# A New Interpretation of The Keller-Segel Model Based on Multiphase Modelling

preprint

posted on 19.07.2005, 16:08 by Helen M. Byrne, Markus R. OwenIn this paper an alternative derivation and interpretation are presented
of the classical Keller-Segel model of cell migration due to random motion
and chemotaxis. A multiphase modelling approach is used to describe how a population
of cells moves through a fluid containing a diffusible chemical to which
the cells are attracted. The cells and fluid are viewed as distinct components of a
two-phase mixture. The principles of mass and momentum balances are applied
to each phase, and appropriate constitutive laws imposed to close the resulting
equations. A key assumption here is that the stress in the cell phase is influenced
by the concentration of the diffusible chemical.
By restricting attention to one-dimensional cartesian geometry we show how
the model reduces to a pair of nonlinear coupled partial differential equations for
the cell density and the chemical concentration. These equations may be written in
the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between
the random motility and chemotaxis coefficients, both depending in an
inter-related manner on the chemical concentration. We suggest that this may explain
why many chemicals appear to stimulate both chemotactic and chemokinetic
responses in cell populations.
After specialising our model to describe slime mold we then show how the
functional form of the chemical potential that drives cell locomotion influences
the ability of the system to generate spatial patterns. The paper concludes with
a summary of the key results and a discussion of avenues for future research.

## History

## School

- Science

## Department

- Mathematical Sciences