# Harmonic Torsional Diffusion for Protein-Ligand Flexible Docking

arXiv: https://arxiv.org/abs/2608.20366 · HTML: https://mlsanity.com/paper/2608.20366/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 18 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [OpenMM 8: Molecular Dynamics Simulation with Machine Learning Potentials](https://mlsanity.com/paper/2310.03121/) (2023)
- [An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networks](https://mlsanity.com/paper/2501.13986/) (2025)
- [Re-Dock: Towards Flexible and Realistic Molecular Docking with Diffusion Bridge](https://mlsanity.com/paper/2402.11459/) (2024)
- [SigmaDock: Untwisting Molecular Docking With Fragment-Based SE(3) Diffusion](https://mlsanity.com/paper/2511.04854/) (2025)
- [Score Shocks: The Burgers Equation Structure of Diffusion Generative Models](https://mlsanity.com/paper/2604.07404/) (2026)

## Abstract (arXiv)

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, whose noise-level dependence is supplied analytically by the heat semigroup 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.