Representation of molecular structures with persistent homology for machine learning applications in chemistry

Dissecting glial scar formation by spatial point pattern and topological data analysis

Glial scar formation represents a fundamental response to central nervous system (CNS) injuries. It is mainly characterized by a well-defined spatial rearrangement of reactive astrocytes and microglia. The mechanisms underlying glial scar formation have been extensively studied, yet quantitative descriptors of the spatial arrangement of reactive glial cells remain limited. Here, we present a novel approach using point pattern analysis (PPA) and topological data analysis (TDA) to quantify spatial patterns of reactive glial cells after experimental ischemic stroke in mice. We provide open and reproducible tools using R and Julia to quantify spatial intensity, cell covariance and conditional distribution, cell-to-cell interactions, and short/long-scale arrangement, which collectively disentangle the arrangement patterns of the glial scar. This approach unravels a substantial divergence in the distribution of GFAP+ and IBA1+ cells after injury that conventional analysis methods cannot fully characterize. PPA and TDA are valuable tools for studying the complex spatial arrangement of reactive glia and other nervous cells following CNS injuries and have potential applications for evaluating glial-targeted restorative therapies.

Topological Learning Approach to Characterizing Biological Membranes

Biological membranes play key roles in cellular compartmentalization, structure, and its signaling pathways. At varying temperatures, individual membrane lipids sample from different configurations, a process that frequently leads to higher-order phase behavior and phenomena. Here, we present a persistent homology (PH)-based method for quantifying the structural features of individual and bulk lipids, providing local and contextual information on lipid tail organization. Our method leverages the mathematical machinery of algebraic topology and machine learning to infer temperature-dependent structural information on lipids from static coordinates. To train our model, we generated multiple molecular dynamics trajectories of dipalmitoyl-phosphatidylcholine membranes at varying temperatures. A fingerprint was then constructed for each set of lipid coordinates by PH filtration, in which interaction spheres were grown around the lipid atoms while tracking their intersections. The sphere filtration formed a simplicial complex that captures enduring key topological features of the configuration landscape using homology, yielding persistence data. Following fingerprint extraction for physiologically relevant temperatures, the persistence data were used to train an attention-based neural network for assignment of effective temperature values to selected membrane regions. Our persistence homology-based method captures the local structural effects, via effective temperature, of lipids adjacent to other membrane constituents, e.g., sterols and proteins. This topological learning approach can predict lipid effective temperatures from static coordinates across multiple spatial resolutions. The tool, called MembTDA, can be accessed at https://github.com/hyunp2/Memb-TDA.

Utilizing Machine Learning Models for Predicting Diamagnetic Susceptibility of Organic Compounds

This research is centered on examining the magnetic characteristics of organic molecules, with a particular emphasis on magnetic susceptibility, an essential physical property that provides insights into molecular microstructures and reaction processes. Traditional approaches for determining and calculating magnetic susceptibility are generally inefficient and demanding. To overcome these challenges, we have introduced a novel approach using quantitative structure-property relationships, which efficiently elucidates the relationship between the structural properties of molecules and their molar magnetic susceptibility. In our study, we utilized a comprehensive database comprising molar magnetic susceptibility data for 382 organic molecules. We applied six distinct molecular fingerprinting methods-RDKit Fingerprint, Morgan Fingerprint, MACCS Keys, atom pair fingerprint, Avalon Fingerprint, and topology fingerprint-as feature inputs for training seven different machine learning models, namely random forest, AdaBoost, gradient boosting, extra trees, elastic net, support vector machine, and multilayer perceptron (MLP). Our findings revealed that the integration of the atom pair fingerprint with the MLP model yielded R2 values of 0.88 and 0.90 in the validation and test sets, respectively, showcasing exceptional predictive accuracy. This advancement significantly expedites research and development processes related to the magnetic properties of organic molecules. Moreover, by employing this effective predictive method, it is expected to considerably reduce both experimental and computational expenses while maintaining high accuracy. This development represents a breakthrough in the rapid screening and prediction of properties for various compounds, offering a new and efficient pathway in this field of study.

