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Inference-Time Projection for Physically Valid Biomolecular Diffusion Models

Qurat-ul-ain, Yee Whye Teh, Charlotte M. Deane, Matteo Cagiada

arXiv:2610.07037v1cs.AIcs.LG

TL;DR

AF3-style cofolding models can produce structurally accurate complexes that violate physical geometry, while existing correction approaches can require costly steering or finetuning. This paper applies two closed-form projections to denoised coordinates; across five benchmarks and two models, it reports 100% physical validity while preserving structural-accuracy and ligand-position metrics.

  • Problem

    AF3-style models can generate accurate-looking structures with physical violations, while direct correction methods have relied on costly steering or model finetuning.

    Source: §1 · pp. 1–2

  • Method

    Two closed-form projections on denoised coordinates correct inter-chain clashes and ligand geometry without retraining the model.

    Source: §1 · p. 2

  • Results

    Physical validity reaches 100% on five benchmarks for Boltz-2 and OpenFold-3 while structural-accuracy and ligand-position metrics are preserved.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The results suggest bounded inference-time projection can remove targeted physical violations while leaving the base model’s structural prior intact.

    Source: §4 · p. 9

    The bounded partial projection does not guarantee feasibility, and the evaluated criteria are not a general certificate of physical realism.

    Source: §4 · p. 9

Abstract

from arXiv · show

AlphaFold 3-style cofolding models predict biomolecular complexes with high structural accuracy, yet a large fraction of their outputs are physically invalid: chains overlap at interfaces, ligand bond lengths and angles are distorted, rings are non-planar, and stereocentres are inverted. Current approaches either steer the sampler with physics-informed potentials, which multiplies sampling cost and memory overhead making inference impossible on large complexes, or finetune the model, costing time and tying the fix to one architecture. We observe that, unlike structural accuracy, physical validity is fully verifiable at inference time from quantities the sampler already holds. We therefore treat physical validity as a constrained inference problem and introduce two closed-form projection operators applied to the diffusion model's denoised clean-coordinate estimate, $\hat{x}_0$: an inter-chain van der Waals projection that pushes apart the most severely clashing atom pairs, and a ligand distance-geometry projection that restores bond lengths, angles, internal contacts, planarity and chirality. Both operators are local, sparse and displacement-capped, require no network evaluations, gradients or importance sampling, and leave the denoiser and its weights untouched, so they can be dropped into any AF3-style sampler without retraining. Applied to two independently developed models, Boltz-2 and OpenFold-3, across five benchmarks (CASP15, CASP16, the PoseBusters monomer and complex sets, and the Boltz physical-validity test set), our method recovers perfect physical validity while preserving structural-accuracy and ligand-placement metrics. These gains are achieved with negligible runtime and memory overhead, providing a practical, model-agnostic route to physically valid all-atom structure prediction.

1 INTRODUCTION

AF3-style models broaden all-atom complex prediction, but accurate-looking structures can still violate physical geometry needed for downstream use. The paper proposes inference-time projections as an alternative to costly steering or model finetuning, reporting perfect validity across five benchmarks while preserving accuracy metrics.

1 INTRODUCTION

  • Physical validity requires basic all-atom geometry constraints because violations can corrupt binding pockets and interaction placement despite structural accuracy.The described defects include strained bonds, invalid angles, clashes, distorted packing, chirality inversions, and overlapping chains.Source: §1 · pp. 1–2
  • Prior direct-correction approaches use costly Feynman–Kac steering or a finetuned differentiable projection that traded accuracy for efficiency.Boltz-2x uses importance sampling and gradient-based refinement; Gauss–Seidel Projection finetunes the base model.Source: §1 · p. 2
  • The paper asks whether physical validity can be added without costly pretraining or finetuning, or sacrificing speed and structural accuracy.Its projection is demonstrated on the independently developed Boltz-2 and OpenFold-3 models.Source: §1 · p. 2
  • Two closed-form projections correct inter-chain clashes and ligand geometry on the denoised clean-coordinate estimate without retraining.The ligand projection restores bond lengths, angles, internal contacts, planarity, and chirality.Source: §1 · p. 2
  • Physical validity reaches 100% on five benchmarks for both Boltz-2 and OpenFold-3, while structural-accuracy and ligand-position metrics are preserved.The benchmarks are CASP15, CASP16, PoseBusters monomers and complexes, and the Boltz test set.Source: §1 · p. 2

1.1 RELATED WORK

  • Unlike iterative projection solvers, this method replaces the inner solver with a single closed-form, displacement-bounded correction during denoising.Related projected-diffusion approaches perform iterative optimization inside every reverse step, adding substantial per-sample cost.Source: §1.1 · p. 3

2 METHODS

The method treats physical validity as an inference-time constraint verifiable from sampler-held structural information. It applies bounded local projections to denoised coordinates during the low-noise tail, when corrections can accumulate.

