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4 Chemotactic Cell Motion and Biological Pattern Formation
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The phenomenon of finite time blow up of solutions of the classical Keller–
Segel model has been extensively discussed in the bio-mathematical literature.
Clearly, the local pre-blow up behaviour corresponds to biologically reasonable
cell accumulation due to the chemotactic attraction and has been observed in
experiments, e.g. with the slime mold Dictyostelium discoideum. However, con-
centration of the cell density in a single point is clearly biologically unreasonable
and has to be considered a defect of the model. We remark that this defect can be
repaired rather easily, for example by taking a chemotactic sensitivity c = c(r),
whichdecaystozeroasr tendstoinfinity.Thisisreferredtoas“quorumsensing”.
We nowturnour attentiontothe modeling of patternformation in the
context of embryology. Embryology is the area of biology, which is concerned
with the formation and development of a embryos from fertilisation until birth.
Morphogenesisasapartofembryologydeals with thedevelopment of patterns
and forms. One of the major problems in biology is how genetic information is
physically translated into the desired patters and forms. We typically observe
that cells move around within the embryo and finally differentiate according to
their position. But why does this happen?
Positional information is a phenomenological concept of pattern forma-
tion and differentiation introduced by Wolpert [10]. He suggested that cells are
pre-programmed to react to a chemical (“morphogen”) concentration and dif-
ferentiate accordingly. The first step however is the creation of the morphogen
concentration spatial (pre)pattern. The further morphogenesis is then a slave
process. Often, chemotaxis is considered to be a mechanism for density pre-
pattern formation. More precisely, cell differentiation occurs in regions of high
cell density [6], possibly generated by chemotactic attraction. A mathematical
study of spatial patterns and their stability in one-dimensional Keller–Segel
models with small cell diffusivity can be found in [2].
In the sequel we shall discuss (cp. [5]) a model for morphogenesis, based on
2
reaction-diffusion equations, as introduced by A.M. Turing in the year 1952 [9].
Theunknownsare theconcentrationsoftwo chemical species, u> 0and v> 0.
n
We assume that the modeling domain B is a subset of R with the dimension n
either 1, 2 or 3 and that within the set B the two concentrations satisfy the
reaction-diffusion system

u t = Δu + γf (u, v)
v t = dΔv + γg(u, v). (4.6)
Hence both chemicals are subject to diffusion, but with different diffusion co-
efficients. Here, already after non-dimensionalisation, the diffusion coefficient
of the chemical u is set to one, whereas the diffusivity of v is represented by the
constant d. This constant represents the ratio of the diffusion coefficients before
non-dimensionalisation. Furthermore both chemicals are subject to production
and decay respectively. From non-dimensionalisation we obtain that the extent
2
http://www.turing.org.uk/turing/