UC San Diego

We consider a system of coupled reaction-diffusion equations that models the interaction between multiple types of chemical species, particularly the interaction between one messenger RNA and different types of non-coding microRNAs in biological cells. We construct various modeling systems with different levels of complexity for the reaction, nonlinear diffusion, and coupled reaction and diffusion of the RNA interactions, respectively, with the most complex one being the full coupled reaction-diffusion equations. The simplest system consists of ordinary differential equations (ODE) modeling the chemical reaction. We present a derivation of this system using the chemical master equation and the mean-field approximation, and prove the existence, uniqueness, and linear stability of equilibrium solution of the ODE system. Next, we consider a single, nonlinear diffusion equation for one species that results from the slow diffusion of the others. Using variational techniques, we prove the existence and uniqueness of solution to a boundary-value problem of this nonlinear diffusion equation. Finally, we consider the full system of reaction-diffusion equations, both steady-state and time-dependent. We use the monotone method to construct iteratively upper and lower solutions and show that their respective limits are solutions to the reaction-diffusion system. For the time-dependent system of reaction-diffusion equations, we obtain the existence and uniqueness of global solutions. We also obtain some asymptotic properties of such solutions.