Chemosensing in Escherichia coli: two regimes of two-state receptors

Signal integration and adaptive sensory diversity tuning in Escherichia coli chemotaxis

In uncertain environments, phenotypic diversity can be advantageous for survival. However, as the environmental uncertainty decreases, the relative advantage of having diverse phenotypes decreases. Here, we show how populations of E. coli integrate multiple chemical signals to adjust sensory diversity in response to changes in the prevalence of each ligand in the environment. Measuring kinase activity in single cells, we quantified the sensitivity distribution to various chemoattractants in different mixtures of background stimuli. We found that when ligands bind uncompetitively, the population tunes sensory diversity to each signal independently, decreasing diversity when the signal's ambient concentration increases. However, among competitive ligands, the population can only decrease sensory diversity one ligand at a time. Mathematical modeling suggests that sensory diversity tuning benefits E. coli populations by modulating how many cells are committed to tracking each signal proportionally as their prevalence changes.

Signal Integration and Adaptive Sensory Diversity Tuning in Escherichia coli Chemotaxis

In uncertain environments, phenotypic diversity can be advantageous for survival. However, as the environmental uncertainty decreases, the relative advantage of having diverse phenotypes decreases. Here, we show how populations of E. coli integrate multiple chemical signals to adjust sensory diversity in response to changes in the prevalence of each ligand in the environment. Measuring kinase activity in single cells, we quantified the sensitivity distribution to various chemoattractants in different mixtures of background stimuli. We found that when ligands bind uncompetitively, the population tunes sensory diversity to each signal independently, decreasing diversity when the signal ambient concentration increases. However, amongst competitive ligands the population can only decrease sensory diversity one ligand at a time. Mathematical modeling suggests that sensory diversity tuning benefits E. coli populations by modulating how many cells are committed to tracking each signal proportionally as their prevalence changes.

Signal integration in chemoreceptor complexes

Motile bacteria use large receptor arrays to detect chemical and physical stimuli in their environment, process this complex information, and accordingly bias their swimming in a direction they deem favorable. The chemoreceptor molecules form tripod-like trimers of receptor dimers through direct contacts between their cytoplasmic tips. A pair of trimers, together with a dedicated kinase enzyme, form a core signaling complex. Hundreds of core complexes network to form extended arrays. While considerable progress has been made in revealing the hierarchical structure of the array, the molecular properties underlying signal processing in these structures remain largely unclear. Here we analyzed the signaling properties of nonnetworked core complexes in live cells by following both conformational and kinase control responses to attractant stimuli and to output-biasing lesions at various locations in the receptor molecule. Contrary to the prevailing view that individual receptors are binary two-state devices, we demonstrate that conformational coupling between the ligand binding and the kinase-control receptor domains is, in fact, only moderate. In addition, we demonstrate communication between neighboring receptors through their trimer-contact domains that biases them to adopt similar signaling states. Taken together, these data suggest a view of signaling in receptor trimers that allows significant signal integration to occur within individual core complexes.

A nonequilibrium allosteric model for receptor-kinase complexes: The role of energy dissipation in chemotaxis signaling

The Escherichia coli chemotaxis signaling pathway has served as a model system for the adaptive sensing of environmental signals by large protein complexes. The chemoreceptors control the kinase activity of CheA in response to the extracellular ligand concentration and adapt across a wide concentration range by undergoing methylation and demethylation. Methylation shifts the kinase response curve by orders of magnitude in ligand concentration while incurring a much smaller change in the ligand binding curve. Here, we show that the disproportionate shift in binding and kinase response is inconsistent with equilibrium allosteric models. To resolve this inconsistency, we present a nonequilibrium allosteric model that explicitly includes the dissipative reaction cycles driven by adenosine triphosphate (ATP) hydrolysis. The model successfully explains all existing joint measurements of ligand binding, receptor conformation, and kinase activity for both aspartate and serine receptors. Our results suggest that the receptor complex acts as an enzyme: Receptor methylation modulates the ON-state kinetics of the kinase (e.g., phosphorylation rate), while ligand binding controls the equilibrium balance between kinase ON/OFF states. Furthermore, sufficient energy dissipation is responsible for maintaining and enhancing the sensitivity range and amplitude of the kinase response. We demonstrate that the nonequilibrium allosteric model is broadly applicable to other sensor-kinase systems by successfully fitting previously unexplained data from the DosP bacterial oxygen-sensing system. Overall, this work provides a nonequilibrium physics perspective on cooperative sensing by large protein complexes and opens up research directions for understanding their microscopic mechanisms through simultaneous measurements and modeling of ligand binding and downstream responses.

Trade-offs between cost and information in cellular prediction

Living cells can leverage correlations in environmental fluctuations to predict the future environment and mount a response ahead of time. To this end, cells need to encode the past signal into the output of the intracellular network from which the future input is predicted. Yet, storing information is costly while not all features of the past signal are equally informative on the future input signal. Here, we show for two classes of input signals that cellular networks can reach the fundamental bound on the predictive information as set by the information extracted from the past signal: Push-pull networks can reach this information bound for Markovian signals, while networks that take a temporal derivative can reach the bound for predicting the future derivative of non-Markovian signals. However, the bits of past information that are most informative about the future signal are also prohibitively costly. As a result, the optimal system that maximizes the predictive information for a given resource cost is, in general, not at the information bound. Applying our theory to the chemotaxis network of Escherichia coli reveals that its adaptive kernel is optimal for predicting future concentration changes over a broad range of background concentrations, and that the system has been tailored to predicting these changes in shallow gradients.

Spatial biology of Ising-like synthetic genetic networks

BACKGROUND

Understanding how spatial patterns of gene expression emerge from the interaction of individual gene networks is a fundamental challenge in biology. Developing a synthetic experimental system with a common theoretical framework that captures the emergence of short- and long-range spatial correlations (and anti-correlations) from interacting gene networks could serve to uncover generic scaling properties of these ubiquitous phenomena.

RESULTS

Here, we combine synthetic biology, statistical mechanics models, and computational simulations to study the spatial behavior of synthetic gene networks (SGNs) in Escherichia coli quasi-2D colonies growing on hard agar surfaces. Guided by the combined mechanisms of the contact process lattice simulation and two-dimensional Ising model (CPIM), we describe the spatial behavior of bi-stable and chemically coupled SGNs that self-organize into patterns of long-range correlations with power-law scaling or short-range anti-correlations. These patterns, resembling ferromagnetic and anti-ferromagnetic configurations of the Ising model near critical points, maintain their scaling properties upon changes in growth rate and cell shape.

CONCLUSIONS

Our findings shed light on the spatial biology of coupled and bistable gene networks in growing cell populations. This emergent spatial behavior could provide insights into the study and engineering of self-organizing gene patterns in eukaryotic tissues and bacterial consortia.

