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Fold'EM: Direct atomic structure inference from Cryo-EM particles

Advaith Maddipatla, Märt-Erik Mäeots, Marco Pegoraro, Nikolaus Dräger, Roberto Covino, Sanketh Vedula, Martin Pacesa, Alex M. Bronstein

arXiv:2610.01358v1q-bio.BMcs.AI

TL;DR

Cryo-EM typically averages noisy particle images into a density map before atomic modeling, while sequence-derived structural priors have generally been applied only after that reconstruction. Fold’EM guides a protein generative model directly with particle evidence, recovering accurate structures across known-pose, ab initio, and heterogeneous settings using substantially fewer particles.

  • Problem

    Cryo-EM generally requires many noisy particle images for density reconstruction before atomic modeling, while structural priors have typically been applied only after reconstruction.

    Source: §1 · pp. 1–2

  • Method

    Fold’EM uses sequence-conditioned structural priors to jointly infer atomic structures and particle-specific pose or class variables directly from cryo-EM images.

    Source: §1.1 · p. 2, §2 · p. 4

  • Results

    Fold’EM recovers accurate structures with fewer than 1% of conventional particles for known poses, roughly 3% ab initio, and roughly 5% for heterogeneous samples.

    Source: §1.1 · p. 3, §6 · p. 20

  • Takeaways & Limitations

    The results suggest a route to characterizing low-population conformational states directly from cryo-EM particle images.

    Source: §1.1 · p. 4

    Fold’EM currently requires curated particle collections with known CTF parameters, and ab initio inference relies on accurate pose estimates for initialization.

    Source: §6 · p. 20

Abstract

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Single-particle cryo-electron microscopy (cryo-EM) has become a widely adopted technique for biomolecular structure determination. The conventional cryo-EM computational pipeline first combines many particle images to reconstruct an electrostatic potential (ESP) map and then fits an atomic model to the recovered map. Density reconstruction has high sample complexity, requiring large numbers of particle images and making structure determination high-cost and low-throughput, particularly for heterogeneous samples. Downstream atomic model building, in turn, becomes increasingly difficult as the resolution of the reconstructed map deteriorates. Protein structure prediction models provide strong sequence-derived priors on atomic structure, and experiment-guided approaches can use these priors to recover structures consistent with experimental measurements. Yet, in cryo-EM, such priors are typically integrated only after density reconstruction during atomic model fitting. We introduce Fold'EM, an inference-time framework that combines priors from protein generative models directly with cryo-EM particle images to determine atomic models from a small number of single particle images, bypassing both intermediate density reconstruction and downstream model building against the reconstructed map. Across synthetic and experimental cryo-EM datasets, Fold'EM recovers accurate atomic structures both with known particle orientations and in an ab-initio setting where orientations are inferred jointly with structure. In heterogeneous datasets, Fold'EM further resolves distinct conformational states from mixed particle populations without separately reconstructing a density map and building an atomic model for each state. We believe these results open new avenues for structure determination in the low-sample regime and for characterizing low-population conformational states directly from cryo-EM particles.

1 Introduction

Cryo-EM’s noisy particle images usually must be averaged into a density map before atomic modeling, a demanding pipeline for scarce or heterogeneous samples. Fold’EM instead combines sequence-conditioned structural priors with particle-level evidence to infer atomic structures and particle-specific variables directly.

1 Introduction

  • Radiation damage makes individual cryo-EM images too noisy for direct structural interpretation, while scarce or transient conformations may lack enough particles for separate reconstruction.Separating conformations leaves fewer observations per state, and combining them can blur structural features.Source: §1 · pp. 1–2
  • Existing heterogeneous modeling generally requires prior particle orientations, while cryo-EM approaches using structural priors have focused on reconstructed maps and inherit their particle-count requirements.Blush regularization is an example of map-based prior knowledge, using a denoiser trained on experimental cryo-EM maps.Source: §1 · p. 2

