Minimal speed of fronts of reaction-convection-diffusion equations

We study the minimal speed of propagating fronts of convection reaction diffusion equations of the form $u_t + \mu \phi(u) u_x = u_{xx} +f(u)$ for positive reaction terms with $f'(0 >0$. The function $\phi(u)$ is continuous and vanishes at $u=0$. A variational principle for the minimal speed of the waves is constructed from which upper and lower bounds are obtained. This permits the a priori assesment of the effect of the convective term on the minimal speed of the traveling fronts. If the convective term is not strong enough, it produces no effect on the minimal speed of the fronts. We show that if $f''(u)/\sqrt{f'(0)} + \mu \phi'(u) < 0$, then the minimal speed is given by the linear value $2 \sqrt{f'(0)}$, and the convective term has no effect on the minimal speed. The results are illustrated by applying them to the exactly solvable case $u_t + \mu u u_x = u_{xx} + u (1 -u)$. Results are also given for the density dependent diffusion case $u_t + \mu \phi(u) u_x = (D(u)u_x)_x +f(u)$.Comment: revised, new results adde

Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

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

The intersection of theory and application in elucidating pattern formation in developmental biology

We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and vice-versa. The chosen systems - Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns - illustrate the fundamental physical processes of signaling, growth and cell division, and cell movement involved in pattern formation and development. These systems exemplify the current state of theoretical and experimental understanding of how these processes produce the observed patterns, and illustrate how theoretical and experimental approaches can interact to lead to a better understanding of development. As John Bonner said long ago 'We have arrived at the stage where models are useful to suggest experiments, and the facts of the experiments in turn lead to new and improved models that suggest new experiments. By this rocking back and forth between the reality of experimental facts and the dream world of hypotheses, we can move slowly toward a satisfactory solution of the major problems of developmental biology.' © EDP Sciences, 2009

Non-local kinetic and macroscopic models for self-organised animal aggregations

The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient ϵ is varied from ϵ=1 (for 1D transport models) to ϵ=0 (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit ϵ→0, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams

Large scale dynamics of the Persistent Turning Walker model of fish behavior

International audienceThis paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results

The effect of salts on the liquid–liquid phase equilibria of PEG600 + salt aqueous two-phase systems

Six new ATPSs were prepared by combining polyethylene glycol PEG600 with potassium citrate, dipotassium hydrogen phosphate, sodium formate, potassium formate, sodium sulfate, and lithium sulfate. Complete phase diagrams, including the binodal curve and three tie-lines, were determined at 23 °C. The experimental data obtained for the binodal curve were successfully adjusted to the Merchuk equation, and the reliability of tie-line data was confirmed using the equations suggested by Othmer–Tobias and Bancroft. The ability of each ion to induce ATPS formation was investigated. Na+ proved to be more effective in ATPS formation than K+ and Li+. For potassium salts, the order observed for the effectiveness of the anions was: HPO42– > C6H5O73– > HCO2–. Regarding the sodium salts, it was found that SO42– is clearly more effective than HCO2–. The position of the ions in the Hofmeister series and their free energy of hydration (ΔGhyd) were used to explain the ability of the ions to induce PEG salting-out. Furthermore, the effective excluded volume (EEV) of the salts was determined and the following order was found: Na2SO4 > K2HPO4 > Li2SO4 > K3C6H5O7 > NaCHO2 > KCHO2. Similar order was obtained when analyzing the size of the heterogeneous regions, suggesting the practical use of EEV as a comparison parameter between different ATPSs.This work is partially supported by project PEst-C/EQB/LA0020/2011, financed by FEDER through COMPETE-Programa Operacional Factores de Competitividade and by FCT-Fundacao para a Ciencia e a Tecnologia. Sara Silverio acknowledges her Ph.D. grant from FCT (SFRH/BD/43439/2008)

Modelling collective cell behaviour

The classical mean-field approach to modelling biological systems makes a number of simplifying assumptions which typically lead to coupled systems of reaction-diffusion partial differential equations. While these models have been very useful in allowing us to gain important insights into the behaviour of many biological systems, recent experimental advances in our ability to track and quantify cell behaviour now allow us to build more realistic models which relax some of the assumptions previously made. This brief review aims to illustrate the type of models obtained using this approach

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with two examples arising from cancer invasion