A model of excitation and adaptation in bacterial chemotaxis

Evolutionary Stability of Small Molecular Regulatory Networks That Exhibit Near-Perfect Adaptation

Large-scale protein regulatory networks, such as signal transduction systems, contain small-scale modules ('motifs') that carry out specific dynamical functions. Systematic characterization of the properties of small network motifs is therefore of great interest to molecular systems biologists. We simulate a generic model of three-node motifs in search of near-perfect adaptation, the property that a system responds transiently to a change in an environmental signal and then returns near-perfectly to its pre-signal state (even in the continued presence of the signal). Using an evolutionary algorithm, we search the parameter space of these generic motifs for network topologies that score well on a pre-defined measure of near-perfect adaptation. We find many high-scoring parameter sets across a variety of three-node topologies. Of all possibilities, the highest scoring topologies contain incoherent feed-forward loops (IFFLs), and these topologies are evolutionarily stable in the sense that, under 'macro-mutations' that alter the topology of a network, the IFFL motif is consistently maintained. Topologies that rely on negative feedback loops with buffering (NFLBs) are also high-scoring; however, they are not evolutionarily stable in the sense that, under macro-mutations, they tend to evolve an IFFL motif and may-or may not-lose the NFLB motif.

A mathematical model for Escherichia coli chemotaxis to competing stimuli

Chemotactic bacteria sense and respond to temporal and spatial gradients of chemical cues in their surroundings. This phenomenon plays a critical role in many microbial processes such as groundwater bioremediation, microbially enhanced oil recovery, nitrogen fixation in legumes, and pathogenesis of the disease. Chemical heterogeneity in these natural systems may produce numerous competing signals from various directions. Predicting the migration behavior of bacterial populations under such conditions is necessary for designing effective treatment schemes. In this study, experimental studies and mathematical models are reported for the chemotactic response of Escherichia coli to a combination of attractant (α-methylaspartate) and repellent (NiCl₂ ), which bind to the same transmembrane receptor complex. The model describes the binding of chemoeffectors and phosphorylation of the kinase in the signal transduction mechanism. Chemotactic parameters of E. coli (signaling efficiency σ , stimuli sensitivity coefficient γ , and repellent sensitivity coefficient κ ) were determined by fitting the model with experimental results for individual stimuli. Interestingly, our model naturally identifies NiCl₂ as a repellent for κ > 1 . The model is capable of describing quantitatively the response to the individual attractant and repellent, and correctly predicts the change in direction of bacterial population migration for competing stimuli with a twofold increase in repellent concentration.

Statistical signatures of a targeted search by bacteria

Chemoattractant gradients are rarely well-controlled in nature and recent attention has turned to bacterial chemotaxis toward typical bacterial food sources such as food patches or even bacterial prey. In environments with localized food sources reminiscent of a bacterium's natural habitat, striking phenomena-such as the volcano effect or banding-have been predicted or expected to emerge from chemotactic models. However, in practice, from limited bacterial trajectory data it is difficult to distinguish targeted searches from an untargeted search strategy for food sources. Here we use a theoretical model to identify statistical signatures of a targeted search toward point food sources, such as prey. Our model is constructed on the basis that bacteria use temporal comparisons to bias their random walk, exhibit finite memory and are subject to random (Brownian) motion as well as signaling noise. The advantage with using a stochastic model-based approach is that a stochastic model may be parametrized from individual stochastic bacterial trajectories but may then be used to generate a very large number of simulated trajectories to explore average behaviors obtained from stochastic search strategies. For example, our model predicts that a bacterium's diffusion coefficient increases as it approaches the point source and that, in the presence of multiple sources, bacteria may take substantially longer to locate their first source giving the impression of an untargeted search strategy.