1.1 Contributions and main results

  • Fold’EM jointly infers atomic structures and particle-specific pose and conformational assignment directly from sparse images, bypassing density reconstruction and model building against a map.Its ab initio variant alternates structure inference with pose and class refinement.Source: §1.1 · pp. 2, 4
  • Particle-level guidance preserves the experiment-induced metric for known poses and retains particle-specific information that consensus density can lose under pose or class uncertainty.Standard density-space L2 mismatch does not generally reproduce the particle likelihood.Source: §1.1 · p. 2
  • For integrin–BIIG2, ab initio Fold’EM recovers the bound complex from 1,000 particles—0.26% of those used for reconstruction—with 2.49 Å RMSD, TM-score 0.956, and 3.52 Å FSC resolution.AlphaFold3 predicts the components but places the Fab at the wrong binding site; the deposited map has 2.6 Å resolution.Source: Figure 1 · p. 3
  • Optimizing intermediate Pairformer states passes experimental updates through the remaining blocks, inducing a sequence- and state-dependent geometry that strengthens guidance.The authors identify this reparameterization as a key enabling component of direct particle-level inference.Source: §1.1 · p. 3
  • Fold’EM recovers accurate structures with fewer than 1% of conventional particles for known poses, roughly 3% ab initio, and roughly 5% for heterogeneous state resolution.These results suggest a route to characterizing low-population conformational states directly from particle images.Source: §1.1 · pp. 3–4
  • In continuously heterogeneous systems, conformational organization in particle space is reflected in AlphaFold embedding space, allowing a trajectory to be represented by a small number of optimized anchor states.Together, the results suggest a route to characterizing low-population states from cryo-EM particles.Source: §1.1 · pp. 3–4

2 Problem formulation

Fold'EM represents structures through intermediate Pairformer states and jointly infers structures and particle-specific nuisance variables from raw cryo-EM images. Its particle-level formulation preserves information that density fitting can lose, especially when poses or classes are uncertain.

2 Problem formulation

  • Fold'EM parameterizes an ensemble of structures with intermediate Pairformer states and jointly infers particle poses and class assignments from raw images.The formulation assumes imaging parameters other than pose and class are known.Source: §2 · p. 4
  • The sequence-conditioned prior is relaxed around the precursor Pairformer state, while the inference objective combines experimental likelihood with structural and nuisance-variable priors.The generator is stochastic, so Jensen's inequality yields a tractable ELBO-like surrogate for the objective.Source: §2 · p. 5
  • Optimizing intermediate Pairformer states induces sequence- and state-dependent preconditioning in terminal conditioning space, amplifying sensitive directions and suppressing insensitive ones.The local preconditioner is JJ^T, determined by the Pairformer's Jacobian.Source: §2 · p. 5
  • With known poses, fitting structures to particle likelihoods is equivalent to density fitting only when the density loss uses the experiment-induced metric Q.The ordinary density-space L2 norm generally does not provide this metric.Source: §2 · p. 6
  • With uncertain poses or classes, particle-level fitting retains particle-specific information that can be lost when observations are collapsed into a reconstructed map.With uncertain poses, the sample mean is not a sufficient statistic because the likelihood also depends on the sum of squared observations.Source: §2 · p. 6

3 Methods

Fold’EM jointly infers atomic structures and particle-specific variables directly from cryo-EM images, alternating structural updates with pose and class refinement. Its setup uses sequence as the only structural input, estimates poses and classes when unknown, and treats coordinate ascent as an organizing principle rather than a guarantee of monotonic improvement.

3 Methods

  • Fold’EM jointly infers structural states and particle-specific nuisance variables from raw images, alternating structure and nuisance updates.For heterogeneous data, the nuisance variables include pose and discrete class.Source: §3 · p. 6, §3.2 · pp. 7–8

3.1 Inputs, initialization, and reference frame

  • The sequence is the only structural input; pixel and box size, CTF parameters, and centered particles are assumed known.The number of classes is specified, while poses and class labels are supplied only in known-pose and known-class evaluations.Source: §3.1 · p. 7
  • Ab-initio pose estimation uses VDAM, Wiener back-projection, and AlphaFold3 docking, then fixes the global SE(3) gauge relative to the docked structure.VDAM is a variable-metric gradient descent algorithm with adaptive moments estimation.Source: §3.1 · p. 7
  • B-RECOVAR classifies particles by combining residual-subspace estimates, cross-validated class comparisons, and votes across randomized subsamples and frequency bands.The vote margin supplies per-particle confidence; the method also extracts successive modes for more than two states and interprets them as variability directions for continuous heterogeneity.Source: §3.1 · p. 7

3.2 Ensemble inference

  • Structural updates backpropagate likelihood gradients into intermediate Pairformer states, which condition subsequent reverse-diffusion steps and persist across stochastic runs.The update optimizes H rather than the terminal Pairformer representation; resampling diffusion noise encourages the state to explain observations across realizations.Source: §3.2 · p. 8
  • Nuisance updates refine each particle’s pose and class against the current structural ensemble, with the two update blocks alternated.The structures and poses remain in a docked reference frame, fixing the otherwise arbitrary global SE(3) gauge.Source: §3.2 · pp. 8–9
  • Monotonic improvement is not guaranteed because structural updates use stochastic diffusion samples, particle minibatches, and finite gradient steps.The coordinate-ascent interpretation is used primarily to organize joint inference.Source: §3.2 · p. 9

