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

q-bio.BMOct 1, 2026

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

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

Organizations: Institute of Science and Technology Austria · University of Zurich · Frankfurt Institute for Advanced Studies · Princeton University · Broad Institute of MIT and Harvard

Abstract

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.

Figures & tables

Figure 1: Fold’EM infers atomic structures directly from sparse cryo-EM particle images. (A) Fold’EM alternates structural inference with pose and class refinement. (B) Particle evidence is back-propagated into intermediate Pairformer states $\mathbf{H}$ , filtering experimental updates through the remaining Pairformer blocks before AlphaFold3 diffusion. (C) For the integrin–BIIG2 Fab complex (PDB 9B9J ), AlphaFold3 predicts the individual components but places the Fab at the wrong binding site. (D) Ab initio Fold’EM recovers the bound complex and epitope from only $1{,}000$ particles— $0.26\%$ of those used for the deposited reconstruction—with RMSD $2.49\,\text{\AA}$ , TM-score $0.956$ , and FSC resolution $3.52\,\text{\AA}$ (deposited map: $2.6\,\text{\AA}$ ).

Panel of Figure 2 — Figure 2: Guiding with the particles recovers the structure, whereas guiding with their reconstructed ESP fails. Ab initio inference on synthetic 8U4E ( $3{,}304$ residues) from $500$ particles at SNR $0.005$ . The grey surface is the consensus density reconstructed from these particles at the estimated poses, and the numbers are C $\alpha$ RMSD to the deposited model. (A) The deposited model. (B) AlphaFold3 misplaces the domains of the assembly. (C) ESP guidance optimizes the same Pairformer state with the same particles, poses and optimizer, but replaces the particle likelihood by a loss against the consensus density, and improves the prediction without recovering the arrangement. (D) Fold’EM, guided by the particle likelihood, recovers it, with a TM-score of $0.81$ against $0.48$ for AlphaFold3. As argued in Section 2 , a density loss reproduces the particle likelihood only under the experiment-induced metric, and with estimated poses the consensus density is not a sufficient statistic for the particles, so the information needed to refine the poses together with the istructure is lost.

Figure 4: Fold’EM recovers atomic structures from $0.03\%$ of the deposited particles with known poses and from $0.27\%$ ab initio . Each column is one target: a synthetic dataset on the left (A, C, E, G) and an experimental one on the right (B, D, F, H). Rows show the structures (A, B; AlphaFold3 in blue, Fold’EM in red, deposited model in grey), representative particles (C, D), FSC resolution (E, F) and TM-score against the deposited model (G, H) as a function of the number of particles; filled markers denote known poses, open markers ab initio , and the dotted line the AlphaFold3 prediction. Left: Pma1, the yeast plasma-membrane H + -ATPase, in its activated (E2P) state ( 9UGB ); synthetic particles at SNR $0.005$ . AlphaFold3 predicts the transmembrane domain but misplaces the cytosolic A-, N- and P-domains, whose arrangement changes along the catalytic cycle (TM-score $0.70$ ). Fold’EM recovers the deposited state, reaching $3.3\,\text{\AA}$ and a TM-score of $0.97$ from $100$ particles with known poses, and $3.3\,\text{\AA}$ from $500$ particles ab initio . Right: Integrin $\alpha_{5}\beta_{1}$ –BIIG2 ( 9B9J , EMPIAR-12552), where AlphaFold3 predicts both component folds but places the Fab at the wrong site (TM-score $0.78$ ); antibody–antigen pairs provide little paired coevolutionary information. Fold’EM recovers the bound complex and epitope, although not the interface variable loops, reaching $3.5\,\text{\AA}$ and a TM-score of $0.97$ from $100$ particles with known poses and $3.5\,\text{\AA}$ from $1{,}000$ particles ab initio ; no baseline reaches $6\,\text{\AA}$ , and Fold’EM improves over AlphaFold3 even at the smallest particle counts. Sparse particle evidence therefore corrects the global arrangement largely supplied by the structural prior. Resolution ( Å ; FSC $=0.143$ ).

