Inverse problems with experiment-guided AlphaFold
Authors: Sai Advaith Maddipatla
Nadav Bojan
Meital Bojan
Sanketh Vedula
Paul Schanda
Ailie Marx
Alexander Bronstein
Venue: ICML 2025 poster (2025)
Keywords: protein structure prediction
alphafold
protein generative models
experiment-grounded generative models
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We develop experiment-guided AlphaFold-3 to solve inverse problems in structural biology, leading to faster experimental cycles and improved modeling in X-ray crystallography and NMR spectroscopy.

Abstract: Proteins exist as a dynamic ensemble of multiple conformations, and these motions are often crucial for their functions. However, current structure prediction methods predominantly yield a single  conformation, overlooking the conformational heterogeneity revealed by diverse experimental modalities. Here, we present a framework for building experiment-grounded protein structure generative models that infer conformational ensembles consistent with measured experimental data. The key idea is to treat state-of-the-art protein structure predictors (e.g., AlphaFold3) as sequence-conditioned structural priors, and cast ensemble modeling as posterior inference of protein structures given experimental measurements. Through extensive real-data experiments, we demonstrate the generality of our method to incorporate a variety of experimental measurements. In particular, our framework uncovers previously unmodeled conformational heterogeneity from crystallographic densities, generates high-accuracy NMR ensembles orders of magnitude faster than status quo, and incorporates pairwise cross-link constraints. Notably, we demonstrate that our ensembles outperform AlphaFold3 and sometimes better fit experimental data than publicly deposited structures to the protein database (PDB). We believe that this approach will unlock building predictive models that fully embrace experimentally observed conformational diversity.

Inverse problems with experiment-guided AlphaFold
Advaith Maddipatla 12 Nadav Bojan Sellam 13 Meital Bojan 13
Sanketh Vedula 13f Paul Schanda 3
Ailie Marx 45t
Alex M. Bronstein 13
Abstract
1. Introduction
Proteins exist as dynamic ensemble of multi- ple conformations, and these motions are often crucial for their functions. However, current struc- ture prediction methods predominantly yield a single conformation, overlooking the conforma- tional heterogeneity revealed by diverse exper- imental modalities. Here, we present frame- work for building experiment-groundde protein structure generative models that infer conforma- tional ensembles consistent with measured ex- perimental data. The key idea is to treat state- of-the-art protein structure predictors (e.g., Al- phaFold3) as sequence-conditione structural pri- ors, and cast ensemble modeling as posterior in- ference of protein structures given experimental measurements. Through extensive periments, demonstrate the generality of our method to incorporate variety of experimental measurements, In particular, our framework un- covers previously unmodeled conformational het- erogeneity from crystallographic densities, and generates high-accuracy NMR ensembles orders of magnitude faster than the status quo. Notably, we demonstrate that our ensembles outperform AlphaFold3 (Abramson et al., 2024) and some- times better fit experimental data than publicly de- posited structures to the Protein Data Bank (PDB, Burley et al. (2017)). We believe that this ap- proach will unlock building predictive models that fully embrace experimentally observed con- formational diversity.
Proteins are inherently dynamic entities, sampling con- tinuum of conformational states to fulfill their biological roles. Experimental techniques such as X-ray crystallogra- phy, nuclear magnetic resonance (NMR) spectroscopy, and cryo-electron microscopy (cryo-EM) inherently report on ansemble-averagee data rather than singular static snapshots. In X-ray crystallography, the resolved electron density map represents a spatial and temporal average over all molecules in the crystal lattice, with regions of flexibility manifesting as diffuse or poorly resolved density. NMR spectroscopy measures the interaction between nuclear spins (e.g.. mag- netization transfer due to nuclear Overhauser effect, NOE) and spins and electrons (c.g.. chemical shifts) arising from dynamic conformational ensembles in solution, with these experimental restraints used computationally to resolve com- patible structural states. Cryo-EM similarly resolves mul- tiple conformational states, as individual particles frozen in vitreous ice adopt distinct orientations and conforma- tions, which are computationally classified into discrete or continuous flexibility ranges.
ex-
On the computational front, ab initio protein structure de- termination based on modeling the molecule's free energy and its subsequent minimization (e.g.. Rosetta and many of its variants (Baek et al., 2021; 2024)) have been only partially successful and computationally expensive. A giant leap in protein structure prediction resulted from the funda- mental discovery of the cocvolution of contacting residues (Göbel et al., 1994; Hopf et al., 2014), underlying deep learning- based models such as AlphaFold (Jumper et al., 2021; Abramson et al., 2024), which had a groundbreaking impact on structural biology and was awarded the recent 2024 Nobel Prize Chemistry.
Protein structure predictors are trained exclusively on X- ray crystallographic models, which are themselves firted to electron density maps averaged over trillions of molecule instances. While it has been recognised several decades ago that the conformations of proteins in crystals are het- erogencous (Smith et al., 1986; Furnham et al., 2006), early crystallographic refinements prioritized linll--ooffrmer models. Advances in resolution, the more widespread ap- plication of room-temperature crystallographic experiments
"Equally contributing first author. 'Equally con- tributing senior author. Correspondence sai. maddimattal @ @cc.@cs.oooooo ac.uukk n.sellam @wwa@ecccaeceeeeeeeee aechnion.ac.il, meital. bojan @ campus. acc.nion.ac.iil -Technion-Israel Institute of Technology. Israel. University of Oxford, UK. 'Institute of Science and Technology, Austria. *Tel Hai Academic College, Isracl. "MIGAL- Galilee Research Institute, Isracl.
contact:
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Inverse problems with experiment-guided AlphaFold
Experimental inputs
log Ikellhood
Likelhoodds
guidance score
Generated
subsstructure
AlphaFold3
ensemble
Ensemble statistics
dlectron,density
Diftusion model
sequence
MSA
Pairformer
F
MQI--RGG
latents
Average distances
sequence
Force field relaxation
Ensemble pruning
Output ensemble
Figure 1. Schematic depiction of the proposed method. AlphaFold3 allows the sampling of protein structures given amino acid sequence. To further condition the model by experimental observations, each time step of the AlphaFold3 diffusion model, an ensemble of structures is generated. Likelihoods of experimental observations calculated given each individual ensemble member (e.g.. enforce substructure) and on ensemble averages (e.g.. calculated electron density F. and average inter-atomic distances). The gradient of the combined likelhoodd terms is used the guidance score. At the final diffusion step, the generated ensemble is refined by force field relaxation and pruned by an orthogonal matching pursuit-like procedure to imprave the likelihood terms.
(as opposed to those performed 100 K), and progress in refinement tools permit explicit modeling of alterna- tive conformations ("altlocs") within overlapping density regions (Furnham et al., 2006; van den Bedem & Fraser, 2015; Wankowicz et al., 2024). Recent studies analyzing the PDB reveal that such multi-conformer annotations are widespread, reflecting inherent structural variability cap- tured in crystallography (Rosenberg et al., 2024a). In NMR spectroscopy, the experimental observables, such inter- atomic distances or bond-vector orientations reflect the time- and ensemble average, and NMR structures are always re- ported as bundles of conformations. However, AlphaFold's training objective to predict single "most probable" struc- ture biases its output toward statie snapshots, effectively marginalizing conformational heterogeneity encoded in training data.
for new models that can explicitly model protein ensembles that are faithful experimental measurements, Developing such models is the focus of the present work.
2. Contributions
In this work, we introduce experiment- guided AlphaFold3, computational framework that integrates experimental data with deep learning priors generate structural sembles consistent with experimental observables. Our key insight is that AlphaFold3 can be viewed as strong dequence-conditiones protein structure prior that may be further leveraged to solve inverse problems the space of protein structures. By solving these inverse problems under the prior imposed by AlphaFold3, we bridge the gap between data-driven predictions and experimental evidence, yielding ensembles that are both physically plausible and experimentally consistent.
Over the past year, multiple sequence-conditioned protein structure generative models like AlphaFlow (Jing et al., 2024), and the recent AlphaFold3 (Abramson et al., 2024) have been proposed to move beyond the one-sequence-one- structure paradigm. However, since these approaches re- main trained unimedally-modeled PDB entries derived predominantly from crystallographic data, the generated ensembles fail to capture the full heterogeneity implied by experimental measurements, thus limiting their practical utility (Rosenberg et al., 2024b). This emphasizes the need
Experiment-guider AlphaFold3. Our primary technical contribution a three-stage sacemble-fitting pipeline (Fig- ure 1). First, we present Guided AlphaFold3, where we adapt the diffusion-based structure module of AlphaFold3 to incorporate experimental measurements during sampling. To properly handle ensemble measurements, introduce non-i.i.d sampling scheme that jointly samples the ensem- ble, directing conformational exploration toward regions
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Inverse problems with pnteriment-guidee AlphaFold
7F5F
7JX6
38
38-31
72-78
72-78
7JX6
102-112
7F5F 102-112
A.
C.
AlphaFolca
AlphaFokts
Guided
Figure 2. Two crystallographic observations of the SARS-CoV-2 ORF8 protein 1.62À resolution (PDB: 7JX6 and 7F5F color coded as purple and green, respectively). The two structures exhibit major differences at three sites despite sharing a nearly identical main acid sequence deferring by single mutation L67S (A). AlphaFold3 prediction appears practically identical for the two sequences mispredicting the structure of the turn at site 38-41, correctly predicting the structure of 7F5F and mispredicting that of 7 JX6 at site 72-78, and correctly predicting the structure of 7 mispredicting that of /F5F site 102-112 (B). Electron density-guidee AlphaFold3 corrects these predictions by producing ensembles fitting well into the observed electron densities F, of both structures depicted as the 0.3 [e /À " J-isosurfaces (C).
compatible with the experimental constraints. We show that this approach effectively captures multi-modal ensem- ble measurements, where standard i.i.d. sampling methods fail (Figure A2). To our knowledge, this represents the first application of guided sampling within AlphaFold3 for exper- imental structural resolution. Second, we address artifacts introduced during guided sampling by using AlphaFold2's computationally efficient force-field relaxation step, effec- tively projecting candidate structures onto physically realis- tic conformations. Finally, we develop a matching-pursuit ensemble selection algorithm to iteratively refine the en- semble by maximizing agreement with experimental data while preserving structural diversity. We validate our frame- work through case studies on two foundational challenges in structural biology: (1) X-ray crystallographic structure modeling, where we recover conformational heterogeneity obscured in static electron density maps, and (2) NMR struc- ture determination, where we resolve ensembles that obey NOE-derived distance restraints.
multimodality in the density that cannot be fully captured by models like AlphaFold3 that predict single structures. This limitation, recognized early on in protein crystallog- raphy (Smith et al., 1986), is particularly evident in altloc regions (Rosenberg et al., 2024b), where multiple conforma- tions coexist in the same lattice (van den Bedem & Fraser, 2015; Furnham et al., 2006). This inadequacy presents compelling case for protein generative models to improve crystallographic structural modeling.
Hence, we introduce Density-guided AlphaFold3, which guides MlphaFol33-generaii structural ensembles to be faithful to experimental electron density maps. Density- guided AlphaFold3 renders structures that are consistently more faithful to the observed electron density maps than unguided AlphaFold3. In some cases, the guided structure outperforms PDB-deposited structure's faithfulness to the density (Table A3). Additionally, guided structures capture structural heterogeneity better than AlphaFold3 (Figures 2, 3). In some cases, guided structures capture the structural heterogeneity that PDB-deposited structures fail to model (Figure A1). Lastly, we are able to leverage the strong prior learned by AlphaFold3 to generate density-faithful ensembles in a fraction of the time required by conventional X-ray crystallography pipelines (Adams et al., 2010) (Table A5). our opinion, this advancement not only improves the accuracy of computational structural modeling but also has the potential to automate workflows for crystallographers
Improved crystal density modeling. X-ray crystallography is one of the most accurate techniques for protein structure determination. A typical pipeline involves the crystalliza- tion of protein samples and the subsequent fitting of atomic structures to electron density maps generated from X-ray diffraction patterns. However, this procedure is expensive, time-consuming, and often requires manual intervention by crystallographers (Doudna, 2000). As a result, several struc- human-induced induced biases that can degrade structural accuracy. Another limitation of crystallographic pipelines is the misleading notion of "sin- gle crystal and single structure". However, the PDB exhibits
tures deposited in the PDB exhibit human-
Accelerated NMR ensemble structure determination. Solution-state NMR enables the study of proteins in near- physiological aqueous environments, capturing conforma-
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Inverse problems with experiment-guides AlphaFold
Conformation
Conformetion B
423-131
PDB (4OLE)
AlphaFold3
Guided
Figure 3. Crystallographic observation of the human NBR1 protein at 2.52A resolution (PDB: 40L8) exhibits multi-modal backbone distribution at 423-431 (conformation modes A and color-coded in green and purple, respectively). AlphaFold3 predicts only conformation while completely missing the helical conformation A. Electron density-guidee AlphaFold3 predicts a bi m-ddalll distributed ensemble better describing the observed electron density Fs. Light blue surfaces and gray meshes in the zoomed-in inserts depict the 0.3 [e /A* Jiisosuurfs of the observed and calculated electron densities, F. and Fe,, respectively. Side chains in the inserts omitted for clarity.
tional heterogeneity through nuclear interaction restraints such nuclear Overhauser effects (NOEs) and scalar cou- plings (J-couplings). NMR-based structure determination typically employs restrained molecular dynamics (MD) sim- ulations, requiring hundreds of independent trajectories to adequately sample conformational spaces consistent with experimental data-a computationally intensive process that struggles to balance accuracy, efficiency, and ensemble di- versity (Lindorff-Larssee et al., 2005; Lange et al., 2008).
all atoms as X =X Xm). where x, denotes the i-th atom in the structure. Note that X implicitly depends on as the atom configuration is dependent on the amino acid identities.
Problem statement. Given protein sequence a and an experimental observation y, sample non-i.i.d ensemble of structures X' (XL X") from the posterior distri- bution p(X y.
Using Bayes' rule, the posterior distribution can be factor- ized as, p(X'' a, x p(y X) p((X' a), where p(y X, a) is the data term representing the likelihood of the experimental observation given the structural ensemble and amino acid sequence. The knowledge of the instru- ment's forward model is embodied by the likelihood term. On the other hand, p(X) a) is the prior term representing the probability of a structural ensemble given the amino acid sequence. A key distinction between these two terms is that the prior can be factorized into a product of indepen- dent priors for each sample in the ensemble X, whereas the likelihood is conditioned on the entire ensemble and hence is inseparable.
Here, we propose NOE-guided AlphaFold3, which refines SahhaFFol-generaate structural ensembles to satisfy NOE- derived distance restraints. The resulting ensembles adhere experimental NOE data more faithfully than AlphaFold3 predictions and, in some cases, even surpass the accuracy of existing PDB- deppsiisd NMR ensembles (see Table A4). In particular, we demonstrate that the ensembles produced by NOE-guided AlphaFold3 on ubiquitin, a benchmark sys- tem for NMR structure and dynamics, accurately capture experimentally observed conformational flexibility, as inde- pendently validated against experimentally measured S order parameters (Figure 5; Lienin et al. (1998)). In con- trast, standard AlphaFold3 predictions generate overly rigid ensembles inconsistent with ubiquitin's 's dynamic behavior. Finally, we note that our method dramatically improves the NMR structure determination process from many hours to a few minutes, while retaining the accuracy obtained through MD. We believe this will enable new experimental work- flows for NMR structural biologists.
meeeeuurr N-H
As the prior, we use AlphaFold3 (Abramson et al., 2024) to generate ensembles. To model the data term, we consider three distinct experimental modalities: crystallographic elec- tron density maps (Section 3.1). nuclear Overhauser effect (NOE) restraints (Section 3.2), and sub-structure condition- ing using known atom locations (Section 3.3).
3. Protein structure inverse problems
3.1. Crystallographic electron densities
This section introduces the forward model of crystallo- graphic electron density observables. Electron densities are volumetric images of the spatial charge distribution within
Notation. We denote the amino-acid sequence of protein as a and the corresponding 3D Cartesian coordinates of

