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A Generative Model for Molecular Distance Geometry

Gregor N. C. Simm, José Miguel Hernández-Lobato

arXiv:1909.11459v4stat.MLcs.LG

TL;DR

Sampling molecular equilibrium states with methods such as MCMC can be computationally expensive. This paper introduces a graph-based generative model using a CVAE and Euclidean distance geometry, reports closer-to-ground-truth conformations than other methods on a new benchmark, and uses the model in importance sampling to estimate molecular properties.

  • Problem

    MCMC and molecular dynamics can sample equilibrium states accurately but become computationally expensive for larger molecules.

    Source: §1 · p. 1

  • Method

    A conditional variational autoencoder and Euclidean distance geometry generate one-shot molecular-conformation samples from graph representations.

    Source: §1 · p. 2, §2.1 · p. 2, §2.2 · p. 3

  • Results

    On the new benchmark, conformations generated by GRAPHDG are closer to ground truth than those generated by other methods.

    Source: §7 · p. 8

  • Takeaways & Limitations

    GRAPHDG can be used as an importance-sampling proposal to estimate molecular properties, including orbital energies and dipole moments predicted well in the reported study.

    Source: §1 · p. 2, §7 · p. 8

    The CONF17 benchmark covers only a small portion of chemical space.

    Source: §6 · p. 8

Abstract

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Great computational effort is invested in generating equilibrium states for molecular systems using, for example, Markov chain Monte Carlo. We present a probabilistic model that generates statistically independent samples for molecules from their graph representations. Our model learns a low-dimensional manifold that preserves the geometry of local atomic neighborhoods through a principled learning representation that is based on Euclidean distance geometry. In a new benchmark for molecular conformation generation, we show experimentally that our generative model achieves state-of-the-art accuracy. Finally, we show how to use our model as a proposal distribution in an importance sampling scheme to compute molecular properties.

1. Introduction

The introduction motivates generating molecular conformations from graphs as an alternative to costly equilibrium sampling. It introduces a probabilistic graph-based model, a benchmark and evaluation approach, and importance-sampling use for estimating properties.

1. Introduction

  • MCMC and molecular dynamics can sample equilibrium states accurately, but become computationally expensive for larger molecules.Conformation probabilities are governed by the Boltzmann distribution under a wide range of conditions.Source: §1 · p. 1
  • The new publicly available benchmark is presented as the first of its kind for molecular conformation generation.The authors describe it as challenging.Source: §1 · p. 2
  • A conditional variational autoencoder with Euclidean distance geometry provides state-of-the-art one-shot sampling of conformations for unseen molecules, independent of their size and shape.The model is based on molecular graph representations.Source: §1 · p. 2
  • The authors propose a rigorous experimental approach for comparing conformation-generation accuracy using mean maximum deviation distance.Source: §1 · p. 2
  • The model can serve as a proposal distribution in an importance-sampling scheme for estimating molecular properties.Source: §1 · p. 2

2. Method

The method learns molecular conformational distributions from graph-based distances, then maps generated distances to Cartesian coordinates. It also uses the model as a proposal distribution for estimating molecular properties through importance sampling.

2. Method

  • The model generates distances from a molecular graph and maps them to coordinates, with its samples used in an importance-sampling scheme for molecular-property estimates.The method describes these stages as graph-conditioned distance generation, Euclidean distance geometry, and property estimation using a quantum-chemical method.Source: §2 · p. 2, §2.4 · p. 4

2.1. Extended Molecular Graphs and Distance Geometry

  • The extended graph adds second-neighbor edges to constrain angles and selected third-neighbor edges to constrain dihedrals; distances are invariant to rotation and translation.Bond edges alone do not suffice to describe a conformation, and auxiliary edges can represent longer-range interactions.Source: §2.1 · p. 2, Figure 2 · p. 3

2.2. Generative Model

  • A conditional variational autoencoder models distances given a graph, using message-passing neural networks and a latent space scaled to molecule size but smaller than 3N_v.The encoder maps graph and distance data into latent variables, and the decoder maps latent variables back to distances; training maximizes the ELBO.Source: §2.2 · pp. 2–3

2.3. Conformation Generation through Euclidean Distance Geometry

  • Euclidean distance geometry converts CVAE-predicted distance intervals, set to each distance’s mean plus or minus one standard deviation, into Cartesian coordinates.The coordinates satisfy the predicted bounds and provide the format required by quantum-chemical calculations.Source: §2.3 · p. 4