Figure 5: Fold’EM resolves both states of a synthetic 9UGB / 9UGC mixture from $250$ particles per state with known poses, and from $500$ ab initio . Pma1 is the plasma-membrane proton pump of yeast, a P-type ATPase whose cytosolic A-, N- and P-domains rearrange along the catalytic cycle; 9UGB is the activated (E2P) state and 9UGC the autoinhibited state. Particles are rendered from the two deposited conformations, approximately $9$ Å apart in C $\alpha$ RMSD, and mixed in equal proportion at SNR $0.005$ . In every panel, 9UGB is on the left and 9UGC on the right. (A) AlphaFold3 (blue) over the deposited models (grey) recovers neither state (TM-score $0.70$ and $0.66$ ): the transmembrane domain is predicted correctly and the error is confined to the cytosolic domains, whose arrangement depends on the state of the cycle. (B) Fold’EM (red) with known poses resolves both from the mixed stack of $1{,}000$ particles ( $500$ per state). (C) Fold’EM ab initio resolves both from $2{,}000$ particles ( $1{,}000$ per state). (D) Representative particles of each state. (E, F) FSC resolution (top) and TM-score against the deposited model (bottom) as a function of the number of particles per state, for 9UGB (E) and 9UGC (F); filled markers denote known poses and open markers ab initio , the dashed grey line is the resolution of the deposited model and the dotted blue line the AlphaFold3 prediction.

Figure 6: Fold’EM resolves both UvrA–DNA conformations from $0.5\%$ of the deposited particles with known poses and from $5\%$ ab initio . EMPIAR-13396 contains two states of UvrA bound to DNA carrying an abasic site (PDB 9GXQ / 9GXR ); damage recognition involves insertion of the Y62 / Q65 / F66 wedge into the duplex. (A) AlphaFold3 recovers 9GXR but not 9GXQ (TM-score $0.734$ ). (B,C) Fold’EM resolves both states from $1{,}000$ particles per state with known poses and from $10{,}000$ particles per state ab initio (left: 9GXQ , right: 9GXR ). (D) Representative particles. (E) FSC versus particles per state (filled: known poses; open: ab initio ). With known poses Fold’EM reaches $3.1/3.0\,\text{\AA}$ from $1{,}000$ particles per state; ab initio it reaches $3.0\,\text{\AA}$ on 9GXR and $4.6\,\text{\AA}$ on 9GXQ , while RELION and cryoDRGN-AI remain above $8\,\text{\AA}$ . (F) TM-score against the deposited models. (G) AlphaFold3 retains a fully paired duplex, whereas Fold’EM recovers the wedge insertion and disrupted DNA, but not the flipped-out nucleotide, as expected because the lesion was supplied to both models as intact DNA. Resolution ( Å ; FSC $=0.143$ ).

Figure 7: B-RECOVAR assigns particles to conformational states more accurately and more reliably than RECOVAR and cryoDRGN at low particle counts, and its confidence score orders the assignments by their reliability. Label-free class assignment on the UvrA protein–DNA complex (EMPIAR-13396, 9GXQ vs. 9GXR ), scored against the deposited labels. (A) Accuracy as a function of the number of particles per state, from

250

5{,}000

, that is,

0.13

–

2.6\%

of the particles deposited for each state. Boxes summarize ten random draws per size, shared by B-RECOVAR and RECOVAR; triangles mark cryoDRGN. The RECOVAR estimator is ill-conditioned at these sample sizes and its accuracy ranges from chance to near-perfect between draws of the same size; B-RECOVAR trails it at

250

particles but is both more accurate and markedly more stable from

500

particles onward, and cryoDRGN remains below B-RECOVAR throughout. (B) Accuracy of B-RECOVAR stratified by its confidence score

\kappa_{k}

, the normalized margin of the ensemble vote (Appendix D.4 ). The percentage in each bar is the fraction of particles in that bin; the solid line is the accuracy over all particles and the dashed line chance. Accuracy increases monotonically with confidence, so the score ranks the particles by the reliability of their label; Fold’EM uses this ordering to guide each ensemble member with the more confident half of its particles (Table A26 ). (C, D) Ablation of the estimator at

1000

particles per state. The base estimator is a single rank-one fit to the consensus-subtracted residuals, a one-dimensional RECOVAR fit without noise correction, and is improved in turn by restricting the mode to a solvent mask, by the cross-validated refinement of the labels, and by ensembling replicates over particle subsamples, frequency bands and initializations. (C) Accuracy of each stage, with error bars over repeated draws. (D) Distribution of the per-particle class score at each stage for the particles of either state, with their separation

d^{\prime}

, the difference of the two means in units of the pooled standard deviation. Every step widens the separation between the states, and the ensemble accounts for the largest gain.