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Inverse problems with experiment-guided AlphaFold
NMR structure 2K52
AlphaFold3
Guided (0.2)
Guided (0.5)
Constraint violation [Å]
-3
-2
10.6% 0.32À
23.1% 00.7AA
11.8% .85 0.50A
9.6% 0.33A
Figure 4. NOE constraint violations in the Methanocaldoeoccus jannaschii MJ1198 protein in the NMR structure ensemble (PDB: 2K52) and ensembles predicted by AlphaFold3 and using NOE-guidane, with strength 0.2 and 0.5. Violated constraints are depicted as lines color-coded by the amount of violation. Percentage of violated constraints (out of total 1212) and their median violation are reported helow each structure. A single best-fitting structure from each ensemble is shown for clarity.
the unit cell of a protein crystal lattice (Riley et al., 2021; van Zundert et al., 2018). The process of obtaining these maps begins with purifying and crystallizing the protein sample, followed by X-ray diffraction analysis (Smyth & Martin, 2000). When the crystal is exposed to an X-ray beam, it generates multiple diffraction patterns encoding the Fourier transform of the electron density function that is periodic on the crystal lattice (the Fourier transform is thus discrete on the reciprocal lattice). However, during this process phase information is lost, and molecular replacement procedure is required to impute the missing phase angles, after which a 3D electron density distribution is reconstructed. The re- sulting electron density map reflects the average density of trillions of protein molecules in the crystal rather than that of an individual molecule.
NOE arises from through-sper space dipolar interactions between nuclear spins, usually involving hydrogen atoms. The inter- actions are also dependent spatial proximity, with NOE effects observed typically when atoms are less than 6Λ apart. NOE mcasurements represent an ensemble average over all conformations of the protein in solution, capturing its intrin- sie structural heterogeneity.
Interatomic distances are usually measured in NOE spec- troscopy (NOESY) experiments, which consist of two- dimensional correlation spectrum. The proximity between two atoms is evidenced by a correlation peak at position in the spectrum that correspond to the two resonance fre- quencies of the spins. The intensity of the NOE signals is proportional to the temporal average of the inverse sixth power of the interatomic distance and can be used to es- timate spatial constraints within the protein. NMR signal intensities depend several other factors, which obscures the dependency on the inverse sixth power of the distance to extent. While quantitative distances can be obtained (Vögeli, 2014), it is common to use the NOE signal intensity only semi- coni-quaniitaiieely. With some abuse of NMR physics, we henceforth assume that the distance average observed directly, and define the NOE constraints as set D { (dly dıy) (i,j) P) of pairs of lower and upper bounds on the ensemble average, (X)
We denote the observed electron density map as Fo : R R without explicitly distinguishing between the continuous map and its discretized version. Given F. and a protein structure ensemble X, the log-likelihood log p(F. X, a) quantifies the agreement between the experimental data and the electron density predicted by the ensemble. To compute this likelihood, we calculate the predicted electron density (X(X) for each structure X in the ensemble (see Appendix A.2, A.6 for details). The log- ligellhooh is given by
Ex= day (X*) of the distances di (X*) = ||x x5| || between pairs of atoms j in individual structures X The log-likelihood is given by,
(L) where we pragmatically choose the L1 norm to quantify the discrepancy between the observed density and its calculated counterpart. Note that both F. and (X) are functions of the spatial coordinate & as explicated in the Appendix.
logp(F'. x) Fo
1
Σ
Xx
&
log p(D) X, Σ
[day - di (2)) + [di((x) -dulit) (2) (LS)EP
3.2. Nuclear Overhauser effect restraints
This section introduces the forward model for the nuclear Overhauser effect (NOE) restraints as measured using NMR spectroscopy. NOE restraints provide essential informa- tion about interatomic distances between biomolecules. The
where [x]4 = max(x, 0). Additional computational details, noise model, and limitations are provided in App. A.2, A.12.
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Inverse problems with experiment-guided AlphaFold
3.3. Substructure conditioning
bution, further plug in the guidance score, obtaining,
In many cases, especially when refining crystallographic structures, it is useful to determine the protein structure only for subset of amino acids while keeping the rest frozen. This can be achieved by specialized likelihood term, that parallels the BubsurructueConde iiner in Chroma (Ingraham et al., 2023), which incorporates a reference structure to constrain and guide the optimization process during inference.
X
X
Vxi p
(X'
aa
d
=
+
3dt
2
X
X
Vx- lo X
a)
N
1 x logp (y/x².
X" a)) Bdt + VB.
(5)
As the input, we will assume to be given collection of reference atom locations Y {y, i E A} for some sub- set of atom indices A. Using a quadratic penalty on the deviation, the log-likelihood assumes the form
where N ~ (), In the above equation, unconditional score term Vxx log Pa (X* a) is separable in the ensemble members. However, the guidance score term is not sepa- rable due to the non-i.id nature of the likelihood function. Additionally, the hyperparameter v) can be used scale the guidance score and direct the diffusion model to generate samples with high posterior likelihood. The pseudocode for guided AlphaFold3 and other implementation details are presented in Appendix A.1.
po a)
ΣΣ
|x - k=liEA
(3)
Note that unlike the previously discussed forward models, this term is separable with respect to the individual ensemble constituents. See A.2 for the noise model.
4.2, Force-field relaxation
After performing non-ii.d guided diffusion, we noticed that when sampling large ensemble, the conformations tend to overfit to the noise in the experimental observations. This can significantly degrade the ensemble quality, often leading to artifacts such as broken bands and atomic clashes (Shapo- valov & Dunbrack, 2011). 1). To eliminate these artifacts, we remove from the ensembles structures with broken bonds (distance between any pair of bonded atoms exceeding the threshold Thond A) or steric clashes (distance between any two atoms is less than Telash Å). Specific threshold values and implementation details are provided in Appendix A.7.
4. Experiment-ggooundel AlphaFold3
4.1. Guiding AlphaFold3
One of the key features distinguishing AlphaFold3 from its predecessors (Jumper et al., 2021), is the introduction of diffusion-barte based (Ho et al., 2020) generative model for protein structure prediction. This model acts as a prior over the all-atom distribution of protein structures and en- forces structural coherence. AlphaFold3's forward diffusion process is modeled as a uuriance-preservngg stochastic dif- ferential equation (SDE) (Song& Ermon, 2019; Weiss et al., 2023), whose backward SDE simplified variant of the formulation in (Karras al., 2022),
Post-filtering, subtle bond length violations and geometric inconsistencies may persist. To accurately model molecular interactions and eliminate geometric violations, relax the structures with off-the-shelf harmonic force-field. such as AMBER (Hornak et al., 2006). This ensures that all remaining structures are physically plausible with no structural artifacts while still maximizing the log- likelioeddood of the experimental observations.
dx = V (X a) Bydt + VBN. (4)
Vx
Here X containing the atomic coordinates acts diffusion variable, 3e defines the noise schedule parameters, and N (0, I) is sampled from isotropic normal distribution. The score function V log Pt (X a) is modeled using atom-transformmr based denoising network (Jumper et al., 2021; Abramson et al., 2024; Vaswani, 2017). AlphaFold3 samples random latent vectors from the base distribution Xr (O, BoI) and numerically integrate equation (4) from t = T down to t 0.
4.3. Ensemble filtering using matching pursuit
Post relaxation, we employ a matching pursuit-based ap- proach (Mallat & Zhang, 1993) to greedily select a subset of the relaxed ensemble, Xx (X* fits the observation y. Starting with = 0. during every iteration of the matching pursuit algorithm. we seek max- imize log p(y Xzu(k): a) over all l: & I. The optimal element A is then added to the support set I and the process continues until the likelihood no longer increases or the maximum allowed ensemble size Amax is reached. de- tailed explanation of the matching pursuit-ban based ensemble
I}. that best
Here, rather than predicting a single structure, we propose to sample an ensemble X' = (x1,.. X") of n structures. The SDE for single-structure generation in equation (4) can be straightforwardly generalized for ensemble sampling. In order to sample a non-i.i.d sample from the posterior distri-
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Inverse problems with experiment-guided AlphaFold
filtering procedure, along with its pseudocode, is provided in Appendix A.8.
5TEU conformation variant is mispredicted by AlphaFold3, adding electron density guidance reproduces both variants faithfully. We also observe more heterogeneous ensemble generated in the latter case, including flipped carbonyl oxy- gen and sidechain flexibility that better explain the density. A quantitative evaluation of this and additional homologous pairs is presented in Table A2.
5. Modeling crystallographic ensembles
In what follows, we investigate three distinct cases of con- formational heterogeneity evident in X-ray crystal struc- tures that are consistently mispredicted by the unguided AI- phaFold3 and demonstrate that AlphaFold3 guidance with electron densities significantly improves these predictions.
Heterogeneity within the same crystal. In the last case we consider, the heterogeneous conformations are intrinsic to the protein itself and are observed in single crystal mea- surement, with the electron density appearing markedly bi- or multi-modal. While the phenomenon is very common in flexible side chains, the heterogeneity in the backbone conformations has remained underappreciated. In fact, com- mon visualization platforms (e.g., PyMOL (DeLano et al., 2002) and ChimeraX (Pettersen et al., 2021)) and structural modeling tools (e.g.. GROMACS (Van Der Spoel et al., 2005)) frequently reading the first listed conformation as de- fault and disregarding alternate conformations (Gutermuth et al., 2023). Nevertheless, the Protein Data Bank is actu- ally riddled with such altlocs. A recent study by Rosenberg et al. (2024a) compiled comprehensive catalog of alternate conformations from PDB structures and the same group showed that even for regions with well-separated and sta- ble alternate conformations, structural ensemble predictors such as AlphaFold3 fail to reproduce the experimentally determined distributions or capture the bimodal nature of backbone conformations (Rosenberg et al., 2024b). Here we demonstrate that this same set of separated and stable con- formations is well-modeled using electron density guidance. use protein decoded from NBRI,
Structurally heterogeneous homologous proteins. The first is the most obvious case where a specific protein is captured in different conformations over multiple experi- ments. Apart from different interactions with molecular partners, these altered conformations can result from dif- ferences in the expression or purification processes or dif- use the
ferent solvent conditions. As illustration, SARS-CoV-2 accessory protein that is encoded by the open reading frame ORF8 and facilitates immune evasion in in- fected host cells. This small (104 amino acids) protein has been structurally resolved in several independent works that demonstrate significant structural variability in the loop re- gions. PDB structures 7F5F and 7 differing only by single point mutation, crystallized individually without the presence of molecular partners and diffracting to the same resolution (1.6A) exhibit major conformational variations in three loops as depicted in Figure 2. Due to essentially identical sequence conditioning, AlphaFold3 fails to capture both global conformations well, predicting a tight ensemble that is similar to one of the variants one site and to the other the other. Guiding the ensemble generation with the observed electron density allows to better capture the physical reality of these different structures. A quantitative evaluation of this example is presented in Table A1.
As an illustration, the neighbor of the human BRCAI gene I implicated in breast cancer. The PDB structure 40LE resolved to 2.52A contains a region of 9 amino acids that was modeled as a superposition of two alternate conformations (Figure 3). While AlphaFold3 accurately predicts only one of the con- formations, electron density guidance generates an ensem- ble capturing the bi-modal nature of the backbone and better explaining the density. A quantitative evaluation of this an other 11 cases is presented in Table A3. Figure A2 presents a quantitative evaluation of ensemble bi-modality (refer to Appendix A.10 for details) and demonstrates that density- guided AlphaFold3 consistently produces bi-modally dis- tributed ensembles, while its unguided counterpart typically produces a single mode.
Heterogeneity in equivalent environments. The second case we explore is more local and subtle version of the former, highlighting that the same amino acid segment in distinct but homologous proteins can adopt different confor- mations, even when the immediately surrounding protein environment is equivalent (and, thus, the local segments have identical contact networks). A key component of Al- phaFold3 prediction is based on cocvolutionary signals from multiple sequence alignment (MSA) of homologous pro- teins in order to discern contact maps (which, in the form of MSA embedding, condition the diffusion model as depicted in Figure 1). It is, therefore, unsurprising then that homol- ogous proteins harboring regions of identical amino acid sequence embedded into an equivalent spatially adjacent amino acid environment are predicted to have identical local conformation even from distinct protein sequences. As an il- lustration, we show in Figure Al pair of such homologous proteins (PDB 4 4 and 5TEU) featuring a sequence of 4 amino acids in two distinct turn conformations. While the
An extended analysis is presented in Table A9, with addi- tional 15 proteins where regions up to 22 residues in some cases (5V2M & 6E2S) are optimized. In all cases, cosine similarity between observed and calculated electron density maps matched or exceeded that of the PDB entries. Similar results were observed for polypeptide chains.
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Inverse problems with experiment-guidee AlphaFold
K11
AlphaFold3
Guided (ii.d.)
Guided (non-ii.d.)
1.0
Ensemble 5t
K11
0.8
0.6
AlphaFold3
K29
0.4
C.
D.
1.0
Ensemble 52
Flexible turn
Infiexible helix
0.8
1.0
N-H order parameter St
0.6
rw0.52
Guided (L..d.)
0.4
0.6
1.0
Ensemble 52
0.4
0.8
0.2
Experiment
Guided (l..d.)
0.6
r-0.72
AlphaFold3
Guided (non-i.i.d.)
Guided (non-4.1..)
0.0
0.4
A.
76
0.6
1.0
Residue number
B.
Experimental 5
Figure 5. Comparison of conformational flexibility ubiquitin ensembles predicted by AlphaFold3 and our i.i.d and non-i.i.d NOE- guided method. The N-H order parameter S is calculated on the ensembles and compared to the experimental NMR observables. For guided predictions, plotted are the mean and standard deviation
independent runs. Green and purple shadings indicate a-helices and f-strands as predicted by DSSP (Frishman & Argos, 1995) (A). AlphaFold3 exhibits the lowest correlation to the experimental measurement (r 0.47), while guiding AlphaFold3 with NOE measurements improves it to r = 0.52 for i.i.d guidance and r = 0.72 with non-i.i.d. ensembles (B). For two lysines site 11 (entrance into the second 3-strand from flexible loop) and 29 (middle of a-helix) highlighted in (C), guided ensembles exhibit more variability of the amide N-H bond direction in the flexible turn, while showing less variability in the inflexible helix, in agreement with the experimental observation (D).
6. Modeling NMR ensembles
prediction, its training on static X-ray crystallography data biases its ensembles toward rigid conformations, failing to capture conformational heterogeneity satisfy NMR- derived restraints (Figure A7 and Table A4). This limits its utility for modeling protein dynamics. In what follows, we demonstrate that NOF-guided AlphaFold3 recovers en- sembles that (i) rigorously obey NOE distance restraints; (ii) reproduce experimentally observed flexibility; and (iii) achieve this in minutes, overcoming the computational bot- tleneck of traditional ensemble methods.
NMR exploits the magnetic resonance of atomic nuclei to probe protein structure and dynamics. NOEs acquired from solution-state NMR, in particular, provide distances between atoms averaged over the ensemble of molecules in the sam- ple and over time scales up to milliseconds. NOE-derived distances, thus, comprise the conformational heterogeneity.
Status quo. NMR structures are determined by integrating molecular dynamics (MD) simulations with NMR- restraints, using biomolecular force fields (Wang et al., 2004; Schwieters et al., 2006; Güntert, 2004). However, since NMR observables inherently reflect ensemble-averaged measurements, simulating single conformers often leads to mode collapse, producing rigid ensembles that poorly capture true conformational dynamics (Figure 4 in Lindorff- Larsen et al. (2005)). To address this, ensemble-based MD pponnerees-pioneeeeee by (Lindorff-Larsen et al., 2005) and (Lange et al., 2008) simultaneously to satisfy experimental restraints. While ef- fective, these methods remain computationally prohibitive, requiring days even for small systems like the 76-residue ubiquitin. While AlphaFold has revolutionized structure
Ubiquitin, widely regarded as the benchmark system for NMR- based protein dynamics studies, served as a critical test case for our method. To incorporate experimental con- straints, we applied NOE-based guidance by integrating the likelihood term derived in Section 3.2 into the ensem- ble refinement framework defined by equation (5). This framework was explicitly parameterized with NOE-derived distance restraints obtained from the NMR structure PDB 1D32. NMR experiments based on 1 spin relaxation are a well-established way to assess protein dynamics (Palmer, 2004). Particularly, these experiments, which are entirely in- dependent from NOE measurements, provide the amplitude
multiple conformers
8

