Mathematical description of bacterial traveling pulses

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on {\em E. coli} have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with {\em E. coli}. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Graph Theoretical Model of a Sensorimotor Connectome in Zebrafish

Mapping the detailed connectivity patterns (connectomes) of neural circuits is a central goal of neuroscience. The best quantitative approach to analyzing connectome data is still unclear but graph theory has been used with success. We present a graph theoretical model of the posterior lateral line sensorimotor pathway in zebrafish. The model includes 2,616 neurons and 167,114 synaptic connections. Model neurons represent known cell types in zebrafish larvae, and connections were set stochastically following rules based on biological literature. Thus, our model is a uniquely detailed computational representation of a vertebrate connectome. The connectome has low overall connection density, with 2.45% of all possible connections, a value within the physiological range. We used graph theoretical tools to compare the zebrafish connectome graph to small-world, random and structured random graphs of the same size. For each type of graph, 100 randomly generated instantiations were considered. Degree distribution (the number of connections per neuron) varied more in the zebrafish graph than in same size graphs with less biological detail. There was high local clustering and a short average path length between nodes, implying a small-world structure similar to other neural connectomes and complex networks. The graph was found not to be scale-free, in agreement with some other neural connectomes. An experimental lesion was performed that targeted three model brain neurons, including the Mauthner neuron, known to control fast escape turns. The lesion decreased the number of short paths between sensory and motor neurons analogous to the behavioral effects of the same lesion in zebrafish. This model is expandable and can be used to organize and interpret a growing database of information on the zebrafish connectome

A Bayesian approach to modelling heterogeneous calcium responses in cell populations

Calcium responses have been observed as spikes of the whole-cell calcium concentration in numerous cell types and are essential for translating extracellular stimuli into cellular responses. While there are several suggestions for how this encoding is achieved, we still lack a comprehensive theory. To achieve this goal it is necessary to reliably predict the temporal evolution of calcium spike sequences for a given stimulus. Here, we propose a modelling framework that allows us to quantitatively describe the timing of calcium spikes. Using a Bayesian approach, we show that Gaussian processes model calcium spike rates with high fidelity and perform better than standard tools such as peri-stimulus time histograms and kernel smoothing. We employ our modelling concept to analyse calcium spike sequences from dynamically-stimulated HEK293T cells. Under these conditions, different cells often experience diverse stimuli time courses, which is a situation likely to occur in vivo. This single cell variability and the concomitant small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances. Our results therefore pave the way towards a statistical description of heterogeneous calcium oscillations in a dynamic environmen

Logarithmic sensing in Bacillus subtilis aerotaxis

Aerotaxis, the directed migration along oxygen gradients, allows many microorganisms to locate favorable oxygen concentrations. Despite oxygen’s fundamental role for life, even key aspects of aerotaxis remain poorly understood. In Bacillus subtilis, for example, there is conflicting evidence of whether migration occurs to the maximal oxygen concentration available or to an optimal intermediate one, and how aerotaxis can be maintained over a broad range of conditions. Using precisely controlled oxygen gradients in a microfluidic device, spanning the full spectrum of conditions from quasi-anoxic to oxic (60 n mol/l–1 m mol/l), we resolved B. subtilis’ ‘oxygen preference conundrum’ by demonstrating consistent migration towards maximum oxygen concentrations (‘monotonic aerotaxis’). Surprisingly, the strength of aerotaxis was largely unchanged over three decades in oxygen concentration (131 n mol/l–196 μ mol/l). We discovered that in this range B. subtilis responds to the logarithm of the oxygen concentration gradient, a rescaling strategy called ‘log-sensing’ that affords organisms high sensitivity over a wide range of conditions. In these experiments, high-throughput single-cell imaging yielded the best signal-to-noise ratio of any microbial taxis study to date, enabling the robust identification of the first mathematical model for aerotaxis among a broad class of alternative models. The model passed the stringent test of predicting the transient aerotactic response despite being developed on steadystate data, and quantitatively captures both monotonic aerotaxis and log-sensing. Taken together, these results shed new light on the oxygen-seeking capabilities of B. subtilis and provide a blueprint for the quantitative investigation of the many other forms of microbial taxis

The effect of noisy flow on endothelial cell mechanotransduction: A computational study

International audienceFlow in the arterial system is mostly laminar, but turbulence occurs in vivo under both normal and pathological conditions. Turbulent and laminar flow elicit significantly different responses in endothelial cells (ECs), but the mechanisms allowing ECs to distinguish between these different flow regimes remain unknown. The authors present a computational model that describes the effect of turbulence on mechanical force transmission within ECs. Because turbulent flow is inherently "noisy" with random fluctuations in pressure and velocity, our model focuses on the effect of signal noise (a stochastically changing force) on the deformation of intracellular transduction sites including the nucleus, cell-cell adhesion proteins (CCAPs), and focal adhesion sites (FAS). The authors represent these components of the mechanical signaling pathway as linear viscoelastic structures (Kelvin bodies) connected to the cell surface via cytoskeletal elements. The authors demonstrate that FAS are more sensitive to signal noise than the nucleus or CCAP. The relative sensitivity of these various structures to noise is affected by the nature of the cytoskeletal connections within the cell. Finally, changes in the compliance of the nucleus dramatically affect nuclear sensitivity to noise, suggesting that pathologies that alter nuclear mechanical properties will be associated with abnormal EC responsiveness to turbulent flow. © 2010 The Author(s)

Model of Bacterial Band Formation in Aerotaxis

Aerotaxis is a particular form of “energy taxis”. It is based on a largely elusive signal transduction machinery. In aerotaxis, oxygen dissolved in water plays the role of both attractant (at moderate concentrations) and repellent (at high and low concentrations). Cells swimming from favorable oxygen concentrations into regions with unfavorable concentrations increase the frequency of reversals, turn back into the favorable domain, and become effectively trapped there. At the same time, bacteria consume oxygen, creating an oxygen gradient. This behavior leads to a pattern formation phenomenon: bacteria self-organize into a dense band at a certain distance from the air-water interface. We incorporate experimental observations of the aerotactic bacterium, Azospirillum brasilense, into a mathematical model. The model consists of a system of differential equations describing swimming bacterial cells and diffusing oxygen. The cells' frequency of reversals depends on the concentration of oxygen and its time derivative while oxygen is depleted by the bacteria. We suggest a hypothetical model of energy sensing mediated by aerotactic receptors Aer and Tsr. Computer simulations and analysis of the model equations allow comparisons of theoretical and experimental results and provide insight into the mechanisms of bacterial pattern formation and underlying signal transduction machinery. We make testable predictions about position and density of the bacterial band