Membrane fluctuations in migrating mesenchymal cells preclude instantaneous velocity definitions

The dynamics of single-cell migration on flat surfaces are usually modeled by a Langevin-like problem for particle velocity, consisting of a ballistic motion for short time intervals and a diffusive regime for long time intervals. However, experiments and simulations have revealed an additional diffusive motion at very short time intervals, which rules out the definition of cells’ instantaneous velocities. The experimental consequence is that one cannot estimate velocity by decreasing time intervals in a sequence of measurements. While for ballistic motion, the ratio of displacement over time intervals converges to a constant value (the limit value is taken as the velocity estimate), for diffusive motion this ratio diverges. Here, instead of cell velocity, we consider cell polarization as the dynamical variable and assume evolution equations for polarization modulus (a biased Langevin equation) and direction (a Wiener process). This circumvents the measurement problems for cell velocity when the system presents short-time diffusion. That means we take cell speed and displacement as consequences of cell polarization. For displacements, we also considered an additional noise term to account for membrane fluctuations that, in fact, cause the short-time-interval diffusive behavior. As novel results, we obtain memory preservation in polarization modulus for short time intervals and a displacement distribution function that may present a maximum at finite values of polarization modulus, for adequate time intervals. We also obtain a bi-exponential displacement autocorrelation function, improving the agreement between theoretical fits and experiments or simulations. Overall, this model contributes to the robustness of experimental measurements in cell migration experiments, besides providing a sound basis for theoretical models. For wet laboratory procedures, our findings provide a warning in what regards velocity and speed estimates and we propose experimental protocols to obtain these estimates at different levels of precision.