Kinetic modeling of the chemotactic process in run-and-tumble bacteria

The chemotactic process of run-and-tumble bacteria results from modulating the tumbling rate in response to changes in chemoattractant gradients felt by the bacteria. The response has a characteristic memory time and is subject to important fluctuations. These ingredients are considered in a kinetic description of chemotaxis, allowing the computation of the stationary mobility and the relaxation times needed to reach the steady state. For large memory times, these relaxation times become large, implying that finite-time measurements give rise to nonmonotonic currents as a function of the imposed chemoattractant gradient, contrary to the stationary regime where the response is monotonic. The case of an inhomogeneous signal is analyzed. Contrary to the usual Keller-Segel model, the response is nonlocal, and the bacterial profile is smoothed with a characteristic length that grows with the memory time. Finally, the case of traveling signals is considered, where appreciable differences appear compared to memoryless chemotactic descriptions.

Collective chemotaxis in a Voronoi model for confluent clusters

Collective chemotaxis, where single cells cannot climb a biochemical signaling gradient but clusters of cells can, has been observed in different biological contexts, including confluent tissues where there are no gaps or overlaps between cells. Although particle-based models have been developed that predict important features of collective chemotaxis, the mechanisms in those models depend on particle overlaps, and so it remains unclear if they can explain behavior in confluent systems. Here, we develop an open-source code that couples a two-dimensional Voronoi simulation for confluent cell mechanics to a dynamic chemical signal that can diffuse, advect, and/or degrade and use the code to study potential mechanisms for collective chemotaxis in cellular monolayers. We first study the impact of advection on collective chemotaxis and delineate a regime where advective terms are important. Next, we investigate two possible chemotactic mechanisms, contact inhibition of locomotion and heterotypic interfacial tension, and demonstrate that both can drive collective chemotaxis in certain parameter regimes. We further demonstrate that the scaling behavior of cluster motion is well captured by simple analytic theories.

Bacterial chemotaxis to saccharides is governed by a trade-off between sensing and uptake

To swim up gradients of nutrients, E. coli senses nutrient concentrations within its periplasm. For small nutrient molecules, periplasmic concentrations typically match extracellular concentrations. However, this is not necessarily the case for saccharides, such as maltose, which are transported into the periplasm via a specific porin. Previous observations have shown that, under various conditions, E. coli limits maltoporin abundance so that, for extracellular micromolar concentrations of maltose, there are predicted to be only nanomolar concentrations of free maltose in the periplasm. Thus, in the micromolar regime, the total uptake of maltose from the external environment into the cytoplasm is limited not by the abundance of cytoplasmic transport proteins but by the abundance of maltoporins. Here we present results from experiments and modeling suggesting that this porin-limited transport enables E. coli to sense micromolar gradients of maltose despite having a high-affinity ABC transport system that is saturated at these micromolar levels. We used microfluidic assays to study chemotaxis of E. coli in various gradients of maltose and methyl-aspartate and leveraged our experimental observations to develop a mechanistic transport-and-sensing chemotaxis model. Incorporating this model into agent-based simulations, we discover a trade-off between uptake and sensing: although high-affinity transport enables higher uptake rates at low nutrient concentrations, it severely limits the range of dynamic sensing. We thus propose that E. coli may limit periplasmic uptake to increase its chemotactic sensitivity, enabling it to use maltose as an environmental cue.

Hexameric rings of the scaffolding protein CheW enhance response sensitivity and cooperativity in Escherichia coli chemoreceptor arrays

The Escherichia coli chemoreceptor array is a supramolecular assembly that enables cells to respond to extracellular cues dynamically and with great precision and sensitivity. In the array, transmembrane receptors organized as trimers of dimers are connected at their cytoplasmic tips by hexameric rings of alternating subunits of the kinase CheA and the scaffolding protein CheW (CheA-CheW rings). Interactions of CheW molecules with the members of receptor trimers not directly bound to CheA-CheW rings may lead to the formation of hexameric CheW rings in the chemoreceptor array. Here, we detected such CheW rings with a cellular cysteine-directed cross-linking assay and explored the requirements for their formation and their participation in array assembly. We found that CheW ring formation varied with cellular CheW abundance, depended on the presence of receptors capable of a trimer-of-dimers arrangement, and did not require CheA. Cross-linking studies of a CheA~CheW fusion protein incapable of forming homomeric CheW oligomers demonstrated that CheW rings were not essential for the assembly of CheA-containing arrays. Förster resonance energy transfer (FRET)-based kinase assays of arrays containing variable amounts of CheW rings revealed that CheW rings enhanced the cooperativity and the sensitivity of the responses to attractants. We propose that six-membered CheW rings provide the additional interconnectivity required for optimal signaling and gradient tracking performance by chemosensory arrays.

Effect of receptor cooperativity on methylation dynamics in bacterial chemotaxis with weak and strong gradient

We study methylation dynamics of the chemoreceptors as an Escherichia coli cell moves around in a spatially varying chemoattractant environment. We consider attractant concentration with strong and weak spatial gradient. During the uphill and downhill motion of the cell along the gradient, we measure the temporal variation of average methylation level of the receptor clusters. Our numerical simulations we show that the methylation dynamics depends sensitively on the size of the receptor clusters and also on the strength of the gradient. At short times after the beginning of a run, the methylation dynamics is mainly controlled by short runs which are generally associated with high receptor activity. This results in demethylation at short times. But for intermediate or large times, long runs play an important role and depending on receptor cooperativity or gradient strength, the qualitative variation of methylation can be completely different in this time regime. For weak gradient, both for uphill and downhill runs, after the initial demethylation, we find methylation level increases steadily with time for all cluster sizes. Similar qualitative behavior is observed for strong gradient during uphill runs as well. However, the methylation dynamics for downhill runs in strong gradient show highly nontrivial dependence on the receptor cluster size. We explain this behavior as a result of interplay between the sensing and adaptation modules of the signaling network.

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).

Non-Genetic Diversity in Chemosensing and Chemotactic Behavior

Non-genetic phenotypic diversity plays a significant role in the chemotactic behavior of bacteria, influencing how populations sense and respond to chemical stimuli. First, we review the molecular mechanisms that generate phenotypic diversity in bacterial chemotaxis. Next, we discuss the functional consequences of phenotypic diversity for the chemosensing and chemotactic performance of single cells and populations. Finally, we discuss mechanisms that modulate the amount of phenotypic diversity in chemosensory parameters in response to changes in the environment.

Effect of receptor clustering on chemotactic performance of E. coli: Sensing versus adaptation

We show how the competition between sensing and adaptation can result in a performance peak in Escherichia coli chemotaxis using extensive numerical simulations in a detailed theoretical model. Receptor clustering amplifies the input signal coming from ligand binding which enhances chemotactic efficiency. But large clusters also induce large fluctuations in total activity since the number of clusters goes down. The activity and hence the run-tumble motility now gets controlled by methylation levels which are part of adaptation module rather than ligand binding. This reduces chemotactic efficiency.

Connection between the Bacterial Chemotactic Network and Optimal Filtering

The chemotactic network of Escherichia coli has been studied extensively both biophysically and information theoretically. Nevertheless, connection between these two aspects is still elusive. In this work, we report such a connection. We derive an optimal filtering dynamics under the assumption that E. coli's sensory system optimally infers the binary information whether it is swimming up or down along an exponential ligand gradient from noisy sensory signals. Then we show that a standard biochemical model of the chemotactic network is mathematically equivalent to this information-theoretically optimal dynamics. Moreover, we demonstrate that an experimentally observed nonlinear response relation can be reproduced from the optimal dynamics. These results suggest that the biochemical network of E. coli chemotaxis is designed to optimally extract the binary information along an exponential gradient in a noisy condition.

