Harmonic Torsional Diffusion for Protein-Ligand Flexible Docking

Maksim Zhdanov; Pavel Strashnov; Vladislav Kurenkov

q-bio.BMJun 18, 2026

Harmonic Torsional Diffusion for Protein-Ligand Flexible Docking

Authors: Maksim Zhdanov, Pavel Strashnov, Vladislav Kurenkov

Organizations: AXXX

Abstract

Molecular docking requires reasoning jointly about ligand pose and protein flexibility. Most diffusion-based docking models predict torsional updates with generic Euclidean heads that ignore the periodic geometry of angular variables. This mismatch is especially limiting in flexible docking, where ligand conformations and pocket side chains co-adapt to form the bound complex. Here, we introduce Harmony, a harmonic torsional diffusion framework for flexible protein-ligand docking. Harmony parameterizes ligand and side-chain torsional score fields as derivatives of learned harmonic potentials on the circle, which are noised mode by mode in the Fourier basis owing to the properties of variance-exploding diffusion on the torus. This construction makes periodicity explicit and gives the model a frequency-aware inductive bias over rotameric motion. On the PDBBind benchmark, Harmony improves ligand pose accuracy and pocket all-atom reconstruction over recent flexible docking methods. On PoseBusters, it improves the physical validity of generated complexes. Case studies on EBNA1 and KRAS G12D illustrate the method's behavior on a polar and a shallow binding site, respectively. Together, these results indicate that aligning the score parameterization with the geometry of the diffusion process is a simple and effective lever for improving flexible docking.

Figures & tables

Figure 1: Overview of Harmony . Starting from a representation produced by any graph-based neural backbone, Harmony replaces direct torsion regression with a structured periodic parameterization. Instead of predicting a single angular update, the model learns harmonic coefficients that define a Fourier potential, whose derivative yields the torsional score field for ligand and flexible side-chain dihedrals. This construction makes periodicity explicit and enables the model to naturally represent difficult torsional configurations. See more in Section 4.3 .

Figure 2: Frequency damping effect of VE noising. As noise level increases high-frequency components saturate faster, but low-frequency modes remain. Thus, Harmony learns torsional motion naturally by filtering out different scales of molecular motion during training.

Method	Ligand RMSD		All-Atom	Runtime (s)
Method	% $<2$ Å $\uparrow$	Med. Å $\downarrow$	% $<1$ Å $\uparrow$	Runtime (s)
SMINA (rigid)	6.6	7.7	N.A.	258
SMINA	3.6	7.3	5.2	1914
GNINA (rigid)	6.7	7.1	N.A.	260
GNINA	8.4	7.9	4.5	1575
DiffDock-Pocket	41.8 $\pm$ 1.7	2.5	32.4 $\pm$ 1.5	17

Table 1: Top-1 PDBBind ESMFold Docking Performance . Harmony samples 10 complexes and selects one best on the highest confidence. For Harmony , DiffDock-Pocket, FIGRDock and DiffBindFR, we report mean

\pm

std over three training seeds. ReDock and FlexDock results are taken from prior work. The best results are highlighted in bold . N.A. indicates results missing due to baselines taken from previous work where output structures were not provided. Runtime of Harmony was measured for regular Clebsch-Gordan tensor product implementation and with OpenEquivariance CUDA kernels utilized in our base model.

Figure 3: PoseBusters validity on the PDBBind test set with no relaxation .

Figure 4: Ablations of Harmony . Left : Ablation of the number of harmonic modes

K

on PDBBind success rate. A moderate number of modes gives the best performance. Middle : Ablation of the VE heat-flow damping term. Removing damping degrades docking success, showing that explicit noise-dependent suppression of high-frequency torsional modes is important. Right : Applying harmonic learning only to ligand torsions improves side-chain modeling, while applying it only to side chains improves ligand success rate. This indicates that Harmony improves the joint representation used for flexible docking.

Panel of Figure 5 — Figure 5: Case studies on EBNA1 (6NPP) and KRAS G12D (6GJ8). Ground truth ligand is black and predicted ligand is red . Receptor is in its predicted configuration. In both examples, Harmony produces a more accurate ligand pose and improved local all-atom reconstruction than the baseline.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
Complex representation
$\mathcal{G}$ , $\mathcal{G}_{\mathrm{pocket}}$	Heterogeneous complex graph and its pocket-centered subgraph
$\mathcal{V}^{l}$ , $\mathcal{V}^{b}$ , $\mathcal{V}^{a}$	Ligand atoms, receptor residue nodes (at $C_{\alpha}$ ), receptor atoms
$\mathcal{E}^{l}$ , $\mathcal{E}^{b}$ , $\mathcal{E}^{a}$	Intra-graph ligand, residue and receptor-atom edges
$\mathcal{E}^{lb}$ , $\mathcal{E}^{la}$ , $\mathcal{E}^{ab}$	Ligand–residue, ligand–atom and atom–residue edges
Docking state

Appendix

Table 2: Summary of notation.

Figure 6: PoseBusters check-wise breakdown for the baseline model and Harmony . Each bar reports the pass rate for one PoseBusters criterion, highlighting that Harmony improves several physically meaningful checks, particularly those related to internal steric validity, ligand strain, and protein–ligand separation.

