The de Rham-Hodge Analysis and Modeling of Biomolecules

Hodge Decomposition of Single-Cell RNA Velocity

RNA velocity has the ability to capture the cell dynamic information in the biological processes; yet, a comprehensive analysis of the cell state transitions and their associated chemical and biological processes remains a gap. In this work, we provide the Hodge decomposition, coupled with discrete exterior calculus (DEC), to unveil cell dynamics by examining the decomposed curl-free, divergence-free, and harmonic components of the RNA velocity field in a low dimensional representation, such as a UMAP or a t-SNE representation. Decomposition results show that the decomposed components distinctly reveal key cell dynamic features such as cell cycle, bifurcation, and cell lineage differentiation, regardless of the choice of the low-dimensional representations. The consistency across different representations demonstrates that the Hodge decomposition is a reliable and robust way to extract these cell dynamic features, offering unique analysis and insightful visualization of single-cell RNA velocity fields.

Persistent Dirac for molecular representation

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.

TIDAL: Topology-Inferred Drug Addiction Learning

Drug addiction is a global public health crisis, and the design of antiaddiction drugs remains a major challenge due to intricate mechanisms. Since experimental drug screening and optimization are too time-consuming and expensive, there is urgent need to develop innovative artificial intelligence (AI) methods for addressing the challenge. We tackle this challenge by topology-inferred drug addiction learning (TIDAL) built from integrating multiscale topological Laplacians, deep bidirectional transformer, and ensemble-assisted neural networks (EANNs). Multiscale topological Laplacians are a novel class of algebraic topology tools that embed molecular topological invariants and algebraic invariants into its harmonic spectra and nonharmonic spectra, respectively. These invariants complement sequence information extracted from a bidirectional transformer. We validate the proposed TIDAL framework on 22 drug addiction related, 4 hERG, and 12 DAT data sets, which suggests that the proposed TIDAL is a state-of-the-art framework for the modeling and analysis of drug addiction data. We carry out cross-target analysis of the current drug addiction candidates to alert their side effects and identify their repurposing potentials. Our analysis reveals drug-mediated linear and bilinear target correlations. Finally, TIDAL is applied to shed light on relative efficacy, repurposing potential, and potential side effects of 12 existing antiaddiction medications. Our results suggest that TIDAL provides a new computational strategy for pressingly needed antisubstance addiction drug development.

PERSISTENT PATH LAPLACIAN

Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.

Persistent Homology for RNA Data Analysis

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.

Biomolecular Topology: Modelling and Analysis

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.

Methodology-Centered Review of Molecular Modeling, Simulation, and Prediction of SARS-CoV-2

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.

Hodge theory-based biomolecular data analysis

Hodge theory reveals the deep intrinsic relations of differential forms and provides a bridge between differential geometry, algebraic topology, and functional analysis. Here we use Hodge Laplacian and Hodge decomposition models to analyze biomolecular structures. Different from traditional graph-based methods, biomolecular structures are represented as simplicial complexes, which can be viewed as a generalization of graph models to their higher-dimensional counterparts. Hodge Laplacian matrices at different dimensions can be generated from the simplicial complex. The spectral information of these matrices can be used to study intrinsic topological information of biomolecular structures. Essentially, the number (or multiplicity) of k-th dimensional zero eigenvalues is equivalent to the k-th Betti number, i.e., the number of k-th dimensional homology groups. The associated eigenvectors indicate the homological generators, i.e., circles or holes within the molecular-based simplicial complex. Furthermore, Hodge decomposition-based HodgeRank model is used to characterize the folding or compactness of the molecular structures, in particular, the topological associated domain (TAD) in high-throughput chromosome conformation capture (Hi-C) data. Mathematically, molecular structures are represented in simplicial complexes with certain edge flows. The HodgeRank-based average/total inconsistency (AI/TI) is used for the quantitative measurements of the folding or compactness of TADs. This is the first quantitative measurement for TAD regions, as far as we know.

Neighborhood hypergraph model for topological data analysis

Hypergraph, as a generalization of the notions of graph and simplicial complex, has gained a lot of attention in many fields. It is a relatively new mathematical model to describe the high-dimensional structure and geometric shapes of data sets. In this paper,we introduce the neighborhood hypergraph model for graphs and combine the neighborhood hypergraph model with the persistent (embedded) homology of hypergraphs. Given a graph,we can obtain a neighborhood complex introduced by L. Lovász and a filtration of hypergraphs parameterized by aweight function on the power set of the vertex set of the graph. Theweight function can be obtained by the construction fromthe geometric structure of graphs or theweights on the vertices of the graph. We show the persistent theory of such filtrations of hypergraphs. One typical application of the persistent neighborhood hypergraph is to distinguish the planar square structure of cisplatin and transplatin. Another application of persistent neighborhood hypergraph is to describe the structure of small fullerenes such as C20. The bond length and the number of adjacent carbon atoms of a carbon atom can be derived from the persistence diagram. Moreover, our method gives a highly matched stability prediction (with a correlation coefficient 0.9976) of small fullerene molecules.

EVOLUTIONARY DE RHAM-HODGE METHOD

The de Rham-Hodge theory is a landmark of the 20^th 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.

HERMES: PERSISTENT SPECTRAL GRAPH SOFTWARE

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.