Adaptive tuning of cell sensory diversity without changes in gene expression

In the face of uncertainty, cell populations tend to diversify to enhance survival and growth. Previous studies established that cells can optimize such bet hedging upon environmental change by modulating gene expression to adapt both the average and diversity of phenotypes. Here, we demonstrate that cells can tune phenotypic diversity also using posttranslational modifications. In the chemotaxis network of Escherichia coli, we find, for both major chemoreceptors Tar and Tsr, that cell-to-cell variation in response sensitivity is dynamically modulated depending on the presence or absence of their cognate chemoeffector ligands in the environment. Combining experiments with mathematical modeling, we show that this diversity tuning requires only the environment-dependent covalent modification of chemoreceptors and a standing cell-to-cell variation in their allosteric coupling. Thus, when environmental cues are unavailable, phenotypic diversity enhances the population's readiness for many signals. However, once a signal is perceived, the population focuses on tracking that signal.

Quantitative dissection of transcription in development yields evidence for transcription-factor-driven chromatin accessibility

Thermodynamic models of gene regulation can predict transcriptional regulation in bacteria, but in eukaryotes, chromatin accessibility and energy expenditure may call for a different framework. Here, we systematically tested the predictive power of models of DNA accessibility based on the Monod-Wyman-Changeux (MWC) model of allostery, which posits that chromatin fluctuates between accessible and inaccessible states. We dissected the regulatory dynamics of hunchback by the activator Bicoid and the pioneer-like transcription factor Zelda in living Drosophila embryos and showed that no thermodynamic or non-equilibrium MWC model can recapitulate hunchback transcription. Therefore, we explored a model where DNA accessibility is not the result of thermal fluctuations but is catalyzed by Bicoid and Zelda, possibly through histone acetylation, and found that this model can predict hunchback dynamics. Thus, our theory-experiment dialogue uncovered potential molecular mechanisms of transcriptional regulatory dynamics, a key step toward reaching a predictive understanding of developmental decision-making.

Maximal information transmission is compatible with ultrasensitive biological pathways

Cells are often considered input-output devices that maximize the transmission of information by converting extracellular stimuli (input) via signaling pathways (communication channel) to cell behavior (output). However, in biological systems outputs might feed back into inputs due to cell motility, and the biological channel can change by mutations during evolution. Here, we show that the conventional channel capacity obtained by optimizing the input distribution for a fixed channel may not reflect the global optimum. In a new approach we analytically identify both input distributions and input-output curves that optimally transmit information, given constraints from noise and the dynamic range of the channel. We find a universal optimal input distribution only depending on the input noise, and we generalize our formalism to multiple outputs (or inputs). Applying our formalism to Escherichia coli chemotaxis, we find that its pathway is compatible with optimal information transmission despite the ultrasensitive rotary motors.

Evolutionary and functional insights into the mechanism underlying body-size-related adaptation of mammalian hemoglobin

Hemoglobin (Hb) represents a model protein to study molecular adaptation in vertebrates. Although both affinity and cooperativity of oxygen binding to Hb affect tissue oxygen delivery, only the former was thought to determine molecular adaptations of Hb. Here, we suggest that Hb affinity and cooperativity reflect evolutionary and physiological adaptions that optimized tissue oxygen delivery. To test this hypothesis, we derived the relationship between the Hill coefficient and the relative affinity and conformational changes parameters of the Monod-Wymann-Changeux allosteric model and graphed the 'biophysical Hill landscape' describing this relation. We found that mammalian Hb cooperativity values all reside on a ridge of maximum cooperativity along this landscape that allows for both gross- and fine-tuning of tissue oxygen unloading to meet the distinct metabolic requirements of mammalian tissues for oxygen. Our findings reveal the mechanism underlying body size-related adaptation of mammalian Hb. The generality and implications of our findings are discussed.

New Twists and Turns in Bacterial Locomotion and Signal Transduction

Prokaryotic organisms occupy the most diverse set of environments and conditions on our planet. Their ability to sense and respond to a broad range of external cues remain key research areas in modern microbiology, central to behaviors that underlie beneficial and pathogenic interactions of bacteria with multicellular organisms and within complex ecosystems. Advances in our understanding of the one- and two-component signal transduction systems that underlie these sensing pathways have been driven by advances in imaging the behavior of many individual bacterial cells, as well as visualizing individual proteins and protein arrays within living cells. Cryo-electron tomography continues to provide new insights into the structure and function of chemosensory receptors and flagellar motors, while advances in protein labeling and tracking are applied to understand information flow between receptor and motor. Sophisticated microfluidics allow simultaneous analysis of the behavior of thousands of individual cells, increasing our understanding of how variance between individuals is generated, regulated and employed to maximize fitness of a population. In vitro experiments have been complemented by the study of signal transduction and motility in complex in vivo models, allowing investigators to directly address the contribution of motility, chemotaxis and aggregation/adhesion on virulence during infection. Finally, systems biology approaches have demonstrated previously uncharted areas of protein space in which novel two-component signal transduction pathways can be designed and constructed de novo These exciting experimental advances were just some of the many novel findings presented at the 15^th Bacterial Locomotion and Signal Transduction conference (BLAST XV) in January 2019.

Predictive shifts in free energy couple mutations to their phenotypic consequences

Mutation is a critical mechanism by which evolution explores the functional landscape of proteins. Despite our ability to experimentally inflict mutations at will, it remains difficult to link sequence-level perturbations to systems-level responses. Here, we present a framework centered on measuring changes in the free energy of the system to link individual mutations in an allosteric transcriptional repressor to the parameters which govern its response. We find that the energetic effects of the mutations can be categorized into several classes which have characteristic curves as a function of the inducer concentration. We experimentally test these diagnostic predictions using the well-characterized LacI repressor of Escherichia coli, probing several mutations in the DNA binding and inducer binding domains. We find that the change in gene expression due to a point mutation can be captured by modifying only the model parameters that describe the respective domain of the wild-type protein. These parameters appear to be insulated, with mutations in the DNA binding domain altering only the DNA affinity and those in the inducer binding domain altering only the allosteric parameters. Changing these subsets of parameters tunes the free energy of the system in a way that is concordant with theoretical expectations. Finally, we show that the induction profiles and resulting free energies associated with pairwise double mutants can be predicted with quantitative accuracy given knowledge of the single mutants, providing an avenue for identifying and quantifying epistatic interactions.

