Two-dimensional Multiscale Model of Cell Motion in a Chemotactic Field

Abstract

The Cellular Potts Model (CPM) has been used at a cellular scale for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. Continuous models in the form of partial differential, integral or integro-differential equations are used for studying biological problems at large scale. It is crucial for developing multiscale biological models to establish a connection between discrete microscopic stochastic models, including CPM, and macroscopic continuous models. To demonstrate multiscale approach we derive in this paper continuous limit of a two-dimensional CPM with the chemotactic interactions in the form of a Fokker-Planck equation describing evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. We demonstrate that CPM Monte Carlo simulations are in excellent agreement with the numerics for the continuous macroscopic model with different forms of the chemical field term.