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/ The predictions of biomolecular functions and properties from biomolecular structures are of fundamental importance in computational biophysics. The structural and biological complexities of biomolecules and their interactions hinder successful predictions. Machine learning has become an important tool for such predictions. Recent advances in deep learning architectures, particularly convolutional neural network (CNN), have profoundly impacted a number of disciplines, such as image classification and voice recognition. Though CNN can be directly applied to molecular sciences by using a three-dimensional (3D) image-like brute-force representation, it is computationally intractable when applied to large biomolecules and large datasets. We propose a topological strategy to significantly reduce the structural and biological complexity of biomolecules and provide an efficient topology based CNN architecture. Element-specific persistent homology, a new algebraic topology, has been developed to cast biomolecules in a multichannel image-like representation suitable for CNN. The power of the proposed topology based neural network (TopologyNet) is further enhanced by auxiliary descriptors and a multi-task deep learning architecture. It has been demonstrated that TopologyNet framework outperforms other methods in the predictions of protein-ligand binding affinities and mutation induced protein stability changes.

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. Conventional persistent homology neglects chemical and biological information during the topological abstraction and thus has limited representational power for complex chemical and biological systems. In terms of methodological development, we introduce advanced persistent homology approaches for the characterization of small molecular structures which can capture subtle structural difference. We also introduce electrostatic persistent homology to embed physics in topological invariants. These approaches encipher physics, chemistry and biology, such as hydrogen bonds, electrostatics, van der Waals interactions, hydrophobicity and hydrophilicity, into topological fingerprints which, although cannot literally recast into physical interpretations, are ideally suitable for machine learning, particularly deep learning, rendering topological learning algorithms. In terms of applications, we construct a structure-based virtual screening model which outperforms other existing methods. This competitive model on the DUD database is derived by assessing the performance of a comprehensive collection of topological approaches proposed in this work and introduced in our earlier work, on the PDBBind database. The topological features constructed in this work can readily be applied to other biomolecular problems where the characterization of proteins or small molecules is needed.