Cell responses only partially shape cell-to-cell variations in protein abundances in Escherichia coli chemotaxis

Cell-to-cell variations in protein abundance in clonal cell populations are ubiquitous in living systems. Because protein composition determines responses in individual cells, it stands to reason that the variations themselves are subject to selective pressures. However, the functional role of these cell-to-cell differences is not well understood. One way to tackle questions regarding relationships between form and function is to perturb the form (e.g., change the protein abundances) and observe the resulting changes in some function. Here, we take on the form-function relationship from the inverse perspective, asking instead what specific constraints on cell-to-cell variations in protein abundance are imposed by a given functional phenotype. We develop a maximum entropy-based approach to posing questions of this type and illustrate the method by application to the well-characterized chemotactic response in Escherichia coli. We find that full determination of observed cell-to-cell variations in protein abundances is not inherent in chemotaxis itself but, in fact, appears to be jointly imposed by the chemotaxis program in conjunction with other factors (e.g., the protein synthesis machinery and/or additional nonchemotactic cell functions, such as cell metabolism). These results illustrate the power of maximum entropy as a tool for the investigation of relationships between biological form and function.

Data-driven quantification of the robustness and sensitivity of cell signaling networks

Robustness and sensitivity of responses generated by cell signaling networks has been associated with survival and evolvability of organisms. However, existing methods analyzing robustness and sensitivity of signaling networks ignore the experimentally observed cell-to-cell variations of protein abundances and cell functions or contain ad hoc assumptions. We propose and apply a data-driven maximum entropy based method to quantify robustness and sensitivity of Escherichia coli (E. coli) chemotaxis signaling network. Our analysis correctly rank orders different models of E. coli chemotaxis based on their robustness and suggests that parameters regulating cell signaling are evolutionary selected to vary in individual cells according to their abilities to perturb cell functions. Furthermore, predictions from our approach regarding distribution of protein abundances and properties of chemotactic responses in individual cells based on cell population averaged data are in excellent agreement with their experimental counterparts. Our approach is general and can be used to evaluate robustness as well as generate predictions of single cell properties based on population averaged experimental data in a wide range of cell signaling systems.

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.

Reverse engineering biomolecular systems using -omic data: challenges, progress and opportunities

Recent advances in high-throughput biotechnologies have led to the rapid growing research interest in reverse engineering of biomolecular systems (REBMS). 'Data-driven' approaches, i.e. data mining, can be used to extract patterns from large volumes of biochemical data at molecular-level resolution while 'design-driven' approaches, i.e. systems modeling, can be used to simulate emergent system properties. Consequently, both data- and design-driven approaches applied to -omic data may lead to novel insights in reverse engineering biological systems that could not be expected before using low-throughput platforms. However, there exist several challenges in this fast growing field of reverse engineering biomolecular systems: (i) to integrate heterogeneous biochemical data for data mining, (ii) to combine top-down and bottom-up approaches for systems modeling and (iii) to validate system models experimentally. In addition to reviewing progress made by the community and opportunities encountered in addressing these challenges, we explore the emerging field of synthetic biology, which is an exciting approach to validate and analyze theoretical system models directly through experimental synthesis, i.e. analysis-by-synthesis. The ultimate goal is to address the present and future challenges in reverse engineering biomolecular systems (REBMS) using integrated workflow of data mining, systems modeling and synthetic biology.

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.

Mathematical modeling and experimental validation of chemotaxis under controlled gradients of methyl-aspartate in Escherichia coli

Escherichia coli has evolved an intracellular pathway to regulate its motion termed as chemotaxis so as to move towards a favorable environment such as regions with higher concentration of nutrients. Chemotaxis is a response to temporal and spatial variation of extracellular ligand concentration and randomness in motion induced by collisions with solvent molecules. Previous studies have reported average drift velocities for a given gradient and do not measure drift velocities as a function of time and space. To address this issue, a novel experimental technique was developed to quantify the motion of E. coli cells to varying concentrations and gradients of methyl-aspartate so as to capture the spatial and temporal variation of the drift velocity. A two-state receptor model accounting for the intracellular signaling pathway predicted the experimentally observed increase in drift velocity with gradient and the subsequent adaptation. Our study revealed that the rotational diffusivity induced by the extracellular environment is crucial in determining the drift velocity of E. coli. The model predictions matched with experimental observations only when the response of the intracellular pathway was highly ultra-sensitive to overcome the extracellular randomness. The parametric sensitivity of the pathway indicated that the dissociation constant for the binding of the ligand and the rate constants of the methylation/demethylation of the receptor are key to predict the performance of the chemotactic behavior. The study also indicates a possible role of oxygen in the chemotaxis response and that the response to a ligand may have to account for effects of oxygen.