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Inverse problems with experiment-guided AlphaFold
of motion of the amide bond vector on time scales shorter than a few nanoseconds. This amplitude is expressed as (I-S²), where S² is the so-called squared order parameter. This parameter is, thus. a convenient independent to vali- date the heterogeneity found in the ensembles determined by our NOE-guided AlphaFold approach. We computed the N-H S bond order parameters and compared them against experimental measurements reported in (Lienin et al., 1998) (see Appendix A.14 for details). As illustrated in Figure 5, AlphaFold3 predictions yield ensembles dominated by rigid conformations, exhibiting only moderate correlation with the experimental S² (r = 0.47). In contrast, NOE- guided AlphaFold significantly improves agreement: i.id. guidance achieves r = 0.52 for i.i.d guidance, whereas non-i.idd ensemble sampling elevates it to r 0.72. No- tably, the refined ensembles better replicate the dynamic behavior observed in both flexible and structured regions, as evidenced by the distribution of N-H bond orientations across conformers.
Patiny, 2008). For each protein, the guided ensemble better satisfies NOE restraints than unguided AlphaFold3 struc- tures, and sometimes, even outperforms the corresponding NMR PDB structure.
7. Related Work
This work presents the first application of guided sampling with AlphaFold3 resolve protein structural ensembles using crystallographic and NMR experimental data. Prior approaches have attempted similar goals but exhibit key limitations. Toth-Petroczy et al. (2016) uses Potts model (Wu, 1982) to predict structured regions in intrinsically dis- ordered proteins from evolutionary data. However, they do not leverage experimental measurements. Fadini et al. (2025) fit a single conformer to density maps by optimizing MSA contact maps but cannot model conformational hetero- geneity. Liu et al. (2024) use str2str protein diffusion model (Lu et al., 2014) to generate cryo-EM -MMidguiedd struc- tural ensembles. While related, their focus is on a different modality. Maddipatla et al. (2024) fits ensembles to crys- tallographic density maps using Chroma (Ingraham et al., 2023) and captures multiple conformers some extent. However, it fails to capture conformational heterogeneity over a long residue range due to Chroma's 's hierarchical for- mulation. Also, Chroma's 's sequence conditioning is only promoted, limiting enforcement of NMR distances. Finally, Levy et al. (2024) uses Chroma as their diffusion model and inherit similar limitations. Additionally, they use synthetic data instead of raw experimental observations.
Peptides. We used the benchmark from McDonald et al. (2023) from which we selected 20 peptides worst predicted by AlphaFold3, for which the NMR structures result vi- olate less than 10% distance restraints. This resulted in three structures, namely, 1DEC, 2LI3, 3B3G (Figure A7), for which AlphaFold3 produces a partially wrong fold than what is suggested by NMR measurements. This is visually depicted in Figure A7, and is evidenced by the violation of restraints of AlphaFold3 baseline presented in Table A4, We observed that NOE-guide AlphaFold3, both produces ensembles that obey > 15% more restraints and order of magnitude lower eтror (see Table A4) and fixes the mispre- diction made by unguided AlphaFold3 (Figure A7).
8. Conclusion
100 NMR spectra database. We evaluated the proposed method on 9 proteins (55- - 102 residues) from a recently compiled database of NMR structures (Klukowski et al., 2024). While AlphaFold3 largely retains the correct fold for these systems, its baseline predictions exhibit, aver- age, 8% more NOE restraint violations than the deposited NMR structures (Table A4). In all cases, NOF-guided Al- phaFold reduced violations, and in half of the cases, the guided ensembles even outperformed PDB -eeeossttee NMR structures restraint compliance. Visual alignment with NMR ensembles (Figure A7) and quantitative agreement in p-RMSF (Table A4) confirm that guidance recovers confor- mational heterogeneity consistent with NMR structures. In- creasing the guidance scale progressively enforces restraint satisfaction, as shown in Figure 4. Critically, these improve- ments are achieved within minutes (Table A6) - reaching the accuracy of MD-derived ensembles at a fraction of the computational cost.
In this paper, we present a general methodology for guid- ing AlphaFold3 using experimental observables and sys- tematically evaluated its utility in fitting structural ensem- bles to erystallographic and NMR data. In crystallography, the robustness of our method demonstrated by the wide range of conditions of the evaluated targets: electron density maps covering wide range of resolutions (from sub-A to medium-low resolution) and qualities (covering high and low B-factors), protein segments with different lengths, vari- able amino acid identities, different secondary structure contexts, and AlphaFold3 mispredictions of severity rang- ing from the subtle conformation of the carbonyl oxygen to loop regions predicted tens of Angstroms away from the experimentally observed location. Likewise, in NMR we show a great variability of the number and quality of NOE constraints in the evaluated structures. In the future, we in- tend to generalize the method to jointly sampling ensembles representing multiple molecules to model protein complexes and protein-ligant interactions. We also plan to extend the model to single- and multi-particke cryoEM where handling conformational heterogeneity constitutes major challenge.
A supplementary analysis is presented in Table A10, fea- turing 27 additional NMR proteins from NMRDb (Banfi &

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Inverse problems with experiment-guided AlphaFold
Acknowledgments
D., Arnold, E., al. International tables for crystallog- raphy volume A: Space-group symmetry.. Wiley Online Library, 2016.
This work was supported by the Israeli Science Founda- tion (ISF) grant number 1834/24. We acknowledge sup- port from the Austrian Science Fund (FWF grant numbers 15812-B and 16223) and the financial support of the Helms- ley Fellowships Program for Sustainability and Health. This research uses resources of the Institute of Science and Tech- nology Austria's scientific computing cluster.
Burley, S. K., Berman, H. M., Kleywegt, G. J., Markley, J. L.. Nakamura, H., and Velankar, S. Protein data bank (pdb): the single global macromolecular structure archive. Protein crystallography methods and protocols, pp. 627- 641, 2017.
Impact statement
Chen, X., Zhang, Y., Lu, C., Ma, W., Guan, J., Gong, C., Yang, J., Zhang, H., Zhang, K., Wu, S., Zhou, K., Yang, Y., Liu, Z., Wang, L.,, Shi, B., Shi, S., and Xiao, W. Protenix advancing structure prediction through a com- prehensive alphafold3 reproduction. bioRxiv, 2025. doi: 10.1101/2025.011 00..0.......666....
The work presented here attempts to advance the modeling of protein structure and dynamies using experiment-guise AlphaFold3. The developed methods lead to solving ex- isting tasks in structural biology significantly faster and more accurately. They may further permit the utilization of the wealth of experimental structural and dynamic mea- surements currently unused for the generative modeling of protein ensembles. Since proteins fundamental ingre- dients of life and implicated in health and disease, the potential impact on basic and applied research may be pro- found. We do not see any special ethical concerns worth highlighting.
DeLano, W. L. et al. Pymol: An open-source molecular graphics tool. CCP4 Newsl. Protein Crystallogr, 40(1): 82-92, 2002.
Diederik, P. K. Adam: A method for stochastic optimization. International Conference on Learning Representations, 2015.
Doudna, J. A. Structural genomics of mna Nature Structural Biology. 7(11):954-956, Nov 2000. ISSN 1545-9985.
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Inverse problems with apperieet--guidee AlphaFold
STEU
D83
4NE4
B.
4NE4
A8280
STEU
AlphaFold3
73-76
AlphaFold3
A.
C.
Guided
Figure AI. Crystallographic observations of a pair of homologous proteins (PDB: 4NE4 at 1.73A resolution and 5TEU at 1.62A). The pair features a distinct turn conformation at a corresponding site residues 73-76 4NE4 and 91-94 in 5TEU despite identical local amino acid sequences and contacts with the environment (A). AlphaFold3 predicts the conformation of 4NK4 in both sequences mispredicting the conformation of STEU (B). Electron density- density-guided AlphaFold3 corrects these predictions by producing ensembles fitting well into the observed electron densities F. of both structures depicted the 0.3 [e [e-/ I-sssususss (C). The ensemble predicted for STEU exhibits conformation heterogeneity with two flipped states of the carbonyl oxygen in A82 (highlighted red) and flexible side chain in A82 (dark green) better explaining the Fo (D).
1.00
Uncond
Guided (l.id)
0.75
AF3 Guided (non-l.i.d)
0.50
Normalized distance
0.25
-0225
6F2
2YNT-
7ECH
2016
4DLE
201A
600F
5051
3V35
INPU
7R7W
Figure A2. Distribution of normalized distances to conformations (-1) and (+-1) in generated ensembles for 12 protein struetures.
Compared are ensembles generated by AlphaFold3 and our electron density i.i.d. and non-i.i.d guidance. While unguided AlphaFold3 typically fails to capture the multi-modal nature of these structures, non-i.i.d consistently produces multi-modal ensembles.
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Inverse problems with experiment-guidee AlphaFold
Alphaflow Unguided
ESMFlow Unguided
AF3 Guided
Normalized distance
0.25
0.00
-0.25
-0.50
-0.75
-1.00
6/2-
20N
7CCB
203G
40LE
201A
600r
5653
3435
4NPU
7W7W
30NE
Figure A.3. Distribution of normalized distances to conformations A (-1) and B (+1) in generated ensembles for 12 protein structures. Compared ensembles generated by Alphaßlow ESMFllw (Jing et al., 2024) and our electron density non-i.idd guidance. While Alphaflow and ESMFlow typically fail to capture the multi- nature of these structures, non-i.i.d consistently produces multi-modal ensembles.
1.00
Chroma Guided Chroma Unguided AF3 Guided
0.75
0.50
Normalized distance
0.25
-0.25
6F2-
21NT-
7ECH
2016
4DLE
201A
600F
5651
3V35
BOME
Figure A4. Distribution of normalized distances to conformations (-1) and B (+1) in generated ensembles for 12 protein structures.
Compared are ensembles generated by Chroma (Ingraham et al., 2023), non-i.i.d guided Chroma (Maddipatla et al., 2024) and our electron density non-i.i.d. guidance. While non-i.i.d. guided Chroma captures multimodal distributions some extent, our approach achieves more consistent bimodal across different proteins.
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Inverse problems with experiment-guisn AlphaFold
cosine similarity (Fo.F) # bond length violations
200
150
100
50
0.8
Guided AF3
Unguided AF3
0.7
Guided AF3 + Relaxation
Unguided AF3 Relaxation
0.6
0.5
25
50
75
100
125
150
175
200
Diffusion timestep
Figure A5. Quantitative assessment of generated structure validity (top plot) and agreement with the experimental observation (bottom plot) during the progress of diffusion iterations using an electron density likelihood for guidance. Structural validity is quantified as the number of violated bond lengths, while the agreement to experiment measured using the cosine similarity between F. and Fe.
set overall cosine similarity
0.0
-05
1.5
-2.0
OMP
OMP set size
Figure A6. Ablation study depicting faithfulness to F, as we more samples the chosen ensemble X.. Faithfulness to experimental observation measured using the normalized cosine similarity between F. and Fe.
15

