Quantitative modeling of bacterial chemotaxis: signal amplification and accurate adaptation

Information Processing in Biochemical Networks

Living systems are characterized by controlled flows of matter, energy, and information. While the biophysics community has productively engaged with the first two, addressing information flows has been more challenging, with some scattered success in evolutionary theory and a more coherent track record in neuroscience. Nevertheless, interdisciplinary work of the past two decades at the interface of biophysics, quantitative biology, and engineering has led to an emerging mathematical language for describing information flows at the molecular scale. This is where the central processes of life unfold: from detection and transduction of environmental signals to the readout or copying of genetic information and the triggering of adaptive cellular responses. Such processes are coordinated by complex biochemical reaction networks that operate at room temperature, are out of equilibrium, and use low copy numbers of diverse molecular species with limited interaction specificity. Here we review how flows of information through biochemical networks can be formalized using information-theoretic quantities, quantified from data, and computed within various modeling frameworks. Optimization of information flows is presented as a candidate design principle that navigates the relevant time, energy, crosstalk, and metabolic constraints to predict reliable cellular signaling and gene regulation architectures built of individually noisy components.

Role of a single MCP in evolutionary adaptation of Shewanella putrefaciens for swimming in planktonic and structured environments

Bacteria can adapt to their environments by changing phenotypic traits by mutations. However, improving one trait often results in the deterioration of another one, a trade-off that limits the degree of adaptation. The gammaproteobacterium Shewanella putrefaciens CN-32 has an elaborate motility machinery comprising two distinct flagellar systems and an extensive chemotaxis array with 36 methyl-accepting chemotaxis sensor proteins (MCPs). In this study, we performed experimental selection on S. putrefaciens for increased spreading through a porous environment. We readily obtained a mutant that showed a pronounced increase in covered distance. This phenotype was almost completely caused by a deletion of 24 bp from the chromosome, which leads to a moderately enhanced production of a single MCP. Accordingly, chemotaxis assays under free-swimming conditions and cell tracking in soft agar showed that the mutation improved navigation through nutritional gradients. In contrast, further increased levels of the MCP negatively affected spreading. The study demonstrates how moderate differences in the abundance of a single MCP can lead to an efficient upgrade of chemotaxis in specific environments at a low expense of cellular resources.IMPORTANCEExperimental evolution experiments have been used to determine the trade-offs occurring in specific environments. Several studies that have used the spreading behavior of bacteria in structured environments identified regulatory mutants that increase the swimming speed of the cells. While this results in a higher chemotaxis drift, the growth fitness decreases as the higher swimming speed requires substantial cellular resources. Here we show that rapid chemotaxis adaptation can also be achieved by modifying the chemotaxis signal input at a low metabolic cost for the cell.

Temporal Contrastive Learning through implicit non-equilibrium memory

The backpropagation method has enabled transformative uses of neural networks. Alternatively, for energy-based models, local learning methods involving only nearby neurons offer benefits in terms of decentralized training, and allow for the possibility of learning in computationally-constrained substrates. One class of local learning methods contrasts the desired, clamped behavior with spontaneous, free behavior. However, directly contrasting free and clamped behaviors requires explicit memory. Here, we introduce 'Temporal Contrastive Learning', an approach that uses integral feedback in each learning degree of freedom to provide a simple form of implicit non-equilibrium memory. During training, free and clamped behaviors are shown in a sawtooth-like protocol over time. When combined with integral feedback dynamics, these alternating temporal protocols generate an implicit memory necessary for comparing free and clamped behaviors, broadening the range of physical and biological systems capable of contrastive learning. Finally, we show that non-equilibrium dissipation improves learning quality and determine a Landauer-like energy cost of contrastive learning through physical dynamics.

The main sources of molecular organization in the cell. Atlas of self-organized and self-regulated dynamic biostructures

One of the most important goals of contemporary biology is to understand the principles of the molecular order underlying the complex dynamic architecture of cells. Here, we present an overview of the main driving forces involved in the cellular molecular complexity and in the emergent functional dynamic structures, spanning from the most basic molecular organization levels to the complex emergent integrative systemic behaviors. First, we address the molecular information processing which is essential in many complex fundamental mechanisms such as the epigenetic memory, alternative splicing, regulation of transcriptional system, and the adequate self-regulatory adaptation to the extracellular environment. Next, we approach the biochemical self-organization, which is central to understand the emergency of metabolic rhythms, circadian oscillations, and spatial traveling waves. Such a complex behavior is also fundamental to understand the temporal compartmentalization of the cellular metabolism and the dynamic regulation of many physiological activities. Numerous examples of biochemical self-organization are considered here, which show that practically all the main physiological processes in the cell exhibit this type of dynamic molecular organization. Finally, we focus on the biochemical self-assembly which, at a primary level of organization, is a basic but important mechanism for the order in the cell allowing biomolecules in a disorganized state to form complex aggregates necessary for a plethora of essential structures and physiological functions. In total, more than 500 references have been compiled in this review. Due to these main sources of order, systemic functional structures emerge in the cell, driving the metabolic functionality towards the biological complexity.