2.4. Calculation of Molecular Properties

  • Chirality constraints are fixed by the molecular graph and are not modeled by the method.Source: §2.4 · p. 4
  • For property estimation, the CVAE serves as an importance-sampling proposal, while conformational energies provide unnormalized Boltzmann probabilities for weighting samples.The estimate uses a finite sample set and evaluates properties with a quantum-chemical method.Source: §2.4 · p. 4

3. Related Works

Prior methods either sample by costly trajectory-based approaches, target idealized low-energy conformations, or are specialized to protein structures. The authors position their approach as a fast, distance-based alternative for general molecular graphs.

Trajectory-based methods

  • MCMC and molecular dynamics can sample equilibrium states accurately but become computationally expensive, especially for large molecules or quantum-mechanical Hamiltonians.The cited approaches include Shim & MacKerell (2011), Ballard et al. (2015), and De Vivo et al. (2016).Source: §3 · p. 5

Distance geometry

  • EDG fixes distance bounds from idealized bonds, angles, and torsions to seek low-energy conformations, not unbiased samples at a specified temperature.The passage cites Havel (2002), Blaney & Dixon (2007), Lagorce et al. (2009), and Riniker & Landrum (2015).Source: §3 · p. 5

Protein-focused models

  • Protein-focused generative models learn protein-specific coordinate transformations that the authors say cannot transfer to other molecules.The passage discusses Noé et al. (2019) and Lemke & Peter (2019).Source: §3 · p. 5

Protein structure prediction

  • Sequence-based protein-structure methods are described as unsuitable for general molecules because molecular graphs are branched and often contain cycles.The passage cites AlQuraishi (2019), Ingraham et al. (2019), and Senior et al. (2020).Source: §3 · p. 5

Molecular-graph methods

  • Mansimov et al. (2019) predict a most-likely conformation in Cartesian coordinates, with reported structural differences that required force-field refinement.The authors contrast this with their proposed fast alternative to MCMC or MD, using pairwise distances and not restricting the approach to a particular molecular structure.Source: §3 · p. 5

4. The CONF17 Benchmark

CONF17 is introduced as a benchmark for molecular conformation sampling, built from ISO17 conformations and split by molecular graph. Its molecules have complex structures and context-dependent distance distributions.

Benchmark construction

  • CONF17 contains 430,692 valid graph–conformation pairs and 197 unique molecular graphs, with 176 graphs for training and 30 for testing.The splits ensure no molecular graph appears in both training and test sets.Source: §4 · p. 5
  • The benchmark uses C7H10O2 molecules from ISO17, whose conformations were generated by ab initio molecular dynamics at 500 K.ISO17 conformations were drawn from the QM9 dataset.Source: §4 · p. 5

Benchmark characteristics

  • The benchmark’s distances are multimodal and highly context-dependent, while its strained, complex 3D structures make it challenging.The passage attributes the challenge to structural complexity rather than the number of degrees of freedom; Figure 4 groups distances by edge type and element.Source: §4 · p. 5

5. Experiments

The experiments compare GRAPHDG with RDKIT and DL4CHEM on conformation-distance distributions, generated structures, and molecular properties. Results favor GRAPHDG for distance distributions and structural similarity, while property accuracy remains limited for electronic energy.

5.1. Distributions Over Distances

  • GRAPHDG samples are significantly closer to the ground-truth distance distributions than RDKIT and DL4CHEM; RDKIT is slightly worse, while DL4CHEM struggles with molecular complexity and limited training graphs.The comparison uses median MMDs and mean rankings over marginal, pairwise, and joint C–O distance distributions across three train-test splits.Source: §5.1 · p. 6, Table 1 · p. 7
  • For the illustrated test molecule, GRAPHDG most accurately predicts the C–O distance means and variances in most cases; RDKIT underestimates variances and DL4CHEM often misses the mean.Figure 5 plots ground-truth and predicted marginal distance distributions.Source: §5.1 · p. 6, Figure 5 · p. 7

