Generative Modeling of Full-Atom Protein Conformations using Latent Diffusion on Graph Embeddings
Authors: Aditya Sengar
Ali Hariri
Daniel Probst
PATRICK BARTH
Pierre Vandergheynst
Venue: NeurIPS 2025 poster ()
Keywords: Latent Diffusion
Graph Neural Networks
Protein Structure Generation
All-atom modeling
Molecular Dynamics
Primary area: machine_learning_for_sciences

Abstract: Generating diverse, all‐atom conformational ensembles of dynamic proteins such as G‐protein‐coupled receptors (GPCRs) is critical for understanding their function, yet most generative models simplify atomic detail or ignore conformational diversity altogether. We present latent diffusion for full protein generation (LD-FPG), a framework that constructs complete all‐atom protein structures, including every side‐chain heavy atom, directly from molecular dynamics (MD) trajectories. LD-FPG employs a Chebyshev graph neural network (ChebNet) to obtain low‐dimensional latent embeddings of protein conformations, which are processed using three pooling strategies: blind, sequential and residue‐based. A diffusion model trained on these latent representations generates new samples that a decoder, optionally regularized by dihedral‐angle losses, maps back to Cartesian coordinates. Using D2R-MD, a $2\mu\text{s}$ MD trajectory (12 000 frames) of the human dopamine D$2$ receptor in a membrane environment, the sequential and residue-based pooling strategies reproduce the reference ensemble with high structural fidelity (all‐atom lDDT \~ $0.7$; $C\alpha$-lDDT \~ $0.8$) and recovers backbone and side‐chain dihedral‐angle distributions with a Jensen–Shannon divergence $<0.03$ compared to the MD data. LD-FPG thereby offers a practical route to system‐specific, all‐atom ensemble generation for large proteins, providing a promising tool for structure‐based therapeutic design on complex, dynamic targets. The D2R-MD dataset and our implementation are freely available to facilitate further research.

Generative Modeling of Full-Atom Protein Conformations using Latent Diffusion on Graph
Embeddings
Aditya Sengar aditya. werrraasnnnrrr.l.c.om
Ali Hariri
Daniel Probst
ali ali.hariri@epfl.co
daniel Shiiel--robst@wurr
Patrick Barth 1t patrick. bartheepfl.ch ch
Pierre Vandergheynst* w vanderghhenstt@epfl.co
Abstract
Generating diverse, all-atom conformational ensembles of dynamic proteins such as G-protein-coupled receptors (GPCRs) is critical for understanding their func- tion, yet most generative models simplify atomic detail ignore conformational diversity altogether. We present latent diffusion for full protein generation (LDD- FPG), framework that constructs complete all-atom protein structures, including every side-chain heavy atom, directly from molecular dynamics (MD) trajectories. LD-FPG employs a Chebyshev graph neural network (ChebNet) obtain low- dimensional latent embeddings of protein conformations, which are processed using three pooling strategies: blind, sequential and residue-based A diffusion model trained on these latent representations generates new samples that de- coder, optionally regularized by dihedral-angle losses, maps back to Cartesian coordinates. Using D2R-MD, a 2 yus MD trajectory (12 000 frames) of the human dopamine D2 receptor in a membrane environment, the sequential and residue- based pooling strategies reproduce the reference ensemble with high structural fidelity (all-atom IDDT ~ 0.7: Ca-IDDT ~ 0.8) and recovers backbone and side- chain dihedral-ange angle distributions with Jensen-Shannon divergence < 0.03 com- pared to the MD data. LD-FPG thereby offers a practical route to system-specific, all-atom ensemble generation for large proteins, providing promising tool for structure-base therapeutic design on complex, dynamic targets. The D2R-MD dataset and our implementation are freely available to facilitate further research.
Introduction
Proteins function as dynamic molecular machines whose biological activities critically depend on transitioning between distinct conformational states 2 Landmark artificial intelligence methods like AlphaFold2 3 and others 7 have advanced structure prediction but predominantly pre- dict single static conformations, limiting their utility for systems with conformational heterogeneity. Accurate modeling of an ensemble of accessible conformations is essential elucidate protein func- tion and guide therapeutic design [ [8-1 Crucially, these ensembles must explicitly represent all
Signal Processing Laboratory (LTS2), EPFL, Lausanne, Switzerland
Institute of Bioengineering, EPFL. Lausanne, Switzerland
I *Conresoondigg author Bioinformatics Group, University Research Wageningen, The Netherlands Ludwig Institute for Cancer Research, Lausanne, Switzerland
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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atomic details, particularly side chains, whose subtle conformational rearrangements often govern
molecular recognition and catalytic mechanisms [2].
Despite rapid advancements, existing generative models frequently fall short in capturing the de-
tailed dynamics of side-chain movements specific particular proteins [1]] Many powerful methodologies have focused either on de novo backbone designs [14 or on generating static all-
atom structures [17 [8]. but neither produce comprehensive conformational ensembles. Ensemble-
generating approaches-vis perturbations of static predictors [19] 20 flow-matched variants (21),
23 -typccally operate at the backbone or coarse- coarse-grained level. Consequently, they fail to capture the intricate all-atomm rearrangements, particularly involving side chains, that are critical for function. Although promising latent space [24 and physics- informed models [27] (8) have emerged, their capability to generate high-resolution all-atom en- sembles from MD data reflecting functional transitions remains unproven, highlighting significant unmet need for specialized generative frameworks.
or general MD-trained generators 22
protein-coupled receptors (GPCRs) provide a compelling example of dynamic systems wherein precise all-atom modeling is indispensable (29]. This large family of transmembrane receptors, com- prising over 800 human members, is responsible for mediating most known transmembrane signal transduction [30 31 and is targeted by approximately 50% of all marketed drugs (32 33]. GPCR signaling involves conformational transitions among multiple states, frequently induced by ligand binding (34), and occurs through intricate allosteric mechanisms where specific side-chain interac- tions are crucial 33 36). Capturing such dynamic, atomically detailed landscapes is vital to under- standing receptor signaling specificity [37]. biased agonism [38]. and the design of drugs targeting unique allosteric sites 39 4 yet studying these events computationally remains formidable [42]
44
Current predictive methods for GPCR dynamics [45] do not generate the comprehensive all-
atom conformational landscapes essential for mechanistic understanding.
To address this critical requirement, we introduce Latent Diffusson for Full Protein Generation (LD- FPG). generative framework designed learn and generate diverse, all-atom conformational ensembles from existing MD simulation data of a target protein, explicitly including side-chain details. Rather than simulating new trajectories, LD-FPG leverages extensive MD datasets. Our approach employs a Chebyshev Spectral Graph Convolutional Network (ChebNet) 46 to encode all-atom MI) snapshots into a compact latent representation. Denoising Diffusion Probabilistic Model (DDPM) 4 is then trained to explore this learned latent manifold, and latent representa- tions are decoded back into full all-atom Cartesian coordinates. We demonstrate our framework on extensive MD simulations of the human dopamine D2 receptor (D2R), systematically evaluating distinct decoder pooling strategies. Our primary contributions in this paper are:
1. To the best of our knowledge, we present the first latent diffusion modeling framework specifically tailored to generate complete, all-atom protein conformational ensembles, cap- turing both backbone and side-chain dynamics directly from MD simulations.
2. We introduce and eritically evaluate novel graph- autoencoder architecture utilizing ChebNet combined with distinct decoder pooling strategies, offering insights into dynamic protein ensemble generation.
3. Using the D2R system, we demonstrate our method's ability to generate high-fidelity en- sembles, highlight the advantages of residue-based pooling, and assess the impact of aux- iliary dihedral loss terms on generative accuracy.
Our approach provides a computationally efficient tool for exploring complex dynamics in switch- able proteins, supporting both fundamental mechanistic studies and drug discovery applications. The remainder of this paper is structured as follows: Section 2 reviews related work; Section B de- tails our proposed methodology;
the experimental setup and results; and Section summarizes our findings and outlines future directions.
2 Related Work
Generative Models for Protein Design and Static Structure Prediction. Deep generative mod- els have made significant strides in protein science, employing techniques such as diffusion [47), flow-matching [88 49), and learned latent spaces (24 (5) to tackle complex structural tasks. For de
novo backbone design, various methods, often inspired by the success of predictors like AlphaFold2

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3] and diffusion- based approaches like RFdiffusion [14] have emerged. Notable examples include
DoldingDiff [15], alongside Now-matching models such as 2oldFlow2 49 Latent space strategies have also been pivotal in this domain; for instance, LatentDiff 24 generates novel protein back- bones using an equivariant diffusion model within a condensed latent space, while Ig- VA-VAE [50] utilizes variational autoencoder for class-specific backbone generation (e.g., for immunoglobu- lins). While powerful for creating new folds (e.g., Proteina (T6) or specific components, these methods typically target soluble proteins, often generate only backbone coordinates, and are not pri- marily designed for sampling multiple conformations of a specific, existing protein. The generation of complete static all-atom structures has also seen considerable progress. Models like Protpardelle [17 and Chroma [8] can produce full static structures from sequence information. reffuusion-basee generative models such as AlphaFold3 [6] and Boltz-1 1 (7] also provide detailed single-state predic- tions of all-atom structures and complexes. Other approaches, like PLAID (25) integrate predictors with diffusion samplers. To complement backbone or static generation, methods including Flow- Packer [51] and SidechainDiff (52) focus on side-chain packing or prediction. However, these tools predominantly yield single static structures. Furthermore, decoupling backbone and side-chain gen- eration risks overlooking their critical interplay during the complex dynamic transitions [29] relevant to the conformational landscapes our work aims capture.
Modeling Protein Conformational Diversity: From General Strategies to MD-Informed Ap- proaches. Beyond single static structures, capturing protein's ' conformational diversity is cru- cial for understanding its function. Initial strategies to explore this diversity include learning from structural variations in experimental databases (c.g., Str2Str (20) or perturbing static predictions to sample conformational space, as seen with AF2-RAVE 19 for GPCRs and AlphaFlow//EMMilog 21 more broadly. While these methods effectively broaden sampling, capturing -pstei-peecffic, native-like dynamics often benefits from models trained directly on simulation data, which can pro- vide richer representation of given protein's 's accessible states. Among such simulation-informed approaches, latent space models have shown promise. For example, Ensemble VAE (53), trained on MD snapshots and crystal structures of the soluble protein K-Ras, generates C., ensembles from sampled latent features with full-atom picture subsequently produced by RosFold 4. Simi- larly, idpSAM (26). trained on extensive simulations of intrinsically disordered regions (IDRs) with an implicit solvent model, produces C. trace ensembles that can then be converted to all-atom repre- sentations. Such approaches demonstrate the power of leveraging simulation data within generative frameworks, though their application has often focused on particular protein classes (e.g.. soluble proteins, IDRs) or involved multi-stage processes for generating final all-atom structures. Further advancements in learning ensembles directly from MD simulations encompass range of tech- niques. These include methods like ConfDiff (27 (force-guided diffusion), P2DFlow [54] (SE(3) flow matching), and MD-Gen (55) (continuous trajectories). Larger-scale models such as BioEmu [22 and Distributional Graphormer (DiG) 23 aim to learn from vast MD datasets or equilibrium distributions, while experimentalll-guuine approaches like EGDiff (28) integrate diverse data types. Collectively, these MD-informed methods signify substantial progress in modeling protein dynam- ics. However, persistent challenge, particularly when these methods applied generally to very large datasets, is the consistent generation of high-resolution, all-atom ensembles that fully capture system-specific details. This is especially true for intricate side-chain rearrangements within native environments, such as lipid membranes for G protein-couples receptors (GPCRs), which are essential for elucidating their functional transitions 22) The development of generative models that can directly learn and sample such specific, all-atom conformational landscapes from relevant MD data, particularly for complex targets like GPCRs 44 thus remains important frontier.
3 Methodology
The LD-FPG framework (Fig. D models an equilibrium ensemble as a distribution of internal de- formations around a fixed, aligned reference Xyt- Rather than predicting absolute coordinates, we factor generation into two parts: (i) learning a score co over a compact pooled latent that summa- rizes per-atom embeddings, and (ii) reconstructing full all-atom coordinates by decoding under a fixed reference context derived from X Training jointly optimizes a denoising objective in the pooled space and reconstruction losses (Cartesian with optional dihedral terms) on decoder outputs; at inference, individual conformational frames are sampled independently by drawing a latent vector from the learned distribution and decoding with the same reference context, without modeling tem- poral correlations between frames. Module enccfees-encoder,, pooling choices, diffusion sampler,
3

