Critical biological processes, such as cellular physiology or neuronal transmission have very different spatial scales are due to small binding sites inside or on the cell boundary, or narrow passages between large compartments. The great disparity in spatial scales can be resolved by singular perturbation analysis of their mathematical models. Deriving the function of neuronal synapses from their molecular organization falls precisely in the class of problems associated with diffusion and it constitutes the inherent daunting hurdle of multiple scales. We propose here to construct mathematical models of neuronal microdomains, starting from the molecular to the cellular level and to develop stochastic modeling and singular perturbation methods for asymptotic analysis of the model equations, to use the analytical and numerical approximate solutions to extract properties from newly available molecular data and from Brownian dynamics simulations. A major application of the proposed molecular level modeling is the resolution of the spatiotemporal dynamics regulating neuronal microdomains and synapses. The results of this research will be methods in mathematical modeling, data analysis, asymptotic analysis, and in the designing of stochastic simulations of subcellular processes.
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