Dynamical phase transitions in the nonreciprocal Ising model

Nonreciprocal interactions in many-body systems lead to time-dependent states, commonly observed in biological, chemical, and ecological systems. The stability of these states in the thermodynamic limit and the critical behavior of the phase transition from static to time-dependent states are not yet fully understood. To address these questions, we study a minimalistic system endowed with nonreciprocal interactions: an Ising model with two spin species having opposing goals. The mean-field equation predicts three stable phases: disorder, static order, and a time-dependent swap phase. Large-scale numerical simulations support the following: (i) in two dimensions, the swap phase is destabilized by defects; (ii) in three dimensions, the swap phase is stable and has the properties of a time crystal; (iii) the transition from disorder to swap in three dimensions is characterized by the critical exponents of the 3D XY model and corresponds to the breaking of a continuous symmetry, time translation invariance; (iv) when the two species have fully antisymmetric couplings, the static-order phase is unstable in any finite dimension due to droplet growth; and (v) in the general case of asymmetric couplings, static order can be restored by a droplet-capture mechanism preventing the droplets from growing indefinitely. We provide details on the full phase diagram, which includes first- and second-order-like phase transitions, and study how the system coarsens into swap and static-order states.

Lattice ultrasensitivity amplifies signals in E. coli without fine-tuning

The E. coli chemosensory lattice, consisting of receptors, kinases, and adaptor proteins, is an important test case for biochemical signal processing. Kinase output is characterized by precise adaptation to a wide range of background ligand levels and large gain in response to small relative changes in concentration. Existing models of this lattice achieve their gain through allosteric interactions between either receptors or core units of receptors and kinases. Here we introduce a model which operates through an entirely different mechanism in which receptors gate inherently far from equilibrium enzymatic reactions between neighboring kinases. Our lattice model achieves gain through a mechanism more closely related to zero-order ultrasensitivity than to allostery. Thus, we call it lattice ultrasensitivity (LU). Unlike other lattice critical models, the LU model can achieve arbitrarily high gain through time-scale separation, rather than through finetuning. The model also captures qualitative experimental results which are difficult to reconcile with existing models. We discuss possible implementations in the lattice's baseplate where long flexible linkers could potentially mediate interactions between neighboring core units.

Run-and-tumble chemotaxis using reinforcement learning

Bacterial cells use run-and-tumble motion to climb up attractant concentration gradient in their environment. By extending the uphill runs and shortening the downhill runs the cells migrate towards the higher attractant zones. Motivated by this, we formulate a reinforcement learning (RL) algorithm where an agent moves in one dimension in the presence of an attractant gradient. The agent can perform two actions: either persistent motion in the same direction or reversal of direction. We assign costs for these actions based on the recent history of the agent's trajectory. We ask the question: which RL strategy works best in different types of attractant environment. We quantify efficiency of the RL strategy by the ability of the agent (a) to localize in the favorable zones after large times, and (b) to learn about its complete environment. Depending on the attractant profile and the initial condition, we find an optimum balance is needed between exploration and exploitation to ensure the most efficient performance.

Chemotaxing E. coli do not count single molecules

Understanding biological functions requires identifying the physical limits and system-specific constraints that have shaped them. In Escherichia coli chemotaxis, gradient-climbing speed is information-limited, bounded by the sensory information they acquire from real-time measurements of their environment. However, it remains unclear what limits this information. Past work conjectured that E. coli's chemosensing is limited by the physics of molecule arrivals at their sensors. Here, we derive the physical limit on behaviorally-relevant information, and then perform single-cell experiments to quantify how much information E. coli's signaling pathway encodes. We find that E. coli encode two orders of magnitude less information than the physical limit due to their stochastic signal processing. Thus, system-specific constraints, rather than the physical limit, have shaped the evolution of this canonical sensory-motor behavior.

Receptor polarization through localized activity and global sensitization

Eukaryotic cells chemosense concentration gradients of extracellular ligands using membrane-bound receptors that polarize their activity. Receptors from several chemosensing families are preferentially degraded after activation and undergo significant lateral diffusion, both of which may blunt their polarization. To explore the combined role of these two seemingly detrimental phenomena on active receptor polarization, we use a reaction/diffusion model. The model elucidates a counterintuitive principle that governs receptor polarization under external gradients: Localized Activity and Global Sensitization (LAGS). In LAGS, receptor activity is localized through receptor degradation or ligand unbinding. In contrast, uniform sensitivity to ligands is maintained over the plasma membrane through lateral receptor diffusion. Surprisingly, increasing preferential degradation of active receptors and increasing lateral diffusion of all receptors both sharpen active receptor polarization. Additionally, when combined with receptor oligomerization, an increase in preferential degradation allows cells to sense relative ligand gradients over a larger range of background ligand concentrations. An analytical model identifies parameter regimes that dictate which processes dominate receptor polarization. A survey of kinetic parameters suggests that receptor polarization in many mammalian pathways can be modeled using LAGS.