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Training
Bilad Pooling
Securuite
L-NW Gropp
Conutroction
ChociNe Eronder
Zea BTOke
Relecation
Generation
Shde-
Backhone
Revorntrretinn roattio Apri
Figure Overview LD-FPG. Training (top, green): each MD frame ( and the fixed ref- erence Xze are converted to k-NN graphs and encoded with ChebNet into atom-wise embed- dings (Z(0, Z. A pooling module P-chosen from blind, sequential, or residue-based strate- comppeopessss Z(() to low-dimensional latent hg = P(Z(()) A diffusion model is trained in this pooled space using denoising loss to learn the score Eæ over hg. The decoder is trained con- currently as reconstruction head: conditioned the reference context C = Zyeti it maps (ho, C) to full all-atom Cartesian coordinates Xpred, with losses on backbone and side chains (including optional dihedral-angle terms). Generation (bottom, blue): the trained diffusion sampler produces h hen which, together with the same conditioning C, is passed through the same decoder to yield an all-atom conformation Xgen- Frames generated independently (no temporal conditioning).
decoder, conditioning, and loss functions are detailed Sections 3.2 3.3 Appendix A details the overall algorithm and notation; Appendices B provide component-level details, and Appendix consolidates architectural and hyperparameter settings.
3.1 Input Representation and Preprocessing
Each MD snapshot represented by graph ( (V, (V))")), where the node set V comprises the N heavy atoms and the node features are their 3D coordinates X(t) Nx3, For each frame, the edge index E())) is built on the fly by applying k--eerest Neighbors search the aligned coordinates (9) with k = 4. The node positions serve both input features and regression targets. Prior to graph construction, the raw MD coordinates (t) are rigid-body aligned to the first frame using the Kabsch algorithm [56) to remove global rotation and translation.
3.2 Latent learning of conformations
For each MD frame f, the encoder θ maps the Kabsch-aligned heavy-atom coordinates (t) and their N--N graph E(9 to latent embeddings Z eR Nxd. one ddddimensional
Multi-hop encoding:
vector per heavy atom. We implement as a four-layer ChebNet [57] (see Appendix G.2 for archi- tectural rationale) whose layers perform spectral graph convolutions with Chebyshev polynomials of order K 4:
K-1
H(+
Σ t0 TT( R=
where H E NxF the node features at layer I (H () (()),, L is the scaled graph Lapla- cian, (.) is the tthh Chebyshev polynomial, @ E Fix are learnable weights, and a de- notes a Leaky/ReLU non-linearity. Each layer is followed by BatchNorm, and the final output is

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Ln-normalised per atom to yield Z(4) The embedding dimension de was tuned; the best trade-off was obtained with d,=16 for blind pooling, for sequential pooling, and 4 for residue pooling.
Conditioning Mechanism: Each generated conformation is expressed as a deformation of a ref- erence structure: the first Kabsch-aligned MID frame, (Xee, Feer). Rather than conditioning on raw Cartesian coordinates, we feed the decoder the reference's latent representation C' = Zie = O' (Xiet, Exet) where O+ denotes the frozen, pre-trained encoder parameters. This embedding com- pactly summarizes both 3-D geometry and graph topology, and in ablation studies outperformed using Xne directly. At generation time we copy C' to every sample in the batch, Cezi the diffusion model then predicts only the atomic displacements from this common reference. This simplifies learning and guarantees that all sampled conformations stay anchored in the same chemical frame.
3.3 Decoder Architectures and Pooling Strategies
The decoder maps atom-wise latent embeddings 2 E RRNxx (representing conformational de- formations from X output by the encoder detailed in Section (32 and conditioner C. E RR-x to all-atom coordinates Xnred E RN While these z+) embeddings are information-rich (as demonstrated in Section 4.2) their high dimensionality (e.g., up to 35k for D2R with d2 = 16) makes them computationally challenging for direct use input to diffusion model. Therefore, z()) is processed via a pooling strategy to yield much more compact latent representation, be (typically d, R 60 100 for Blind and Sequential strategies), which serves as the substrate for the diffusion model (Section 3.4). This compression is crucial, preliminary experiments showed that dp > 200 300 hampered diffusion training (for blind and sequential), while dp < 50 degraded re- construction quality. The efficacy of LD-FPG thus relies the pooling strategy's ability generate an informative yet compact ho- We investigate three strategies: Blind pooling, sequential pooling, and rased pooling.
Atom-wise embeddings are globally pooled across all N atoms using 2D adaptive average pooling Peioeal (reshaping
Blind pooling:
(t)) as an image-like like tensor of size N x dz), yielding one context vector Zglobal E Rdd per sample in the batch (where dp = H W from the pooling dimensions). This global vector is tiled for each atom and concatenated with the corresponding broadcast conditioner vector (++) to form the input ( for shared MLP, MLP blind which predicts all atom coordinates X ppod simultancously.
Sequential pooling: Decoding is split into two stages. A BackboneDecoder first processes (4) and C' to output backbone coordinates Xbh. It typically pools backbone-specific embeddings to form a backbone context. Subsequently, a SidechainDecoodrr predicts side-chain coordinates Xsc using z(, C, and the predicted Nbb- This stage often involves pooling sidechain-specific embeddings and combining this with backbone information and parts of the conditioner to form features for an MLP. The final structure pred = [xbb X.e]. Three SidechainDDccode variants (arch-types (-2) explore different feature constructions for the sidechain prediction MLP.
Residue-based pooling: This strategy models conformational changes as residue-level deforma- tions relative to X For each residue Ry, its constituent atom embeddings 7 (a subset of the overall atom-wise deformations (()) represent its specific deformation from the reference state im- plicitly provided by Z These z are pooled via Pies into local context vector 2res.j E Rdr, which summarizes residue Ry 's deformation. Each atom (in residue Ry(z)) then receives f()), concatenated with its reference latent C(i)) (from Zıet) as input to res for coordinate prediction. Thus, the decoder reconstructs atom positions from these summaries of residue-specific deforma- tions relative to the reference.
3.4 Latent Diffusion Model for Generation
operates on the pooled latent embeddings ho derived from the chosen decoder pooling strategy. The model trained to predict the noise that was added during a forward diffusion process. This training minimizes the standard DDPM loss function, Laiiiusionn (Eq. 2. New latent representations, h are then sampled by iteratively applying the learned denoising network in reverse diffusion process.
A Denoising Diffusion Probabilistic Model (DDPM) 47
5

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Laiffusion (0)
[lle Eolvā,ho + vI ā,e, ]
3.5 Loss Functions
The LD-FPG framework uses a series of Mean Squared Error (MSE) based losses, detailed in Ap- pendix F train its components. The pre-training phase for the encoder O minimizes coordinate reconstruction MSE (ChNO. Eq. H. For all decoder strategies, the primary objective is coordinate accuracy, defined by the coordinate MSE loss Leoord (Eq. 5
Die Rd Lecordd = (Xpos) [ pred X
(3)
Specific loss configurations vary by decoder: Blind pooling begins with Liee Looord fine- tuned, it applies a weighted composite loss Liee Whese Goood + S (Amse dih Aai din) where Liee (Eq. 6 and Laiv (Eq. ( represent dihedral angle MSE and distribution diver- gence terms respectively, applied stochastically (controlled by S) only for this strategy. Residue- based Pooling uses only coordinate MSE: LDe Leoond Sequential Pooling optimizes backbone (X predbs) and full structure (Xpee)) predictions in stages via separate coordinate MSE losses CBB and Lsc, which correspond to applying Looord to only backbone atoms or the full structure, respectively.
4 Experimental Setup
4.1 Experimental Setup
D2 Receptor Dynamics dataset: We perform extensive all-atom Molecular Dynamics (MD) sim- ulations of the human Dopamine D2 receptor (D2R) to generate the input dataset. The system, comprising the D2R (2191 heavy atoms) embedded in a POPC membrane with water and ions, was simulated using GROMACS. The final dataset consists of 12,241 frames sampled every 100 ps after discarding initial equilibration. All frames were aligned the first frame using the Kabsch algo- rithm. The data was then split into training (90%) and test (10%) sets. Static topology information, including atom indexing and dihedral angle definitions, was pre-processed. Further details on the simulation protocol, system preparation, and data pre-processing are provided in Appendix G. and Appendix B
Evaluation Metries. Model performance is assessed using metrics evaluating coordinate accuracy (e.g., MSE, IDDT, TM-score), dihedral angle distributional accuracy (ISD)) physical plausibility (steric clashes), and conformational landscape sampling (e.g., A100 activation index [58]. PCA of latent embeddings). Auxiliary dihedral training losses are also reported where applicable for specific decoder configurations. Detailed definitions and calculation methods for all evaluation metrics are provided in Appendix G.3
Implementation Details. All models were implemented in PyTorch [59] and trained using the Adam optimizer [60]. The overall three-phase training and generation workflow (encoder pre- training. decoder training, and diffusion model training) is detailed in Appendix A (Algorithm
4.2 Results and Analysis
We evaluated LD-FPG's ability generate high-fidelity all-atom protein conformational ensembles via a multi-stage analysis: (1) assessing ChebNet encoder quality to establish an upper fidelity bound; (2) analyzing decoder reconstruction from ground-truth latents isolate decoder errors; and (3) evaluating conformational ensembles from the full latent diffusion pipeline. Model IDDT scores (vs. Xxet) are interpreted against the "Ground Truth (MD) Ref" IDDT (Tables [ Appendix HH), which reflects the MD ensemble's average internal diversity relative to X Scores near this MD benchmark suggest good balance of structural fidelity and diversity; significantly lower scores imply poorer fidelity, while substantially higher scores (approaching 1.0) might indicate insufficient diversity and over-similarity to X
Multi-hop encoding fidelity of ChebNet The ChebNet encoder generates high-fidelity atom-wiss wise latent representations (2(9) directly from input conformations. Reconstruction from these unpooled embeddings is excellent (details in Appendix H): for the D2R system (N = 2191 atoms), an encoder

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Figure 2: Comparison of dihedral angle distributions for different pooling strategies. (Top row) Ramachandran plots (ø vs vb, log probability) comparing Decoder Reconstruction (left) and Diffu- sion Generation (right) outputs. (Bottom row) Violin plots comparing 1D sidechain dihedral angle (1-X5) distributions for Ground Truth (MD, blue), Decoder Reconstruction (orange), and Diffu- sion Generation (green). (a) Blind pooling. (b) Sequential pooling. (c) Residue pooling.
configuration with a latent dimension d_ 16 achieves backbone MSE (MSE of 0.0008 and dihedral JSDs around 0.00016. This high-dimensional Z (up to 2191 16 35, 056 dimen- sions in this case) establishes strong upper benchmark for the atomic detail initially captured. For efficient downstream diffusion modeling, these rich Z embeddings are significantly compressed via a pooling step (Section 3.3) into much more compact latent vector, hg- This pooling focuses the learning process on con conformational deformations relative to a reference structure (Xref, provided the decoder via Ziet), rather than encoding the entire static fold. While this necessary compres- sion means the final generative pipeline's ' coordinate accuracy may not fully match the encoder's standalone reconstruction capabilities, the high intrinsic fidelity of the initial 7() ensures that ho is distilled from robust, deformation-rich representation.
Decoder Reconstruction Fidelity We next assessed decoder reconstruction of all-atom coordinates from these ground-truth ChebNet latents (2(+), isolating decoder-specific errors (Table []. The blind pooling decoder (using d_ = 16 atom features) achieved good coordinate accuracy (IDDTAI 0.714) but had limited dihedral precision, evidenced by blurred Ramachandran plot (visualiz- ing backbone Φ, 1b angles) and smoothed X-angle distributions ISD 0.0290; Figure 24.. orange sices//ditributionss) Optional dihedral fine-tuning yielded minimal JSD improvement while slightly reducing IDDTs. contrast, sequential pooling (from de = atom features) yielded excellent coordinate accuracy (IDDTA 0.718) and superior backhone geometry, marked by sharp Ramachan- dran plots (Figure orange trace) and the lowest backbone dihedral divergence ISD 0.0026). Its sidechain x-agle distributions were also well-reproduced IESD 0.0192; Figure 2p, orange distributions). Intriguingly, residue pooling (from d₂ = 4 atom features) excelled locally, achiev- ing the lowest backbone/sidechain MSES (0.083/0.2257) and the best sidechain distributional fi- delity JSD = 0,0125) with outstanding x-angle reproduction (Figure 2 orange distributions). This local strength, despite broader global backbone dihedral distribution JSDD 0.0078) and a "hazier" Ramachandran plot (Figure 2 orange trace) partly due to its smaller per-atom latent di-
Table 1: Decoder Reconstruction Performance
Decoder
IDDT IDDTuB
TMAI
ISD
ISD,,
MSE.
MSE,
Σ Can
Configuration
4 MSE
Blind pooling (d, = 16)
0.714
0.792
0.961
0.0032
0.0290
0.1102
0.3934
0.3802
Dih. Fine-tuning
0.698
0.776
0.960
0.0029
0.0279
0.0971
0.3564
0.2849
Seq. pooling (d. 8)
0.718
0.800
0.961
0.0026
0.0192
0.1291
0.5130
0.5164
Residue pooling (d, 4)
0.704
0.777
0.962
0.0078
0.0125
0.083
0.2257
0.2163
Ground Truth (MD) Ref
0.698
0.779
0.959

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(a)
(b)
(c)
Figure 3: Examples of generated D2R conformations using different pooling strategies. Panels likely correspond (a) Blind pooling, (b) sequential pooling, and (c) residue pooling. Structures are visualized to show overall fold and sidechain placement.
mension, stems from its architecture. Pooling features within each of D2R's 's Nuws 273 residues into dp = 4 local contexts (Zres.)) provides the MLP with access to rich information space (effectively Nies dp 1.1k dimensions describing overall residue deformations), boosting local performance. The quality of this decoder stage is key, as the diffusion model samples the distribution defined by these pooled latent embeddings hg.
Table 2: Diffusion Generation Performance.
Model Configuration
IDDT
IDDT
TMAI
ISDD
>IS, +
Avg. Clashes J
Blind pooling
0.719
0.792
0.964
0.006582
0.04185
1350.5
+ Dih. Fine-tuning
0.683
0.748
0.9321
0.00648
0.0409
1340.9
Sequential pooling
0.712
0.801
0.942
0.0029
0.02895
1220.5
Residue pooling
0.6880
0.7575
0.9570
0.0117
0.0224
1145.6
Diffusion Generation Quality The ultimate test is the quality of conformational ensembles from the full LD-FPG pipeline (Table 2). where pooling strategies yield distinct ensemble characteris- tics. Blind pooling produces structures with the highest global coordinate accuracy scores (IDDTAn 0.719, TMAI 0.964). However, this global fidelity, likely impacted by the aggressive compression to its final latent hg, sacrifices finer details: its Ramachandran trace (Figure 24, green trace) is some- what blurred, side-chain distributions are over-smoothed ISD 0.04185), notably higher than
-0.030
-2.7
6.55
-0.015
-0.020
-200
(a)
(b)
212
J00R
K3
Toginio meraag
10350 -355
-203
-015 -1.0
-335
-000
-52 -50
-40
-40
-44
-30
Figure 4: Latent space and collective variable (A100) analysis for Blind (a), Sequential (b), and Residue (c) pooling. Top rows: PCA of dataset (blue) vs. diffusion-generaaeed (red) pooled embeddings (Sequential shows separate backbone/sidechain PCAs). Bottom rows: A100 Value distributions (histograms) and PMFs (-log(Probability Density)) for dataset: ground truth (blue), autoencoder (purple), decoder (green), and generated ensembles (red).

