TY - JOUR

T1 - From individual to collective behavior in bacterial chemotaxis

AU - Erban, Radek

AU - Othmer, Hans G.

PY - 2005

Y1 - 2005

N2 - Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these very different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time- and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time, we obtain a single second-order hyperbolic equation for the population density, and using a parabolic scaling, we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

AB - Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these very different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time- and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time, we obtain a single second-order hyperbolic equation for the population density, and using a parabolic scaling, we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

KW - Aggregation

KW - Bacterial chemotaxis

KW - Chemotaxis equations

KW - Internal dynamics

KW - Transport equations

KW - Velocity-jump process

UR - http://www.scopus.com/inward/record.url?scp=19944408985&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19944408985&partnerID=8YFLogxK

U2 - 10.1137/S0036139903433232

DO - 10.1137/S0036139903433232

M3 - Article

AN - SCOPUS:19944408985

VL - 65

SP - 361

EP - 391

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -