A coupled bulk-surface model for cell polarisation

TL;DR

This paper extends a minimal cell polarisation model to three dimensions using coupled bulk-surface PDEs, analyzing how local perturbations lead to stable polarization patterns influenced by geometry.

Contribution

It introduces a 3D coupled bulk-surface PDE framework for cell polarisation, incorporating protein compartmentalisation and complex geometries, advancing mathematical modeling of cellular processes.

Findings

Local surface perturbations trigger propagating reactions.

Geometry influences pattern formation and pinning.

Numerical simulations confirm theoretical analysis.

Abstract

Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behavior of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions.