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Equivariant Scalar Fields for Molecular Docking with Fast Fourier Transforms

Bowen Jing, Tommi Jaakkola, Bonnie Berger

TL;DR

The paper tackles the bottleneck of expensive pose optimization in structure-based virtual screening by introducing a scoring function defined as the cross-correlation of SE(3)-equivariant scalar fields carried by protein and ligand graphs. It develops equivariant scalar-field networks (ESFs) and derives efficient FFT-based workflows over translations and rotations, enabling rapid, amortizable pose evaluation. Empirical results on decoy-pose scoring and rigid conformer docking show competitive accuracy with Gnina/Vina and superior robustness to predicted structures (ESFold) while achieving substantial runtime gains, including a 45–50x speedup on PDE10A when pocket-level amortization is exploited. The approach promises practical impact for high-throughput docking by enabling large-scale screening with modest resources, and it opens avenues for integration with refinement and torsion-aware strategies in full docking pipelines.

Abstract

Molecular docking is critical to structure-based virtual screening, yet the throughput of such workflows is limited by the expensive optimization of scoring functions involved in most docking algorithms. We explore how machine learning can accelerate this process by learning a scoring function with a functional form that allows for more rapid optimization. Specifically, we define the scoring function to be the cross-correlation of multi-channel ligand and protein scalar fields parameterized by equivariant graph neural networks, enabling rapid optimization over rigid-body degrees of freedom with fast Fourier transforms. The runtime of our approach can be amortized at several levels of abstraction, and is particularly favorable for virtual screening settings with a common binding pocket. We benchmark our scoring functions on two simplified docking-related tasks: decoy pose scoring and rigid conformer docking. Our method attains similar but faster performance on crystal structures compared to the widely-used Vina and Gnina scoring functions, and is more robust on computationally predicted structures. Code is available at https://github.com/bjing2016/scalar-fields.

Equivariant Scalar Fields for Molecular Docking with Fast Fourier Transforms

TL;DR

The paper tackles the bottleneck of expensive pose optimization in structure-based virtual screening by introducing a scoring function defined as the cross-correlation of SE(3)-equivariant scalar fields carried by protein and ligand graphs. It develops equivariant scalar-field networks (ESFs) and derives efficient FFT-based workflows over translations and rotations, enabling rapid, amortizable pose evaluation. Empirical results on decoy-pose scoring and rigid conformer docking show competitive accuracy with Gnina/Vina and superior robustness to predicted structures (ESFold) while achieving substantial runtime gains, including a 45–50x speedup on PDE10A when pocket-level amortization is exploited. The approach promises practical impact for high-throughput docking by enabling large-scale screening with modest resources, and it opens avenues for integration with refinement and torsion-aware strategies in full docking pipelines.

Abstract

Molecular docking is critical to structure-based virtual screening, yet the throughput of such workflows is limited by the expensive optimization of scoring functions involved in most docking algorithms. We explore how machine learning can accelerate this process by learning a scoring function with a functional form that allows for more rapid optimization. Specifically, we define the scoring function to be the cross-correlation of multi-channel ligand and protein scalar fields parameterized by equivariant graph neural networks, enabling rapid optimization over rigid-body degrees of freedom with fast Fourier transforms. The runtime of our approach can be amortized at several levels of abstraction, and is particularly favorable for virtual screening settings with a common binding pocket. We benchmark our scoring functions on two simplified docking-related tasks: decoy pose scoring and rigid conformer docking. Our method attains similar but faster performance on crystal structures compared to the widely-used Vina and Gnina scoring functions, and is more robust on computationally predicted structures. Code is available at https://github.com/bjing2016/scalar-fields.
Paper Structure (33 sections, 2 theorems, 37 equations, 8 figures, 11 tables, 4 algorithms)

This paper contains 33 sections, 2 theorems, 37 equations, 8 figures, 11 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Method
  4. Equivariant Scalar Fields
  5. FFT Over Translations
  6. FFT Over Rotations
  7. Training and Inference
  8. Experiments
  9. Scoring Decoys
  10. Docking Conformers
  11. Conclusion
  12. Mathematical Details
  13. Proof of Proposition 1
  14. Derivations
  15. Equation \ref{['eq:tr_fourier']}
  16. ...and 18 more sections

Key Result

Proposition 1

Suppose the scoring function is parameterized as in Equation eq:field and for any $R\in SO(3), \mathbf{t} \in \mathbb{R}^{3}$ we have $A_{c n j \ell m}(G, R.\mathbf{X} + \mathbf{t}) = \sum_{m'} D^\ell_{mm'}(R)A_{c n j \ell m'}(G, \mathbf{X})$ where $D^\ell(R)$ are the (real) Wigner D-matrices, i.e.,

Figures (8)

  • Figure 1: Overview of the scalar field-based scoring function and docking procedure. The translational FFT procedure is shown here; the rotational FFT is similar, albeit harder to visualize. (A) The protein pocket and ligand conformer are independently passed through equivariant scalar field networks (ESFs) to produce scalar fields. (B) The fields are cross-correlated to produce heatmaps over ligand translations. (C) The ligand coordinates are translated to the argmax of the heatmap. Additional scalar field visualizations are in Appendix \ref{['app:fields']}.
  • Figure 2: Visualizations of learned scalar fields. All five channels of the ESF-N learned scalar fields $\phi^L$ (top row) and $\phi^P$ (bottom row) are shown on the $xy$-plane passing through the center of mass of the ligand, with a box diameter of 20 Å. Positive values of the field are in blue and negative values in red. At left, the ligand and pocket structures are shown looking down the $z$-axis. Note that as the fields are only 2D slices, not all 3D features visible in the structures are visible in the fields.
  • Figure 3: Visualizations of learned scalar fields, continued.
  • Figure 4: Decoy set statistics.Top left: histogram of RMSDs across all decoys sets (12 M total). Bottom left: histogram of minimum RMSDs among the decoy sets. All sets have a pose less than RMSD $<$1 Å from the true pose. Right: cumulative density function of RMSDs in each decoy set. Bottom right inset: all decoy sets have at least 23 poses with RMSD $<$2 Å.
  • Figure 5: PDE10A ligands aligned on 5SFS
  • ...and 3 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 1
  • proof