Physically Valid Biomolecular Interaction Modeling with Gauss-Seidel Projection

TL;DR

This work tackles the problem that all-atom diffusion-based biomolecular predictors often produce physically invalid structures. It introduces a differentiable Gauss-Seidel projection that enforces physical validity during both training and inference, projecting provisional coordinates to the nearest feasible configuration. By decoupling validity from the denoiser and using only two denoising steps, the approach achieves competitive structural accuracy with roughly a 10x speedup over 200-step baselines while guaranteeing physical validity across six benchmarks. The method integrates as a differentiable layer with implicit differentiation and GPU-accelerated sweeps, enabling end-to-end fine-tuning within existing diffusion frameworks.

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 approximately 10 times faster wall-clock speed while guaranteeing physical validity.

Figures (9)

• Figure 1: Comparison on PDB 8B3E (protein-ligand complex): global structure (top) and two zoomed views of the binding pocket (bottom). Protenix-Mini with 5 denoising steps exhibits backbone-ligand clashing. With 200 steps, Boltz-1 resolves the backbone but leaves clashes between the side chain and the ligand. Boltz-1-Steering removes clashes by using physics-informed potentials, but at the cost of large sampling steps. Ours yields physically valid results with only $2$ denoising steps.
• Figure 2: Enforcing physical validity during both training and inference via our Gauss-Seidel projection module. 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.
• Figure 3: Importance of differentiable projection and finetuning. A post-hoc projection without finetuning applied to a $2$-step sampling, while ensuring physical validity, fails to recover the $\alpha$-helical secondary structure (left). Integrating the projection as a differentiable layer and finetuning the diffusion module restores the helix and improves overall structural accuracy (right).
• Figure 4: Qualitative comparison with baseline methods. The red color highlights physically invalid predictions, such as atomic clashes. Our approach consistently guarantees physical validity.
• Figure 5: Convergence and runtime. Left: Potential energy vs. iteration on PDB 8X51 under four noise levels ($\sigma\!\in\!\{160,120,80,40\}$). Gauss-Seidel projection (orange) converges to tolerance within $20$ iterations; gradient descent guidance (blue) oscillates and decays slowly. Right: Wall-clock inference time by atom-count bin on CASP15. Ours is ${\sim}10\times$ faster than baselines while maintaining validity. Runtime includes diffusion and validity module; measured on the same hardware.

Theorems & Definitions (4)

• Remark 4.1: Fast Convergence and Constraint Satisfaction of Gauss-Seidel Projection
• Theorem D.1
• Theorem E.1
• Theorem F.1