Protein docking refinement by convex underestimation in the low-dimensional subspace of encounter complexes

We propose a novel stochastic global optimization algorithm with applications to the refinement stage of protein docking prediction methods. Our approach can process conformations sampled from multiple clusters, each roughly corresponding to a different binding energy funnel. These clusters are obtained using a density-based clustering method. In each cluster, we identify a smooth “permissive” subspace which avoids high-energy barriers and then underestimate the binding energy function using general convex polynomials in this subspace. We use the underestimator to bias sampling towards its global minimum. Sampling and subspace underestimation are repeated several times and the conformations sampled at the last iteration form a refined ensemble. We report computational results on a comprehensive benchmark of 224 protein complexes, establishing that our refined ensemble significantly improves the quality of the conformations of the original set given to the algorithm. We also devise a method to enhance the ensemble from which near-native models are selected.Published versio

Manifold optimization methods for macromolecular docking

Thesis (Ph.D.)--Boston UniversityThis thesis develops efficient algorithms for local optimization problems encountered in predictive docking of biological macromolecules. Predictive docking, defined as computationally obtaining a model of the bound complex from the coordinates of the two component molecules, is one of the fundamental and challenging problems in computational structural biology. Docking methods generally search for the minima of an energy or scoring function that estimates the binding free energy or, more frequently, the interaction energy, of the two molecules. These energy functions generally have large numbers of local minima, resulting in extremely rugged energy landscapes. Therefore, independently of the algorithm used for sampling the conformational space, virtually all docking algorithms include some type of local continuous minimization of the energy function. Most state-of-the-art algorithms allow for the free movement of all atoms of the two molecules and rely on the minimization of the energy function to enforce structural constraints of the molecules. In contrast this thesis exploits the partial or complete rigidity of the molecules when defining the conformational space. As a result, the local optimization problems are formulated as optimization problems on appropriately defined manifolds. In the case of rigid docking, a novel manifold representation of rigid motions of a body is introduced that resolves many of the optimization difficulties associated with the commonly used manifold for this purposed , the so-called Special Euclidean group, SE(3). These difficulties arise from a coupling that SE(3) introduces between the rotational and translational move of the body. The new representation decouples these moves and results in a more appropriate and flexible optimization algorithm. Experimental results show that the proposed algorithm is an order of magnitude more efficient than the current state-of-the-art algorithms. The proposed manifold optimization approach is then extended to the case of flexible docking. The novel manifold representation of rigid motions is combined with the so-called internal coordinate representation of flexible moves to define a new manifold to which the original manifold optimization algorithm can be directly extended. Computational results show that the resulting optimization algorithm is substantially more efficient than energy minimization using a traditional all-atom optimization algorithm while producing solutions of comparable quality. It is shown that the application of the proposed local optimization algorithm as one of the components of a multi-stage refinement protocol for protein-protein docking contributes significantly to the refinement stage by helping to move the distribution of docking decoys closer to the corresponding bound structures. Finally, it is shown that the approach of the thesis can be substantially generalized to address the problem of minimization of a cost function that depends on the location and poses of one or more rigid bodies, or bodies that consist of rigid parts hinged together. This is a formulation used in a number of engineering applications other than molecular docking

LightDock: a new multi-scale approach to protein–protein docking

Computational prediction of protein–protein complex structure by docking can provide structural and mechanistic insights for protein interactions of biomedical interest. However, current methods struggle with difficult cases, such as those involving flexible proteins, low-affinity complexes or transient interactions. A major challenge is how to efficiently sample the structural and energetic landscape of the association at different resolution levels, given that each scoring function is often highly coupled to a specific type of search method. Thus, new methodologies capable of accommodating multi-scale conformational flexibility and scoring are strongly needed. We describe here a new multi-scale protein–protein docking methodology, LightDock, capable of accommodating conformational flexibility and a variety of scoring functions at different resolution levels. Implicit use of normal modes during the search and atomic/coarse-grained combined scoring functions yielded improved predictive results with respect to state-of-the-art rigid-body docking, especially in flexible cases.B.J-G was supported by a FPI fellowship from the Spanish Ministry of Economy and Competitiveness. This work was supported by I+D+I Research Project grants BIO2013-48213-R and BIO2016-79930-R from the Spanish Ministry of Economy and Competitiveness. This work is partially supported by the European Union H2020 program through HiPEAC (GA 687698), by the Spanish Government through Programa Severo Ochoa (SEV-2015-0493), by the Spanish Ministry of Science and Technology (TIN2015-65316-P) and the Departament d’Innovació, Universitats i Empresa de la Generalitat de Catalunya, under project MPEXPAR: Models de Programaciói Entorns d’Execució Paral·lels (2014-SGR-1051).Peer ReviewedPostprint (author's final draft

