Modeling Single Cell Migration: Polarization as a Solution to Short-Time Diffusion Challenges

Understanding single-cell migration is essential for explaining wound healing, tissue growth, tumor metastasis, and many other biological phenomena. It is frequently modeled as a Langevin process that produces a short-time ballistic motion and long-time random walks due to orientation changes. Nonetheless, recent experiments revealed random walk behaviors in smaller time scales caused by cell pseudopod fluctuations, undermining the definition of an instantaneous velocity as the infinitesimal ratio of displacement and time. Moreover, Langevin models do not produce net forward motion observed in cell migration essays.



In this study, we propose a model based on the experimental observations of coupling between displacement and persistence. It consists of a punctual cell whose displacement is proportional to polarization, a quantity calculated geometrically, which makes it well-defined. Polarization models a cell's short-term structural memory and bypasses the lack of velocity definition.



We assume that the polarization follows a Langevin process with added drift that favors forward motion and that displacement follows the polarization with an added noise representing membrane fluctuations. Analytical solutions yield mean squared displacement (MSD) and mean Velocity Autocorrelation (mVACF) functions that agree with numerical results and cellular Potts model simulation. They also fit experimental data better than previous models, showing a normalized Root Mean Square Error (nRMSE) ranging from 0.18% to 0.08%. This corresponds to an error level that is equivalent to previous models in the worst-case scenario and reduced by 2--3 times in the best-case scenario. The model and results provide a robust basis for theoretical models and experimental measurements that wet lab scientists can apply to study single-cell migration.