Multicompartment diffusion and reaction modeling system applied to transmitter-mediated photoreception

An integrated system of programs has been developed with broad applicability to the numerical solution of models with parabolic partial differential equations coupled to ordinary differential equations, as arise, for example, in diffusive transport bulk- or surface-limited by reaction rate processes. The programs have been designed to run optimally in various minicomputer environments and to be as portable as possible. The difference scheme for the parabolic equations is new, and competes favorably with several commonly used implicit schemes. Convergence is proved, and conditions on stability are given. To solve the two sets of coupled diffusion + reaction equations, a new numerical method is developed which digitally filters the second space differences of the diffusion difference equations, making the simple, explicit difference scheme convergent for arbitrary time step size. The method competes favorably with the Cranl-Nicolson scheme in speed and accuracy. The modeling system is applied to certain photoresponsive cells of the Aplysia californica (R₂ giant neuron and the ventral photoresponsive neuron) which hyperpolarize when illuminated, due to an increase of membrane potassium permeability. It has been hypothesized that light releases an internal transmitter from cytoplasmic granules. Three model compartments parallel cellular morphology: the first represents the granule component with bulk-limited diffusion; the second corresponds to the cytoplasm and involves simple diffusion; and the third is situated near the plasma membrane where the transmitter concentration is directly related to membrane conductance. The bimolecular binding succesfully predicts the dynamic non-linearity.