Directional persistence and the optimality of run-and-tumble chemotaxis

Anchoring-induced nonmonotonic velocity versus temperature dependence of motile bacteria in a lyotropic nematic liquid crystal

The elastic and viscous properties of lyotropic chromonic liquid crystals have a very sharp, often exponential temperature dependence. Self-propelled bacteria swimming in this viscoelastic medium induce director deformations which can strongly influence their velocity, and we study the temperature behavior of their motility in the whole range of the nematic phase. We observe experimentally that, with increasing temperature, while the viscosity drops exponentially and the frequency of the flagellum rotation grows linearly, the swimmers' speed first conventionally increases but then, above some crossover temperature, slows down and at the same time bacteria-induced director distortions become visible. It is shown that the physics behind this temperature-driven effect is in a sharp rise in the ability of the bacterium's flagellum to induce director deformations. As temperature increases, the splay and bend elastic constants sharply decrease and the anchoring extrapolation length of the flagellum surface gets shorter and shorter. At the crossover temperature the resulting effective anchoring effect dominates the fast dropping viscosity and the distortion strengthens. As a result, a fraction of the torque the flagellum applies for the propulsion is spent for the elastic degrees of freedom, which results in a bacterium slowdown. To find the director distortions, the flagellum is presented as a collection of anchoring-induced elastic monopoles, and the bacterium velocity is found from the balance of the energy spent for the propulsion and the viscous drag and nematodynamic dissipation.

Reconstructing the genotype-to-fitness map for the bacterial chemotaxis network and its emergent behavioural phenotypes

The signal-transduction network responsible for chemotaxis in Escherichia coli has been characterised in extraordinary detail. Yet, relatively little is known about eco-evolutionary aspects of chemotaxis, such as how the network has been shaped by selection and to what extent natural populations may fine-tune their chemotactic behaviour to the ecological conditions. To address these questions, we here develop an evolutionary-systems-biology model of the chemotaxis network of E. coli, which we apply to estimate the resource accumulation rate (here used as a proxy for fitness) of wildtype and a large number of potential mutant genotypes. Mutant genotypes differ from the wildtype in the concentrations of one or more constituent proteins of the chemotaxis signalling network or in one or more of its kinetic parameters. To guarantee model consistency across the genotype space, we explicitly incorporated biochemical constraints that underly observed phenotypic trade-offs. The model was validated by reconstructing the phenotypic properties of several known mutant genotypes. We also characterised differences in the fitness distribution between genotypes, and reconstructed adaptive walks in genotype space for populations exposed to different environmental conditions. We found that the local fitness landscape is rugged, due to non-additive interactions between mutations. When selection has a consistent direction, just a few adaptive mutations are required to reach a local peak, and different local peaks can be reached by adaptive walks starting from the same initial genotype. However, when the direction of selection is fluctuating, evolutionary paths are much longer and genotype space is explored further. Longer adaptive walks were also observed when evolution was started from a low-fitness genotype such as a CheZ knockout mutant. In line with empirical observations, the initial ΔcheZ mutant did not respond to a step-down stimulus, but a dynamic response similar to the wildtype was recovered following the fixation of compensatory mutations.

Using Synthetic Biology to Engineer Living Cells That Interface with Programmable Materials

We have developed an abiotic-biotic interface that allows engineered cells to control the material properties of a functionalized surface. This system is made by creating two modules: a synthetically engineered strain of E. coli cells and a functionalized material interface. Within this paper, we detail a protocol for genetically engineering selected behaviors within a strain of E. coli using molecular cloning strategies. Once developed, this strain produces elevated levels of biotin when exposed to a chemical inducer. Additionally, we detail protocols for creating two different functionalized surfaces, each of which is able to respond to cell-synthesized biotin. Taken together, we present a methodology for creating a linked, abiotic-biotic system that allows engineered cells to control material composition and assembly on nonliving substrates.

Fast, high-throughput measurement of collective behaviour in a bacterial population

Swimming bacteria explore their environment by performing a random walk, which is biased in response to, for example, chemical stimuli, resulting in a collective drift of bacterial populations towards 'a better life'. This phenomenon, called chemotaxis, is one of the best known forms of collective behaviour in bacteria, crucial for bacterial survival and virulence. Both single-cell and macroscopic assays have investigated bacterial behaviours. However, theories that relate the two scales have previously been difficult to test directly. We present an image analysis method, inspired by light scattering, which measures the average collective motion of thousands of bacteria simultaneously. Using this method, a time-varying collective drift as small as 50 nm s(-1) can be measured. The method, validated using simulations, was applied to chemotactic Escherichia coli bacteria in linear gradients of the attractant α-methylaspartate. This enabled us to test a coarse-grained minimal model of chemotaxis. Our results clearly map the onset of receptor methylation, and the transition from linear to logarithmic sensing in the bacterial response to an external chemoeffector. Our method is broadly applicable to problems involving the measurement of collective drift with high time resolution, such as cell migration and fluid flows measurements, and enables fast screening of tactic behaviours.