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Inverse problems with experiment-guider AlphaFold
1DEC
2L13
2JRM
6SOW
2JVD
2KIF
AlphaFold3
31.1% 0.35A
31.9% 0.59À
NMA structure AlphaFold3
16.7% 0.05A
21.1% 0.09À
26.2% 0.14A
51.8% 0.29A
NOE NOE-guided guided
13.8% 0.02À
5.9% 0.024
14.4% | 0.04A
NMR structure Guided
43.4% 0.13A
9.4% 0.02Á
17.7% 0.06Á
Figure A7. Conformation ensembles generated for six proteins using AlphaFold3 (first row) and the proposed NOE guidance (second row). Ensembles are visualized in blue overlaid on corresponding NMR structures solved from the same NOESY data. PDB identifiers are indicated above each structure. The numbers below report percentage violated constraints and the median violation.
16

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Inverse problems with experiment-guidee AlphaFold
PDB ID
Residue region
Region sequence
Resolution (A)
PDB
AlphaFold3
Guided (non-i.i..)
7F5F A
102 - 112
1.62
0.717
0.562
0.699
CSFYEDFLEYH
:
102 112
1.61
0.698
0.561
0.696
7E5F A
38 - 41
1.62
0.788
0.645
0.785
PIHF
:
38
1.61
0.739
0.579
0.736
7F5F A
72 - 78
1.62
0.668
0.660
0.736
QYIDIGN
72 78
1.61
0.652
0.552
0.657
Table AI. Quantitative evaluation of cosine similarity between the observed and calculated electron density maps (higher is better) on three structurally dissimilar crystallographic structures the SARS-CoV-2 ORF8 protein. Colored in green are the cases in which the ensemble produced by the censity-guided AlphaFold3 better fits the observed electron density better than the unguided counterpart, and blue the cases which the generated ensemble also outperforms structure deposited the PDB.
PDB ID)
Residue region
Region sequence
Resolution [Å]
PDB
AlphaFold3
Guided (non-i.i.d.)
4QTD:A AA
225 228
1.5
0.825
0.666
0.810
FPGR
7
263 266
1.75
0.831
0.627
0.819
4QTD A
280
1.5
0.792
0.626
0.771
FPADSEHNKLKA
A
318 329
1.75
0.783
0.575
0.780
:
238 241
241
1.40
0.855
0.754
0.830
CGVY
:A
283 286
1.60
0.767
0.625
0.749
A
110 113
1.75
0.782
0.685
0.785
RDHA
2F9L:A
110 113
1.55
0.701
0.693
0.739
3FIL:A
-4
0.95
0.886
0.744
0.859
SDGI
3
21 24
1.70
0.859
0.810
0.850
5XNE: A
211 214
1.50
0.820
0.775
0.821
EDCT
6J3D A
211 - 214
1.70
0.849
0.798
0.841
4NE4: A
73 76
1.73
0.734
0.628
0.728
ADDP
5
91
1.62
0.754
0.666
0.763
A
17
1.36
0.699
0.626
0.709
PPAQ
1
17-20
1.10
0.751
0.561
0.728
A
29
1.36
0.710
0.648
0.696
VGDD
1
26 - 29
1.10
0.761
0.611
0.745
A
113 - 119
1.36
0.774
0.712
0.777
PNPDDPL
A
113 - 119
1.10
0.799
0.635
0.778
A
321 - 324
1.80
0.654
0.404
0.560
VPPF
321 - 324
1.50
0.864
0.518
0.851
2IE8: A
284 - 287
1.80
0.697
0.483
0.671
PVPY
1V6S:0 A
284 - 287
1.50
0.812
0.540
0.817
2IE8: A
354 - 363
1.80
0.639
0.454
0.640
VNRLGLKERF
1V6S:0 A
354 - 363
1.50
0.725
0.487
0.731
Table A2. Quantitative evaluation of cosine similarity between the observed and calculated electron density maps (higher is better) on homologous protein pairs harboring locally identical amino acid sequences in the identical environment (same contacts). Colored in green the cases in which the ensemble produced the density-guided AlphaFold3 better fits the observed electron density better than the unguided counterpart, and blue the cases in which the generated ensemble also outperforms the structure deposited in the PDB.
17

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Inverse problems with experiment-guides AlphaFold
Guided AlphaFold3
PDB ID
Residue region
Region sequence
Resolution [Á]
PDB
AlphaFold3
Non-i..d.
i..d.
:
98 - 103
YOGDPAW
1.20
0.773
0.706
0.761
0.755
:
183 - 185
GNG
1.60
0.750
0.702
0.745
0.733
A
24
FNVP
1.11
0.755
0.687
0.737
0.716
B
245 250
KAQERD
1.90
0.798
0.736
0.800
0.804
7EC8:A AA
187 190
DGGI
1.35
0.878
0.839
0.860
0.855
201A:A
50 53
KONN
1.60
0.759
0.739
0.757
0.746
5G51:A
290 295
GSASDQ
1.45
0.748
0.447
0.749
0.742
B
423 - 431
STEKKDVLV
2.52
0.880
0.809
0.882
0.854
129 - 133
VTAGG
2.00
0.862
0.752
0.857
0.845
B
133 136
FEEI
1.50
0.795
0.730
0.797
0.784
BB
46 50
IEKVE
1.17
0.760
0.718
0.761
0.748
6QQF:
68 75
RTPGSRNT.
1.95
0.833
0.814
0.832
0.825
Table A3. Quantitative evaluation of cosine similarity between the observed and calculated electron density maps (higher is better) on structures with separated in which the ensemble produced by the density- density-guided guided AlphaFold3 better fits the observed electron density better than the unguided counterpart, and in blue the which the generated ensemble also outperforms the structure deposited in the PDB.
backbone conformations (altlocs) from (Rosenberg al., 2024a). Colored in green the
NMR (PDB)
AlphaFold3
Guided AlphaFold3
PDB ID
Source
len
#NOEs
Viol. %
Viol.
Viol. %
Viol.
p-MMSF
Viol. %
Viol.
R-MSSF
1DEC
pept.
39
602
11.4%
0.019
31.2%
0.673
0.80
15.0%
0.062
0.86
2L13
pept.
30
354
1.0%
0.003
29.3%
1.104
0.57
5.3%
0.023
0.66
3BBG
pept.
40
535
3.5%
0.023
33.3%
1.253
0.63
21.7%
0.174
0.49
1YEZ
NMRDb
68
1512
11.4%
0.074
12.4%
0.097
0.76
7.8%
0.046
0.79
2JRM
NMRDb
60
2706
12.2%
0.069
19.7%
0.168
0.83
13.0%
0.072
0.86
2JVD
NMRDb
48
1324
6.5%
0.024
13.1%
0.079
0.60
7.9%
0.032
0.70
2K52
NMRDb
74
1212
14.9%
0.092
27.6%
0.295
0.60
14.3%
0.060
0.80
2K57
NMRDb
55
1200
8.9%
0.070
17.8%
0.135
0.70
10.1%
0.058
0.89
2KIF
NMRDb
102
3124
11.8%
0.077
19.3%
0.168
0.57
13.3%
0.081
0.91
2KRS
NMRDb
74
1305
19.8%
0.145
13.8%
0.117
0.82
10.3%
0.057
0.83
2MA6
NMRDb
61
1077
10.4%
0.070
9.7%
0.076
0.93
8.4%
0.053
0.89
6SOW
NMRDb
58
1589
31.4%
0.201
42.6%
0.422
0.30
36.5%
0.197
0.54
Table Quantitative evaluation of restraint violation and backbone flexibility for protein structures sourced database (NMRDb; Klukowski al. (2024)) and benchmark peptides (pept.. McDonald et al. (2023)). Violation percentages (Viol. %) quantify fraction experimental NOE restraints the modeled ensemble deviates from these restraints. Correlation with ground-truth RMSF (p-RMSF) measures the accuracy of the ensemble's backbone flexibility. Colored blue are cases in which the ensemble produced by NOE-guided AlphaFold3 better satisfies the distance restraints than the corresponding NMR structure deposited in the PDB. Colored in green the cases in which the NOE-guided AlphaFold3 achieves at least 15% fewer restraints than AlphaFold3. Colored in black are cases in which NOE-guided AlphaFold3better agrees with the NMR structure in terms of conformational flexibility compared to AlphaFold3
NMR spectra
are not satisfied, while violation distances (Viol. Å) report by how far
18

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Inverse problems with experiment-guided AlphaFold
Runtime (seconds)
PDB
Residue range length
Seq. length
Guided AlphaFold3
AlphaFold3
6JF2
145
99
31
2YNT
3
221
183
43
7EC8
265
225
52
40LE
120
73
25
2B3P
228
181
45
6E2S
20
311
324
82
5SUJ
4
380
427
96
5V2M
20
311
311
60
Table A5. Runtime comparison (in seconds) for Non-I..D, crystallographic guidance with AlphaFold3versus unguided AlphaFold3 across multiple PDB entries.
Ensemble size
Runtime (seconds)
16
503
32
847
64
1545
128
2978
Table A6. Runtime comparison (in seconds) of performing Non-L.LD. NOE guidance with AlphaFold3 across different ensemble sizes.
NMR (PDB)
AlphaFlow
ESMFllw
Guided AlphaFold3
PDB ID
#NOEs
Viol. %
VioLÁ
Viol. %
VioLA
Viol. %
VioL.
Viol. %
Viol.
1DEC
602
11.4%
0.019
48.7%
1.687
40.2%
0.863
15.0%
0.062
2L13
354
1.0%
0.003
28.8%
1.150
29.2%
1.274
5.3%
0.023
3BBG
535
3.5%
0.023
34.8%
0.654
23.5%
0.169
21.7%
0.174
1YEZ
1512
11.4%
0.074
12.4%
0.092
12.3%
0.075
7.8%
0.046
2JRM
2706
12.2%
0.069
18.2%
0.162
17.9%
0.165
13.0%
0.072
2JVD
1324
6.5%
0.024
17.8%
0.117
18.4%
0.129
7.9%
0.032
2K52
1212
14.9%
0.092
37.6%
0.471
35.8%
0.427
14.3%
0.060
2K57
1200
8.9%
0.070
19.3%
0.152
19.9%
0.137
10.1%
0.058
2K1F
3124
11.8%
0.077
21.9%
0.147
21.3%
0.143
13.3%
0.081
2KRS
1305
19.8%
0.145
18.1%
0.127
15.6%
0.097
10.3%
0.057
2MA6
1077
10.4%
0.070
18.0%
0.156
18.9%
0.173
8.4%
0.053
6S0W
1589
31.4%
0.201
47.5%
0.460
44.1%
0.401
36.5%
0.197
Table A7. Table A4 extended with more basclines including AlphaFlow and ESMFlow. Colored in blue are cases in which the ensemble produced by NOE-guided AlphaFold better satisfies the distance restraints than the corresponding NMR structure deposited the PDB. Colored in green are the cases in which the NOE-guided AlphaFold achieves at least 15% fewer restraints than AlphaFold.
19

