Stability Analysis of Gene Expression Patterns by Dynamical Systems and a Combinatorial Algorithm

The emerging field of modern biosciences and biotechnologies ask for appropriate and refined methods from applied mathematics. Modeling and prediction of gene-expression patterns have an important place here. In our paper, we deepen the analytical understanding and the algorithmical treatment by including affine terms on the right-hand side of the nonlinear differential equations and using Runge-Kutta (e.g., Heun) rather than Euler discretization. In the center of our research, there is the investigation of stability which we motivate in terms of biology and medicine, and detect by modified Brayton and Tong algorithm applied to the corresponding time-discrete dynamics in an extended space. This paper pays special attention to the motivating and analytical preparations of the combinatorial algorithm which based on the observation of polyhedra.