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Talk abstract:
Reaction--Diffusion Equations and Mechanochemical Coupling
on Irregular Deforming Domains
Dean Bottino, University of Utah
When numerically modeling the motion of crawling ameboid cells,
the approximation of spatiotemporal intracellular signal dynamics
(such as Ca2+ dynamics) requires solving the relevant
reaction--diffusion equations on the deforming domain bounded
by the cell membrane. Similarly, when modeling mound and slug--level
dictyostelium interactions, or the coordinated movements of
cells in early embryonic development, representing the coupling
between the motion of individual cells to the intercellular
signals that the cells generate, a method to solve reaction--diffusion
equations on the deforming cell mass is required.
A rather straightforward numerical method is presented in
which the (two-dimensional) domain in question is subdivided
by a Delaunay triangulation, the geometric dual of which is
a Voronoi tesselation. One then obtains expressions for approximating
the reaction diffusion equations among the voronoi polygons.
The Delaunay neighbor relationships can also be used to mediate
active mechanical forces that can result from rules based on
the local signal concentrations. Furthermore, a penalty method
can be used to maintain the incompressibility of each Voronoi
polygon.
Preliminary tests of the method indicate that it is spatially
first--order accurate when solving the diffusion equation on
a circular domain. While more accurate methods might exist for
the generalized problem of solving PDE's on a moving, deforming
domain, the method presented here is suitable for biological
applications for two reasons: (1) the lack of accuracy of the
numerical method is likely to be insignificant compared to the
inaccuracy of biological data; (2) the method itself is straightforward,
and in some cases the voronoi polygons can be literally interpreted
as individual cells, so that the interaction rules among the
voronoi polygons can be obtained directly.
Future possiblities for the method will be discussed, including
applications to individual cell movement and multicellular interactions,
extension of the method to three dimensions, and stablization
of the method by iterative, implicit time-stepping.
Back to Workshop Schedule
1998-1999
Mathematics in Biology
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