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Description

Dynamics of biological cells and bacteria at the level of individual cell (microscopic scale) is usually complicated and involves self-propelled random walk, motion in chemotactic gradient or directed motion with quasi-periodic reversals. It makes the task of the modeling of the dynamics of large ensembles of cells quite challenging. We used the microscopic motion of cells to derive the macroscopically averaged equations for the dynamics of the cellular density (coupled with the dynamics of chemoattractant if relevant) in two particular cases. First case is the dynamics of the cells with strongly fluctuating shape with their centers of mass experiencing the random walk relevant e.g. for the the dynamics of Dicty amoeba or mesenchymal cells. Second case is the dynamics of Myxobacteria which move with near constant speed and experience periodic reversals of the direction of their motion. In all cases cells interacts through the excluded volume constraint which does not allow cells to penetrate into each other. The resulting partial differential equations (PDEs) for the cellular density have the form of the nonlinear diffusion equation (although the nonlinear diffusion coefficients are different in these two cases). The derivation of these macroscopic equations is the loose analog of the derivation of the Navier-Stokes equations from the dynamics of individual molecules (except that the individual cell dynamics is highly overdamped and includes different types of self-propelled motion while the molecular dynamics is Newtonian). We show that perturbation theory breaks for quite small cellular densities and instead non-perturbative approaches were developed. The resulting PDEs have no fitting parameters and depend only on the parameters of the individual cellular motion which can be easily measured in experiments. We demonstrated very good quantitative agreement between large scale Monte Carlo simulations of the microscopic stochastic dynamics of the ensembles of cells and the solutions of the derived PDEs.