Learning Neural Generative Dynamics for Molecular Conformation Generation

1 INTRODUCTION . Recently , we have witnessed the success of graph-based representations for molecular modeling in a variety of tasks such as property prediction ( Gilmer et al. , 2017 ) and molecule generation ( You et al. , 2018 ; Shi et al. , 2020 ) . However , a more natural and intrinsic representation of a molecule is its 3D structure , commonly known as the molecular geometry or conformation , which represents each atom by its 3D coordinate . The conformation of a molecule determines its biological and physical properties such as charge distribution , steric constraints , as well as interactions with other molecules . Furthermore , large molecules tend to comprise a number of rotatable bonds , which may induce flexible conformation changes and a large number of feasible conformations in nature . Generating valid and stable conformations of a given molecule remains very challenging . Experimentally , such structures are determined by expensive and time-consuming crystallography . Computational approaches based on Markov chain Monte Carlo ( MCMC ) or molecular dynamics ( MD ) ( De Vivo et al. , 2016 ) are computationally expensive , especially for large molecules ( Ballard et al. , 2015 ) . Machine learning methods have recently shown great potential for molecular conformation generation by training on a large collection of data to model the probability distribution of potential conformations R based on the molecular graph G , i.e. , p ( R|G ) . For example , Mansimov et al . ∗Equal contribution . Work was done during Shitong ’ s internship at Mila . 1Code is available at https : //github.com/DeepGraphLearning/CGCF-ConfGen . ( 2019 ) proposed a Conditional Variational Graph Autoencoders ( CVGAE ) for molecular conformation generation . A graph neural network ( Gilmer et al. , 2017 ) is first applied to the molecular graph to get the atom representations , based on which 3D coordinates are further generated . One limitation of such an approach is that by directly generating the 3D coordinates of atoms it fails to model the rotational and translational invariance of molecular conformations . To address this issue , instead of generating the 3D coordinates directly , Simm & Hernández-Lobato ( 2020 ) recently proposed to first model the molecule ’ s distance geometry ( i.e. , the distances between atoms ) —which are rotationally and translationally invariant—and then generate the molecular conformation based on the distance geometry through a post-processing algorithm ( Liberti et al. , 2014 ) . Similar to Mansimov et al . ( 2019 ) , a few layers of graph neural networks are applied to the molecular graph to learn the representations of different edges , which are further used to generate the distances of different edges independently . This approach is capable of more often generating valid molecular conformations . Although these new approaches have made tremendous progress , the problem remains very challenging and far from solved . First , each molecule may have multiple stable conformations around a number of states which are thermodynamically stable . In other words , the distribution p ( R|G ) is very complex and multi-modal . Models with high capacity are required to model such complex distributions . Second , existing approaches usually apply a few layers of graph neural networks to learn the representations of nodes ( or edges ) and then generate the 3D coordinates ( or distances ) based on their representations independently . Such approaches are necessarily limited to capturing a single mode of p ( R|G ) ( since the coordinates or distances are sampled independently ) and are incapable of modeling multimodal joint distributions and the form of the graph neural net computation makes it difficult to capture long-range dependencies between atoms , especially in large molecules . Inspired by the recent progress with deep generative models , this paper proposes a novel and principled probabilistic framework for molecular geometry generation , which addresses the above two limitations . Our framework combines the advantages of normalizing flows ( Dinh et al. , 2014 ) and energy-based approaches ( LeCun et al. , 2006 ) , which have a strong model capacity for modeling complex distributions , are flexible to model long-range dependency between atoms , and enjoy efficient sampling and training procedures . Similar to the work of Simm & Hernández-Lobato ( 2020 ) , we also first learn the distribution of distances d given the graph G , i.e. , p ( d|G ) , and define another distribution of conformations R given the distances d , i.e. , p ( R|d , G ) . Specifically , we propose a novel Conditional Graph Continuous Flow ( CGCF ) for distance geometry ( d ) generation conditioned on the molecular graph G. Given a molecular graph G , CGCF defines an invertible mapping between a base distribution ( e.g. , a multivariate normal distribution ) and the molecular distance geometry , using a virtually infinite number of graph transformation layers on atoms represented by a Neural Ordinary Differential Equations architecture ( Chen et al. , 2018 ) . Such an approach enjoys very high flexibility to model complex distributions of distance geometry . Once the molecular distance geometry d is generated , we further generate the 3D coordinates R by searching from the probability p ( R|d , G ) . Though the CGCF has a high capacity for modeling complex distributions , the distances of different edges are still independently updated in the transformations , which limits its capacity for modeling long-range dependency between atoms in the sampling process . Therefore , we further propose another unnormalized probability function , i.e. , an energy-based model ( EBM ) ( Hinton & Salakhutdinov , 2006 ; LeCun et al. , 2006 ; Ngiam et al. , 2011 ) , which acts as a tilting term of the flow-based distribution and directly models the joint distribution of R. Specifically , the EBM trains an energy function E ( R , G ) , which is approximated by a neural network . The flow- and energy-based models are combined in a novel way for joint training and