2 METHODS

  • Physical validity is verifiable at inference time from sampler-held quantities, including ligand distance bounds, ring planarity, and assigned stereochemistry.The projection constraints also use structural metadata such as atom masks, chain indices, connectivity, elements, and ligand bounds.Source: §2 · pp. 3–4
  • Two projections act in sequence on the same denoised estimate: van der Waals pushes apart severe inter-chain clashes, while ligand distance geometry restores molecular-graph constraints.The ligand operator handles bond and angle intervals, non-bonded clashes, ring and double-bond planarity, and stereocentre handedness.Source: §2 · p. 4
  • Each operator caps the atom’s combined displacement and computes corrections in closed form without network evaluations, gradients, or weight updates.The net displacement is capped after summing contributions from selected violated constraints.Source: §2 · p. 4
  • Projection during churn leaves validity at 0.02, whereas projection in the deterministic tail reduces violations and reaches 1.00.During churn, injected noise overwrites corrections; in the tail, denoiser motion falls below the projection cap so corrections accumulate.Source: §2 · p. 4
  • Projection is activated in the tail on the coherent clean-coordinate estimate, because noisy coordinates are not yet meaningful for steric constraints.The paper sets projection start t = 0.2, around step 160 of 200.Source: §2 · p. 4

3 RESULTS

Across two independently developed AF3-style models, projection achieves perfect physical validity while preserving structural accuracy. The results also show that corrections accumulate during the sampler’s tail, with little added computational cost, and that applying them in-loop is important for retaining accuracy.

3.1 HOW PROJECTION INTERACTS WITH THE LEARNED PRIOR

  • The model’s unprojected predictions become nearly clash-free during projection, as penetration falls 80.3% across the window while the seed-matched control rises 12.0%.The authors interpret this as the model internalizing corrections rather than being held apart by force at the final step.Source: §3.1 · p. 6

3.2 EXPERIMENTAL SETUP

  • Evaluation covers five benchmark sets and compares projected Boltz-2 and OpenFold-3 with their unmodified bases, plus Boltz-2x as an inference-time method.A direct comparison with GSP was not possible because publicly released model checkpoints were unavailable.Source: §3.2 · p. 6

3.3 ACCURACY AND VALIDITY

  • Both projected models reach 100% physical-validity pass rates on every evaluated dataset, with structural accuracy unchanged.The two independently trained samplers support the method’s model-agnostic application.Source: §3.3 · p. 6

3.4 EFFICIENCY

  • Runtime with projection stays within 3.5% of the base model across size bins, while peak memory shows no measurable increase.Boltz-2x costs 1.5–2.1× the base model; the comparison covers CASP16 and PoseBusters Complex.Source: §3.4 · pp. 6–7, Figure 3 · p. 8

3.5 ABLATIONS

  • Projecting only during churn reaches 0.02 validity on x̂0 and 0.04 on xt, whereas repeated post-hoc projection reaches 0.90 but lowers LDDT from 0.81 to 0.79.The final configuration projects in-loop after churn ends; the table covers 134 targets with at least one invalid Boltz-2 sample.Source: Table 2 · p. 8, §3.5 · p. 9

4 DISCUSSION AND LIMITATIONS

The discussion frames most physical-validity failures as residual local violations rather than failures of global structural reasoning, and argues that bounded projection can remove them while retaining the base model’s structural prior. It cautions that validity is empirical and limited to the enforced and evaluated constraint families.

4 DISCUSSION AND LIMITATIONS

  • On the studied benchmarks, bounded inference-time projection removes residual validity failures while leaving the base model’s structural prior intact.The discussion presents learned models as supplying global structural hypotheses and constrained operators as enforcing local physical constraints.Source: §4 · p. 9
  • The projection is partial and bounded, so reaching validity is empirical rather than guaranteed.Each step caps atom displacement, and only the K most deeply penetrating pairs are corrected.Source: §4 · p. 9
  • The reported validity criteria cover enforced constraint families, not general physical realism; torsional strain, protein backbone geometry, and hydrogen placement are neither corrected nor assessed.The paper cites independent checks of covalent geometry, van der Waals energies, and alpha-complex contacts, but notes these do not make the evaluation a general certificate.Source: §4 · p. 9

A CHARACTERIZING PHYSICAL VIOLATIONS

The section characterizes physical-validity failures as local violations that can occur even when overall fold or ligand placement appears plausible. It relates these failures to all-atom Cartesian diffusion, where geometry constraints are less built in than in earlier constrained representations.