4 Experiments

The experiments test particle-level guidance against density guidance, compare guidance variables, and evaluate structure recovery across homogeneous, discrete heterogeneous, and continuous-heterogeneity settings. Fold’EM recovers accurate structures from sparse particles, while performance depends on pose and class inference and on how well the unguided prediction matches the target.

4 Experiments

  • The experiments span synthetic and EMPIAR particles, evaluating particle-versus-density guidance, homogeneous and heterogeneous structure inference, label-free class assignment, and continuous conformational variation.Baselines use the same particles; pose and class settings vary by experiment, and the reported comparisons include AlphaFold3 and density-reconstruction methods.Source: §4 · p. 9, §4.3 · p. 11, §4.4 · p. 13

4.1 Particle guidance vs. density guidance

  • At equal particle count on 8U4E, Fold’EM’s particle guidance recovers the assembly while density guidance does not.From 500 particles at SNR 0.005, Fold’EM reaches 6.88 Å Cα RMSD and 0.809 TM-score, versus 17.95 Å RMSD for density guidance.Source: §4.1 · p. 9, Figure 2 · p. 10

4.2 Optimizing H vs. Z vs. coordinate-space guidance

  • On 8U4B, optimizing the intermediate Pairformer state recovers the deposited structure, whereas terminal-embedding and coordinate-space guidance degrade the initial prediction.The three methods use the same particle likelihood, poses, and particle subsets; on 9UGB all correct the AlphaFold3 prediction, with Fold’EM best on RMSD, TM-score, and FSC resolution.Source: Figure 3 · p. 10, §4.2 · p. 11

4.3 Guidance on homogeneous samples

  • Fold’EM recovers accurate structures in homogeneous samples from few particles, while 8PWH is an exception where density methods achieve finer FSC resolution at larger counts.On synthetic Pma1, 100 known-pose particles yield 3.25 Å FSC resolution and a 0.965 TM-score; on integrin α5β1–BIIG2, 400 ab-initio particles recover the epitope, though interface variable loops are not atomic-accurate.Source: §4.3 · pp. 11, 13, Figure 4 · p. 12

4.4 Guidance under two-state heterogeneity

  • Fold’EM resolves both states in synthetic Pma1 and experimental UvrA mixtures, with larger ab-initio particle requirements for the UvrA state absent from AlphaFold3.For UvrA, 9GXQ requires 10,000 particles per state ab initio and reaches 4.63 Å; the recovered wedge insertion distorts DNA, but the flipped-out nucleotide is not recovered because intact DNA was supplied.Source: §4.4 · pp. 13, 16, Figure 5 · p. 14, Figure 6 · p. 15

4.5 Label-free class assignment with B-RECOVAR

  • From 500 particles per state onward, B-RECOVAR assignments are more accurate and stable than RECOVAR and cryoDRGN, reaching 92.5% accuracy at 5,000 particles per state.Its normalized ensemble-vote margin orders labels by reliability, although it is not a calibrated probability; Fold’EM uses that ordering to guide with more confident particles.Source: Figure 7 · p. 16, §4.5 · p. 17
  • The ensemble over particle subsets, frequency bands, and initializations yields 80.6% class-assignment accuracy and raises score separation from d′ = 1.13 to d′ = 2.33.The preceding solvent-mask and cross-validated label-refinement stages reach 69.1% and 72.1% accuracy, respectively.Source: §4.5 · p. 17

4.6 Guidance under continuous heterogeneity

  • Eight optimized anchors suffice to represent the continuous IgG hinge trajectory: interpolated B-RECOVAR structures attain median TM-score 0.93 and 92% of ground-truth NCC.Anchors are optimized independently; the leading Pairformer components nevertheless preserve hinge-angle ordering across all three particle coordinates.Source: Figure 8 · p. 18, Figure 9 · p. 18, §4.6 · p. 19

5 Related work

Prior cryo-EM methods reconstruct densities or guide protein predictions using reconstructed maps; Fold’EM’s closest prior work instead constrains predictions with individual particle images.

