A fast validation test of gene regulatory network models via the Fokker-Planck equation

Since Waddington proposed the concept of the "epigenetic landscape" in 1957, researchers have developed various methodologies to represent it in diverse processes. Studying the epigenetic landscape provides valuable qualitative information regarding cell development and the stability of phenotypic and morphogenetic patterns. Although Waddington's original idea was a visual metaphor, a contemporary perspective relates it to the landscape formed by the basins of attraction of a dynamical system describing the temporal evolution of protein concentrations driven by a gene regulatory network. Transitions among these attractors can be driven by stochastic perturbations, with the cell state more likely to transition to the nearest attractor or to the one that presents the path of least resistance. In this study, we define the epigenetic landscape using the free energy potential obtained from the solution of the Fokker-Planck equation on the regulatory network. Specifically, we obtained a numerical approximate solution of the Fokker-Planck equation describing the Arabidopsis thaliana flower morphogenesis process. We observed good agreement between the coexpression matrix obtained from the Fokker-Planck equation and the experimental coexpression matrix. This paper proposes a method for obtaining this landscape by solving the Fokker-Planck equation (FPE) associated with a dynamical system describing the temporal evolution of protein concentrations involved in the process of interest. As these systems are high-dimensional and analytical solutions are often unfeasible, we propose a gamma mixture model to solve the FPE, transforming this problem into an optimization problem. This methodology can enhance the analysis of gene regulatory networks by directly relating theoretical mathematical models with experimental observations of coexpression matrices, thus providing a discriminating technique for competing models.