Page 62 -

55
4. Chemotactic Cell Motion
and Biological Pattern Formation

Peter A. Markowich and Dietmar Ölz 1

One of the most important principles governing the movement of biological
cells is represented by chemotaxis, which refers to cell motion in direction of the
gradient of a chemical substance. In some cases the chemical is externally pro-
duced, in others the cells themselves generate the chemical in order to facilitate
cell aggregation. In certain biological processes more than one chemical is actu-
ally responsible for the chemotactic cell motion. Typical examples of chemotaxis
occur in embryology, in immunology, tumor biology, aggregation of bacteria or
amoeba etc.
The most basic and most famous mathematical model for chemotaxis was
originally derived in 1953 by C.S. Patlak [7] and then in 1970 by E. Keller and
L.A. Segel [4]. Meanwhile, this so called Keller–Segel model has become one of
the most well analyzed systems of partial differential equations in mathematical
biology, giving many insights into cell biology as well as into the analysis of
nonlinear partial differential equations.
The main unknowns of the Keller–Segel system are the nonnegative cell
density r = r(x, t) and the chemical concentration S = S(x, t), where x denotes
the one, two or three dimensional space variable and t> 0 the time variable.
Then, based on the hypothesis that cell motion is driven by diffusion on one
hand and by the gradient of the chemical as driving force on the other hand, the
cell density satisfies the (parabolic) partial differential equation of convection-
diffusion or Fokker/Planck type:

r t = div(D 0 grad r − cr grad S) (4.1)
where D 0 is the positive cell diffusivity and c the positive chemotactic sensitivity.
In many realistic modeling situations, D 0 and c have to be allowed to depend on
the cell density r and on the chemical concentration S.Weremarkthatdiffusion
corresponds to undirected random (Brownian) motion of the cells, while the
convection by the chemo-attractant stems from the reorientation phase of the
cell motion, in direction of the gradient of the chemical concentration. These
two phases in the cell motion have been observed very well for the slime mold
Dictyostelium discoideum.
The temporal variation of the chemical is also determined by diffusion on
one hand and, on the other hand, by its production (by external sources or by
the cells themselves) and its degradation (e.g. due to chemical reactions). This

1
http://homepage.univie.ac.at/dietmar.oelz/