PaperlayerView exact revision on arXiv

Source-linked AI summary

Unsupervised Protein-Ligand Binding Energy Prediction via Neural Euler's Rotation Equation

Wengong Jin, Siranush Sarkizova, Xun Chen, Nir Hacohen, Caroline Uhler

arXiv:2301.10814v2q-bio.BMcs.LG

TL;DR

Affinity prediction is difficult to extend from small molecules to antibodies because labeled antibody data are limited. This paper trains an energy-based model with SE(3) denoising score matching and NERE rotation prediction, outperforming unsupervised baselines and supervised models in the antibody setting.

  • Problem

    Supervised affinity prediction relies on labeled small-molecule complexes, but antibody affinity data are limited and their structures differ substantially from small molecules.

    Source: §1 · p. 1

  • Method

    The paper trains an energy-based model on protein–ligand structures using SE(3) denoising score matching, with NERE predicting rotation noise from energy-gradient forces and torque.

    Source: §1 · pp. 1–2

  • Results

    NERE outperforms unsupervised baselines across evaluated settings and surpasses supervised models in the antibody setting.

    Source: §1 · p. 2, §6 · p. 9

  • Takeaways & Limitations

    The results highlight the benefit of unsupervised learning for affinity prediction when labeled binding data are limited.

    Source: §1 · p. 2

    The antibody model uses only Cα backbone atoms, and the models omit protein side-chain flexibility and ligand rotatable-bond flexibility.

    Source: §6 · p. 9

Abstract

from arXiv · show

Protein-ligand binding prediction is a fundamental problem in AI-driven drug discovery. Prior work focused on supervised learning methods using a large set of binding affinity data for small molecules, but it is hard to apply the same strategy to other drug classes like antibodies as labelled data is limited. In this paper, we explore unsupervised approaches and reformulate binding energy prediction as a generative modeling task. Specifically, we train an energy-based model on a set of unlabelled protein-ligand complexes using SE(3) denoising score matching and interpret its log-likelihood as binding affinity. Our key contribution is a new equivariant rotation prediction network called Neural Euler's Rotation Equations (NERE) for SE(3) score matching. It predicts a rotation by modeling the force and torque between protein and ligand atoms, where the force is defined as the gradient of an energy function with respect to atom coordinates. We evaluate NERE on protein-ligand and antibody-antigen binding affinity prediction benchmarks. Our model outperforms all unsupervised baselines (physics-based and statistical potentials) and matches supervised learning methods in the antibody case.

1 Introduction

NERE targets the limited availability of antibody affinity labels with an unsupervised energy-based framework for protein–ligand complexes. Across the evaluated settings, it outperforms unsupervised baselines and surpasses supervised models for antibodies.

1 Introduction

  • Supervised affinity models rely on labeled small-molecule structures, while antibody affinity data are scarce and their structures differ substantially from small molecules.The largest antibody affinity dataset cited contains 566 data points.Source: §1 · p. 1
  • The framework learns a complex energy model from unlabeled PDB structures, treating structure likelihood as a potential indicator of binding affinity.The approach uses neural networks to learn an expressive energy function and does not require expensive molecular-dynamics simulations.Source: §1 · p. 1
  • NERE predicts rotation noise for SE(3) denoising score matching by converting energy-gradient forces into torque, angular velocity, and a ligand rotation matrix.It is compatible with SE(3)-invariant networks and guarantees equivariance without further architectural requirements.Source: §1 · p. 2
  • NERE outperforms all unsupervised baselines across the evaluated settings and surpasses supervised models in the antibody setting.The benchmarks include crystallized and docked complexes; the authors cite limited antibody affinity data as a motivation for unsupervised learning.Source: §1 · p. 2

2 Related Work

Related approaches include supervised affinity predictors, physics-based and statistical scoring functions, and denoising score matching. NERE distinguishes itself in rotation prediction and uses context-aware neural energy scores.

2 Related Work

  • Supervised models learn from affinity-labeled complexes, whereas unsupervised approaches use physics-based or statistical potentials; DrugScore2018 scores atom pairs using atom types and distance alone.The paper contrasts this pairwise scoring with its context-aware neural energy function.Source: §2 · p. 2
  • Antibody-affinity prediction has few supervised approaches because data are scarce; existing models use hand-crafted features, while docking scores combine tuned physical interaction terms.The paper also implements neural supervised baselines for comparison.Source: §2 · p. 2
  • This work uses rigid-transformation noise rather than standard Gaussian coordinate noise and parameterizes the score as an energy gradient to predict energy and score together.The paper situates denoising score matching among methods previously applied to protein structure prediction and ligand docking.Source: §2 · p. 2
  • Prior equivariant rotation methods use keypoint alignment with Kabsch or tensor field networks and spherical harmonics; NERE directly predicts a rotation matrix without either requirement.NERE can be plugged into SE(3)-invariant encoder architectures.Source: §2 · p. 2
  • The model has separate input designs: a binding pocket and MPN-encoded molecular graph for small molecules, versus antibody CDR and epitope residues with ESM2 features.The small-molecule and antibody architectures are shown on the left and right, respectively.Source: Figure 1 · p. 3