Panel of Figure 8 — Figure 8: Fold’EM resolves continuous heterogeneity of an IgG antibody complex. (A) Shown as ensembles are $10$ of the $100$ ground-truth CryoBench IgG-1D synthetic models, colored by hinge angle; and (B) the corresponding structures recovered by Fold’EM with B-RECOVAR particle classification. The inset depicts the two leading principal components of the optimized Pairformer states, overlaid on the B-RECOVAR embedding of the particles. (C) TM-score and (D) Fourier NCC plotted as a function of hinge angle, for the three conformational coordinates.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

PDB ID	EMPIAR ID	EMD ID	Resolution(Å)	# Particles	Protein size (kDa)
9B9J	$12552$	$44386$	$2.60$	$374814$	$115.11$
9B77	$12002$	$44302$	$2.74$	$490807$	$141.72$
8PWH	$11665$	$17993$	$3.17$	$708609$	$162.75$
9GXR	$13396$	$51671$	$3.04$	$197058$	$162.70$
9GXQ	$13396$	$51672$	$2.74$	$195233$	$162.70$

Appendix

Table A1: Experimental cryo-EM datasets from EMPIAR [ Iudin et al., 2016 ] . For each PDB entry, we report the corresponding EMPIAR and EMDB entries [ wwp, 2024 ] , reported resolution, molecular weight (kDa), and number of particles in original dataset.

PDB ID	EMD ID	Resolution(Å)	Seq Length	Protein size (kDa)
9UGB	$64135$	$3.25$	$818$	$88.97$
9UGC	$64136$	$3.52$	$818$	$99.62$
8U4B	$41877$	$3.80$	$2764$	$312.70$
8U4E	$41880$	$4.20$	$3304$	$373.10$
1HZH	CryoBench (IgG- $1$ D)	$2.70$	$1331$	$146.96$

Appendix

Table A2: Synthetic datasets constructed using PDB-deposited structures [ Burley et al., 2017 ] . For each entry, we report the corresponding EMDB entry [ wwp, 2024 ] , reported resolution, molecular weight (kDa), and amino acid sequence length.

Fraction of dataset	Method	9B9J	9B77	8PWH	9GXQ	9GXR
$1/1000$	Fold’EM	$\mathbf{3.73}$	$\mathbf{2.88}$	$\mathbf{6.21}$	$\mathbf{3.08}$	$\mathbf{3.18}$
	cryoDRGN	$8.40$	$5.70$	$7.56$	$9.87$	$4.33$
	RELION	$21.94$	$11.48$	$14.12$	$14.15$	$17.92$
	AlphaFold3	$21.90$	$5.96$	$64.96$	$11.69$	$3.54$
	PDB	$2.38$	$2.49$	$3.06$	$2.64$	$2.89$
$1/500$	Fold’EM	$\mathbf{3.73}$	$\mathbf{2.85}$	$\mathbf{6.63}$	$\mathbf{3.02}$	$\mathbf{3.20}$

Appendix

Table A3: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow,\text{\AA}

) on the experimental cryo-EM datasets from Table A1 . For Fold’EM, AlphaFold3 and PDB structures, FSC is computed using phenix.model_map . All baselines at different fraction of the original dataset

500

particles with known poses and are evaluated against the deposited EMD map. Entries highlighted in bold indicate the best result (excluding PDB).

Fraction of dataset	9B9J	9B77	8PWH	9GXQ	9GXR	9B9J	9B77	8PWH	9GXQ	9GXR
	C $\alpha$ RMSD ( $\downarrow$ , Å)					TM-score ( $\uparrow$ )
$1/1000$	$1.96$	$1.90$	$4.340$	$2.20$	$1.87$	$0.969$	$0.972$	$0.900$	$0.877$	$0.899$
$1/500$	$1.97$	$1.77$	$4.989$	$\mathbf{1.99}$	$1.39$	$0.970$	$0.979$	$0.897$	$\mathbf{0.893}$	$0.902$
$1/250$	$\mathbf{1.88}$	$\mathbf{1.71}$	$4.597$	$2.48$	$\mathbf{1.23}$	$\mathbf{0.972}$	$\mathbf{0.981}$	$0.885$	$0.880$	$\mathbf{0.904}$
$1/100$	$1.92$	$1.74$	$5.349$	$2.02$	$1.35$	$0.971$	$0.980$	$0.884$	$\mathbf{0.893}$	$0.903$
$1/50$	$\mathbf{1.88}$	$\mathbf{1.71}$	$\mathbf{4.174}$	$2.09$	$1.47$	$\mathbf{0.972}$	$\mathbf{0.981}$	$\mathbf{0.909}$	$0.885$	$0.897$

Appendix

Table A4: Quantitative evaluation of C

\alpha

RMSD (

\downarrow

, Å) and TM-score (

\uparrow

) on the experimental cryo-EM datasets from Table A1 . Fold’EM structures are obtained using different fractions of the original dataset. Entries highlighted in bold indicate the best result.