Secondary bacterial flagellar system improves bacterial spreading by increasing the directional persistence of swimming

As numerous bacterial species, Shewanella putrefaciens CN-32 possesses a complete secondary flagellar system. A significant subpopulation of CN-32 cells induces expression of the secondary system under planktonic conditions, resulting in formation of one, sometimes two, filaments at lateral positions in addition to the primary polar flagellum. Mutant analysis revealed that the single chemotaxis system primarily or even exclusively addresses the main polar flagellar system. Cells with secondary filaments outperformed their monopolarly flagellated counterparts in spreading on soft-agar plates and through medium-filled channels despite having lower swimming speed. While mutant cells with only polar flagella navigate by a "run-reverse-flick" mechanism resulting in effective cell realignments of about 90°, wild-type cells with secondary filaments exhibited a range of realignment angles with an average value of smaller than 90°. Mathematical modeling and computer simulations demonstrated that the smaller realignment angle of wild-type cells results in the higher directional persistence, increasing spreading efficiency both with and without a chemical gradient. Taken together, we propose that in S. putrefaciens CN-32, cell propulsion and directional switches are mainly mediated by the polar flagellar system, while the secondary filament increases the directional persistence of swimming and thus of spreading in the environment.

Limits of feedback control in bacterial chemotaxis

Inputs to signaling pathways can have complex statistics that depend on the environment and on the behavioral response to previous stimuli. Such behavioral feedback is particularly important in navigation. Successful navigation relies on proper coupling between sensors, which gather information during motion, and actuators, which control behavior. Because reorientation conditions future inputs, behavioral feedback can place sensors and actuators in an operational regime different from the resting state. How then can organisms maintain proper information transfer through the pathway while navigating diverse environments? In bacterial chemotaxis, robust performance is often attributed to the zero integral feedback control of the sensor, which guarantees that activity returns to resting state when the input remains constant. While this property provides sensitivity over a wide range of signal intensities, it remains unclear how other parameters such as adaptation rate and adapted activity affect chemotactic performance, especially when considering that the swimming behavior of the cell determines the input signal. We examine this issue using analytical models and simulations that incorporate recent experimental evidences about behavioral feedback and flagellar motor adaptation. By focusing on how sensory information carried by the response regulator is best utilized by the motor, we identify an operational regime that maximizes drift velocity along chemical concentration gradients for a wide range of environments and sensor adaptation rates. This optimal regime is outside the dynamic range of the motor response, but maximizes the contrast between run duration up and down gradients. In steep gradients, the feedback from chemotactic drift can push the system through a bifurcation. This creates a non-chemotactic state that traps cells unless the motor is allowed to adapt. Although motor adaptation helps, we find that as the strength of the feedback increases individual phenotypes cannot maintain the optimal operational regime in all environments, suggesting that diversity could be beneficial.

The biased random walk is a fundamental strategy used by many organisms to navigate their environment. Drift along the desired direction is achieved by reducing the probability to reorient whenever conditions improve. In the chemotaxis system of Escherichia coli, this is accomplished with a sensory module that implements negative integral feedback control, the output of which is relayed to the flagellar motors (the actuators) by a response regulator to control the probability to change direction. The proper dynamical coupling between sensor and actuator is critical for the performance of the random walker. Here, we identify an optimal regime for this coupling that maximizes drift velocity in the direction of the gradient in multiple environments. Our analysis reveals that feedback of the behavior onto the system in steep gradients can constrain individual cell performance, by causing bi-stable behavior that can trap cells in non-chemotactic states. These limitations are inherent in the biased random walk strategy with integral feedback control, but can be alleviated if the output of the pathway adapts, as recently characterized for the flagellar motors in Escherichia coli.

Chemotactic clusters in confined run-and-tumble bacteria: a numerical investigation

We present a simulation study of pattern formation in an ensemble of chemotactic run-and-tumble bacteria, focussing on the effect of spatial confinement, either within traps or inside a maze. These geometries are inspired by previous experiments probing pattern formation in chemotactic strains of E. coli under these conditions. Our main result is that a microscopic model of chemotactic run-and-tumble particles which themselves secrete a chemoattractant is able to reproduce the main experimental observations, namely the formation of bacterial aggregates within traps and in dead ends of a maze. Our simulations also demonstrate that stochasticity plays a key role and leads to a hysteretic response when the chemotactic sensitivity is varied. We compare the results of run-and-tumble particles with simulations performed with a simplified version of the model where the active particles are smooth swimmers which respond to chemotactic gradients by rotating towards the source of chemoattractant. This class of models leads again to aggregation, but with quantitative and qualitative differences in, for instance, the size and shape of clusters.

Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak-Keller-Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.