Run-and-tumble motion with steplike responses to a stochastic input

We study a simple run-and-tumble random walk whose switching frequencies between run mode and tumble mode depend on a stochastic signal. We consider a particularly sharp, steplike dependence, where the run-to-tumble switching probability jumps from zero to one as the signal crosses a particular value (say y_{1}) from below. Similarly, tumble-to-run switching probability also shows a jump like this as the signal crosses another value (y_{2}<y_{1}) from above. We are interested in characterizing the effect of signaling noise on the long-time behavior of the random walker. We consider two different time-evolutions of the stochastic signal. In one case, the signal dynamics is an independent stochastic process and does not depend on the run-and-tumble motion. In this case we can analytically calculate the mean value and the complete distribution function of the run duration and tumble duration. In the second case, we assume that the signal dynamics is influenced by the spatial location of the random walker. For this system, we numerically measure the steady state position distribution of the random walker. We discuss some similarities and differences between our system and Escherichia coli chemotaxis, which is another well-known run-and-tumble motion encountered in nature.

Long-term positioning and polar preference of chemoreceptor clusters in E. coli

The bacterial chemosensory arrays are a notable model for studying the basic principles of receptor clustering and cellular organization. Here, we provide a new perspective regarding the long-term dynamics of these clusters in growing E. coli cells. We demonstrate that pre-existing lateral clusters tend to avoid translocation to pole regions and, therefore, continually shuttle between the cell poles for many generations while being static relative to the local cell-wall matrix. We also show that the polar preference of clusters results fundamentally from reduced clustering efficiency in the lateral region, rather than a developmental-like progression of clusters. Furthermore, polar preference is surprisingly robust to structural alterations designed to probe preference due to curvature sorting, perturbing the cell envelope physiology affects the cluster-size distribution, and the size-dependent mobility of receptor complexes differs between polar and lateral regions. Thus, distinct envelope physiology in the polar and lateral cell regions may contribute to polar preference.

Bacterial chemoreceptors form clusters, preferably at  the cell poles. Here, Koler et al. show that polar and lateral clusters exhibit distinct long-term positional dynamics   and that polar bias may be due to differences in mobility of receptor complexes between the polar and lateral cell regions.

Optimal methylation noise for best chemotactic performance of E. coli

In response to a concentration gradient of chemoattractant, E. coli bacterium modulates the rotational bias of flagellar motors which control its run-and-tumble motion, to migrate towards regions of high chemoattractant concentration. Presence of stochastic noise in the biochemical pathway of the cell has important consequences on the switching mechanism of motor bias, which in turn affects the runs and tumbles of the cell in a significant way. We model the intracellular reaction network in terms of coupled time evolution of three stochastic variables-kinase activity, methylation level, and CheY-P protein level-and study the effect of methylation noise on the chemotactic performance of the cell. In presence of a spatially varying nutrient concentration profile, a good chemotactic performance allows the cell to climb up the concentration gradient quickly and localize in the nutrient-rich regions in the long time limit. Our simulations show that the best performance is obtained at an optimal noise strength. While it is expected that chemotaxis will be weaker for very large noise, it is counterintuitive that the performance worsens even when noise level falls below a certain value. We explain this striking result by detailed analysis of CheY-P protein level statistics for different noise strengths. We show that when the CheY-P level falls below a certain (noise-dependent) threshold the cell tends to move down the concentration gradient of the nutrient, which has a detrimental effect on its chemotactic response. This threshold value decreases as noise is increased, and this effect is responsible for noise-induced enhancement of chemotactic performance. In a harsh chemical environment, when the nutrient degrades with time, the amount of nutrient intercepted by the cell trajectory is an effective performance criterion. In this case also, depending on the nutrient lifetime, we find an optimum noise strength when the performance is at its best.

Behavioral Variability and Phenotypic Diversity in Bacterial Chemotaxis

Living cells detect and process external signals using signaling pathways that are affected by random fluctuations. These variations cause the behavior of individual cells to fluctuate over time (behavioral variability) and generate phenotypic differences between genetically identical individuals (phenotypic diversity). These two noise sources reduce our ability to predict biological behavior because they diversify cellular responses to identical signals. Here, we review recent experimental and theoretical advances in understanding the mechanistic origin and functional consequences of such variation in Escherichia coli chemotaxis-a well-understood model of signal transduction and behavior. After briefly summarizing the architecture and logic of the chemotaxis system, we discuss determinants of behavior and chemotactic performance of individual cells. Then, we review how cell-to-cell differences in protein abundance map onto differences in individual chemotactic abilities and how phenotypic variability affects the performance of the population. We conclude with open questions to be addressed by future research.

Tuning Transcriptional Regulation through Signaling: A Predictive Theory of Allosteric Induction

Allosteric regulation is found across all domains of life, yet we still lack simple, predictive theories that directly link the experimentally tunable parameters of a system to its input-output response. To that end, we present a general theory of allosteric transcriptional regulation using the Monod-Wyman-Changeux model. We rigorously test this model using the ubiquitous simple repression motif in bacteria by first predicting the behavior of strains that span a large range of repressor copy numbers and DNA binding strengths and then constructing and measuring their response. Our model not only accurately captures the induction profiles of these strains, but also enables us to derive analytic expressions for key properties such as the dynamic range and [EC₅₀]. Finally, we derive an expression for the free energy of allosteric repressors that enables us to collapse our experimental data onto a single master curve that captures the diverse phenomenology of the induction profiles.

Adaptation of Living Systems

Adaptation refers to the biological phenomenon where living systems change their internal states in response to changes in their environments in order to maintain certain key functions critical for their survival and fitness. Adaptation is one of the most ubiquitous and arguably one of the most fundamental properties of living systems. It occurs throughout all biological scales, from adaptation of populations of species over evolutionary time to adaptation of a single cell to different environmental stresses during its life span. In this article, we review some of the recent progress made in understanding molecular mechanisms of cellular level adaptation. We take the minimalist (or the physicist) approach and study the simplest systems that exhibit generic adaptive behaviors. We focus on understanding the basic biochemical interaction networks in living matter that are responsible for adaptation dynamics. By combining theoretical modeling with quantitative experimentation, we demonstrate universal features in adaptation as well as important differences in different cellular systems, including chemotaxis in bacterium cells (Escherichia coli) and eukaryotic cells (Dictyostelium). Future work in extending the modeling framework to study adaptation in more complex systems such as sensory neurons are discussed.

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.

Multiple sources of slow activity fluctuations in a bacterial chemosensory network

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

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

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

Many sophisticated computer programs use random number generators to help solve challenging problems. These problems range from achieving secure communication across the Internet to deciding how best to invest in the stock market. Much research in recent years has found that randomness is also widespread in living cells, where it is often called “noise”. For example, the activity of some genes is so unpredictable to the extent that it appears random. Yet, relatively little is known about how such gene-expression noise propagates up to change how the cell behaves. Many open questions also remain about how cells might exploit these or other fluctuations to achieve complex tasks, like people use random number generators.

Bacteria perform a number of complex tasks. Some bacteria will swim toward chemicals that suggest a potential reward, such as food. Yet they swim away from chemicals that could lead them to harm. This ability is called chemotaxis and it relies on a network of interacting enzymes and other proteins that coordinates a bacterium’s movements with the input from its senses.

