1
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Ferrà A, Cecchini G, Nobbe Fisas FP, Casacuberta C, Cos I. A topological classifier to characterize brain states: When shape matters more than variance. PLoS One 2023; 18:e0292049. [PMID: 37782651 PMCID: PMC10545107 DOI: 10.1371/journal.pone.0292049] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2023] [Accepted: 08/04/2023] [Indexed: 10/04/2023] Open
Abstract
Despite the remarkable accuracies attained by machine learning classifiers to separate complex datasets in a supervised fashion, most of their operation falls short to provide an informed intuition about the structure of data, and, what is more important, about the phenomena being characterized by the given datasets. By contrast, topological data analysis (TDA) is devoted to study the shape of data clouds by means of persistence descriptors and provides a quantitative characterization of specific topological features of the dataset under scrutiny. Here we introduce a novel TDA-based classifier that works on the principle of assessing quantifiable changes on topological metrics caused by the addition of new input to a subset of data. We used this classifier with a high-dimensional electro-encephalographic (EEG) dataset recorded from eleven participants during a previous decision-making experiment in which three motivational states were induced through a manipulation of social pressure. We calculated silhouettes from persistence diagrams associated with each motivated state with a ready-made band-pass filtered version of these signals, and classified unlabeled signals according to their impact on each reference silhouette. Our results show that in addition to providing accuracies within the range of those of a nearest neighbour classifier, the TDA classifier provides formal intuition of the structure of the dataset as well as an estimate of its intrinsic dimension. Towards this end, we incorporated variance-based dimensionality reduction methods to our dataset and found that in most cases the accuracy of our TDA classifier remains essentially invariant beyond a certain dimension.
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Affiliation(s)
- Aina Ferrà
- Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Catalonia, Spain
| | - Gloria Cecchini
- Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Catalonia, Spain
| | - Fritz-Pere Nobbe Fisas
- Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Catalonia, Spain
| | - Carles Casacuberta
- Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Catalonia, Spain
- Institut de Matemàtica de la Universitat de Barcelona (IMUB), Barcelona, Spin
| | - Ignasi Cos
- Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Catalonia, Spain
- Institut de Matemàtica de la Universitat de Barcelona (IMUB), Barcelona, Spin
- Serra-Húnter Fellow Programme, Barcelona, Catalonia, Spain
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2
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Wee J, Bianconi G, Xia K. Persistent Dirac for molecular representation. Sci Rep 2023; 13:11183. [PMID: 37433870 DOI: 10.1038/s41598-023-37853-z] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 06/28/2023] [Indexed: 07/13/2023] Open
Abstract
Molecular representations are of fundamental importance for the modeling and analysing molecular systems. The successes in drug design and materials discovery have been greatly contributed by molecular representation models. In this paper, we present a computational framework for molecular representation that is mathematically rigorous and based on the persistent Dirac operator. The properties of the discrete weighted and unweighted Dirac matrix are systematically discussed, and the biological meanings of both homological and non-homological eigenvectors are studied. We also evaluate the impact of various weighting schemes on the weighted Dirac matrix. Additionally, a set of physical persistent attributes that characterize the persistence and variation of spectrum properties of Dirac matrices during a filtration process is proposed to be molecular fingerprints. Our persistent attributes are used to classify molecular configurations of nine different types of organic-inorganic halide perovskites. The combination of persistent attributes with gradient boosting tree model has achieved great success in molecular solvation free energy prediction. The results show that our model is effective in characterizing the molecular structures, demonstrating the power of our molecular representation and featurization approach.