A predictive computational model of the kinetic mechanism of stimulus-induced transducer methylation and feedback regulation through CheY in archaeal phototaxis and chemotaxis

Background

Photo- and chemotaxis of the archaeon Halobacterium salinarum is based on the control of flagellar motor switching through stimulus-specific methyl-accepting transducer proteins that relay the sensory input signal to a two-component system. Certain members of the transducer family function as receptor proteins by directly sensing specific chemical or physical stimuli. Others interact with specific receptor proteins like the phototaxis photoreceptors sensory rhodopsin I and II, or require specific binding proteins as for example some chemotaxis transducers. Receptor activation by light or a change in receptor occupancy by chemical stimuli results in reversible methylation of glutamate residues of the transducer proteins. Both, methylation and demethylation reactions are involved in sensory adaptation and are modulated by the response regulator CheY.

Results

By mathematical modeling we infer the kinetic mechanisms of stimulus-induced transducer methylation and adaptation. The model (deterministic and in the form of ordinary differential equations) correctly predicts experimentally observed transducer demethylation (as detected by released methanol) in response to attractant and repellent stimuli of wildtype cells, a cheY deletion mutant, and a mutant in which the stimulated transducer species is methylation-deficient.

Conclusions

We provide a kinetic model for signal processing in photo- and chemotaxis in the archaeon H. salinarum suggesting an essential role of receptor cooperativity, antagonistic reversible methylation, and a CheY-dependent feedback on transducer demethylation.

Biomimetic control based on a model of chemotaxis in Escherichia coli

In the field of molecular biology, extending now to the more comprehensive area of systems biology, the development of computer models for synthetic cell simulation has accelerated extensively and has begun to be used for various purposes, such as biochemical analysis. These models, describing the highly efficient environmental searching mechanisms and adaptability of living organisms, can be used as machine-control algorithms in the field of systems engineering. To realize this biomimetic intelligent control, we require a stripped-down model that expresses a series of information-processing tasks from stimulation input to movement. Here we selected the bacterium Escherichia coli as a target organism because it has a relatively simple molecular and organizational structure, which can be characterized using biochemical and genetic analyses. We particularly focused on a motility response known as chemotaxis and developed a computer model that includes not only intracellular information processing but also motor control. After confirming the effectiveness and validity of the proposed model by a series of computer simulations, we applied it to a mobile robot control problem. This is probably the first study showing that a bacterial model can be used as an autonomous control algorithm. Our results suggest that many excellent models proposed thus far for biochemical purposes can be applied to problems in other fields.

Implementation of a Hebbian chemoreceptor model for diffusive source localization

While new approaches to chemical localization have been proposed, animals are still widely used for locating landmines and illegal substances. Existing electronic noses still do not have the necessary sensitivity and accuracy. By modeling a cell's chemical detection system, we can gain insight into the basic "olfactory" system. We use an inspiration from chemotaxis and Hebbian learning to enhance localization and tracking of gradient sources, which can be applied to both chemicals and heat. The eukaryotic receptor clustering model shows improvement over previous prokaryotic chemotaxis-inspired methods that do not take into account receptor clustering. Receptor clustering essentially adapts receptors spatio-temporally. For a mobile simulation, our method locates the source in less convergence time than the other chemotaxis algorithms and insignificantly less time compared to no spatio-temporal filtering (e.g. a single-sensor memoryless case). We then show that local regions of receptor cooperation have the best performance reflecting observations of receptor behavior in biology. To demonstrate the performance of this system in real-time, a stationary 4/8-sensor version of the array is implemented, and the algorithm improves the convergence time, mean, and variance of the Direction-of-Arrival calculation in diffusive, turbulent, and noisy environments.