\mathbf{t}_{\lambda}=\mathbf{t}_{0}+\sigma_{\mathrm{tr}}(\lambda)\boldsymbol{\epsilon}_{\mathrm{tr}},\qquad\mathbf{R}_{\lambda}=\mathrm{Exp}(\boldsymbol{\omega}_{\lambda})\mathbf{R}_{0},

Appendix

Algorithm 1 Training of Harmony

\mathbf{t}_{N}\sim\mathcal{N}(\mathbf{0},\sigma_{\mathrm{tr},\max}^{2}\mathbf{I}_{3}),\qquad\mathbf{R}_{N}\sim\mathcal{U}(SO(3)),

Appendix

Algorithm 2 Inference with Harmony

\mathcal{F}\leftarrow\mathcal{F}\cup\{(e_{\mathrm{rot}},a):a\in C_{\mathrm{rot}}\}

Appendix

Algorithm 3 Finding Flexible Bonds and Rotated Fragments

\hat{\boldsymbol{\tau}}=\arg\min_{\boldsymbol{\tau}\in[-\pi,\pi]^{|\mathcal{B}^{l}|}}\mathrm{RMSD}\!\left(\mathrm{Rotate}(\mathbf{X}^{l,0},\boldsymbol{\tau}),\mathbf{X}^{l,\star}\right)

Appendix

Algorithm 4 Conformer Matching for Ligands and Side Chains

Method	Top-1 $\uparrow$	Top-5 $\uparrow$	Oracle-10 $\uparrow$
FlexDock	39.7 $\pm$ 1.1	47.7 $\pm$ 0.6	53.9 $\pm$ 1.2
Harmony	64.2 $\pm$ 0.7	70.2 $\pm$ 0.3	73.5 $\pm$ 0.8

Appendix

Table 3: Ligand success rate (L-RMSD

<2

Å, %) on PDBBind ESMFold under different sample selection rules (mean

\pm

std over three training seeds).

Model	L-RMSD $<2$ Å (%) $\uparrow$	AA-RMSD $<1$ Å (%) $\uparrow$
DiffDock-Pocket	41.8	32.4
DiffDock-Pocket + Harmony head	59.9	45.2

Appendix

Table 4: Replacing only the torsional head of DiffDock-Pocket with the Harmony head on PDBBind ESMFold (top-1).

Method	Circular MAE ( ∘ ) $\downarrow$	Acc@15 ∘ (%) $\uparrow$	Acc@30 ∘ (%) $\uparrow$	All-ligand@30 ∘ (%) $\uparrow$
FlexDock	57.7	28.0	45.0	10.0
Harmony	50.2	33.0	49.0	16.0

Appendix

Table 5: Ligand torsion accuracy on PoseBusters (top-1).

Method	$\chi_{1}$ MAE ( ∘ ) $\downarrow$	$\chi_{2}$ MAE ( ∘ ) $\downarrow$	$\chi_{1}$ @20 ∘ (%) $\uparrow$	$\chi_{2}$ @20 ∘ (%) $\uparrow$	Rotamer@20 ∘ (%) $\uparrow$
FlexDock	23.7	23.0	76.4	70.0	64.0
Harmony	14.6	18.8	89.0	82.0	73.0

Appendix

Table 6: Side-chain torsion accuracy on PoseBusters (top-1).

Method	L-RMSD $<2$ Å and PB-valid (%) $\uparrow$
FlexDock + OpenMM	34.2
Harmony + OpenMM	44.2

Appendix

Table 7: Success rate (L-RMSD

<2

Å and PB-valid) on the PoseBusters benchmark set after OpenMM relaxation.

Full-backbone model	L-RMSD $<2$ Å (%) $\uparrow$	AA-RMSD $<1$ Å (%) $\uparrow$
Riemannian flow matching baseline	38.9	39.9
+ harmonic torsional head	41.5	41.0

Appendix

Table 8: Harmonic torsional head in a Riemannian flow matching model with flexible backbone on PDBBind ESMFold (top-1).

Panel of Figure 7 — Figure 7: Extended results.

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Curriculum Learning for Generative Flexible Docking

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KTH Royal Institute of Technology · Science for Life Laboratory

Structure-Based Drug Design Curriculum Learning

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Flexible docking is a highly practical task, aiming to predict the joint configuration of a protein-ligand complex from a protein apo structure and a ligand SMILES string. Diffusion and flow-based generative models have made this task tractable by modeling the posterior distribution over the ligand pose and the protein conformation conditioned on the apo input. The central challenge, however, remains the long-tailed apo-to-holo structure shift, where a small fraction of protein undergo large conformational changes. In this work, motivated by the practical observation that downweighting large shift examples improves generalization, and by the mixed nature of the long tail consisting of genuinely difficult examples and noisy ones, we propose to use curriculum learning for this task, gradually incorporating large conformational change examples during training. With a linear pacing function, we show that an existing method can be strongly enhanced. On the PDBBind ESMFold apo benchmark, with relaxation and confidence-based pose selection, the resulting model GCLDock achieves a 46.52% top-1 success rate (ligand RMSD less than 2A) and 40.26% when additionally requiring PoseBusters validity, outperforming existing baselines.

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