Biology Reference
In-Depth Information
experiments that would not be possible in reality due to ethical reasons.
Although we focus on applications of spatiotemporal computer simula-
tions in biology, the employed concepts and methods are more gener-
ally valid.
Resolving a dynamic process in space greatly increases the number of
degrees of freedom (variables) that need to be tracked. Consider, for
example, a biochemical heterodimerization reaction. This reaction can be
modeled by its chemical kinetics using three variables: the concentrations
of the two monomers and the concentration of dimers. Assume now that
monomers are produced at certain locations in space and freely diffuse
from there. Their concentration thus varies in space in such a way that it
is higher close to the source and lower farther away, which greatly
increases the number of variables we have to track in the simulation.
If we are, say, interested in the local concentrations at 1000 positions, we
already have to keep track of 3000 variables. Moreover, the reactions tak-
ing place at different points in space are not independent. Each local
reaction can influence the others through diffusive transport of
monomers and dimers. The complexity of spatiotemporal models thus
rapidly increases. In fact, there is no theoretical limit to the number of
points in space that we may use to resolve the spatial patterns in the con-
centration fields. Using infinitely many points corresponds to modeling
the system as a continuum.
A number of powerful mathematical tools are available to efficiently
deal with spatiotemporal models and to simulate them. While it is not
possible within the scope of this chapter to describe each of them in
detail, we will give an overview with references to specialized literature.
We then review in detail one particular method that is well suited for
applications in biology. But before we start, we revisit some of the moti-
vations and particularities of spatiotemporal modeling in the life sciences.
In spatiotemporal modeling, nature is mostly described in four
dimensions: time plus three spatial dimensions. While time and the pres-
ence of reservoirs (integrators) are essential for the existence of dynam-
ics, three-dimensional (3D) spatial aspects also play important roles in
many biological processes. Think, for example, of predators hunting their
prey in a forest, of blood flowing through our arteries, of the electro-
magnetic fields in the brain, or of such an unpleasant phenomenon as the