Predicting perfect adaptation motifs in reaction kinetic networks

Adaptation and compensation mechanisms are important to keep organisms fit in a changing environment. "Perfect adaptation" describes an organism's response to an external stepwise perturbation by resetting some of its variables precisely to their original preperturbation values. Examples of perfect adaptation are found in bacterial chemotaxis, photoreceptor responses, or MAP kinase activities. Two concepts have evolved for how perfect adaptation may be understood. In one approach, so-called "robust perfect adaptation", the adaptation is a network property (due to integral feedback control), which is independent of rate constant values. In the other approach, which we have termed "nonrobust perfect adaptation", a fine-tuning of rate constant values is needed to show perfect adaptation. Although integral feedback describes robust perfect adaptation in general terms, it does not directly show where in a network perfect adaptation may be observed. Using control theoretic methods, we are able to predict robust perfect adaptation sites within reaction kinetic networks and show that a prerequisite for robust perfect adaptation is that the network is open and irreversible. We applied the method on various reaction schemes and found that new (robust) perfect adaptation motifs emerge when considering suggested models of bacterial and eukaryotic chemotaxis.

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.

An allosteric model for transmembrane signaling in bacterial chemotaxis

Bacteria are able to sense chemical gradients over a wide range of concentrations. However, calculations based on the known number of receptors do not predict such a range unless receptors interact with one another in a cooperative manner. A number of recent experiments support the notion that this remarkable sensitivity in chemotaxis is mediated by localized interactions or crosstalk between neighboring receptors. A number of simple, elegant models have proposed mechanisms for signal integration within receptor clusters. What is a lacking is a model, based on known molecular mechanisms and our accumulated knowledge of chemotaxis, that integrates data from multiple, heterogeneous sources. To address this question, we propose an allosteric mechanism for transmembrane signaling in bacterial chemotaxis based on the "trimer of dimers" model, where three receptor dimers form a stable complex with CheW and CheA. The mechanism is used to integrate a diverse set of experimental data in a consistent framework. The main predictions are: (1) trimers of receptor dimers form the building blocks for the signaling complexes; (2) receptor methylation increases the stability of the active state and retards the inhibition arising from ligand-bound receptors within the signaling complex; (3) trimer of dimer receptor complexes aggregate into clusters through their mutual interactions with CheA and CheW; (4) cooperativity arises from neighboring interaction within these clusters; and (5) cluster size is determined by the concentration of receptors, CheA, and CheW. The model is able to explain a number of seemingly contradictory experiments in a consistent manner and, in the process, explain how bacteria are able to sense chemical gradients over a wide range of concentrations by demonstrating how signals are integrated within the signaling complex.

Necessary conditions for a minimal model of receptor to show adaptive response over a wide range of levels of stimulus

Sensory systems respond to temporal changes in the stimulus and adapt to the new level when it persists, this pattern of response being maintained in a wide range of levels of stimulus. Here we use a simple model of adaptation developed by Segel et al. (J. Theor. Biol. 120 (1986) 151-179) and extended by Hauri and Ross (Biophys. J. 68 (1995) 708-722) to study the conditions in which it shows wide range of response. The model consists of a receptor that switches between a variable number of states, either by mass action law or by covalent modification. Using a global optimization procedure, we have optimized the adaptive response of the alternatives of the model with different number of states. We find that it is impossible to obtain a wide range of response if the receptor switches between states following mass-action laws, irrespective of the number of states. Instead, a wide range (of five orders of magnitude of ligand concentration) can be obtained if the receptor switches between several states by irreversible covalent modification, in agreement with previous models. Therefore, in this model, expenditure of energy to maintain a large number of covalent modification cycles operating outside equilibrium is necessary to achieve a wide range of response. The optimal values of the parameters present similar patterns to those reported for specific receptors, but there is no quantitative agreement. For instance, ligand affinity varies several orders of magnitude between the different states of the receptor, what is unlikely to be fulfilled by real systems. To see if the minimal model can show adaptive response and range with quantitatively plausible parameter values a sub-optimal receptor was studied, finding that adaptive response of high intensity can still be obtained in at least three orders of magnitude.

