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persistent homology for the Quantitative Analysis of the Structure and Stability of Carboranes

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Chinese Journal of Structural Chemistry 2020年第6期39卷 999-1008页

作者：陈冬张明正陈海标谢作伟魏国卫潘锋 School of Advanced Materials Peking University Shenzhen Graduate School Shenzhen 518055China Department of Chemistry and State Key Laboratory of Synthetic Chemistry The Chinese University of HongKongShatinNew TerritoriesHong KongChina Department of Mathematics Michigan State UniversityMI 48824 USA Department of Electrical and Computer Engineering Michigan State UniversityMI 48824USA Department of Biochemistry and Molecular Biology Michigan State UniversityMl 48824USA

persistent homology is a powerful and novel tool for quantifying the inherent topological features of structure. In this work, we used the persistent homology for the first time to study the closo-carboranes C2Bn-2Hn(n = 5~20) and their parent structures closo-boranes dianions BnHn2-(n = 5~20), where multiple elements are present. All these structures are first investigated with the standard Vitoris-Rips complex. We interpret all barcodes representation and associate them with structural details. By means of average bar length, a linear regression model was established to construct the relationship between persistent homology features and molecular stability, which was expressed by the relative energies. For closo-boranes dianions, we only use B atom set since B and H atoms are in pairs. The average lengths of β0, β1 and β2 bars are used as the features for linear regression, and excellent correlation coefficient(0.977) between the values predicted by persistent homology and those by quantum calculations was achieved. For closo-carboranes, C–B atom set(ignore the differences in the atoms), B atom set and C atom set were considered to get the persistent homology features(since there were only two C atoms in C2Bn-2Hn, only β0 bars were considered), and seven average bar lengths were calculated, respectively. Pearson coefficient of 0.937 was obtained. We found that the stability of carboranes showed a high linear correlation with the characteristics generated from topological bars in H0, H1 and H2. The results show that the topological information generated by persistent homology can be extended and applied to multi-element systems.

关键词： topology persistent homology carboranes filtration stability relative energy

Computational Tools in Weighted persistent homology

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Chinese Annals of Mathematics,Series B 2021年第2期42卷 237-258页

作者： Shiquan REN Chengyuan WU Jie WU Yau Mathematical Sciences Center Tsinghua UniversityBeijing 100084China Department of Mathematics National University of SingaporeSingapore 119076Singapore Institute of High Performance Computing A*STARSingapore 138632Singapore School of Mathematical Sciences Hebei Normal UniversityShijiazhuang 050024China Center for Topology and Geometry based Technology Hebei Normal University050024China.

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p^(2) weighted persistent homology provided with some information on the mod p weighted persistent homology.

关键词： Algebraic topology persistent homology Weighted persistent homology Bockstein spectral sequence

Biomolecular Topology:Modelling and Analysis

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Acta Mathematica Sinica,English Series 2022年第10期38卷 1901-1938页

作者： Jian LIU Ke-Lin XIA Jie WU Stephen Shing-Toung YAU Guo-Wei WEI School of Mathematical Sciences Hebei Normal UniversityShijiazhuang 050024P.R.China Yanqi Lake Beijing Institute of Mathematical Sciences and Applications Beijing 101408P.R.China School of Physical and Mathematical Sciences Nanyang Technological UniversitySingapore 639798 Department of Mathematical Sciences Tsinghua UniversityBeijing 100084P.R.China Department of Mathematics&Department of Biochemistry and Molecular Biology&Department of Electrical and Computer Engineering Michigan State UniversityWells Hall 619 Red Cedar Road East LansingMI 48824-1027US

With the great advancement of experimental tools,a tremendous amount of biomolecular data has been generated and accumulated in various databases.The high dimensionality,structural complexity,the nonlinearity,and entanglements of biomolecular data,ranging from DNA knots,RNA secondary structures,protein folding configurations,chromosomes,DNA origami,molecular assembly,to others at the macromolecular level,pose a severe challenge in their analysis and characterization.In the past few decades,mathematical concepts,models,algorithms,and tools from algebraic topology,combinatorial topology,computational topology,and topological data analysis,have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge.In this work,we introduce biomolecular topology,which concerns the topological problems and models originated from the biomolecular systems.More specifically,the biomolecular topology encompasses topological structures,properties and relations that are emerged from biomolecular structures,dynamics,interactions,and functions.We discuss the various types of biomolecular topology from structures(of proteins,DNAs,and RNAs),protein folding,and protein assembly.A brief discussion of databanks(and databases),theoretical models,and computational algorithms,is presented.Further,we systematically review related topological models,including graphs,simplicial complexes,persistent homology,persistent Laplacians,de Rham-Hodge theory,Yau-Hausdorff distance,and the topology-based machine learning models.

关键词： persistent homology topological data analysis biomolecular topology protein structure machine learning

Analyzing architectural diversity in maize plants using the skeletonimage-based method

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Journal of Integrative Agriculture 2023年第12期22卷 3804-3809页

作者： LIU Min-guo Thomas CAMPBELL LI Wei WANG Xi-qing College of Biological Sciences China Agricultural UniversityBeijing 100193P.R.China Frontier Technology Research Institute of China Agricultural University in Shenzhen Shenzhen 518000P.R.China Center for Crop Functional Genomics and Molecular Breeding China Agricultural UniversityBeijing 100193P.R.China Department of Biology Northeastern Illinois UniversityChicago 60625USA

Shoot architecture in maize is critical since it determines resource use,impacts wind and rain damage tolerance,and affects yield stability.Quantifying the diversity among inbred lines in heterosis breeding is essential,especially when describing germplasm resources.However,traditional geometric description methods oversimplify shoot architecture and ignore the plant’s overall architecture,making it difficult to reflect and illustrate diversity.This study presents a new method to describe maize shoot architecture and quantifies its diversity by combining computer vision algorithms and persistent homology.Our results reveal that persistent homology can capture key characteristics of shoot architecture in maize and other details often overlooked by traditional geometric analysis.Based on this method,the morphological diversity of shoot architecture can be mined(quantified),and the main shoot architecture types can be obtained.Consequently,this method can easily describe the diversity of shoot architecture in many maize materials.

关键词： maize shoot architecture persistent homology phenotyping technology morphological diversity

The Realization Problem for Discrete Morse Functions on Trees

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Algebra Colloquium 2020年第3期27卷 455-468页

作者： Yuqing Liu Nicholas A.Scoville Department of Mathematics and Computer Science Ursinus College CollegevillePA 19426USA

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.

关键词： persistent homology barcode discrete Morse theory tree