# particles	Method	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC		8U4B		8U4E
$100$	Fold’EM	$\mathbf{3.24}$	$\mathbf{3.25}$	$\mathbf{3.59}$	$\mathbf{3.45}$	$\mathbf{3.76}$	$\mathbf{3.80}$	$\mathbf{4.04}$	$\mathbf{4.18}$
	cryoDRGN	$3.63$	$3.95$	$3.81$	$3.96$	$7.85$	$9.80$	$9.47$	$10.03$
	RELION	$9.79$	$9.79$	$9.79$	$9.79$	$17.78$	$16.84$	$19.20$	$18.29$
	AlphaFold3	$8.21$		$19.29$		$7.29$		$8.84$
	PDB	$3.39$		$3.47$		$3.50$		$3.67$

Appendix

Table A5: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the synthetic cryo-EM datasets from Table A2 . For Fold’EM, AlphaFold3 and PDB structures, FSC is computed using phenix.model_map . Fold’EM structures are generated using

100,250,500,\&\,1000

particles at SNR

0.01

and

0.005

with known poses and are evaluated against the deposited EMD map. Entries highlighted in bold indicate the best result (excluding PDB).

Metric	# particles	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC		8U4B		8U4E
C $\alpha$ RMSD ( $\downarrow$ , Å)	$100$	$1.05$	$1.01$	$2.78$	$1.45$	$1.681$	$1.907$	$1.769$	$2.116$
	$250$	$0.88$	$0.88$	$1.34$	$\mathbf{1.37}$	$1.362$	$1.350$	$1.798$	$1.512$
	$500$	$\mathbf{0.81}$	$1.12$	$1.44$	$1.41$	$1.248$	$\mathbf{1.227}$	$\mathbf{1.503}$	$\mathbf{1.479}$
	$1000$	$0.84$	$\mathbf{0.86}$	$\mathbf{1.33}$	$1.44$	$\mathbf{1.132}$	$1.228$	$1.686$	$1.711$
	AlphaFold3	$8.96$		$11.22$		$7.780$		$26.670$

Appendix

Table A6: Quantitative evaluation of C

\alpha

RMSD (

\downarrow

, Å) and TM-score (

\uparrow

) on the synthetic cryo-EM datasets from Table A2 . Fold’EM structures are obtained using

100,250,500,\&\,1000

particles with known poses at SNR

0.01

and

0.005

; AlphaFold3 is the unguided baseline. Scores are computed with respect to the experimentally resolved structures deposited in the PDB. Entries highlighted in bold indicate the best result.

MolProbity statistic	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
	9UGB		9UGC		8U4B		8U4E
MolProbity score	$1.01$	$1.05$	$1.36$	$1.44$	$1.35$	$1.35$	$1.56$	$1.54$
Clashscore	$2.29$	$2.61$	$3.38$	$2.85$	$1.59$	$1.49$	$3.01$	$3.02$
Ramachandran outliers	$0.00$	$0.00$	$0.28$	$0.55$	$1.49$	$1.40$	$1.72$	$1.26$

Appendix

Table A7: MolProbity evaluation of Fold’EM structures over synthetic targets with known poses . We report median MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

), computed using phenix.molprobity [ Williams et al., 2018 ] over all relaxed structures in Table A6 .

MolProbity statistic	9B9J	9B77	8PWH	9GXQ	9GXR
MolProbity score	$0.84$	$0.81$	$1.71$	$1.08$	$1.07$
Clashscore	$0.38$	$0.60$	$5.42$	$1.21$	$0.99$
Ramachandran outliers	$0.10$	$0.24$	$0.43$	$0.16$	$0.40$

Appendix

Table A8: MolProbity evaluation of Fold’EM structures over real targets with known poses . We report median MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

), computed using phenix.molprobity [ Williams et al., 2018 ] over all structures in Table A4 .

