Reaction–diffusion systems in protein networks: Global existence and identification

Abstract

Spatio-temporal biochemical signaling in a large class of protein–protein interaction networks is well modeled by a reaction–diffusion system. The global existence of the solution to the reaction–diffusion system is determined by the reaction kinetics model and the protein network topology. We propose a novel reaction kinetics model that guarantees that the reaction–diffusion system with this model has a nonnegative invariant global classical solution for any network topology. We then present a computational method to identify the unknown parameters and initial values for a reaction–diffusion system with this reaction kinetics model. The identification approach solves an optimization problem that minimizes the cost function defined as the L2-norm of the difference between the data and the solution of the reaction–diffusion system. We utilize an adjoint-based optimal control method to obtain the gradients of the cost function with respect to the parameters and initial values. The regularity of the global classical solutions of the reaction–diffusion system and its corresponding adjoint system avoids situations in which the gradients blow up, and therefore guarantees the success of the identification method for any network structure. Utilizing this gradient information, an efficient algorithm to solve the optimization problem is proposed and applied to estimate the mass diffusivities, rate constants and initial values of a reaction–diffusion system that models protein–protein interactions in a signaling network that regulates the actin cytoskeleton in a malignant breast cell.