Keegstra et al. set out to find sources of noise that might act as random number generators and help the bacterium E. coli to best perform chemotaxis. An improved version of a technique called in vivo Förster resonance energy transfer (or in vivo FRET for short) was used to give a detectable signal when two proteins involved in the chemotaxis network interacted inside a single bacterium. The experiments showed that this protein network amplifies gene-expression noise for some genes while lessening it for others. In addition, the interactions between proteins encoded by genes acted as an extra source of noise, even when gene-expression noise was eliminated.

Keegstra et al. found that the amount of signaling within the chemotaxis network, as measured by in vivo FRET, varied wildly over time. This revealed two sources of noise at the level of protein signaling. One was due to randomness in the activity of the enzymes involved in tuning the cell’s sensitivity to changes in its environment. The other was due to protein interactions within a large complex that acts as the cell’s sensor. Unexpectedly, this second source of noise under some conditions could be so strong that it flipped the output of the cell’s signaling network back and forth between just two states: “on” and “off”.

Together these findings uncover how signaling networks can not only amplify or lessen gene-expression noise, but can themselves become a source of random events. The new knowledge of how such random events interact with a complex trait in a living cell – namely chemotaxis – could aid future antimicrobial strategies, because many bacteria use chemotaxis to help them establish infections. More generally, the new insights about noise in protein networks could help engineers seeking to build synthetic biochemical networks or produce useful compounds in living cells.

Drift and Behavior of E. coli Cells

Chemotaxis of the bacterium Escherichia coli is well understood in shallow chemical gradients, but its swimming behavior remains difficult to interpret in steep gradients. By focusing on single-cell trajectories from simulations, we investigated the dependence of the chemotactic drift velocity on attractant concentration in an exponential gradient. Whereas maxima of the average drift velocity can be interpreted within analytical linear-response theory of chemotaxis in shallow gradients, limits in drift due to steep gradients and finite number of receptor-methylation sites for adaptation go beyond perturbation theory. For instance, we found a surprising pinning of the cells to the concentration in the gradient at which cells run out of methylation sites. To validate the positions of maximal drift, we recorded single-cell trajectories in carefully designed chemical gradients using microfluidics.

Mechanism of bidirectional thermotaxis in Escherichia coli

In bacteria various tactic responses are mediated by the same cellular pathway, but sensing of physical stimuli remains poorly understood. Here, we combine an in-vivo analysis of the pathway activity with a microfluidic taxis assay and mathematical modeling to investigate the thermotactic response of Escherichia coli. We show that in the absence of chemical attractants E. coli exhibits a steady thermophilic response, the magnitude of which decreases at higher temperatures. Adaptation of wild-type cells to high levels of chemoattractants sensed by only one of the major chemoreceptors leads to inversion of the thermotactic response at intermediate temperatures and bidirectional cell accumulation in a thermal gradient. A mathematical model can explain this behavior based on the saturation-dependent kinetics of adaptive receptor methylation. Lastly, we find that the preferred accumulation temperature corresponds to optimal growth in the presence of the chemoattractant serine, pointing to a physiological relevance of the observed thermotactic behavior.

Many bacteria can move towards or away from chemicals, heat and other stimuli in their environment. The ability of bacteria to move in response to nutrients and other chemicals, known as chemotaxis, is the best understood of these phenomena. Bacteria generally swim in a fairly random way and frequently change direction. During chemotaxis, however, the bacteria sense changes in the concentrations of a chemical in their surroundings and this biases the direction in which they swim so that they spend more time swimming towards or away from the source of the chemical. The bacteria have various receptor proteins that can detect different chemicals. For example, the Tar and Tsr receptors can recognize chemicals called aspartate and serine, respectively, which are – amongst other things – nutrients that are used to build proteins.

Tar and Tsr are also involved in the response to temperature, referred to as thermotaxis. At low temperatures, a bacterium Escherichia coli will move towards sources of heat. Yet when the bacteria detect both serine and aspartate they may reverse the response and move towards colder areas instead. However, it was not clear why the bacteria do this, and what roles Tar and Tsr play in this response.

Paulick et al. have now combined approaches that directly visualise signalling inside living bacteria and that track the movements of individual bacterial cellswith mathematical modelling to investigate thermotaxis in E. coli. The experiments show that the bacteria’s behaviour could be explained by interplay between the responses mediated by Tar and Tsr. In the absence of both serine and aspartate, both receptors stimulate heat-seeking responses, causing the bacteria to move towards hotter areas. When only aspartate is present, Tsr continues to stimulate the heat-seeking response, but the aspartate causes Tar to switch to promoting a cold-seeking response instead. This leads to the bacteria accumulating in areas of intermediate temperature. In the presence of serine only, the bacteria behave in a similar way because the receptors swap roles so that Tsr stimulates the cold-seeking response, while Tar promotes the heat-seeking one.

The intermediate temperature at which the bacteria accumulate in response to serine is also around the optimal temperature for E.coli growth in presence of this chemical, suggesting that thermotaxis might play an important role in allowing bacteria to survive and grow in many different environments, including in the human body. Thus, understanding how chemotaxis and thermotaxis are regulated may lead to new ways to control how bacteria behave in patients and natural environments.

Noise-Induced Increase of Sensitivity in Bacterial Chemotaxis

Flagellated bacteria, like Escherichia coli, can swim toward beneficial environments by modulating the rotational direction of their flagellar motors through a chemotaxis signal transduction network. The noise of this network, the random fluctuation of the intracellular concentration of the signal protein CheY-P with time, has been identified in studies of single cell behavioral variability, and found to be important in coordination of multiple motors in a bacterium and in enhancement of bacterial drift velocity in chemical gradients. Here, by comparing the behavioral difference between motors of wild-type E. coli and mutants without signal noise, we measured the magnitude of this noise in wild-type cells, and found that the noise increases the sensitivity of the bacterial chemotaxis network downstream at the level of the flagellar motor. This provided a simple mechanism for the noise-induced enhancement of chemotactic drift, which we confirmed by simulating the E. coli chemotactic motion in various spatial profiles of chemo-attractant concentration.

Feedback between motion and sensation provides nonlinear boost in run-and-tumble navigation

Many organisms navigate gradients by alternating straight motions (runs) with random reorientations (tumbles), transiently suppressing tumbles whenever attractant signal increases. This induces a functional coupling between movement and sensation, since tumbling probability is controlled by the internal state of the organism which, in turn, depends on previous signal levels. Although a negative feedback tends to maintain this internal state close to adapted levels, positive feedback can arise when motion up the gradient reduces tumbling probability, further boosting drift up the gradient. Importantly, such positive feedback can drive large fluctuations in the internal state, complicating analytical approaches. Previous studies focused on what happens when the negative feedback dominates the dynamics. By contrast, we show here that there is a large portion of physiologically-relevant parameter space where the positive feedback can dominate, even when gradients are relatively shallow. We demonstrate how large transients emerge because of non-normal dynamics (non-orthogonal eigenvectors near a stable fixed point) inherent in the positive feedback, and further identify a fundamental nonlinearity that strongly amplifies their effect. Most importantly, this amplification is asymmetric, elongating runs in favorable directions and abbreviating others. The result is a “ratchet-like” gradient climbing behavior with drift speeds that can approach half the maximum run speed of the organism. Our results thus show that the classical drawback of run-and-tumble navigation—wasteful runs in the wrong direction—can be mitigated by exploiting the non-normal dynamics implicit in the run-and-tumble strategy.

