Voronoi tessellations have been used to model the geometric arrangement of
cells in morphogenetic or cancerous tissues, however so far only with flat
hypersurfaces as cell-cell contact borders. In order to reproduce the
experimentally observed piecewise spherical boundary shapes, we develop a
consistent theoretical framework of multiplicatively weighted distance
functions, defining generalized finite Voronoi neighborhoods around cell bodies
of varying radius, which serve as heterogeneous generators of the resulting
model tissue. The interactions between cells are represented by adhesive and
repelling force densities on the cell contact borders. In addition, protrusive
locomotion forces are implemented along the cell boundaries at the tissue
margin, and stochastic perturbations allow for non-deterministic motility
effects. Simulations of the emerging system of stochastic differential
equations for position and velocity of cell centers show the feasibility of
this Voronoi method generating realistic cell shapes. In the limiting case of a
single cell pair in brief contact, the dynamical nonlinear Ornstein-Uhlenbeck
process is analytically investigated. In general, topologically distinct tissue
conformations are observed, exhibiting stability on different time scales, and
tissue coherence is quantified by suitable characteristics. Finally, an
argument is derived pointing to a tradeoff in natural tissues between cell size
heterogeneity and the extension of cellular lamellae.Comment: v1: 34 pages, 19 figures v2: reformatted 43 pages, 21 figures, 1
table; minor clarifications, extended supplementary materia