Continuous macroscopic limit of a discrete stochastic model for interaction of living cells

Abstract

In the development of multiscale biological models it is crucial to establish a connection between discrete microscopic or mesoscopic stochastic models and macroscopic continuous descriptions based on cellular density. In this paper a continuous limit of a two-dimensional Cellular Potts Model (CPM) with excluded volume is derived, describing cells moving in a medium and reacting to each other through both direct contact and long range chemotaxis. The continuous macroscopic model is obtained as a Fokker-Planck equation describing evolution of the cell probability density function. All coefficients of the general macroscopic model are derived from parameters of the CPM and a very good agreement is demonstrated between CPM Monte Carlo simulations and numerical solution of the macroscopic model. It is also shown that in the absence of contact cell-cell interactions, the obtained model reduces to the classical macroscopic Keller-Segel model. General multiscale approach is demonstrated by simulating spongy bone formation from loosely packed mesenchyme via the intramembranous route suggesting that self-organizing physical mechanisms can account for this developmental process.

4 pages, 3 figures

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• Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description
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• Logarithmic scaling of the collapse in the critical Keller-Segel equation
• A Keller-Segel model for C elegans L1 aggregation
• Statistical Dynamics of Regional Populations and Economies
• Soliton approximation in continuum models of leader-follower behavior
• Mathematical models for chemotaxis and their applications in self-organisation phenomena
• Integrodifference master equation describing actively growing blood vessels in angiogenesis