Kinetic and hydrodynamic models of chemotactic aggregation

Abstract

We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.

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References in corpus (6)

• Logotropic distributions
• Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
• Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions
• Jeans type instability for a chemotactic model of cellular aggregation
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot
• Critical mass of bacterial populations in a generalized Keller-Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars

Cited by in corpus (6)

• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Jeans type analysis of chemotactic collapse
• Two-dimensional Brownian vortices
• Critical mass of bacterial populations in a generalized Keller-Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars