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Ye X, Sun F, Xiang S. TREPH: A Plug-In Topological Layer for Graph Neural Networks. ENTROPY (BASEL, SWITZERLAND) 2023; 25:331. [PMID: 36832697 PMCID: PMC9954936 DOI: 10.3390/e25020331] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Revised: 02/04/2023] [Accepted: 02/08/2023] [Indexed: 06/18/2023]
Abstract
Topological Data Analysis (TDA) is an approach to analyzing the shape of data using techniques from algebraic topology. The staple of TDA is Persistent Homology (PH). Recent years have seen a trend of combining PH and Graph Neural Networks (GNNs) in an end-to-end manner to capture topological features from graph data. Though effective, these methods are limited by the shortcomings of PH: incomplete topological information and irregular output format. Extended Persistent Homology (EPH), as a variant of PH, addresses these problems elegantly. In this paper, we propose a plug-in topological layer for GNNs, termed Topological Representation with Extended Persistent Homology (TREPH). Taking advantage of the uniformity of EPH, a novel aggregation mechanism is designed to collate topological features of different dimensions to the local positions determining their living processes. The proposed layer is provably differentiable and more expressive than PH-based representations, which in turn is strictly stronger than message-passing GNNs in expressive power. Experiments on real-world graph classification tasks demonstrate the competitiveness of TREPH compared with the state-of-the-art approaches.
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Affiliation(s)
- Xue Ye
- National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
- School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 101408, China
| | - Fang Sun
- School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
| | - Shiming Xiang
- National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
- School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 101408, China
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2
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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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3
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Pun CS, Lee SX, Xia K. Persistent-homology-based machine learning: a survey and a comparative study. Artif Intell Rev 2022. [DOI: 10.1007/s10462-022-10146-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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4
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Chen J, Zhao R, Tong Y, Wei GW. EVOLUTIONARY DE RHAM-HODGE METHOD. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES B 2021; 26:3785-3821. [PMID: 34675756 DOI: 10.3934/dcdsb.2020257] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
Abstract
The de Rham-Hodge theory is a landmark of the 20th Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.
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Affiliation(s)
- Jiahui Chen
- Department of Mathematics, Michigan State University, MI 48824, USA
| | - Rundong Zhao
- Department of Computer Science and Engineering, Michigan State University, MI 48824, USA
| | - Yiying Tong
- Department of Computer Science and Engineering, Michigan State University, MI 48824, USA
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, MI 48824, USA
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5
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Padellini T, Brutti P. Supervised learning with indefinite topological Kernels. STATISTICS-ABINGDON 2021. [DOI: 10.1080/02331888.2021.1976777] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
- Tullia Padellini
- Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Rome, Italy
| | - Pierpaolo Brutti
- Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Rome, Italy
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Yen PTW, Xia K, Cheong SA. Understanding Changes in the Topology and Geometry of Financial Market Correlations during a Market Crash. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1211. [PMID: 34573837 PMCID: PMC8467365 DOI: 10.3390/e23091211] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/19/2021] [Revised: 09/05/2021] [Accepted: 09/06/2021] [Indexed: 12/24/2022]
Abstract
In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.
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Affiliation(s)
- Peter Tsung-Wen Yen
- Center for Crystal Researches, National Sun Yet-Sen University, No. 70, Lien-hai Rd., Kaohsiung 80424, Taiwan;
| | - Kelin Xia
- Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore;
| | - Siew Ann Cheong
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
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7
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Cang Z, Munch E, Wei GW. Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis. ACTA ACUST UNITED AC 2020; 4:481-507. [PMID: 34179350 DOI: 10.1007/s41468-020-00057-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
While the spatial topological persistence is naturally constructed from a radius-based filtration, it has hardly been derived from a temporal filtration. Most topological models are designed for the global topology of a given object as a whole. There is no method reported in the literature for the topology of an individual component in an object to the best of our knowledge. For many problems in science and engineering, the topology of an individual component is important for describing its properties. We propose evolutionary homology (EH) constructed via a time evolution-based filtration and topological persistence. Our approach couples a set of dynamical systems or chaotic oscillators by the interactions of a physical system, such as a macromolecule. The interactions are approximated by weighted graph Laplacians. Simplices, simplicial complexes, algebraic groups and topological persistence are defined on the coupled trajectories of the chaotic oscillators. The resulting EH gives rise to time-dependent topological invariants or evolutionary barcodes for an individual component of the physical system, revealing its topology-function relationship. In conjunction with Wasserstein metrics, the proposed EH is applied to protein flexibility analysis, an important problem in computational biophysics. Numerical results for the B-factor prediction of a benchmark set of 364 proteins indicate that the proposed EH outperforms all the other state-of-the-art methods in the field.