Design and diversity in bacterial chemotaxis: a comparative study in Escherichia coli and Bacillus subtilis

Comparable processes in different species often involve homologous genes. One question is whether the network structure, in particular the feedback control structure, is also conserved. The bacterial chemotaxis pathways in E. coli and B. subtilis both regulate the same task, namely, excitation and adaptation to environmental signals. Both pathways employ many orthologous genes. Yet how these orthologs contribute to network function in each organism is different. To investigate this problem, we propose what is to our knowledge the first computational model for B. subtilis chemotaxis and compare it to previously published models for chemotaxis in E. coli. The models reveal that the core control strategy for signal processing is the same in both organisms, though in B. subtilis there are two additional feedback loops that provide an additional layer of regulation and robustness. Furthermore, the network structures are different despite the similarity of the proteins in each organism. These results demonstrate the limitations of pathway inferences based solely on homology and suggest that the control strategy is an evolutionarily conserved property.

Computational modeling reveals some important differences in the networks that regulate chemotaxis in E. coli and B. subtilis, differences that are hard to predict on the basis of sequence homology alone

Perfect and near-perfect adaptation in a model of bacterial chemotaxis

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its prestimulus level exactly, and this perfect adaptability is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalizations of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near-perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near-perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis.

Control motifs for intracellular regulatory networks

A number of technological innovations are yielding unprecedented data on the networks of biochemical, genetic, and biophysical reactions that underlie cellular behavior and failure. These networks are composed of hundreds to thousands of chemical species and structures, interacting via nonlinear and possibly stochastic physical processes. A central goal of modern biology is to optimally use the data on these networks to understand how their design leads to the observed cellular behaviors and failures. Ultimately, this knowledge should enable cellular engineers to redesign cellular processes to meet industrial needs (such as optimal natural product synthesis), aid in choosing the most effective targets for pharmaceuticals, and tailor treatment for individual genotypes. The size and complexity of these networks and the inevitable lack of complete data, however, makes reaching these goals extremely difficult. If it proves possible to modularize these networks into functional subnetworks, then these smaller networks may be amenable to direct analysis and might serve as regulatory motifs. These motifs, recurring elements of control, may help to deduce the structure and function of partially known networks and form the basis for fulfilling the goals described above. A number of approaches to identifying and analyzing control motifs in intracellular networks are reviewed.

Multi-stage regulation, a key to reliable adaptive biochemical pathways

A general "multi-stage" regulation model, based on linearly connected regulatory units, is formulated to demonstrate how biochemical pathways may achieve high levels of accuracy. The general mechanism, which is robust to changes in biochemical parameters, such as protein concentration and kinetic rate constants, is incorporated into a mathematical model of the bacterial chemotaxis network and provides a new framework for explaining regulation and adaptiveness in this extensively studied system. Although conventional theories suggest that methylation feedback pathways are responsible for chemotactic regulation, the model, which is deduced from known experimental data, indicates that protein interactions downstream of the bacterial receptor complex, such as CheAs and CheZ, may play a crucial and complementary role.

Neural network model of gene expression

Many natural processes consist of networks of interacting elements that, over time, affect each other's state. Their dynamics depend on the pattern of connections and the updating rules for each element. Genomic regulatory networks are networks of this sort. In this paper we use artificial neural networks as a model of the dynamics of gene expression. The significance of the regulatory effect of one gene product on the expression of other genes of the system is defined by a weight matrix. The model considers multigenic regulation including positive and/or negative feedback. The process of gene expression is described by a single network and by two linked networks where transcription and translation are modeled independently. Each of these processes is described by different network controlled by different weight matrices. Methods for computing the parameters of the model from experimental data are discussed. Results computed by means of the model are compared with experimental observations. Generalization to a 'black box' concept, where the molecular processes occurring in the cell are considered as signal processing units forming a global regulatory network, is discussed.-Vohradský, J. Neural network model of gene expression.

