Equivariant 3D-Conditional Diffusion Models for Molecular Linker Design

Ilia Igashov; Hannes Stärk; Clément Vignac; Victor Garcia Satorras; Pascal Frossard; Max Welling; Michael Bronstein; Bruno Correia

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Equivariant 3D-Conditional Diffusion Models for Molecular Linker Design

Ilia Igashov, Hannes Stärk, Clément Vignac, Victor Garcia Satorras, Pascal Frossard, Max Welling, Michael Bronstein, Bruno Correia

arXiv:2210.05274v1Oct 11, 2022cs.LGq-bio.BM

TL;DR

Molecular linker design must connect disconnected fragments into chemically relevant molecules, while existing approaches are limited in fragment count and pocket conditioning. DiffLinker uses an E(3)-equivariant 3D-conditional diffusion model to generate linkers, and it outperforms other methods while supporting arbitrary fragment counts and protein-pocket conditioning. Its translation-handling construction and equivariant 3D context define important scope and modeling constraints.

Problem
Molecular linker design must combine disconnected fragments into a connected molecule, but no prior computational method incorporated target protein-pocket information.
Method
DiffLinker predicts linker size and generates linker atom types and positions with an E(3)-equivariant diffusion model conditioned on 3D fragments and optionally protein-pocket atoms.
Results
DiffLinker outperforms other methods on standard benchmarks, produces more chemically relevant and diverse molecules, links more than two fragments, and generates valid pocket-conditioned linkers.
Takeaways & Limitations
The model provides a molecular linker-generation tool applicable to realistic drug-design scenarios involving arbitrary fragment sets and target protein pockets.
Takeaways & Limitations
The probabilistic model is not translation equivariant, so the method centers data at a context-dependent point and samples initial noise from N(0, I).

Abstract

from arXiv · show

Fragment-based drug discovery has been an effective paradigm in early-stage drug development. An open challenge in this area is designing linkers between disconnected molecular fragments of interest to obtain chemically-relevant candidate drug molecules. In this work, we propose DiffLinker, an E(3)-equivariant 3D-conditional diffusion model for molecular linker design. Given a set of disconnected fragments, our model places missing atoms in between and designs a molecule incorporating all the initial fragments. Unlike previous approaches that are only able to connect pairs of molecular fragments, our method can link an arbitrary number of fragments. Additionally, the model automatically determines the number of atoms in the linker and its attachment points to the input fragments. We demonstrate that DiffLinker outperforms other methods on the standard datasets generating more diverse and synthetically-accessible molecules. Besides, we experimentally test our method in real-world applications, showing that it can successfully generate valid linkers conditioned on target protein pockets.

1 INTRODUCTION

DiffLinker addresses molecular linker design by using an E(3)-equivariant, fragment-conditioned diffusion model that generates linkers for disconnected fragments. It supports arbitrary fragment counts, automatically handles linker size and attachment points, and can condition generation on protein-pocket atoms.

Motivation: Fragment-based drug design reduces the search space by starting from small compounds, but disconnected fragments must still be combined into one molecule.The paper frames fragment linking as a central challenge after relevant fragments are identified and docked to a target protein.
Prior limitations: Existing linker methods use syntactic or autoregressive approaches, but they lack permutation equivariance and generally connect only pairs of fragments.The cited approaches differ in whether they use SMILES or 3D fragment information.
Method: DiffLinker generates molecular linkers with an E(3)-equivariant diffusion model conditioned on a 3D atomic point cloud of input fragments.It predicts linker size, samples initial atom types and positions, and iteratively denoises them using a fragment-conditioned neural network.
Capabilities: The model is equivariant to translations, rotations, reflections, and permutations, while supporting arbitrary fragment counts, unspecified attachment atoms, and unconstrained linker sizes.Its conditioning mechanism can also incorporate surrounding protein-pocket atoms.
Results: DiffLinker outperforms previous methods in chemical relevance, synthetic accessibility, drug-likeness, and diversity, and it links more than two fragments.The paper also reports valid pocket-conditioned linkers that respect geometric constraints and minimize clashes with protein pockets.