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the = 1023 clashes averaged by the ground truth MD ensemble. It is crucial to note that dynamic MD ensembles naturally exhibit transient sterie clashes, making the MD simulation itself, rather than an idealized zero-clash structure, the appropriate physical baseline for evaluat- ing plausibility. Rendered structures (Figure show well-placed helices but wandering X-angles, leading to protruding or clashing side chains. Sequential pooling offers an excellent all-round compromise, It maintains sharp backbone geometry (lowest ISD 0.0029; clean Ramachandran in Figure 2b. green trace), respectable global scores (IDDTAnl 0.712), and good side-chain realism IESD se 0.02895), with an average of 1220.5 clashes. Visually (Figure (D)) models are largely tidy. However, some surface side-chains may still flare out or collide. This aligns with its ISD,, being better than blind but not residue pooling, suggesting some X-angles outside dense rotamer clouds, possibly due to the side-chain decoder's pooled backbone context leading to ambiguity that diffusion "averages". Thus, while delivering a sharp backbone and strong side-chain stats, a few rotamers (preferred side-chain conformations) may remain in incorrect basins. Its global backbone statistics are lower after diffusion (IDDTA Al 0.6880; IS 0.0117; Figure 2 green trace), with occasional backbone kinks (Figure 3p), partly due to its use of smaller (dg 4) per-atom latent dimension. However, its per-residue focus excels at x-angle distributions (best ISD 0.0224) and yields the lowest average clash count (1145.6, closest to the MD reference). producing the tightest side-chain packing (Figure 3+). In essence, Figure 2 these strategies' distinct behaviors: blind pooling prioritizes global fold over side-chain details; se- quential pooling balances these, with minor outliers; residue pooling excels at rotamers, sometimes with less regular global backbones.
(dihedrals) and Figure [3] (structures) reveal
Figure 4 further explores conformational landscape coverage using PCA of latent embeddings and the A 100 collective variable, an index of D2R activation states that proxies the MD-sampled land- scape. It provides a dual analysis: PCA plots (top row) compare tions (hg red) against MD-derived ones (blue), while A100 value distributions (bottom row) assess the replication of the MD conformational landscape (and its autoencoderrddecoder capture) by the full diffusion pipeline. The diffusion models' ability to cover major PCA regions of the MD latents posi- tively indicates effective learning of the underlying data distribution. When examining the A100 dis- tributions, Blind and Sequential Pooling. operating on more compact, coobally-pooled latent spaces (he with dp Re 100), show reasonable landscape coverage, with Sequential closely tracing the MD distribution. However, residue pooling (Figure 4 red curve) achieves the most comprehensive A100 landscape capture, populating both primary and subtler MDD-oberrved states often missed by other methods. This superior recovery, despite moderate global backbone metrics, stems from its distinct latent space configuration. Unlike the compact, global latents of Blind/Sequential strategies, residue pooling's 's diffusion model leverages a significantly larger effective latent space formed by all per-residue contexts (Nres x dp ~ 1.1k for D2R with dp 4). This richer, structured space, even if derived from a smaller d2 = per-atom encoding, enables more nuanced representation of conformational substates. Its proficiency in local rotamer modeling is likely a key contributor, as accurate side-chain placement is crucial for the subtle cooperative shifts defining A100. The decoder's A100 distribution for residue pooling (Figure 4p. green curve) already indicates robust MD landscape coverage, which the diffusion model effectively samples. Visualizations for residue pooling in Figure Temployed multi-epoch sampling-aggregatingg aggregating samples from checkpoints across different DDPM training stages, Appendix !
latent distribu-
4.3 Benchmark
To contextualize LD-FPG, we benchmarked against leading ensemble generators applicable to (21), BioEmu (22). and Boltz-2 (MDD-ondditioinng 77. We evaluate on the
D2R system using two complementary axes: (i) local geometric accuracy, via the divergence (JSD) of backbone dihedrals (ø, v). and (ii) dynamic flexibility, via the mean Ca root- mean-square fluctuation (RMSF), which quantifies ensemble diversity.
Superior local geometry. LD-FPG attains a backbone dihedral JSD of 0.007, improving over base- lines by 3-5x, indicating state-of-the recovery of local conformational preferences. Flexibility vs. static fidelity. BioEmu and Boltz-2 produce high- sighlfidliity static structures (IDDT ~ 1.0) but exhibit very low ensemble flexibility (RMSF 0.07-0.09 nm). AlphaFlow shows higher flexibility (RMSF x 0.84 nm) yet still under-represents the ground-truth dynamics. LD-FPG matches MD- level flexibility (RMSF 1.22 nm vs. MD 1. 3.34 nm) while preserving superior local geometry. Con- sequently, its IDDT (~ 0.80) is lower than static predictors by design: for an ensemble generator,

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Table 3: Benchmark comparison of LD-FPG against state-of-the-arr models on the D2R ensemble. LD-FPG's IDDT is lower as it models the full, flexible ensemble, whereas baselines produce overly rigid structures, resulting in artificially high IDDT against the static reference. All baseline metrics were recomputed using their publicly available code. The Ground Truth MD (Ref) row indicates the properties of the original simulation data.
Model
Backbone JSD
Backhone IDDT
Backbone TM-score
Backbone RMSF (nm)
(0.4)
LD-FPG (Ours)
0.007
~0.80
00.9
1.22
BioEmu
~0,022
0,999
0.925
0.09
AlphaFlow
~0.023
0.859
0.993
0.84
Boltz-2 (MD-cond)
~0.03
0.997
0.975
0.07
Ground Truth MD (Ref)
(Ref)
(Ref)
(Ref)
1.34
IDDT to a single reference frame decreases as realistic conformational spread increases. All-atom advantage. Unlike backbone-centric baselines, LD-FPG generates full side-chain detail, enabling side-chain validation: we obtain low summed JSD over X angles (~~ 0.022) and all-atom RMSF (1.36 nm) close to MD (1.60 nm).
5 Conclusion and Future Work
We introduced LD-FPG, latent diffusion framework generating all-atom protein conformational ensembles from MD data, demonstrated D2R GPCR. Crucially, benchmarking showed LD-FPG uniquely matches MD-level dynamic flexibility (RMSF) while achieving superior local geometry, unlike more rigid baseline methods. It captures system-specific dynamics, including side-chain details, via learned deformations from a reference. Blind pooling offered global fidelity but com- promised side-chain detail and clashes. Sequential pooling provided strong balance, especially for backbone geometry. Residue pooling excelled in local side-chain accuracy and landscape coverage, despite some global backbone trade-offs and needing multi-epoch sampling for full diversity.
Future work includes targeted enhancements. For Blind and Sequential pooling, exploring larger pooled latent dimensions (dp)) is immediate, potentially improving detail capture but needing more extensive, diverse training data (e.g., from multiple related protein systems). For Residue pool- ing, with its promising high-dimensional effective latent space (Nres x d,.). better denoisers are key to overcome multi-epoch sampling limits. This might use complex P/conooluutionall archi- tectures, attention for its structured latent vector, alternative generative models basere-based, flow-matching). For Sequential pooling, improving denoiser coverage of the MD-derived la- tent space (Fig 4p, backbone/sidechain overlap) could resolve side-chain misplacements and boost performance. Broader initiatives will enhance physical realism and architectural sophistication by incorporating lightweight energy surrogates, physics-guided diffusion schemes [61 62 28), and advanced architectures like attention- bbasd fusion mechanisms (63) SE(3)-equivarian GNNs/transformers, or flow-matching generators [48 [66) A long-term goal is generalizing LD- FPG by training on data from multiple related proteins (e.g., Class GPCRs in various states) for foundational models akin specialized "LLMM for protein dynamics." " Efforts will also cover tailoring pooling for applications, scaling to larger systems 18], and rigorous benchmarking.
Societal Impact: LD-FPG offer benefits by accelerating drug discovery, improving biological understanding, and enabling protein engineering for medicine/biotechnllogy. Ethical issues include dual-use, equitable access, data privacy for sensitive data, and rigorous validation to prevent misdi- rected efforts.
Acknowledgmentss
A.S. was supported by grant from the Center of Intelligence Systems (EPFL) P.B. and P.V. This work was also supported by the Institute of Bioengineering (EPFL), the Signal Processing Laboratory (LTS2, EPFL). the Ludwig Institute for Cancer Research, and the Swiss National Science Foundation (grants 3.003A_ 182663 and 310030_208179)
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[83] Yi-Lun Liao and Tess Smidt. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs. arXiv preprint arXiv:2206.11990, 2022.
[84] Fabian Fuchs, Daniel Worrall, Volker Fischer, and Max Welling, Se (3)-transformers: 3d roto- translation cquivariant attention networks. Advances in neural information processing systems, 13:1970-1981, 2020.
[85] Valerio Mariani, Marco Biasini, Alessandro Barbato, and Torsten Schwede. Iddt: a local superposition-free score for comparing protein structures and models using distance difference tests. Bioinformatics, 29(21):2722- 27228, 2013.
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A Methodology Overview and Notation
This section provides a consolidated overview of the LD-FPG framework's workflow, key mathe- matical notation used throughout the paper, and supplementary details regarding data processing, model architectures, and training procedures.
A.1 Notation
We define the following notation used throughout the main text and appendix:
k: Number nearest neighbors used for graph construction.
(+) E N33 Raw (unaligned) Molecular Dynamics coordinates for frame
+: Encoder neural network function; denotes frozen, pre-trained encoder parameters.
K: Order of Chebyshev polynomials in the ChebNet encoder.
de: Dimensionality of the conditioner tensor C.
H, W: Output dimensions (height and width respectively) for pooling layers, where dp
W.
Mayers: Number of layers Multi-Layer Perceptron (MLP).
Laifiusion: Loss function for training the Denoising Diffusion Probabilistic Model (DDPM).
N: Total heavy atoms per structure.
V = (1...., N): Atom nodes.
(V, E): Graph representation.
Tab C V: Backbone atom indices; No [bbll.
Isc C V: Sidechain atom indices; Nsc- Iscl. (V Tub U.x)).
Eres.j TR:::: Local pooled context vector summarizing the deformation of residue R, (this
corresponds to hR, as used in Appendix D.3..
Xure IR Nx3, Ground truth aligned coordinates. x(0.i) for atom i, sample b.
Xpred E N33, Predicted coordinates.
Xref N23 Reference structure coordinates (e.g., first frame).
dzi Encoder latent embedding dimensionality.
Z Nxd=; Latent atom embeddings from encoder: Z Encoder( (uuee E).
Zref E Nxd:: Reference structure latent embeddings: Zref Encoder (ercoder(xief, Eret).
R. Nxde; General conditioning tensor (typically C Z,er). Can is batch-expanded.
dp: Pooled context vector dimension.
P: Generic pooling operator.
* R= -RR,,... RN,. }: Partition of atoms into residues.
f.V (1,..., Nrs)): Atom-to-residue mapping.
di vh, Xx: Dihedral angles; A is generic type.
Ma(i): Mask for valid angle residue j.
P(appa) P(amwe: Empirical angle distributions.
D(-lI-): Divergence function (KL, JS).
Whases Amser Adiv: Loss weights.
fan: Probability of applying dihedral-based loss terms during stochastic fine-tuning.
||v Squared Euclidean norm.
T: Diffusion timesteps.
Bi, ar āy: Diffusion schedule parameters.

