# A moving grid finite element method applied to a mechanobiochemical model for 3D cell migration

## TL;DR

This paper develops a moving grid finite element method to numerically solve a complex 3D mechanobiochemical model of cell migration, capturing cell deformation, protrusion, and contraction driven by biochemical and mechanical interactions.

## Contribution

It introduces a novel moving grid finite element approach for solving highly nonlinear 3D cell migration models involving reaction-diffusion and biomechanical forces.

## Key findings

- Numerical simulations show diverse cell deformation behaviors.
- Linear stability analysis supports early migration dynamics.
- The method enables studying complex biochemical and mechanical interactions.

## Abstract

This work presents the development, analysis and numerical simulations of a biophysical model for 3D cell deformation and movement, which couples biochemical reactions and biomechanical forces. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled by a force balancing equation for the displacements, the pressure and concentration forces are driven by actin and myosin dynamics, and these are in turn modelled by a system of reaction-diffusion equations on a moving cell domain. The biophysical model consists of highly non-linear partial differential equations whose analytical solutions are intractable. To obtain approximate solutions to the model system, we employ the moving grid finite element method. The numerical results are supported by linear stability theoretical results close to bifurcation points during the early stages of cell migration. Numerical simulations exhibited show both simple and complex cell deformations in 3-dimensions that include cell expansion, cell protrusion and cell contraction. The computational framework presented here sets a strong foundation that allows to study more complex and experimentally driven reaction-kinetics involving actin, myosin and other molecular species that play an important role in cell movement and deformation.