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Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Docking

Octavian-Eugen Ganea, Xinyuan Huang, Charlotte Bunne, Yatao Bian, Regina Barzilay, Tommi Jaakkola, Andreas Krause

arXiv:2111.07786v2cs.AIcs.LG

TL;DR

Protein docking seeks complex structures from unbound proteins, with rigid docking assuming no conformational change. EquiDock predicts the docking transformation using equivariant graph matching, keypoint alignment, optimal transport, and differentiable Kabsch superimposition. It is reported to be much faster than baselines and often competitive or better without heavy candidate sampling, templates, or refinement.

  • Problem

    Rigid protein-protein docking predicts a complex from unbound structures, while existing methods can require costly candidate sampling, ranking, and refinement.

  • Method

    EquiDock uses pairwise-independent SE(3)-equivariant graph matching, binding-pocket keypoint alignment with optimal transport, and differentiable rigid superimposition.

  • Results

    EquiDock is competitive and often outperforms baselines, while avoiding heavy candidate sampling, re-ranking, templates, task-specific features, and structure fine-tuning.

  • Takeaways & Limitations

    EquiDock provides an extremely fast end-to-end rigid docking approach suitable for intensive screening and larger biological modeling pipelines.

  • Takeaways & Limitations

    The model does not incorporate protein flexibility, and its soft steric-clash constraint can permit artifacts.

Abstract

from arXiv · show

Protein complex formation is a central problem in biology, being involved in most of the cell's processes, and essential for applications, e.g. drug design or protein engineering. We tackle rigid body protein-protein docking, i.e., computationally predicting the 3D structure of a protein-protein complex from the individual unbound structures, assuming no conformational change within the proteins happens during binding. We design a novel pairwise-independent SE(3)-equivariant graph matching network to predict the rotation and translation to place one of the proteins at the right docked position relative to the second protein. We mathematically guarantee a basic principle: the predicted complex is always identical regardless of the initial locations and orientations of the two structures. Our model, named EquiDock, approximates the binding pockets and predicts the docking poses using keypoint matching and alignment, achieved through optimal transport and a differentiable Kabsch algorithm. Empirically, we achieve significant running time improvements and often outperform existing docking software despite not relying on heavy candidate sampling, structure refinement, or templates.

1 INTRODUCTION

Protein interactions motivate computational docking, but rigid body docking seeks complex structures from unbound proteins while existing methods can be slow and sampling-intensive. EquiDock directly predicts the rigid transformation and targets input-independent complex predictions.

  • Protein interactions support processes including signaling, protein synthesis, and DNA repair, making docking relevant to therapeutic intervention.
  • Rigid body docking predicts a protein-complex 3D structure from two unbound protein structures while assuming no within-protein deformation.
  • Existing docking methods can take minutes to hours per pair and may not guarantee accurate structures.
  • These methods commonly sample millions of candidates, rank them, and refine top structures with energy models.
  • EquiDock directly predicts the ligand’s SE(3) transformation and is designed to return the same complex regardless of initial placements or constituent roles.

2 RELATED WORK

Related work spans geometric deep learning, protein folding, docking, interaction prediction, and drug-target modeling. EquiDock differs by deriving pairwise-independent SE(3)-equivariance constraints for rigid docking rather than relying on conventional sampling-heavy pipelines.

  • Geometric Deep Learning: Geometric neural networks incorporate symmetries such as SE(3)-equivariance to model data in three-dimensional Euclidean space without costly augmentation.
  • Geometric Deep Learning: EquiDock derives pairwise-independent SE(3)-equivariance constraints for pairs of 3D objects and applies them to rigid body docking.
  • Protein Folding: Prior protein-folding systems predict structural constraints or directly infer structures using sequence-derived information and geometric neural networks.
  • Protein-Protein Docking and Interaction: Protein docking commonly samples thousands or millions of candidates, ranks them, and refines top-ranked structures with energy or geometric models.
  • Protein-Protein Docking and Interaction: Protein-interface prediction uses graph neural networks, while AlphaFold 2 and RosettaFold have been used as subroutines for protein-interaction tasks.
  • Drug-Target Interaction: Drug-target interaction methods often use heavy sampling and ranking, focus on small drug-like ligands, and may assume a known binding pocket.

3 MATHEMATICAL CONSTRAINTS FOR RIGID BODY DOCKING

The rigid docking formulation predicts a rotation and translation that place one unbound protein into its bound position. The model is constrained so predicted complexes remain invariant to initial transformations and consistent when protein roles are exchanged.

