Pade Approximation Of Single-Channel Calcium Nanodomains In The Presence Of Cooperative Calcium Buffers

Ca2+ elevations produced in the vicinity of single open Ca2+ channels are termed Ca2+ nanodomains, and are known to trigger secretory vesicle exocytosis, myocyte contraction and other fundamental physiological processes. Ca2+ nanodomains are shaped by the interplay between Ca2+ influx, diffusion and binding to Ca2+ buffers, and can be approximated with reasonable accuracy by analytic approximations of quasi-stationary solutions of the corresponding reaction-diffusion equations. Such closed-form approximations help to reveal the qualitative dependence of nanodomain characteristics on buffering and diffusion parameters without resorting to computationally expensive numerical simulations. Although several nanodomain approximations had been developed for the case of buffers with a single Ca2+ binding site, most biological buffers have more complex Ca2+-binding stoichiometry. Further, Ca2+ buffers such as calretinin and calmodulin consist of distinct EF-hand domains, each possessing two Ca2+ binding sites exhibiting cooperativity in binding, whereby the affinity of the second Ca2+ binding is much higher than the first. While the Rapid Buffering Approximation (RBA) has been recently generalized to such cooperative buffers, its performance is limited by the complex interplay between the condition of slow diffusion required for RBA accuracy, and the slow rate of the first Ca2+ binding reaction characterizing cooperative Ca2+ binding. To resolve this problem, we extend the recently developed alternative method, the Padé approximation method, to the case of 2-to-1 Ca2+ buffers. The Padé approximation interpolates between the short-range and long-range distance-dependence of nanodomain concentration using a rational function Ansatz. We examine in detail the parameter-dependence of the lowest-order Padé approximation accuracy, and show that this method is superior to RBA for a wide ranges of buffering parameter values. The limitations of the Padé method, in particular its algebraic complexity, are also discussed. Supported in part by NSF DMS-1517085 (V.M)