Enhanced chemotaxis efficiency of Escherichia coli in viscoelastic solutions

Bacteria inhabit complex environments rich in macromolecular polymers that exhibit viscoelastic properties. While the influence of viscoelasticity on bacterial swimming is recognized, its impact on chemotaxis-a critical behavior for bacterial survival and colonization-remains elusive. In this study, we employed a microfluidic device to establish attractant gradients and observed the chemotactic behavior of Escherichia coli in both viscoelastic solutions containing carboxymethyl cellulose (CMC) and Newtonian buffers. Our results reveal that E. coli demonstrates markedly enhanced chemotactic efficiency in viscoelastic media. Notably, bacteria achieved faster migration velocities and higher steady-state accumulation in areas with higher attractant concentrations compared to those in Newtonian conditions. Through 3D tracking, we determined that changes in bulk motility parameters alone do not account for the observed enhancements. Further investigations through theoretical analysis and stochastic simulations suggested that the main enhancement mechanisms are mitigation of surface hydrodynamic hindrance resulting from solid surfaces commonly present in bacterial habitats, and the induction of a lifting force in viscoelastic solutions. These findings highlight the significant role of the rheological properties of bacterial habitats in shaping their chemotactic strategies, offering deeper insights into bacterial adaptive mechanisms in both natural and clinical settings.

A modified variational approach to noisy cell signaling

Signaling in cells is full of noise and, hence, described with stochastic biochemical models. Thus, an efficient computation algorithm for these fluctuating reactions is much needed. Apart from the very popular Monte Carlo simulation, methods based on probability distributions are frequently desired due to their analytical tractability and possible numerical advantages in diverse circumstances, among which the variational approach is the most notable. In this paper, new basis functions are proposed to better depict possibly complex distribution profiles, and an extra regularization scheme is supplied to the variational equation to remove occasional degeneracy-induced singularities during the evolution. The new extension is applied to four typical biochemical reaction models and restores the Gillespie results accurately but with greatly reduced simulation time. This modified variational approach is expected to work in a wide range of cell signaling networks.

Data-driven discovery of chemotactic migration of bacteria via coordinate-invariant machine learning

BACKGROUND

E. coli chemotactic motion in the presence of a chemonutrient field can be studied using wet laboratory experiments or macroscale-level partial differential equations (PDEs) (among others). Bridging experimental measurements and chemotactic Partial Differential Equations requires knowledge of the evolution of all underlying fields, initial and boundary conditions, and often necessitates strong assumptions. In this work, we propose machine learning approaches, along with ideas from the Whitney and Takens embedding theorems, to circumvent these challenges.

RESULTS

Machine learning approaches for identifying underlying PDEs were (a) validated through the use of simulation data from established continuum models and (b) used to infer chemotactic PDEs from experimental data. Such data-driven models were surrogates either for the entire chemotactic PDE right-hand-side (black box models), or, in a more targeted fashion, just for the chemotactic term (gray box models). Furthermore, it was demonstrated that a short history of bacterial density may compensate for the missing measurements of the field of chemonutrient concentration. In fact, given reasonable conditions, such a short history of bacterial density measurements could even be used to infer chemonutrient concentration.

CONCLUSION

Data-driven PDEs are an important modeling tool when studying Chemotaxis at the macroscale, as they can learn bacterial motility from various data sources, fidelities (here, computational models, experiments) or coordinate systems. The resulting data-driven PDEs can then be simulated to reproduce/predict computational or experimental bacterial density profile data independent of the coordinate system, approximate meaningful parameters or functional terms, and even possibly estimate the underlying (unmeasured) chemonutrient field evolution.

Time-reversal symmetry breaking in the chemosensory array reveals a general mechanism for dissipation-enhanced cooperative sensing

The Escherichia coli chemoreceptors form an extensive array that achieves cooperative and adaptive sensing of extracellular signals. The receptors control the activity of histidine kinase CheA, which drives a nonequilibrium phosphorylation-dephosphorylation reaction cycle for response regulator CheY. Cooperativity and dissipation are both important aspects of chemotaxis signaling, yet their consequences have only been studied separately. Recent single-cell FRET measurements revealed that kinase activity of the array spontaneously switches between active and inactive states, with asymmetric switching times that signify time-reversal symmetry breaking in the underlying dynamics. Here, we present a nonequilibrium lattice model of the chemosensory array, which demonstrates that the observed asymmetric switching dynamics can only be explained by an interplay between the dissipative reactions within individual core units and the cooperative coupling between neighboring units. Microscopically, the switching time asymmetry originates from irreversible transition paths. The model shows that strong dissipation enables sensitive and rapid signaling response by relieving the speed-sensitivity trade-off, which can be tested by future single-cell experiments. Overall, our model provides a general framework for studying biological complexes composed of coupled subunits that are individually driven by dissipative cycles and the rich nonequilibrium physics within.

