Date of Award

8-31-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Victor Victorovich Matveev

Second Advisor

Cyrill B. Muratov

Third Advisor

Casey Diekman

Fourth Advisor

Denis L. Blackmore

Fifth Advisor

Arthur Stewart Sherman

Abstract

Mathematical and computational modeling plays an important role in the study of local Ca2+ signals underlying many fundamental physiological processes such as synaptic neurotransmitter release and myocyte contraction. Closed-form approximations describing steady-state distribution of Ca2+ in the vicinity of an open Ca2+ channel have proved particularly useful for the qualitative modeling of local Ca2+ signals. This dissertation presents several simple and efficient approximants for the equilibrium Ca2+ concentration near a point source in the presence of a mobile Ca2+ buffer, which achieve great accuracy over a wide range of model parameters. Such approximations provide an efficient method for estimating Ca2+ and buffer concentrations without resorting to numerical simulations and allow to study the qualitative dependence of nanodomain Ca2+ distribution on the buffer’s Ca2+ binding properties and its diffusivity. The new approximants presented here for the case of a simple, one-to-one Ca2+ buffer 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 methods. One of the methods involves interpolation between the short-range Taylor series of the buffer concentration and its long-range asymptotic series in inverse powers of distance from the channel. 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. Extensive parameter sensitivity analysis is presented, comparing approximants found using these two methods with the previously developed approximants. Apart from increased accuracy, the strength of the new approximants is that they can be extended to more realistic buffers with multiple Ca2+ binding sites, such as calmodulin and calretinin. In the second part of the dissertation, the series interpolation method is extended to buffers with two Ca2+ binding sites, yielding closed-form interpolants combining exponential and rational functions that achieve reasonable accuracy even in the case of buffers characterized by significant Ca2+ binding cooperativity. Finally, open challenges and potential future extensions of this work are discussed in detail.

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