  • Rigid Protein-Protein Docking – Problem Setup: Inputs are two proteins represented as 3D point clouds, with the ligand randomly rotated and translated while the receptor remains in the bound location.
  • Rigid Protein-Protein Docking – Problem Setup: The task predicts R ∈ SO(3) and t ∈ R3 such that RX1 + t equals the docked ligand structure.
  • Rigid Protein-Protein Docking – Problem Setup: The formulation assumes rigid backbones and side chains and therefore does not address flexible docking.
  • The desired prediction is independent of both proteins’ initial locations, orientations, and assigned roles.
  • SE(3)-equivariance Constraints: Under separate SE(3) transformations, the predicted rotation changes by undoing the ligand rotation, applying the original rotation, and applying the receptor rotation.
  • Commutativity: Commutativity requires docking either protein relative to the other to produce superimposable final complexes with zero RMSD.
  • Point Permutation Invariance: The predicted rotation and translation must also be invariant to the ordering of residues within either protein.

4 EQUIDOCK MODEL

EquiDock represents proteins as geometric graphs and processes paired structures with pairwise-independent SE(3)-equivariant message passing. It predicts docking transformations by aligning learned binding-pocket keypoints, while adding losses and features to support pocket localization and non-intersection.

  • Protein representation: Each protein is represented as a residue graph whose nodes use α-carbon coordinates and whose edges form a Euclidean k-nearest-neighbor graph.
  • Equivariant graph matching: IEGMNs update coordinates and features using intra- and inter-protein messages while preserving pairwise-independent SE(3)-equivariance for coordinates and invariance for features.The network uses cross-messages to model pairwise interactions and computes the rotation and translation for docking.
  • Equivariant graph matching: Proposition 4 establishes that the IEGMN architecture satisfies the pairwise independent SE(3)-equivariance property.
  • Keypoint alignment: Attention discovers K keypoints per protein, which are softly trained to represent binding-pocket locations and then superimposed with a differentiable Kabsch procedure.Optimal transport recovers the unknown alignment between learned keypoints and bound-state pocket points.
  • Geometric inductive biases: The model includes a non-intersection objective and a surface-aware node feature that distinguishes interior residues from surface residues using local neighbor geometry.Interior residues tend toward values near 0, whereas surface residues have neighbors concentrated in a narrower cone; the feature correlates with more expensive residue-depth estimates.

5 EXPERIMENTS

Experiments evaluate EquiDock on DB5.5 and DIPS using docking-quality metrics, runtime distributions, and qualitative visualizations. The method is reported as competitive or often better than baselines while using direct prediction rather than extensive sampling and refinement.

  • Datasets: DB5.5 contains 253 structures, while DIPS is larger and is split by protein family to separate similar proteins.The final evaluation uses the full DB5.5 test set and 100 DIPS test pairs from different protein families.
  • Evaluation metrics: Complex-RMSD and Interface-RMSD evaluate predicted structures after Kabsch superposition, with Interface-RMSD restricted to residues within 8Å of the other protein.Both metrics use only α-carbon coordinates for fair baseline comparison.
  • Complex prediction: EquiDock is competitive and often outperforms the baselines despite avoiding heavy candidate sampling, task-specific hand-crafted features, and structure fine-tuning.The method aims to predict the ligand’s SE(3) transformation directly in one shot.
  • Experimental caveat: Baseline comparisons may be optimistic because proprietary methods might have used parts of the test sets for templates or feature extraction.A fairer comparison would require rebuilding those baselines without test-set information.
  • Computational efficiency: 80-500 times faster than the baselines, EquiDock shows a substantial inference-time advantage.Inference-time distributions are reported in Figure 5 on a log10 scale.
  • Visualization: EquiDock significantly outperforms all baselines on a visualized DIPS test pair, whereas the other methods identify the interface on the wrong side of the black protein.The example is presented in Figure 6.

6 CONCLUSION

The conclusion presents EquiDock as a fast end-to-end rigid docking method built around geometric priors and residue-level protein representations. It also identifies protein flexibility and soft steric-clash prevention as important limitations for future extensions.

  • Conclusion: EquiDock performs end-to-end rigid protein docking without candidate sampling, templates, task-specific features, or pre-computed meshes.Its design incorporates commutativity and pairwise-independent SE(3)-equivariance to reduce the computational burden of data augmentation.
  • Limitations: The method does not incorporate protein flexibility, which is necessary for various protein families such as antibodies.The authors note that DB5 and DIPS are biased toward rigid-body docking.
  • Limitations: Soft steric-clash prevention can allow artifacts, motivating future hard constraints against such clashes.This limitation is linked to the soft constraint used in Eq. (15).
  • Protein representation: The model represents proteins at residue level, using α-carbon coordinates, amino-acid identity features, nearest-neighbor graphs, and local residue coordinate systems.The local systems support relative position, orientation, and distance features between residues.
  • Geometric features: Distance-based edge features use 15 radial-basis-function features with scale parameters {1.5^x | x = 0, 1, 2, ..., 14}.The model also computes five surface-aware node features using λ ∈ {1., 2., 5., 10., 30.}.

