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Abstract
Eukaryotic cell crawling is a highly complex biophysical and biochemical process,
where deformation and motion of a cell are driven by internal, biochemical regulation
of a poroelastic cytoskeleton. One challenge to building quantitative models that
describe crawling cells is solving the reaction-diffusion-advection dynamics for the
biochemical and cytoskeletal components of the cell inside its moving and deforming
geometry. Here we develop an algorithm that uses the level set method to move the
cell boundary and uses information stored in the distance map to construct a finite
volume representation of the cell. Our method preserves Cartesian connectivity of
nodes in the finite volume representation while resolving the distorted cell geometry.
Derivatives approximated using a Taylor series expansion at finite volume interfaces
lead to second order accuracy even on highly distorted quadrilateral elements. A modified,
Laplacian-based interpolation scheme is developed that conserves mass while interpolating
values onto nodes that join the cell interior as the boundary moves. An implicit time-stepping
algorithm is used to maintain stability. We use the algoirthm to simulate two simple
models for cellular crawling. The first model uses depolymerization of the cytoskeleton
to drive cell motility and suggests that the shape of a steady crawling cell is strongly
dependent on the adhesion between the cell and the substrate. In the second model,
we use a model for chemical signalling during chemotaxis to determine the shape of
a crawling cell in a constant gradient and to show cellular response upon gradient
reversal.