Topological Characteristics of the Pore Network in the Tight Sandstone Using Persistent Homology

Tight sandstone reservoirs have become important areas for unconventional reservoir development, and their pore network is a key feature for identifying tight sandstone, which affects fluid migration path and reservoir development efficiency. However, the connectivity characteristics of the pore network at different scales have remained unclear owing to the numerous pores and uneven pore shape. Here, using pore size distributions from many hundreds of tight sandstone samples and subsequent topological data analysis, we construct the topological structure of the pore network in the Yanchang Formation tight sandstone of the Ordos Basin in China and visualize the topological characteristics of the pore network with distances. We show that there are three connected groups within the pore structure of the tight sandstone. The topology of the pore network resides on a trident ring manifold, suggesting that the pore network in the tight sandstone encompasses three obvious dominant connection paths. One prominent bar on the H0 dimension in the barcode indicates a two-point connection from nanoscale to microscale in the pore network. Three prominent bars with varying durations on the H1 dimension indicate the presence of three separate multipoint connections within a limited extent in the pore network. Connectivity of combined pores is good and controlled by the topological structure of the pore network. This demonstration of pore connections on a trident ring manifold provides a population-level visualization of the pore network in the tight sandstone.

Accurate Electronic and Optical Properties of Organic Doublet Radicals Using Machine Learned Range-Separated Functionals

Luminescent organic semiconducting doublet-spin radicals are unique and emergent optical materials because their fluorescent quantum yields (Φfl) are not compromised by the spin-flipping intersystem crossing (ISC) into a dark high-spin state. The multiconfigurational nature of these radicals challenges their electronic structure calculations in the framework of single-reference density functional theory (DFT) and introduces room for method improvement. In the present study, we extended our earlier development of ML-ωPBE [J. Phys. Chem. Lett., 2021, 12, 9516-9524], a range-separated hybrid (RSH) exchange-correlation (XC) functional constructed using the stacked ensemble machine learning (SEML) algorithm, from closed-shell organic semiconducting molecules to doublet-spin organic semiconducting radicals. We assessed its performance for a new test set of 64 doublet-spin radicals from five categories while placing all previously compiled 3926 closed-shell molecules in the new training set. Interestingly, ML-ωPBE agrees with the nonempirical OT-ωPBE functional regarding the prediction of the molecule-dependent range-separation parameter (ω), with a small mean absolute error (MAE) of 0.0197 a0-1, but saves the computational cost by 2.46 orders of magnitude. This result demonstrates an outstanding domain adaptation capacity of ML-ωPBE for diverse organic semiconducting species. To further assess the predictive power of ML-ωPBE in experimental observables, we also applied it to evaluate absorption and fluorescence energies (Eabs and Efl) using linear-response time-dependent DFT (TDDFT), and we compared its behavior with nine popular XC functionals. For most radicals, ML-ωPBE reproduces experimental measurements of Eabs and Efl with small MAEs of 0.299 and 0.254 eV, only marginally different from those of OT-ωPBE. Our work illustrates a successful extension of the SEML framework from closed-shell molecules to doublet-spin radicals and will open the venue for calculating optical properties for organic semiconductors using single-reference TDDFT.

Machine Learning Models for Predicting Zirconocene Properties and Barriers

Zr metallocenes have significant potential to be highly tunable polyethylene catalysts through modification of the aromatic ligand framework. Here we report the development of multiple machine learning models using a large library (>700 systems) of DFT-calculated zirconocene properties and barriers for ethylene polymerization. We show that very accurate machine learning models are possible for HOMO-LUMO gaps of precatalysts but the performance significantly depends on the machine learning algorithm and type of featurization, such as fingerprints, Coulomb matrices, smooth overlap of atomic positions, or persistence images. Surprisingly, the description of the bonding hapticity, the number of direct connections between Zr and the ligand aromatic carbons, only has a moderate influence on the performance of most models. Despite robust models for HOMO-LUMO gaps, these types of machine learning models based on structure connectivity type features perform poorly in predicting ethylene migratory insertion barrier heights. Therefore, we developed several relatively robust and accurate machine learning models for barrier heights that are based on quantum-chemical descriptors (QCDs). The quantitative accuracy of these models depends on which potential energy surface structure QCDs were harvested from. This revealed a Hammett-type principle to naturally emerge showing that QCDs from the π-coordination complexes provide much better descriptions of the transition states than other potential-energy structures. Feature importance analysis of the QCDs provides several fundamental principles that influence zirconocene catalyst reactivity.

