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Harmonic Torsional Diffusion for Protein-Ligand Flexible Docking

The paper introduces Harmony, a harmonic torsional diffusion framework that explicitly models the periodic geometry of angular variables to significantly improve the accuracy and physical validity of flexible protein-ligand docking compared to existing methods.

Original authors: Maksim Zhdanov, Pavel Strashnov, Vladislav Kurenkov

Published 2026-08-24

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CC BY 4.0

Original authors: Maksim Zhdanov, Pavel Strashnov, Vladislav Kurenkov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

In the microscopic world of drug discovery, scientists are constantly trying to solve a three-dimensional puzzle. They need to fit a small molecule, known as a ligand, into a specific pocket on a much larger protein, much like finding the right key for a lock. For decades, computer programs designed to solve this puzzle operated under a simplifying assumption: that the protein lock remains perfectly rigid and unmoving. This made the math easier, but it ignored a fundamental truth of biology. Proteins are not static statues; they are dynamic machines that shift, twist, and reshape themselves when a drug molecule arrives. This movement is often essential for the drug to bind effectively. When researchers try to predict how a drug will attach to a protein while ignoring these movements, the computer models often produce results that look plausible on a screen but fail in the real world, leading to wasted time and resources in the laboratory.

A team of researchers has now introduced a new approach called Harmony, which changes how computers visualize this molecular interaction. Instead of treating the protein as a fixed object, Harmony allows the computer to simulate the natural flexibility of the protein's surface while the drug finds its place. The core innovation lies in how the software handles the angles of rotation. In a protein, many parts move by spinning around chemical bonds, similar to how a door swings on its hinges. These angles are circular; if you turn a door 360 degrees, you end up back where you started. Previous computer models often treated these angles as straight lines, which caused them to make errors when the rotation crossed the boundary between zero and 360 degrees. Harmony fixes this by building the mathematics of the model directly around the circular nature of these movements. It uses a method that understands that these angles wrap around, allowing the software to predict the most stable and accurate positions for both the drug and the shifting protein parts.

The researchers tested this new method against existing tools using a large collection of known protein-drug pairs from a database called PDBBind. They compared Harmony's predictions to those of other advanced computer models, including some that also attempt to handle protein flexibility. The results showed a clear improvement. When Harmony predicted where a drug would sit, it was correct within a very tight margin of error in 64.2 percent of the cases, a significant jump from the roughly 40 percent accuracy achieved by the best competing flexible docking methods. Furthermore, the model was better at reconstructing the exact shape of the protein's binding pocket, getting the position of individual atoms right more often than previous attempts. This suggests that by respecting the true geometry of how molecules rotate, the computer can learn the rules of binding more effectively without needing extra steps to fix its mistakes later.

To ensure these improvements were not just lucky guesses, the team also checked the physical validity of the generated structures using a separate set of tests called PoseBusters. These tests look for common physical errors, such as atoms crashing into each other or bonds stretching to impossible lengths. Harmony produced structures that passed these physical checks more often than its competitors, indicating that the predicted drug-protein combinations are not only geometrically accurate but also chemically realistic. The researchers demonstrated the method's power on two specific, difficult targets: a viral protein involved in maintaining the Epstein-Barr virus and a mutated protein linked to cancer. In both cases, where other models struggled to find the correct shape, Harmony successfully predicted a stable arrangement that closely matched the known biological reality.

The success of Harmony suggests that the way a computer model is built matters as much as the data it learns from. By aligning the mathematical rules of the simulation with the actual physical geometry of molecular rotation, the researchers created a tool that is simpler in its design but more powerful in its results. This approach does not require the computer to guess the rules of movement; instead, it builds those rules directly into the system. The findings indicate that acknowledging the circular nature of molecular angles is a key step forward for flexible docking. While the method currently focuses on the movement of side chains on the protein surface rather than the entire protein backbone, it offers a promising path toward more accurate drug design. The researchers have made their code available to the public, allowing other scientists to build upon this geometric insight to further refine how we understand the dance of molecules in the body.

Technical Summary: Harmonic Torsional Diffusion for Protein-Ligand Flexible Docking

Problem Statement
Molecular docking requires jointly reasoning about ligand pose and protein flexibility. While recent diffusion-based models have advanced the field, most predict torsional updates using generic Euclidean heads that ignore the periodic geometry of angular variables. This mismatch is particularly limiting in flexible docking, where ligand conformations and pocket side chains co-adapt to form the bound complex. Existing deep learning methods often rely on basic Riemannian generative machinery that fails to utilize the rich geometric structure inherent in conformational degrees-of-freedom, specifically the discrete rotameric states of protein side chains arising from rotations around $\chi$ dihedral angles.