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Affiliation(s)
- Junjie Wee
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore.
| | - Ginestra Bianconi
- School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK
- The Alan Turing Institute, London, NW1 2DB, UK
| | - Kelin Xia
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore
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3
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Xia K, Liu X, Wee J. Persistent Homology for RNA Data Analysis. Methods Mol Biol 2023; 2627:211-229. [PMID: 36959450 DOI: 10.1007/978-1-0716-2974-1_12] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/25/2023]
Abstract
Molecular representations are of great importance for machine learning models in RNA data analysis. Essentially, efficient molecular descriptors or fingerprints that characterize the intrinsic structural and interactional information of RNAs can significantly boost the performance of all learning modeling. In this paper, we introduce two persistent models, including persistent homology and persistent spectral, for RNA structure and interaction representations and their applications in RNA data analysis. Different from traditional geometric and graph representations, persistent homology is built on simplicial complex, which is a generalization of graph models to higher-dimensional situations. Hypergraph is a further generalization of simplicial complexes and hypergraph-based embedded persistent homology has been proposed recently. Moreover, persistent spectral models, which combine filtration process with spectral models, including spectral graph, spectral simplicial complex, and spectral hypergraph, are proposed for molecular representation. The persistent attributes for RNAs can be obtained from these two persistent models and further combined with machine learning models for RNA structure, flexibility, dynamics, and function analysis.
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Affiliation(s)
- Kelin Xia
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore.
| | - Xiang Liu
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore
- Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, China
| | - JunJie Wee
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore
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4
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Rana MM, Nguyen DD. EISA-Score: Element Interactive Surface Area Score for Protein–Ligand Binding Affinity Prediction. J Chem Inf Model 2022; 62:4329-4341. [DOI: 10.1021/acs.jcim.2c00697] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Md Masud Rana
- Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, United States
| | - Duc Duy Nguyen
- Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, United States
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5
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Gao K, Wang R, Chen J, Cheng L, Frishcosy J, Huzumi Y, Qiu Y, Schluckbier T, Wei X, Wei GW. Methodology-Centered Review of Molecular Modeling, Simulation, and Prediction of SARS-CoV-2. Chem Rev 2022; 122:11287-11368. [PMID: 35594413 PMCID: PMC9159519 DOI: 10.1021/acs.chemrev.1c00965] [Citation(s) in RCA: 31] [Impact Index Per Article: 15.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
Abstract
Despite tremendous efforts in the past two years, our understanding of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), virus-host interactions, immune response, virulence, transmission, and evolution is still very limited. This limitation calls for further in-depth investigation. Computational studies have become an indispensable component in combating coronavirus disease 2019 (COVID-19) due to their low cost, their efficiency, and the fact that they are free from safety and ethical constraints. Additionally, the mechanism that governs the global evolution and transmission of SARS-CoV-2 cannot be revealed from individual experiments and was discovered by integrating genotyping of massive viral sequences, biophysical modeling of protein-protein interactions, deep mutational data, deep learning, and advanced mathematics. There exists a tsunami of literature on the molecular modeling, simulations, and predictions of SARS-CoV-2 and related developments of drugs, vaccines, antibodies, and diagnostics. To provide readers with a quick update about this literature, we present a comprehensive and systematic methodology-centered review. Aspects such as molecular biophysics, bioinformatics, cheminformatics, machine learning, and mathematics are discussed. This review will be beneficial to researchers who are looking for ways to contribute to SARS-CoV-2 studies and those who are interested in the status of the field.