Method # particles	9B9J $1000$	9B77 $1000$	8PWH $5000$	9GXQ $5000$	9GXR $5000$
Fold’EM	$\mathbf{3.52}$	$\mathbf{2.85}$	$10.83$	$\mathbf{4.48}$	$\mathbf{3.22}$
cryoDRGN-AI	$35.90$	$31.01$	$32.80$	$24.44$	$26.88$
RELION ( ab initio )	$11.62$	$7.95$	$\mathbf{9.84}$	$6.25$	$4.98$
AlphaFold3	$21.90$	$4.40$	$64.96$	$11.70$	$3.50$
PDB	$2.38$	$2.49$	$3.06$	$2.64$	$2.89$

Appendix

Table A9: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the experimental cryo-EM datasets from Table A1 , with poses inferred ab initio . For Fold’EM ab initio , AlphaFold3 and PDB structures, FSC is computed using phenix.model_map . The number of particles used for the experiment is given below each identifier. Entries highlighted in bold indicate the best result.

Method # particles	9B9J $1000$	9B77 $1000$	8PWH $5000$	9GXQ $5000$	9GXR $5000$	9B9J $1000$	9B77 $1000$	8PWH $5000$	9GXQ $5000$	9GXR $5000$
	C $\alpha$ RMSD ( $\downarrow$ , Å)					TM-score ( $\uparrow$ )
Fold’EM	$\mathbf{2.49}$	$\mathbf{2.25}$	$\mathbf{18.91}$	$\mathbf{2.87}$	$\mathbf{1.54}$	$\mathbf{0.956}$	$\mathbf{0.967}$	$\mathbf{0.602}$	$\mathbf{0.857}$	$\mathbf{0.898}$
AlphaFold3	$21.90$	$3.87$	$33.16$	$5.96$	$2.01$	$0.776$	$0.916$	$0.397$	$0.734$	$0.886$

Appendix

Table A10: Quantitative evaluation of TM-score (

\uparrow

) and

C\alpha

-RMSD (

\downarrow

) on the experimental cryo-EM datasets from Table A1 , with poses inferred ab initio . The number of particles used is given below each identifier, and scores are computed with respect to the experimentally resolved structures deposited in the PDB. Entries highlighted in bold indicate the best result.

# particles	Method	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC		8U4B		8U4E
$500$	Fold’EM	$\mathbf{3.47}$	$\mathbf{3.25}$	$\mathbf{3.59}$	$\mathbf{3.91}$	$\mathbf{5.87}$	$\mathbf{7.23}$	$11.38$	$11.53$
	cryoDRGN-AI	$12.80$	$27.70$	$33.30$	$33.20$	$22.59$	$38.40$	$38.40$	$38.40$
	RELION ( ab initio )	$7.23$	$8.93$	$6.93$	$7.56$	$10.97$	$16.00$	$11.64$	$15.36$
	AlphaFold3	$8.21$		$19.29$		$7.29$		$\mathbf{8.84}$
	PDB	$3.39$		$3.47$		$3.5$		$3.67$

Appendix

Table A11: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the synthetic cryo-EM datasets from Table A2 , with poses inferred ab initio . For Fold’EM ab initio , AlphaFold3 and PDB structures, FSC is computed using phenix.model_map . All methods use

500

particles at SNR

0.01

and

0.005

. Entries highlighted in bold indicate the best result (excluding PDB).

Metric	Method	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC		8U4B		8U4E
C $\alpha$ RMSD ( $\downarrow$ , Å)	Fold’EM	$\mathbf{1.78}$	$\mathbf{1.57}$	$\mathbf{2.35}$	$\mathbf{3.07}$	$\mathbf{3.948}$	$\mathbf{3.941}$	$\mathbf{7.684}$	$\mathbf{6.885}$
C $\alpha$ RMSD ( $\downarrow$ , Å)	AlphaFold3	$8.96$		$11.22$		$7.780$		$26.670$
TM-score ( $\uparrow$ )	Fold’EM	$\mathbf{0.969}$	$\mathbf{0.976}$	$\mathbf{0.948}$	$\mathbf{0.917}$	$\mathbf{0.9174}$	$\mathbf{0.9182}$	$\mathbf{0.810}$	$\mathbf{0.809}$
TM-score ( $\uparrow$ )	AlphaFold3	$0.699$		$0.661$		$0.7981$		$0.4772$

Appendix

Table A12: Quantitative evaluation of TM-score (

\uparrow

) and

C\alpha

-RMSD (

\downarrow,\text{\AA}

) on the synthetic cryo-EM datasets from Table A2 , with poses inferred ab initio . Fold’EM ab initio results are obtained using

500

particles at SNR

0.01

and

0.005

, and scores are computed with respect to the experimentally resolved structures deposited in the PDB. Entries highlighted in bold indicate the best result.