Understanding and Designing a High-Performance Ultrafiltration Membrane Using Machine Learning

Ultrafiltration (UF) as one of the mainstream membrane-based technologies has been widely used in water and wastewater treatment. Increasing demand for clean and safe water requires the rational design of UF membranes with antifouling potential, while maintaining high water permeability and removal efficiency. This work employed a machine learning (ML) method to establish and understand the correlation of five membrane performance indices as well as three major performance-determining membrane properties with membrane fabrication conditions. The loading of additives, specifically nanomaterials (A_wt %), at loading amounts of >1.0 wt % was found to be the most significant feature affecting all of the membrane performance indices. The polymer content (P_wt %), molecular weight of the pore maker (M_Da), and pore maker content (M_wt %) also made considerable contributions to predicting membrane performance. Notably, M_Da was more important than M_wt % for predicting membrane performance. The feature analysis of ML models in terms of membrane properties (i.e., mean pore size, overall porosity, and contact angle) provided an unequivocal explanation of the effects of fabrication conditions on membrane performance. Our approach can provide practical aid in guiding the design of fit-for-purpose separation membranes through data-driven virtual experiments.

Topological deep learning based deep mutational scanning

High-throughput deep mutational scanning (DMS) experiments have significantly impacted protein engineering, drug discovery, immunology, cancer biology, and evolutionary biology by enabling the systematic understanding of protein functions. However, the mutational space associated with proteins is astronomically large, making it overwhelming for current experimental capabilities. Therefore, alternative methods for DMS are imperative. We propose a topological deep learning (TDL) paradigm to facilitate in silico DMS. We utilize a new topological data analysis (TDA) technique based on the persistent spectral theory, also known as persistent Laplacian, to capture both topological invariants and the homotopic shape evolution of data. To validate our TDL-DMS model, we use SARS-CoV-2 datasets and show excellent accuracy and reliability for binding interface mutations. This finding is significant for SARS-CoV-2 variant forecasting and designing effective antibodies and vaccines. Our proposed model is expected to have a significant impact on drug discovery, vaccine design, precision medicine, and protein engineering.

Gradient Boosted Machine Learning Model to Predict H2, CH4, and CO2 Uptake in Metal-Organic Frameworks Using Experimental Data

Predictive screening of metal-organic framework (MOF) materials for their gas uptake properties has been previously limited by using data from a range of simulated sources, meaning the final predictions are dependent on the performance of these original models. In this work, experimental gas uptake data has been used to create a Gradient Boosted Tree model for the prediction of H2, CH4, and CO2 uptake over a range of temperatures and pressures in MOF materials. The descriptors used in this database were obtained from the literature, with no computational modeling needed. This model was repeated 10 times, showing an average R2 of 0.86 and a mean absolute error (MAE) of ±2.88 wt % across the runs. This model will provide gas uptake predictions for a range of gases, temperatures, and pressures as a one-stop solution, with the data provided being based on previous experimental observations in the literature, rather than simulations, which may differ from their real-world results. The objective of this work is to create a machine learning model for the inference of gas uptake in MOFs. The basis of model development is experimental as opposed to simulated data to realize its applications by practitioners. The real-world nature of this research materializes in a focus on the application of algorithms as opposed to the detailed assessment of the algorithms.

Artificial Intelligence in Physical Sciences: Symbolic Regression Trends and Perspectives

Symbolic regression (SR) is a machine learning-based regression method based on genetic programming principles that integrates techniques and processes from heterogeneous scientific fields and is capable of providing analytical equations purely from data. This remarkable characteristic diminishes the need to incorporate prior knowledge about the investigated system. SR can spot profound and elucidate ambiguous relations that can be generalizable, applicable, explainable and span over most scientific, technological, economical, and social principles. In this review, current state of the art is documented, technical and physical characteristics of SR are presented, the available programming techniques are investigated, fields of application are explored, and future perspectives are discussed.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11831-023-09922-z.

