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) <<epsilon (c)). To leading order, the EBA and RBA presented here recover results previously obtained by Neher ( 1986) and Keizer and coworkers ( Wagner and Keizer, 1994; Smith, 1996), respectively while the IBA corresponds to unbuffered diffusion of Ca2+. However, the asymptotic formalism allows derivation for the rst time of higher order terms, which are shown numerically to significantly extend the range of validity of these approximations. We show that another approximation, derived by linearization rather than by asymptotic approximation ( Stern [ Cell Calcium 13 (1992), pp. 183-192], Pape Jong, and Chandler [ J. Gen. Physiol. 106 (1995), pp. 259-336], and Naraghi and Neher [ J. Neurosci. 17 (1997), pp. 6961-6973]), interpolates between the EBA and IBA solutions. Finally we indicate where in the (epsilon (c),epsilon (b))-plane each of the approximations is accurate and show how the validity of each depends not only on buffer parameters but also on source strength.