# Numerical analysis of a mechanotransduction dynamical model reveals homoclinic bifurcations of extracellular matrix mediated oscillations of the mesenchymal stem cell fate

## TL;DR

This study uses numerical bifurcation analysis to explore how extracellular matrix stiffness influences mesenchymal stem cell differentiation, revealing complex oscillatory behaviors and bifurcations that affect cell fate transitions.

## Contribution

It introduces a detailed bifurcation analysis of a mechanotransduction model, uncovering homoclinic bifurcations and oscillation regimes linked to cell differentiation.

## Key findings

- Oscillations of adhesion area and transcription factors depend on substrate stiffness.
- Homoclinic bifurcations cause abrupt loss of oscillations, influencing cell fate.
- Oscillatory regimes favor neurogenic and adipogenic differentiation on soft substrates.

## Abstract

We perform one and two-parameter numerical bifurcation analysis of a mechanotransduction model approximating the dynamics of mesenchymal stem cell differentiation into neurons, adipocytes, myocytes and osteoblasts. For our analysis, we use as bifurcation parameters the stiffness of the extracellular matrix and parameters linked with the positive feedback mechanisms that up-regulate the production of the YAP/TAZ transcriptional regulators (TRs) and the cell adhesion area. Our analysis reveals a rich nonlinear behaviour of the cell differentiation including regimes of hysteresis and multistability, stable oscillations of the effective adhesion area, the YAP/TAZ TRs and the PPAR$\gamma$ receptors associated with the adipogenic fate, as well as homoclinic bifurcations that interrupt relatively high-amplitude oscillations abruptly. The two-parameter bifurcation analysis of the Andronov-Hopf points that give birth to the oscillating patterns predicts their existence for soft extracellular substrates ($<1kPa$), a regime that favours the neurogenic and the adipogenic cell fate. Furthermore, in these regimes, the analysis reveals the presence of homoclinic bifurcations that result in the sudden loss of the stable oscillations of the cell-substrate adhesion towards weaker adhesion and high expression levels of the gene encoding Tubulin beta-3 chain, thus favouring the phase transition from the adipogenic to the neurogenic fate.