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Hodge theory-based biomolecular data analysis

Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-ba...

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Autores principales: Wei, Ronald Koh Joon, Wee, Junjie, Laurent, Valerie Evangelin, Xia, Kelin
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Nature Publishing Group UK 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9188576/
https://www.ncbi.nlm.nih.gov/pubmed/35690623
http://dx.doi.org/10.1038/s41598-022-12877-z
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author Wei, Ronald Koh Joon
Wee, Junjie
Laurent, Valerie Evangelin
Xia, Kelin
author_facet Wei, Ronald Koh Joon
Wee, Junjie
Laurent, Valerie Evangelin
Xia, Kelin
author_sort Wei, Ronald Koh Joon
collection PubMed
description Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know.
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spelling pubmed-91885762022-06-13 Hodge theory-based biomolecular data analysis Wei, Ronald Koh Joon Wee, Junjie Laurent, Valerie Evangelin Xia, Kelin Sci Rep Article Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know. Nature Publishing Group UK 2022-06-11 /pmc/articles/PMC9188576/ /pubmed/35690623 http://dx.doi.org/10.1038/s41598-022-12877-z Text en © The Author(s) 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Wei, Ronald Koh Joon
Wee, Junjie
Laurent, Valerie Evangelin
Xia, Kelin
Hodge theory-based biomolecular data analysis
title Hodge theory-based biomolecular data analysis
title_full Hodge theory-based biomolecular data analysis
title_fullStr Hodge theory-based biomolecular data analysis
title_full_unstemmed Hodge theory-based biomolecular data analysis
title_short Hodge theory-based biomolecular data analysis
title_sort hodge theory-based biomolecular data analysis
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9188576/
https://www.ncbi.nlm.nih.gov/pubmed/35690623
http://dx.doi.org/10.1038/s41598-022-12877-z
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