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Spatio-temporal biochemical signaling in a large class of protein-proteininteraction networks is well modeled by a reaction-diffusion system. The globalexistence of the solution to the reaction-diffusion system is determined by thereaction kinetics model and the protein network topology. We propose a novelreaction kinetics model that guarantees that the reaction-diffusion system withthis model has a nonnegative invariant global classical solution for anynetwork topology. We then present a computational method to identify theunknown parameters and initial values for a reaction-diffusion system with thisreaction kinetics model. The identification approach solves an optimizationproblem that minimizes the cost function defined as the $L^2$-norm of thedifference between the data and the solution of the reaction-diffusion system.We utilize an adjoint-based optimal control method to obtain the gradients ofthe cost function with respect to the parameters and initial values. Theregularity of the global classical solutions of the reaction-diffusion systemand its corresponding adjoint system avoids situations in which the gradientsblow up, and therefore guarantees the success of the identification method forany network structure. Utilizing this gradient information, an efficientalgorithm to solve the optimization problem is proposed and applied to estimatethe mass diffusivities, rate constants and initial values of areaction-diffusion system that models protein-protein interactions in asignaling network that regulates the actin cytoskeleton in a malignant breastcell.

Identification of Parameters and Initial Values for Reaction-Diffusion Systems in Protein Networks (Extended Version)