5 Related work

  • CryoDRGN learns continuous structural variation and RECOVAR estimates volume covariance, but both use poses from a prior consensus reconstruction.CryoDRGN-AI jointly estimates poses and structural distributions from images; CryoDRGN, RECOVAR, and CryoDRGN-AI.Source: §5 · p. 19
  • Map-guided approaches include CryoBoltz, which needs an approximately aligned input structure, and methods that guide AlphaFold3 ab initio from sequence.Other approaches optimize AlphaFold latent representations against synthetic ESP maps; CryoBoltz (Raghu et al., 2026), Maddipatla et al. (2026b), Embed-Opt (Li et al., 2026), and IT-Opt (Maddipatla et al., 2026a).Source: §5 · p. 19
  • CoCoFold constrains AlphaFold using individual cryo-EM particles but assumes upstream pose and CTF estimates and a coarse reconstruction for its coordinate frame.It fine-tunes the AlphaFold-Multimer structure module with an FRC loss between particles and differentiable projections.Source: §5 · p. 19

6 Conclusion

Fold’EM guides AlphaFold3 with particle-level evidence to infer atomic structures without density reconstruction or model building. It reports accurate structures from small particle fractions, while noting that inputs and local-detail recovery remain constrained.

6 Conclusion

  • Fold’EM infers atomic structures directly from particle images, bypassing density reconstruction and downstream model building.It jointly infers structural representations and particle-specific latent variables, including pose and conformational assignment.Source: §6 · p. 20
  • Fold’EM recovers accurate structures with fewer than 1% of conventional particles for known poses, roughly 3% ab initio, and roughly 5% for heterogeneous samples.Reported recovered features include the Integrin α5β1–BIIG2 binding site and distorted DNA of UvrA; continuous motions can be represented by a few anchor states.Source: §6 · p. 20
  • Fold’EM is not yet an end-to-end pipeline replacement: it uses curated particles with known CTF parameters, and ab initio inference relies on accurate pose initialization.Unmodelled components appear as unexplained signal, and fine local details such as interface loops are harder to recover than global assemblies and major conformational changes.Source: §6 · p. 20

AI use statement

The authors report using generative AI tools during manuscript preparation for language editing, mathematical exposition, code assistance, and literature formatting.

AI use statement

  • ChatGPT and Claude were used for language editing and improving manuscript clarity and organization.The tools also assisted with mathematical exposition, code development and debugging, and literature and reference formatting.Source: AI use statement · p. 20
  • The authors state that they reviewed and verified all scientific claims, derivations, code, experimental results, interpretations, and citations.Source: AI use statement · p. 20
  • The authors take full responsibility for the paper’s content and full credit for its scientific claims.Source: AI use statement · p. 20

Ethics statement

The work uses synthetic or publicly available cryo-EM datasets and reports no human-subject or sensitive-personal-data involvement. The appendix outlines additional results, algorithms, and implementation details.

Ethics statement

  • The work reports no human subjects or sensitive personal data; datasets were synthetically generated from PDB structures or derived from public EMPIAR datasets.Source: Ethics statement · p. 20

A Overview of the appendix

  • Appendix B lists experimental and synthetic datasets and quantitative results for homogeneous, ab initio, heterogeneous, and continuous-heterogeneity experiments.Source: A Overview of the appendix · p. 25
  • Appendix C provides pseudocode for Fold’EM and the B-RECOVAR residual-subspace estimator, while Appendix D covers preprocessing, baselines, metrics, image formation, and implementation details.Source: A Overview of the appendix · p. 25

B.1 Datasets

The appendices describe the datasets and evaluation scopes for homogeneous, heterogeneous, and continuous-conformation experiments, followed by implementation algorithms.

B.1 Datasets

  • The experimental and synthetic dataset tables identify entries and report dataset characteristics, including resolution and molecular weight; the synthetic table also reports sequence length.The captions do not provide the table cell values.Source: Table A1 · p. 26, Table A2 · p. 26

B.2 Homogeneous reconstruction with known poses

  • Known-pose homogeneous evaluations cover experimental and synthetic datasets, measuring FSC resolution and structural accuracy with Cα-RMSD and TM-score.The experimental and synthetic tables also report MolProbity measures for resulting structures.Source: Table A3 · p. 27, Table A4 · p. 27, Table A5 · p. 28

B.3 Homogeneous ab initio reconstruction

  • Homogeneous ab-initio evaluations report FSC resolution, TM-score, and Cα-RMSD for experimental and synthetic datasets, with poses inferred from images.Additional tables assess MolProbity properties of the structures.Source: Table A9 · p. 29, Table A10 · p. 29, Table A11 · p. 30