5.2. Generation of Conformations

  • For 83% of molecular graphs, EDG successfully converted GRAPHDG distances into conformations, 7 percentage points above RDKIT; 99.9% of distance sets satisfied all triangle inequalities.Generating 50 conformations per graph took each method hundreds of milliseconds per molecule on average, compared with about a minute for one ISO17 conformation.Source: §5.2 · p. 6
  • With 50 latent samples, overlap between p(d|z,G) distributions is small, which the authors explain by the distance vector’s average dimensionality of about 60.Source: §5.2 · p. 6
  • Across six illustrated molecules, GRAPHDG conformations are closest to ground truth despite slightly overestimating structural variance; RDKIT shows too little variance and DL4CHEM mostly invalid structures.The paper attributes some DL4CHEM invalidity to incorrect interatomic-angle predictions.Source: §5.2 · p. 6, Figure 6 · p. 7

5.3. Calculation of Molecular Properties

  • RDKIT and GRAPHDG perform similarly for estimated properties, but both remain highly inaccurate for total electronic energy; GRAPHDG overestimates variance in some strongly constrained distances.The authors associate those distance variances, such as for triple bonds, with drastically increased conformational energies; DL4CHEM was excluded because its conformations were too poor for property calculations.Source: §5.3 · p. 8, Table 2 · p. 8

6. Limitations

The authors identify three limitations: the CVAE can generate unrealizable distance sets, CONF17 covers only a small portion of chemical space, and capturing long-range correlations would require many auxiliary edges.

Limitations

  • The CVAE can sample distance sets with no corresponding 3D structure, and the CONF17 benchmark covers only a small portion of chemical space.Capturing long-range correlations, as in proteins, would require a large set of auxiliary edges.Source: §6 · p. 8

7. Conclusions

The paper presents GRAPHDG, a transferable generative model for molecular conformations, and reports that its samples are closer to ground truth than those of other methods. It also uses the model as an importance-sampling proposal for molecular properties, while identifying open sampling and data needs.

7. Conclusions

  • GRAPHDG is a transferable generative model that samples molecular conformations represented through interatomic distances.The paper reports that its generated conformations are closer to ground truth than those from other methods.Source: §7 · p. 8
  • The authors propose a benchmark for comparing molecular conformation generators and report that GRAPHDG samples are closer to ground truth than those of other methods.The model is used as a proposal distribution in an importance-sampling integration scheme to estimate molecular properties.Source: §7 · p. 8
  • Orbital energies and dipole moments were predicted well, but meaningful estimates of electronic energies require a larger, more diverse dataset.Source: §7 · p. 8
  • Methods are still needed to estimate how many conformations must be generated to sample all important conformations.Source: §7 · p. 8
  • The model could be trained on conformational distributions at different temperatures in a transfer-learning-type setting.Source: §7 · p. 8

A.1. Data Generation

The ISO17 data processing discarded conformations that XYZ2MOL could not parse and augmented molecular graphs with auxiliary edges. The authors note that this augmentation can over-specify geometry, but did not cause problems in their experiments.

A.1. Data Generation

  • Conformations that XYZ2MOL could not parse were discarded during ISO17 processing.Source: §A.1 · p. 11
  • Molecular graphs were augmented with auxiliary edges during preprocessing.Source: §A.1 · p. 11
  • Auxiliary edges can over-specify system geometry, but this did not pose a problem in the reported experiments.Source: §A.1 · p. 11

A.2. Input Features

The CONF17 benchmark’s input features are listed in separate node-feature and edge-feature tables.

A.2. Input Features

B. Model Architecture

The supplied material covers the model architecture, quantum-chemical calculations, Euclidean distance geometry, example generated conformations, and predicted distance distributions. It reports a quantum-chemical calculation failure for DL4CHEM conformations with atoms that were too close, and describes the scope of the visual comparisons.

B. Model Architecture

C.1. Quantum-Chemical Calculations

  • Quantum-chemical calculations used PySCF 1.5 with PBE and def2-SVP, reporting electronic energy, HOMO and LUMO energies, and dipole-moment norm.Source: §C.1 · p. 11
  • DL4CHEM conformations failed quantum-chemical calculations because atoms were too close for PySCF self-consistent-field algorithms to converge.Source: §C.1 · p. 11

C.2. Euclidean Distance Geometry

  • Euclidean distance geometry proceeds through bound smoothing, embedding, and coordinate optimization against distance and chirality constraints.Source: §C.2 · pp. 11–12

E. Distributions over Distances

D. Generation of Conformations

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