Continuous limit of a discrete stochastic model of cell migration
We analytically derive the continuous limit of the Cellular Potts Model (CPM) for a one-dimensional cell subjected to constant and run-and-tumble driving forces. By coarse-graining the discrete lattice dynamics, we obtain the Fokker-Planck equations governing the cell's size and center-of-mass position. We show that in the low-force regime, the cell dynamics are accurately described by an overdamped Langevin equation. Beyond this regime, we expose intrinsic algorithmic artifacts, including a force-dependent diffusion coefficient, a non-linear force-velocity relationship, and the breakdown of the Einstein relation. We demonstrate that replacing the conventional Metropolis update rule with Glauber dynamics significantly mitigates these artifacts, broadening the physically valid parameter space. Our exact results bridge the gap between lattice-based simulations and continuous active matter models.