EnhancerNet: a predictive model of cell identity dynamics through enhancer selection

Understanding how cell identity is encoded by the genome and acquired during differentiation is a central challenge in cell biology. I have developed a theoretical framework called EnhancerNet, which models the regulation of cell identity through the lens of transcription factor-enhancer interactions. I demonstrate that autoregulation in these interactions imposes a constraint on the model, resulting in simplified dynamics that can be parameterized from observed cell identities. Despite its simplicity, EnhancerNet recapitulates a broad range of experimental observations on cell identity dynamics, including enhancer selection, cell fate induction, hierarchical differentiation through multipotent progenitor states and direct reprogramming by transcription factor overexpression. The model makes specific quantitative predictions, reproducing known reprogramming recipes and the complex haematopoietic differentiation hierarchy without fitting unobserved parameters. EnhancerNet provides insights into how new cell types could evolve and highlights the functional importance of distal regulatory elements with dynamic chromatin in multicellular evolution.

The dual role of a novel Sinorhizobium meliloti chemotaxis protein CheT in signal termination and adaptation

Sinorhizobium meliloti senses nutrients and compounds exuded from alfalfa host roots and coordinates an excitation, termination, and adaptation pathway during chemotaxis. We investigated the role of the novel S. meliloti chemotaxis protein CheT. While CheT and the Escherichia coli phosphatase CheZ share little sequence homology, CheT is predicted to possess an α-helix with a DXXXQ phosphatase motif. Phosphorylation assays demonstrated that CheT dephosphorylates the phosphate-sink response regulator, CheY1~P by enhancing its decay two-fold but does not affect the motor response regulator CheY2~P. Isothermal Titration Calorimetry (ITC) experiments revealed that CheT binds to a phosphomimic of CheY1~P with a KD of 2.9 μM, which is 25-fold stronger than its binding to CheY1. Dissimilar chemotaxis phenotypes of the ΔcheT mutant and cheT DXXXQ phosphatase mutants led to the hypothesis that CheT exerts additional function(s). A screen for potential binding partners of CheT revealed that it forms a complex with the methyltransferase CheR. ITC experiments confirmed CheT/CheR binding with a KD of 19 μM, and a SEC-MALS analysis determined a 1:1 and 2:1 CheT/CheR complex formation. Although they did not affect each other's enzymatic activity, CheT binding to CheY1~P and CheR may serve as a link between signal termination and sensory adaptation.

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.

Chemosensory systems interact to shape relevant traits for bacterial plant pathogenesis

Chemosensory systems allow bacteria to respond and adapt to environmental conditions. Many bacteria contain more than one chemosensory system, but knowledge of their specific roles in regulating different functions remains scarce. Here, we address this issue by analyzing the function of the F6, F8, and alternative (non-motility) cellular functions (ACF) chemosensory systems of the model plant pathogen Pseudomonas syringae pv. tomato. In this work, we assign PsPto chemoreceptors to each chemosensory system, and we visualize for the first time the F6 and F8 chemosensory systems of PsPto using cryo-electron tomography. We confirm that chemotaxis and swimming motility are controlled by the F6 system, and we demonstrate how different components from the F8 and ACF systems also modulate swimming motility. We also determine how the kinase and response regulators from the F6 and F8 chemosensory systems do not work together in the regulation of biofilm, whereas both components from the ACF system contribute together to regulate these traits. Furthermore, we show how the F6, F8, and ACF kinases interact with the ACF response regulator WspR, supporting crosstalk among chemosensory systems. Finally, we reveal how all chemosensory systems play a role in regulating virulence.

IMPORTANCE

Chemoperception through chemosensory systems is an essential feature for bacterial survival, as it allows bacterial interaction with its surrounding environment. In the case of plant pathogens, it is especially relevant to enter the host and achieve full virulence. Multiple chemosensory systems allow bacteria to display a wider plasticity in their response to external signals. Here, we perform a deep characterization of the F6, F8, and alternative (non-motility) cellular functions chemosensory systems in the model plant pathogen Pseudomonas syringae pv. tomato DC3000. These chemosensory systems regulate key virulence-related traits, like motility and biofilm formation. Furthermore, we unveil an unexpected crosstalk among these chemosensory systems at the level of the interaction between kinases and response regulators. This work shows novel results that contribute to the knowledge of chemosensory systems and their role in functions alternative to chemotaxis.

E. coli do not count single molecules

Organisms must perform sensory-motor behaviors to survive. What bounds or constraints limit behavioral performance? Previously, we found that the gradient-climbing speed of a chemotaxing Escherichia coli is near a bound set by the limited information they acquire from their chemical environments (1). Here we ask what limits their sensory accuracy. Past theoretical analyses have shown that the stochasticity of single molecule arrivals sets a fundamental limit on the precision of chemical sensing (2). Although it has been argued that bacteria approach this limit, direct evidence is lacking. Here, using information theory and quantitative experiments, we find that E. coli's chemosensing is not limited by the physics of particle counting. First, we derive the physical limit on the behaviorally-relevant information that any sensor can get about a changing chemical concentration, assuming that every molecule arriving at the sensor is recorded. Then, we derive and measure how much information E. coli's signaling pathway encodes during chemotaxis. We find that E. coli encode two orders of magnitude less information than an ideal sensor limited only by shot noise in particle arrivals. These results strongly suggest that constraints other than particle arrival noise limit E. coli's sensory fidelity.