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Inverse problems with experiment-guider AlphaFold
PDB ID
PDB
Guided Chroma
ESMFlow
AlphaFlow
Guided AlphaFold3
:
0.773
0.730
0.674
0.711
0.761
2 A
0.759
0.740
0.725
0.714
0.745
A
0.755
0.743
0.646
0.642
0.737
A
0.798
0.774
0.749
0.698
0.800
7EC8: A
0.878
0.869
0.841
0.829
0.860
2
0.759
0.754
0.736
0.737
0.757
5G51:A :
0.748
0.749
0.486
0.513
0.749
: B
0.880
0.863
0.792
0.793
0.882
6JF2: A
0.833
0.848
0.795
0.735
0.832
4NPU: B
0.796
0.795
0.756
0.772
0.797
7R7W: B
0.760
0.758
0.692
0.698
0.761
6QQF
0.833
0.832
0.773
0.775
0.832
Table A8. Table A3 extended with baselines. Colored in green the cases in which the ensemble produced by the density- guidedd AlphaFold3 better fits the observed electron density better than other baselines (barring the PDB structure), and in blue the cases which the generated ensemble also outperforms the structure deposited in the PDB.
PDB ID
Residue Range
PDB
AlphaFold3
AlphaFlow
ESMFlow
Guided AlphaFold3
2
189
195
0.674
0.534
0.568
0.566
0.697
A
208
213
0.853
0.745
0.751
0.782
0.853
3
1050
1053
0.781
0.695
0.723
0.724
0.770
BSUJ:B
198 201
0.753
0.551
0.509
0.539
0.720
8A4A:A
41 49
0.623
0.575
0.643
0.585
0.592
5V2M:A
293 312
0.672
0.643
0.654
0.573
0.690
7
225 232
0.689
0.648
0.655
0.650
0.665
:A
200 218
0.785
0.713
0.716
0.709
0.760
7A7M:A
189 195
0.806
0.761
0.777
0.783
0.791
SJUD:A
251 263
0.808
0.762
0.789
0.802
0.797
6
292 313
0.709
0.683
0.633
0.629
0.742
1AWR:G
1 6
0.442
0.360
0.344
0.310
0.436
2DF6:C
1
0.515
0.364
0.404
0.320
0.481
6I42:B
1 13
0.506
0.430
0.456
0.418
0.491
7
1 8
0.522
0.403
0.359
0.368
0.504
Table A9. Quantitative evaluation of cosine similarity between the observed and calculated electron density maps (higher is better) on an expanded of proteins, including those with extended alternative conformations (altloc regions) and polypeptides (second half of table). Colored in green are the cases in which the ensemble produced by the density-gun guided AlphaFold3 better fits the observed electron density hetter than the unguided counterpart, and in blue the in which the generated ensemble also outperforms the structure deposited in the PDB.
20

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Inverse problems with apprriment-guides AlphaFold
NMR (PDB)
AlphaFold3
Guided AlphaFold3
PDB ID
Seq. Len
#NOEs
Viol. %
VioLÁ
Viol. %
Viol.Å
Viol. %
Viol.A
1FQX
91
1395
0.93%
0.0003
5.30%
0.093
4.44%
0.045
2JRM
60
993
10.27%
0.052
19.44%
0.184
16.62%
0.133
2JT1
71
1202
14.23%
0.057
19.97%
0.132
15.64 15.64%
0.072
JJVD
48
1029
9.91%
0.037
16.23%
0.103
12.63%
0.048
2KOM
104
1834
12.76%
0.059
19.79%
0.177
15.87%
0.112
2K3A
155
1872
18.59%
0.101
21.53%
0.162
16.08%
0.080
2K3D
87
291
33.68%
1.525
5.50%
0.013
4.47%
0.009
2K5D
110
1615
14.92%
0.067
20.43%
0.208
16.41%
0.121
2K75
106
1227
15.40%
0.068
14.75%
0.097
11.08%
0.049
2KD1
118
2142
13.50%
0.069
21.66%
0.181
18.07%
0.120
2KJR
95
1281
21.70%
0.101
26.54%
0.229
20.61%
0.114
2KKZ
134
264
34.09%
0.856
9.85%
0.057
9.85%
0.047
2 2KZV
92
808
15.47%
0.083
18.44%
0.166
12.75%
0.075
2L33
91
2310
15.19%
0.066
23.46%
0.200
20.30%
0.129
2L82
162
4582
16.54%
0.078
36.77%
0.719
25.56%
0.170
2L8V
143
1167
19.71%
0.090
23.74%
0.188
18.85%
0.092
2LF2
175
1825
18.25%
0.091
19.62%
0.159
15.89%
0.081
2LGH
144
1479
14.27%
0.053
18.80%
0.132
14.13%
0.045
2LK2
89
845
13.37%
0.063
14.56%
0.107
11.72%
0.044
2 LLL
87
1038
13.58%
0.061
14.16%
0.088
11.18%
0.038
2LTL
119
2391
10.96%
0.050
18.32%
0.154
12.80%
0.075
2LTM
107
2598
13.32%
0.059
21.29%
0.166
15.94%
0.088
2LX7
60
407
6.88%
0.029
7.86%
0.061
6.14%
0.022
2MA6
61
716
10.75%
0.057
15.64%
0.151
11.45%
0.057
2MK2
109
1268
10.02%
0.043
14.43%
0.132
9.07%
0.050
2MQL
105
48
22.92%
0.818
25.00%
0.788
22.92%
0.609
6F3K
353
109
43.12%
0.700
38.53%
0.624
13.76%
0.046
Table A10. Quantitative evaluation of restraint violation and backbone flexibility on an expanded set of proteins from NMRDb. Violation
percentages (Viol. %)) quantify fraction experimental NOE restraints that are not satisfied. while violation distances (Viol. Á) report by how far the modeled ensemble deviates from these restraints. Colored in blue are cases in which the ensemble produced by -gE-gedded AlphaFold3 better satisfies the distance restraints than the corresponding NMR structure deposited in PDB. Colored green are the cases where the ensemble abides by NOEs better than counterparts (excluding PDB).
21

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Inverse problems with experiment-guided AlphaFold
Algorithm 1 AlphaFold3 guidance
Input: {f*). (s,"""i), J. {s! J. (2 j. Noise Schedule [Bu Bi,..., 3r],
,r]. "oo = 0.8, Ymin 1.0, noise scale À 1.003,
step scale x = 1.5, experimental observation y, guidance scale 7, reference structure substructure conditioner flag b, substructure indices I, batch size n, number of atoms
Output: X X ~ 30- [N' N
Guided Ensemble N (O 13), E R Ruxmx3
for 3- E [ 3T| do
X + ContrRRaddomAuggnennttiiiii
70 if 3- Yimin else
3--1 (7+
1 8418y-1
E1 4 \/ 82 - 187 IN. [N'....N')
N ~ (N), E R2xmx3
X 2 X &
X denoised
Diffusion (2 J, (f*), (ss (ssppuss }. {s!munk
}. { (2tnin)
if b then
for i E I do X denaised
[ [r.....,
Repeated times
end for
end
+ (X, X.lenoisedd 1/8
log X, a)
UL
9 a.
Guidance Score is with respect to the ensemble
|
+ 9 1
Gradient Scaling
6, g n
dt 8
X + X X
dt 6)
end for return X:
Algorithm 2 Selecting samples using matching pursuit (Mallat & Zhang, 1993)
Input: samples in ensemble (X.,..,,"") experimental observation y; amino acid sequence a; likelihood function be maximized log p(y a)) maximum samples select Nymxi
0
Scurrent 0
while | < mmax do
L = {Ex- log p(y X XzJ(N), a) k E }
E: <- arg max, L S" + max; L
Maximize log p (Tu())). a)
< IU }
Add best sample
Seurrent then break
end if
Scurrent f- 5*
end while
return X' = {x* k E IZ
22

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Inverse problems with experiment-guided AlphaFold
A. Appendix and Supplemental Material
A.1. Guidance using AlphaFold3
In this section, we extend Algorithm 18 from the Supplemental Information of AlphaFold3 (Abramson et al., 2024) to incorporate additive guidance using experimental observations. The modified Algorithm is detailed in Algorithm 1.
Below, we describe the hyperparameters and log-likelihood formulations used for density-guidance and NOE-guidance.
For density-guidance, we used equation (I) as the primary log-likelihood function. However, due to the local nature of density-guidannee we apply the density loss function to atoms within a continuous residue region of the amino acid a: r ['mn, ). where 1 S Tmin < Timx I |a.. To ensure stability outside this region, we used the Substructure Conditioner loss equation (3) to anchor the remaining atoms. Hence, we used the following log-likelihood for -unndat--uidance..
logp(F. X. - F. Σ
Fx a)
| -ill
k=1iEA
Where A is the set of reference atom locations within the residue region r. The specific choice of the r depends on the protein and is detailed in the latter sections. We used À 0.1 scale the substructure conditioner. For guidance, we used
77 0.1 in equation (5).
Unlike guiidace which focuses local fitting, NOE-derived restraints are global in nature. Therefore, we do not apply the substructure conditioner when using equation (2) for NOE guidance. For guidance, we evaluated 0.3, 0.5 in equation (5), and selected the parameter based on the number of restrained obeyed.
To ensure numerical stability during guided diffusion, we apply gradient clipping to clip guidance score. This prevents instability due to large gradients, ensuring a smooth integration of experimental constraints into the diffusion process.
A.2. Noise model underlying log likelihood
This section discusses the underlying noise models L that lead to the log-likelihood functions Equations 1, 2, and
Density Loss: Laplace noise model is used, where the difference between F. and Fe is drawn from Laplace distribution centered zero with unit scaling. This given by,
= Fo - Fe ~ Laplace(O, 1)
(6)
This model, along with the Gaussian model, is used model electron density. However, a more realistic noise model involves complex physics, as noise is introduced at the level of Fourier intensities. We will address this in future works.
NOE Loss: Instead of typical noise model, piecewise function is used to model the underlying noise function.
0. L (x) diy)² (di)()) -.y)² df (x) > d.j
if d, I d: (X) $ dij if df (X) < dij
In Equation 7, if the pairwise distances (2) is between the bounds [d.j- d.y]. then we assume a uniform noise distribution (0 loss). Otherwise, we assume a Gaussian-like noise distribution (quadratic).
Substructure Loss: A Gaussian distribution with fixed isotropic covariance is used as the noise model.
C= N Nyy,, I)
(8)
Here, each atom x is drawn from multivariate Gaussian distribution centered at reference atom location y; with identity covariance matrix I (isotropic).
A.3. Baselines
We extend the evaluation presented in Tables A3 and A4 for both X-ray crystallography and NMR experimental observations by incorporating additional baselines, specifically AlphaFlow and ESMFlow (Jing et al.,, 2024). For the X-ray crystallography,
23

---

Inverse problems with experiment-guided AlphaFold
include comparison with Chroma ensembles generated using non-i.i.d. guidance (Maddipatla al., 2024). Table A8 presents direct comparison between non-i.i.d. guided X-ray erystallography (which consistently outperforms other baselines in Table A3) and the newly added baselines. In all cases, non-i.i.d. guided AlphaFold3 demonstrates superior performance compared to other methods.
Similarly, Table A4 has been extended to include AlphaFlow and ESMFlow in the NMR structure benchmark, as shown in Table A7. Our results indicate that the structural ensembles generated by NOE-guided AlphaFold3 adhere more closely to experimental constraints than those produced by any other baseline. ably, in half of the cases, these ensembles show better agreement with the constraints than the deposited NMR structures resolved using molecular dynamics. important to note that due the absence of explicit sequence conditioning in Chroma, the resulting ensembles diverged significantly from the true structures.
A.4. Runtime Analysis
In addition to predicting experimentally faithful protein ensembles, the proposed approach renders samples in computation- ally efficient manner. As shown in Table A5, an ensemble of 16 proteins with over 300 residues is sampled in approximately 7 minutes using density guidance, with minimal added latency compared to unguided AlphaFold3. Similarly, in Table A6, we generate ensembles of 32 conformations in approximately 14 minutes while modeling distance restraints as ensemble statistics. Our approach is significantly faster than restrained molecular dynamics methods like CYANA (Güntert, 2004), which is the current state-of-the-art technique for NMR structure determination.
A.5. AlphaFold3 model and hardware resources
Across all experiments in this paper, we used the open-sourced Protenix (Chen et al., 2025) model, a PyTorch-basee (Paszke et al., 2019) re-implementation of AlphaFold3. However, for the AlphaFold3 baseline comparisons, we report predictions generated using the official AlphaFold3 weights and source code (Abramson et al., 2024). All computations were performed on NVIDIA H100 and L40S GPUs.
A.6. Calculating Fe from a protein structure
In equation (1), we compute the L norm of the difference between F, and expected value of F. along the protein ensemble X. Here, Pa.: R IR is the calculated electron density determine using the 3D Cartesian coordinates of specifie protein structure X (x1, Xm). Formally, Fo can be computed using a sum over finite number of kernel density estimates,
(E)
m
4x
exp
bis + B R2
4x
+ ty - -E12)
(9)
bis B
where N₂ is the number of symmetry operations (Brock et al., 2016), m is the number of atoms in the asymmetric unit, R.g is the rotation matrix of the q-th symmetry operation, tą is the translation vector of the k-th symmetry operation, x; E IR3 is the location of the i-th atom, dij and bij are tabulated form factors defined for every heavy atom (Prince, 2004), B is the B-factor, and & E R2 is the point in Euclidean space where density is calculated. In standard crystallographic pipelines, the B-factor is used to model experimental electron density as a mixture of Gaussians. However, we consider the f-factor to be a bandwidth parameter in kernel density estimation (KDE) techniques (Terrell & Scott, 1992). Ideally, we would like to optimize the B-factor when guiding the diffusion process; however, due to the stochastic nature of the diffusion process, optimizing B B-factor proved to be quite unstable and often pushed the diffusion variable off of the diffusion manifold. Hence, used uniform B-factor that is inversely related to the size of the ensemble, B 4 for all the atoms in the ensemble. We consistently performed density guidance ensemble of size 16. For some bigger proteins, used batch size 12 to avoid out of memory exceptions.
A.7. Filtering& & relaxation after density guidance
Following the guided (non-i.i.d.) diffusion sampling procedure using density maps, we apply a two-stage filtering and refinement procedure to ensure the physical plausibility of the generated structures. To remove structurally invalid samples, we first identify and eliminate structures with broken covalent bonds and/or steric clashes.
To check for structures with broken bonds, we determine all bonded atom pairs within the protein structure using Gemmi (Wojdyr, 2022) and compute their Fuclidean distances. A structure is considered to have broken bonds if the distance
24