---

hy: Initial pooled latent embedding for diffusion.
h Noisy latent at step 1.
(0, I): Standard Gaussian noise.
( f): Denoising network predicting noise e.
A.2 Overall Training and Generation Workflow
The LD-FPG framework follows a multi-phase training and generation procedure, outlined in Algo-
rithm 1
25

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Algorithm Overall Training Generation Workflow (LD-FPG)
Input: Aligned coordinates file (my wwy-protein.oo json),
Topology
file
chain.pdb). 2: Parameters: Encoder config K...)), Decoder config (Pooling Type, H, W, Magerss.....
(condensed resieeseedrreiduuss residues..ssonn. Reference PDB (heavy chain pdb).
Optional fine-tuning weights (Amser Adin), Diffusion config (T, Pstarts Bend, cg type).
// Preprocessing (Details in Appendix
3: Load aligned coordinates {_x11) f=1 from my my-protein. proonnn.json.
Nimms
4: Load topology (atom indices Inb.Isa residue map S, dihedral defs.) condensed_resideee json.
from
5: Build graph dataset GraphDataset = (BuildGraph( X
irue h = y
// - Phase 1: Encoder Pre-training (Optional) (Details in Appendix ??)
6: Initialize Encodero (ChebNet model).
7: Train Encoder, on GraphDatasel Dalaset using coordinate reconstruction loss LINO (Eq. in Ap- pendix
8: Save best encoder parameters d"
V/-Phase Decoder Training (Details in Appendix D
9: Load pre-trained Encoderg, and freeze weights.
10: Generate latent embeddings z Encoderg- (X for all frames f-
11: Create Decoder Input Dataset Déec = {(Z(), ( I L..mmermmes }.
12: Define conditioner C Zuel Encoderg- ( (Details in Appendix ??).
13: Select Decoder Variation:
if PoolingTyppe is Blind then
15:
Initialize Decoder, & BinndPooingDDecoder(.-)
16:
Train Decoder, Dacc using coordinate loss Lixe Lecord
17:
Optional Fine-tuning: Continue training with Lee = Whase Fooord + Amse mmse + Adto dii (stochastically, see Appendix Save best decoder parameters o²
18:
else if PoolingType is Sequential then Initialize BBDecoder SCDecoder (...).
20:
21:
Train BBDecoder das on Ddec using LBS = coard Freeze dhe
22:
Train SCDecoder on Daee using Lsc coord (requires frozen BBDecoderg;)
Cfull
Save best parameters Lig d'c.
24: else if PoolingType is Residue-Based then
Initialize Decoder, +-
26: Train Decoder, on Ddec using coordinate loss Liee
Leoord-
Save best decoder parameters o"
28: end if
// Phase Latent Diffusion Training (Details in Appendix E
29: Load best Encoderg- and Decoder,- (or relevant pooling part).
30: Generate pooled latent embeddings hg (o))(Z(n)) for all f. using the specific pooling
mechanism of the chosen decoder.
31: Create f fusionInpuu Datoset = {bg (.)
32: Initialize éo (Denoising model).
33: Train eo on Di fusion Inciion/npp.DDiia rpput Dataset using Laimusion (Eq. 2
34: Save best diffusion model parameters 07
// Output Generation (Sampling)
35: Load best models: Encoderg- Decoderg- (or BBDecodera;, SCDecoder ), Caats"
36: Sample novel pooled latent(s) hōp using Gair (Algorithm in Appendix
37: Decode generated latent(s): (Feed h into appropriate decoder stage)
38: X-0aa Decoder, (..., context h C))
39: Output: Generated coordinates (X
gen pred
26