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.

Role of Topology in Relaxation of One-Dimensional Stochastic Processes

Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions.

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.

Signal amplification in growth cone gradient sensing by a double negative feedback loop among PTEN, PI(3,4,5)P3 and actomyosin

Axon guidance during neural wiring involves a series of precisely controlled chemotactic events by the motile axonal tip, the growth cone. A fundamental question is how neuronal growth cones make directional decisions in response to extremely shallow gradients of guidance cues with exquisite sensitivity. Here we report that nerve growth cones possess a signal amplification mechanism during gradient sensing process. In neuronal growth cones of Xenopus spinal neurons, phosphatidylinositol-3,4,5-trisphosphate (PIP3), an important signaling molecule in chemotaxis, was actively recruited to the up-gradient side in response to an external gradient of brain-derived neurotrophic factor (BDNF), resulting in an intracellular gradient with approximate 30-fold amplification of the input. Furthermore, a reverse gradient of phosphatase and tensin homolog (PTEN) was induced by BDNF within the growth cone and the increased PTEN activity at the down-gradient side is required for the amplification of PIP3 signals. Mechanistically, the establishment of both positive PIP3 and reverse PTEN gradients depends on the filamentous actin network. Together with computational modeling, our results revealed a double negative feedback loop among PTEN, PIP3 and actomyosin for signal amplification, which is essential for gradient sensing of neuronal growth cones in response to diffusible cues.

A multi-state dynamic process confers mechano-adaptation to a biological nanomachine

Adaptation is a defining feature of living systems. The bacterial flagellar motor adapts to changes in the external mechanical load by adding or removing torque-generating (stator) units. But the molecular mechanism behind this mechano-adaptation remains unclear. Here, we combine single motor eletrorotation experiments and theoretical modeling to show that mechano-adaptation of the flagellar motor is enabled by multiple mechanosensitive internal states. Dwell time statistics from experiments suggest the existence of at least two bound states with a high and a low unbinding rate, respectively. A first-passage-time analysis of a four-state model quantitatively explains the experimental data and determines the transition rates among all four states. The torque generated by bound stator units controls their effective unbinding rate by modulating the transition between the bound states, possibly via a catch bond mechanism. Similar force-mediated feedback enabled by multiple internal states may apply to adaptation in other macromolecular complexes.

Combining experiments with modeling, Wadhwa et al. propose a model for mechano-adaptation in the bacterial flagellar motor, finding that load-dependent transitions between multiple internal states govern the binding and unbinding of subunits.

A negative feedback loop in the GPCR pathway underlies efficient coding of external stimuli

Efficient navigation based on chemical cues is an essential feature shared by all animals. These cues may be encountered in complex spatiotemporal patterns and with orders of magnitude varying intensities. Nevertheless, sensory neurons accurately extract the relevant information from such perplexing signals. Here, we show how a single sensory neuron in Caenorhabditis elegans animals can cell-autonomously encode complex stimulus patterns composed of instantaneous sharp changes and of slowly changing continuous gradients. This encoding relies on a simple negative feedback in the G-protein-coupled receptor (GPCR) signaling pathway in which TAX-6/Calcineurin plays a key role in mediating the feedback inhibition. This negative feedback supports several important coding features that underlie an efficient navigation strategy, including exact adaptation and adaptation to the magnitude of the gradient's first derivative. A simple mathematical model explains the fine neural dynamics of both wild-type and tax-6 mutant animals, further highlighting how the calcium-dependent activity of TAX-6/Calcineurin dictates GPCR inhibition and response dynamics. As GPCRs are ubiquitously expressed in all sensory neurons, this mechanism may be a general solution for efficient cell-autonomous coding of external stimuli.

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.

Protein rings are critical to the remarkable signaling properties of bacterial chemotaxis nanoarrays

Bacteria build an extensive sensory nanoarray to collect information that guides their swimming. In this issue of Science Signaling, Piñas et al. demonstrate that a key element of these arrays that enhances chemotaxis responses are hexameric rings of CheW, one of two types of rings that couple the responses of core signaling units to achieve remarkable signaling properties such as single-molecule detection.

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.

Escherichia coli chemotaxis is information limited

Organisms acquire and use information from their environment to guide their behaviour. However, it is unclear whether this information quantitatively limits their behavioural performance. Here, we relate information to the ability of Escherichia coli to navigate up chemical gradients, the behaviour known as chemotaxis. First, we derive a theoretical limit on the speed with which cells climb gradients, given the rate at which they acquire information. Next, we measure cells' gradient-climbing speeds and the rate of information acquisition by their chemotaxis signaling pathway. We find that E. coli make behavioural decisions with much less than the one bit required to determine whether they are swimming up-gradient. Some of this information is irrelevant to gradient climbing, and some is lost in communication to behaviour. Despite these limitations, E. coli climb gradients at speeds within a factor of two of the theoretical bound. Thus, information can limit the performance of an organism, and sensory-motor pathways may have evolved to efficiently use information acquired from the environment.

