Padé Approximation of a Stationary Single-Channel Ca²⁺ Nanodomain

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.