A CHARACTERIZING PHYSICAL VIOLATIONS

  • Physical-validity failures include inter-chain clashes, ligand bond and angle distortions, internal clashes, stereochemical errors, and non-planar rings or double-bond groups.Figure 4 illustrates these categories; such defects can occur despite plausible overall folds or ligand placement and can compromise downstream use.Source: Figure 4 · p. 13, A CHARACTERIZING PHYSICAL VIOLATIONS · p. 13
  • Table 3 catalogs physical-validity failure modes for Boltz-2 and OpenFold-3, including per-target violation totals, chain-clash counts, and oracle physical-validity percentages.Its oracle selects a valid sample when available among five, otherwise the sample with fewest violations.Source: Table 3 · p. 14, A CHARACTERIZING PHYSICAL VIOLATIONS · p. 14

B.1 BIOMOLECULAR DIFFUSION MODELS

  • All-atom Cartesian generation removes many built-in constraints on bonds, angles, steric exclusion, and stereochemistry found in earlier constrained representations.This flexibility supports heterogeneous complexes with arbitrary ligands and modified residues, while making physical validity a more explicit inference problem.Source: §B.1 · pp. 14–15
  • AF3-style models iteratively denoise noisy all-atom coordinates into a final complex structure after forming a trunk representation from sequence, MSA, templates, and molecular features.The section situates Boltz-2 and OpenFold-3 among open-source models following this cofolding paradigm.Source: §B.1 · p. 15

C METHODS

The method projects the denoised estimate with a sparse, bounded van der Waals correction for inter-chain clashes and a ligand distance-geometry correction for internal geometry. Both operators limit per-atom motion and apply late in the sampling trajectory.

C.1 VAN DER WAALS PROJECTION

  • The van der Waals projection acts on denoised coordinates during the late sampling window, returning them unchanged outside it.With T = 200 and projection start t = 0.2, the active window is the final 40 steps.Source: §C.1 · p. 16
  • It flags candidate non-covalently connected atom pairs only when their distance falls below 85% of their summed van der Waals radii, then selects the K = 200 deepest overlaps.The β = 0.15 buffer admits mild contacts; the operator targets clear inter-chain overlaps.Source: §C.1 · p. 16
  • Selected pairs receive damped, symmetric push-apart updates, whose accumulated per-atom displacement is capped to keep corrections local and minimal.Boltz-2 uses α = 0.25 and OpenFold-3 uses α = 1; the cap is applied after pairwise displacements are accumulated.Source: §C.1 · p. 16

C.2 LIGAND DISTANCE-GEOMETRY PROJECTION

  • The ligand projection builds widened RDKit distance bounds for bonded and angle pairs, lower bounds for other heavy-atom pairs, and sets for planar groups and assigned stereocentres.Bond and angle pairs are corrected as two-sided intervals, while other pairs receive only a lower bound.Source: §C.2 · p. 17
  • Ligand sweeps correct chirality, bonds and angles, internal clashes, then planarity; a latch keeps inverted stereocentres active until they reach a handedness margin.Bonds and angles are re-seated after clash correction, and planarity runs last because it moves whole ring systems.Source: §C.2 · p. 19

D SAMPLER DYNAMICS AND THE PROJECTION WINDOW

Tail projection reaches full physical validity in the reported benchmark comparison, unlike projection confined to churn, while accuracy remains unchanged. The sampler analysis explains why corrections to the denoised estimate accumulate after churn ends.