Methodology: Harmony
The authors introduce Harmony, a harmonic torsional diffusion framework designed to address these geometric mismatches. The core innovation lies in parameterizing ligand and side-chain torsional score fields as derivatives of learned harmonic potentials on the circle, rather than predicting direct angular updates.

Variance-Exploding (VE) Diffusion on the Torus: The method adopts a VE-SDE framework where the forward noising process acts as a heat flow on the torus. For torsional angles $\vartheta$ on the circle $\mathbb{T}$ , the noise is applied as a wrapped Gaussian. The authors demonstrate that this process coincides with the heat kernel on the torus, satisfying a heat equation with time-dependent diffusivity.
Harmonic Parameterization: Leveraging the fact that the heat semigroup acts diagonally on the Fourier basis (eigenfunctions of the Laplace-Beltrami operator on the circle), Harmony represents the torsional score field using a finite Fourier expansion:
$\psi_e(\vartheta) = \sum_{k=1}^K [a_{e,k} \cos(k\vartheta) + b_{e,k} \sin(k\vartheta)]$
The model predicts the harmonic coefficients ( $a, b$ ) via a neural network. The score field is then derived analytically by differentiating the heat-smoothed harmonic expansion with respect to the angle.
Frequency-Aware Inductive Bias: The heat-semigroup damping factor, $\exp(-\frac{1}{2}k^2\sigma^2(\lambda))$ , is explicitly built into the score function. This ensures that high-frequency angular structures are suppressed at high noise levels and gradually re-emerge as the noise level decreases ( $\lambda \to 0$ ). This construction makes periodicity explicit and provides a frequency-aware inductive bias over rotameric motion.
Architecture: Harmony utilizes a heterogeneous SE(3)-equivariant graph neural network (based on e3nn) that processes ligand atoms, receptor residues (C $\alpha$ ), and receptor atoms within a binding pocket. The network jointly predicts rigid-body scores (translation/rotation) and the harmonic coefficients for both ligand torsions and flexible pocket side-chain torsions.

Key Contributions

Geometric Alignment: The paper proposes aligning the score parameterization with the intrinsic geometry of the diffusion process (the torus) rather than forcing torsional angles into Euclidean space.
Analytical Score Formulation: By using harmonic potentials, the method derives an analytical score function that naturally respects the periodicity of torsional angles and the noise-level dependence of the diffusion process.
Joint Modeling: The framework naturally models the coupled relationship between ligand conformation and protein side-chain orientations, capturing multi-frequency motion features essential for induced fit.

Results
The authors evaluate Harmony on the PDBBind benchmark (using ESMFold-predicted apo structures) and the PoseBusters set.

PDBBind Performance: Harmony achieves a ligand success rate (L-RMSD < 2 Å) of 64.2%, a significant improvement over recent flexible docking methods like FlexDock (41.7%) and DiffDock-Pocket (32.4%). It also improves pocket all-atom reconstruction (AA-RMSD < 1 Å) from 41.7% to 47.3%.
PoseBusters: On the PoseBusters benchmark, which tests physical validity without post-hoc relaxation, Harmony outperforms baselines, showing improvements in chemical validity, internal steric clash reduction, and ligand strain.
Case Studies: The method demonstrates robust behavior on challenging targets:
- EBNA1 (6NPP): Successfully maintains the enclosing loop structure in a broad binding site where baselines fail.
- KRAS G12D (6GJ8): Recovers accurate conformations in a shallow, highly exposed binding site where the baseline model produces large pose errors.
Ablation Studies: Experiments confirm that a small number of harmonic modes ( $K=4$ ) is optimal, and that the explicit frequency damping term is crucial for performance. Furthermore, applying harmonic learning to both ligand and side chains simultaneously yields better results than applying it to only one, validating the benefit of joint representation.

Significance and Claims
The paper claims that the primary contribution is demonstrating that "aligning the score parameterization with the geometry of the diffusion process is a simple and effective lever for improving flexible docking." The authors assert that respecting the intrinsic manifold structure of torsional motion is not merely a modeling detail but a key ingredient for accurate diffusion-based flexible docking.

The results indicate that Harmony improves ligand pose accuracy and pocket reconstruction without relying on auxiliary relaxation modules or post-processing steps that other methods (like FlexDock) require. The authors conclude that their approach provides a natural representation for the specific degrees-of-freedom central to molecular binding, leading to higher structural validity and physical plausibility in generated complexes. The source code is made publicly available to facilitate further research.

Technical Summary: Harmonic Torsional Diffusion for Protein-Ligand Flexible Docking

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