Using cell potential energy to model the dynamics of adhesive biological cells

Developing a continuous mathematical model of a physical phenomenon which is based on a discrete model of the same system is not straightforward. Yet such a process is useful in illustrating the link between the individual behavior of the elements comprising a system and its macroscopic behavior. Collections of biological cells can exhibit phenomena such as pattern formation, aggregation, and invasion, and mathematics has proven useful in elucidating the underlying dynamics of these phenomena. The continuous models formulated are frequently of reaction-diffusion form, and central to their application is a knowledge of the diffusion coefficient of a collection of the elements comprising the system. Cohen and Murray [J. Math Biol. 12, 237 (1981)] developed a means of deriving this quantity which has since been largely neglected by model developers, and which is based on a knowledge of the potential energy associated with the mutual interaction between the cells. In this work, we begin by deriving the energy of interaction of biological cells modeled as adhesive, deformable spheres. In so doing, we are able to quantify the equilibrium density of a biological cell aggregate, and also obtain a quantitative estimate of the diffusion coefficient of a collection of cells modeled in this way. In so doing, we are able to use experimental data from single-cell studies of the adhesiveness and cell membrane elasticity of a biological cell to derive the diffusion coefficient of a cell mass composed of a collection of identical cells. This allows us to better inform the parameter values used in reaction-diffusion models of biological systems. We go on to apply this technique to a particular situation: modeling the dynamics of a collection of biological cells which experience strong cell-cell adhesion. In so doing, we derive a nonlinear fourth-order partial differential equation to model this system. We conclude by discussing the practical utility of this work in illuminating the link between the microscopic behavior of individual biological cells and the macroscopic behavior of the aggregate to which they give rise, and also by giving some insights into how the modeling of cell-cell adhesion may be treated mathematically.