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.

SuHAN: Substructural hierarchical attention network for molecular representation

Recently, molecular representation and property exploration, with the combination of neural network, play a critical role in the field of drug design and discovery for assisting in drug related research. However, previous research in molecular representation relies heavily on artificial extraction of features based on biological experiments which may result in a manually introduced noise of molecular information with high cost in time and money. In this paper, a novel method named Substructural Hierarchical Attention Network (SuHAN) is proposed to discover inherent characteristics of molecules for representation learning. Specifically, SuHAN is composed of the cascaded layer: atom-level layer and substructure-level layer. Molecule in the SMILES format is divided into several substructural fragments by predefined partition rules, and then they are fed into atom-level layer and substructure-level layer successively to obtain feature representation from different perspective: atomic view and substructural view. In this way, the prominent structural features that may be omitted in global extraction are excavated from a fine-grained viewpoint and fused to reconstruct representative pattern in an overall view. Experiments on biophysics and physiology datasets demonstrate that our model is competitive with a significant improvement of both accuracy and stability in performance. We confirmed that the substructural segments and progressive hierarchical networks lead to an effective molecular representation for downstream tasks. These results provide a novel perspective about reconstructing overall pattern through local prominent structure.

Path Topology in Molecular and Materials Sciences

The structures of molecules and materials determine their functions. Understanding the structure and function relationship is the holy grail of molecular and materials sciences. However, the rational design of molecules and materials with desirable functions remains a grand challenge despite decades of efforts. A major obstacle is the lack of an intrinsic mathematical characteristic that attributes to a specific function. This work introduces persistent path topology (PPT) to effectively characterize directed networks extracted from functional units, such as constitutional isomers, cis-trans isomers, chiral molecules, Jahn-Teller isomerism, and high-entropy alloy catalysts. Path homology (PH) theory is utilized to decipher the role of mirror-symmetric sublattices that hinder the formation of periodic unit cells in amorphous solids. Topological perturbation analysis (TPA) is proposed to reveal the critical target in the blood coagulation system. The proposed topological tools can be directly applied to systems biology, omics sciences, topological materials, and machine learning study of molecular and materials sciences.

Heterogeneous catalysis mediated by light, electricity and enzyme via machine learning: Paradigms, applications and prospects

Energy crisis and environmental pollution have become the bottleneck of human sustainable development. Therefore, there is an urgent need to develop new catalysts for energy production and environmental remediation. Due to the high cost caused by blind screening and limited valuable computing resources, the traditional experimental methods and theoretical calculations are difficult to meet with the requirements. In the past decades, computer science has made great progress, especially in the field of machine learning (ML). As a new research paradigm, ML greatly accelerates the theoretical calculation methods represented by first principal calculation and molecular dynamics, and establish the physical picture of heterogeneous catalytic processes for energy and environment. This review firstly summarized the general research paradigms of ML in the discovery of catalysts. Then, the latest progresses of ML in light-, electricity- and enzyme-mediated heterogeneous catalysis were reviewed from the perspective of catalytic performance, operating conditions and reaction mechanism. The general guidelines of ML for heterogeneous catalysis were proposed. Finally, the existing problems and future development trend of ML in heterogeneous catalysis mediated by light, electricity and enzyme were summarized. We highly expect that this review will facilitate the interaction between ML and heterogeneous catalysis, and illuminate the development prospect of heterogeneous catalysis.

Machine learning, artificial intelligence, and chemistry: how smart algorithms are reshaping simulation and the laboratory

Machine learning and artificial intelligence are increasingly gaining in prominence through image analysis, language processing, and automation, to name a few applications. Machine learning is also making profound changes in chemistry. From revisiting decades-old analytical techniques for the purpose of creating better calibration curves, to assisting and accelerating traditional in silico simulations, to automating entire scientific workflows, to being used as an approach to deduce underlying physics of unexplained chemical phenomena, machine learning and artificial intelligence are reshaping chemistry, accelerating scientific discovery, and yielding new insights. This review provides an overview of machine learning and artificial intelligence from a chemist’s perspective and focuses on a number of examples of the use of these approaches in computational chemistry and in the laboratory.