Robust perfect adaptation in bacterial chemotaxis through integral feedback control

Integral feedback control is a basic engineering strategy for ensuring that the output of a system robustly tracks its desired value independent of noise or variations in system parameters. In biological systems, it is common for the response to an extracellular stimulus to return to its prestimulus value even in the continued presence of the signal-a process termed adaptation or desensitization. Barkai, Alon, Surette, and Leibler have provided both theoretical and experimental evidence that the precision of adaptation in bacterial chemotaxis is robust to dramatic changes in the levels and kinetic rate constants of the constituent proteins in this signaling network [Alon, U., Surette, M. G., Barkai, N. & Leibler, S. (1998) Nature (London) 397, 168-171]. Here we propose that the robustness of perfect adaptation is the result of this system possessing the property of integral feedback control. Using techniques from control and dynamical systems theory, we demonstrate that integral control is structurally inherent in the Barkai-Leibler model and identify and characterize the key assumptions of the model. Most importantly, we argue that integral control in some form is necessary for a robust implementation of perfect adaptation. More generally, integral control may underlie the robustness of many homeostatic mechanisms.

Models of cell cycle control in eukaryotes

The molecular mechanisms of cell cycle control are now known in enough detail to warrant mathematical modeling by kinetic equations. Despite the repetitive nature of the cell division cycle, the most appropriate models emphasize steady-state solutions rather than limit cycle oscillations, because cells progress toward division by passing a series of checkpoints (steady states).

Robustness in bacterial chemotaxis

Networks of interacting proteins orchestrate the responses of living cells to a variety of external stimuli, but how sensitive is the functioning of these protein networks to variations in their biochemical parameters? One possibility is that to achieve appropriate function, the reaction rate constants and enzyme concentrations need to be adjusted in a precise manner, and any deviation from these 'fine-tuned' values ruins the network's performance. An alternative possibility is that key properties of biochemical networks are robust; that is, they are insensitive to the precise values of the biochemical parameters. Here we address this issue in experiments using chemotaxis of Escherichia coli, one of the best-characterized sensory systems. We focus on how response and adaptation to attractant signals vary with systematic changes in the intracellular concentration of the components of the chemotaxis network. We find that some properties, such as steady-state behaviour and adaptation time, show strong variations in response to varying protein concentrations. In contrast, the precision of adaptation is robust and does not vary with the protein concentrations. This is consistent with a recently proposed molecular mechanism for exact adaptation, where robustness is a direct consequence of the network's architecture.

Chemotaxis receptors: a progress report on structure and function

Recent biochemical and structural studies have provided many new insights into the structure and function of bacterial chemoreceptors. Aspects of their ligand binding, conformational changes, and interactions with other members of the signaling pathway are being defined at the structural level. It is anticipated that the combined effort will soon provide a detailed, unified view of an entire response system.

CheZ has no effect on flagellar motors activated by CheY13DK106YW

The behaviors of both cheZ-deleted and wild-type cells of Escherichia coli were found to be very sensitive to the level of expression of CheZ, a protein known to accelerate the dephosphorylation of the response regulator CheY-phosphate (CheY-P). However, cells induced to run and tumble by the unphosphorylated mutant protein CheY13DK106YW (CheY**) failed to respond to CheZ, even when CheZ was expressed at high levels. Therefore, CheZ neither affects the flagellar motors directly nor sequesters CheY**. In in vitro cross-linking studies, CheY** promoted trimerization of CheZ to the same extent as wild-type CheY but failed to induce the formation of complexes of higher molecular weight observed with CheY-P. Also, CheY** could be cross-linked to FliM, the motor receptor protein, nearly as well as CheY-P. Thus, to CheZ, CheY** looks like CheY, but to FliM, it looks like CheY-P.

Response regulator output in bacterial chemotaxis

Chemotaxis responses in Escherichia coli are mediated by the phosphorylated response-regulator protein P-CheY. Biochemical and genetic studies have established the mechanisms by which the various components of the chemotaxis system, the membrane receptors and Che proteins function to modulate levels of CheY phosphorylation. Detailed models have been formulated to explain chemotaxis sensing in quantitative terms; however, the models cannot be adequately tested without knowledge of the quantitative relationship between P-CheY and bacterial swimming behavior. A computerized image analysis system was developed to collect extensive statistics on freeswimming and individual tethered cells. P-CheY levels were systematically varied by controlled expression of CheY in an E.coli strain lacking the CheY phosphatase, CheZ, and the receptor demethylating enzyme CheB. Tumbling frequency was found to vary with P-CheY concentration in a weakly sigmoidal fashion (apparent Hill coefficient approximately 2.5). This indicates that the high sensitivity of the chemotaxis system is not derived from highly cooperative interactions between P-CheY and the flagellar motor, but rather depends on nonlinear effects within the chemotaxis signal transduction network. The complex relationship between single flagella rotation and free-swimming behavior was examined; our results indicate that there is an additional level of information processing associated with interactions between the individual flagella. An allosteric model of the motor switching process is proposed which gives a good fit to the observed switching induced by P-CheY. Thus the level of intracellular P-CheY can be estimated from behavior determinations: approximately 30% of the intracellular pool of CheY appears to be phosphorylated in fully adapted wild-type cells.