Statistic	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
	9UGB		9UGC		8U4B		8U4E
MolProbity score	$1.18$	$1.15$	$1.25$	$1.15$	$1.50$	$1.53$	$1.58$	$1.79$
Clashscore	$3.72$	$3.64$	$4.83$	$3.56$	$1.21$	$2.35$	$1.56$	$2.21$
Ramachandran outliers	$0.00$	$0.25$	$0.12$	$0.00$	$1.26$	$0.71$	$0.98$	$0.66$

Appendix

Table A13: MolProbity evaluation of the ab initio Fold’EM structures in Table A12 . For each relaxed structure, we report MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

), computed using phenix.molprobity [ Williams et al., 2018 ] .

Statistic	9B9J	9B77	8PWH	9GXQ	9GXR
MolProbity score	$0.93$	$0.80$	$1.48$	$1.22$	$1.11$
Clashscore	$0.82$	$1.04$	$4.01$	$2.52$	$1.57$
Ramachandran outliers	$0.19$	$0.00$	$0.27$	$0.16$	$0.24$

Appendix

Table A14: MolProbity evaluation of the ab initio Fold’EM structures in Table A10 . For each relaxed structure, we report MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

), computed using phenix.molprobity [ Williams et al., 2018 ] . Particle counts are as in Table A9 .

# particles per conf.	Method	9GXQ	9GXR
$500$	Fold’EM	$6.25$	$\mathbf{3.02}$
	RELION	$8.67$	$8.40$
	cryoDRGN	$\mathbf{5.60}$	$4.89$
$1000$	Fold’EM	$\mathbf{3.09}$	$\mathbf{2.95}$
	RELION	$6.89$	$10.75$
	cryoDRGN	$4.34$	$4.41$

Appendix

Table A15: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the heterogeneous experimental dataset EMPIAR-13396, with known poses. The two conformational states 9GXQ and 9GXR are resolved from the same mixed particle stack, and the sweep is over the number of particles per state. For Fold’EM, AlphaFold3 and PDB structures, FSC is computed using phenix.model_map . Entries highlighted in bold indicate the best result.

# particles per conf.	9GXQ	9GXR	9GXQ	9GXR
	C $\alpha$ RMSD ( $\downarrow$ , Å)		TM-score ( $\uparrow$ )
$500$	$3.30$	$1.85$	$0.840$	$0.890$
$1000$	$1.63$	$\mathbf{1.45}$	$0.891$	$0.899$
$2000$	$1.76$	$1.71$	$0.886$	$0.894$
$5000$	$\mathbf{1.48}$	$\mathbf{1.45}$	$\mathbf{0.893}$	$\mathbf{0.900}$
$10000$	$1.82$	$1.73$	$0.885$	$0.894$

Appendix

Table A16: Quantitative evaluation of C

\alpha

RMSD (

\downarrow

, Å) and TM-score (

\uparrow

) on the heterogeneous experimental dataset EMPIAR-13396, with known poses. Fold’EM resolves the two conformational states 9GXQ and 9GXR from the same mixed particle stack, and the sweep is over the number of particles per state. Entries highlighted in bold indicate the best result.

# particles per conf.	Method	9GXQ	9GXR
$500$	Fold’EM	$\mathbf{8.67}$	$3.58$
	RELION ( ab initio )	$22.40$	$12.80$
	cryoDRGN-AI	$38.40$	$26.90$
$1000$	Fold’EM	$\mathbf{7.47}$	$4.20$
	RELION ( ab initio )	$38.40$	$10.75$
	cryoDRGN-AI	$26.90$	$22.40$

Appendix

Table A17: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the heterogeneous experimental dataset EMPIAR-13396, with poses inferred ab initio . The two conformational states 9GXQ and 9GXR are resolved from the same mixed particle stack, and the sweep is over the number of particles per state. For Fold’EM ab initio , AlphaFold3 and PDB structures, FSC is computed using phenix.model_map . Entries highlighted in bold indicate the best result (excluding PDB).