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Inverse problems with experiment-guided AlphaFold
between any bonded atoms exceeds Toood 2.1 A. In addition, check for structures with steric clashes, compute all pairwise interatomic distances and classify structure as exhibiting steric clashes if the distance between any atoms is less than Telash
1.1 Å.
After the initial filtering, we further refine the remaining samples by relaxing them with AMBER force field (Wang et al., 2004). This process resolves minor bond length deviations and improves structural consistency by ensuring that atomic interactions conform to physically realistic energy landscapes. For this, we use the publicly available ColabFold implementation (Mirdita et al., 2022).
A.8. Ensemble pruning with matching pursuit
The matching pursuit-based (Mallat & Zhang, 1993) ensemble selection procedure is detailed in Algorithm 2. Below, we describe additional optimizations and hyperparameters used for ensemble selection following density guidance.
Before selecting the ensemble, we optimize scalar B-factor B to maximize the log-likelihood in equation (1). This step adjusts the bandwidth of the ensemble's theoretical electron density F. to best fit the observed density F.. While optimizing B during the diffusion process can introduce numerical instabilities, we avoid the issue here because B is optimized after structure generation, filtering, and relaxation is complete. Hence, we do not encounter similar instabilities here. We optimized using Adam (Diederik, 2015) optimizer with a step size of 1.0 over 100 iterations. The optimized B-factor B" used uniformly across all atoms the remaining structures. Following this, we apply the matching pursuit algorithm to select the best-fit ensemble.
Across all experiments, we set the maximum ensemble size to Ammax 5. We heuristically found that the density is well explained by at most 5 samples and adding more samples to the ensemble would overfit the noise in the density map without yielding a considerable increase in cosine similarity. In Figure A6, we plot the normalized cosine similarity against the number of samples in an ensemble of size 16. The cosine similarity plateaus at 5 samples after which we either get a deteriorated fit to the density, or overfit to noise- - both undesirable.
For NMR guided ensembles, we not employ ensemble selection and our results are evaluated on the full ensemble.
A.9. Electron density pre-processing
Since the electron density maps available in the PDB are mean-centered and lack an absolute scale, we converted them to physical units of density [e 1 using the method described (Lang et al., 20.4). A comprehensive list of proteins used in our experiments and their corresponding residue regions is provided in Tables A1-A3. While most density maps in our dataset are of high resolution, our method performs exceptionally well density maps with relatively lower resolution like PDB entries 40LE, 6JF2, and 6QQF.
A.10. Ensemble bi-modal distribution evaluation
In Section 5, we demonstrated that density-guiide diffusion effectively captures structural heterogeneity present in the protein crystal. Specifically, we evaluated this method on altloc regions of proteins listed in Table A3 and observed that our non-i.i.d. density guidance framework consistently outperforms i.i.d. density guidance across most proteins in the dataset. Also, both methods consistently outperform AlphaFold3.
In this section, we describe a quantitative evaluation demonstrating that non-i.idd density guidance better captures the bimodality inherent to altloc regions compared other methods. To this end, we use samples from three sets of experiments, density-guided (i.i.d.), density-guided (non-i.i.d.), and AlphaFold3 (unguided). In each case, we filter, relax (Appendix A.7), and refine the generated ensemble using matching pursuit-based selection (Appendix A.10). To quantify bimodality, we compute the normalized distance of each sample in the ensemble X' relative to the known altloc configurations (A or B). Consider an individual sample X E m from the ensemble X' and the reference altloc A and B structures X,, X, E m respectively. Following Rosenberg et al. (2024b), we define the signed normalized distance between X
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Inverse problems with experiment-guided AlphaFold
and the reference conformations as
dg = X Xall2
dg = |X- Xx
db da
dz dz ] sign(d, db).
Normalized Distance
min
(10)
This metric provides an interpretable method for quantifying ensemble distributions. Specifically, samples closer to reference conformation A have the normalized distance approaching -1, while those closer to reference conformation B have the normalized distance approaching +1. Consequently, negative values indicate proximity to mode A, while positive values indicate proximity to mode B. The resulting normalized distance distributions are visualized in Figure A2, A3, and A4. We observe that non-i.i.d. guided sampling with both Chroma and AlphaFold3 achieves bimodal and multimodal distributions behavior with proximity to both reference conformations (positive and negative modes in the plot). Notably, our electron consity-guidee approach with AlphaFold3 achieves bimodality more consistently than Chroma- guidance. In contrast, ii.d. guided sampling with AlphaFold3 achieves a moderate degree of bimodality but is less effective than non-ii.d. sampling in terms of accurately capturing the full extent of conformational heterogeneity present in the electron density. Furthermore, unguided methods- - including AlphaFold3, AlphaFlow, ESMFlow, and Chroma frequently fail to recover the correct bimodal behavior.
This demonstrates that non-i.i.d. guidance significantly outperforms i.i.d. guidance in modeling bimodal distributions in electron density maps. In addition, both guided approaches perform significantly better than unguided models.
A.11. Diffusion guidance landscape
In this section, analyze the loss curves during the density-guided diffusion process, as depicted in Figure A5, over batch of 16 ensembles of 2ESK structures (Residue Range: 113-119) At each diffusion timestep, we log the cosine similarity between F, and F. to quantify the alignment between the observed and predicted electron density maps.
We also log the number of bond length violations across all structures in the ensemble, computed using the same methodology described in Appendix A.7. From Figure A5, we observe that without density guidance, the cosine similarity between Fo and F. remains significantly lower than in the guided case. This demonstrates that incorporating density guidance during diffusion leads to significantly improved density alignment. Furthermore, we identify a critical phase of the diffusion process, between iterations 130 and 175, where the cosine similarity increases the most. This period coincides with a shapr reduction in bond length violations. This suggests that during these iterations, the diffusion model corrects high-frequency structural features. During iterations 100 to 135, we observe significant fluctuations in the confidence bands of the cosine similarity plots. This suggests high variability in bond length violations across different structures in the ensemble, likely indicating that the diffusion model is correcting lowffrequency structural features during this phase.
Lastly, after relaxation and filtering, the number of bond length violations reduces to zero, confirming that relaxation effectively resolves structural inconsistencies. However, we note a slight dip in cosine similarity post-relaxation. This could be attributed to overfitting of the guided diffusion process to the electron density maps, leading to subtle structural violations that are later corrected during relaxation.
A.12. Estimating NOE restraints from ensemble
Measuring distance restraints on an ensemble. Given an ensemble X' and NOE restraints D [dd.. d:y) (i,5) P
of pairs of lower and upper bounds, the NOE restraints are employed on the ensemble average, d. (x) = Ek= di (
of the distances dey(X (X*) ||* x/|
between pairs of atoms i,j in individual structures X written in equation (2).
Heavy atom approximation. AlphaFold3 models only heavy atoms (i.e., it does not include hydrogen atoms), whereas all NOE restraints based on internuclear interactions- are defined between hydrogen atoms. To address this discrepancy during guidance, each NOE restraint in experimental data D, initially specified between hydrogen atoms, is approximated by applying an equivalent distance constraint to their covalently bonded heavy atoms (c.g.. N for NH or C for CH groups). Given fixed bond lengths (N-H: 1.0Å, C-H: I,1 the maximum error introduced by this substitution is 4.4A = 2x (111A) During evaluation, hydrogen atoms are placed into the model at the relaxation stage (Section 4.2), enabling correct evaluation of NOE restraints against explicit hydrogen positions without employing the heavy atom approximation.
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Inverse problems with experiment-guided AlphaFold
Distances vs. peak intensities NOE measurements in NMR arises from dipolar interactions between nuclei (typically protons) within 6Å. The intensity of NOE cross-peaks in NOESY spectrum is inversely proportional to the sixth power of the interatomic distance (I x 1/r"). Post peak assignment, the cross-peak intensities are converted into distance restraints via r Tref 11 where Tier and Let are the distance and peak intensity of a reference nuclei pair. To ensure physical accuracy, the NOE-implied distance between two atoms in an ensemble should be computed by first averaging the calibrated peak intensities, and then converting the resulting mean intensity back into distance. We will adopt this more rigorous approach in our follow-up work.
A.13. NMR data collection and pre-processing
NOE-based distance restraints were extracted from NMR STAR files corresponding to the PDB entries. The pynmrstar library was employed parse these restraints, with explicit selection of NOE-derived constraints (cat- egorized under _Genddist_const drogen bonds, dihedral angles, or RDC-derived constraints) were excluded to focus specifically distance geome- try derived from NOEs. Ambiguous NOE assignments involving multiple proton pairs were retained without filter- ing to reflect inherent NMR uncertainty and are considered for evaluation. Distance lower and upper bounds were directly obtained from the STAR file fields _Genndiinnn sst.constrrantte. its..sst.............ttt...tttst Distiisssss www.annnewpprrrbboddeeee
All non-NOE structural restraints (e.g.. hy-
wowndrrboundvee and respectively. Missing lower bounds were explicitly set to Å enforce a physically meaningful minimum distance. Parsed bounds were retained Angström units as provided, with no additional thresholding or normalization applied to preserve the experimental restraint set. These bounds were used directly inputs to the likelihood term (Section 3.2), which models NOE-derived distance uncertainties through truncated super- Gaass-na potential acting between the parsed lower and upper bounds.
A.14. Computational estimation of N-H bond order
The N-H bond order parameter 2 captures the backbone flexibility. Physically, S measures the long-tim hong-time limit of the autocorreation function of the N-H bond vector. Given structural ensemble X, the N-H bond order can be computationally estimated as follows (Palmer, 2004):
1. Align the ensemble to a reference structure X to account for rotational and translational symmetries.
2. For every structure X E X' in the ensemble, compute the normalized N-H bond-vectors {d } at all residues i.
3. Given the normalized N-H bond-vector, the calculated bond-order S is given by
1
$ n(n
P2(d: di),
kelt
where | the ensemble size, and P2(x) (3x² - is the Legendre polynomial of order 2.
A.15. Evaluation metrics used in NMR structure determination experiments
In what follows, we describe how we compute quantitative metrics reported in Table A4 and Figure A7. Given structural ensemble X' and restraint list D = [dd. d.y) (i.j) E P), we measure the quality of X in terms of its adherence to D and as a secondary measure, we evaluate the recovered conformational fiexibility with respect to the NMR ensemble X' NMR- To evaluate restraint violation across the ensemble, we first compute the ensemble distance matrix D(X), whose entries are given by (x) = +. Eh= (
Restraint violation percentage (Viol. %) For each experimental restraint in D, a violation occurs if the ensemble averaged distance dıy (X') lies outside the interval (d. d.y]. Viol. % is the percentage of restraints in / that are violated.
Restraint violation distance (Viol. Å). For each restraint, the violation magnitude is the absolute deviation of from the nearest bound (lower or upper) if outside the interval; otherwise, it is zero. Viol. Á is the average of these deviations across all restraints in D.
27

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Inverse problems with experiment-guided AlphaFold
Handling ambiguous restraint groups. In NMR experiments, distance restraints are often organized into restraii groups 9 _ D with intra-group OR conditions to account for assignment ambiguity. For violation metrics, compute the effective violation of group G as:
Viol(g) = min (Viol(g) 9 E g)
where Viol(g) is the violation of constituent restraint g (measured via Viol. or Viol. A). This ensures group is considered satisfied if at least one constituent restraint complies with the experimental bounds.
RMSE with respect to X' NMR- Given X' this metric measures the correlation in conformational flexibility with respect to NMR. We first align X' and NMR to a reference structure X to account for rotational and translational symmetries. Then, for each residue location in the ensemble X, the root mean square fluctuation (RMSF) computed as,
n
RMSF
x
X ||?
where X; =
n
and x represent the coordinates of the CA atom in residue i in sample &t of the ensemble. The p-RMSF is the Pearson correlation coefficient between the RMSF profiles of X' and NMR. This effectively measures their similarity in residue-wise conformational flexibility, Higher p-RMSF values indicate better agrecment flexibility patterns.
28


