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LOCALIZATION AND PSEUDOSPECTRA OF TWISTED TOEPLITZ MATRICES WITH APPLICATIONS TO ION CHANNELS.

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Title: LOCALIZATION AND PSEUDOSPECTRA OF TWISTED TOEPLITZ MATRICES WITH APPLICATIONS TO ION CHANNELS.
Authors: BURRAGE, KEVIN1 kevin.burrage@qut.edu.au, BURRAGE, PAMELA2 pmburrage@gmail.com, MACNAMARA, SHEV3 shev.mac@gmail.com
Source: SIAM Journal on Matrix Analysis & Applications. 2021, Vol. 42 Issue 4, p1656-1679. 24p.
Subjects: Toeplitz matrices, Ion channels, Pseudospectrum, Sodium channels, Enzyme kinetics, Anderson localization
Abstract: We study pseudospectra of matrices arising in models of ion channel dynamics, bimolecular reactions, and Michaelis--Menten enzyme kinetics. We show that matrices for ion channel models and for bimolecular reactions are examples of a special class known as twisted Toeplitz matrices. Using Michaelis--Menten enzyme kinetics as an example, we suggest more complicated reactions cannot be modeled simply by the twisted Toeplitz ideas currently in the literature, and thus we propose a generalization to what we call block twisted Toeplitz matrices. Furthermore, we show that certain simple cases of ion channel models always exhibit a phenomenon known as localization. This result comes by studying the symbol of the operator and an application of a theorem of Trefethen and Chapman. We further observe that a larger magnitude of the so-called twist-ratio, a number that comes from the symbol, is associated with stronger localization. For certain bimolecular reactions, specialized to a dimerization, we show by the same theorem that there are large regions of parameter space where localization is guaranteed and that this is always guaranteed in the limit as the matrix size becomes very large. However, we also find examples for these reactions where the hypotheses of the theorem of Trefethen and Chapman do not hold, and yet in numerical experiments, we still observe localization. Finally we return to the first of our matrix models for a more detailed study to give further insight into important applications to cardiac ion channels and in particular sodium channels. We demonstrate behavior similar to the so-called cutoff phenomenon of Markov processes, and we show that the stationary distribution, which is so crucial to applications of these models to ion channels and especially sodium channels, can be extremely localized. [ABSTRACT FROM AUTHOR]
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Abstract:We study pseudospectra of matrices arising in models of ion channel dynamics, bimolecular reactions, and Michaelis--Menten enzyme kinetics. We show that matrices for ion channel models and for bimolecular reactions are examples of a special class known as twisted Toeplitz matrices. Using Michaelis--Menten enzyme kinetics as an example, we suggest more complicated reactions cannot be modeled simply by the twisted Toeplitz ideas currently in the literature, and thus we propose a generalization to what we call block twisted Toeplitz matrices. Furthermore, we show that certain simple cases of ion channel models always exhibit a phenomenon known as localization. This result comes by studying the symbol of the operator and an application of a theorem of Trefethen and Chapman. We further observe that a larger magnitude of the so-called twist-ratio, a number that comes from the symbol, is associated with stronger localization. For certain bimolecular reactions, specialized to a dimerization, we show by the same theorem that there are large regions of parameter space where localization is guaranteed and that this is always guaranteed in the limit as the matrix size becomes very large. However, we also find examples for these reactions where the hypotheses of the theorem of Trefethen and Chapman do not hold, and yet in numerical experiments, we still observe localization. Finally we return to the first of our matrix models for a more detailed study to give further insight into important applications to cardiac ion channels and in particular sodium channels. We demonstrate behavior similar to the so-called cutoff phenomenon of Markov processes, and we show that the stationary distribution, which is so crucial to applications of these models to ion channels and especially sodium channels, can be extremely localized. [ABSTRACT FROM AUTHOR]
ISSN:08954798
DOI:10.1137/20M1384920