model these processes, we developed algorithms to simulate the diffusion, membrane interactions, and reactions of individual molecules, and implemented these in the Smoldyn program. Compared to the popular MCell and ChemCell simulators, we found that Smoldyn was in many cases more accurate, more computationally efficient, and easier to use. Using Smoldyn, we modeled pheromone response system signaling among yeast cells of opposite mating type. This model showed that secreted Bar1 protease might help a cell identify the fittest mating partner by sharpening the pheromone concentration gradient. This model involved about 200,000 protein molecules, about 7000 cubic microns of volume, and about 75 minutes of simulated time; it took about 10 hours to run. Over the next several years, as faster computers become available, Smoldyn will allow researchers to model and explore systems the size of entire bacterial and smaller eukaryotic cells.

Accurate particle-based simulation of adsorption, desorption and partial transmission

Particle-based simulators represent molecules of interest with point-like particles that diffuse and react in continuous space. These simulators are often used to investigate spatial or stochastic aspects of biochemical systems. This paper presents new particle-based simulation algorithms for modeling interactions between molecules and surfaces; they address irreversible and reversible molecular adsorption to, desorption from and transmission through membranes. Their central elements are: (i) relationships between adsorption, desorption and transmission coefficients on the one hand, and simulator interaction probabilities on the other, and (ii) probability densities for initial placements of desorbed molecules. These algorithms, which were implemented and tested in the Smoldyn simulator, are accurate, easy to implement and computationally efficient. They allow longer time steps and better address reversible processes than an algorithm that Erban and Chapman recently presented (Physical Biology 4:16-28, 2007). This paper also presents a method for simulating unbounded diffusion in a limited spatial domain using a partially absorbing boundary, as well as new solutions to the diffusion differential equation with reversible Robin boundary conditions.

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.

Protein domains and residues involved in the CheZ/CheAS interaction

CheZ localizes to chemoreceptor patches by binding CheA-short (CheA(S)). Residues 70 to 134 of CheZ, constituting the apical loops and part of the dimerization domain, suffice for localization. Replacements of Tyr-118, Ile-119, Leu-123, Arg-124, and Leu-126 of CheA interfere with localization. These residues are exposed in the 'P1 domain of CheA(S).

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.

Dynamic map of protein interactions in the Escherichia coli chemotaxis pathway

Protein-protein interactions play key roles in virtually all cellular processes, often forming complex regulatory networks. A powerful tool to study interactions in vivo is fluorescence resonance energy transfer (FRET), which is based on the distance-dependent energy transfer from an excited donor to an acceptor fluorophore. Here, we used FRET to systematically map all protein interactions in the chemotaxis signaling pathway in Escherichia coli, one of the most studied models of signal transduction, and to determine stimulation-induced changes in the pathway. Our FRET analysis identified 19 positive FRET pairs out of the 28 possible protein combinations, with 9 pairs being responsive to chemotactic stimulation. Six stimulation-dependent and five stimulation-independent interactions were direct, whereas other interactions were apparently mediated by scaffolding proteins. Characterization of stimulation-induced responses revealed an additional regulation through activity dependence of interactions involving the adaptation enzyme CheB, and showed complex rearrangement of chemosensory receptors. Our study illustrates how FRET can be efficiently employed to study dynamic protein networks in vivo.

Adaptive dynamics with a single two-state protein

An important step towards understanding biological systems is to relate simple biochemical elements to dynamics. Here, we present the arguably simplest dynamical element in biochemical networks. It consists of a single protein with two states (active and inactive) and an external signal that catalyses the conversion between these two states. Further, there is steady synthesis and degradation of the inactive and active forms, respectively. As this element captures both structural and dynamical features of biochemical networks at the lowest level, we refer to it as a biochemical network unit (BioNetUnit). Using both simulations and mathematical analysis, we find that BioNetUnit shows perfect adaptation that leads to temporal responses to step changes in the incoming signal. Compared with a well-described adaptive system, which is found in bacterial chemotaxis, BioNetUnit has lower sensitivity and its adaptation time is less robust to the base signal levels. We show that these dynamical limitations lead to 'once-and-only-once' responses for certain signal sequences. These findings demonstrate that BioNetUnit is relevant in adaptive and cyclic processes. In particular, it could be seen as a generic representation for ligand-activated receptors that are desensitized upon continuous activation. The analysis of coupled BioNetUnits will show how the presented dynamics at single unit will change upon increased system complexity and how such systems would mediate biological functions.

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.

