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.　(C)　2014　Elsevier　Inc.　All　rights　reserved.