GLOBAL SOLUTIONS OF NONLINEAR TRANSPORT EQUATIONS FOR CHEMOSENSITIVE MOVEMENT

A widespread phenomenon in moving microorganisms and cells is their ability to reorient themselves depending on changes of concentrations of certain chemical signals. In this paper we discuss kinetic models for chemosensitive movement, which also takes into account evaluations of gradient fields of chemical stimuli which subsequently influence the motion of the respective microbiological species. The basic type of model was discussed by Alt [J. Math. Biol., 9 (1980), pp. 147-177], [J. Reine Angew. Math., 322 (1981), pp. 15-41] and by Othmer, Dunbar, and Alt [J. Math. Biol., 26 (1988), pp. 263-298]. Chalub et al. rigorously proved that, in three dimensions, these kinds of kinetic models lead to the classical Keller-Segel model as its drift-diffusion limit when the equation for the chemo-attractant is of elliptic type [Monatsh. Math., 142 (2004), pp. 123-141], [On the Derivation of Drift-Diffusion Model for Chemotaxis from Kinetic Equations, ANUM preprint 14/02, Vienna Technical University, 2002]. In [H. Hwang, K. Kang, and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B., to appear] it was proved that the macroscopic diffusion limit exists in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type. So far in the rigorous derivations, only the density of the chemo-attractant was supposed to influence the motion of the chemosensitive species. Here we show that in the macroscopic limit some types of evaluations of gradient fields of the chemical stimulus result in a change of the classical parabolic Keller-Segel model for chemotaxis. Under suitable structure conditions, global solutions for the kinetic models can be shown.open1128Nsciescopu

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

Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of 'run and tumble'. The existence of solutions has been obtained in several papers [Chalub et al. Monatsh. Math. 142, 123--141 (2004), Hwang et al. SIAM J. Math. Anal. 36, 1177--1199 (2005)] using direct and strong dispersive effects. Here, we use the weak dispersion estimates of [Castella et al. C. R. Acad. Sci. Paris 322, 535--540 (1996)] to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.Comment: 19 pages, 2 figure

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

On the hydrodynamical limit for a one dimensional kinetic model of cell aggregation by chemotaxis

International audienceThe hydrodynamic limit of a one dimensional kinetic model describing chemotaxis is investigated. The limit system is a conservation law coupled to an elliptic problem for which the macroscopic velocity is possibly discontinuous. Therefore, we need to work with measure-valued densities. After recalling a blow-up result in finite time of regular solutions for the hydrodynamic model, we establish a convergence result of the solutions of the kinetic model towards solutions of a problem limit defined thanks to the flux. Numerical simulations illustrate this convergence result

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 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 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