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ABSTRACT Protein structural fluctuations, measured by Debye‐Waller factors or B‐factors, are known to be closely associated with protein flexibility and function. Theoretical approaches have also been developed to predict B‐factor values, which reflect protein flexibility. Previous models have made significant strides in analyzing B‐factors by fitting experimental data. In this study, we propose a novel approach for B‐factor prediction using differential geometry theory, based on the assumption that the intrinsic properties of proteins reside on a family of low‐dimensional manifolds embedded within the high‐dimensional space of protein structures. By analyzing the mean and Gaussian curvatures of a set of low‐dimensional manifolds defined by kernel functions, we develop effective and robust multiscale differential geometry (mDG) models. Our mDG model demonstrates a 27% increase in accuracy compared to the classical Gaussian network model (GNM) in predicting B‐factors for a dataset of 364 proteins. Additionally, by incorporating both global and local protein features, we construct a highly effective machine‐learning model for the blind prediction of B‐factors. Extensive least‐squares approximations and machine learning‐based blind predictions validate the effectiveness of the mDG modeling approach for B‐factor predictions. 

Artificial intelligence-assisted drug design is revolutionizing the pharmaceutical industry. Effective molecular features are crucial for accurate machine learning predictions, and advanced mathematics plays a key role in designing these features. Persistent homology theory, which equips topological invariants with persistence, provides valuable insights into molecular structures. The standard homology theory is based on a differential rule for the boundary operator that satisfies [Formula: see text] = 0. Our recent work has extended this rule by employing Mayer homology with generalized differentials that satisfy [Formula: see text] = 0 for [Formula: see text] 2, leading to the development of persistent Mayer homology (PMH) theory and richer topological information across various scales. In this study, we utilize PMH to create a novel multiscale topological vectorization for molecular representation, offering valuable tools for descriptive and predictive analyses in molecular data and machine learning prediction. Specifically, benchmark tests on established protein-ligand datasets, including PDBbind-v2007, PDBbind-v2013, and PDBbind-v2016, demonstrate the superior performance of our Mayer homology models in predicting protein-ligand binding affinities. 

In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein–ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area—knot deep learning—in data science. 

Abstract Pain is a significant global health issue, and the current treatment options for pain management have limitations in terms of effectiveness, side effects, and potential for addiction. There is a pressing need for improved pain treatments and the development of new drugs. Voltage-gated sodium channels, particularly Nav1.3, Nav1.7, Nav1.8, and Nav1.9, play a crucial role in neuronal excitability and are predominantly expressed in the peripheral nervous system. Targeting these channels may provide a means to treat pain while minimizing central and cardiac adverse effects. In this study, we construct protein–protein interaction (PPI) networks based on pain-related sodium channels and develop a corresponding drug–target interaction network to identify potential lead compounds for pain management. To ensure reliable machine learning predictions, we carefully select 111 inhibitor data sets from a pool of more than 1000 targets in the PPI network. We employ 3 distinct machine learning algorithms combined with advanced natural language processing (NLP)–based embeddings, specifically pretrained transformer and autoencoder representations. Through a systematic screening process, we evaluate the side effects and repurposing potential of more than 150,000 drug candidates targeting Nav1.7 and Nav1.8 sodium channels. In addition, we assess the ADMET (absorption, distribution, metabolism, excretion, and toxicity) properties of these candidates to identify leads with near-optimal characteristics. Our strategy provides an innovative platform for the pharmacological development of pain treatments, offering the potential for improved efficacy and reduced side effects.