Countless bacteria, larvae and even larger organisms (and robots) navigate gradients by alternating periods of straight motion (runs) with random reorientation events (tumbles). Control of the tumble probability is based on previously-encountered signals. A drawback of this run-and-tumble strategy is that occasional runs in the wrong direction are wasteful. Here we show that there is an operating regime within the organism’s internal parameter space where run-and-tumble navigation can be extremely efficient. We characterize how the positive feedback between behavior and sensed signal results in a type of non-equilibrium dynamics, with the organism rapidly tumbling after moving in the wrong direction and extending motion in the right ones. For a distant source, then, the organism can find it fast.

Barrier Crossing in Escherichia coli Chemotaxis

We study cell navigation in spatiotemporally complex environments by developing a microfluidic racetrack device that creates a traveling wave with multiple peaks and a tunable wave speed. We find that while the population-averaged chemotaxis drift speed increases with wave speed for low wave speed, it decreases sharply for high wave speed. This reversed dependence of population-averaged chemotaxis drift speed on wave speed is caused by a "barrier-crossing" phenomenon, where a cell hops backwards from one peak attractant location to the peak behind by crossing an unfavorable (barrier) region with low attractant concentrations. By using a coarse-grained model of chemotaxis, we map bacterial motility in an attractant field to the random motion of an overdamped particle in an effective potential. The observed barrier-crossing phenomenon of living cells and its dependence on the spatiotemporal profile of attractant concentration are explained quantitatively by Kramers reaction rate theory.

Monod-Wyman-Changeux Analysis of Ligand-Gated Ion Channel Mutants

We present a framework for computing the gating properties of ligand-gated ion channel mutants using the Monod-Wyman-Changeux (MWC) model of allostery. We derive simple analytic formulas for key functional properties such as the leakiness, dynamic range, half-maximal effective concentration ([EC₅₀]), and effective Hill coefficient, and explore the full spectrum of phenotypes that are accessible through mutations. Specifically, we consider mutations in the channel pore of nicotinic acetylcholine receptor (nAChR) and the ligand binding domain of a cyclic nucleotide-gated (CNG) ion channel, demonstrating how each mutation can be characterized as only affecting a subset of the biophysical parameters. In addition, we show how the unifying perspective offered by the MWC model allows us, perhaps surprisingly, to collapse the plethora of dose-response data from different classes of ion channels into a universal family of curves.

Networked Chemoreceptors Benefit Bacterial Chemotaxis Performance

Motile bacteria use large receptor arrays to detect and follow chemical gradients in their environment. Extended receptor arrays, composed of networked signaling complexes, promote cooperative stimulus control of their associated signaling kinases. Here, we used structural lesions at the communication interface between core complexes to create an Escherichia coli strain with functional but dispersed signaling complexes. This strain allowed us to directly study how networking of signaling complexes affects chemotactic signaling and gradient-tracking performance. We demonstrate that networking of receptor complexes provides bacterial cells with about 10-fold-heightened detection sensitivity to attractants while maintaining a wide dynamic range over which receptor adaptational modifications can tune response sensitivity. These advantages proved especially critical for chemotaxis toward an attractant source under conditions in which bacteria are unable to alter the attractant gradient.

Chemoreceptor arrays are found in many motile bacteria. However, although our understanding of bacterial chemotaxis is quite detailed, the signaling and behavioral advantages of networked receptor arrays had not been directly studied in cells. We have recently shown that lesions in a key interface of the E. coli receptor array diminish physical connections and functional coupling between core signaling complexes while maintaining their basic signaling capacity. In this study, we exploited an interface 2 mutant to show, for the first time, that coupling between core complexes substantially enhances stimulus detection and chemotaxis performance.

Non-genetic diversity modulates population performance

Biological functions are typically performed by groups of cells that express predominantly the same genes, yet display a continuum of phenotypes. While it is known how one genotype can generate such non-genetic diversity, it remains unclear how different phenotypes contribute to the performance of biological function at the population level. We developed a microfluidic device to simultaneously measure the phenotype and chemotactic performance of tens of thousands of individual, freely swimming Escherichia coli as they climbed a gradient of attractant. We discovered that spatial structure spontaneously emerged from initially well-mixed wild-type populations due to non-genetic diversity. By manipulating the expression of key chemotaxis proteins, we established a causal relationship between protein expression, non-genetic diversity, and performance that was theoretically predicted. This approach generated a complete phenotype-to-performance map, in which we found a nonlinear regime. We used this map to demonstrate how changing the shape of a phenotypic distribution can have as large of an effect on collective performance as changing the mean phenotype, suggesting that selection could act on both during the process of adaptation.

Sugar Influx Sensing by the Phosphotransferase System of Escherichia coli

The phosphotransferase system (PTS) plays a pivotal role in the uptake of multiple sugars in Escherichia coli and many other bacteria. In the cell, individual sugar-specific PTS branches are interconnected through a series of phosphotransfer reactions, thus creating a global network that not only phosphorylates incoming sugars but also regulates a number of cellular processes. Despite the apparent importance of the PTS network in bacterial physiology, the holistic function of the network in the cell remains unclear. Here we used Förster resonance energy transfer (FRET) to investigate the PTS network in E. coli, including the dynamics of protein interactions and the processing of different stimuli and their transmission to the chemotaxis pathway. Our results demonstrate that despite the seeming complexity of the cellular PTS network, its core part operates in a strikingly simple way, sensing the overall influx of PTS sugars irrespective of the sugar identity and distributing this information equally through all studied branches of the network. Moreover, it also integrates several other specific metabolic inputs. The integrated output of the PTS network is then transmitted linearly to the chemotaxis pathway, in stark contrast to the amplification of conventional chemotactic stimuli. Finally, we observe that default uptake through the uninduced PTS network correlates well with the quality of the carbon source, apparently representing an optimal regulatory strategy.

The bacterial phosphotransferase system (PTS) mediates uptake of multiple sugars from the environment and also controls cell physiology and swimming behavior in sugar gradients. In Escherichia coli and other bacteria, the PTS consists of a number of sugar-specific branches, interconnected via shared components through a series of phosphotransfer reactions. Whereas most previous studies have focused on understanding individual PTS branches, the holistic function of the entire PTS network in the cell remained elusive. In this study we address this question by investigating the dynamics of multiple protein interactions within the cellular PTS network upon stimulation with sugars and other metabolites. We demonstrate that despite its seeming complexity, the core part of the PTS network operates in a strikingly simple way, sensing the overall influx of PTS sugars and key metabolites into the cell and utilizing this information to control bacterial behavior. We further show that the default influx of the carbon source correlates with its quality, and we use computer simulations to demonstrate that this correlation apparently represents an optimal regulatory strategy.