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Affiliation(s)
- Zixuan Cang
- Department of Mathematics, Michigan State University
| | - Elizabeth Munch
- Department of Computational Mathematics, Science and Engineering, Michigan State University
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University
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Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis. Sci Rep 2020; 10:9685. [PMID: 32546801 PMCID: PMC7297731 DOI: 10.1038/s41598-020-66710-6] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2019] [Accepted: 05/20/2020] [Indexed: 12/24/2022] Open
Abstract
It has long been observed that trimethylamine N-oxide (TMAO) and urea demonstrate dramatically different properties in a protein folding process. Even with the enormous theoretical and experimental research work on these two osmolytes, various aspects of their underlying mechanisms still remain largely elusive. In this paper, we propose to use the weighted persistent homology to systematically study the osmolytes molecular aggregation and their hydrogen-bonding network from a local topological perspective. We consider two weighted models, i.e., localized persistent homology (LPH) and interactive persistent homology (IPH). Boltzmann persistent entropy (BPE) is proposed to quantitatively characterize the topological features from LPH and IPH, together with persistent Betti number (PBN). More specifically, from the localized persistent homology models, we have found that TMAO and urea have very different local topology. TMAO is found to exhibit a local network structure. With the concentration increase, the circle elements in these networks show a clear increase in their total numbers and a decrease in their relative sizes. In contrast, urea shows two types of local topological patterns, i.e., local clusters around 6 Å and a few global circle elements at around 12 Å. From the interactive persistent homology models, it has been found that our persistent radial distribution function (PRDF) from the global-scale IPH has same physical properties as the traditional radial distribution function. Moreover, PRDFs from the local-scale IPH can also be generated and used to characterize the local interaction information. Other than the clear difference of the first peak value of PRDFs at filtration size 4 Å, TMAO and urea also shows very different behaviors at the second peak region from filtration size 5 Å to 10 Å. These differences are also reflected in the PBNs and BPEs of the local-scale IPH. These localized topological information has never been revealed before. Since graphs can be transferred into simplicial complexes by the clique complex, our weighted persistent homology models can be used in the analysis of various networks and graphs from any molecular structures and aggregation systems.
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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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10
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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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11
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Nguyen DD, Gao K, Wang M, Wei GW. MathDL: mathematical deep learning for D3R Grand Challenge 4. J Comput Aided Mol Des 2020; 34:131-147. [PMID: 31734815 PMCID: PMC7376411 DOI: 10.1007/s10822-019-00237-5] [Citation(s) in RCA: 45] [Impact Index Per Article: 11.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2019] [Accepted: 10/14/2019] [Indexed: 12/17/2022]
Abstract
We present the performances of our mathematical deep learning (MathDL) models for D3R Grand Challenge 4 (GC4). This challenge involves pose prediction, affinity ranking, and free energy estimation for beta secretase 1 (BACE) as well as affinity ranking and free energy estimation for Cathepsin S (CatS). We have developed advanced mathematics, namely differential geometry, algebraic graph, and/or algebraic topology, to accurately and efficiently encode high dimensional physical/chemical interactions into scalable low-dimensional rotational and translational invariant representations. These representations are integrated with deep learning models, such as generative adversarial networks (GAN) and convolutional neural networks (CNN) for pose prediction and energy evaluation, respectively. Overall, our MathDL models achieved the top place in pose prediction for BACE ligands in Stage 1a. Moreover, our submissions obtained the highest Spearman correlation coefficient on the affinity ranking of 460 CatS compounds, and the smallest centered root mean square error on the free energy set of 39 CatS molecules. It is worthy to mention that our method on docking pose predictions has significantly improved from our previous ones.
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Affiliation(s)
- Duc Duy Nguyen
- Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA
| | - Kaifu Gao
- Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA
| | - Menglun Wang
- 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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12
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Zhao R, Cang Z, Tong Y, Wei GW. Protein pocket detection via convex hull surface evolution and associated Reeb graph. Bioinformatics 2019; 34:i830-i837. [PMID: 30423105 DOI: 10.1093/bioinformatics/bty598] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
Motivation Protein pocket information is invaluable for drug target identification, agonist design, virtual screening and receptor-ligand binding analysis. A recent study indicates that about half holoproteins can simultaneously bind multiple interacting ligands in a large pocket containing structured sub-pockets. Although this hierarchical pocket and sub-pocket structure has a significant impact to multi-ligand synergistic interactions in the protein binding site, there is no method available for this analysis. This work introduces a computational tool based on differential geometry, algebraic topology and physics-based simulation to address this pressing issue. Results We propose to detect protein pockets by evolving the convex hull surface inwards until it touches the protein surface everywhere. The governing partial differential equations (PDEs) include the mean curvature flow combined with the eikonal equation commonly used in the fast marching algorithm in the Eulerian representation. The surface evolution induced Morse function and Reeb graph are utilized to characterize the hierarchical pocket and sub-pocket structure in controllable detail. The proposed method is validated on PDBbind refined sets of 4414 protein-ligand complexes. Extensive numerical tests indicate that the proposed method not only provides a unique description of pocket-sub-pocket relations, but also offers efficient estimations of pocket surface area, pocket volume and pocket depth. Availability and implementation Source code available at https://github.com/rdzhao/ProteinPocketDetection. Webserver available at http://weilab.math.msu.edu/PPD/.