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B Input Data Processing and Representation
The raw Molecular Dynamics (MD) simulation data (trajectory processing detailed in Ap- pendix G.Ip is transformed into structured format suitable for the machine learning pipeline. This involves generating two key JSON files using custom Python scripts (see Supplementary Code for extract.residues py and condense-json : one defining the static topology of the pro- tein with a consistent indexing scheme (condensed residues. json), and another containing per- frame coordinate and dihedral angle data.
py):
B.1 Static Topology and Consistent Indexing File bronddense_residddee json)
To provide a consistent structural map for the machine learning models, a static JSON file, typically named condensed residues. json, is generated. This file is crucial as it establishes definitive and model-centric representation of the protein's topology:
Zero-Based Contiguous Atom Indexing: A new, zero-based, and contiguous indexing scheme (0, ,,....,N, N - 1) is created for all N heavy atoms in the protein system. This re-indexing maps original PDB atom identifiers to consistent integer range, essential for constructing graph inputs and feature matrices for the neural network.
Residue Definitions: Residues are also re-indexed contiguously (e.g.. 0, O...,, ......,,es.. - -1). For each re-indexed residue, the file stores:
The residue type (e.g.. 'ALA', 'LYS').
Lists of atom indices (using the new zero-based scheme) that constitute the backbone atoms of that residue.
-istss of atom indices (using the new zero-based scheme) that constitute the sidechain heavy atoms of that residue.
Dihedral Angle Definitions (Atom Quadruplets): This is critical component for calcu- lating dihedral-based losses and analyses. For each residue, the file stores definitions for all applicable standard backbone angles (ø, vh) and sidechain angles (X1, X2, X3, X4; X5). Each dihedral angle is defined by an ordered quadruplet atom indices adhere the new, zero-based, contiguous indexing scheme. For example, o angle for a specific residue would be defined by four specific integer indices from the O... 1 range. This allows for unambiguous calculation of any dihedral angle directly from a set of N atomic coordinates, whether they are ground truth or model-predicted. The definitions also account for residue types where certain x angles are not present (e.g., Alanine has no X angles, Glycine has no sidechain).
four atom indices. Crucially, these
B.2 Primary Per-Frame Data File
This second JSON file stores the dynamic information extracted from each frame of the MD trajec- tory. For every snapshot:
Heavy Atom Coordinates: The 3D Cartesian coordinates all heavy atoms are recorded after rigid-body alignment to common reference frame (the first frame of the trajectory, as described in Section 3.1). These coordinates are stored an order that corresponds to the new zero-based indexing defined in the static topology file.
Dihedral Angle Values: The values for standard backbone dihedral angles (ø, v/) and sidechain dihedral angles (x1 through Xs, where applicable for each residue type) are pre- calculated. These calculations initially use atom identifications based on the original PDB residue and atom naming conventions but are stored in a way that can be mapped to the new indexing if needed for direct comparison or analysis.
This primary data file is typically organized on a per-residue basis (using original PDB residue numbering for initial organization if helpful during generation), associating each residue with its constituent atoms' names, original PDB indices, and the time series of their coordinates and caleu- lated dihedral angles.
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B.3
Usage in Models
The two JSON files are used in conjunction:
The static topology file (condensee_reeicduus,, residues...d . serves as the definitive reference for all structural metadata used by the model during training and inference. This in- cludes identifying which atoms belong to the backbone versus sidechain (using their new zero-based indices) and, most importantly, providing the specific quadruplets of new zero- based atom indices required to calculate any dihedral angle from a given set of 3D coor- dinates. This capability is essential for implementing the dihedral angle- baseed loss terms (Lmse dih, daiv dihh mentioned in Section 3.3 as these losses operate on dihedral angles com- puted from the model's predicted coordinates (X pred).
The 3D coordinates for each heavy atom, required as input features (X(0) for the encoder at each frame t, are drawn from the primary per-frame data file. These coordinates must be arranged and ordered according to the new zero-based indexing scheme established by the static topology file to ensure consistency with the model's s internal graph representation.
This separation of dynamic coordinate data from static, re-indexed topological information allows for efficient data loading and consistent geometric calculations within the LD-FPG framework.
C Other validation studies
C.1 Cross-system coverage diverse proteins
Setup. To test generality beyond GPCRs, we evaluated LD-FPG on twenty proteins spanning a wide range of lengths and secondary-structure composition (helical, 8-sheet, and mixed a/3). The systems are: 3dan_A, 1af7_A, _A, 3tdt_A, 1kop_A, 5jak 1sdi_A, 1zt5_A, 1u7k_D, 1jyo 2ppp_A A, 3zcb_B, B, 5187 AA, 3mmy_D, D, 1ab1 _A, few_sec, alpha-beta, antistatin, villin, kinase_lptq, 7jfl_C. All reference trajectories are public MD simulations from the ATLAS database. We Kabsch-align each trajectory to its first frame using Ca atoms to eliminate rigid-body motion. LD- FPG is trained per system on the aligned frames and then used to sample new all-atom conformations that are decoded back Cartesian coordinates.
PCA protocol (per system). For an interpretable 2D visualization of the conformational manifold, we compute principal components only from the reference MD: (i) flatten the aligned Co coordinates for each frame vector in R3N (iii) fit PCA and keep the top two components. We then project (a) a downsample of MD frames (blue points) and (b) an equal number of LD-FPG samples (red points) into that fixed basis. This avoids any leakage from model samples into the PCA axes.
Observation. Across folds and sizes, LD-FPG reproduces the global geometry of the MD mani- folds: elongated, anisotropic clouds (e.g., 1af7 _A, 3tdt_A, 7jf1_C), multi-lobed structures (3dan_A 1k)p_A), and more compact clusters (3mmy D. villin). In some systems the model slightly con- tracts the most distant extremes conservative bias), while in others it marginally broadens dense cores.
C.2 Validation on GPCR activation: TM-distance free-energy surface
Dataset and preprocessing. We use a publicly released all-atom, explicit-solvent molecular dy- namics (MD) trajectory of the human adenosine A₁ receptor embedded a lipid bilayer [67]. Prior to encoding/decoding. solvent, ions, and lipids stripped; all protein frames are Kabsch- aligned to a reference using Ca atoms from the transmembrane (TM) bundle to remove higid-body drift. The same alignment and atom selections are applied to both the MD reference and the LD- FPG-generated structures.
Activation coordinates. Following standard GPCR analyses, we quantify activation by two intra- cellular opening distances: TM3-6 and TM3-7. Each distance is computed between the centers of mass of Ca atoms at the cytoplasmic ends of the respective helices (helix assignments by DSSP; residue windows as in Amore al. (7).. These coordinates track the outward swing of TM6 and the cavity opening toward the G-protein, i.c., the hallmark GPCR activation motion.
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1af7
1kop
Sjak
1sdi A
1x15
1u7k
1jyo
2ppp
3zcb
5187 A
3mmy D
1ab1 A
few,sec Sec
alpha-beta
antistatin
villin
kinase 1ptq
7jfl
Figure 5: Diverse-protein PCA coverage. Top-2 PCA of Co coordinates computed from MD only for each system; MD frames (blue) and LD-FPG samples (red) are projected into the same basis. Panels span proteins of different lengths and secondary-strucet content.
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Free-energy estimation. For both the MD reference and LD-FPG samples, we estimate the joint distribution P(d36. d37) by binning the two distances on regular grid and converting to potential of mean force (PMF) via F(d3s, d37) = - kB7 In P(d36. d37) + C. One-dimensional profiles are obtained by marginalization, e.g., F F(d36)
- Inf P(dze; d37) dd37.
Result. Figure 6 (left) shows the MD PMF (heatmap) overlaid with LD-FPG contours in the (da6, d37) plane; the right panels plot the corresponding one-dimensional PMFs along each coor- dinate. The reference surface forms diagonal valley connecting inactive- and active-like regions. LD-FPG reproduces this valley and the position/curvature of the principal minimum, with modest broadening at the valley shoulders. The agreement of the ID profiles indicates that the model sam- ples along the activation corridor with near-refcrence occupancy, supporting that LD-FPG captures functionally relevant GPCR motions beyond local geometry.
TM3-7 distance (A)
A1AR MD (Ref)
LD-FPG
7
Free Energy (KT)
16
3.
TM3-6 distance (A)
Free Energy (KT)
13
12-
15
16
20
12
13
15
16
TM3-7 distance (A)
TM3-6 distance (Å)
Figure 6: A1AR activation surface. Left: Two-dimensional PMF over TM3-6 and TM3-7 dis- tances; MD reference heatmap. LD-FPG as overlaid contours. Right: dinedimnniional PMFs along each coordinate (MD in green; LD-FPG in orange). LD-FPG tracks the diagonal activation valley and the principal minimum with minor broadening at the flanks.
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D Decoder Architectures and Pooling Strategies
This section provides detailed description of the three primary pooling strategies employed within the decoder architectures: Blind pooling, sequential pooling, and residue-based pooling. For each strategy, we delineate how atom-wise latent embeddings Z E _BxNxx, (where B is batch size, N is the number of heavy atoms, and dN is the latent dimension per atom) and conditioner C' E IR xd. (lypically the latent representation of the reference structure, Zeer) processed generate the input for the final coordinate prediction MLP. Specific hyperparameter configurations for representative models are detailed the Extended Technical Appendix (Section TT.
D.1 Blind Pooling Strategy
The blind pooling strategy aims to capture global context from all atom embeddings for the entire protein structure. Let g(6) E xdd be the latent atom embeddings for a single sample a batch.
1. Global Pooling: The atom embeddings (b) treated
image- tensor and pro- cessed by a 2D adaptive average pooling layer, Petobal (typically nn. AdaptiveAvgPoll with output size H W). This operation pools across all N atoms for each sample in the batch:
(h) global Polahaa (7()) E 1
where dp = H W is the dimension of the pooled global context vector. For batch, this results in Halobal E BBxd,
2. Context Expansion: This global context vector h (b)
globall is then expanded (tiled) to match the number of atoms N, resulting in ((() globalex RNNd, where each row i (for atom i) is h (b)
identical to global- For a batch, this is H globaleex E BxNxx,
3. Conditioner Expansion: The conditioner C' E IR xde is expanded for the batch to Can
IR. xd.
4. MLP Input Formulation:
For each atom in sample 6, the input to the final MLP, MLLP blind formed by concatenating its corresponding expanded global context and its conditioner vector:
M concat (b)
global ((b,)) B.
Note that h (6) global the same for all atoms within sample
5. Coordinate Prediction: shared MLPPoind processes (b(b.1) for each atom to predict its 3D coordinates:
x(h.a) (h,a) pred = MLPplind (b.1) (1))) E R3
This results in the full predicted structure X Xpred E R BxNx3
D.2 Sequential Pooling Strategy
The sequential pooling strategy decodes the protein structure in stages: first the backbone atoms (Ia)), then the sidechain atoms (Işe), using information from the preceding stage.
D.2.1 Backbone Decoder Stage
Let ( E N.Nxx be the full latent atom embeddings and C1b) E Nxd be the full conditioner for sample
1. Backbone Embedding Selection: Latent embeddings z E R Nux and conditioner
vectors c(6) E No corresponding to backbone atoms Iab are selected.
2. Backbone Pooling: 2!0h is pooled using 2D adaptive average pooling layer Pbs (c.g., BlindPooling!/ from the implementation, with output size Hob x Wor) to obtain a bookbone-speciffe context vector:
(b) h bb
Poo (Z(()) TRd
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where dp,06 = Has Wos- For a batch, this is Hob E R Bxd
3. Context and Conditioner Expansion: (()) is expanded to H bblex E R Ni
R.-aadppi xdp.
Cbb is
used directly.
4. MLPP Input for Backbone Atoms: For each backbone atom j E Tab in sample b, the input to the backbone MLP, MLP is:
[b,] M M concat(h (b)
c(b Rd..ttd.
5. Backbone Coordinate Prediction: A shared MLP predicts backbone coordinates:
(b,j) MLP (b,d)) R phod
This yields the predicted backbone structure X predy R BxXx3x
D.2.2 Sidechain Decoder Stage
This stage predicts sidechain atom coordinates Xpredse using the full latent embeddings Z E RNxd, the conditioner (2 R RNx,, and the predicted backbone coordinates Xpredaa R Nxx3 from the Backbone Decoder Stage.
(b)
1. Sidechain Embedding Selection: Latent embeddings z RNNx corresponding to sidechain atoms Iae are selected from the full latent embeddings 2
2. Sidechain Pooling:
The selected sidechain embeddings _ are pooled using a 2D adap- tive average pooling layer Psx (c.g., BlindPool ing2D with output size Hac Wee to obtain sidechain-speciff context vector for each sample in the batch:
h (b) sc (2()
RRd..s
where dp.sc Hsc Wsc- For a batch, this results in H E Bxd,....
3. Feature Construction for Sidechain MLP Input: The input the sidechain MLP, MLP a global vector (() constructed per sample b. The construction varies based in
the ar arch_type:
(b) Let Xreedhh N.- be the flattened predicted backbone coordinates for sample b.
Let C(l)) Ne be the sidechain portion of the reference conditioner. If de dg. this is z(0) se This is flattened to (.(b) E .e.
flat Arch-Type 0: The input consists of the flattened predicted backbone coordinates and the
pooled sidechain context from the current frame's embeddings.
in, [Arch 0] concat( (X prac (b)
The dimension of M() Arch 0) is (Noo 3) + dp.ec- Arch-Type 1: This architecture adds a reduced representation of the sidechain portion of the reference conditioner.
The flattened sidechain conditioner (b)
sc.flat passed through a linear reduction layer:
scrreducdd = Linear (c.(b) )c. E R
(e.g., d' 128 in the implementation).
The MLP input is then:
(b) _h Arch 1] concat (X pt flat h( (()))
(b)
reduced)
The dimension of (0 Arch I] is (Nbb 3) dp, + d' Arch-Type 2: This architecture uses reduced representation of both the predicted back- bone coordinates and the sidechain portion of the reference conditioner.
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The flattened predicted backbone coordinates X peed are passed through linear reduction layer:
(b)
X peeuud =
(6)
Linearat (- Aprcchbottaal E R
(e.g., dbbb 128 in the implementation).
(()) The sidechain conditioner is reduced as in Arch- Arch-Type Type I to reduced E RR
The MLP input is then:
M [Arch 2] concat( predfs reduced h Ce reduced)
(b)
The dimension of M Arch 2] is d'e + dp,sc &essc-
4. Sidechain Coordinate Prediction: The sidechain MIP, MLPx> processes the constructed input vector M se (corresponding to the chosen arch_type) to predict all sidechain coor- dinates for sample b simultaneously:
peedge - MLP MM.PssMM E R.N...3
(b)
in, sc
This flattened output then reshaped Xpredse (b) RRR x.
5. Full Structure Assembly: The final predicted structure X X E RNx33 for sample is assembled by combining the predicted backbone coordinates Xpredho sidechain coordinates Xpredse (b)
(b) and the predicted
Residue-based Pooling Strategy D.3
The Residue-basse Pooling strategy generates a context vector specific to each residue and uses this local context for predicting the coordinates of atoms within that residue. Let 7 E xxx, be the atom indices belonging residue Rj, and f: V + {1, N be the mapping from global atom index to its residue index.
latent atom embeddings for sample b. Let Vj be the
1. Per-Residue Pooling: For each residue R, in sample b:
Select atom embeddings for residue Ry: z(6) E RR/..
Pool these embeddings using 2D adaptive average pooling layer Pres (e.g., nn AdaptiveAvgPool2c with output H W):
(b) hR = Pres Pre. (Z E T
where dp = H W.
This results in set of Nis pooled vectors for sample b, which be represented as (b) E X For a batch, this is Hees E R Bx Nos xd2,
2. Atom-Specific Context Assembly: For each atom in sample b, its specific context vector is the pooled vector its parent residue Ry(3)
(b.z)
(b) Ryo
E Rd4
atom context
This can be gathered for all atoms to form H
atom context E R NNd,,
3. Conditioner Expansion: The conditioner C R Nxd, is expanded for the batch Cen IR BxN xd.
