# Mathematical modelling of mechanotransduction via RhoA signalling pathways

## TL;DR

This paper develops a mathematical model to study how cells convert mechanical signals into chemical ones through RhoA signaling, revealing how cell shape and stiffness affect this process.

## Contribution

The novel contribution is a two-way coupled model of RhoA signaling and cell mechanics using bulk-surface finite elements, showing robustness in cell deformation.

## Key findings

- Cell deformation is robust to changes in substrate stiffness due to signaling-mechanics coupling.
- Cell shape strongly influences mechanotransduction dynamics.
- A threshold-like response occurs with changes in substrate stiffness.

## Abstract

We derive and simulate a mathematical model for mechanotransduction related to the Rho GTPase signalling pathway. The model addresses the bidirectional coupling between signalling processes and cell mechanics. A numerical method based on bulk-surface finite elements is proposed for the approximation of the coupled system of nonlinear reaction-diffusion equations, defined inside the cell and on the cell membrane, and the equations of elasticity. Our simulation results illustrate novel emergent features such as the strong dependence of the dynamics on cell shape, a threshold-like response to changes in substrate stiffness, and the fact that coupling mechanics and signalling can lead to the robustness of cell deformation to larger changes in substrate stiffness, ensuring mechanical homeostasis in agreement with experiments.

Mechanotransduction, a process by which cells convert mechanical stimuli into chemical signals, plays a crucial role in cell functions. To better understand this phenomenon, we need to analyse how signalling processes and cell mechanics work together. For this purpose we derive and simulate a mathematical model of mechanotransduction related to the Rho GTPase signalling pathways, central to almost all fundamental cellular processes including cell polarity, movement, division, and cytoskeleton reorganization. The model introduces a two-way coupling between the signalling processes and cell mechanics. We use a numerical method based on bulk-surface finite elements to solve model equations numerically. Our simulation results illustrate novel emergent features such as a strong dependence of the dynamics on cell shape, a threshold-like response to changes in substrate stiffness, and the fact that the two-way coupling between mechanics and signalling can lead to the robustness of cell deformation to larger changes in substrate stiffness, ensuring mechanical homeostasis in agreement with experiments. These interesting insights help us unravel the underlying mechanisms in mechanotransduction.