A traveling-wave solution for bacterial chemotaxis with growth

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemotaxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes.

Multiple functions of flagellar motility and chemotaxis in bacterial physiology

Most swimming bacteria are capable of following gradients of nutrients, signaling molecules and other environmental factors that affect bacterial physiology. This tactic behavior became one of the most-studied model systems for signal transduction and quantitative biology, and underlying molecular mechanisms are well characterized in Escherichia coli and several other model bacteria. In this review, we focus primarily on less understood aspect of bacterial chemotaxis, namely its physiological relevance for individual bacterial cells and for bacterial populations. As evident from multiple recent studies, even for the same bacterial species flagellar motility and chemotaxis might serve multiple roles, depending on the physiological and environmental conditions. Among these, finding sources of nutrients and more generally locating niches that are optimal for growth appear to be one of the major functions of bacterial chemotaxis, which could explain many chemoeffector preferences as well as flagellar gene regulation. Chemotaxis might also generally enhance efficiency of environmental colonization by motile bacteria, which involves intricate interplay between individual and collective behaviors and trade-offs between growth and motility. Finally, motility and chemotaxis play multiple roles in collective behaviors of bacteria including swarming, biofilm formation and autoaggregation, as well as in their interactions with animal and plant hosts.

Theoretical Constraints Imposed by Gradient Detection and Dispersal on Microbial Size in Astrobiological Environments

The capacity to sense gradients efficiently and acquire information about the ambient environment confers many advantages such as facilitating movement toward nutrient sources or away from toxic chemicals. The amplified dispersal evinced by organisms endowed with motility is possibly beneficial in related contexts. Hence, the connections between information acquisition, motility, and microbial size are explored from an explicitly astrobiological standpoint. By using prior theoretical models, the constraints on organism size imposed by gradient detection and motility are elucidated in the form of simple heuristic scaling relations. It is argued that environments such as alkaline hydrothermal vents, which are distinguished by the presence of steep gradients, might be conducive to the existence of "small" microbes (with radii of ≳0.1 μm) in principle, when only the above two factors are considered; other biological functions (e.g., metabolism and genetic exchange) could, however, regulate the lower bound on microbial size and elevate it. The derived expressions are potentially applicable to a diverse array of settings, including those entailing solvents other than water; for example, the lakes and seas of Titan. The article concludes with a brief exposition of how this formalism may be of practical and theoretical value to astrobiology.

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.

Production and Characterization of Motile and Chemotactic Bacterial Minicells

Minicells are nanosized membrane vesicles produced by bacteria. Minicells are chromosome-free but contain cellular biosynthetic and metabolic machinery, and they are robust due to the protection provided by the bacterial cell envelope, which makes them potentially highly attractive in biomedical applications. However, the applicability of minicells and other nanoparticle-based delivery systems is limited by their inefficient accumulation at the target. Here we engineered the minicell-producing Escherichia coli strain to overexpress flagellar genes, which enables the generation of motile minicells. We subsequently performed an experimental and theoretical analysis of the minicell motility and their responses to gradients of chemoeffectors. Despite important differences between the motility of minicells and normal bacterial cells, minicells were able to bias their movement in chemical gradients and to accumulate toward the sources of chemoattractants. Such motile and chemotactic minicells may thus be applicable for an active effector delivery and specific targeting of tissues and cells according to their metabolic profiles.

Integrative Neuroscience of Paramecium, a "Swimming Neuron"

Paramecium is a unicellular organism that swims in fresh water by beating thousands of cilia. When it is stimulated (mechanically, chemically, optically, thermally…), it often swims backward then turns and swims forward again. This "avoiding reaction" is triggered by a calcium-based action potential. For this reason, some authors have called Paramecium a "swimming neuron." This review summarizes current knowledge about the physiological basis of behavior of Paramecium.

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.

Swimming Escherichia coli Cells Explore the Environment by Lévy Walk

Escherichia coli cells swim in aqueous environment in a random walk of alternating runs and tumbles. The diffusion characteristics of this random walk remains unclear. In this study, by tracking the swimming of wild-type cells in a three-dimensional (3D) homogeneous environment, we found that their trajectories are superdiffusive, consistent with Lévy walk behavior. For comparison, we tracked the swimming of mutant cells that lack the chemotaxis signaling noise (the steady-state fluctuation of the concentration of the chemotaxis response regulator CheY-P) and found that their trajectories are normal diffusive. Therefore, wild-type E. coli cells explore the environment by Lévy walk, which originates from the chemotaxis signaling noise. This Lévy walk pattern enhances their efficiency in environmental exploration.IMPORTANCE E. coli cells explore the environment in a random walk of alternating runs and tumbles. By tracking the 3D trajectories of E. coli cells in an aqueous environment, we found that their trajectories are superdiffusive, with a power-law shape for the distribution of run lengths, which is characteristics of Lévy walk. We further show that this Lévy walk behavior is due to the random fluctuation of the output level of the bacterial chemotaxis pathway, and it enhances the efficiency of the bacteria in exploring the environment.

Collective motion enhances chemotaxis in a two-dimensional bacterial swarm

In a dilute liquid environment in which cell-cell interaction is negligible, flagellated bacteria, such as Escherichia coli, perform chemotaxis by biased random walks alternating between run-and-tumble. In a two-dimensional crowded environment, such as a bacterial swarm, the typical behavior of run-and-tumble is absent, and this raises the question whether and how bacteria can perform chemotaxis in a swarm. Here, by examining the chemotactic behavior as a function of the cell density, we showed that chemotaxis is surprisingly enhanced because of cell crowding in a bacterial swarm, and this enhancement is correlated with increase in the degree of cell body alignment. Cells tend to form clusters that move collectively in a swarm with increased effective run length, and we showed analytically that this resulted in increased drift velocity toward attractants. We also explained the enhancement by stochastically simulating bacterial chemotaxis in a swarm. We found that cell crowding in a swarm enhances chemotaxis if the cell-cell interactions used in the simulation induce cell-cell alignment, but it impedes chemotaxis if the interactions are collisions that randomize cell moving direction. Therefore, collective motion in a bacterial swarm enhances chemotaxis.

Bootstrapped Motion of an Agent on an Adaptive Resource Landscape

We theoretically show that isolated agents that locally and symmetrically consume resources and sense positive resource gradients can generate constant motion via bootstrapped resource gradients in the absence of any externally imposed gradients, and we show a realization of this motion using robots. This self-generated agent motion can be coupled with neighboring agents to act as a spontaneously broken symmetry seed for emergent collective dynamics. We also show that in a sufficiently weak externally imposed gradient, it is possible for an agent to move against an external resource gradient due to the local resource depression on the landscape created by an agent. This counter-intuitive boot-strapped motion against an external gradient is demonstrated with a simple robot system on an light-emitting diode (LED) light-board.

Short-term prediction through ordinal patterns

Prediction in natural environments is a challenging task, and there is a lack of clarity around how a myopic organism can make short-term predictions given limited data availability and cognitive resources. In this context, we may ask what kind of resources are available to the organism to help it address the challenge of short-term prediction within its own cognitive limits. We point to one potentially important resource: ordinal patterns, which are extensively used in physics but not in the study of cognitive processes. We explain the potential importance of ordinal patterns for short-term prediction, and how natural constraints imposed through (i) ordinal pattern types, (ii) their transition probabilities and (iii) their irreversibility signature may support short-term prediction. Having tested these ideas on a massive dataset of Bitcoin prices representing a highly fluctuating environment, we provide preliminary empirical support showing how organisms characterized by bounded rationality may generate short-term predictions by relying on ordinal patterns.

Multi-bit Boolean model for chemotactic drift of Escherichia coli

Dynamic biological systems can be modelled to an equivalent modular structure using Boolean networks (BNs) due to their simple construction and relative ease of integration. The chemotaxis network of the bacterium Escherichia coli (E. coli) is one of the most investigated biological systems. In this study, the authors developed a multi-bit Boolean approach to model the drifting behaviour of the E. coli chemotaxis system. Their approach, which is slightly different than the conventional BNs, is designed to provide finer resolution to mimic high-level functional behaviour. Using this approach, they simulated the transient and steady-state responses of the chemoreceptor sensory module. Furthermore, they estimated the drift velocity under conditions of the exponential nutrient gradient. Their predictions on chemotactic drifting are in good agreement with the experimental measurements under similar input conditions. Taken together, by simulating chemotactic drifting, they propose that multi-bit Boolean methodology can be used for modelling complex biological networks. Application of the method towards designing bio-inspired systems such as nano-bots is discussed.

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.

Sequential modification of bacterial chemoreceptors is key for achieving both accurate adaptation and high gain

Many regulatory and signaling proteins have multiple modification sites. In bacterial chemotaxis, each chemoreceptor has multiple methylation sites that are responsible for adaptation. However, whether the ordering of the multisite methylation process affects adaptation remains unclear. Furthermore, the benefit of having multiple modification sites is also unclear. Here, we show that sequentially ordered methylation/demethylation is critical for perfect adaptation; adaptation accuracy decreases as randomness in the multisite methylation process increases. A tradeoff between adaptation accuracy and response gain is discovered. We find that this accuracy-gain tradeoff is lifted significantly by having more methylation sites, but only when the multisite modification process is sequential. Our study suggests that having multiple modification sites and a sequential modification process constitute a general strategy to achieve both accurate adaptation and high response gain simultaneously. Our theory agrees with existing data and predictions are made to help identify the molecular mechanism underlying ordered covalent modifications.