Model for Protein Concentration Gradients in the Cytoplasm

Intracellular protein concentration gradients are generally thought to be unsustainable at steady-state due to diffusion. Here we show how protein concentration gradients can theoretically be sustained indefinitely through a relatively simple mechanism that couples diffusion to a spatially segregated kinase-phosphatase system. Although it is appreciated that such systems can theoretically give rise to phosphostate gradients, it has been assumed that they do not give rise to gradients in the total protein concentration. Here we show that this assumption does not hold if the two forms of protein have different diffusion coefficients. If, for example, the phosphorylated state binds selectively to a second larger protein or protein complex then a steady state gradient in total protein concentration will be created. We illustrate the principle with an analytical solution to the diffusion-reaction problem and by stochastic individual-based simulations using the Smoldyn program. We argue that protein gradients created in this way need to be considered in experiments using fluorescent probes and could in principle encode spatial information in the cytoplasm.

Robust effects of Tsr–CheBp and CheA–CheYp affinity in bacterial chemotaxis

OBJECTIVE

Cell motility and chemotaxis play a role in the virulence of pathogenic bacteria, such as escape from host immune responses. Escherichia coli chemotaxis provides a well-characterized model system for the bacterial chemotaxis network. Two features of E. coli chemotaxis include signal amplification and robustly accurate adaptation. Recent simulation studies with models considering the effects of other receptors have suggested possible mechanisms for signal amplification. Although precise adaptation to aspartate has been explained by conventional kinetic models, the adaptation behavior of models incorporating the effects of other receptors remains unclear.

METHODS

We concentrated on how receptor crosstalk affects minimization of adaptation error and compared models in which the contribution of other receptors varied.

RESULTS

We demonstrated that the model is adaptable to attractant concentrations ranging from 0.1microM to 10mM with a decreased error rate (from 8% to 2%) when the kinetic constant of CheA and phosphorylated CheY dissociation is increased.

CONCLUSION

The results suggest that accurate adaptation is maintained through control of both the interaction of cytoplasmic Che proteins and the activity of the receptor complex.

Computational methods for diffusion-influenced biochemical reactions

MOTIVATION

We compare stochastic computational methods accounting for space and discrete nature of reactants in biochemical systems. Implementations based on Brownian dynamics (BD) and the reaction-diffusion master equation are applied to a simplified gene expression model and to a signal transduction pathway in Escherichia coli.

RESULTS

In the regime where the number of molecules is small and reactions are diffusion-limited predicted fluctuations in the product number vary between the methods, while the average is the same. Computational approaches at the level of the reaction-diffusion master equation compute the same fluctuations as the reference result obtained from the particle-based method if the size of the sub-volumes is comparable to the diameter of reactants. Using numerical simulations of reversible binding of a pair of molecules we argue that the disagreement in predicted fluctuations is due to different modeling of inter-arrival times between reaction events. Simulations for a more complex biological study show that the different approaches lead to different results due to modeling issues. Finally, we present the physical assumptions behind the mesoscopic models for the reaction-diffusion systems.

AVAILABILITY

Input files for the simulations and the source code of GMP can be found under the following address: http://www.cwi.nl/projects/sic/bioinformatics2007/

Assays for CheC, FliY, and CheX as Representatives of Response Regulator Phosphatases

Much study of two-component systems deals with the excitation of the histidine kinase, activation of the response regulator, and the ultimate target of the signal. Removal of the message is of great importance to these signaling systems. Many methods have evolved in two-component systems to this end. These include autodephosphorylation of the response regulator, hydrolysis of the phosphoryl group by the kinase, or a dedicated phosphatase protein. It has long been known that CheZ is the phosphatase in the chemotaxis system of Escherichia coli and related bacteria. Most bacteria and archaea, however, do not have a cheZ gene, but instead rely on the CheC, CheX, and FliY family of CheY-P phosphatases. Here, we describe assays to test these chemotactic phosphatases, applicable to many other response regulator phosphatases.

Accommodating space, time and randomness in network simulation

Interest in the possibility of dynamically simulating complex cellular processes has escalated markedly in recent years. This interest has been fuelled by three factors: the generally accepted value in understanding living processes as integrated systems; the dramatic increase in computational capability; and the availability of new or improved technology for making the quantitative measurements that are needed to drive and validate cellular simulations. Between the extremes of atom-scale and organism-scale simulation is a vast middle-ground requiring simulation strategies that are capable of dealing with a range of spatial, temporal and molecular abundance scales that are crucial for a comprehensive understanding of integrative cell biology. Although at an early stage, methodological improvements and the development of computational platforms provide some hope that simulations will emerge that can bridge the gap between network models and the true operation of the cell as a complex machine.