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Affiliation(s)
- Kaifu Gao
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Rui Wang
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Jiahui Chen
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Limei Cheng
- Clinical
Pharmacology and Pharmacometrics, Bristol
Myers Squibb, Princeton, New Jersey 08536, United States
| | - Jaclyn Frishcosy
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Yuta Huzumi
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Yuchi Qiu
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Tom Schluckbier
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Xiaoqi Wei
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
| | - Guo-Wei Wei
- Department
of Mathematics, Michigan State University, East Lansing, Michigan 48824, United States
- Department
of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824, United States
- Department
of Biochemistry and Molecular Biology, Michigan
State University, East Lansing, Michigan 48824, United States
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6
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Affiliation(s)
- Chengyuan Wu
- Data Analytics Consulting Centre, Department of Statistics and Applied Probability, Faculty of Science, National University of Singapore, Singapore
- Institute of High Performance Computing, A*STAR, Singapore
| | - Carol Anne Hargreaves
- Data Analytics Consulting Centre, Department of Statistics and Applied Probability, Faculty of Science, National University of Singapore, Singapore
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7
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Loughrey C, Fitzpatrick P, Orr N, Jurek-Loughrey A. The topology of data: Opportunities for cancer research. Bioinformatics 2021; 37:3091-3098. [PMID: 34320632 PMCID: PMC8504620 DOI: 10.1093/bioinformatics/btab553] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2021] [Revised: 06/14/2021] [Accepted: 07/28/2021] [Indexed: 01/20/2023] Open
Abstract
Motivation Topological methods have recently emerged as a reliable and interpretable framework for extracting information from high-dimensional data, leading to the creation of a branch of applied mathematics called Topological Data Analysis (TDA). Since then, TDA has been progressively adopted in biomedical research. Biological data collection can result in enormous datasets, comprising thousands of features and spanning diverse datatypes. This presents a barrier to initial data analysis as the fundamental structure of the dataset becomes hidden, obstructing the discovery of important features and patterns. TDA provides a solution to obtain the underlying shape of datasets over continuous resolutions, corresponding to key topological features independent of noise. TDA has the potential to support future developments in healthcare as biomedical datasets rise in complexity and dimensionality. Previous applications extend across the fields of neuroscience, oncology, immunology and medical image analysis. TDA has been used to reveal hidden subgroups of cancer patients, construct organizational maps of brain activity and classify abnormal patterns in medical images. The utility of TDA is broad and to understand where current achievements lie, we have evaluated the present state of TDA in cancer data analysis. Results This article aims to provide an overview of TDA in Cancer Research. A brief introduction to the main concepts of TDA is provided to ensure that the article is accessible to readers who are not familiar with this field. Following this, a focussed literature review on the field is presented, discussing how TDA has been applied across heterogeneous datatypes for cancer research.
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Affiliation(s)
- Ciara Loughrey
- School of Electronics, Electrical Engineering and Computer Science, Queen's University Belfast, BT9 5BN, United Kingdom
| | - Padraig Fitzpatrick
- School of Electronics, Electrical Engineering and Computer Science, Queen's University Belfast, BT9 5BN, United Kingdom
| | - Nick Orr
- Patrick G Johnston Centre for Cancer Research, Queen's University Belfast, BT9 7AE, United Kingdom
| | - Anna Jurek-Loughrey
- School of Electronics, Electrical Engineering and Computer Science, Queen's University Belfast, BT9 5BN, United Kingdom
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8
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Abstract
Recently, machine learning (ML) has established itself in various worldwide benchmarking competitions in computational biology, including Critical Assessment of Structure Prediction (CASP) and Drug Design Data Resource (D3R) Grand Challenges. However, the intricate structural complexity and high ML dimensionality of biomolecular datasets obstruct the efficient application of ML algorithms in the field. In addition to data and algorithm, an efficient ML machinery for biomolecular predictions must include structural representation as an indispensable component. Mathematical representations that simplify the biomolecular structural complexity and reduce ML dimensionality have emerged as a prime winner in D3R Grand Challenges. This review is devoted to the recent advances in developing low-dimensional and scalable mathematical representations of biomolecules in our laboratory. We discuss three classes of mathematical approaches, including algebraic topology, differential geometry, and graph theory. We elucidate how the physical and biological challenges have guided the evolution and development of these mathematical apparatuses for massive and diverse biomolecular data. We focus the performance analysis on protein-ligand binding predictions in this review although these methods have had tremendous success in many other applications, such as protein classification, virtual screening, and the predictions of solubility, solvation free energies, toxicity, partition coefficients, protein folding stability changes upon mutation, etc.
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Affiliation(s)
- Duc Duy Nguyen
- Department of Mathematics, Michigan State University, MI 48824, USA.
| | - Zixuan Cang
- Department of Mathematics, Michigan State University, MI 48824, USA.