2 RELATED WORK

Prior molecular linker methods include computationally expensive search, simulation, and learning-based approaches. DiffLinker is positioned against methods limited by data, representation, permutation handling, or the number of fragments they can connect.

Early methods: Early linker-design methods relied on predefined libraries, search procedures, genetic algorithms, tabu search, or force-field optimization.These methods are described as computationally expensive and substantially limited by available data.
Learning-based methods: SyntaLinker treats linker design as SMILES sentence completion, inheriting the absence of 3D structure and SMILES consistency limitations.SMILES can place nearby molecular atoms far apart in the sequence.
DiffLinker: DiffLinker’s workflow computes linker-size probabilities, then samples and denoises linker atoms with a fragment-conditioned equivariant diffusion model.The generated linker atoms are incorporated with the input fragments into a connected molecule.
Learning-based methods: DeLinker and 3DLinker use 3D information but are autoregressive and not permutation equivariant, limiting sample efficiency and scalability to large molecules.They also connect only pairs of fragments and are difficult to extend to larger fragment sets.
Diffusion-model context: Other 3D diffusion models address conformers, molecule generation, protein scaffolds, or antibody design rather than molecular linker point clouds.The antibody-design conditioning mechanism is described as closest to DiffLinker but targets amino-acid chains.

3 PRELIMINARIES

Diffusion models transform data into noise and learn a reverse denoising process for generation. For molecules, the data are represented as attributed 3D atomic point clouds with continuous coordinates and noisy one-hot atom types, while equivariance preserves geometric symmetries.

Diffusion models: A diffusion model progressively adds noise to data and learns a generative denoising process that approximates the reverse trajectory.Sampling starts from Gaussian noise and iteratively produces an intermediate state before generating a data point.
Diffusion models: The forward process is Markovian, with α_t controlling retained signal and σ_t controlling added noise.Gaussian transitions permit closed-form intermediate-state distributions and analytically tractable reverse distributions when conditioned on the original data.
Training and sampling: The denoising network predicts Gaussian noise, which is used to estimate the underlying data point during reverse sampling.The network’s prediction is written as ˆϵ_t = ϕ(z_t, t), and ˆx is then computed from the noisy state.
Training and sampling: Training uses a simplified mean-squared error between predicted and true noise, optimized with a mini-batch estimator over diffusion time steps.This objective is equivalent to the evidence-lower-bound objective up to a time-dependent prefactor.
Molecular representation: A molecular data point is an attributed point cloud with atom coordinates r and one-hot atom-type features h.The representation is written as x = [r, h] for M atoms and n_f feature dimensions.
Molecular representation: Discrete atom types are lifted to a continuous space by adding Gaussian noise to their one-hot encodings during diffusion.This lets the molecular diffusion model process categorical atom features alongside continuous coordinates.
Equivariance: E(3)-equivariance requires transformations of 3D inputs and outputs to remain consistent under translations, rotations, and reflections.The corresponding conditional-distribution definition applies the same geometric transformation to both molecular data and context.

4 DIFFLINKER: EQUIVARIANT 3D-CONDITIONAL DIFFUSION MODEL FOR MOLECULAR LINKER DESIGN

DiffLinker is an E(3)-equivariant diffusion model that generates linker atoms conditioned on 3D fragments and optionally protein-pocket atoms. Its pipeline predicts linker size, denoises linker coordinates and features while keeping context fixed, and supports multiple fragments without predefined attachment atoms or linker size.

Conditional diffusion: DiffLinker conditions linker generation on a fixed 3D context containing input fragments and optionally protein-pocket atoms.The noisy linker is updated while the context remains fixed.
Equivariance: The model’s denoising process is designed to be equivariant to joint transformations of the noisy linker and context.If the prior function is O(3)-equivariant and the denoising network is equivariant, the generated conditional distribution is O(3)-equivariant.
Equivariant denoising network: DiffLinker uses an EGNN-based learnable function to predict coordinate and feature noise from the noisy linker, context, and diffusion time.The predicted coordinate noise is translation-invariant because the initial coordinates are subtracted from its coordinate component.
Equivariant denoising network: Only linker nodes receive non-zero coordinate displacements during EGNN updates, while context coordinates remain unchanged.This reflects the model’s role in modifying the linker around fixed fragments and pocket atoms.
Linker-size prediction: A separately trained GNN predicts linker size from fragment atom types, relative positions, and orientations.The fragments are represented as a fully connected graph with one-hot atom features and distance-based edge features.
Model capabilities: DiffLinker can handle an arbitrary number of input fragments, requires neither predefined attachment atoms nor linker size, and can incorporate protein-pocket atoms as conditioning information.This conditioning supports structure-based drug-design applications; unconditioned samples can produce multiple pocket clashes.

