Analyzing the bistability of the minimally bistable ERK network using the discriminant locus

The extracurricular signal regulated kinase (ERK) system is a biological signaling network with “important roles in regulating cellular activity." For this project, we will analyze the bistability, i.e, its capacity to hold two or more positive steady states that are stable to small perturbations, of the minimally bistable ERK network by analyzing its number of real positive steady states for different parameterizations of the network. Previous research used mixed volume computations to determine that the possible range of positive steady states is 1 to 5, however, it has been conjectured that the maximum number of positive steady states is 3. While this system has been analyzed from a convex geometry perspective, our goal is to analyze the ERK system from an algebraic perspective by generalizing the behavior of the steady states of the ERK system over a two-dimensional parameter space in terms of parameters {kcat, kon}, which primarily impact bistability. We use homotopy continuation to discretely sample the discriminant locus and inflection curves of the parameter space, which separates the parameter space into distinct regions each corresponding to a constant number of positive real solutions. Our results demonstrate that in two different parameterizations of the network that the maximum number of positive steady states was 3 and that the relationship between the rate constants was the primary factor in determining this upper bound.