# particles per conf.	9GXQ	9GXR	9GXQ	9GXR
	C $\alpha$ RMSD ( $\downarrow$ , Å)		TM-score ( $\uparrow$ )
$500$	$4.54$	$2.61$	$0.799$	$0.874$
$1000$	$5.48$	$2.80$	$0.781$	$0.868$
$2000$	$5.84$	$2.39$	$0.737$	$0.880$
$5000$	$5.06$	$\mathbf{1.63}$	$0.776$	$0.895$
$10000$	$\mathbf{3.72}$	$1.66$	$\mathbf{0.836}$	$\mathbf{0.896}$

Appendix

Table A18: Quantitative evaluation of C

\alpha

RMSD (

\downarrow

, Å) and TM-score (

\uparrow

) on the heterogeneous experimental dataset EMPIAR-13396, with poses inferred ab initio . Fold’EM ab initio resolves the two conformational states 9GXQ and 9GXR from the same mixed particle stack, and the sweep is over the number of particles per state. Entries highlighted in bold indicate the best result.

# particles per conf.	Method	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC
$250$	Fold’EM	$\mathbf{3.25}$	$\mathbf{3.28}$	$\mathbf{3.43}$	$\mathbf{3.47}$
	RELION	$9.24$	$10.40$	$9.24$	$11.09$
	cryoDRGN	$4.38$	$6.16$	$6.40$	$8.76$
$500$	Fold’EM	$\mathbf{3.24}$	$\mathbf{3.25}$	$\mathbf{3.39}$	$\mathbf{3.43}$
	RELION	$7.92$	$8.76$	$8.32$	$9.24$

Appendix

Table A19: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the heterogeneous synthetic 9UGB / 9UGC mixture, with known poses. The two conformational states are resolved from the same mixed particle stack at SNR

0.01

and

0.005

, and the sweep is over the number of particles per state. Entries highlighted in bold indicate the best result.

# particles per conf.	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
	C $\alpha$ RMSD ( $\downarrow$ , Å)				TM-score ( $\uparrow$ )
	9UGB		9UGC		9UGB		9UGC
$250$	$\mathbf{0.76}$	$0.99$	$1.55$	$1.38$	$\mathbf{0.964}$	$0.960$	$\mathbf{0.987}$	$0.982$
$500$	$0.99$	$0.89$	$1.36$	$\mathbf{1.28}$	$0.960$	$0.962$	$0.982$	$0.983$
$1000$	$0.93$	$\mathbf{0.81}$	$\mathbf{1.18}$	$1.59$	$0.961$	$\mathbf{0.963}$	$0.986$	$\mathbf{0.985}$
AlphaFold3	$8.96$		$11.22$		$0.699$		$0.661$

Appendix

Table A20: Quantitative evaluation of C

\alpha

RMSD (

\downarrow

, Å) and TM-score (

\uparrow

) on the heterogeneous synthetic 9UGB / 9UGC mixture, with known poses. Fold’EM resolves the two conformational states from the same mixed particle stack at SNR

0.01

and

0.005

, and the sweep is over the number of particles per state. Entries highlighted in bold indicate the best result.

# particles per conf.	Method	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC
$250$	Fold’EM	$\mathbf{3.22}$	$\mathbf{7.61}$	$\mathbf{3.63}$	$\mathbf{4.81}$
	RELION ( ab initio )	$8.32$	$8.32$	$23.77$	$8.76$
	cryoDRGN-AI	$33.28$	$33.28$	$41.60$	$18.49$
$500$	Fold’EM	$\mathbf{3.25}$	$\mathbf{3.35}$	$\mathbf{3.51}$	$\mathbf{3.55}$
	RELION ( ab initio )	$6.93$	$7.92$	$7.23$	$8.32$

Appendix

Table A21: Resolution at Fourier shell correlation (FSC)

=0.143

(

\downarrow

) on the heterogeneous synthetic 9UGB / 9UGC mixture, with poses inferred ab initio . The two conformational states are resolved from the same mixed particle stack at SNR

0.01

and

0.005

, and the sweep is over the number of particles per state. Entries highlighted in bold indicate the best result.