Reviews:
[{"review_id": "o3EHsDg7F9", "reviewer": "Reviewer_v7F4", "summary": "This paper examines the inverse problem of resolving protein structure from experimental data and capturing the heterogeneity arising from the dynamic nature of proteins as an ensemble. To do so, it guides the diffusion module of AF3 with experimental data to satisfy NOE constraints and substructure likelihoods.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We hope our answers explain the purpose and novelty of the proposed framework and invite the reviewer to ask further questions.\n\n**Clarification 1: The problem is not trivial**\n\nRaw experimental observations do not contain the atomic model. **In crystallography**, the raw experimental data are intensities of the X-ray diffraction patterns collected at different orientations. Phases, required to reconstruct the electron density, cannot be directly measured and are estimated to generate a rough electron density map that is iteratively refined (partially manually) by improving the fit of the model to the density map that is improved by better phase estimates from the improved model.\n\nTraditionally, the structure is represented as a product of marginal distributions, i.e., Gaussian distributions centered at atom locations with B-factors representing the uncertainty within the atom position. This representation is limited since: 1. ~15% of protein structures exhibit multiple backbone conformers (modes) reinforcing the fact the electron density is an ensemble measurement, 2. It loses information crucial for mechanistic structural biological studies.\n\nWe claim to be probably the first approach to sample from the joint distribution of the entire set of atoms. This is possible because of AF3 sequence-conditioned diffusion model. We believe that ensembles of realizations sampled from the joint distribution are necessary to study complicated effects like allostery. We are currently working on refitting the structures on the PDB with ensembles instead of B-factors.\n\nIn **biomolecular NMR**, through-space dipolar couplings are measured and assigned to imply pairwise distance restraints within the structure/ensemble. Recovering the structure/ensemble from distance restraints requires restrained molecular dynamics simulations, taking hours to days to produce one sample. Typically hundreds of proteins are simulated, and the ~20 lowest energy ones are selected. Not only is this approach time-consuming, it also fails to treat the NOE restraint as an ensemble measurement [Fowler NJ et al., 2022 Structure]. Attempts at ensemble MD [Lindorff-Larsen et al., 2005,  Lange et al., 2008], are extremely computationally expensive and have been tested on only a handful of proteins. Our approach accelerates this workflow by at least two orders of magnitude. We are currently running NMR-based structure determination for the entire PDB. We believe downstream analyses of the refitted NMR structures could potentially lead to scientific discoveries.\n\n**Major concern 2: Baselines.**\n\nOur method is not an “improved version of AlphaFold” and solves a different problem. AF3 is an inductive model capable of predicting the structure from the sequence in the absence of any experimental input. We use AF3 as a prior in solving the inverse problem of recovering the structure given the experimental input (transductive method). AF3 performs well on many protein regions and our benchmarks were chosen as “difficult” and interesting cases where AF3 fails consistently. We show that guiding with experimental data produces structures consistent with the experiment and often fits the electron density better than the PDB structures.\n\nWe compare the PDB structure as the state of the art solution and unconditional AF3, to quantify the information gained due to the conditioning by the experimental input. “Simple” cases in which AF3 already performs well, emphasize that the experimental guidance does not deteriorate performance (Tables in the Appendix). We extended our experiments to include additional baselines and proteins. The updated quantitative results are available at the following link - X-ray: https://postimg.cc/XZt5h686, NMR: https://postimg.cc/Wtws9D06.\n\n*No real “groundtruth” exists*. The observables contain only indirect information about the underlying atomic structural ensemble and cannot be compared to directly. The electron density does not contain atom labels, and in NMR only distance constraints are observed directly. Hence there is no real “ground truth”. However, we still quantify and report quality of fit criteria as customary in the field.\n\n**Minor comments:**\n> CryoDRGN\n\nCryoDRGN solves inverse problems in cryoEM, outputting electrostatic potential maps, not atomic models and is not relevant to the proposed methods.\n> NOE\n\nSec. 3.2 explains how NOE is manifested in the form of interatomic distance constraints. We will add a brief explanation about the physics of the underlying phenomenon.\n> “inverse problems” feels too broad \n\nWe can refine it to “Experiment-guided AlphaFold for characterizing protein structure ensembles” or similar.\n>Need to flip to appendix for results\n\nWith more results than can possibly fit into the space limitations, we had to relegate some part to the Appendix. For the text we favored visuals demonstrating conceptually the different test cases. It is not ideal, and we will attempt a better distribution of the results.", "claims_and_evidence": "I think the key claim here is that we can get a better ensemble of structures out of AF3 that obeys structural heterogeneity if we already have our hands on some x-ray crystollography or NMR data, or known atomic substructures. On the case study proteins, shown in the Appendix, the experimental data guided results have better cosine similarity. The authors also introduce an algorithm to select samples based on the matching pursuit algorithm.", "methods_and_evaluation_criteria": "**Method**: The method \"hacks\" the diffusion module in AF3, which outputs an ensemble of structures. The method \"refines\" this output based on given experimental data. The likelihood of the experimental observation is calculated given each individual ensemble member (for substructures) and for the ensemble average (for electron density and interatomic distances). At the end, AMBER relaxation is used, similar to AF2. Most of the results are in the Appendix. Quantitative metrics include cosine similarity. Results are calculated on selected case study proteins.\n\n**Evaluation**: The evaluation criteria is frankly pretty unclear to me -- as I'll describe throughout this review, I'm not sure why the baseline of AF3 is chosen since it accepts such different inputs (ie. it seems trivial to compare a version of the structure prediction model _with_ the structural experiment data as input and a version _without_ it).", "theoretical_claims": "n/a", "experimental_designs_or_analyses": "Three data terms are considered:\n* crystallographic electron density maps\n* nuclear Overhauser effect (NOE) restraints\n* sub-structure conditioning using known atom locations\n\n**Question**: what is used as ground truth here? Do you still solve the inverse problem in some other way in order to assess the solution from the AF3-guided method?\n\nFrom the way I understand it right now, the experiment is almost saying \"if you have the result, and guide the prediction with the result, you'll preform better\", which feels trivial -- I'll be coming back to this point throughout the review, but I think it's hard for me to appreciate the rest of this paper without clearing up this confusion.", "relation_to_broader_scientific_literature": "not qualified to comment, but my (possibly uninformed) hunch is that actual methods for solving structure from NMR/etc. is more useful, like CryoDRGN etc.?", "essential_references_not_discussed": "not qualified to comment, but baselines are generally too vague / doesn't seem to match the problem statement in Section 3", "other_strengths_and_weaknesses": "Strengths:\n* As I'll mention throughout this review, I don't fundamentally understand why we need to predict the structure if we already have structure experimental results, though I think this could just be due to me not being familiar enough with this subject area. I think guiding AF3 is a pretty cool engineering feat, but I don't understand this problem enough to comment on how this is better than previous methods.\n\nWeaknesses:\n* As I understand it right now, this problem is sort of a trivial one from the ML perspective (i.e. if you guide the prediction with the result, then of course it should do better than not having the result, right?) -- I'm hoping maybe this is only because I'm missing some key piece of understanding, so I'm happy to have a conversation about this during the discussion period.\n* Related to my overall confusion about the problem setting: the presentation of the paper is not very well-suited for a ML conference. For example, there aren't any baselines except what is already deposited in the PDB, and it's pretty hard to find what's used for ground truth for the evaluations in Section 6", "other_comments_or_suggestions": "Nitpicks:\n* using the world \"inverse problems\" in the title feels too broad; can we use something more specific here?\n* There isn't really a \"results\" section in the main text. We have a description of the experiment, and the results described in words all in the same paragraph, and need to flip to the appendix to find the results. From a subjective presentation perspective, I'd prefer to see a smaller / less developed Figure 2 and see more of the graphs in the original text.\n* Related works are sprinkled throughout the paper; this makes it hard for the ML audience to directly understand what the comparable problems are.\n* The article makes frequent allusions to NOE in the introduction without explaining what it is except the abbrv. -- this is a simple fix that can really make this work more accessible to more parts of the ML audience.\n\nOverall: this seems like a really interesting paper! With the current presentation, though, it seems more useful at a journal where it can reach the right domain experts", "questions_for_authors": "I've sprinkled forms of these questions through out the review, but for completeness:\n1. Why we can't just get the structure from the experimental NMR /x-ray crystallography data using the existing ways of solving this inverse problem, rather than go through AF3-guidance?\n2. Why were these baselines (AF3, existing PDB deposits) chosen? These seem to be solving a very different problem setting, with very different input information available.\n3. What is the ground truth that's used for the results in Section 6?\n4. Am I correct in understanding that the problem setting we're looking at is that of resolving the original heterogeneous protein structure from the experimental NMR/x-ray crystallography data? If this is the case, it feels like that we should be comparing to CryoDRGN etc.\n\nAgain -- I don't feel fully confident in my understanding of this area, so I'm happy to have a discussion about this with the authors.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3yXaNKQLQP", "reviewer": "Reviewer_GYEr", "summary": "This paper proposes to use AlphaFold3 as a structure prior for protein crystal structure determination from cryo-EM or NMR experiments. Specifically, it uses AlphaFold3 to predict the initial structure and then use the gradient from the electron densities to guide the diffusion module of AlphaFold3 to refine its structure to agree with experimental observation. The method is evaluated on crystallographic electron density data in PDB and shows good performance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their review.\n\n**More baselines**\n\nFollowing the reviewer’s suggestion, we extended the evaluation for both the X-ray crystallography and the NMR experimental observations. \nWe include the following additional baselines:\n- AlphaFlow (Jing et al., 2024)\n- ESMFlow (Jing et al., 2024)\n\nAnd specifically for X-ray crystallography altlocs benchmark (Table A3) we also compared against: \n- Chroma (Ingraham et al., 2022)\n- Chroma-guided (Maddipatla et al., 2024)\n\nThe updated tables for X-ray have been included in this [link](https://postimg.cc/6yhdQS9C).\n\n**X-ray Crystallography.**  We particularly focused on Table A3 as it captures structural heterogeneity (altlocs). We note that, in all cases, guided AlphaFold3 (AF3) outperforms all other baselines, and in some cases, we are able to fit structures to density better than the structures deposited to the PDB.\n\nWe further extended the violin plot in Figure A2 to include the aforementioned baselines ([link1](https://postimg.cc/F794Vc04), [link2](https://postimg.cc/dh5rV8jt), [link3](https://postimg.cc/VS5MrdK9)). We notice that our method captures bimodality better than all other methods (including recent experiment-guided approaches from Maddipatla et al. 2024). We hope that these baselines help justify our claims even further.\n\n*Regarding Terwillinger et al. 2023 [1]*: We attempted to include the comparison to [1] as the reviewer suggested, however, we did not find a publicly available codebase for [1]. We hope that the reviewer can understand that it is difficult to replicate a comprehensive study in this timeframe. Additionally, based on our understanding of the paper, the goal of this paper was not to recover structural heterogeneity given the density.\n\n**NMR.** In a similar vein, we added comparison to AlphaFlow and ESMFlow (Jing et al. 2024) for the NMR structure determination benchmark (Table A4).  We attempted running Chroma (both unconditional and NOE-conditioned), however, we observed that due to lack of explicit sequence conditioning in Chroma, the produced ensembles completely deviated from the true structures. Our results suggest that the structural ensembles produced by NOE-guided AlphaFold adhere to the constraints better than all other baselines, and in half of the cases, they produce a better agreement to the constraints compared to the deposited NMR structures resolved with MD, while taking a tiny fraction of the runtime ([link](https://postimg.cc/FYQKksH6)).\n \n**Note Regarding “Relation to Broader Scientific Literature:”**\n\nWhile we are the first to use classifier guidance to refine AF3 predictions, we would like to emphasize that we are proposing a new approach to perform classifier guidance on diffusion models. Specifically, regular classifier guided diffusion is i.i.d. in that it encourages each sample in the batch to independently reduce a loss function. However, we are proposing a non-i.i.id. classifier guided diffusion that encourages the entire batch to reduce a loss function.\n\n**Not citing Terwillinger et al. 2023 [1]**\n\nWe apologize to the reviewer for oversight. We will add a comprehensive discussion to [1], along with references suggested by Reviewer 4.\n\nWe hope this clears all concerns.", "claims_and_evidence": "The claims made in this paper looks convincing.", "methods_and_evaluation_criteria": "The evaluation criteria make sense, but the experiment section would benefit from more baseline.", "theoretical_claims": "There is no theoretical claims in this paper", "experimental_designs_or_analyses": "The experimental design would benefit from more baselines, such as [1].\n\n[1] Accelerating crystal structure determination with iterative AlphaFold prediction, Acta Crystallogr D Struct Biol, 2023", "relation_to_broader_scientific_literature": "This paper is the first method that uses classifier guidance to refine AlphaFold3 predicted structure. This contribution is original.", "essential_references_not_discussed": "[1] This paper should cite Accelerating crystal structure determination with iterative AlphaFold prediction, Structure Biology 2023", "other_strengths_and_weaknesses": "The proposed method is the first method that uses classifier guidance to refine AlphaFold3 predicted structure. This contribution is important.", "other_comments_or_suggestions": "Suggestion: the author should move some important figures to the main paper rather than in the supplementary material.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eJrt9YxOJ5", "reviewer": "Reviewer_kpSM", "summary": "This paper introduces Experiment-Guided AlphaFold3, a framework that integrates experimental data with deep learning priors to generate structural ensembles aligned with experimental observables. Standard protein structure predictors like AlphaFold3 produce single static structures, failing to capture conformational heterogeneity. The proposed method adapts AlphaFold3’s diffusion-based sampling to incorporate experimental constraints, refines structures via force-field relaxation, and selects ensembles maximizing agreement with experimental data. Density-Guided AlphaFold3 improves crystallographic modeling by generating structures more faithful to electron density maps. NOE-Guided AlphaFold3 refines ensembles to satisfy NMR-derived distance restraints, better capturing protein dynamics. The approach significantly reduces computational time compared to traditional crystallographic and NMR workflows. Results show enhanced structural accuracy over standard AlphaFold3 and, in some cases, even PDB-deposited structures. This work advances protein structure modeling by bridging the gap between deep learning predictions and experimental measurements.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments.