Asymptotic Analysis of Buffered Calcium Diffusion near a Point Source

Abstract

The "domain" calcium (${\rm Ca}^{\rm 2+}$) concentration near an open ${\rm Ca}^{\rm 2+}$ channel can be modeled as buffered diffusion from a point source. The concentration profiles 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, ${\varepsilon}_c$ and ${\varepsilon}_b$, that correspond to the diffusion coefficients of dimensionless ${\rm Ca}^{\rm 2+}$ and buffer, respectively. Using perturbation methods, we derive approximations for the ${\rm Ca}^{\rm 2+}$ and buffer profiles in three asymptotic limits: (1) an "excess buffer approximation" (EBA), where the mobility of buffer exceeds that of ${\rm Ca}^{\rm 2+}$ (${\varepsilon}_b \gg {\varepsilon}_c$) and the fast diffusion of buffer toward the ${\rm Ca}^{\rm 2+}$ 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 ${\rm Ca}^{\rm 2+}$ and buffer are comparable, but slow compared to reaction (${\varepsilon}_c \ll 1$, ${\varepsilon}_b \ll 1$, and ${\varepsilon}_c / {\varepsilon}_b = O(1)$), resulting in saturation of buffer near the ${\rm Ca}^{\rm 2+}$ 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 ${\rm Ca}^{\rm 2+}$ (${\varepsilon}_b \ll {\varepsilon}_c$). 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 ${\rm Ca}^{\rm 2+}$. However, the asymptotic formalism allows derivation for the first 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 (${\varepsilon}_c$,${\varepsilon}_b$)-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.