3 Unsupervised Binding Energy Prediction

The section formulates unsupervised binding-energy prediction as learning an energy function from crystal structures without affinity labels. It combines an energy-based model with rigid-transformation denoising score matching, including a rotation-score predictor for SE(3).

3.1 EBM Architecture

  • The energy function sums pairwise atom-interaction energies predicted from SE(3)-invariant atom representations; inputs are tailored separately for small molecules and antibodies.Small-molecule ligands use jointly trained message-passing features, whereas antibody interfaces use frozen ESM-2 embeddings and Cα atoms.Source: §3.1 · p. 3

3.2 Training EBMs with Denoising Score Matching

  • Maximizing crystal-structure likelihood is motivated by the hypothesis that crystal structures are the lowest-energy states of protein-ligand complexes.Because p(A, X) is proportional to exp(−E(A, X)), the likelihood objective seeks to minimize their energy.Source: §3.2 · p. 4
  • DSM trains the energy-based model by matching its energy-gradient score to the perturbation-noise score, with the crystal structure at a local energy minimum.The standard Gaussian-coordinate-noise formulation drives the gradient to zero at the unperturbed crystal pose.Source: §3.2 · p. 4

3.3 Perturbing a Complex with Rigid Transformation Noise

  • Rigid rotation and translation perturbations are used instead of Gaussian coordinate noise, which can create conformations that violate physical constraints.Figure 2 depicts NERE's rotation prediction as force calculation, Euler-equation torque and angular-velocity prediction, then conversion to a rotation.Source: §3.3 · p. 4, Figure 2 · p. 5

3.4 Training EBMs with SE(3) Denoising Score Matching

  • SE(3) DSM matches model scores to rotation and translation noise scores, mapping atom-coordinate gradients into rotation space because their dimensions differ.Its objective combines separate translation and rotation losses; the rotation predictor is based on NERE.Source: §3.4 · p. 5

4 Neural Euler’s Rotation Equation

NERE is introduced to infer rotational scores from the energy gradient for SE(3) denoising score matching. It predicts rotation through a physics-based force-and-torque formulation and is equivariant under SO(3).

4 Neural Euler’s Rotation Equation

  • NERE maps energy gradients to rotational scores for SE(3) denoising score matching, using a rotation predictor derived from Euler’s rotation equation.In this application, the energy gradient with respect to atom coordinates is treated as force.Source: §4 · p. 5, §4.2 · p. 6

4.1 Euler’s Rotation Equations

  • Euler’s rotation equation relates ligand forces and torque to angular motion, with the ligand’s inertia matrix describing its mass distribution.The section approximates the angular velocity after a short time interval from torque and inertia.Source: §4.1 · pp. 5–6

4.2 Applying NERE to SE(3) DSM

  • Under SO(3), NERE is equivariant: rotating a ligand gives angular velocity ω′ = Rω; SE(3) equivariance follows by placing the rotation center at the origin using μ, applying NERE, then adding μ back.Source: §4.2 · p. 6

5 Experiments

Experiments evaluate NERE on PDBBind protein–ligand and SAbDab antibody–antigen benchmarks using crystal and docked structures. NERE outperforms unsupervised baselines across the reported settings and also outperforms supervised models on antibody crystal structures.

5.1 Protein-Ligand Binding; 5.2 Antibody-Antigen Binding

  • Evaluation used 285 PDBBind core test complexes and 566 labeled SAbDab test complexes, with separate unlabeled training sets.The final unlabeled training sets contained 4,806 PDBBind complexes and 3,416 SAbDab complexes.Source: §5.1 · p. 6, §5.2 · p. 7

5.1 Protein-Ligand Binding

  • On crystal PDBBind complexes, NERE beat physics-based and unsupervised methods, ran in 100ms versus MM/GBSA's 1hr, and recovered almost 80% of supervised performance without labels.The authors interpret the result as evidence that binding affinity is closely related to an input complex's geometric structure.Source: §5.1 · p. 7
  • On docked PDBBind complexes, NERE continued to outperform baselines, with performance close to crystal-structure evaluation at median test RMSD=3.55.The authors attribute the small difference to relatively low docking error.Source: §5.1 · p. 7, Table 1 · p. 7

