Transport equations and indices for random and biased cell migration based on single cell properties

A mathematical-modeling-and-analysis framework is presented to predict quantitative indices for random and biased cell migration based on mechanistic parameters describing the receptor-mediated motility of an individual cell. A general set of stochastic differential equations is derived to model cell movement on the time scale of the molecular processes that govern cell locomotion. Then, by adiabatic elimination of the fast variables with projector operator formalism, we derive approximate Fokker-Planck equations (FPEs) for the resultant cell movement on longer time scales. Analysis of these FPEs provides expressions for statistical indices that are commonly used to characterize cell movement, such as root-mean-squared cell speed, directional persistence time, mean-squared displacement, random motility coefficient, and drift velocity, in terms of the mechanistic parameters. As specific examples, we apply this approach to adhesion-mediated directed cell migration (haptotaxis) and chemoattractant-mediated directed cell migration (chemotaxis).