Physically Valid Biomolecular Interaction Modeling With Gauss-Seidel Projection

Siyuan Chen*1, Minghao Guo*†2, Caoliwen Wang1, Anka He Chen3, Yikun Zhang4, Jingjing Chai5, Yin Yang6, Wojciech Matusik2, Peter Yichen Chen†1
1University of British Columbia, 2MIT CSAIL, 3NVIDIA, 4Peking University, 5Foundry Biosciences, 6University of Utah
arXiv Code (Coming soon)
Teaser image

Our 2-step model achieves the same structural accuracy as state-of-the-art 200-step diffusion baselines, delivering ∼10× wall-clock speedups while guaranteeing physical validity.

Abstract

Biomolecular interaction modeling has been substantially advanced by foundation models, yet they often produce all-atom structures that violate basic steric feasibility. We address this limitation by enforcing physical validity as a strict constraint during both training and inference with a uniffed module. At its core is a differentiable projection that maps the provisional atom coordinates from the diffusion model to the nearest physically valid conffguration. This projection is achieved using a Gauss-Seidel scheme, which exploits the locality and sparsity of the constraints to ensure stable and fast convergence at scale. By implicit differentiation to obtain gradients, our module integrates seamlessly into existing frameworks for end-to-end ffnetuning. With our Gauss-Seidel projection module in place, two denoising steps are sufffcient to produce biomolecular complexes that are both physically valid and structurally accurate. Across six benchmarks, our 2-step model achieves the same structural accuracy as state-of-the-art 200-step diffusion baselines, delivering ∼10× wall-clock speedups while guaranteeing physical validity.



Provisional all-atom coordinates from the diffusion model are corrected by a Gauss-Seidel projection that sequentially resolves local constraints, each acting on a small set of atoms and updating coordinates in place. The module is differentiable via implicit differentiation, allowing seamless integration into training. The same projection is applied at inference, ensuring physical validity and enabling accurate predictions with as few as two denoising steps.

Evaluation

Visual comparison with the ground truth

We visualize our predicted protein structures together with their corresponding ground truth (pink colored) across a diverse set of PDB entries. Each protein's name is indicated in the top-left corner. Our method shows consistently close agreement with the ground truth, capturing the overall structural features with high fidelity.

7TPU

7XRX

7XS3

7Y1S

7Y4H

8B5W

8BBR

8BRY

8BVI

8CLZ

8CR7

8DAJ

8G0N

8G53

7TPU

8IUB

8JQE

8K5K

8OWF

8SUT


Qualitative Results

The red color highlights physicallyinvalid predictions, such as atomic clashes. Our approach consistently guarantees physical validity.



Cnvergence and Runtime

Convergence Speed: Gauss-Seidel quickly reduces potential, converging within 20 iterations; gradient descent oscillates and converges slowly (left). Runtime: On CASP15, our method is ∼9-10× faster than 200-step baselines and ∼23-46× faster than Boltz-1-Steering.



BibTeX

@misc{chen2025physicallyvalidbiomolecularinteraction,
      title={Physically Valid Biomolecular Interaction Modeling with Gauss-Seidel Projection}, 
      author={Siyuan Chen and Minghao Guo and Caoliwen Wang and Anka He Chen and Yikun Zhang and Jingjing Chai and Yin Yang and Wojciech Matusik and Peter Yichen Chen},
      year={2025},
      eprint={2510.08946},
      archivePrefix={arXiv},
      primaryClass={q-bio.BM},
      url={https://arxiv.org/abs/2510.08946}, 
}