Efficient Approximations for Stationary Single-Channel Ca²⁺ Nanodomains across Length Scales

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 Ca²⁺ buffer with 1:1 Ca²⁺ binding. We present computationally efficient approximants that estimate stationary single-channel Ca²⁺ nanodomains with great accuracy in broad regions of parameter space. The presented approximants have a functional form that combines rational and exponential functions, which is similar to that of the well-known excess buffer approximation and the linear approximation but with parameters estimated using two novel, to our knowledge, methods. One of the methods involves interpolation between the short-range Taylor series of the free buffer concentration and its long-range asymptotic series in inverse powers of distance from the channel. Although this method has already been used to find Padé (rational-function) approximants to single-channel Ca²⁺ and buffer concentrations, extending this method to interpolants combining exponential and rational functions improves accuracy in a significant fraction of the relevant parameter space. A second method is based on the variational approach and involves a global minimization of an appropriate functional with respect to parameters of the chosen approximations. An extensive parameter-sensitivity analysis is presented, comparing these two methods with previously developed approximants. Apart from increased accuracy, the strength of these approximants is that they can be extended to more realistic buffers with multiple binding sites characterized by cooperative Ca²⁺ binding, such as calmodulin and calretinin.