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Affiliation(s)
- Rundong Zhao
- Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, USA
| | - Zixuan Cang
- Department of Mathematics, Michigan State University, East Lansing, MI, USA
| | - Yiying Tong
- Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, USA
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, MI, USA
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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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14
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Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges. J Comput Aided Mol Des 2018; 33:71-82. [PMID: 30116918 DOI: 10.1007/s10822-018-0146-6] [Citation(s) in RCA: 99] [Impact Index Per Article: 16.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2018] [Accepted: 08/03/2018] [Indexed: 12/18/2022]
Abstract
Advanced mathematics, such as multiscale weighted colored subgraph and element specific persistent homology, and machine learning including deep neural networks were integrated to construct mathematical deep learning models for pose and binding affinity prediction and ranking in the last two D3R Grand Challenges in computer-aided drug design and discovery. D3R Grand Challenge 2 focused on the pose prediction, binding affinity ranking and free energy prediction for Farnesoid X receptor ligands. Our models obtained the top place in absolute free energy prediction for free energy set 1 in stage 2. The latest competition, D3R Grand Challenge 3 (GC3), is considered as the most difficult challenge so far. It has five subchallenges involving Cathepsin S and five other kinase targets, namely VEGFR2, JAK2, p38-α, TIE2, and ABL1. There is a total of 26 official competitive tasks for GC3. Our predictions were ranked 1st in 10 out of these 26 tasks.
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Xia K. Persistent homology analysis of ion aggregations and hydrogen-bonding networks. Phys Chem Chem Phys 2018; 20:13448-13460. [PMID: 29722784 DOI: 10.1039/c8cp01552j] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Despite the great advancement of experimental tools and theoretical models, a quantitative characterization of the microscopic structures of ion aggregates and their associated water hydrogen-bonding networks still remains a challenging problem. In this paper, a newly-invented mathematical method called persistent homology is introduced, for the first time, to quantitatively analyze the intrinsic topological properties of ion aggregation systems and hydrogen-bonding networks. The two most distinguishable properties of persistent homology analysis of assembly systems are as follows. First, it does not require a predefined bond length to construct the ion or hydrogen-bonding network. Persistent homology results are determined by the morphological structure of the data only. Second, it can directly measure the size of circles or holes in ion aggregates and hydrogen-bonding networks. To validate our model, we consider two well-studied systems, i.e., NaCl and KSCN solutions, generated from molecular dynamics simulations. They are believed to represent two morphological types of aggregation, i.e., local clusters and extended ion networks. It has been found that the two aggregation types have distinguishable topological features and can be characterized by our topological model very well. Further, we construct two types of networks, i.e., O-networks and H2O-networks, for analyzing the topological properties of hydrogen-bonding networks. It is found that for both models, KSCN systems demonstrate much more dramatic variations in their local circle structures with a concentration increase. A consistent increase of large-sized local circle structures is observed and the sizes of these circles become more and more diverse. In contrast, NaCl systems show no obvious increase of large-sized circles. Instead a consistent decline of the average size of the circle structures is observed and the sizes of these circles become more and more uniform with a concentration increase. As far as we know, these unique intrinsic topological features in ion aggregation systems have never been pointed out before. More importantly, our models can be directly used to quantitatively analyze the intrinsic topological invariants, including circles, loops, holes, and cavities, of any network-like structures, such as nanomaterials, colloidal systems, biomolecular assemblies, among others. These topological invariants cannot be described by traditional graph and network models.
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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, 637371, Singapore.