B.4 Heterogeneous reconstruction

  • Heterogeneous evaluations assess recovery of two states from mixed experimental or synthetic particle stacks, with known-pose and ab-initio tables reporting resolution and structural metrics.The experimental states are identified as 9GXQ and 9GXR; the synthetic mixture is 9UGB/9UGC.Source: Table A15 · p. 31, Table A16 · p. 32, Table A17 · p. 32

B.5 Continuous heterogeneity on IgG-1D

  • The IgG-1D continuous-heterogeneity evaluation reports MolProbity, clash, and Ramachandran-outlier measures averaged across eight anchors and 100 interpolated states.The caption describes these measures for each experimental arm.Source: Table A25 · p. 36

B.6 Confidence-based particle selection

  • High-confidence particle selection typically yields better-separated conformations and higher TM-scores to the corresponding conformations in the EMPIAR-13396 experiment.The table defines separation by comparing true and predicted conformational separation; smaller differences indicate closer agreement.Source: Table A26 · p. 36

C Algorithms

  • Algorithm 1 alternates guided reverse-diffusion structure inference with nuisance-variable refinement, while Algorithm 2 initializes heterogeneous classes using a residual-subspace estimator.B-RECOVAR estimates residual modes, fits particle coefficients, clusters them, and refines labels by cross-validated updates.Source: C Algorithms · pp. 36–37

D.1 Pre-processing

The appendix specifies particle preparation and the density-reconstruction baselines used for comparison, including separate reconstruction of heterogeneous classes.

D.1 Pre-processing

  • Synthetic particles are rendered from PDB structures and receive Fourier-domain white Gaussian noise; experiments use SNR 0.01 or 0.005.Experimental particles are retained from final STAR files after junk-particle exclusion, then centered, Fourier-cropped, phase-flipped, and softly masked.Source: §D.1 · p. 38

D.2 Baselines

  • For known-pose comparisons, RELION and CryoDRGN reconstruct 3D density from particle images.The appendix also specifies their training or refinement procedures for the known-pose setting.Source: §D.2 · p. 38
  • For ab-initio comparisons, RELION and CryoDRGN-AI jointly estimate poses and 3D structure without externally supplied orientations.Fold’EM ab-initio experiments use first-stage poses and an initial map for subsequent refinement.Source: §D.2 · p. 39
  • For heterogeneous baselines, particles are first clustered into structural groups, then each group is reconstructed separately to estimate a density for each class.This procedure is described as avoiding additional heterogeneity-modelling assumptions.Source: §D.2 · p. 39

D.2.1 AlphaFold3 and hardware resources

The appendix records the software and hardware environment, stereochemical regularizers, and post-inference relaxation used for Fold’EM structures.

D.2.1 AlphaFold3 and hardware resources

  • Experiments use Protenix v1.0.4, a PyTorch reimplementation of AlphaFold3, with RELION 5.0.0 for map and pose estimation and Phenix v1.21.2 for validation.Runs used NVIDIA H100 and L40S GPUs with Debian GNU/Linux 12.Source: §D.2.1 · p. 39

D.2.2 Collision and bond length loss

  • Stereochemical losses penalize elongated covalent bonds and clashes between non-bonded atoms.The bond-length term uses a 0.2 Å tolerance margin, and collision padding is 0.4 Å.Source: §D.2.2 · p. 40

D.2.3 Relaxation

  • Because minor clashes can persist after regularization, all Fold’EM outputs are relaxed with the AMBER14 force field in vacuum.Energy minimization runs for at most 2,000 iterations.Source: §D.2.3 · p. 40

D.2.4 Metrics

The appendix defines structural evaluation metrics and describes the image-formation and noise models used to compare observed particles with simulated images.

D.2.4 Metrics

  • Cα-RMSD measures aligned structural deviation, TM-score measures global structural similarity, and MolProbity summarizes stereochemical quality.TM-score is less sensitive than RMSD to a small number of large local deviations; lower MolProbity scores indicate better stereochemical quality.Source: §D.2.4 · pp. 40–41
  • FSC resolution is reported at the 0.143 threshold against the experimental map.The FSC curve is computed with phenix.map_box.Source: §D.2.4 · p. 41

D.3.1 Image formation model

  • The image-formation model represents molecular potential as a sum of atomic potentials and forms a pose-dependent projection by sampling a central Fourier plane.The model applies the particle’s CTF in Fourier space to produce the noiseless image.Source: §D.3.1 · pp. 41–42