Modeling novelty habituation during exploratory activity in Drosophila

Habituation is a common form of non-associative learning in which the organism gradually decreases its response to repeated stimuli. The decrease in exploratory activity of many animal species during exposure to a novel open field arena is a widely studied habituation paradigm. However, a theoretical framework to quantify how the novelty of the arena is learned during habituation is currently missing. Drosophila melanogaster display a high mean absolute activity and a high probability for directional persistence when first introduced to a novel arena. Both measures decrease during habituation to the arena. Here, we propose a phenomenological model of habituation for Drosophila exploration based on two principles: Drosophila form a spatial representation of the arena edge as a set of connected local patches, and repeated exposure to these patches is essential for the habituation of the novelty. The level of exposure depends on the number of visitations and is quantified by a variable referred to as "coverage". This model was tested by comparing predictions against the experimentally measured behavior of wild type Drosophila. The novelty habituation of wild type Canton-S depends on coverage and is specifically independent of the arena radius. Our model describes the time dependent locomotor activity, ΔD, of Canton-S using an experimentally established stochastic process Pn(ΔD), which depends on the coverage. The quantitative measures of exploration and habituation were further applied to three mutant genotypes. Consistent with a requirement for vision in novelty habituation, blind no receptor potential A(7) mutants display a failure in the decay of probability for directional persistence and mean absolute activity. The rutabaga(2080) habituation mutant also shows defects in these measures. The kurtz(1) non-visual arrestin mutant demonstrates a rapid decay in these measures, implying reduced motivation. The model and the habituation measures offer a powerful framework for understanding mechanisms associated with open field habituation.

Emergent run-and-tumble behavior in a simple model of Chlamydomonas with intrinsic noise

Recent experiments on the green alga Chlamydomonas that swims using synchronized beating of a pair of flagella have revealed that it exhibits a run-and-tumble behavior similar to that of bacteria such as E. coli. Using a simple purely hydrodynamic model that incorporates a stroke cycle and an intrinsic Gaussian white noise, we show that a stochastic run-and-tumble behavior could emerge due to the nonlinearity of the combined synchronization-rotation-translation dynamics. Our study suggests that nonlinear mechanics could be a significant contributing factor to how the trajectories of the microorganism are selected.

Anomalous diffusion in run-and-tumble motion

A random walk scheme, consisting of alternating phases of regular Brownian motion and Lévy walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and short-time behavior of the mean squared displacement of the walker as depending on the properties of the dwelling time distribution in each phase. Depending on these distributions, normal diffusion, superdiffusion, and ballistic spreading may arise.

Modeling local interactions during the motion of cyanobacteria

Synechocystis sp., a common unicellular freshwater cyanobacterium, has been used as a model organism to study phototaxis, an ability to move in the direction of a light source. This microorganism displays a number of additional characteristics such as delayed motion, surface dependence, and a quasi-random motion, where cells move in a seemingly disordered fashion instead of in the direction of the light source, a global force on the system. These unexplained motions are thought to be modulated by local interactions between cells such as intercellular communication. In this paper, we consider only local interactions of these phototactic cells in order to mathematically model this quasi-random motion. We analyze an experimental data set to illustrate the presence of quasi-random motion and then derive a stochastic dynamic particle system modeling interacting phototactic cells. The simulations of our model are consistent with experimentally observed phototactic motion.

Soluble electron shuttles can mediate energy taxis toward insoluble electron acceptors

Shewanella species grow in widely disparate environments and play key roles in elemental cycling, especially in environments with varied redox conditions. To obtain a system-level understanding of Shewanella's robustness and versatility, the complex interplay of cellular growth, metabolism, and transport under conditions of limiting carbon sources, energy sources, and electron acceptors must be elucidated. In this paper, population-level taxis of Shewanella oneidensis MR-1 cells in the presence of a rate-limiting, insoluble electron acceptor was investigated. A novel mechanism, mediated energy taxis, is proposed by which Shewanella use riboflavin as both an electron shuttle and an attractant to direct cell movement toward local sources of insoluble electron acceptors. The cells secrete reduced riboflavin, which diffuses to a nearby particle containing an insoluble electron acceptor and is oxidized. The oxidized riboflavin then diffuses away from the particle, establishing a spatial gradient that draws cells toward the particle. Experimental and modeling results are presented to support this mechanism. S. oneidensis MR-1 cells inoculated into a uniform dispersion of MnO(2) particles in dilute agar exhibited taxis outward, creating a clear zone within which riboflavin was detected by mass spectrometry. Cells inoculated into dilute agar containing oxidized riboflavin similarly exhibited taxis, rapidly forming an expanding zone of reduced riboflavin. A mathematical model based on the proposed mechanism was able to predict experimental trends, including how concentrations of riboflavin and insoluble electron acceptors (e.g., MnO(2)) affected tactic cell migration.

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

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