5.1 Protein-Ligand Binding; 5.2 Antibody-Antigen Binding

  • Across crystal and docked evaluations, NERE outperformed unsupervised baselines on both PDBBind and SAbDab, and outperformed supervised models in antibody crystal evaluation.The table covers Pearson correlations for both test sets and both structure settings.Source: §5.1 · p. 7, Table 1 · p. 7, §5.2 · p. 8

5.2 Antibody-Antigen Binding

  • On antibody crystal structures, NERE outperformed physics-based methods, protein language models, and supervised models, including FANNtransfer trained on 5,427 labeled points.The gain over FANNtransfer was small.Source: Table 1 · p. 7, §5.2 · p. 8
  • On docked antibody-antigen complexes with median RMSD 19.4, NERE outperformed baselines, and EBMs remained robust across the tested training algorithms.The tested algorithms were contrastive learning, Gaussian DSM, and NERE DSM.Source: Table 1 · p. 7, §5.2 · p. 8

5.3 Ablation Studies and Visualization

  • Rotation landscapes place crystal poses relatively near local minima, while residue-energy maps emphasize CDR-H3/L3 interactions and leave pairs beyond 20Å blank.Darker heatmap entries indicate stronger interactions.Source: §5.3 · p. 8, Figure 3 · p. 9
  • Removing SE(3) invariance or either rotation or translation DSM substantially hurts performance; one-hot features retain reasonable antibody accuracy, though ESM-2 helps.The one-hot comparison applies only to the antibody model.Source: §5.3 · pp. 8–9

6 Discussion

The discussion presents an unsupervised energy-based model whose learned energy correlates with experimental binding affinity and outperforms supervised models for antibodies. It also identifies omitted molecular flexibility as a limitation.

6 Discussion

  • The learned energy correlates with experimental binding affinity and outperforms supervised models in the antibody case.The authors describe the model as trained with SE(3) denoising score matching, with rotation scores predicted by NERE.Source: §6 · p. 9
  • The antibody model uses only backbone Cα atoms, and the models omit protein side-chain flexibility and ligand rotatable bonds.The authors identify these interactions as crucial for protein-ligand binding prediction.Source: §6 · p. 9
  • The authors state that the method applies to drug discovery, immunology, and structural biology, and report no potential negative societal impacts.They describe the aim as accelerating scientific discovery.Source: §6 · p. 9
  • The paper reports that its code and data are available at github.com/wengong-jin/DSMBind.Source: §6 · p. 9

A NERE Rotational Equivariance

The proposition considers how NERE quantities transform when a protein-ligand complex is rotated. Its proof uses the encoder's SE(3)-invariant energy and rotation-compatible cross products to establish layer equivariance.

A NERE Rotational Equivariance

B Experimental Details

NERE’s experimental setup uses a frame-averaging protein encoder, evaluates additional antibody-binding baselines, and specifies docking protocols and model-selection settings. Additional physics-based potentials perform much worse than NERE on the antibody task.

Protein encoder architecture

  • The protein encoder uses frame averaging to make the energy function SE(3)-invariant and differentiable with respect to atom coordinates.It projects coordinates onto eight frames, encodes them with atom features, and averages the frame representations.Source: B Experimental Details · p. 13

Model hyperparameters

  • The protein encoder has 256 hidden units; validation Pearson correlation selects depth and distance threshold from task-specific candidate values.Small-molecule depths are 1, 2, or 3 with thresholds 5.0 or 10.0; antibody depths are 1, 2, or 3 with thresholds 10.0 or 20.0.Source: B Experimental Details · p. 14

Docking protocol

  • Autodock Vina uses default parameters with a 20Å grid, 0.375Å interval, and exhaustiveness 32; ZDOCK generates 2,000 poses that are rescored by ZRANK2.The best ZRANK2-scored pose is selected for each antibody-antigen pair.Source: B Experimental Details · p. 14

Additional results

  • Additional physics-based potentials evaluated on the SAbDab test set perform much worse than NERE.The additional baselines include ZRANK, RosettaDock, PyDock, SIPPER, AP_PISA, CP_PIE, FIREDOCK, and FIREDOCK_AB.Source: B Experimental Details · p. 14, Table 2 · p. 14

Additional supervised baselines

  • On 273 SKEMPI antibody-antigen complexes, 3D CNN and GCN baselines perform much worse than FANN, which is used as the default supervised architecture.The GCN implementation includes ESM-2 residue embeddings.Source: B Experimental Details · p. 14

Additional results

  • Figure 4 plots the correlation between predicted and experimental binding energy.Source: Figure 4 · p. 15
Loading 2301.10814v2…