mutual enhancement . First , energy-based methods are usually difficult to train due to the slow sampling process . In addition , the distribution of conformations is usually highly multi-modal , and the sampling procedures based on Gibbs sampling or Langevin Dynamics ( Bengio et al. , 2013a ; b ) tend to get trapped around modes , making it difficult to mix between different modes ( Bengio et al. , 2013a ) . Here we use the flow-based model as a proposal distribution for the energy model , which is capable to generate diverse samples for training energy models . Second , the flow-based model lacks the capacity to explicitly model the long-range dependencies between atoms , which we find can however be effectively modeled by an energy function E ( R , G ) . Our sampling process can be therefore viewed as a two-stage dynamic system , where we first take the flow-based model to quickly synthesize realistic conformations and then used the learned energy E ( R , G ) to refine the generated conformations through Langevin Dynamics . We conduct comprehensive experiments on several recently proposed benchmarks , including GEOM-QM9 , GEOM-Drugs ( Axelrod & Gomez-Bombarelli , 2020 ) and ISO17 ( Simm & Hernández-Lobato , 2020 ) . Numerical evaluations show that our proposed framework consistently outperforms the previous state-of-the-art ( GraphDG ) on both conformation generation and distance modeling tasks , with a clear margin . 2 PROBLEM DEFINITION AND PRELIMINARIES . 2.1 PROBLEM DEFINITION . Notations . Following existing work ( Simm & Hernández-Lobato , 2020 ) , each molecule is represented as an undirected graph G = 〈V , E〉 , where V is the set of nodes representing atoms and E is the set of edges representing inter-atomic bonds . Each node v in V is labeled with atomic properties such as element type . The edge in E connecting u and v is denoted as euv , and is labeled with its bond type . We also follow the previous work ( Simm & Hernández-Lobato , 2020 ) to expand the molecular graph with auxiliary bonds , which is elaborated in Appendix B . For the molecular 3D representation , each atom in V is assigned with a 3D position vector r ∈ R3 . We denote duv = ‖ru − rv‖2 as the Euclidean distance between the uth and vth atom . Therefore , we can represent all the positions { rv } v∈V as a matrix R ∈ R|V|×3 and all the distances between connected nodes { duv } euv∈E as a vector d ∈ R|E| . Problem Definition . The problem of molecular conformation generation is defined as a conditional generation process . More specifically , our goal is to model the conditional distribution of atomic positions R given the molecular graph G , i.e. , p ( R|G ) . 2.2 PRELIMINARIES . Continuous Normalizing Flow . A normalizing flow ( Dinh et al. , 2014 ; Rezende & Mohamed , 2015 ) defines a series of invertible deterministic transformations from an initial known distribution p ( z ) to a more complicated one p ( x ) . Recently , normalizing flows have been generalized from discrete number of layers to continuous ( Chen et al. , 2018 ; Grathwohl et al. , 2018 ) by defining the transformation fθ as a continuous-time dynamic ∂z ( t ) ∂t = fθ ( z ( t ) , t ) . Formally , with the latent variable z ( t0 ) ∼ p ( z ) at the start time , the continuous normalizing flow ( CNF ) defines the transformation x = z ( t0 ) + ∫ t1 t0 fθ ( z ( t ) , t ) dt . Then the exact density for pθ ( x ) can be computed by : log pθ ( x ) = log p ( z ( t0 ) ) − ∫ t1 t0 Tr ( ∂fθ ∂z ( t ) ) dt ( 1 ) where z ( t0 ) can be obtained by inverting the continuous dynamic z ( t0 ) = x + ∫ t0 t1 fθ ( z ( t ) , t ) dt . A black-box ordinary differential equation ( ODE ) solver can be applied to estimate the outputs and inputs gradients and optimize the CNF model ( Chen et al. , 2018 ; Grathwohl et al. , 2018 ) . Energy-based Models . Energy-based models ( EBMs ) ( Dayan et al. , 1995 ; Hinton & Salakhutdinov , 2006 ; LeCun et al. , 2006 ) use a scalar parametric energy function Eφ ( x ) to fit the data distribution . Formally , the energy function induces a density function with the Boltzmann distribution pφ ( x ) = exp ( −Eφ ( x ) ) /Z ( φ ) , where Z = ∫ exp ( −Eφ ( x ) ) dx denotes the partition function . EBM can be learned with Noise contrastive estimation ( NCE ) ( Gutmann & Hyvärinen , 2010 ) by treating the normalizing constant as a free parameter . Given the training examples from both the dataset and a noise distribution q ( x ) , φ can be estimated by maximizing the following objective function : J ( φ ) = Epdata [ log pφ ( x ) pφ ( x ) + q ( x ) ] + Eq [ log q ( x ) pφ ( x ) + q ( x ) ] , ( 2 ) which turns the estimation of EBM into a discriminative learning problem . Sampling from Eφ can be done with a variety of methods such as Markov chain Monte Carlo ( MCMC ) or Gibbs sampling ( Hinton & Salakhutdinov , 2006 ) , possibly accelerated using Langevin dynamics ( Du & Mordatch , 2019 ; Song et al. , 2020 ) , which leverages the gradient of the EBM to conduct sampling : xk = xk−1 − 2 ∇xEφ ( xk−1 ) + √ ω , ω ∼ N ( 0 , I ) , ( 3 ) where refers to the step size . x0 are the samples drawn from a random initial distribution and we take the xK withK Langevin dynamics steps as the generated samples of the stationary distribution . 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Summary: The authors of this manuscript proposed a generative dynamics system for the modelling and generation of 3D conformations of molecules. Specifically, there are three components: (1) conditional graph continuous flow (CGCF) to transform random noise to distances,  (2)a closed-form distribution p(R|d, G), and (3) an energy-based tilting model (ETM) to capture long-range interactions and correct the position matrix distribution. The proposed framework was compared with two deep learning methods for conformation generations -- CVGAE & GraphDG, as well as the computational chemistry tool RDKit on GEOM-QM9, GEOM-Drugs, and ISO17 data sets. Comparisons in terms of COV and MAT scores show that the proposed method (particularly the one enhanced with ETM) can outperform baselines. Further comparisons of distances densities show that CGCF (but without ETM) worked best over baselines.