4. MLP Input Formulation: For each atom i in sample b, the input the final MLP, MLP, resp is formed by concatenating its residue's 's pooled context and its specific conditioner vector:
(b,4) concat ( (b,i) atomecomtet:: C(b.))) Rd
5. Coordinate Prediction: shared MLP processes (b.i)) for each atom to predict its 3D in, res
coordinates:
((b.)) MLP (M, pred
res E R
This results in the full predicted structure Xpea E R BxNx3
This strategy allows the model to learn representations that are localized the residue level, poten- tially capturing residuc-specific conformational preferences more directly.
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E Latent Diffusion Model Details
This section details the Denoising Diffusion Probabilistic Model (DDPM) 47 work. The DDPM operates on the pooled latent embeddings ha E RR (where dp is the dimension of the pooled latent space, dependent on the pooling strategy) obtained from the encoder and pooling stages. Specific architectures and hyperparameters for the denoising network ep are discussed in the Extended Technical Appendix (Section
utilized in our frame-
E.1 DDPM Formulation
The DDPM consists of predefined forward noising process and learned reverse denoising process.
Forward Process (Noising): The forward process gradually adds Gaussian noise to an initial latent embedding ho over T discrete timesteps. At each timestep the transition is defined by:
q( ( = Whr / 1..
where { \T= is a predefined variance schedule (e.g., linear, cosine) that controls the noise level each step. A useful property of this process is that we sample h, any arbitrary timestep t directly from ho:
((, hb) Nhh vārho, II)
where Or = 3e and a. = II -I===100 Os.- As f - T, if the schedule is chosen appropriately, hr approaches an isotropic Gaussian distribution (0, I).
Reverse Process (Denoising): The reverse process aims to learn the transition g( [he), which intractable directly. Instead, neural network, eg (h,, t), is trained to predict the noise component e that was added to hg to produce h, / La + v1 âjc, where ~ (O.01) The network is optimized by minimizing the simplified DDPM loss function (as shown Eq. in the main text):
diillusion (0) = Eg,b [||e co(vá,bu vI- 1
where t is sampled uniformly from {1...., TT.
E.2
Sampling New Latent Embeddings
Once the denoising network eg trained, new latent embeddings he" can be generated by start- ing with a sample from the prior distribution, hy ~ A I). and iteratively applying the reverse denoising step:
Be
h
en (be: o,
Vot
VI at
where z ~ ((O, I) for > 1, and 0 for = 1. The variance of is typically set to Br or 3.
1- B. The full sampling procedure is outlined in Algorithm T-
Algorithm Reverse Diffusion Sampling for Latent Embeddings
1: Input: Trained denoising model éa, number of generation samples Been dimension pooled latent dz. diffusion timesteps T, schedule parameters (e.g., .1,..., Br).
2: Calculate o₁
3 and ā; =II'.= as for all
3: Set o² = 3: (or alternative like 30).
4: Sample initial noise hy N( I) of shape (Byens dz).
5: for = T, T,... do
6: Sample z ~ A/ I) of shape (Byen d,) ift 1, else 0.
7: Predict noise: (pred + co (h,, t)
8: Calculate conditional mean: (o 1) = 1 (he
vi Cpred)
9: Update latent sample: h < μg (h,, 1) + 0,2
10: end for
11: Output: Generated latent embeddings h = ho
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Loss Function Formulations
This section provides the detailed mathematical formulations for the loss functions used in the LD- FPG framework. The notation used is consistent with Appendix A.1
F.1 Encoder Pre-training Loss
The pre-training phase for the ChebNet encoder (Encoder,) can be performed using a direct coor- dinate reconstruction head (MLPHNo). The loss function LhNO is the Mean Squared Error (MSE) between the predicted coordinates and the ground truth coordinates:
LHNO (X.oxx [[MLPPHNO(Z) Xime P
(4)
where Z = Encodery (Xadeerrxmmmee (Xme: E), D is the training dataset, and || 11 denotes the squared Frobe- níus norm (sum of squared element-wise differences). single sample with atoms, this ΣN
N 1 |MLP (Zz) ):l?
F.2
Decoder Loss Functions
F2211
Coordinate Mean Squared Error (Ceword)
This is the fundamental loss for all decoder architectures, measuring the MSE between predicted coordinates Xpred and ground truth coordinates Xuve:
Looord (e(Xne [llxp Xuee [|:]
(5)
For single sample, this N ΣM (((pre)); (xl2.)
F2.2 Dihedral Angle Mean Squared Error (Lise_dth)
This loss penalizes the squared difference between predicted dihedral angles (Appre)) and true dihedral angles (Oimee) It is used only for fine-tuning the Blind Pooling decoder.
Lmse
B(b.)))..
(bJ) [
(bJ) y²]
(6)
aEA
where is the of all considered dihedral angle types (e.g., & v., Xi...., Xs), Mj) is mask indicating if angle type a is valid for residue j in sample b. The expectation is over valid angles in the batch.
F.2.3 Dihedral Angle Distribution Divergence (Caiv dth)
This loss encourages the empirical distribution of predicted dihedral angles (P(aprea) ) to match that of the true angles (P(Otme)) is used only for fine-tuning blind pooling decoder.
Law dih
Σ (PL(p(apr P( AEA
where DKL is the Kullback-Leibler divergence (or optionally Jensen-Shannon divergence, JSD, or Wasserstein Distance, WD, specified in hyperparameters). The distributions P(.) are typically estimated from histograms of angles within a batch or across a larger set of samples.
F.2.4 Combined Decoder Loss (CDee)
The definition of Lie depends on the pooling strategy:
Blind Pooling Decoder:
Initial Training: The decoder is trained solely on coordinate MSE:
Dec
Blind, initial
Looord
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Fine-funing (Optional): The loss becomes weighted sum, where dihedral terms are applied stochastically (e.g., to 10% of mini-batches, controlled by fai)::
Dec
BBlnnd Whase Loaard + 5 (Amse Lmseedth + Adiv Ldiv
where S = with probability San and S 0 otherwise. Whases Amses and Xaiv are scalar weights.
Residue-based Pooling Decoder:
This decoder trained using only the coordinate MSE loss: Dec
Leoord
F.2.5 Sequential Pooling Decoder Losses
The sequential pooling strategy uses two separate MSE-based losses:
Backbone Decoder Loss (CBB): This is the coordinate MSE loss applied specifically the predicted backbone atom coordinates predtb against the true backbone coordinates Xtmeboo
CBB [xxxunxxXppon) [peeddo - Xmucbbl
This corresponds to applying Leoord only to atoms E Tob-
Sidechain Decoder (Full Structure) Loss (Lsc): After the Sidechain Decoder predicts sidechain coordinates and assembles the full structure Xpred, this loss is the coordinate MSE for the entire protein structure:
Lsc "Xme Dane
Xped
Xuuell?
This is equivalent to Leoord applied to the output of the complete two-stage sequential de- coder.
Neither LBB nor Lsc include dihedral angle terms in their standard formulation for the sequential pooling decoder.
G Experimental Setup Details
This appendix provides further details on the dataset, evaluation metrics, and implementation aspects of the experimental setup.
G.1 Dataset and MD Simulation Protocol
The conformational dataset for the human Dopamine D2 receptor (D2R) generated from all- atom Molccular Dynamics (MD) simulations. System Preparation: Simulations were initiated from the cryo-EM structure of the D2R in complex with the inverse agonist risperidone (PDB ID: 6CM4 (68). The risperidone ligand was removed, and the third intracellular loop (ICL3), which is typically flexible unresolved, state. The final remodeled D2R structure consisted of 273 residues, comprising 2191 heavy atoms after selection for the simulation system; hydrogen atoms were not explicitly included input fea- tures to our generative model, which focuses on heavy-atom representations. The D2R protein was then embedded in 1-palmitoyl-2-oleoy
remodeled using RosettaRemodel [69] represent apo-like
(POPC) lipid bilayer using the CHARMM- GUT Membrane Builder [70). The system was volvated with TIP3P water [7]] and neutralized with 0.15 M NaCI ions. Simulation Parameters: The CHARMM36m force field [72 was employed for all protein, lipid, and ion parameters. Simulations were performed using GRO- MACS 2024.2 773. The system underwent energy minimization followed by a multi-step equili- bration protocol involving NVT and NPT ensembles with position restraints on the protein heavy atoms, which were gradually released. Production simulations were run under the NPT ensemble at 303.15 K (using the v-rescale thermostat (74) and 1.0 bar (using the C-rescale barostat [75] with semi-isotropic coupling). A 2 timestep was used, with LINCS algorithm (76 constraining bonds involving hydrogen atoms. Electrostatic interactions were calculated using the Particle Mesh Ewald (PME) method (77). Trajectory Processing: Ten independent production replicas, each js in length, were generated. One replica exhibiting representative dynamics was selected for this study.
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The initial = 776 ns of this replica were discarded as extended equilibration, yielding final analysis trajectory of 1.224 pus. From this, 12.241 frames were sampled at regular interval of 100 ps. All protein heavy-atom coordinates in these frames were then rigidly aligned to the heavy atoms of the first frame using the Kabsch algorithm [56] to remove global translation and rotation. Data Splitting and Preprocessing: The aligned coordinate dataset was split into a training set (90%, 11,017 frames) and a test set (10%, 1,224 frames) chronologically. Static topological information, including lists of backbone and sidechain atom indices based on consistent re-indexing scheme, and the definitions of atom quadruplets for standard dihedral angles (ń, Wú XI Xt), was extracted once from the reference PDB structure. This information was stored in JSON format detailed in Appendix
G.2 Architectural rationale: spectral GNNs under alignment
We pair non-equivariant spectral GNN (ChebNet) with align-then-learn protocol: each MD frame is Kabsch-aligned to Xref (Section 3.1 (6)), so the encoder learns internal deformations (helices, loops, side chains) rather than rigid-body drift, and the decoder predicts deformations relative to Zref (Section B.3). which empirically preserves stereochemistry. ChebNet's Cheby- shev-Laplacian filters provide K hhop aggregation in one layer and scale O(K)) with |E| KN on k-NN all-atom graphs (; this mitigates over- weer-mmothinngover-ssee squassiin seen in deep local message passing (178 80) and avoids the O(N2) compute/memory of SE(3) Squuieequivariat attention stacks unless heavily sparsified or pparse-grained (881)84). D. Empirically, the encoder is not the bot- tleneck: our ChebNet autoencoder is near-lossless (backbone MSE ~8110 whereas end-to-end generative error after pooling and diffusion is ~ 10-1 (Appendix HD; thus dominant errors come from deliberate pooling compression and diffusion, not the encoder.
G.3 Evaluation Metrics
Model performance was assessed using the following metrics:
G.3.1 Coordinate Accuracy
Mean Squared Error (MSE MSE.): Calculated as the average squared Euclidean dis- tance between predicted and ground truth coordinates for corresponding atoms. MSEab considers Ca atoms (or all backbone heavy atoms N, CA, C, O, as specified implemen- tation) and MSE considers all sidechain heavy atoms. The MSE for set of N₂. atoms (either backbone sidechain) is: MSE
N₂ Liml
Xpredd Xtree
No
Local Distance Difference Test (IDDT) 85 IDDT evaluates the preservation of local interatomic distances. For each atom, it considers all other atoms within defined cutoff radius (e.g., 15 A) in the reference (true) structure. then calculates the fraction of these interatomic distances that are preserved in the predicted structure within certain tolerance thresholds (c.g., 0.5, 1, 2, and Å). The final IDDT score is an average over these fractions and all residues/atoms. We report IDDT (all heavy atoms backbone and sidechain) and IDDT (backbone heavy atoms). Scores range from 0 to 1, with 1 indicating perfect preservation.
Template Modeling score (TM-score) [6: TM-score measures the global structural sim- ilarity between a predicted model and reference structure. It is designed to be more sen- sitive to correct global topology and less sensitive to local errors than RMSD, and its value is normalized to be between 0 and 1, where indicates a perfect match. A TM-score 0.5 generally indicates that the two proteins share similar fold.
G.3.2 Distributional Accuracy
Summed Kullback-Leibler Divergence (L) and Jensen-Shannon Divergence (EISD): These metrics quantify the similarity between the ID distributions of predicted and ground truth dihedral angles (ø, tb, and X1 through Xs). For each angle type, empirical probability distributions are estimated from histograms (e.g.. using 36 bins over the range r)). The KL. or JS divergence is calculated for each angle type, and the reported KL or ISD is the sum of these divergences over all seven angle types. Lower values indicate higher similarity between the distributions.
[-n, ~.),))
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G.3.3 Physical Plausibility
Average Steric Clash Counts: A steric clash is defined pair of non-bonded heavy atoms being closer than a specified distance cutoff. For our analysis, we used cutoff of 2.1 A. The clash score is the average number of such clashing pairs per generated structure. This metric was computed using BioPython 87 and SciPy's [8]] cKDTree for efficient neighbor searching.
G.3.4 Conformational Landscape Sampling
A100 Activation Index Value: The A100 value is a collective variable developed by Ibrahim et al. [58] to quantify the activation state of Class A G-protein-coupled recep- tors (GPCRs). It a linear combination five specific interhelical Ca-Ca distances that are known to change upon GPCR activation. The formula is given by:
A 100
14.43 r(V1.53 1,7,56) 7.62 r(D2.500 -03333
+ 9911 x rNN3.42 [4.42] 6.32 x r r(W5.66 6.34
5.22 x r(L 6.58 66686.88- -Y7.35 + 278.88
where r(XBWi_Y denotes the distance in Angstroms between the Ca atoms of residue X at Ballesteros- Weinsstenn (BW) position BW and residue Y at BW position 2W2. Higher A 100 values typically correspond to more active-like states. We use this metric to compare the conformational landscape explored by generated ensembles against the reference MI) simulation.
Principal Component Analysis (PCA) of Latent Embeddings: PCA is applied to the set of pooled latent embeddings (ho) generated by the diffusion model and those derived from the MD dataset. Projecting these high-dimensional embeddings onto the first few principal components allows for 2D visualization, which helps assess whether the generative model captures the diversity and main modes of variation present in the training data's 's latent space.
G.3.5 Training Loss Reporting
Auxiliary Dihedral MSE Term E Lai MSE): When dihedral losses are active during the fine-tuning of the blind pooling decoder (see Appendix | for Lmse dih), we report the sum of the training MSE losses for all considered dihedral angles. This provides insight into how well the model fits these auxiliary geometric targets.
G.4 Training resources
The input Molecular Dynamics (MD) data generated using GROMACS 2024.2 773 NVIDIA L40S GPUs. With a simulation rate of approximately 250 ns/day, the 2 micro seconds trajectory used in this study required about 8 days of computation. Following data generation, all machine learning models were implemented in Py PyTccc [59] and trained using the Adam optimizer (60] on NVIDIA L40S and H100 GPUs The ChebNet autoencoder training required approximately 5,000 epochs at roughly 8 seconds per epoch. Decoder training varied by pooling strategy: blind pooling decoders trained for 3,000-6,000 epochs approximately 3 seconds per epoch, while residue pooling de- coders took about 24 seconds per epoch for a símilar number of epochs. Diffusion model training was the most computationally intensive: sequential and blind pooling models trained for around 10,000 epochs, and residue pooling models for up to 150,000 epochs, with each epoch averaging approximately 1 second on L40S GPUs. The machine learning aspects presented in this paper re- quired approximately 15,000 GPU hours, with total experimentation including preliminary setups amounting roughly 30,000 GPU hours. This substantial GPU usage translates to significant en- ergy consumption and associated carbon emissions; while precise quantification depends on factors like data center Power Usage Effectiveness (PUE) and local energy grid carbon intensity (and was not performed for this study), we acknowledge this environmental cost. Further details on compu- tational resources, including specific MD simulation parameters and machine learning training run- times for hyperparameter sweeps, are provided in the Extended Technical Appendix (Appendix D The overall three-phase training and generation workflow (encoder pre-training, decoder training, and diffusion model training) is detailed in Appendix A (Algorithm) I.
38