D SAMPLER DYNAMICS AND THE PROJECTION WINDOW

  • The denoised estimate’s RMSD drops from about 1 Å to about 0.1 Å when churn switches off, while its all-atom LDDT levels off within a few steps.The iterate xt tracks noise level σ until the churn boundary; the remaining approximately 45 steps refine the structure by less than 0.1 Å.Source: D SAMPLER DYNAMICS AND THE PROJECTION… · p. 19, Figure 5 · p. 20
  • Projecting x0 targets the current best structural estimate rather than noise, and after churn is off projecting either x0 or xt gives the same validity and accuracy.Once churn ends, x0 and xt are within σ ≤ 1 Å of each other.Source: D SAMPLER DYNAMICS AND THE PROJECTION… · p. 19
  • Churn-only corrections are overwritten by injected noise, whereas corrections in the tail accumulate after fresh noise stops and x0 changes by less than 0.1 Å RMSD.The diagnostic’s remaining pairs largely fall between the projection’s 0.85 and evaluation’s 0.75 van der Waals thresholds; 19 of 20 projected structures are clash-free.Source: D SAMPLER DYNAMICS AND THE PROJECTION… · pp. 19–20
  • 1.00 physical validity with tail projection contrasts with 0.02 for churn-only projection on x0 and 0.04 on xt, with unchanged accuracy.The passage reports the benchmark comparison for the two projection windows.Source: D SAMPLER DYNAMICS AND THE PROJECTION… · p. 20

D.1 PROJECTION DYNAMICS

Projection is activated late, after the denoiser has largely settled the fold, to repair remaining local steric clashes. The corrections stay small and capped while van der Waals penetration decreases toward the terminal structure.

D.1 PROJECTION DYNAMICS

  • The ˆx0 backbone is close to its final accuracy from the first step, while xt becomes meaningful only after churn switches off.Source: Figure 7 · p. 21
  • Van der Waals penetration consistently decreases toward zero across CASP15 targets as sampling approaches the terminal structure.Penetration is nonzero only for atom pairs that are too close; zero indicates the pair satisfies the van der Waals separation constraint.Source: §D.1 · p. 21, Figure 8 · p. 21
  • RMS and maximum per-atom projection displacements shrink over the final steps as clashes are resolved, with maximum displacement capped at δmax.The RMS correction decreases throughout the trajectory, indicating progressively smaller required corrections.Source: §D.1 · p. 21
  • Late projection targets residual local clashes after the denoiser has settled the fold, acting as a sparse repair rather than a global energy minimization.In the described experiment, projection starts at t = 0.2, corresponding to the final 40 steps of the 200-step sampler.Source: §D.1 · pp. 20, 22

E.1 EVALUATION CRITERIA

The evaluation compares structural accuracy, physical validity, runtime, and GPU memory under matched inference settings. Physical validity requires passing all considered geometry and clash checks.

E.1 EVALUATION CRITERIA

  • All methods use the same target sets, MSA settings, random seed, post-processing, and metric scripts, with five samples per target by default.The projection method uses the same model weights and sampling schedule, enabling comparison with projection as the only added inference-time operation.Source: §E.1 · p. 22
  • A prediction counts as physically valid only if it passes every considered check, including ligand geometry, stereochemistry, planarity, and inter-chain clashes.The planarity check contains a corrected bug, so the reported Boltz validity values are higher than those in the original Boltz paper.Source: §E.1 · p. 22
  • Runtime is measured per target excluding preprocessing, and peak GPU memory is measured on A100 80GB GPUs; reported means are across targets.Uncertainty is computed by bootstrap resampling over targets.Source: §E.1 · p. 22
  • DockQ and ligand-placement metrics are scored only on targets with an interface or ligand score, so those columns use subsets.Source: §E.1 · p. 23

E.2 BENCHMARK TARGETS

The benchmark evaluation covers five target sets and reports a qualitative comparison of projection with steering on two PoseBusters monomer targets. Projection preserves the base-model oracle pose in those examples, while steering yields tighter ensembles with worse oracle RMSD.

E.2 BENCHMARK TARGETS

  • The five benchmark sets total 966 targets, and comparisons use only targets completed by every Boltz method.Boltz-2 and Ours (Boltz) completed all targets; Boltz-2x ran out of memory on 9 targets on an 80 GB A100.Source: §E.2 · p. 23

E.3 QUALITATIVE RESULTS

  • On 7n7h and 7p5t, projection preserves the base-model oracle RMSD at 0.31 Å and 1.37 Å, respectively, while achieving physical validity.Source: §E.3 · p. 24
  • On 7n7h and 7p5t, Boltz-2x has σ values of 0.05 and 0.21 Å versus 1.58 and 4.89 Å, but oracle RMSD worsens to 3.71 Å and 11.16 Å.On 7p5t, every steered sample places the ligand tail outside the crystallographic channel.Source: §E.3 · p. 24

F TIME AND MEMORY

Projection adds negligible inference cost and leaves peak GPU memory unchanged. Structural checks also find that it removes steric violations while leaving accuracy, covalent geometry, and local packing statistically indistinguishable from the base model.

