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: This paper presents an approach to generate diverse small molecule conformations given its graph by combining a conditional flow-based model with an energy-based model. Sampling is performed in two separate stages: 1) a normalizing flow produces a distribution over interatomic distances (which is then postprocessed into cartesian coordinates), 2) sampled coordinates are refined by Langevin dynamics with gradient signal produced from an energy-based model. The models are trained separately.