Data-driven mathematical modelling and simulation of Rho-Myosin dynamics

In this thesis, a full repertoire of model formulation, model analysis, numerical analysis, sensitivity analysis and Bayesian method for parameter identification are presented, that seek to describe faithfully the temporal dynamics of GEF–Rho– Myosin signalling pathway as observed experimentally. The thesis is based on rigorous mathematical and numerical analysis to provide robust models and numerical results that exhibit the temporal dynamics as observed in experiments. We also explore the effect of spatial inhomogeneity on two of the models formulated. The modelling is based on experimental observations, and therefore three different mathematical models are formulated from first principles depending on the constitutive laws for the interaction between chemical species, entailing that new mathematical models are obtained. Detailed mathematical analysis of the stability of uniform steady states using nullcline theory, linear stability theory and sign pattern analysis is carried out, to characterise mathematically the key temporal dynamics of stability, oscillations, excitability and bistability as observed in experiments. Numerical bifurcation analysis using Matcont and numerical simulations carried using MATLAB illustrate theoretical analytical results through parameter variations for the key temporal dynamics. Rigorous sensitivity analysis provides a powerful tool for investigating the effects of parameter variations through local and global sensitivity. In particular, we use local sensitivity theory to characterise the limit cycle behaviour of an oscillatory dynamical system in terms of parameter variations and therefore, the thesis provides premises to characterise or study amplitude and period sensitivity to parameter variations. A full Bayesian approach is applied to the model for the identification of parameters that best-fits the model to experimental results. Therefore, the thesis provides a new framework for incorporating prior knowledge about parameters, which results in obtaining full probability distribution for parameters. Finally, the thesis explores and studies the spatially extended version on the ODE models. We analyse the existence of Turing instability for some parameter values. This proof-of-concept set premises to extend the temporal models to include spatial variations in the form of coupled bulk-surface reaction-diffusion systems through compartmentalisation of the spatial domain.