Biology Reference
In-Depth Information
in space. All chemical reactions are described by the source
terms
f
i
.
8.1. Previous Approaches and Applications in Biology
Coupled reaction-diffusion systems exhibit nontrivial stability properties
that can give rise to the formation of stable concentration patterns called
Turing patterns,
1
or traveling waves called Fisher waves.
126
Twenty years after the seminal work of Turing,
1
Gierer and
Meinhardt used reaction-diffusion systems to formulate their theory of
pattern formation in biology.
127
They introduced the Gierer-Meinhardt
model, which has become one of the most widely used pattern formation
models in biology, with later applications also in computer graphics.
128
The first biological applications of reaction-diffusion models con-
sidered morphogenesis,
1
following the idea that reaction-diffusion pat-
terns of growth factors could explain the geometries and shapes found
in nature. Computer simulations linking pattern formation to growth
and morphogenesis were done using, e.g. hybrid cellular automata-
PDE simulations to explain stalk formation and cell differentiation in
slime mold.
2
Simulations of reaction-diffusion patterns on surfaces first considered
spherical objects such as globular tumors.
18
Morphogenesis of more complex
surfaces was simulated using an FE method to solve the reaction-
diffusion equation on triangulated surfaces.
3
This method allowed treat-
ing shapes as complex as branched unicellular algae. Moving-mesh FE
techniques were later used to directly couple the motion of the boundary
to reaction-diffusion patterns on continuously deforming 2D domains.
129
A different approach uses the solution of an interior Poisson problem to
evolve the surface shape.
20
Besides morphogenesis, reaction-diffusion models also have impor-
tant applications in cell motility,
130
cell modeling,
131
and cell culture pat-
tern formation.
132
Emerging applications also include spatiotemporal
simulations of cell signaling pathways.
133
Since the first ODE model of
the chemotaxis pathway in
Escherichia coli
was published by Bray
et al
. in
1993,
134
computer simulations have become increasingly more sophisti-
cated in resolving spatial phenomena. A recent model explicitly includes