5 EXPERIMENTS

Experiments evaluate DiffLinker on standard pair-fragment benchmarks, a new multiple-fragment benchmark, and a protein-pocket dataset. The results show stronger chemical relevance, successful multi-fragment linking, and pocket-conditioned control of clashes.

Datasets: The evaluation uses ZINC and CASF pair-fragment benchmarks, a GEOM benchmark containing three or more fragments, and a protein-pocket dataset.ZINC uses computationally generated conformations, CASF uses experimentally verified conformations, GEOM targets multiple-fragment linking, and the Pockets Dataset tests pocket conditioning.
Metrics: Table 1 reports chemical relevance using QED, synthetic accessibility, and linker ring counts, alongside validity, uniqueness, and novelty.These are followed by additional measures of standard generative properties.
ZINC and CASF: DiffLinker outperforms other methods on ZINC and CASF in synthetic accessibility, drug-likeness, linker ring counts, and chemical relevance, while validity remains on par.The generated molecules also show higher chemical and geometric similarity to reference molecules according to SCRDKit scores.
Multiple Fragments: On the multiple-fragment benchmark, DiffLinker achieves 94% validity and recovers more than 50% of reference molecules, whereas adapted 3DLinker fails in almost 84% of cases.The comparison concerns linkers connecting more than two fragments, where iterative pairwise generation cannot use the full context at once.
Protein Pocket Conditioning: Pocket conditioning yields an average of 7 clashes per molecule with full-atomic pockets, compared with 14 for backbone-only conditioning and 21 without conditioning.Reference complexes average 6 clashes per molecule, providing the comparison point for full-atomic conditioning.

6 CONCLUSION

DiffLinker is an E(3)-equivariant 3D-conditional diffusion model that generates linkers connecting disconnected fragments. It scales beyond fragment pairs, predicts linker size and attachment points, performs strongly on benchmarks, and can condition generation on protein pockets.

6 CONCLUSION: DiffLinker generates an atomic point cloud linker that interconnects a set of disconnected molecular fragments.The denoised linker atoms and input fragment atoms form one connected molecule.
6 CONCLUSION: Unlike previous pairwise methods, DiffLinker naturally scales to an arbitrary number of fragments without requiring specified attachment points.It predicts the distribution of linker size from the input fragments.
6 CONCLUSION: DiffLinker outperforms other models on standard benchmarks and produces more chemically relevant molecules.
6 CONCLUSION: Conditioning on protein pockets enables DiffLinker to generate linkers with a minimum number of clashes.

A APPENDIX

The training algorithm perturbs linker data with Gaussian noise at a sampled diffusion time. The sampling algorithm then denoises a context-conditioned latent from Gaussian initialization to produce a linker.

Algorithm 1 Training: Training samples a diffusion time t and Gaussian noise, then forms the noisy state z_t = α_t x + σ_t ε_t.
Algorithm 2 Sampling: Sampling centers the context, initializes z_T from a standard normal distribution, and iteratively updates it for t = T, T − 1, …, 1.
Algorithm 2 Sampling: The final linker is sampled from p(x|z_0, u) after the reverse diffusion loop.

A.1 PROOF OF PROPOSITION 1

The proof establishes O(3)-equivariance of the conditional diffusion model by combining an equivariant prior with equivariant transition probabilities and marginalizing intermediate states.

