  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 $L^2$-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.