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16
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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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17
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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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18
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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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19
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Multiscale Persistent Functions for Biomolecular Structure Characterization. Bull Math Biol 2017; 80:1-31. [PMID: 29098540 DOI: 10.1007/s11538-017-0362-6] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2016] [Accepted: 10/19/2017] [Indexed: 10/18/2022]
Abstract
In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions (MRFs) with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy (MPE) model is discussed in great detail. Mathematically, unlike the previous persistent entropy (Chintakunta et al. in Pattern Recognit 48(2):391-401, 2015; Merelli et al. in Entropy 17(10):6872-6892, 2015; Rucco et al. in: Proceedings of ECCS 2014, Springer, pp 117-128, 2016), a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our MPE can be used in conformational entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of an MRF built from angular distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the intermediate range. Moreover, by comparing with traditional entropies from various grid sizes, bond angle-based methods and a persistent homology-based support vector machine method (Cang et al. in Mol Based Math Biol 3:140-162, 2015), we find that our MPE method gives the best results in terms of average true positive rate in a classic protein structure classification test. More interestingly, all-alpha and all-beta protein classes can be clearly separated from each other with zero error only in our model. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Basically, a protein structure is deemed as regular if it has a consistent and orderly configuration. Our PSI model is tested on a database of 110 proteins; we find that structures with larger portions of loops and intrinsically disorder regions are always associated with larger PSI, meaning an irregular configuration, while proteins with larger portions of secondary structures, i.e., alpha-helix or beta-sheet, have smaller PSI. Essentially, PSI can be used to describe the "regularity" information in any systems.
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Otter N, Porter MA, Tillmann U, Grindrod P, Harrington HA. A roadmap for the computation of persistent homology. EPJ DATA SCIENCE 2017; 6:17. [PMID: 32025466 PMCID: PMC6979512 DOI: 10.1140/epjds/s13688-017-0109-5] [Citation(s) in RCA: 124] [Impact Index Per Article: 17.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/30/2017] [Accepted: 06/07/2017] [Indexed: 05/21/2023]
Abstract
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking. ELECTRONIC SUPPLEMENTARY MATERIAL The online version of this article (doi:10.1140/epjds/s13688-017-0109-5) contains supplementary material.
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Affiliation(s)
- Nina Otter
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK
- The Alan Turing Institute, 96 Euston Road, London, NW1 2DB UK
| | - Mason A Porter
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK
- CABDyN Complexity Centre, University of Oxford, Oxford, OX1 1HP UK
- Department of Mathematics, UCLA, Los Angeles, CA 90095 USA
| | - Ulrike Tillmann
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK
- The Alan Turing Institute, 96 Euston Road, London, NW1 2DB UK
| | - Peter Grindrod
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK
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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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Liu B, Wang B, Zhao R, Tong Y, Wei GW. ESES: Software for Eulerian solvent excluded surface. J Comput Chem 2017; 38:446-466. [PMID: 28052350 DOI: 10.1002/jcc.24682] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2016] [Revised: 11/02/2016] [Accepted: 11/09/2016] [Indexed: 12/17/2022]
Abstract
Solvent excluded surface (SES) is one of the most popular surface definitions in biophysics and molecular biology. In addition to its usage in biomolecular visualization, it has been widely used in implicit solvent models, in which SES is usually immersed in a Cartesian mesh. Therefore, it is important to construct SESs in the Eulerian representation for biophysical modeling and computation. This work describes a software package called Eulerian solvent excluded surface (ESES) for the generation of accurate SESs in Cartesian grids. ESES offers the description of the solvent and solute domains by specifying all the intersection points between the SES and the Cartesian grid lines. Additionally, the interface normal at each intersection point is evaluated. Furthermore, for a given biomolecule, the ESES software not only provides the whole surface area, but also partitions the surface area according to atomic types. Homology theory is utilized to detect topological features, such as loops and cavities, on the complex formed by the SES. The sizes of loops and cavities are measured based on persistent homology with an evolutionary partial differential equation-based filtration. ESES is extensively validated by surface visualization, electrostatic solvation free energy computation, surface area and volume calculations, and loop and cavity detection and their size estimation. We used the Amber PBSA test set in our electrostatic solvation energy, area, and volume validations. Our results are either calibrated by analytical values or compared with those from the MSMS software. © 2017 Wiley Periodicals, Inc.
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Affiliation(s)
- Beibei Liu
- Department of Computer Science and Engineering, Michigan State University, East Lansing, Michigan, 48824
| | - Bao Wang
- Department of Mathematics, Michigan State University, East Lansing, Michigan, 48824
| | - Rundong Zhao
- Department of Computer Science and Engineering, Michigan State University, East Lansing, Michigan, 48824
| | - Yiying Tong
- Department of Computer Science and Engineering, Michigan State University, East Lansing, Michigan, 48824
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, Michigan, 48824.,Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan, 48824.,Department of Biochemistry and, Molecular Biology, Michigan State University, East Lansing, Michigan, 48824
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