# Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration

## TL;DR

This paper introduces and analyzes a non-local, non-linear Fokker-Planck model for cell crawling migration, capturing the universal coupling between cell speed and persistence, with results on solution behavior and convergence.

## Contribution

It presents a novel non-local, non-linear Fokker-Planck model for cell motility, including analysis of solution existence, blow-up conditions, and convergence properties.

## Key findings

- Solutions are global below critical mass
- Solutions blow up above critical mass
- Quantitative convergence to equilibrium

## Abstract

Cell movement has essential functions in development, immunity and cancer. Various cell migration patterns have been reported and a general rule has recently emerged, the so-called UCSP (Universal Coupling between cell Speed and cell Persistence), [30]. This rule says that cell persistence, which quantifies the straightness of trajectories, is robustly coupled to migration speed. In [30], the advection of polarity cues by a dynamic actin cytoskeleton undergoing flows at the cellular scale was proposed as a first explanation of this universal coupling. Here, following ideas proposed in [30], we present and study a simple model to describe motility initiation in crawling cells. It consists of a non-linear and non-local Fokker-Planck equation, with a coupling involving the trace value on the boundary. In the one-dimensional case we characterize the following behaviours: solutions are global if the mass is below the critical mass, and they can blow-up in finite time above the critical mass. In addition, we prove a quantitative convergence result using relative entropy techniques.