Microscopic Langevin model for the modified Fürth equation*

Recent experimental research on the mobility of living cells indicate that their diffusion is not completely described by Fürth's equation since they display a normal diffusive regime before the onset of the ballistic motion. Moreover, results from a cellular Potts model simulation for the motion of cells in two dimensions present the same type of behavior and a modification of Fürth's equation has been proposed to fit the mean squared displacements (MSD) found both experimentally and computationally (Fortuna et al., submitted). Here, we propose a microscopic Langevin model to explain this anomaly in the MSD: we show that a polarization direction for the persistent motion as well as at least two time scales are needed. The model reproduces well the MSD obtained from the modified Fürth equation and is robust to changes in the microscopic dynamics of the polarization as a function of cell migration speed, which is in agreement with experimental data which suggests that persistence of trajectories is coupled to cell migration speed mediated by actin flows (Maiuri et al., 2015).