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基于 Hodge 理论的生物分子数据分析。

Hodge theory-based biomolecular data analysis.

机构信息

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore.

出版信息

Sci Rep. 2022 Jun 11;12(1):9699. doi: 10.1038/s41598-022-12877-z.

Abstract

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.

摘要

霍奇理论揭示了微分形式的深刻内在关系,并在微分几何、代数拓扑和泛函分析之间架起了桥梁。在这里,我们使用 Hodge Laplacian 和 Hodge 分解模型来分析生物分子结构。与传统的基于图的方法不同,生物分子结构被表示为单纯复形,它可以被看作是图模型到其更高维对应物的推广。可以从单纯复形中生成不同维数的 Hodge Laplacian 矩阵。这些矩阵的谱信息可用于研究生物分子结构的内在拓扑信息。本质上,k 维零特征值的数量(或重数)等同于 k 维贝蒂数,即 k 维同调群的数量。相关的特征向量表示同调生成元,即分子基单纯复形中的环或孔。此外,基于 Hodge 分解的 HodgeRank 模型用于描述分子结构的折叠或紧致性,特别是在高通量染色体构象捕获(Hi-C)数据中的拓扑相关域(TAD)。从数学上讲,分子结构用具有特定边流的单纯复形来表示。基于 HodgeRank 的平均/总不一致性(AI/TI)用于定量测量 TAD 的折叠或紧致性。据我们所知,这是 TAD 区域的第一个定量测量。

https://cdn.ncbi.nlm.nih.gov/pmc/blobs/0cde/9188576/93312e9b9d21/41598_2022_12877_Fig1_HTML.jpg

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