Predicting energetically favorable 3-dimensional conformations of organic molecules frommolecular graph plays a fundamental role in computer-aided drug discovery research.However, effectively exploring the high-dimensional conformation space to identify (meta) stable conformers is anything but trivial.In this work, we introduce RMCF, a novel framework to generate a diverse set of low-energy molecular conformations through samplingfrom a regularized molecular conformation field.We develop a data-driven molecular segmentation algorithm to automatically partition each molecule into several structural building blocks to reduce the modeling degrees of freedom.Then, we employ a Markov Random Field to learn the joint probability distribution of fragment configurations and inter-fragment dihedral angles, which enables us to sample from different low-energy regions of a conformation space.Our model constantly outperforms state-of-the-art models for the conformation generation task on the GEOM-Drugs dataset.We attribute the success of RMCF to modeling in a regularized feature space and learning a global fragment configuration distribution for effective sampling.The proposed method could be generalized to deal with larger biomolecular systems.

Predicting energetically favorable 3-dimensional conformations of organic molecules frommolecular graph plays a fundamental role in computer-aided drug discovery research.However, effectively exploring the high-dimensional conformation space to identify (meta) stable conformers is anything but trivial.In this work, we introduce RMCF, a novel framework to generate a diverse set of low-energy molecular conformations through samplingfrom a regularized molecular conformation field.We develop a data-driven molecular segmentation algorithm to automatically partition each molecule into several structural building blocks to reduce the modeling degrees of freedom.Then, we employ a Markov Random Field to learn the joint probability distribution of fragment configurations and inter-fragment dihedral angles, which enables us to sample from different low-energy regions of a conformation space.Our model constantly outperforms state-of-the-art models for the conformation generation task on the GEOM-Drugs dataset.We attribute the success of RMCF to modeling in a regularized feature space and learning a global fragment configuration distribution for effective sampling.The proposed method could be generalized to deal with larger biomolecular systems.

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in topological tools such as persistent homology. However, on the one hand, topological tools are inherently limited to providing only coarse information about the underlying space of the data. On the other hand, more geometric approaches rely predominately on the manifold hypothesis, which asserts that the underlying space is a smooth manifold. This assumption fails for many physical models where the underlying space contains singularities. In this paper we develop a machine learning pipeline that captures fine-grain geometric information without having to rely on any smoothness assumptions. Our approach involves working within the scope of algebraic geometry and algebraic varieties instead of differential geometry and smooth manifolds. In the setting of the variety hypothesis, the learning problem becomes to find the underlying variety using sample data. We cast this learning problem into a Maximum A Posteriori optimization problem which we solve in terms of an eigenvalue computation. Having found the underlying variety, we explore the use of Gr\"obner bases and numerical methods to reveal information about its geometry. In particular, we propose a heuristic for numerically detecting points lying near the singular locus of the underlying variety.

Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view and, as an application, we describe a dimensionality reduction algorithm based on topological inference for vector bundles. The algorithm takes as input a dataset together with an initial representation in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations obtained through local linear dimensionality reduction. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.

EARTHMOVER-BASED MANIFOLD LEARNING FOR ANAL YZING MOLECULAR CONFORMA TION SPACES Nathan Zelesko Amit Moscovich Joe Kileel Amit Singer, Department of Mathematics, Brown University Program in Applied and Computational Mathematics, Princeton University Department of Mathematics, Princeton University ABSTRACT In this paper, we propose a novel approach for manifold learning that combines the Earthmover's distance (EMD) with the diffusion maps method for dimensionality reduction. We demonstrate the potential benefits of this approach for learning shape spaces of proteins and other flexible macromolecules using a simulated dataset of 3-D density maps that mimic the nonuniform rotary motion of A TP synthase. Our results show that EMD-based diffusion maps require far fewer samples to recover the intrinsic geometry than the standard diffusion maps algorithm that is based on the Euclidean distance. To reduce the computational burden of calculating the EMD for all volume pairs, we employ a wavelet-based approximation to the EMD which reduces the computation of the pairwise EMDs to a computation of pairwise weighted-null 1 distances between wavelet coefficient vectors. Index T erms -- Shape space, dimensionality reduction, Wasserstein metric, diffusion maps, Laplacian eigenmaps, cryo-electron microscopy 1. INTRODUCTION Proteins and other macromolecules are elastic structures that may deform in various ways.

Docking is an important tool in computational drug discovery that aims to predict the binding pose of a ligand to a target protein through a combination of pose scoring and optimization. A scoring function that is differentiable with respect to atom positions can be used for both scoring and gradient-based optimization of poses for docking. Using a differentiable grid-based atomic representation as input, we demonstrate that a scoring function learned by training a convolutional neural network (CNN) to identify binding poses can also be applied to pose optimization. We also show that an iteratively-trained CNN that includes poses optimized by the first CNN in its training set performs even better at optimizing randomly initialized poses than either the first CNN scoring function or AutoDock Vina.

Rosetta is one of the leading algorithms for protein structure prediction today. It is a Monte Carlo energy minimization method requiring many random restarts to find structures with low energy. In this paper we present a resampling technique for structure prediction of small alpha/beta proteins using Rosetta. From an initial round of Rosetta sampling, we learn properties of the energy landscape that guide a subsequent round of sampling toward lower-energy structures. Rather than attempt to fit the full energy landscape, we use feature selection methods--both L1-regularized linear regression and decision trees--to identify structural features that give rise to low energy. We then enrich these structural features in the second sampling round. Results are presented across a benchmark set of nine small alpha/beta proteins demonstrating that our methods seldom impair, and frequently improve, Rosetta's performance.

Rosetta is one of the leading algorithms for protein structure prediction today. It is a Monte Carlo energy minimization method requiring many random restarts to find structures with low energy. In this paper we present a resampling technique for structure prediction of small alpha/beta proteins using Rosetta. From an initial round of Rosetta sampling, we learn properties of the energy landscape that guide a subsequent round of sampling toward lower-energy structures. Rather than attempt to fit the full energy landscape, we use feature selection methods--both L1-regularized linear regression and decision trees--to identify structural features that give rise to low energy. We then enrich these structural features in the second sampling round. Results are presented across a benchmark set of nine small alpha/beta proteins demonstrating that our methods seldom impair, and frequently improve, Rosetta's performance.