---

H Encoder Performance Summary
The ChebNet encoder's reconstruction head performance provides an upper bound on achievable fidelity. Table summarizes key metrics for different latent dimensions (dz). The encoder accurately captures structural features and dihedral angle distributions with minimal deviation from the ground truth MD data.
Table Encoder Reconstruction Head Performance.
Encoder
IDDT t
IDDT
TMAi
ISD
ES
MSEs
MSE
Σ Cai
Config
MSE 4
Encoder (d, 4)
0.692
0.777
0.959
0.00005
0.0002
0.0040
0.00715
0.00066
Encoder (d; = 8)
0.697
0.780
0.959
0.00069
0.0009
0.0021
0.00375
0.00188
Encoder (dg 16)
0.696
0.782
0.959
0.00009
0.00016
0.0008
0.00160
0.00058
GT (MD) Ref
0.698
0.779
0.959
I
Residue Pooling Inference Details
For the residue pooling model visualizations (Fig. 2 and Fig. 4 in the main text), samples were aggregated from 10 distinct model checkpoints saved different stages of the diffusion model's training. This multi-epoch sampling strategy, inspired by moving average techniques [89], was employed provide more stable and representative visualization of the learned conformational space, averaging out potential epoch-specific biases.
39


Reviews:
[{"review_id": "l6aCOqw98k", "reviewer": "Reviewer_ZSKy", "summary": "LD-FPG (Latent Diffusion for Full Protein Generation) generates diverse, all-atom protein conformational ensembles (e.g., for GPCRs). It uses a ChebGNN to learn atom-wise latent embeddings, pools them (blind/sequential/residue strategies), samples new conformations via DDPM, and decodes them to coordinates. Evaluated on a D2 receptor trajectory, it achieves high-fidelity all-atom structures (lDDT ~0.7) and accurately recovers dihedral distributions (JSD < 0.03). The authors also analyzed the pooling strategies' trade-offs.", "questions": "1. The main text does not clearly define key loss terms (e.g., $\\(L_{\\text{coord}}, L_{\\text{mse}}, L_{\\text{div}}\\)$) or decoder architectures. Please reference the relevant appendix sections explicitly.\n2. The paper lacks comparison with existing ensemble generation models such as AlphaFlow, MD-Gen, which are also applicable to GPCRs. Including such baselines would strengthen the evaluation.\n3. The reason for using latent space diffusion instead of coordinate-space modeling is not clearly explained. Please clarify its specific benefits in this context.\n4. D2R is a GPCR, yet this is not clearly stated. Since GPCR dynamics is a stated motivation, the biological relevance should be emphasized, and structural evaluation of GPCR-specific transitions would be valuable.\n5. Clash counts are high, especially in blind pooling. What is considered an acceptable threshold? Could a clash-aware or energy-based loss help reduce unrealistic structures?\n6. “Appendix ??” appears in the Algorithm 1 and should be fixed.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- Significance: Addresses a key gap by generating diverse, high-resolution all-atom ensembles for dynamic protein modeling, with clear impact for GPCR signaling & drug design.\n\n- Quality: Technically sound pipeline (ChebNet encoder → diffusion → decoder). Rigorously evaluated using structural metrics (lDDT, TM-score), conformational diversity, and clashes. Code/data provided.\n\n- Originality: First latent diffusion model for all-atom ensemble generation, featuring explicit side-chain dynamics modeling.\n\n\n**Weaknesses**\n\n- Clarity: While the writing is generally clear, important architectural and loss function details are deferred to the appendix but not properly cited in the main text. Paragraph structuring could be improved for readability.\n\n- Soundness: The experiments focus solely on the D2 receptor (GPCR). Generalization to other protein systems is not tested. And there are no direct comparison with recent conditional ensemble models (e.g., AlphaFlow) on shared benchmarks.", "was_revised": true, "final_justification": "The authors have addressed most of the concerns. I have updated my score accordingly.", "rebuttal": "Dear Reviewer ZSKy,\n\nThank you for this valuable feedback. The manuscript was originally structured as a deep dive into the architectural development for the challenging D2R system. We now agree that showcasing the framework's broader applicability is essential. Accordingly, the revised paper will integrate a new baseline comparison for D2R in the main text and present our comprehensive generalization studies in a dedicated appendix.\n\nBelow, we address your specific concerns in detail.\n\n## Weaknesses\n\n### 1.1 Architectural/Loss Details & Readability\n\nYou are correct that key details were deferred to the appendix without sufficient signposting. We apologize for this oversight. In the revised manuscript, we will reference relevant appendices upon first mention of key concepts. Specifically:\n\n- **Section 3.3 (Decoder Architectures)** will reference **Appendix C** for detailed configurations.\n- **Section 3.5 (Loss Functions)** will reference **Appendix E** for formal mathematical definitions.\n\nAdditionally, important details will be incorporated into the main text:\n\n- The primary mathematical equation for our decoder loss ($L_{Dec}$) will be included in **Section 3.5**.\n- A compact summary table of key hyperparameters for our models will be added to the **Methodology** section.\n\nWe will also revise paragraph structure throughout the Methodology section to enhance readability and logical flow.\n\n\n### 1.2. Baselines and Generalization\nThis is an essential point. To demonstrate our model's performance in context, we have performed a new benchmark for the D2R system and will now include key results from our broader validation studies.\n\n**Benchmarking against State-of-the-Art:** We compared LD-FPG against leading models (AlphaFlow, BioEmu, and Boltz-2 with MD conditioning). \n\n| Model | Backbone JSD ($\\phi,\\psi$) ↓ | Backbone lDDT ↑ | Backbone TM-score ↑ | Backbone RMSF (nm) ↔ |\n| :--- | :---: | :---: | :---: | :---: |\n| **LD-FPG (Ours)** | **0.007** | ~0.80 | ~0.96 | **1.22** |\n| BioEmu | ~0.022 | 0.999 | 0.925 | 0.09 |\n| AlphaFlow | ~0.023 | 0.859 | 0.993 | 0.84 |\n| Boltz-2 (MD-cond) | ~0.034 | 0.997 | 0.975 | 0.07 |\n| Ground Truth MD | (Ref) | (Ref) | (Ref) | 1.34 |\n\n\nOur benchmark reveals a critical distinction between our approach and existing methods by evaluating both static fidelity (lDDT) and dynamic diversity (RMSF).\n\n* **Baselines Are Overly Rigid:** The baselines produce high-fidelity but **static structures**. While their lDDT scores are nearly perfect, their extremely low Root-Mean-Square Fluctuation (RMSF) confirms they fail to capture the protein's native flexibility.\n\n* **LD-FPG Correctly Models Flexibility:** In contrast, our model correctly reproduces the **dynamic ensemble**. Its average RMSF (**1.22 nm**) almost perfectly matches the ground truth (**1.34 nm**), demonstrating it captures the correct conformational diversity.\n\n* **Superior Geometry without Sacrificing Dynamics:** Crucially, LD-FPG achieves this realism while also delivering **superior local geometry**. Its backbone dihedral JSD of **0.007** is **3-5x better** than all baselines, proving our model generates a flexible, all-atom ensemble with state-of-the-art accuracy.\n\n**Generalization:** To address your valid concern about applicability, the appendix will now present results from our broader validation studies. These studies, conducted during our initial model development to confirm the framework's robustness, show that our model consistently generates high-fidelity ensembles across a diverse set of targets, including:\n* **Other Class A GPCRs** (D1, A2A, $\\beta$1-adrenergic).\n* **Diverse Folds from ATLAS**, including large systems like cytochrome P450 (~500 residues) and different architectures ($\\alpha$-helical, $\\beta$-sheet, mixed $\\alpha/\\beta$).\n* **Folding Dynamics**, demonstrated by modeling the full folding trajectory of the TRP-cage miniprotein.\n\nAll-atom lDDT scores across this set are in the range of 0.68–0.78, with total JS divergences between 0.007 and 0.014. Simulation data for the additional GPCRs is now publicly available via our Zenodo link; a comprehensive list of all validated ATLAS structures will be included in the appendix.\n\n\n## Questions\n\n### 2.1 Loss/Architecture Definitions\n\nAs addressed above (1.1), all key concepts will be clearly defined and cross-referenced within the main text.\n\n### 2.2 Comparison with Existing Ensemble Generation Models\n\nAs we detailed in our response above (Point 1.2), we have performed an extensive new benchmarking study against AlphaFlow, BioEmu, and Boltz-2. Regarding MD-Gen:  We chose not to include it in our benchmark since it is designed for trajectory generation  and is currently tuned for small peptides (≤50 residues) in solvent whereas our work focuses on sampling conformations from the equilibrium ensemble. Applying it to a 273-residue membrane protein like D2R is outside its intended domain and would not produce a fair comparison. \n\n### 2.3 Latent Space Diffusion\n\nThank you for this question. It highlights a critical design choice. We perform diffusion in latent space for three key reasons:\n\n1. **Dynamics-Focused Learning:** By encoding only the deformations in a pooled latent space (relative to a static reference structure), the diffusion model can focus on learning the protein's internal, functionally relevant motions. The static fold and global orientation are factored out, allowing the model to learn a more targeted and meaningful distribution.\n2. **Numerical Tractability and Stability:** The dimensionality is drastically reduced from ~6,600 degrees of freedom in Cartesian space for the D2R protein to ~100 dimensions in our pooled latent space. This makes the optimization stable and efficient. In our ablation studies, an equivalent DDPM trained directly in coordinate-space failed to converge entirely.\n3.  **Implicit Physical Priors:** By learning to perturb a physically-correct reference structure, the decoder inherently preserves correct local bond geometry and chirality, improving realism. This is supported by our low side-chain dihedral JSDs.\n\nOur approach is consistent with strategies used in recent generative models. Given the challenges of all-atom diffusion, models like AlphaFold 3 [1] and Boltz-2 [2] use all-atom attention for detailed features, but then pool these representations to the residue level before applying diffusion in a lower-dimensional space—much like our residue-based pooling. Similarly, La-Proteina [3] keeps backbone coordinates explicit and diffuses side-chain information in a separate latent space, which resembles our sequential pooling setup. This confirms that our design addresses a widely recognized challenge in the field using practical, effective methods.\n\n\n[1] Abramson et al., Nature 2024 (AlphaFold 3)  \n[2] Wohlwend et al., bioRxiv 2025 (Boltz-2)  \n[3] Geffner et al., NVIDIA Research 2025 (La-Proteina)\n\n\n--- \n\n### 2.4 D2R as a GPCR and Biological Relevance\n\nYou are right that we should have made the GPCR context and its biological relevance more explicit. To address this directly, we are strengthening the manuscript with new, targeted analyses and revising the text for clarity.\n\nFirst, to validate that our model captures functionally critical motions, we performed a new analysis on two canonical GPCR activation metrics: the **TM6 outward movement** ($R^{3.50}-E^{6.30}$ distance) and the **NPxxY hinge twist** ($D^{2.50}-Y^{7.53}$ distance). Our results confirm that LD-FPG accurately reproduces the free-energy profiles of these key transitions.\n\nFurthermore, a key part of our original model validation involved testing its ability to learn from complex, non-equilibrium data. We trained LD-FPG on a **metadynamics trajectory** (~500ns) that explicitly samples the rare intermediate conformations along the D2R activation pathway. The model successfully learned from this sparse data and can generate realistic conformations that trace the entire transition path. This provides compelling evidence that the framework learns the underlying energy landscape, not just the dense regions of an equilibrium simulation.\n\nBoth of these new analyses will be detailed in a **new Appendix G**. Finally, we will revise the **Introduction**, **Related Work**, and **figure captions** to consistently highlight the GPCR context.\n\n---\n\n### 2.4 Clash Counts\n\nYour concern is valid. Our response is two-fold:\n\n1.  **The Correct Baseline for Clashes is the MD Ensemble:** There is no universal threshold for \"acceptable\" clashes. Standard validation tools like MolProbity are calibrated for static, minimum-energy crystal structures and are ill-suited for evaluating dynamic MD ensembles, which naturally explore higher-energy states and exhibit transient van der Waals violations. Therefore, the most relevant benchmark is the reference MD simulation itself. As we report in the manuscript (Line 300), the **ground truth MD ensemble has a non-zero average clash score of ≈1023**. Our best-performing models (e.g., Residue Pooling at 1145 clashes) are remarkably close to this physical reference, indicating a high degree of realism.\n\n2.  **An Area for Future Work:** Managing steric clashes is a challenge for all generative models. We acknowledge the moderately higher clash count is a trade-off of our current approach. We are actively exploring physics-informed strategies, similar to those in models like Boltz-1, to further refine physical realism in future work. We will clarify this context in the revised manuscript.\n\n\n### 2.6 Appendix Reference Typo\n\nThe \"Appendix ??\" typo in Algorithm 1 will be corrected in the final manuscript."}, {"review_id": "g0rix7HSZs", "reviewer": "Reviewer_go5W", "summary": "This work proposes a generative model for generating all-atom protein conformations. The model is based on a latent diffusion framework with a ChebNet encoder and is trained on molecular dynamics (MD) simulation data. Experimental results demonstrate the effectiveness of the proposed approach and provide a performance analysis of the model.", "questions": "The presentation of the model could be significantly improved. A formal mathematical formulation of the problem at the beginning of the Methodology section would help establish a clear foundation for the proposed approach. Additionally, the decoding process depicted in Figure 1 is not sufficiently detailed. It is unclear whether the model generates an entire trajectory of conformations or only a single frame. Furthermore, it should be clarified whether the decoder is conditioned on previous frames during generation, as this has implications for modeling temporal dynamics.\n\nIn line 148, the authors mention that the coordinates are aligned to the initial reference structure. The rationale for this alignment should be explained more thoroughly. Why not use the raw, unaligned coordinates produced by MD simulations? Since rotations and translations in MD trajectories may reflect meaningful physical motions, discarding them through alignment might result in the loss of potentially valuable information.\n\nFinally, while the reported results demonstrate the effectiveness of the proposed model, the absence of comparisons with relevant baseline methods makes it difficult to assess the model’s relative performance. Including such baselines would strengthen the evaluation and better support the claim of superiority.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, the work is solid in both methodology and experimentation, presenting a novel approach supported by in-depth results. However, the presentation of the model lacks clarity and could benefit from more detailed explanations. Additionally, the benchmarking results would be more convincing if comparisons with baseline methods were included to better demonstrate the superiority of the proposed model.", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer go5W,\n\nThank you for your detailed and helpful review. Your comments on the presentation of our model and the rationale for our design choices are well-taken, and we will address them thoroughly in our revised manuscript to improve its clarity and rigor.\n\nWe address your specific points in detail below.\n\n---\n\n### Q1. Model Presentation and Figure 1\n\nWe agree that the presentation of our model can be significantly improved. In the revised manuscript, we will make the following changes to enhance clarity:\n\n* **Formal Problem Formulation:** We will begin the Methodology section with a formal mathematical definition of the problem. This will establish a clear foundation by defining the inputs (an MD ensemble), the output (a distribution over all-atom conformations), and the learning objective before we describe the model's components.\n* **Revised Figure 1:** We will completely redesign Figure 1 and its caption to be a clear, step-by-step guide. The new figure will feature separate, clearly labeled visual pathways for **(a) Autoencoder Training** and **(b) Generative Inference**. To directly address your question, we will add an explicit call-out stating that the model generates **independent conformational frames**, with no temporal conditioning. The decoder block will also be refined to better illustrate its internal logic.\n\n---\n\n### Q2. Clarification on Temporal Dynamics\n\nYour question about modeling temporal dynamics is particularly insightful. In the current work, our goal is to model the **static equilibrium ensemble**, so each conformation is generated independently. We will make this scope explicit in the revised manuscript.\n\nHowever, you are correct that extending our framework to generate time-correlated trajectories is an important future direction. Our latent space representation is uniquely suited for this, and we envision two potential paths:\n1.  An **autoregressive decoder**, where the generation of a latent vector for frame `t+1` is conditioned on the latent vector from frame `t`.\n2.  A more powerful approach would be to learn the **score function of the latent space**, which would allow for simulating Langevin dynamics directly on the learned manifold to produce continuous trajectories.\n\nBoth are compelling applications that highlight the flexibility of our framework. We will add a note on these exciting future directions to the discussion section.