UV-Visible Absorption Spectra of Solvated Molecules by Quantum Chemical Machine Learning

Predicting UV-visible absorption spectra is essential to understand photochemical processes and design energy materials. Quantum chemical methods can deliver accurate calculations of UV-visible absorption spectra, but they are computationally expensive, especially for large systems or when one computes line shapes from thermal averages. Here, we present an approach to predict UV-visible absorption spectra of solvated aromatic molecules by quantum chemistry (QC) and machine learning (ML). We show that a ML model, trained on the high-level QC calculation of the excitation energy of a set of aromatic molecules, can accurately predict the line shape of the lowest-energy UV-visible absorption band of several related molecules with less than 0.1 eV deviation with respect to reference experimental spectra. Applying linear decomposition analysis on the excitation energies, we unveil that our ML models probe vertical excitations of these aromatic molecules primarily by learning the atomic environment of their phenyl rings, which align with the physical origin of the π →π* electronic transition. Our study provides an effective workflow that combines ML with quantum chemical methods to accelerate the calculations of UV-visible absorption spectra for various molecular systems.

Machine-Learning-Guided Identification of Coordination Polymer Ligands for Crystallizing Separation of Cs/Sr

Separation of Cs/Sr is one of many coordination-chemistry-centered processes in the grand scheme of spent nuclear fuel reprocessing, a critical link for a sustainable nuclear energy industry. To deploy a crystallizing Cs/Sr separation technology, we planned to systematically screen and identify candidate ligands that can efficiently and selectively bind to Sr2+ and form coordination polymers. Therefore, we mined the Cambridge Structural Database for characteristic structural information and developed a machine-learning-guided methodology for ligand evaluation. The optimized machine-learning model, correlating the molecular structures of the ligands with the predicted coordinative properties, generated a ranking list of potential compounds for Cs/Sr selective crystallization. The Sr2+ sequestration capability and selectivity over Cs+ of the promising ligands identified (squaric acid and chloranilic acid) were subsequently confirmed experimentally, with commendable performances, corroborating the artificial-intelligence-guided strategy.

0-Dimensional Persistent Homology Analysis Implementation in Resource-Scarce Embedded Systems

Persistent Homology (PH) analysis is a powerful tool for understanding many relevant topological features from a given dataset. PH allows finding clusters, noise, and relevant connections in the dataset. Therefore, it can provide a better view of the problem and a way of perceiving if a given dataset is equal to another, if a given sample is relevant, and how the samples occupy the feature space. However, PH involves reducing the problem to its simplicial complex space, which is computationally expensive and implementing PH in such Resource-Scarce Embedded Systems (RSES) is an essential add-on for them. However, due to its complexity, implementing PH in such tiny devices is considerably complicated due to the lack of memory and processing power. The following paper shows the implementation of 0-Dimensional Persistent Homology Analysis in a set of well-known RSES, using a technique that reduces the memory footprint and processing power needs of the 0-Dimensional PH algorithm. The results are positive and show that RSES can be equipped with this real-time data analysis tool.

The DP5 probability, quantification and visualisation of structural uncertainty in single molecules

Whenever a new molecule is made, a chemist will justify the proposed structure by analysing the NMR spectra. The widely-used DP4 algorithm will choose the best match from a series of possibilities, but draws no conclusions from a single candidate structure. Here we present the DP5 probability, a step-change in the quantification of molecular uncertainty: given one structure and one 13C NMR spectra, DP5 gives the probability of the structure being correct. We show the DP5 probability can rapidly differentiate between structure proposals indistinguishable by NMR to an expert chemist. We also show in a number of challenging examples the DP5 probability may prevent incorrect structures being published and later reassigned. DP5 will prove extremely valuable in fields such as discovery-driven automated chemical synthesis and drug development. Alongside the DP4-AI package, DP5 can help guide synthetic chemists when resolving the most subtle structural uncertainty. The DP5 system is available at https://github.com/Goodman-lab/DP5.

