Multiresolution Topological Simplification

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.

Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis

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.

A review of mathematical representations of biomolecular data

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.

Weighted persistent homology for biomolecular data analysis

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.

Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks

Dramatically different patterns can be observed in the topological fingerprints for hydrogen-bonding networks from two types of osmolyte systems.

Gene Coexpression Network Comparison via Persistent Homology

Persistent homology, a topological data analysis (TDA) method, is applied to microarray data sets. Although there are a few papers referring to TDA methods in microarray analysis, the usage of persistent homology in the comparison of several weighted gene coexpression networks (WGCN) was not employed before to the very best of our knowledge. We calculate the persistent homology of weighted networks constructed from 38 Arabidopsis microarray data sets to test the relevance and the success of this approach in distinguishing the stress factors. We quantify multiscale topological features of each network using persistent homology and apply a hierarchical clustering algorithm to the distance matrix whose entries are pairwise bottleneck distance between the networks. The immunoresponses to different stress factors are distinguishable by our method. The networks of similar immunoresponses are found to be close with respect to bottleneck distance indicating the similar topological features of WGCNs. This computationally efficient technique analyzing networks provides a quick test for advanced studies.

Persistent homology analysis of ion aggregations and hydrogen-bonding networks

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.

Multiscale Persistent Functions for Biomolecular Structure Characterization

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.

TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions

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/.

Flexibility-rigidity index for protein-nucleic acid flexibility and fluctuation analysis

Protein-nucleic acid complexes are important for many cellular processes including the most essential functions such as transcription and translation. For many protein-nucleic acid complexes, flexibility of both macromolecules has been shown to be critical for specificity and/or function. The flexibility-rigidity index (FRI) has been proposed as an accurate and efficient approach for protein flexibility analysis. In this article, we introduce FRI for the flexibility analysis of protein-nucleic acid complexes. We demonstrate that a multiscale strategy, which incorporates multiple kernels to capture various length scales in biomolecular collective motions, is able to significantly improve the state of art in the flexibility analysis of protein-nucleic acid complexes. We take the advantage of the high accuracy and O(N) computational complexity of our multiscale FRI method to investigate the flexibility of ribosomal subunits, which are difficult to analyze by alternative approaches. An anisotropic FRI approach, which involves localized Hessian matrices, is utilized to study the translocation dynamics in an RNA polymerase.

Object-oriented Persistent Homology

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.