B.1 PROOF OF PROPOSITION 1.

The proof establishes that the predicted ligand placement transforms consistently under changes in the proteins’ initial rotations and translations, ensuring an invariant docked complex.

  • The predicted ligand position is defined as R(X1|X2)X1 + t(X1|X2) = ˜X1.
  • Applying the derived transformation to Q1X1 + g1 produces the same placement as transforming it into Q2˜X1 + g2.
  • The resulting transformation docks the transformed ligand with respect to the transformed receptor, by the definitions of R and t.
  • Conversely, the stated constraints imply SE(3)-invariance of the predicted docked complex.

B.2 PROOF OF PROPOSITION 3.

The proof shows that the network’s layer updates preserve pairwise independent SE(3)-equivariance: each graph’s coordinates respond appropriately to its own transformation while cross-graph effects remain invariant.

  • The IEGMN layer is required to satisfy pairwise independent SE(3)-equivariance under separate rotations and translations of the two input graphs.
  • Node coordinate and feature embeddings represent the geometric and latent information propagated through the layer.
  • Message computations are SE(3)-invariant because they use invariant features, latent embeddings, and Euclidean distances.
  • The updated latent node embeddings remain SE(3)-invariant.
  • Coordinate updates are equivariant to nodes in the same graph and invariant to coordinates in the other graph.

C SURFACE FEATURES

The proposed surface features are designed to distinguish interior from surface residues, correlate strongly with established depth estimates, and admit a closed-form interpretation under idealized geometric assumptions.

  • The Spearman correlation with MSMS depth estimates has mean 0.68 and median 0.70 across DIPS test-set proteins.
  • For near-surface points surrounded by infinitely many equally spaced, equally distant points, a closed-form feature expression follows from cancellation of non-bisector components.
  • Interior residues have smaller directionally averaged feature values, whereas surface residues receive contributions from a narrower neighborhood and larger ρ values.
  • In a synthetic unit-circle experiment, the surface feature strongly correlates with depth, with the real-protein and synthetic λ scales differing.

D DATASETS

The datasets support rigid docking evaluation, but their structural coverage is limited: DIPS contains bound structures only, while DB5.5 includes mostly rigid unbound structures and exposes the need for better conformational-change benchmarks.

  • DIPS contains only bound structures and is therefore currently suitable for rigid docking.
  • DB5.5 includes unbound structures, but most proteins show relatively small unbound-to-bound structural changes.
  • Figure 10 summarizes the distribution of per-protein Spearman correlations between MSMS residue depths and the proposed surface features over 100 DIPS test-set proteins.
  • The limited conformational changes in DB5.5 indicate that better datasets are needed to address docking with conformational change.
  • Table 2 provides the dataset overview, with DIPS residue and atom statistics computed on a 200-protein subset.

E MORE EXPERIMENTAL DETAILS AND RESULTS

The experiments report baseline failures, runtime comparisons, ablations, and analyses of EquiDock’s losses on DB5.5 and DIPS. Results emphasize the role of pocket loss for interface accuracy and intersection loss for avoiding protein overlap.

  • Baseline failures: ATTRACT fails on one DB5.5 case and three DIPS cases; the unbound input is used as the prediction for metric calculation.The listed failures are 1N2C in DB5.5 and oi_4oip.pdb1_8, oi_4oip.pdb1_3, and p7_4p7s.pdb1_2 in DIPS.
  • Running times: Runtime comparisons use different CPU hardware for ATTRACT, HDOCK, and EquiDock, while ClusPro and PatchDock run through public web servers.The web-server runtimes are influenced by cluster load.
  • Ablation studies: Removing pocket loss produces lower interface RMSD scores than removing the other evaluated model components.Table 5 reports DIPS test median C-RMSD and I-RMSD values for the corresponding best validation models.
  • Intersection loss: The intersection loss achieves almost perfect values on ground-truth structures and softly constrains predicted proteins to remain non-intersecting.The analysis uses γ = 10 and σ = 25, selected on the DB5 validation set.
  • DB5.5 results: Figure 13 presents DB5.5 complex-RMSD distributions, interface-RMSD distributions, and a C-RMSD versus I-RMSD scatter plot.The accompanying text identifies these as the corresponding DB5.5 result plots.
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