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, MI 48824, USA. and Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA and Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA
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9
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Weighted persistent homology for biomolecular data analysis. Sci Rep 2020; 10:2079. [PMID: 32034168 PMCID: PMC7005716 DOI: 10.1038/s41598-019-55660-3] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2019] [Accepted: 11/29/2019] [Indexed: 11/08/2022] Open
Abstract
In this paper, we systematically review weighted persistent homology (WPH) models and their applications in biomolecular data analysis. Essentially, the weight value, which reflects physical, chemical and biological properties, can be assigned to vertices (atom centers), edges (bonds), or higher order simplexes (cluster of atoms), depending on the biomolecular structure, function, and dynamics properties. Further, we propose the first localized weighted persistent homology (LWPH). Inspired by the great success of element specific persistent homology (ESPH), we do not treat biomolecules as an inseparable system like all previous weighted models, instead we decompose them into a series of local domains, which may be overlapped with each other. The general persistent homology or weighted persistent homology analysis is then applied on each of these local domains. In this way, functional properties, that are embedded in local structures, can be revealed. Our model has been applied to systematically study DNA structures. It has been found that our LWPH based features can be used to successfully discriminate the A-, B-, and Z-types of DNA. More importantly, our LWPH based principal component analysis (PCA) model can identify two configurational states of DNA structures in ion liquid environment, which can be revealed only by the complicated helical coordinate system. The great consistence with the helical-coordinate model demonstrates that our model captures local structure variations so well that it is comparable with geometric models. Moreover, geometric measurements are usually defined in local regions. For instance, the helical-coordinate system is limited to one or two basepairs. However, our LWPH can quantitatively characterize structure information in regions or domains with arbitrary sizes and shapes, where traditional geometrical measurements fail.
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10
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Nguyen DD, Wei GW. DG-GL: Differential geometry-based geometric learning of molecular datasets. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2019; 35:e3179. [PMID: 30693661 PMCID: PMC6598676 DOI: 10.1002/cnm.3179] [Citation(s) in RCA: 47] [Impact Index Per Article: 9.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2018] [Revised: 11/21/2018] [Accepted: 12/06/2018] [Indexed: 05/11/2023]
Abstract
MOTIVATION Despite its great success in various physical modeling, differential geometry (DG) has rarely been devised as a versatile tool for analyzing large, diverse, and complex molecular and biomolecular datasets because of the limited understanding of its potential power in dimensionality reduction and its ability to encode essential chemical and biological information in differentiable manifolds. RESULTS We put forward a differential geometry-based geometric learning (DG-GL) hypothesis that the intrinsic physics of three-dimensional (3D) molecular structures lies on a family of low-dimensional manifolds embedded in a high-dimensional data space. We encode crucial chemical, physical, and biological information into 2D element interactive manifolds, extracted from a high-dimensional structural data space via a multiscale discrete-to-continuum mapping using differentiable density estimators. Differential geometry apparatuses are utilized to construct element interactive curvatures in analytical forms for certain analytically differentiable density estimators. These low-dimensional differential geometry representations are paired with a robust machine learning algorithm to showcase their descriptive and predictive powers for large, diverse, and complex molecular and biomolecular datasets. Extensive numerical experiments are carried out to demonstrate that the proposed DG-GL strategy outperforms other advanced methods in the predictions of drug discovery-related protein-ligand binding affinity, drug toxicity, and molecular solvation free energy. AVAILABILITY AND IMPLEMENTATION http://weilab.math.msu.edu/DG-GL/ Contact: wei@math.msu.edu.
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Affiliation(s)
- Duc Duy Nguyen
- Department of Mathematics, Michigan State University, East Lansing, 48824, Michigan
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, 48824, Michigan
- Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA
- Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, Michigan
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11
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Xia K, Anand DV, Shikhar S, Mu Y. Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks. Phys Chem Chem Phys 2019; 21:21038-21048. [DOI: 10.1039/c9cp03009c] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Dramatically different patterns can be observed in the topological fingerprints for hydrogen-bonding networks from two types of osmolyte systems.