\n> tested on a few case studies (e.g., ubiquitin, selected crystal structures)\n\nThis paper is as a proof of concept that AlphaFold3 (AF3) can be guided by experimental observations to generate heterogeneous ensembles. \n\nWe note that there was extensive quantitative evaluation presented in the Appendix (crystallography: Tab. A1-3 – 44 structures; NMR: Tab. A4 – 13 structures in the manuscript). Our case studies were curated in order to showcase the method in biologically and methodically interesting settings. For e.g., Ubiquitin is the benchmark protein for NMR structure determination, and was the sole subject of study of highly influential works in Nature (Lindorff-Larsen et al., 2005) and Science (Lange et al., 2008). It is essential for any NMR structure determination work to benchmark their performance on Ubiquitin. \n\nIn crystallography, our benchmarks comprise several “difficult” and interesting cases that we curated, which AF3 consistently fails on (Tabs. A1 – 3). As control, we cover “simple” cases, where AF3 performs well, to show our approach does not deteriorate AF3’s performance (Tab. A2).\n\nThat said, considering the reviewer’s request, we curated an additional set of results for both crystallography (+15 proteins) and NMR (+27 proteins). The updated quantitative results are available at these links: https://postimg.cc/XZt5h686, https://postimg.cc/Wtws9D06.\n\nWe added new baselines to existing tables (Tab. A3 and A4): https://postimg.cc/6yhdQS9C, https://postimg.cc/FYQKksH6\n\nWe are currently extending this study to refit the entire PDB.\n> Unclear how well the method generalizes across diverse protein families\n\n- *Flexible proteins:* AF3 structures are generally rigid and do not accurately capture the flexibility of proteins. Protein flexibility is captured by NMR in two ways: (i) solution-state NMR, the proteins are generally more flexible in solution compared to crystals, (ii) NMR order parameters, e.g. N-H S², measure the backbone flexibility of a protein because they capture the time-expectation of the N-H bond vector. In Fig. A3, we compare the backbone flexibility of the recovered ensemble against the experimentally measured order parameters. We also demonstrate that we successfully capture this flexibility, but unguided AF3 does not.\n\n- *Disordered proteins:* Because AF3 was trained on crystal structures, it fails to predict intrinsically disordered proteins. While very interesting, this is beyond the scope of this work and deserves a dedicated study.\n\n- *Multidomain assemblies:*  One of the reasons we chose AF3 as the generative model is its ability to predict the structure of protein-ligand and multi-protein complexes. We currently have promising preliminary results of extending the proposed methods to such cases, however, we believe that this deserves a dedicated study. \n\n> primary audience seems to be structural biologists than ML community\n\nWe understand the reviewer’s concern. We agree that our approach is of immediate use to structural biologists, and our writing style might be unorthodox to the ICML audience. Yet, it was the ML community that pioneered the development of protein structure generative models, such as AlphaFlow (ICML 2024). Our perspective is that one of the primary uses of a protein structure generative model is its usage as a structural prior while solving inverse problems. To our knowledge, ours is the first work that employed these structural priors to solve real-world inverse problems. This is inherently a computational problem (not a biological one) addressed with computational tools. While we believe that the tools developed in our work directly impact structural biology workflows, they also introduce a new benchmark for validating future developments in modeling protein structure priors on different downstream tasks. Furthermore, our work might inspire different ways of solving these important real-world inverse problems.\n> computational efficiency, runtime comparisons\n\nFor crystallography (Table: https://postimg.cc/qgJK5wcR), our approach samples a batch of 16 proteins in ~7 minutes for 300+ residue systems, adding minimal latency vs. unguided AF3. For NMR (Table: https://postimg.cc/jndm3jrZ), we generate ensembles of 20 conformations in ~10 minutes—significantly faster than restrained MD methods like [CYANA](https://link.springer.com/article/10.1007/s10858-015-9924-9), while modeling distance restraints as true ensemble statistics.\n>Broader benchmarking and more rigorous validation would strengthen the conclusions.\n\nTo validate NMR, we evaluate backbone flexibility using N-H S² parameters (Fig. A2). For crystallography, we quantify ensemble heterogeneity via bimodality scores (subsection A3/Fig. A2), outperforming baselines in comparative benchmarks. (https://postimg.cc/F794Vc04, https://postimg.cc/dh5rV8jt, https://postimg.cc/VS5MrdK9). We note that we _don't_ guide our ensembles on either of the validation metrics.", "claims_and_evidence": "The paper primarily validates its approach through specific case studies, such as ubiquitin for NMR and selected crystal structures for X-ray modeling. However, it remains unclear how well the method generalizes across diverse protein families, particularly for flexible or disordered proteins and multi-domain assemblies. Without broader benchmarking, the extent to which Experiment-Guided AlphaFold3 captures conformational heterogeneity across structurally varied proteins is uncertain. Additional validation on a wider range of experimental datasets would strengthen the claim of broad applicability.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are reasonable and relevant, but they currently rely on a limited set of case studies. A more diverse benchmarking strategy, additional validation metrics, and explicit runtime comparisons would improve the strength of the evaluation.", "theoretical_claims": "There are no theoretical claims in this paper.", "experimental_designs_or_analyses": "The experimental design is reasonable but has limitations in scope and validation. The method is tested on a few case studies (e.g., ubiquitin, selected crystal structures), raising concerns about generalizability to diverse protein families. The claim of computational efficiency is not backed by explicit runtime comparisons. Additionally, statistical analyses (e.g., RMSD distributions, significance testing) are missing. Broader benchmarking and more rigorous validation would strengthen the conclusions.", "relation_to_broader_scientific_literature": "This paper builds on recent advances in deep learning-based protein structure prediction, particularly AlphaFold3 (Abramson et al., 2024), and addresses its limitation in capturing conformational heterogeneity. While previous methods like Rosetta (Baek et al., 2021) and MD-based NMR refinement (Lindorff-Larsen et al., 2005) have incorporated experimental data, they are computationally expensive. The work also relates to AlphaFlow (Jing et al., 2024) and ensemble modeling approaches in X-ray crystallography (Furnham et al., 2006; van den Bedem & Fraser, 2015) and NMR. However, it uniquely leverages AlphaFold3 as a structure prior, integrating experimental constraints to generate physically realistic and experimentally consistent structural ensembles. By bridging deep learning-based structure prediction with experimental refinement, this work contributes to both computational and experimental structural biology.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "A key concern is that the paper's primary audience seems to be structural biologists rather than the machine learning (ML) community. The writing style and structure are uncommon for an ML paper, with extensive discussions on case studies and biological validation rather than a focus on methodological advancements or generalizable ML insights. While this depth may make it a high-quality protein design or structural biology application paper, it is unclear whether it aligns with ICML’s core focus on ML innovation. The paper would be more suitable for ICML if it placed greater emphasis on the ML contributions, generalizable techniques, and broader computational impact rather than domain-specific experimental results.", "other_comments_or_suggestions": "I initially feel inappropriate to accept this paper due to the weak experiments and the uncommon writing style. I am not a biologist, so I will consider raise my initial score by carefully reading other reviewers' opinions and seeing my major concerns are well addressed.\n\n## Update after rebuttal\nI appreciate the authors' significant efforts in conducting additional experiments, which have strengthened the paper. As a result, I have decided to raise my score. However, the unconventional writing style—unusual for a machine learning conference—still prevents me from assigning a higher score, despite finding the paper more interesting with the new results.", "questions_for_authors": "No more questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lJirbfUrpf", "reviewer": "Reviewer_LiAh", "summary": "In this paper, the author introduce a method to guide diffusion-based structure prediction models (e.g., AlphaFold 3) with experimental data to sample conformational ensembles.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their review.\n\n**Underlying noise model.**\n\n*Eq. 1.* We assume a Laplace noise model. Here, the difference between $F_{o}$ and $F_{c}$ is drawn from a Laplace distribution centered at zero with unit scaling. This model, along with the Gaussian model, is used in electron density modelling. However, a more realistic noise model involves complex physics, as noise is introduced at the level of Fourier intensities. We will address this in future works.\n\n*Eq. 2*: Instead of a typical noise model, we assume a piecewise noise model. If the distance is between the lower and upper bounds, then we assume a uniform noise distribution (0 loss). Otherwise, we assume a Gaussian-like noise distribution (quadratic loss). This model is common in NMR structure modeling (Lindorff-Larsen et al., 2005). \n\n*Eq. 3*: We assume a Gaussian distribution with a fixed isotropic covariance as the noise model. Each $\\mathbf{x}\\_{i}^{k}$ is assumed to be drawn from $\\mathcal{N} (\\mathbf{y}_{i}, \\mathbf{I})$\n\n**Substructure Conditioner**\n\nDue to space constraints, we could not include the substructure conditioner in the main text. We will move it there in the future. Re the other questions:\n- $y$ is from AF3 predictions.\n- To select the set of anchored atoms, we examine AlphaFold3 (AF3) predicted structures alongside density maps to identify regions that are either not faithful to the map or exhibit structural heterogeneity in the map. We then extract a 3D slice of the map around the region, including all the relevant atoms and applying a padding of 5Å along each axis.\n- No, the method does not yield significantly worse results when long regions are not anchored. For instance, in Figure 2A and 3 (of manuscript), a 10-residue region is not anchored by the conditioner. Yet, we recover conformers that fit the density map well and the cosine similarity is comparable to PDB structures. We conducted additional experiments on 15 structures with conformational heterogeneity ([link](https://postimg.cc/XZt5h686)). For some, we optimized regions spanning up to 22 residues (5v2m and 6e2s), and for all cases, the cosine similarity is on par with the PDB (or better). We also included additional baselines like AlphaFlow and ESMFlow (Jing et al., 2024). However, it must be noted that the residue range length does increase the runtime for fitting an ensemble to our experimental observation ([link](https://postimg.cc/qgJK5wcR)).\n\n**Maximum Samples in Ensemble**\n- No, this method is effective even when the sample has a single mode too. For instance, all structures in Tabs. A1 & A2 (in manuscript) have a single mode. Here, the ensemble (selected using algorithm 2) consists of similar looking structures without the separation we typically observe in structures with multiple conformations. Additionally, the ensemble in Figs. 2 & A1 (single mode structures) were selected using Algorithm 2 (manuscript).\n- We heuristically found that the density is well explained by at most 5 samples and adding more samples to the ensemble would overfit the noise in the density map without yielding a considerable increase in cosine similarity. In the attached Figure ([link](https://postimg.cc/BP2nP0WB)), we plot the normalized cosine similarity against the number of samples in the ensemble of size 15. We see the cosine similarity stabilizes at 5 samples. Post that, we either get a deteriorated fit to the density, or overfit to noise – both undesirable.\n\n**Missing literature review**\n- Fadini et al.: This work was released after the ICML deadline. This work fits a single protein conformer to the static electron density map by optimizing AlphaFold2’s MSA contact maps. However, they are unable to account for conformational ensembles (15% of proteins exhibit non-trivial conformational heterogeneity in the same protein). Proteins are better described as ensembles rather than single conformers. \n- Liu et al.: This study samples structural ensembles using the str2str protein diffusion model, with guidance from cryo-EM experimental observation. While related, their focus is on a different modality than ours and does not address crystallographic density fitting or NOE observations.\n- Maddipatla et al.: This method fits ensembles to crystallographic density maps using Chroma and captures multiple conformers to some extent. We observed that it fails to capture conformational heterogeneity when optimizing over a long residue range due to Chroma’s hierarchical formulation. Moreover, because the sequence conditioning in Chroma is only “promoted” and not imposed, we found it impossible to impose “global” structural constraints as those in the case of NMR.\n- Levy et al.: They also use Chroma as their protein diffusion model and inherit similar problems with packing sidechains. Additionally, they do not use experimental electron density maps or distance restraints, but instead rely on synthetic data.\n\n\nWe will update the manuscript to include all aforementioned points.", "claims_and_evidence": "The claims are well supported by the results shown in the paper.", "methods_and_evaluation_criteria": "The baseline (AlphaFold 3) and the test sets seem relevant to evaluate the method.", "theoretical_claims": "The paper does not introduce new theoretical claims.\n\nIn equations (1), (2) and (3), the authors should specify the underlying noise model, leading to the given log-likelihood functions.", "experimental_designs_or_analyses": "The experimental setup for testing the possibility of guiding AlphaFold 3 with static electron density maps is valid.\n\nHowever, in these experiments, most of the structure is also guided using the Substructure Conditioner loss (eq. 3). This is not, per se, an issue but:\n* I only saw this information in the supplementary material and believe this would important emphasize this point in the main text to avoid confusion regarding what the method is capable of.\n* The authors should discuss:\n  * Where does y (the Cartesian coordinates of the atoms in the anchored region) come from?\n  * How should we choose the set of anchored atoms (A)? It seems that we need to know in advance the regions where \"new\" conformations are likely to appear.\n  * What is the influence of the size (in terms of consecutive residues) of the non-anchored region? Does the method perform significantly worse when long regions are not anchored?\n\nFurthermore, the maximum number of structure in the ensemble is set to 5.\n* Does that limit the applicability of the method to cases where the conformational ensemble only has a few modes?\n* Due to what constraints was this maximum number chosen?", "relation_to_broader_scientific_literature": "To the best of my knowledge, this is the first method that applies the idea of guiding a pretrained diffusion model with *experimental* data to sample *conformational ensembles*.", "essential_references_not_discussed": "However, the authors do not cite important prior work in this space (i.e., guiding a protein diffusion model with experimental data). The following references should be discussed:\n\nFadini, Alisia, et al. \"AlphaFold as a Prior: Experimental Structure Determination Conditioned on a Pretrained Neural Network.\" bioRxiv (2025).\nLiu, Yikai, et al. \"ExEnDiff: An Experiment-guided Diffusion model for protein conformational Ensemble generation.\" bioRxiv (2024): 2024-10.\nMaddipatla, Sai Advaith, et al. \"Generative modeling of protein ensembles guided by crystallographic electron densities.\" arXiv preprint arXiv:2412.13223 (2024).\nLevy, Axel, et al. \"Solving inverse problems in protein space using diffusion-based priors.\" arXiv preprint arXiv:2406.04239 (2024).", "other_strengths_and_weaknesses": "The description of the forward models is clear and accurate. The method is well described and seems reproducible.", "other_comments_or_suggestions": "Typo on L139 (right column): [I]n some cases...", "questions_for_authors": "Please see my questions in \"Experimental Designs Or Analyses\".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors propose a framework for building protein conformational ensembles that are consistent with measured experimental data. The authors use Alphafold 3 as a core building block used to generated conformations, and propose an elaborate architecture on top of this to curate the ensemble that relies on diffusion guidance. Reviewers are all supportive of this work, notably post-rebuttal. Some reviewers initially voiced concerns that this work was too close to applications to be relevant to ICML, but these concerns were alleviated following a discussion with authors that emphasized the involvement of the ML community in the study of these problems. I support acceptance.
Decision: Accept (poster)