F TIME AND MEMORY

  • Ours (Boltz) runs at 1.001–1.007× its time per target with unchanged peak GPU memory, while Ours (OpenFold) runs at 0.99–1.03× its baseline time with memory unchanged.Boltz-2x takes roughly 1.7–2.1× Boltz-2's time, requires up to 78 GiB versus at most 44 GiB, and runs out of memory on 9 targets that Boltz-2 and Ours (Boltz) complete.Source: F TIME AND MEMORY · p. 24
  • Projection reproduces the base-model ligand ensemble almost exactly and retains its oracle pose, whereas steering collapses the ensemble onto a tight but incorrect binding mode.Source: Figure 10 · p. 25

G STRUCTURAL EFFECTS OF THE PROJECTION

G.1 IN-LOOP PROJECTION VERSUS POST-HOC CORRECTION

In-loop projection preserves structural accuracy better than post-hoc correction while restoring physical validity; its van der Waals projection leaves protein-chain covalent geometry essentially unchanged. The Phenix comparison covers only ligand-free CASP16 targets with available restraints.

G.1 IN-LOOP PROJECTION VERSUS POST-HOC CORRECTION

  • The comparison tests in-loop projection against post-hoc correction on final structures, using the same projection operator and hyperparameters for the post-hoc alternative.Phenix minimization is evaluated on 42 of 52 CASP16 targets because restraints could not be generated for ligand-bearing targets.Source: §G.1 · pp. 25–26
  • A single post-hoc projection pass removes 14% of targeted violations and reaches 0.22 validity, versus 1.00 in-loop; 40 passes reach 0.90 validity but lower LDDT to 0.79 versus 0.81.Each pass caps atom movement, and after sampling the denoiser cannot absorb corrections into the structure.Source: §G.1 · pp. 25–26
  • 1.000 validity with in-loop projection versus 0.881 after Phenix minimization, while projection remains statistically indistinguishable from baseline on structural metrics.Projection also significantly outperforms minimization on LDDT (+0.022) and backbone LDDT (+0.027).Source: §G.1 · p. 26, Table 5 · p. 27

G.2 COVALENT GEOMETRY

  • Bonded protein-chain geometry is essentially unchanged: bond RMSD is 0.0062 Å with projection versus 0.0060 Å for Boltz-2, and angle RMSD is 0.955° versus 0.940°.The MolProbity analysis removes ligands, so this assessment covers protein chains; ligand-internal geometry is assessed only by the validity criterion.Source: §G.2 · p. 27

G.3 PACKING AND NON-BONDED ENERGETICS

The projection removes severe contacts while leaving post-relaxation energies and residue contact topology close to the base model. Hyperparameter sensitivity results come from an enriched set of hard, clash-heavy targets, so their accuracy values are relative comparisons rather than benchmark-level estimates.

G.3 PACKING AND NON-BONDED ENERGETICS

  • Figure 11 compares target-wise MolProbity metric distributions for experimental structures, Boltz-2, and projection, with dotted median lines.Metrics include MolProbity score, bond and angle RMSD, and maximum bond, angle, and dihedral deviations.Source: Figure 11 · p. 28
  • The median change in non-covalent neighbours is +0.003 per residue (≤0.01%), and even the largest changes on RNA and large multimers stay below 2%.The reported contact topology remains essentially unchanged without degradation in covalent geometry or energy.Source: §G.3 · p. 28, Table 7 · p. 29
  • 224.5 versus 249.5 kJ/mol per residue is the median pre-relaxation Lennard-Jones energy under projection and Boltz-2; after identical relaxation, medians are 145.1 versus 145.5.On T1272s5, energy drops from approximately 6.4 × 10^6 to 1.1 × 10^4 kJ/mol per residue; after relaxation, the methods agree within a fraction of a kJ/mol on every target.Source: §G.3 · pp. 27–28, Table 7 · p. 29

H HYPERPARAMETER SENSITIVITY

  • The 134-target sensitivity set includes targets where unmodified Boltz-2 produced at least one invalid sample; its accuracy values are lower than benchmark averages and support row-to-row comparisons only.The passage directs readers to main-text Table 1 for benchmark-level comparisons.Source: H HYPERPARAMETER SENSITIVITY · p. 29, Table 8 · p. 30
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