Simulation of prokaryotic genetic circuits

Biochemical and genetic approaches have identified the molecular mechanisms of many genetic reactions, particularly in bacteria. Now a comparably detailed understanding is needed of how groupings of genes and related protein reactions interact to orchestrate cellular functions over the cell cycle, to implement preprogrammed cellular development, or to dynamically change a cell's processes and structures in response to environmental signals. Simulations using realistic, molecular-level models of genetic mechanisms and of signal transduction networks are needed to analyze dynamic behavior of multigene systems, to predict behavior of mutant circuits, and to identify the design principles applicable to design of genetic regulatory circuits. When the underlying design rules for regulatory circuits are understood, it will be far easier to recognize common circuit motifs, to identify functions of individual proteins in regulation, and to redesign circuits for altered functions.

Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets

An integrated understanding of molecular and developmental biology must consider the large number of molecular species involved and the low concentrations of many species in vivo. Quantitative stochastic models of molecular interaction networks can be expressed as stochastic Petri nets (SPNs), a mathematical formalism developed in computer science. Existing software can be used to define molecular interaction networks as SPNs and solve such models for the probability distributions of molecular species. This approach allows biologists to focus on the content of models and their interpretation, rather than their implementation. The standardized format of SPNs also facilitates the replication, extension, and transfer of models between researchers. A simple chemical system is presented to demonstrate the link between stochastic models of molecular interactions and SPNs. The approach is illustrated with examples of models of genetic and biochemical phenomena where the ULTRASAN package is used to present results from numerical analysis and the outcome of simulations.

Origins of individual swimming behavior in bacteria

Cells in a cloned population of coliform bacteria exhibit a wide range of swimming behaviors--a form of non-genetic individuality. We used computer models to examine the proposition that these variations are due to differences in the number of chemotaxis signaling molecules from one cell to the next. Simulations were run in which the concentrations of seven gene products in the chemotaxis pathway were changed either deterministically or stochastically, with the changes derived from independent normal distributions. Computer models with two adaptation mechanisms were compared with experimental results from observations on individuals drawn from genetically identical populations. The range of swimming behavior predicted for cells with a standard deviation of protein copy number per cell of 10% of the mean was found to match closely the experimental range of the wild-type population. We also make predictions for the swimming behaviors of mutant strains lacking the adaptational mechanism that can be tested experimentally.

Kinetic characterization of phosphotransfer between CheA and CheY in the bacterial chemotaxis signal transduction pathway

Phosphorylation of the CheY protein is a crucial step in the chemotaxis signal transduction pathway of Escherichia coli. CheY becomes phosphorylated by acquiring a phosphoryl group from CheA, an autophosphorylating protein kinase. In this study, we utilized a rapid-quench instrument to investigate the kinetics of phosphotransfer in single-turnover experiments. Our results are consistent with a three-step mechanism for the CheA-to-CheY phosphotransfer reaction: (i) reversible binding of CheY to P-CheA; (ii) rapid, reversible phosphotransfer to CheY; (iii) reversible dissociation of the resulting CheA x CheY-P complex. Investigation of the effect of CheY concentration on the observed rate of phosphotransfer demonstrated saturation kinetics; the extrapolated limiting rate constant for phosphotransfer was 650 +/- 200 s(-1), while the Km value indicated from this work was 6.5 +/- 2 microM. We demonstrated that the CheA-CheY phosphotransfer reaction was reversible by observing partial transfer of [32P]phosphate from CheY-P to CheA and by observing the effect of high concentrations of unphosphorylated CheA on the equilibrium: P-CheA + CheY <--> CheA + CheY-P. We found that the rate of phosphotransfer from P-CheA to CheY can be inhibited by unphosphorylated CheA as well as by a fragment of CheA (CheA124-257) that contains the CheY binding site; these results suggest that the unphosphorylated form of CheA can effectively compete with P-CheA for available CheY (Kd approximately 1.5 +/- 0.6 microM for the CheY x CheA124-257 complex and for the CheY x CheA complex).

