Single-cell RNA sequencing (scRNA-seq) enables dissecting cellular heterogeneity in tissues, resulting in numerous biological discoveries. Various computational methods have been devised to delineate cell types by clustering scRNA-seq data, where clusters are often annotated using prior knowledge of marker genes. In addition to identifying pure cell types, several methods have been developed to identify cells undergoing state transitions, which often rely on prior clustering results. The present computational approaches predominantly investigate the local and first-order structures of scRNA-seq data using graph representations, while scRNA-seq data frequently display complex high-dimensional structures. Here, we introduce scGeom, a tool that exploits the multiscale and multidimensional structures in scRNA-seq data by analyzing the geometry and topology through curvature and persistent homology of both cell and gene networks. We demonstrate the utility of these structural features to reflect biological properties and functions in several applications, where we show that curvatures and topological signatures of cell and gene networks can help indicate transition cells and the differentiation potential of cells. We also illustrate that structural characteristics can improve the classification of cell types.

During early mammalian embryo development, a small number of cells make robust fate decisions at particular spatial locations in a tight time window to form inner cell mass (ICM), and later epiblast (Epi) and primitive endoderm (PE). While recent single-cell transcriptomics data allows scrutinization of heterogeneity of individual cells, consistent spatial and temporal mechanisms the early embryo utilize to robustly form the Epi/PE layers from ICM remain elusive. Here we build a multiscale three-dimensional model for mammalian embryo to recapitulate the observed patterning process from zygote to late blastocyst. By integrating the spatiotemporal information reconstructed from multiple single-cell transcriptomic datasets, the data-informed modeling analysis suggests two major processes critical to the formation of Epi/PE layers: a selective cell-cell adhesion mechanism (via EphA4/EphrinB2) for fate-location coordination and a temporal attenuation mechanism of cell signaling (via Fgf). Spatial imaging data and distinct subsets of single-cell gene expression data are then used to validate the predictions. Together, our study provides a multiscale framework that incorporates single-cell gene expression datasets to analyze gene regulations, cell-cell communications, and physical interactions among cells in complex geometries at single-cell resolution, with direct application to late-stage development of embryogenesis.

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/.

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.

MOTIVATION: Site directed mutagenesis is widely used to understand the structure and function of biomolecules. Computational prediction of mutation impacts on protein stability offers a fast, economical and potentially accurate alternative to laboratory mutagenesis. Most existing methods rely on geometric descriptions, this work introduces a topology based approach to provide an entirely new representation of mutation induced protein stability changes that could not be obtained from conventional techniques. RESULTS: Topology based mutation predictor (T-MP) is introduced to dramatically reduce the geometric complexity and number of degrees of freedom of proteins, while element specific persistent homology is proposed to retain essential biological information. The present approach is found to outperform other existing methods in the predictions of globular protein stability changes upon mutation. A Pearson correlation coefficient of 0.82 with an RMSE of 0.92 kcal/mol is obtained on a test set of 350 mutation samples. For the prediction of membrane protein stability changes upon mutation, the proposed topological approach has a 84% higher Pearson correlation coefficient than the current state-of-the-art empirical methods, achieving a Pearson correlation of 0.57 and an RMSE of 1.09 kcal/mol in a 5-fold cross validation on a set of 223 membrane protein mutation samples. AVAILABILITY AND IMPLEMENTATION: http://weilab.math.msu.edu/TML/TML-MP/. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

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/.