Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics

TL;DR

This paper introduces a generalized Voronoi tessellation model incorporating heterogeneous cell sizes, forces, and stochastic effects to realistically simulate two-dimensional tissue dynamics and cell shape variability.

Contribution

It develops a novel theoretical framework for weighted Voronoi diagrams with force interactions and stochastic motility, enabling more realistic tissue modeling.

Findings

Realistic cell shapes generated by the model.

Topologically distinct tissue conformations observed.

Tradeoff between cell size heterogeneity and lamellae extension.

Abstract

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.