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.