Allosteric proteins as logarithmic sensors

Many sensory systems, from vision and hearing in animals to signal transduction in cells, respond to fold changes in signal relative to background. Responding to fold change requires that the system senses signal on a logarithmic scale, responding identically to a change in signal level from 1 to 3, or from 10 to 30. It is an ongoing search in the field to understand the ways in which a logarithmic sensor can be implemented at the molecular level. In this work, we present evidence that logarithmic sensing can be implemented with a single protein, by means of allosteric regulation. Specifically, we find that mathematical models show that allosteric proteins can respond to stimuli on a logarithmic scale. Next, we present evidence from measurements in the literature that some allosteric proteins do operate in a parameter regime that permits logarithmic sensing. Finally, we present examples suggesting that allosteric proteins are indeed used in this capacity: allosteric proteins play a prominent role in systems where fold-change detection has been proposed. This finding suggests a role as logarithmic sensors for the many allosteric proteins across diverse biological processes.

The Role of Adaptation in Bacterial Speed Races

Evolution of biological sensory systems is driven by the need for efficient responses to environmental stimuli. A paradigm among prokaryotes is the chemotaxis system, which allows bacteria to navigate gradients of chemoattractants by biasing their run-and-tumble motion. A notable feature of chemotaxis is adaptation: after the application of a step stimulus, the bacterial running time relaxes to its pre-stimulus level. The response to the amino acid aspartate is precisely adapted whilst the response to serine is not, in spite of the same pathway processing the signals preferentially sensed by the two receptors Tar and Tsr, respectively. While the chemotaxis pathway in E. coli is well characterized, the role of adaptation, its functional significance and the ecological conditions where chemotaxis is selected, are largely unknown. Here, we investigate the role of adaptation in the climbing of gradients by E. coli. We first present theoretical arguments that highlight the mechanisms that control the efficiency of the chemotactic up-gradient motion. We discuss then the limitations of linear response theory, which motivate our subsequent experimental investigation of E. coli speed races in gradients of aspartate, serine and combinations thereof. By using microfluidic techniques, we engineer controlled gradients and demonstrate that bacterial fronts progress faster in equal-magnitude gradients of serine than aspartate. The effect is observed over an extended range of concentrations and is not due to differences in swimming velocities. We then show that adding a constant background of serine to gradients of aspartate breaks the adaptation to aspartate, which results in a sped-up progression of the fronts and directly illustrate the role of adaptation in chemotactic gradient-climbing.

Biological sensory pathways are presumed to evolve for the processing of environmental information, yet quantitative evidence is scant. Chemotaxis allows bacteria to sense chemical gradients but their ecological distribution, e.g. whether natural gradients sensed by E. coli change slowly or rapidly in space and time, is unknown. That distribution matters, as it controls constraints and selective pressure acting on the pathway. We used microfluidic devices to generate controlled chemoattractant gradients and measure the speed of bacterial climbing of those gradients. We could thereby assay the impact of adaptation properties of the chemotaxis pathway onto the progression of gradient climbing. We specifically show that loss of adaptation, induced by adding a background of serine to gradients of aspartate, leads to a faster progression of the bacteria along the chemoattractant gradient. We finally discuss why our experiments suggest that ecological conditions are likely to involve chemoattractant profiles more complex than constant gradients usually considered in the laboratory.

Information processing in bacteria: memory, computation, and statistical physics: a key issues review

Living systems have to constantly sense their external environment and adjust their internal state in order to survive and reproduce. Biological systems, from as complex as the brain to a single E. coli cell, have to process these data in order to make appropriate decisions. How do biological systems sense external signals? How do they process the information? How do they respond to signals? Through years of intense study by biologists, many key molecular players and their interactions have been identified in different biological machineries that carry out these signaling functions. However, an integrated, quantitative understanding of the whole system is still lacking for most cellular signaling pathways, not to say the more complicated neural circuits. To study signaling processes in biology, the key thing to measure is the input-output relationship. The input is the signal itself, such as chemical concentration, external temperature, light (intensity and frequency), and more complex signals such as the face of a cat. The output can be protein conformational changes and covalent modifications (phosphorylation, methylation, etc), gene expression, cell growth and motility, as well as more complex output such as neuron firing patterns and behaviors of higher animals. Due to the inherent noise in biological systems, the measured input-output dependence is often noisy. These noisy data can be analysed by using powerful tools and concepts from information theory such as mutual information, channel capacity, and the maximum entropy hypothesis. This information theory approach has been successfully used to reveal the underlying correlations between key components of biological networks, to set bounds for network performance, and to understand possible network architecture in generating observed correlations. Although the information theory approach provides a general tool in analysing noisy biological data and may be used to suggest possible network architectures in preserving information, it does not reveal the underlying mechanism that leads to the observed input-output relationship, nor does it tell us much about which information is important for the organism and how biological systems use information to carry out specific functions. To do that, we need to develop models of the biological machineries, e.g. biochemical networks and neural networks, to understand the dynamics of biological information processes. This is a much more difficult task. It requires deep knowledge of the underlying biological network-the main players (nodes) and their interactions (links)-in sufficient detail to build a model with predictive power, as well as quantitative input-output measurements of the system under different perturbations (both genetic variations and different external conditions) to test the model predictions to guide further development of the model. Due to the recent growth of biological knowledge thanks in part to high throughput methods (sequencing, gene expression microarray, etc) and development of quantitative in vivo techniques such as various florescence technology, these requirements are starting to be realized in different biological systems. The possible close interaction between quantitative experimentation and theoretical modeling has made systems biology an attractive field for physicists interested in quantitative biology. In this review, we describe some of the recent work in developing a quantitative predictive model of bacterial chemotaxis, which can be considered as the hydrogen atom of systems biology. Using statistical physics approaches, such as the Ising model and Langevin equation, we study how bacteria, such as E. coli, sense and amplify external signals, how they keep a working memory of the stimuli, and how they use these data to compute the chemical gradient. In particular, we will describe how E. coli cells avoid cross-talk in a heterogeneous receptor cluster to keep a ligand-specific memory. We will also study the thermodynamic costs of adaptation for cells to maintain an accurate memory. The statistical physics based approach described here should be useful in understanding design principles for cellular biochemical circuits in general.

Statistical Mechanics of Allosteric Enzymes

The concept of allostery in which macromolecules switch between two different conformations is a central theme in biological processes ranging from gene regulation to cell signaling to enzymology. Allosteric enzymes pervade metabolic processes, yet a simple and unified treatment of the effects of allostery in enzymes has been lacking. In this work, we take a step toward this goal by modeling allosteric enzymes and their interaction with two key molecular players-allosteric regulators and competitive inhibitors. We then apply this model to characterize existing data on enzyme activity, comment on how enzyme parameters (such as substrate binding affinity) can be experimentally tuned, and make novel predictions on how to control phenomena such as substrate inhibition.

