Multiscale analysis of a kinetic equation for mechanotaxis

We present a new kinetic equation for cell migration driven by mechanical interactions with the substrate, an effect not previously captured in kinetic models, and essential for explaining observed collective behaviors such as those in bacterial colonies. The model introduces an acceleration term that accounts for the dynamics of motile cells undergoing mechanotaxis, where extracellular signals modulate the forces arising from cell-substrate interactions. From this formulation, we derive a family of macroscopic limit equations and analyze their principal properties. In particular, we examine linear stability and pattern formation ability through theoretical analysis, supported by numerical simulations.