D.3.2 Noise model and likelihood function

  • Processed particle-image residuals are modeled as additive Gaussian noise with frequency-dependent variance, rather than as raw electron-arrival counts.The paper characterizes this as an effective model for processed images.Source: §D.3.2 · p. 42
  • Inference uses a band-limited, normalized Fourier-space correlation as a scale-invariant surrogate for the Gaussian image log-likelihood.Normalization removes sensitivity to global amplitude and contrast, while the selected frequency band excludes low-frequency background effects and unreliable high frequencies.Source: §D.3.2 · p. 42

D.3.3 Particle image rendering from atomic structure

  • Particle images are rendered directly from atomic structures by projecting atom-centered isotropic Gaussians and integrating them over pixel areas.The projected Gaussian widths are set using the Nyquist resolution.Source: §D.3.3 · p. 43

D.3.4 Contrast transfer function

  • The CTF is applied in Fourier space; phase-flipped experimental stacks are simulated using the magnitude of the CTF.The CTF parameterization accounts for astigmatism through two principal defocus values and their orientation.Source: §D.3.4 · pp. 43–44

D.4 B-RECOVAR: bootstrap-aggregated RECOVAR class assignment

B-RECOVAR uses consensus-subtracted cryo-EM particle residuals to identify conformational classes through a bootstrap-style ensemble, without reference structures or atomic models. It combines coefficient clustering, cross-validated refinement, and aggregated voting to stabilize assignments and report particle-level confidence.

D.4 B-RECOVAR: bootstrap-aggregated RECOVAR class assignment

  • B-RECOVAR aggregates subsampled fits to residual subspaces, particle-coefficient clustering, and cross-validated class-reconstruction comparisons to assign heterogeneous particles.Replicate labelings are aggregated by spectral voting, which also produces a particle-level confidence score.Source: §D.4 · p. 44
  • Class deviations span at most n −1 dimensions, their expected coefficients depend on class rather than pose, and masking suppresses solvent contributions if class-specific density is retained.The mask permits variance reduction under that condition.Source: §D.4 · pp. 44–45
  • Cross-validated refinement scores each particle against class reconstructions from the opposite fold, avoiding direct self-correlation but requiring enough particles from every class in both folds.Fold-specific score centering compensates for class-dependent offsets caused by unequal reconstruction precision.Source: §D.4 · p. 45
  • At low particle counts, class and nuisance directions can have comparable energy, while nuisance variation can rival structural signal and destabilize a single fit.The ensemble tests whether a class subspace recurs across particle subsamples and frequency bands containing structural signal.Source: §D.4 · pp. 44, 46
  • Aligned replicate votes yield class assignments and normalized particle margins, but shared data make conditional independence approximate, so margins and eigengaps are diagnostic rather than calibrated probabilities.Replicate alignments use a reference labeling and the Hungarian algorithm; reversed or chance-level replicates are downweighted.Source: §D.4 · p. 46
  • Unlike RECOVAR’s full noise-whitened covariance PCA, B-RECOVAR fits only n −1 masked, band-limited class-deviation modes, reducing reliance on nuisance-sensitive leading covariance directions.For two classes, it is a rank-one, uncorrected analogue of one-dimensional RECOVAR; only B-RECOVAR uses cross-fitted refinement in the matched comparison.Source: §D.4 · p. 46
  • Experiments use 24 replicates, 80% particle subsamples, eight alternating updates, and four refinement iterations; the reported dominant cost is about 10 CPU-minutes for 10^4 particles on a D = 128 grid with n = 2.With fewer than approximately 5,000 particles per conformation, class and competing nuisance directions each account for about 1.5% of residual energy.Source: §D.4 · p. 47

D.5 Hyperparameters and implementation details

The experiments use a shared set of optimization hyperparameters, while the prior’s role is partly supplied by the parameterization itself.

D.5 Hyperparameters and implementation details

  • All experiments use D = 128 pixels, Adam, a maximum gradient norm of 0.6, and at most 128 guiding particles per optimization step; outer iterations are 3 for real and 5 for synthetic examples.The particle cap reflects memory constraints.Source: §D.5 · p. 47
  • In the present parameterization, reparameterization partly supplies the prior’s role by expressing displacements in H through g in directions the Pairformer trunk natively realizes.The cited prior log p(Z | a) was essential in the earlier formulation when Z was a free variable, to prevent optimization drifting into regions the Pairformer does not produce.Source: §D.5 · p. 47
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