Asymptotic analysis of buffered calcium diffusion near a point source

The domain calcium (Ca2+) concentration near an open Ca2+ channel can be modeled as buffered diffusion from a point source. The concentration pro les can be well approximated by hemispherically symmetric steady-state solutions to a system of reaction-diffusion equations. After nondimensionalizing these equations and scaling space so that both reaction terms and the source amplitude are O(1), we identify two dimensionless parameters, epsilon (c) and epsilon (b) that correspond to the diffusion coefficients of dimensionless Ca2+ and buffer, respectively. Using perturbation methods, we derive approximations for the Ca2+ and buffer pro les in three asymptotic limits: (1) an excess buffer approximation (EBA), where the mobility of buffer exceeds that of Ca2+ (epsilon (b) >> epsilon (c)) and the fast diffusion of buffer toward the Ca2+ channel prevents buffer saturation (cf. Neher [ Calcium Electrogenesis and Neuronal Functioning Exp. Brain Res. 14, Springer-Verlag, Berlin, 1986, pp. 80-96]); (2) a rapid buffer approximation (RBA), where the diffusive time-scale for Ca2+ and buffer are comparable, but slow compared to reaction (epsilon (c) << 1, epsilon (b) << 1, and epsilon (c)/epsilon b = O(1)), resulting in saturation of buffer near the Ca2+ channel (cf. Wagner and Keizer [ Biophys. J. 67 (1994), pp. 447-456] and Smith [ Biophys. J. 71 (1996), pp. 3064-3072]); and (3) a new immobile buffer approximation ( IBA) where the diffusion of buffer is slow compared to that of Ca2+ (epsilon (b) <