Control of bacterial chemotaxis

Bacterial chemotaxis, which has been extensively studied for three decades, is the most prominent model system for signal transduction in bacteria. Chemotaxis is achieved by regulating the direction of flagellar rotation. The regulation is carried out by the chemotaxis protein, CheY. This protein is activated by a stimulus-dependent phosphorylation mediated by an autophosphorylatable kinase (CheA) whose activity is controlled by chemoreceptors. Upon phosphorylation, CheY dissociates from its kinase, binds to the switch at the base of the flagellar motor, and changes the motor rotation from the default direction (counter-clockwise) to clockwise. Phosphorylation may also be involved in terminating the response. Phosphorylated CheY binds to the phosphatase CheZ and modulates its oligomeric state and thereby its dephosphorylating activity. Thus CheY phosphorylation appears to be involved in controlling both the excitation and adaptation mechanisms of bacterial chemotaxis. Additional control sites might be involved in bacterial chemotaxis, e.g. lateral control at the receptor level, control at the motor level, or control by metabolites that link central metabolism with chemotaxis.

Kinetic comparison of the specificity of the vancomycin resistance VanSfor two response regulators, VanR and PhoB

Induction of vancomycin resistance in the Gram-positive Enterococci requires a two-componet regulatory system, VanS and VanR, for transcriptional activation of three genes (vanH, A, X) that encode enzymes for a cell wall biosynthetic pathway that produces an altered peptidoglycan intermediate with lower affinity for the antibiotic. The catalytic efficiency (kcat/KM) has been determined for phosphotransfer from the phosphohistidyl form of VanS to both its homologous partner VanR and the heterologous (Escherichia coli) response regulator Phob. The rate of formation of the phosphoaapartyl forms of VanR and PhoB were determined as well as the rate of appearance of inorganic phosphate. Using PhoB in excess of P-VanS, a pseudo-first-order rate constant (kxfer) of 0.2 min-1 for phosphotransfer and a KM for PhoB of 100 microM were readily determined. The corresponding kxfer of 96 min-1 for phosphotransfer from P-VanS to VanR required quench kinetics. A KM of 3 microM was estimated for VanR, leading to a 10(4)-fold preference in kxfer/KM for phosphotransfer to VanR compared to PhoB. No phosphotransfer was detachable to three other E. coli response regulators, OmpR, ArcB, or CreB, providing some sense of the selectivity against two-component regulatory system cross-talk. In the phosphotransfer from P-VanS to PhoB and VanR, there was evidence of competition between water, to give Pi, and the specific aspartyl beta-COO- moiety of either PhoB or VanR with about 25% of the initial flux generating inorganic phosphate. The kinetics of phosphotransfer from P-VanS to VanR were complicated by inhibition by free VanS but, the inhibition pattern could be modeled to yield at KD of 30 nM for VanR binding to free VanS, an affinity similar to that of the CheA-CheY pair in E. coli chemotaxis.

Circuit simulation of genetic networks

Genetic networks with tens to hundreds of genes are difficult to analyze with currently available techniques. Because of the many parallels in the function of these biochemically based genetic circuits and electrical circuits, a hybrid modeling approach is proposed that integrates conventional biochemical kinetic modeling within the framework of a circuit simulation. The circuit diagram of the bacteriophage lambda lysislysogeny decision circuit represents connectivity in signal paths of the biochemical components. A key feature of the lambda genetic circuit is that operons function as active integrated logic components and introduce signal time delays essential for the in vivo behavior of phage lambda.