Optimization and machine learning methods for Computational Protein Docking

Computational Protein Docking (CPD) is defined as determining the stable complex of docked proteins given information about two individual partners, called receptor and ligand. The problem is often formulated as an energy/score minimization where the decision variables are the 6 rigid body transformation variables for the ligand in addition to more variables corresponding to flexibilities in the protein structures. The scoring functions used in CPD are highly nonlinear and nonconvex with a very large number of local minima, making the optimization problem particularly challenging. Consequently, most docking procedures employ a multistage strategy of (i) Global Sampling using a coarse scoring function to identify promising areas followed by (ii) a Refinement stage using more accurate scoring functions and possibly allowing more degrees of freedom. In the first part of this work, the problem of local optimization in the refinement stage is addressed. The goal of local optimization is to remove steric clashes between protein partners and obtain more realistic score values. The problem is formulated as optimization on the space of rigid motions of the ligand. Employing a recently introduced representation of the space of rigid motions as a manifold, a new Riemannian metric is introduced that is closely related to the Root Mean Square Deviation (RMSD) distance measure widely used in Protein Docking. It is argued that the new metric puts rotational and translational variables on equal footing as far local changes of RMSD is concerned. The implications and modifications for gradient-based local optimization algorithms are discussed. In the second part, a new methodology for resampling and refinement of ligand conformations is introduced. The algorithm is a refinement method where the inputs to the algorithm are ensembles of ligand conformations and the goal is to generate new ensembles of refined conformations, closer to the native complex. The algorithm builds upon a previous work and introduces multiple new innovations: Clustering the input conformations, performing dimensionality reduction using Principle Component Analysis (PCA), underestimating the scoring function and resampling and refinement of new conformations. The performance of the algorithm on a comprehensive benchmark of protein complexes is reported. The third part of this work focuses on using machine learning framework for addressing two specific problems in Protein Docking: (i) Constructing a machine learning model in order to predict whether a given receptor and ligand pair interact. This is of significant importance for constructing the so-called protein interaction networks, an critical step in the Drug Discovery process. The success of the algorithm is verified on a benchmark for discrimination between Biological and Crystallographic Dimers. (ii) A ranking scheme for output predictions of a protein docking server is devised. The machine learning model employs the features of the docking server predictions to produce a ranked list with the top ranked predictions having higher probability of being close to the native solution. Two state-of-the-art approaches to the ranking problem are presented and compared in detail and the implications of using the superior approach for a structural docking server is discussed

Learning Harmonic Molecular Representations on Riemannian Manifold

Molecular representation learning plays a crucial role in AI-assisted drug discovery research. Encoding 3D molecular structures through Euclidean neural networks has become the prevailing method in the geometric deep learning community. However, the equivariance constraints and message passing in Euclidean space may limit the network expressive power. In this work, we propose a Harmonic Molecular Representation learning (HMR) framework, which represents a molecule using the Laplace-Beltrami eigenfunctions of its molecular surface. HMR offers a multi-resolution representation of molecular geometric and chemical features on 2D Riemannian manifold. We also introduce a harmonic message passing method to realize efficient spectral message passing over the surface manifold for better molecular encoding. Our proposed method shows comparable predictive power to current models in small molecule property prediction, and outperforms the state-of-the-art deep learning models for ligand-binding protein pocket classification and the rigid protein docking challenge, demonstrating its versatility in molecular representation learning.Comment: 25 pages including Appendi

DiffDock-PP: Rigid Protein-Protein Docking with Diffusion Models

Understanding how proteins structurally interact is crucial to modern biology, with applications in drug discovery and protein design. Recent machine learning methods have formulated protein-small molecule docking as a generative problem with significant performance boosts over both traditional and deep learning baselines. In this work, we propose a similar approach for rigid protein-protein docking: DiffDock-PP is a diffusion generative model that learns to translate and rotate unbound protein structures into their bound conformations. We achieve state-of-the-art performance on DIPS with a median C-RMSD of 4.85, outperforming all considered baselines. Additionally, DiffDock-PP is faster than all search-based methods and generates reliable confidence estimates for its predictions. Our code is publicly available at $\texttt{https://github.com/ketatam/DiffDock-PP}$Comment: ICLR Machine Learning for Drug Discovery (MLDD) Workshop 202

Encounter complexes and dimensionality reduction in protein-protein association

An outstanding challenge has been to understand the mechanism whereby proteins associate. We report here the results of exhaustively sampling the conformational space in protein–protein association using a physics-based energy function. The agreement between experimental intermolecular paramagnetic relaxation enhancement (PRE) data and the PRE profiles calculated from the docked structures shows that the method captures both specific and non-specific encounter complexes. To explore the energy landscape in the vicinity of the native structure, the nonlinear manifold describing the relative orientation of two solid bodies is projected onto a Euclidean space in which the shape of low energy regions is studied by principal component analysis. Results show that the energy surface is canyon-like, with a smooth funnel within a two dimensional subspace capturing over 75% of the total motion. Thus, proteins tend to associate along preferred pathways, similar to sliding of a protein along DNA in the process of protein-DNA recognition

Quantified Uncertainty of Flexible Protein-Protein Docking Algorithms

The strength or weakness of an algorithm is ultimately governed by the confidence of its result. When the domain of the problem is large (e.g. traversal of a high-dimensional space), an exact solution often cannot be obtained, so approximations must be made. These approximations often lead to a reported quantity of interest (QOI) which varies between runs, decreasing the confidence of any single run. When the algorithm further computes this QOI based on uncertain or noisy data, the variability (or lack of confidence) of the QOI increases. Unbounded, these two sources of uncertainty (algorithmic approximations and uncertainty in input data) can result in a reported statistic that has low correlation with ground truth. In molecular biology applications, this is especially applicable, as the search space is generally large and observations are often noisy. This research applies uncertainty quantification techniques to the difficult protein-protein docking problem, where uncertainties arise from the explicit conversion from continuous to discrete space for protein representation (introducing some uncertainty in the input data), as well as discrete sampling of the conformations. It describes the variability that exists in existing software, and then provides a method for computing probabilistic certificates in the form of Chernoff-like bounds. Finally, this paper leverages these probabilistic certificates to accurately bound the uncertainty in docking from two docking algorithms, providing a QOI that is both robust and statistically meaningful