Singly Flagellated Pseudomonas aeruginosa Chemotaxes Efficiently by Unbiased Motor Regulation

Pseudomonas aeruginosa is an opportunistic human pathogen that has long been known to chemotax. More recently, it has been established that chemotaxis is an important factor in the ability of P. aeruginosa to make biofilms. Genes that allow P. aeruginosa to chemotax are homologous with genes in the paradigmatic model organism for chemotaxis, Escherichia coli. However, P. aeruginosa is singly flagellated and E. coli has multiple flagella. Therefore, the regulation of counterclockwise/clockwise flagellar motor bias that allows E. coli to efficiently chemotax by runs and tumbles would lead to inefficient chemotaxis by P. aeruginosa, as half of a randomly oriented population would respond to a chemoattractant gradient in the wrong sense. How P. aeruginosa regulates flagellar rotation to achieve chemotaxis is not known. Here, we analyze the swimming trajectories of single cells in microfluidic channels and the rotations of cells tethered by their flagella to the surface of a variable-environment flow cell. We show that P. aeruginosa chemotaxes by symmetrically increasing the durations of both counterclockwise and clockwise flagellar rotations when swimming up the chemoattractant gradient and symmetrically decreasing rotation durations when swimming down the chemoattractant gradient. Unlike the case for E. coli, the counterclockwise/clockwise bias stays constant for P. aeruginosa. We describe P. aeruginosa’s chemotaxis using an analytical model for symmetric motor regulation. We use this model to do simulations that show that, given P. aeruginosa’s physiological constraints on motility, its distinct, symmetric regulation of motor switching optimizes chemotaxis.

Chemotaxis has long been known to strongly affect biofilm formation by the opportunistic human pathogen P. aeruginosa, whose essential chemotaxis genes have homologues in E. coli, which achieves chemotaxis by biasing the relative probability of counterclockwise and clockwise flagellar rotation. However, the physiological difference between multiflagellated E. coli and singly flagellated P. aeruginosa implies that biased motor regulation should prevent P. aeruginosa populations from chemotaxing efficiently. Here, we used experiments, analytical modeling, and simulations to demonstrate that P. aeruginosa uses unbiased, symmetric regulation of the flagellar motor to maximize its chemotaxis efficiency. This mode of chemotaxis was not previously known and demonstrates a new variant of a paradigmatic signaling system in an important human pathogen.

Bacterial chemotaxis: information processing, thermodynamics, and behavior

Escherichia coli has long been used as a model organism due to the extensive experimental characterization of its pathways and molecular components. Take chemotaxis as an example, which allows bacteria to sense and swim in response to chemicals, such as nutrients and toxins. Many of the pathway's remarkable sensing and signaling properties are now concisely summarized in terms of design (or engineering) principles. More recently, new approaches from information theory and stochastic thermodynamics have begun to address how pathways process environmental stimuli and what the limiting factors are. However, to fully capitalize on these theoretical advances, a closer connection with single-cell experiments will be required.

Protein Connectivity in Chemotaxis Receptor Complexes

The chemotaxis sensory system allows bacteria such as Escherichia coli to swim towards nutrients and away from repellents. The underlying pathway is remarkably sensitive in detecting chemical gradients over a wide range of ambient concentrations. Interactions among receptors, which are predominantly clustered at the cell poles, are crucial to this sensitivity. Although it has been suggested that the kinase CheA and the adapter protein CheW are integral for receptor connectivity, the exact coupling mechanism remains unclear. Here, we present a statistical-mechanics approach to model the receptor linkage mechanism itself, building on nanodisc and electron cryotomography experiments. Specifically, we investigate how the sensing behavior of mixed receptor clusters is affected by variations in the expression levels of CheA and CheW at a constant receptor density in the membrane. Our model compares favorably with dose-response curves from in vivo Förster resonance energy transfer (FRET) measurements, demonstrating that the receptor-methylation level has only minor effects on receptor cooperativity. Importantly, our model provides an explanation for the non-intuitive conclusion that the receptor cooperativity decreases with increasing levels of CheA, a core signaling protein associated with the receptors, whereas the receptor cooperativity increases with increasing levels of CheW, a key adapter protein. Finally, we propose an evolutionary advantage as explanation for the recently suggested CheW-only linker structures.

Receptor clusters of the bacterial chemotaxis sensory system act as antennae to amplify tiny changes in concentrations in the chemical environment of the cell, ultimately steering the cell towards nutrients and away from toxins. Despite bacterial chemotaxis being the most widely studied sensory pathway, the exact architecture of the receptor clusters remains speculative, with understanding suffering from a number of paradoxical observations. To address these issues with respect to the protein arrangement in the linkers connecting receptors, we present a statistical-mechanics model that combines insights from electron cryotomography on the linker architecture with results from fluorescence imaging of signaling in living cells. Although the signaling data for different expression levels of key molecular components in the linkers seems contradictory at first, our model reconciles these predictions with structural and biochemical data. Finally, we provide an evolutionary explanation for the observation that some of the incorporated linkers do not seem to transmit signals from the receptors.

Theory in Biology: Figure 1 or Figure 7?

The pace of modern science is staggering. The quantities of data now flowing from DNA sequencers, fluorescence and electron microscopes, mass spectrometers, and other mind-blowing instruments leave us faced with information overload. This explosion in data has brought on its heels a concomitant need for efforts at the kinds of synthesis and unification we see in theoretical physics. Often in cell biology, when theoretical modeling takes place, it is as a figure 7 reflection on experiments that have already been done, with data fitting providing a metric of success. Figure 1 theory, by way of contrast, is about living dangerously by turning our thinking into formal mathematical predictions and confronting that math with experiments that have not yet been done.

Notes on stochastic (bio)-logic gates: computing with allosteric cooperativity

Recent experimental breakthroughs have finally allowed to implement in-vitro reaction kinetics (the so called enzyme based logic) which code for two-inputs logic gates and mimic the stochastic AND (and NAND) as well as the stochastic OR (and NOR). This accomplishment, together with the already-known single-input gates (performing as YES and NOT), provides a logic base and paves the way to the development of powerful biotechnological devices. However, as biochemical systems are always affected by the presence of noise (e.g. thermal), standard logic is not the correct theoretical reference framework, rather we show that statistical mechanics can work for this scope: here we formulate a complete statistical mechanical description of the Monod-Wyman-Changeaux allosteric model for both single and double ligand systems, with the purpose of exploring their practical capabilities to express noisy logical operators and/or perform stochastic logical operations. Mixing statistical mechanics with logics, and testing quantitatively the resulting findings on the available biochemical data, we successfully revise the concept of cooperativity (and anti-cooperativity) for allosteric systems, with particular emphasis on its computational capabilities, the related ranges and scaling of the involved parameters and its differences with classical cooperativity (and anti-cooperativity).

Properties of cooperatively induced phases in sensing models

A large number of eukaryotic cells are able to directly detect external chemical gradients with great accuracy and the ultimate limit to their sensitivity has been a topic of debate for many years. Previous work has been done to understand many aspects of this process but little attention has been paid to the possibility of emergent sensing states. Here we examine how cooperation between sensors existing in a two-dimensional network, as they do on the cell's surface, can both enhance and fundamentally alter the response of the cell to a spatially varying signal. We show that weakly interacting sensors linearly amplify the cell's response to an external gradient while a network of strongly interacting sensors form a collective nonlinear response with two separate domains of active and inactive sensors forming what have called a "1/2-state." In our analysis we examine the cell's ability to sense the direction of a signal and pay special attention to the substantially different behavior realized in the strongly interacting regime.