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. |
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| 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] |
| Copyright of SIAM Journal on Matrix Analysis & Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: LOCALIZATION AND PSEUDOSPECTRA OF TWISTED TOEPLITZ MATRICES WITH APPLICATIONS TO ION CHANNELS. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22BURRAGE%2C+KEVIN%22">BURRAGE, KEVIN</searchLink><relatesTo>1</relatesTo><i> kevin.burrage@qut.edu.au</i><br /><searchLink fieldCode="AR" term="%22BURRAGE%2C+PAMELA%22">BURRAGE, PAMELA</searchLink><relatesTo>2</relatesTo><i> pmburrage@gmail.com</i><br /><searchLink fieldCode="AR" term="%22MACNAMARA%2C+SHEV%22">MACNAMARA, SHEV</searchLink><relatesTo>3</relatesTo><i> shev.mac@gmail.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Matrix+Analysis+%26+Applications%22">SIAM Journal on Matrix Analysis & Applications</searchLink>. 2021, Vol. 42 Issue 4, p1656-1679. 24p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Toeplitz+matrices%22">Toeplitz matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Ion+channels%22">Ion channels</searchLink><br /><searchLink fieldCode="DE" term="%22Pseudospectrum%22">Pseudospectrum</searchLink><br /><searchLink fieldCode="DE" term="%22Sodium+channels%22">Sodium channels</searchLink><br /><searchLink fieldCode="DE" term="%22Enzyme+kinetics%22">Enzyme kinetics</searchLink><br /><searchLink fieldCode="DE" term="%22Anderson+localization%22">Anderson localization</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Matrix Analysis & Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1137/20M1384920 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 24 StartPage: 1656 Subjects: – SubjectFull: Toeplitz matrices Type: general – SubjectFull: Ion channels Type: general – SubjectFull: Pseudospectrum Type: general – SubjectFull: Sodium channels Type: general – SubjectFull: Enzyme kinetics Type: general – SubjectFull: Anderson localization Type: general Titles: – TitleFull: LOCALIZATION AND PSEUDOSPECTRA OF TWISTED TOEPLITZ MATRICES WITH APPLICATIONS TO ION CHANNELS. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: BURRAGE, KEVIN – PersonEntity: Name: NameFull: BURRAGE, PAMELA – PersonEntity: Name: NameFull: MACNAMARA, SHEV IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: 2021 Type: published Y: 2021 Identifiers: – Type: issn-print Value: 08954798 Numbering: – Type: volume Value: 42 – Type: issue Value: 4 Titles: – TitleFull: SIAM Journal on Matrix Analysis & Applications Type: main |
| ResultId | 1 |