Hybridization of hierarchical clustering with persistent homology in assessing haze episodes between air quality monitoring stations

Haze has been a major issue afflicting Southeast Asian countries, including Malaysia, for the past few decades. Hierarchical agglomerative cluster analysis (HACA) is commonly used to evaluate the spatial behavior between areas in which pollutants interact. Typically, using HACA, the Euclidean distance acts as the dissimilarity measure and air quality monitoring stations are grouped according to this measure, thus revealing the most polluted areas. In this study, a framework for the hybridization of the HACA technique is proposed by considering the topological similarity (Wasserstein distance) between stations to evaluate the spatial patterns of the affected areas by haze episodes. For this, a tool in the topological data analysis (TDA), namely, persistent homology, is used to extract essential topological features hidden in the dataset. The performance of the proposed method is compared with that of traditional HACA and evaluated based on its ability to categorize areas according to the exceedance level of the particulate matter (PM₁₀). Results show that additional topological features have yielded better accuracy compared to without the case that does not consider topological features. The cluster validity indices are computed to verify the results, and the proposed method outperforms the traditional method, suggesting a practical alternative approach for assessing the similarity in air pollution behaviors based on topological characterizations.

HOMOTOPY CONTINUATION FOR THE SPECTRA OF PERSISTENT LAPLACIANS

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

Custom Tokenization Dictionary, CUSTODI: A General, Fast, and Reversible Data-Driven Representation and Regressor

Custom tokenization dictionary (CUSTODI) is introduced as a novel way for tackling the problem of molecular representations, and especially the challenge of molecular property prediction. Herein, the motivational theory and the actual representation and model are presented and shown to have performance that is in line with benchmark methodologies. The uniqueness of CUSTODI is its applicability on small training sets and the developed theory suggests its possible use for a-priori estimation of future fit quality on any given dataset, regardless of the method used for fitting.

New evidence from exceptionally "well-preserved" specimens sheds light on the structure of the ammonite brachial crown

Ammonite soft body remains are rarely preserved. One of the biggest enigmas is the morphology of the ammonite brachial crown that has, up till now, never been recovered. Recently, mysterious hook-like structures have been reported in multiple specimens of Scaphitidae, a large family of heteromorph Late Cretaceous ammonites. A previous examination of these structures revealed that they belong to the ammonites. Their nature, however, remained elusive. Here, we exploit tomographic data to study their arrangement in space in order to clarify this matter. After using topological data analyses and comparing their morphology, number, and distribution to other known cephalopod structures, in both extant and extinct taxa, we conclude that these hook-like structures represent part of the brachial crown armature. Therefore, it appears that there are at least three independent evolutionary origins of hooks: in belemnoids, oegospids, and now in ammonites. Finally, we propose for the first time a hypothetical reconstruction of an ammonite brachial crown.

Predicting biochemical and physiological effects of natural products from molecular structures using machine learning

Covering: 2016 to 2021Discovery of novel natural products has been greatly facilitated by advances in genome sequencing, genome mining and analytical techniques. As a result, the volume of data for natural products has increased over the years, which started to serve as ingredients for developing machine learning models. In the past few years, a number of machine learning models have been developed to examine various aspects of a molecule by effectively processing its molecular structure. Understanding of the biological effects of natural products can benefit from such machine learning approaches. In this context, this Highlight reviews recent studies on machine learning models developed to infer various biological effects of molecules. A particular attention is paid to molecular featurization, or computational representation of a molecular structure, which is an essential process during the development of a machine learning model. Technical challenges associated with the use of machine learning for natural products are further discussed.

Representations of energy landscapes by sublevelset persistent homology: An example with n-alkanes

Encoding the complex features of an energy landscape is a challenging task, and often, chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e., two- or three-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is much information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane C_mH_2m+2 Potential Energy Landscapes (PELs), for all m, in all homological dimensions. We further compare both the analytical configurational PELs and sampled data from molecular dynamics simulation using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.

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.