A.1 PROOF OF PROPOSITION 1: An O(3)-equivariant function and isotropic Gaussian noise imply an O(3)-equivariant prior distribution.The prior satisfies p(Rz_T|Ru) = p(z_T|u).
A.1 PROOF OF PROPOSITION 1: O(3)-equivariance of the neural network and the transition equation implies equivariance of every transition probability p(z_{t−1}|z_t, u).
A.1 PROOF OF PROPOSITION 1: The conditional data distribution p(z_0|u) is obtained by marginalizing the joint distribution over intermediate states z_1, ..., z_T.
A.1 PROOF OF PROPOSITION 1: Because both the prior and transition distributions are equivariant, the resulting distribution p(z_0|u) is also O(3)-equivariant.

A.2 PROBLEM WITH TRANSLATIONS

The proposed transition distribution is not translation-equivariant when its mean transforms with a scaling factor λ ≠ 1. Translation-invariant network processing does not by itself preserve the required distributional symmetry.

Translation condition: Translation equivariance requires jointly translating z_t, z_{t−1}, and u without changing the transition probability.The stated condition is p(z_{t−1}+t|z_t+t,u+t)=p(z_{t−1}|z_t,u) for all t ∈ R^3.
Network property: The neural network ϕ is translation invariant, so translating z_t and u leaves its output unchanged.The network satisfies ϕ(z_t+t,u+t,t)=ϕ(z_t,u,t).
Failure mode: λ ≠ 1 causes the translated input and output means to differ, violating the required equivariance of the transition distributions.The resulting transformed probability is evaluated at z_{t−1}+(1−λ)t rather than z_{t−1}.

A.3 DIFFUSION MODEL

DiffLinker models use 500 diffusion steps with a polynomial noise schedule. A small precision value is included to avoid numerical instability.

Noise schedule: 500 diffusion steps are used to train all DiffLinker models with a polynomial noise schedule.The schedule is specified with T = 500.
Numerical stability: s = 10^-5 serves as a precision value that helps avoid numerically unstable situations.The precision parameter is part of the polynomial noise-schedule specification.

A.4 DYNAMICS

The dynamics model uses an EGNN to update linker atom features and coordinates from the noisy linker and its context. Inputs can include atom, fragment, time-step, anchor, and pocket information.

EGNN inputs: The EGNN receives linker atoms and context as feature vectors and 3D coordinates.Features include atom types, fragment flags, and the diffusion time step; anchor and pocket flags are optional.
Dynamics updates: Sequential equivariant graph convolutional layers update the molecular representation through learned edge, node, and coordinate functions.The learned components use fully connected layers, batch normalization, and SiLU activations.
Training: Separate EGNN models were trained for the ZINC, Multi-Fragment, and Pocket datasets.Training used Adam with learning rate 2 · 10^-5 and weight decay 10^-13 on a single Tesla V100 GPU.

A.5 LINKER SIZE PREDICTION

DiffLinker predicts linker size from fragment graphs, then generates linker point clouds whose bonds are reconstructed afterward. Evaluation covers multi-fragment generation, chemical and generative metrics, and a documented issue with the baseline SA filter.

Linker size prediction: A graph neural network predicts probabilities for predefined linker sizes from fragment features and inter-atomic squared distances.SizeGNN applies graph convolutional layers and aggregates node embeddings into label probabilities.
Multi-fragment evaluation: 3DLinker evaluation on GEOM excludes inputs with more than three fragments, leaving 1,172 fragment triplets for sequential two-stage linking.The comparison uses 10 samples per pair and then 10 samples per resulting three-fragment input.
Molecule construction: DiffLinker generates atom point clouds and learns chemical rules from data, whereas DeLinker and 3DLinker generate covalent bonds with valency rules.Generated point clouds are converted into covalent-bond molecules using OpenBabel before linker extraction.
Evaluation metrics: Table 5 evaluates chemical relevance and standard generative properties on GEOM and the Pockets test set.GEOM uses 250 samples on the full test set, while Pockets uses 100 samples.
2D filters: The authors exclude the baseline SA-filter metric because dummy anchor atoms make fragment SA-scores appear artificially difficult to synthesize.They instead report average synthetic accessibility scores for complete generated molecules.