A Langevin Model for Cell Speed Calculation Under Actin-Driven Membrane Fluctuations

Single-cell migration is quintessential to wound healing, tissue growth, embryo development, and tumor invasion, among many other domains. It is usually modeled as a Langevin-like problem consisting of short-time ballistic motion and random walks for long periods. Conversely, experiments revealed a random motion at very short intervals, ruling out the definition of instantaneous velocity. Moreover, cells exhibit forward net displacement, which produce displacement distributions peaked at positive values. In this study we propose a model for a punctual cell whose displacement is proportional to polarization, a geometric quantity that bypasses the ill-defined velocity problem because it can be defined in a single instant of time, preserving short-time memory. We consider that polarization follows a modified Langevin equation with a colored noise that favors forward net displacements. Numerical solutions yield displacement distributions that peak at positive values, in agreement with experiments and Potts model simulations. Furthermore, solving the model analytically presents bi-exponentially decaying displacement autocorrelation functions, improving agreement between theoretical and experimental results. The normalized Root Mean Square Error nRMSE is used to estimate errors between experimental data and the resulting fitted model. Results range from 0.18\% to 0.08\%, which at worst show equivalent errors to previous models and at best a 2 to 3 times error decrease. The model and results provide a robust basis for theoretical models and experimental measurements that can be utilized by wet-lab scientists to better quantify cellular trajectories.