Padé Approximation of a Stationary Single-Channel Ca2+ Nanodomain

Abstract

We consider the stationary solution for the Ca2+ concentration near a point Ca2+ source describing a single-channel Ca2+ nanodomain, in the presence of a single mobile buffer with one-to-one Ca2+ binding stoichiometry. Previously, a number of Ca2+ nanodomains approximations have been developed, for instance the excess buffer approximation (EBA), the rapid buffering approximation (RBA), and the linear approximation (LIN), each valid for appropriate buffering conditions. Apart from providing a simple method of estimating Ca2+ and buffer concentrations without resorting to computationally expensive numerical solution of reaction-diffusion equations, such approximations proved useful in revealing the dependence of nanodomain Ca2+ distribution on crucial parameters such as buffer mobility and its Ca2+ binding properties. In this study, we present a different form of analytic approximation, which is based on matching the short-range Taylor series of the nanodomain concentration with the long-range asymptotic series expressed in inverse powers of distance from channel location. Namely, we use a “dual” Padé rational function approximation to simultaneously match terms in the short- and the long-range series, and we show that this provides an accurate approximation to the nanodomain Ca2+ and buffer concentrations. We compare this approximation with the previously obtained approximations and show that it yields a better estimate of the free buffer concentration for a wide range of buffering conditions. The drawback of our method is that it has a complex algebraic form for any order higher than the lowest bilinear order, and cannot be readily extended to multiple Ca2+ channels. However, it may be possible to extend the Padé method to estimate Ca2+ nanodomains in the presence of cooperative Ca2+ buffers with two Ca2+ binding sites, the case that existing methods do not address.