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Affiliation(s)
- Kelin Xia
- Division of Mathematical Sciences
- School of Physical and Mathematical Sciences
- School of Biological Sciences
- Nanyang Technological University
- Singapore
| | - D. Vijay Anand
- Division of Mathematical Sciences
- School of Physical and Mathematical Sciences
- School of Biological Sciences
- Nanyang Technological University
- Singapore
| | - Saxena Shikhar
- School of Biological Sciences
- Nanyang Technological University
- Singapore
| | - Yuguang Mu
- School of Biological Sciences
- Nanyang Technological University
- Singapore
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12
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TopP-S: Persistent homology-based multi-task deep neural networks for simultaneous predictions of partition coefficient and aqueous solubility. J Comput Chem 2018; 39:1444-1454. [DOI: 10.1002/jcc.25213] [Citation(s) in RCA: 39] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2017] [Revised: 01/15/2018] [Accepted: 02/25/2018] [Indexed: 01/09/2023]
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13
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Wu K, Wei GW. Quantitative Toxicity Prediction Using Topology Based Multitask Deep Neural Networks. J Chem Inf Model 2018; 58:520-531. [DOI: 10.1021/acs.jcim.7b00558] [Citation(s) in RCA: 75] [Impact Index Per Article: 12.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Kedi Wu
- Department of Mathematics, ‡Department of Electrical and Computer Engineering, and ¶Department of Biochemistry
and Molecular Biology, Michigan State University, East Lansing, Michigan 48824, United States
| | - Guo-Wei Wei
- Department of Mathematics, ‡Department of Electrical and Computer Engineering, and ¶Department of Biochemistry
and Molecular Biology, Michigan State University, East Lansing, Michigan 48824, United States
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14
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Cang Z, Mu L, Wei GW. Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening. PLoS Comput Biol 2018; 14:e1005929. [PMID: 29309403 PMCID: PMC5774846 DOI: 10.1371/journal.pcbi.1005929] [Citation(s) in RCA: 141] [Impact Index Per Article: 23.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2017] [Revised: 01/19/2018] [Accepted: 12/15/2017] [Indexed: 12/05/2022] Open
Abstract
This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes. In contrast to the conventional persistent homology, multi-component persistent homology retains critical chemical and biological information during the topological simplification of biomolecular geometric complexity. Multi-level persistent homology enables a tailored topological description of inter- and/or intra-molecular interactions of interest. Electrostatic persistence incorporates partial charge information into topological invariants. These topological methods are paired with Wasserstein distance to characterize similarities between molecules and are further integrated with a variety of machine learning algorithms, including k-nearest neighbors, ensemble of trees, and deep convolutional neural networks, to manifest their descriptive and predictive powers for protein-ligand binding analysis and virtual screening of small molecules. Extensive numerical experiments involving 4,414 protein-ligand complexes from the PDBBind database and 128,374 ligand-target and decoy-target pairs in the DUD database are performed to test respectively the scoring power and the discriminatory power of the proposed topological learning strategies. It is demonstrated that the present topological learning outperforms other existing methods in protein-ligand binding affinity prediction and ligand-decoy discrimination.
