Exploring the spatio-temporal dynamics in activator-inhibitor systems through a dual approach of analysis and computation.

Real-world mathematical models often manifest as systems of non-linear differential equations, which presents challenges in obtaining closed-form analytical solutions. In this paper, we study the diffusion-driven instability of an activator-inhibitor-type reaction-diffusion (RD) system modeling the GEF-Rho-Myosin signaling pathway linked to cellular contractility. The mathematical model we study is formulated from first principles using experimental observations. The model formulation is based on the biological and mathematical assumptions. The novelty is the incorporation of Myo9b as a GAP for RhoA, leading to a new mathematical model that describes Rho activity dynamics linked to cell contraction dynamics. Assuming mass conservation of molecular species and adopting a quasi-steady state assumption based on biological observations, model reduction is undertaken and leads us to a system of two equations. We adopt a dual approach of mathematical analysis and numerical computations to study the spatiotemporal dynamics of the system. First, in absence of diffusion, we use a combination of phase-plane analysis, numerical bifurcation and simulations to characterize the temporal dynamics of the model. In the absence of spatial variations, we identified two sets of parameters where the model exhibit different transition dynamics. For some set of parameters, the model transitions from stable to oscillatory and back to stable, while for another set, the model dynamics transition from stable to bistable and back to stable dynamics. To study the effect of parameter variation on model solutions, we use partial rank correlation coefficient (PRCC) to characterize the sensitivity of the model steady states with respect to parameters. Second, we extend the analysis of the model by studying conditions under which a uniform steady state becomes unstable in the presence of spatial variations, in a process known as Turing diffusion-driven instability. By exploiting the necessary conditions for diffusion-driven instability and the sufficient conditions for pattern formation we carry out, numerically, parameter estimation through the use of mode isolation. To support theoretical and computational findings, we employ the pdepe solver in one-space dimension and the finite difference method in two-space dimension.