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SE(3)-Stochastic Flow Matching for Protein Backbone Generation
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Introduces FoldFlow, a family of simulation-free flow-matching generative models over SE(3)^N for protein backbone generation, progressing from deterministic flows (FoldFlow-Base) to Riemannian OT (FoldFlow-OT) to stochastic bridges (FoldFlow-SFM).
Problem
Generating novel, designable protein backbones as SE(3)^N-invariant distributions, where existing diffusion models are slow/unstable to train and cannot start from informative priors.
Approach
Extends Riemannian flow matching to SE(3)^N with a closed-form SO(3) conditional vector field, adds Riemannian minibatch optimal transport couplings for straighter flows, and a simulation-free approximation to Brownian bridges on SO(3) via isotropic Gaussian on SO(3) to yield a stochastic flow.
Key results
- FoldFlow-OT reaches 0.820 designability fraction (scRMSD<2Å) vs 0.612 for retrained FrameDiff and 0.969 for RFDiffusion.
- FoldFlow-SFM achieves the highest novelty (0.544 fraction with TM<0.5, avg max TM 0.411) among compared methods including RFDiffusion and Genie.
- FoldFlow-Base, OT and SFM all train at roughly 2.6-2.7 iters/sec versus 1.278 for FrameDiff-Retrained.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Riemannian OT paths instead of independent coupling (ablation) | FoldFlow-Base · Protein backbone design | Designability fraction: 0.657 ± 0.042 -> 0.820 ± 0.037 | Helped |
| Add stochasticity with guided stochastic bridges (ablation) | FoldFlow-OT · Protein backbone design | Novelty fraction: 0.484 ± 0.068 -> 0.544 ± 0.061 | Helped |
| Add stochasticity with guided stochastic bridges (ablation) | FoldFlow-OT · Protein backbone design | Designability fraction: 0.820 ± 0.037 -> 0.716 ± 0.040 | Hurt |
| Use an informed prior instead of a random prior (ablation) | FoldFlow · BPTI equilibrium conformation generation | W2 (all angles): 4.446 -> 4.379 | Helped |
| Use an informed prior instead of a random prior (ablation) | FoldFlow · BPTI equilibrium conformation generation | W2 @ 56: 0.557 -> 0.406 | Helped |
| Apply inference annealing by scaling the velocity (ablation) | FoldFlow · Protein backbone design | Designability | Helped |
| Use the simulation-free SO(3) bridge approximation instead of a simulated Brownian bridge | FoldFlow-SFM · Synthetic SO(3) multimodal densities | Wasserstein distance | No clear effect |
| Use Riemannian OT instead of independent coupling (ablation) | FoldFlow-Base · Synthetic SO(3) multimodal densities | Wasserstein distance: 5.39 ± 0.88 -> 4.96 ± 0.27 | Helped |
Surprising findings
- FoldFlow-OT is the most designable model but has relatively low diversity and novelty, while adding stochasticity (SFM) trades designability for novelty.
- Inference annealing (velocity scaling i(t)=ct with c≈10) is reported as critical for designability.
Limitations
- Evaluation limited to backbones up to 300 residues with 50 samples per length.
- Designability/novelty comparisons with RFDiffusion and Genie are confounded by differing training sets and pretraining.
- Brownian bridge on SO(3) has no closed form, so SFM relies on a simulation-free approximation.
Experimental setup
Datasets: PDB subset (22,248 proteins) BPTI molecular dynamics trajectory (1 ms)
Benchmarks: PDB subset test set synthetic multimodal SO(3) densities BPTI molecular dynamics trajectory
Metrics: designability (scRMSD < 2 Å fraction) mean scRMSD diversity (avg pairwise TM-score) novelty (max TM-score to PDB, fraction TM<0.5) iters/sec Wasserstein-1 and Wasserstein-2 distances W2 over dihedral angles
Baselines: RFDiffusion Genie FrameDiff-ICML FrameDiff-Improved FrameDiff-Retrained Simulated SDE
Builds on: Riemannian flow matching conditional flow matching FrameDiff AlphaFold2 structure module minibatch optimal transport conditional flow matching
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