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Affiliation(s)
- Zixuan Cang
- Department of Mathematics, Michigan State University, East Lansing, Michigan, United States of America
| | - Lin Mu
- Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States of America
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, Michigan, United States of America
- Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, Michigan, United States of America
- Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan, United States of America
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15
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Sommerfeld M, Heo G, Kim P, Rush ST, Marron JS. Bump hunting by topological data analysis. Stat (Int Stat Inst) 2017. [DOI: 10.1002/sta4.167] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
Affiliation(s)
- Max Sommerfeld
- Felix Bernstein Institute for Mathematical Statistics in the Biosciences; University of Göttingen; Göttingen 37077 Germany
| | - Giseon Heo
- School of Dentistry; University of Alberta; Edmonton Alberta T6G 2R7 Canada
| | - Peter Kim
- Department of Mathematics and Statistics; University of Guelph; Guelph Ontario N1G 2W1 Canada
| | - Stephen T. Rush
- School of Medical Sciences; Örebro Universitet; Örebro SE-701 82 Sweden
| | - J. S. Marron
- Department of Statistics; University of North Carolina at Chapel Hill; Chapel Hill NC 27599 USA
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16
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Cang Z, Wei GW. TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions. PLoS Comput Biol 2017; 13:e1005690. [PMID: 28749969 PMCID: PMC5549771 DOI: 10.1371/journal.pcbi.1005690] [Citation(s) in RCA: 159] [Impact Index Per Article: 22.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2017] [Revised: 08/08/2017] [Accepted: 07/18/2017] [Indexed: 11/18/2022] Open
Abstract
Although deep learning approaches have had tremendous success in image, video and audio processing, computer vision, and speech recognition, their applications to three-dimensional (3D) biomolecular structural data sets have been hindered by the geometric and biological complexity. To address this problem we introduce the element-specific persistent homology (ESPH) method. ESPH represents 3D complex geometry by one-dimensional (1D) topological invariants and retains important biological information via a multichannel image-like representation. This representation reveals hidden structure-function relationships in biomolecules. We further integrate ESPH and deep convolutional neural networks to construct a multichannel topological neural network (TopologyNet) for the predictions of protein-ligand binding affinities and protein stability changes upon mutation. To overcome the deep learning limitations from small and noisy training sets, we propose a multi-task multichannel topological convolutional neural network (MM-TCNN). We demonstrate that TopologyNet outperforms the latest methods in the prediction of protein-ligand binding affinities, mutation induced globular protein folding free energy changes, and mutation induced membrane protein folding free energy changes. AVAILABILITY weilab.math.msu.edu/TDL/.
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Affiliation(s)
- Zixuan Cang
- Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
- Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, USA
- Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA
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17
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Abstract
Flexibility-rigidity index (FRI) has been developed as a robust, accurate, and efficient method for macromolecular thermal fluctuation analysis and B-factor prediction. The performance of FRI depends on its formulations of rigidity index and flexibility index. In this work, we introduce alternative rigidity and flexibility formulations. The structure of the classic Gaussian surface is utilized to construct a new type of rigidity index, which leads to a new class of rigidity densities with the classic Gaussian surface as a special case. Additionally, we introduce a new type of flexibility index based on the domain indicator property of normalized rigidity density. These generalized FRI (gFRI) methods have been extensively validated by the B-factor predictions of 364 proteins. Significantly outperforming the classic Gaussian network model, gFRI is a new generation of methodologies for accurate, robust, and efficient analysis of protein flexibility and fluctuation. Finally, gFRI based molecular surface generation and flexibility visualization are demonstrated.
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Affiliation(s)
- Duc Duy Nguyen
- Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
| | - Kelin Xia
- Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
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18
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Xia K, Zhao Z, Wei GW. Multiresolution persistent homology for excessively large biomolecular datasets. J Chem Phys 2015; 143:134103. [PMID: 26450288 PMCID: PMC4592433 DOI: 10.1063/1.4931733] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2015] [Accepted: 09/08/2015] [Indexed: 12/21/2022] Open
Abstract
Although persistent homology has emerged as a promising tool for the topological simplification of complex data, it is computationally intractable for large datasets. We introduce multiresolution persistent homology to handle excessively large datasets. We match the resolution with the scale of interest so as to represent large scale datasets with appropriate resolution. We utilize flexibility-rigidity index to access the topological connectivity of the data set and define a rigidity density for the filtration analysis. By appropriately tuning the resolution of the rigidity density, we are able to focus the topological lens on the scale of interest. The proposed multiresolution topological analysis is validated by a hexagonal fractal image which has three distinct scales. We further demonstrate the proposed method for extracting topological fingerprints from DNA molecules. In particular, the topological persistence of a virus capsid with 273 780 atoms is successfully analyzed which would otherwise be inaccessible to the normal point cloud method and unreliable by using coarse-grained multiscale persistent homology. The proposed method has also been successfully applied to the protein domain classification, which is the first time that persistent homology is used for practical protein domain analysis, to our knowledge. The proposed multiresolution topological method has potential applications in arbitrary data sets, such as social networks, biological networks, and graphs.
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Affiliation(s)
- Kelin Xia
- Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
| | - Zhixiong Zhao
- Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
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