Tropical Abstraction of Biochemical Reaction Networks with Guarantees

Biochemical molecules interact through modification and binding reactions, giving raise to a combinatorial number of possible biochemical species. The time-dependent evolution of concentrations of the species is commonly described by a system of coupled ordinary differential equations (ODEs). However, the analysis of such high-dimensional, non-linear system of equations is often computationally expensive and even prohibitive in practice. The major challenge towards reducing such models is providing the guarantees as to how the solution of the reduced model relates to that of the original model, while avoiding to solve the original model. In this paper, we have designed and tested an approximation method for ODE models of biochemical reaction systems, in which the guarantees are our major requirement. Borrowing from tropical analysis techniques, dominance relations among terms of each species' ODE are exploited to simplify the original model, by neglecting the dominated terms. As the dominant subsystems can change during the system's dynamics, depending on which species dominate the others, several possible modes exist. Thus, simpler models consisting of only the dominant subsystems can be assembled into hybrid, piecewise smooth models, which approximate the behavior of the initial system. By combining the detection of dominated terms with symbolic bounds propagation, we show how to approximate the original model by an assembly of simpler models, consisting in ordinary differential equations that provide time-dependent lower and upper bounds for the concentrations of the initial models species. Our method provides sound interval bounds for the concentrations of the chemical species, and hence can serve to evaluate the faithfulness of tropicalization-based reduction heuristics for ODE models of biochemical reduction systems. The method is tested on several case studies.