Bacterial chemoreceptors have multiple methylation sites, but whether the order of methylation matters is unclear. Here, the authors show that sequentially ordered methylation is critical for perfect adaptation and for attenuating the trade-off between accurate adaptation and high response gain.

Emergence of collective oscillations in adaptive cells

Collective oscillations of cells in a population appear under diverse biological contexts. Here, we establish a set of common principles by categorising the response of individual cells against a time-varying signal. A positive intracellular signal relay of sufficient gain from participating cells is required to sustain the oscillations, together with phase matching. The two conditions yield quantitative predictions for the onset cell density and frequency in terms of measured single-cell and signal response functions. Through mathematical constructions, we show that cells that adapt to a constant stimulus fulfil the phase requirement by developing a leading phase in an active frequency window that enables cell-to-signal energy flow. Analysis of dynamical quorum sensing in several cellular systems with increasing biological complexity reaffirms the pivotal role of adaptation in powering oscillations in an otherwise dissipative cell-to-cell communication channel. The physical conditions identified also apply to synthetic oscillatory systems.

There are many examples of cell populations exhibiting density-dependent collective oscillatory behaviour. Here, the authors show that sustained collective oscillations emerge when cells anticipate variation in signal and attempt to amplify it, a property that can be linked to adaptation.

Chemotactic behaviour of Escherichia coli at high cell density

At high cell density, swimming bacteria exhibit collective motility patterns, self-organized through physical interactions of a however still debated nature. Although high-density behaviours are frequent in natural situations, it remained unknown how collective motion affects chemotaxis, the main physiological function of motility, which enables bacteria to follow environmental gradients in their habitats. Here, we systematically investigate this question in the model organism Escherichia coli, varying cell density, cell length, and suspension confinement. The characteristics of the collective motion indicate that hydrodynamic interactions between swimmers made the primary contribution to its emergence. We observe that the chemotactic drift is moderately enhanced at intermediate cell densities, peaks, and is then strongly suppressed at higher densities. Numerical simulations reveal that this suppression occurs because the collective motion disturbs the choreography necessary for chemotactic sensing. We suggest that this physical hindrance imposes a fundamental constraint on high-density behaviours of motile bacteria, including swarming and the formation of multicellular aggregates and biofilms.

An Adaptive-Threshold Mechanism for Odor Sensation and Animal Navigation

Identifying the environmental information and computations that drive sensory detection is key for understanding animal behavior. Using experimental and theoretical analysis of AWC^ON, a well-described olfactory neuron in C. elegans, here we derive a general and broadly useful model that matches stimulus history to odor sensation and behavioral responses. We show that AWC^ON sensory activity is regulated by an absolute signal threshold that continuously adapts to odor history, allowing animals to compare present and past odor concentrations. The model predicts sensory activity and probabilistic behavior during animal navigation in different odor gradients and across a broad stimulus regime. Genetic studies demonstrate that the cGMP-dependent protein kinase EGL-4 determines the timescale of threshold adaptation, defining a molecular basis for a critical model feature. The adaptive threshold model efficiently filters stimulus noise, allowing reliable sensation in fluctuating environments, and represents a feedforward sensory mechanism with implications for other sensory systems.

Chemotaxis as a navigation strategy to boost range expansion

Bacterial chemotaxis, the directed movement of cells along gradients of chemoattractants, is among the best-characterized subjects in molecular biology^1-10, but much less is known about its physiological roles¹¹. It is commonly seen as a starvation response when nutrients run out, or as an escape response from harmful situations^12-16. Here we identify an alternative role of chemotaxis by systematically examining the spatiotemporal dynamics of Escherichia coli in soft agar^12,17,18. Chemotaxis in nutrient-replete conditions promotes the expansion of bacterial populations into unoccupied territories well before nutrients run out in the current environment. Low levels of chemoattractants act as aroma-like cues in this process, establishing the direction and enhancing the speed of population movement along the self-generated attractant gradients. This process of navigated range expansion spreads faster and yields larger population gains than unguided expansion following the canonical Fisher-Kolmogorov dynamics^19,20 and is therefore a general strategy to promote population growth in spatially extended, nutrient-replete environments.

An evolutionarily stable strategy to colonize spatially extended habitats

The ability of a species to colonize newly available habitats is crucial to its overall fitness1-3. In general, motility and fast expansion are expected to be beneficial for colonization and hence for the fitness of an organism4-7. Here we apply an evolution protocol to investigate phenotypical requirements for colonizing habitats of different sizes during range expansion by chemotaxing bacteria8. Contrary to the intuitive expectation that faster is better, we show that there is an optimal expansion speed for a given habitat size. Our analysis showed that this effect arises from interactions among pioneering cells at the front of the expanding population, and revealed a simple, evolutionarily stable strategy for colonizing a habitat of a specific size: to expand at a speed given by the product of the growth rate and the habitat size. These results illustrate stability-to-invasion as a powerful principle for the selection of phenotypes in complex ecological processes.