Asymptotic analysis of buffered calcium diffusion near a point source

The domain calcium (Ca2+) concentration near an open Ca2+ channel can be mo deled as buffered diffusion from a point source. The concentration pro les can be well approximated by hemispherically symmetric steady-state solution s to a system of reaction-diffusion equations. After nondimensionalizing th ese equations and scaling space so that both reaction terms and the source amplitude are O(1), we identify two dimensionless parameters, epsilon (c) a nd epsilon (b) that correspond to the diffusion coefficients of dimensionle ss Ca2+ and buffer, respectively. Using perturbation methods, we derive approximations for the Ca2+ and buffe r pro les in three asymptotic limits: (1) an "excess buffer approximation" (EBA), where the mobility of buffer exceeds that of Ca2+ (epsilon (b) >> ep silon (c)) and the fast diffusion of buffer toward the Ca2+ channel prevent s buffer saturation (cf. Neher [ Calcium Electrogenesis and Neuronal Functi oning Exp. Brain Res. 14, Springer-Verlag, Berlin, 1986, pp. 80-96]); (2) a rapid buffer approximation (RBA), where the diffusive time-scale for Ca2and buffer are comparable, but slow compared to reaction (epsilon (c) << 1, epsilon (b) << 1, and epsilon (c)/epsilon b = O(1)), resulting in saturati on 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]); an d (3) a new immobile buffer approximation ( IBA) where the diffusion of buf fer 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, 19 94; Smith, 1996), respectively while the IBA corresponds to unbuffered diff usion of Ca2+. However, the asymptotic formalism allows derivation for the rst time of higher order terms, which are shown numerically to significantl y extend the range of validity of these approximations. We show that anothe r approximation, derived by linearization rather than by asymptotic approxi mation ( Stern [ Cell Calcium 13 (1992), pp. 183-192], Pape Jong, and Chand ler [ 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))-plan e each of the approximations is accurate and show how the validity of each depends not only on buffer parameters but also on source strength.