Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation

Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs)-and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of Escherichia coli bacterial motility. The fine scale, chemomechanical, hybrid (continuum-Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained "Keller-Segel class" chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be black-box (when no prior knowledge about the PDE law structure is assumed) or gray-box when parts of the equation (e.g. the pure diffusion part) is known and "hardwired" in the regression process. More importantly, we discuss data-driven corrections (both additive and functional), to analytically known, approximate closures.

Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer

This special issue on "Multiscale phenomena and patterns in biological systems" is an homage to the seminal contributions of Hans Othmer. He has remained at the forefront of multiscale modelling and pattern formation in biology for over half a century, developing models for molecular signalling networks, the mechanics of cellular movements, the interactions between multiple cells and their contributions to tissue patterning and dynamics. The contributions in this special issue follow Hans' legacy in using advanced mathematics to understand complex biological processes.

The ups and downs of S. aureus nasal carriage

Staphylococcus aureus infections are a growing concern worldwide due to the increasing number of strains that exhibit antibiotic resistance. Recent studies have indicated that some percentage of people carry the bacteria in the nasal cavity and therefore are at a higher risk of subsequent, and more serious, infections in other parts of the body. However, individuals carrying the infection can be classified as only intermittent carriers versus persistent carriers, being able to eliminate the bacteria and later colonized again. Using a model of bacterial colonization of the anterior nares, we investigate oscillatory patterns related to intermittent carriage of S. aureus. Following several studies using global sensitivity analysis techniques, various insights into the model's behaviour were made including interacting effects of the bacteria's growth rate and movement in the mucus, suggesting parameter connections associated with biofilm-like behaviour. Here the bacterial growth rate and bacterial movement are explicitly connected, leading to expanded oscillatory behaviour in the model. We suggest possible implications that this oscillatory behaviour can have on the definition of intermittent carriage and discuss differences in the bacterial virulence dependent upon individual host health. Furthermore, we show that connecting the bacterial growth and movement also expands the region of the parameter space for which the bacteria are able to survive and persist.

Complete genome analysis of Glutamicibacter creatinolyticus from mare abscess and comparative genomics provide insight of diversity and adaptation for Glutamicibacter

Bacteria of the genusGlutamicibacterare considered ubiquitous because they can be found in soil, water and air. They have already been isolated from different habitats, including different types of soil, clinical samples, cheese and plants. Glutamicibacter creatinolyticus is a Gram-positive bacterium important to various biotechnological processes, however, as a pathogen it is associated to urinary tract infections and bacteremia. Recently,Glutamicibacter creatinolyticusLGCM 259 was isolated from a mare, which displayed several diffuse subcutaneous nodules with heavy vascularization. In this study, sequencing, genomic analysis ofG. creatinolyticusLGCM 259 and comparative analyseswere performedamong 4representatives of different members of genusfromdifferent habitats, available in the NCBI database. The LGCM 259 strain's genome carries important factors of bacterial virulence that are essential in cell viability, virulence, and pathogenicity. Genomic islands were predicted for 4 members of genusGlutamicibacter,showing ahigh number of GEIs,which may reflect a high interspecific diversity and a possible adaptive mechanism responsible for the survival of each species in its specific niche. Furthermore,G. creatinolyticusLGCM 259 sharessyntenicregions, albeit with a considerable loss of genes, in relation to the other species. In addition,G. creatinolyticusLGCM 259 presentsresistancegenes to 6 differentclasses ofantibiotics and heavy metals, such as: copper, arsenic, chromium and cobalt-zinc-cadmium.Comparative genomicsanalysescouldcontribute to the identification of mobile genetic elements particular to the speciesG. creatinolyticuscompared to other members of genus. The presence of specific regions inG. creatinolyticuscould be indicative of their rolesin host adaptation, virulence, and the characterization ofastrain that affects animals.

Inferring the Chemotactic Strategy of P. putida and E. coli Using Modified Kramers-Moyal Coefficients

Many bacteria perform a run-and-tumble random walk to explore their surrounding and to perform chemotaxis. In this article we present a novel method to infer the relevant parameters of bacterial motion from experimental trajectories including the tumbling events. We introduce a stochastic model for the orientation angle, where a shot-noise process initiates tumbles, and analytically calculate conditional moments, reminiscent of Kramers-Moyal coefficients. Matching them with the moments calculated from experimental trajectories of the bacteria E. coli and Pseudomonas putida, we are able to infer their respective tumble rates, the rotational diffusion constants, and the distributions of tumble angles in good agreement with results from conventional tumble recognizers. We also define a novel tumble recognizer, which explicitly quantifies the error in recognizing tumbles. In the presence of a chemical gradient we condition the moments on the bacterial direction of motion and thereby explore the chemotaxis strategy. For both bacteria we recover and quantify the classical chemotactic strategy, where the tumble rate is smallest along the chemical gradient. In addition, for E. coli we detect some cells, which bias their mean tumble angle towards smaller values. Our findings are supported by a scaling analysis of appropriate ratios of conditional moments, which are directly calculated from experimental data.

The movement strategies of bacteria have received increasing attention over the past decade, in particular with respect to the tracking of individual cells and the mathematical description of the resulting trajectories. Bacteria typically move in almost straight runs interrupted by sharp turning events (run-and-tumble). In order to characterize their motion on a single cell level, the tumble events in individual trajectories have to be identified. Traditionally, tumble recognition relies on threshold values that are applied to the swimming speed and the reorientation angle. They are chosen in an ad hoc fashion and introduce a certain degree of arbitrariness to the results of statistical motion analyses. Here, we propose a new stochastic model for the orientation angle of a bacterium and formulate conditonal moments, which we determine both in theory and from experimental trajectories. This provides an alternative way of quantifying the bacterial run-and-tumble strategy and of recognizing tumble events. Our approach no longer relies on arbitrarily chosen segmentation thresholds and rigorously quantifies the uncertainty in tumble recognition. We successfully apply our method not only to the paradigmatic case of E. coli but also to trajectories of the soil bacterium Pseudomonas putida, demonstrating that our approach provides a novel way to reliably characterize the tumbling statistics and chemotaxis strategies of bacterial swimmers across different species.

Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway

Kinetic-transport equations are, by now, standard models to describe the dynamics of populations of bacteria moving by run-and-tumble. Experimental observations show that bacteria increase their run duration when encountering an increasing gradient of chemotactic molecules. This led to a first class of models which heuristically include tumbling frequencies depending on the path-wise gradient of chemotactic signal. More recently, the biochemical pathways regulating the flagellar motors were uncovered. This knowledge gave rise to a second class of kinetic-transport equations, that takes into account an intra-cellular molecular content and which relates the tumbling frequency to this information. It turns out that the tumbling frequency depends on the chemotactic signal, and not on its gradient. For these two classes of models, macroscopic equations of Keller-Segel type, have been derived using diffusion or hyperbolic rescaling. We complete this program by showing how the first class of equations can be derived from the second class with molecular content after appropriate rescaling. The main difficulty is to explain why the path-wise gradient of chemotactic signal can arise in this asymptotic process. Randomness of receptor methylation events can be included, and our approach can be used to compute the tumbling frequency in presence of such a noise.

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.

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.