# particles per conf.	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
	C $\alpha$ RMSD ( $\downarrow$ , Å)				TM-score ( $\uparrow$ )
	9UGB		9UGC		9UGB		9UGC
$250$	$1.52$	$4.13$	$2.84$	$5.09$	$0.967$	$0.862$	$0.929$	$0.876$
$500$	$\mathbf{1.50}$	$1.95$	$2.71$	$2.61$	$\mathbf{0.968}$	$\mathbf{0.953}$	$0.936$	$0.939$
$1000$	$1.97$	$\mathbf{1.71}$	$\mathbf{2.42}$	$\mathbf{2.17}$	$0.952$	$0.951$	$\mathbf{0.948}$	$\mathbf{0.957}$
AlphaFold3	$8.96$		$11.22$		$0.699$		$0.661$

Appendix

Table A22: Quantitative evaluation of C

\alpha

RMSD (

\downarrow

, Å) and TM-score (

\uparrow

) on the heterogeneous synthetic 9UGB / 9UGC mixture, with poses inferred ab initio . Fold’EM resolves the two conformational states from the same mixed particle stack at SNR

0.01

and

0.005

, and the sweep is over the number of particles per state. Entries highlighted in bold indicate the best result.

Poses	MolProbity statistic	SNR $0.01$	SNR $0.005$	SNR $0.01$	SNR $0.005$
		9UGB		9UGC
Known	MolProbity score	$1.16$	$1.13$	$1.09$	$1.11$
	Clashscore	$3.72$	$3.40$	$3.01$	$3.16$
	Ramachandran outliers	$0.00$	$0.00$	$0.12$	$0.00$
Ab initio	MolProbity score	$1.39$	$1.48$	$1.44$	$1.46$
	Clashscore	$5.85$	$7.28$	$6.65$	$6.72$

Appendix

Table A23: MolProbity evaluation of Fold’EM structures on the heterogeneous synthetic 9UGB / 9UGC mixture . We report median MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

), computed using phenix.molprobity [ Williams et al., 2018 ] over all structures in Tables A20 and A22 .

Poses	MolProbity statistic	9GXQ	9GXR
Known	MolProbity score	$1.16$	$1.20$
	Clashscore	$1.48$	$1.66$
	Ramachandran outliers	$0.16$	$0.16$
Ab initio	MolProbity score	$1.32$	$1.38$
	Clashscore	$2.74$	$2.61$
	Ramachandran outliers	$0.16$	$0.08$

Appendix

Table A24: MolProbity evaluation of Fold’EM structures on the heterogeneous experimental dataset EMPIAR-13396 . We report median MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

), computed using phenix.molprobity [ Williams et al., 2018 ] over all structures in Tables A16 and A18 .

MolProbity statistic	Anchors ( $n=8$ )	Interpolated ( $n=100$ )	Anchors ( $n=8$ )	Interpolated ( $n=100$ )	Anchors ( $n=8$ )	Interpolated ( $n=100$ )
	cryoDRGN		RECOVAR		B-RECOVAR
MolProbity score	$0.72$	$0.71$	$0.70$	$0.72$	$0.74$	$0.72$
Clashscore	$0.59$	$0.59$	$0.59$	$0.66$	$0.66$	$0.54$
Ramachandran outliers	$0.15$	$0.15$	$0.15$	$0.15$	$0.15$	$0.15$

Appendix

Table A25: MolProbity evaluation of Fold’EM structures on IgG-1D . For each arm of the experiment, we report MolProbity score

(\downarrow)

, clash score (

\downarrow

), and Ramachandran outlier percentage (

\downarrow

) of the relaxed structures, computed using phenix.molprobity [ Williams et al., 2018 ] , averaged over the eight anchors and over the

100

interpolated states.

# particles per conf.	Confidence method	9GXQ	9GXR	Separation ( $\downarrow$ )
		TM-score ( $\uparrow$ )
$7.5$ k	Cut by confidence	$\mathbf{0.835}$	$\mathbf{0.894}$	$\mathbf{0.58}$
$7.5$ k	No confidence	$0.822$	$0.888$	$1.35$
$10$ k	Cut by confidence	$\mathbf{0.832}$	$\mathbf{0.894}$	$\mathbf{0.47}$
$10$ k	No confidence	$0.823$	$0.888$	$1.48$

Appendix

Table A26: Confidence-based particle selection described in Section D.4 prevents mode collapse in the ensemble. On the heterogeneous experimental dataset EMPIAR-13396, embeddings optimized using high-confidence particles typically yield better-separated conformations, accompanied by higher TM-scores to the corresponding conformations. The “separation” column reports the difference between the RMSD of the true conformations 9GXQ and 9GXR and that of the predicted conformations; smaller values indicate closer agreement with the true conformational separation. Bold entries indicate the best result.

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