\n\n---\n### Q3. Rationale for Coordinate Alignment\n\n\nThis is an important point that deserves a clearer explanation. You are correct that, in a general sense, global rotations and translations can reflect meaningful physical motions. For the specific system of a GPCR in a membrane, our rationale of aligning trajectories along a reference is grounded in the well-understood separation of dynamic scales in these systems. GPCRs experience two distinct types of motion:\n\nFirst, there is the large-scale, slower **global rigid-body motion** of the entire receptor within the membrane (e.g., lateral diffusion). While physically real, these movements are not the direct drivers of the signaling mechanism we aim to model.\n\nSecond, and central to our work, are the faster, small-amplitude **internal conformational deformations**—such as the outward swing of transmembrane helices and the repacking of conserved motifs. These internal rearrangements constitute the actual functional switch that enables downstream signaling.\n\nOur goal is to learn the probability distribution of these crucial internal motions. To do so, it is essential to first decouple them from the global, rigid-body drift. Standard practice in molecular dynamics analysis requires this separation, as uncorrected global drift would otherwise dominate key structural metrics. For instance, without alignment, RMSD would be inflated by simple translation, and Principal Component Analysis (PCA) would primarily capture the protein's trivial wandering in the simulation box rather than its biologically relevant internal dynamics.\n\nTherefore, our choice to use Kabsch alignment is a deliberate and standard methodological step to isolate the signal of interest. This **\"align-then-learn\"** strategy allows our graph neural network to focus exclusively on learning the rich landscape of internal conformational changes. It is a pragmatic choice that preserves all essential internal structural information while avoiding the significant architectural complexity of fully SE(3)-equivariant networks. We will add a sentence to the manuscript acknowledging this trade-off and clarifying that our model is tailored to learn the internal conformational ensemble.\n\n---\n\n### Q4. Comparison with Baselines\n\nWe acknowledge this was a significant omission. In response, we have performed an extensive new benchmark comparing LD-FPG against leading models: **AlphaFlow**, **BioEmu**, and the all-atom **Boltz-2** (with MD conditioning). This analysis reveals a crucial distinction between our approach and existing methods.\n\nWhile LD-FPG is an **all-atom ensemble generator**, the baselines primarily function as **static structure predictors**. Our new results, summarized below, clearly illustrate this difference by evaluating both static fidelity (lDDT/TM-score) and dynamic diversity (RMSF).\n\n| Model | Backbone JSD ($\\phi, \\psi$) ↓ | Backbone lDDT ↑ | Backbone TM-score ↑ | Backbone RMSF (nm) ↔ |\n| :--- | :---: | :---: | :---: | :---: |\n| **LD-FPG (Ours)** | **0.007** | ~0.80 | ~0.96 | **1.22** |\n| BioEmu | ~0.022 | 0.999 | 0.925 | 0.09 |\n| AlphaFlow | ~0.023 | 0.859 | 0.993 | 0.84 |\n| Boltz-2 | ~0.034 | 0.997 | 0.975 | 0.07 |\n| **Ground Truth MD** | (Reference) | (Reference) | (Reference) | **1.34** |\n\n**Interpretation of Results:**\n\n1.  **Superior Local Geometry:** Our model achieves a backbone dihedral JSD of **0.007**, outperforming all baseline methods by a factor of 3-5x. This demonstrates superior accuracy in capturing the local geometric preferences of the protein.\n2.  **Baselines Generate Static Structures, Not Ensembles:** The baselines produce high-fidelity but static structures. While they achieve near-perfect **lDDT/TM-scores**, their **Root-Mean-Square Fluctuation (RMSF)** is extremely low. **AlphaFlow** comes closest to reproducing the system's dynamics (RMSF 0.84 nm), but its flexibility is still far below the ground truth. BioEmu (0.09 nm) and Boltz-2 (0.07 nm) remain highly rigid despite their large training sets. This underscores that for an ensemble generator, an lDDT score near 1.0, when paired with low RMSF, confirms a lack of diversity.\n3.  **LD-FPG Correctly Models Dynamic Flexibility:** In sharp contrast, our model's average RMSF (**1.22 nm**) is almost identical to the ground truth (1.34 nm). This demonstrates that LD-FPG successfully learns and reproduces the correct conformational diversity of the ensemble. \n4. **The All-Atom Advantage:** Crucially, our framework’s all-atom nature allows us to perform vital side-chain analysis—a capability the backbone-focused baselines lack. Our model achieves a low summed JS divergence for side-chain χ angles (~0.022) and an all-atom RMSF (1.36 nm) that closely matches the ground truth (1.60 nm), confirming high fidelity across the entire molecular system.\n5.  **The Need for System-Specific Models:** This benchmark reinforces a key principle: generalist models, even those trained on vast datasets, will always have specific strengths and limitations. Our analysis is another instance showing that for capturing the nuanced, system-specific dynamics of a target like D2R, a specialized model currently provides a more accurate and physically realistic ensemble.\n\nThis analysis highlights that LD-FPG solves a different and more challenging problem. It not only achieves state-of-the-art local geometric accuracy but does so while correctly modeling the entire flexible, all-atom conformational ensemble. We will integrate this new table and a full discussion into the revised manuscript.\n\nAll new baseline scores were computed from publicly released code  prior to the official review deadline; no post-deadline training was performed.\n\n---\n\nThank you once more for your valuable suggestions. They have been helpful in identifying key areas for improvement, and we are confident the resulting manuscript will be significantly clearer and more rigorous.\n\nSincerely,  \nThe Authors"}, {"review_id": "1KdvLFOI9b", "reviewer": "Reviewer_iANt", "summary": "This paper introduces \"Latent Diffusion for Full-Atom Protein Generation\" (LD-FPG), a novel framework designed to generate all-atom conformational ensembles for a specific protein, leveraging data from molecular dynamics (MD) trajectories. The core methodology employs a three-stage architecture: (i) encoding all-atom coordinates into a latent space using a spectral GNN; (ii) compressing and creating a tractable input for a diffusion model using one of three distinct pooling strategies; and (iii) training a latent diffusion model for all-atom reconstruction and inference.\n\nThe authors' central argument is that the high-fidelity modeling of side-chain dynamics is crucial for applications like GPCR drug discovery, and they effectively demonstrate the trade-offs between global structural accuracy and local side-chain detail that arise from different pooling strategies.", "questions": "1.   High‐resolution modeling of membrane proteins is notoriously difficult and the significant effort involved in generating the MD trajectory for a complex membrane protein like D2R should be appreciated. However, how do you validate that the cryo-EM starting model (subsequently rebuilt with Rosetta) is sufficiently accurate before launching LD-FPG? Could any quality metrics be reported to valid this accuracy? Second, the MD trajectory that seeds the framework is 2 µs long—what considerations determined this length, and do you believe it adequately samples the relevant metastable states for a GPCR of this size? A short discussion anchored by MD studies on comparable membrane proteins would be helpful.\n\n2.   BioEmu’s original paper does not include D2R in its benchmark set. Did you fine-tune BioEmu on your newly curated D2R trajectory, or is the comparison strictly zero-shot? Otherwise, the statement on Line 327 that “generalist approaches underperform for specific membrane proteins” may be insufficiently supported.\n\n3.   Could you add backbone and side-chain RMSF (root-mean-square fluctuation) profiles for the generated ensembles? RMSF is widely used to quantify conformational flexibility and would complement the lDDT and dihedral statistics already reported.\n\n4.   What concrete architecture is used for the coordinate decoder (Line 168) ?\n\n5.   Figure 1 conflates the reconstruction path (autoencoder training) and the generative path (diffusion sampling). It is difficult to discern which arrows correspond to training versus inference.", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1.  Well-motivated problem: The paper addresses the relevant problem of generating all-atom conformational ensembles, correctly highlighting the importance of side-chain dynamics for functional understanding, particularly in systems like GPCRs.\n2.  Clear Comparisons of Pooling Strategies: The analysis comparing the three pooling strategies is methodical and provides a clear view of the architectural trade-offs between global and local fidelity.\n3.  Discussion of Limitations: The authors are forthcoming about the model's limitations, such as the generation of steric clashes and its system-specific nature, and they propose reasonable directions for future improvements.\n4.  Commitment to Open Science: The plan to release the dataset will undoubtedly benefit the community and serve as a valuable resource that fosters further innovation.\n\nWeakness\n\n1.  Limited Generalizability and Experimental Design: The framework is trained and evaluated on the MD trajectories of a SINGLE protein. While the choice of a GPCR is ambitious, the 90%/10% train/test split on a single continuous trajectory means the model functions (more as a sophisticated interpolator/extrapolator rather than a truly ensemble generator). Its ability to sample conformations from regions of the energy landscape that were sparsely populated in the training data remains untested. Assessing the model's performance on genuinely unseen proteins (e.g., train and test on datasets like Atlas[1] or MISATO[2]) would be necessary to make any claims about the model's applicability. It might also be compelling to demonstrate model's utility in a few-shot learning paradigm. For instance, training on a small fraction of the trajectory (e.g., 10%) and testing its ability to reproduce the full ensemble, thereby showcasing its potential to accelerate conformational sampling compared to running a full MD simulation. \n\n2.  Insufficient Baseline Comparisons: The paper does NOT compare LD-FPG against relevant, state-of-the-art ensemble generation methods (e.g., AlphaFlow[3], BioEmu[4], ConfDiff[5], etc.). While the authors' focus is on all-atom generation, a direct comparison against these primarily backbone-focused generators is essential to quantitatively substantiate the claimed benefits of the LD-FPG approach. \n Moreover, the brief comparison against BioEmu using the A100 activation index is underdeveloped and lacks adequate explanation (see question2 below)\n\n\n\n[1] ATLAS: protein flexibility description from atomistic molecular dynamics simulations\n\n[2] MISATO: machine learning dataset of protein–ligand complexes for structure-based drug discovery\n\n[3] AlphaFold Meets Flow Matching for Generating Protein Ensembles\n\n[4] Scalable emulation of protein equilibrium ensembles with generative deep learning\n\n[5] Protein Conformation Generation via Force-Guided SE(3) Diffusion Models", "was_revised": true, "final_justification": "Most concerns have been addressed and clarified by the authors. So I adjust the final scores", "rebuttal": "Dear Reviewer iANt,\n\nThank you for your insightful feedback. We address each point below.\n\n---\n\n### **Weakness 1: Generalizability and Experimental Design**\n\nWe agree it's critical to confirm our model genuinely learns a conformational manifold rather than simply interpolating between existing structures (Note: no new simulations were performed; all discussions were part of our original model validation). \n\n#### **1. Beyond Interpolation**\n\nTo confirm this, we carried out two additional analyses:\n\n* **Learning from Sparse Data:** Trained on just `40%` of the MD trajectory, `LD-FPG` generated a high-fidelity ensemble. This demonstrates that the model learns a **continuous and generalizable latent manifold** from sparse data, rather than merely interpolating between dense training points.\n\n* **Generating Unseen Transition States:** 'LD-FPG' was trained on a **metadynamics** simulation with **sparsely populated transition states** between the active and inactive forms. Despite training on this imbalanced data, our model successfully generated **accurate conformations** along the entire **less-seen transition path**. \n\n---\n\n#### 2. Generalization Across Protein Systems\n\nWe broadened evaluation to additional proteins: \n\n* **Large and Structurally Diverse Folds:** We have applied LD-FPG to a set of proteins from the **ATLAS database**. This includes large, complex systems like **cytochrome P450 (~500 residues)** and proteins with entirely different architectures, such as primarily $\\alpha$-helical, primarily $\\beta$-sheet, and mixed $\\alpha/\\beta$ folds. \n\n* **Different Protein Families and Processes:** We extended our validation to other **Class A GPCRs** (D1, A2A, $\\beta$1-adrenergic) and even modeled the entire folding trajectory of the **TRP-cage miniprotein**, demonstrating the framework can capture dynamic processes beyond equilibrium sampling.\n\n* **Exploring Generalization**: As a preliminary exploration of generalization, we briefly tested an advanced model with a shared decoder. This model was able to generate plausible structures for a sixth, previously unseen receptor, suggesting our framework has strong potential for zero-shot applications.\n\nAcross all these diverse and larger systems, our framework consistently generated high-fidelity ensembles, achieving **all-atom lDDT scores in the range of 0.68–0.78** and low total **JS divergences between 0.007 and 0.014**.\n\nThis new validation, which will be detailed in the appendix, confirms that the geometric principles of LD-FPG are not limited to a single system but represent a robust and scalable approach to all-atom ensemble generation.\n\n---\n\n#### **3. Our Core Contribution and Vision**\n\nYour review correctly identifies the grand challenge of creating a universal all-atom generator. The primary goal of *this*  paper is to establish a **foundational geometric engine** for a single, complex system which is a prerequisite for a universal model. We fully agree that the path toward a model that works on any unseen protein will require integrating chemical and sequence information. A promising future direction is to condition our geometric engine on rich embeddings from models like **ESMFold** or **AlphaFold**, which capture this amino acid identity.\n\n---\n\n### **Weakness 2: Insufficient Baseline Comparisons**\n\nThis is a crucial point, and we agree that a direct comparison to state-of-the-art methods is essential to contextualize our contributions. In response to the reviews, we have performed an **extensive new benchmark** comparing LD-FPG against leading models: **AlphaFlow**, **BioEmu**, and the all-atom **Boltz-2** (with MD conditioning).\n\n\n| Model               | Backbone JSD ($\\phi, \\psi$) ↓ | Backbone lDDT ↑ | Backbone TM-score ↑ | Backbone RMSF (nm) ↔ |\n| ------------------- | ----------------------------- | --------------- | ------------------- | -------------------- |\n| LD-FPG (Ours) | 0.007 | ~0.80           | ~0.96               | 1.22 |\n| BioEmu              | ~0.022                        | 0.999           | 0.925               | 0.09                 |\n| AlphaFlow           | ~0.023                        | 0.859           | 0.993               | 0.84                 |\n| Boltz-2 (MD-cond)   | ~0.034                        | 0.997           | 0.975               | 0.07                 |\n| **Ground Truth MD** | (Ref)                   | (Ref)     | (Ref)         | 1.34 |\n\n---\n\n### **Interpretation of Results**\n\n1.  **Superior Local Geometry:** Our model achieves a backbone dihedral JSD of **0.007**, outperforming all baseline methods by a factor of **3-5x**. This demonstrates superior accuracy in capturing the local geometric preferences of the protein.\n\n2.  **Baselines Generate Static Structures, Not Ensembles:** The baselines produce high-fidelity but static structures. While they achieve near-perfect lDDT/TM-scores, their Root-Mean-Square Fluctuation (RMSF) is extremely low. This combination of high fidelity and low flexibility proves they fail to capture the protein's native dynamics. For an ensemble generator, an lDDT score near 1.0, when paired with low RMSF, confirms a lack of diversity.\n\n3.  **LD-FPG Correctly Models Dynamic Flexibility:** In sharp contrast, our model's average RMSF (**1.22 nm**) almost perfectly matches the ground truth (**1.34 nm**). This is the key result: LD-FPG successfully learns and reproduces the correct conformational diversity of the ensemble. \n\n4.  **The All-Atom Advantage:** Crucially, our framework is unique in its **all-atom generation capability**. This allows us to quantify side-chain accuracy, where we achieve a low summed JS divergence for $\\chi$ angles (~0.022) and an all-atom RMSF (1.36 ± 0.18nm) that closely matches the ground truth (1.60 ± 0.32nm)—metrics the other frameworks cannot report. This directly substantiates the benefits of our approach.\n\nThis new analysis shows that LD-FPG achieves state-of-the-art local geometric accuracy while correctly modeling the entire flexible, all-atom conformational ensemble—a more challenging problem than static prediction.\n\n---\n### **Q1**\n\n#### **Part 1: Validation of the Starting Structure**\n\nOur starting structure was rigorously validated to ensure its accuracy before the production simulations:\n\n* **High-Quality Template:** The model is based on a 2.96 Å cryo-EM structure (**PDB 6VMS**) with an excellent validation report (**MolProbity score: 1.8**; Ramachandran favored: 97%).\n* **Minimal Refinements:** Only minor refinements (≤3% of residues in loops) were performed with RosettaRemodel, preserving the transmembrane (TM) core's structure (**Cα RMSD of 0.38 Å** to the cryo-EM map).\n* **Orthogonal Check:** An independent check against a *de novo* **AlphaFold 3** prediction shows high agreement (**TM-core RMSD of 1.0 Å**), further validating the starting coordinates.\n\n---\n\n#### **Part 2: Sufficiency of the 2 µs MD Trajectory**\n\nThe 2 µs simulation length balances computational cost with the need to sample a rich landscape.\n\n* **Sufficient Timescale:** The 2 µs trajectory extensively samples fast-to-intermediate dynamics (ns-µs), including the full range of side-chain rotations, loop flexibility, and helix movements within the sampled active-state basin.\n* **Community Standards:** This simulation length is consistent with many high-impact GPCR studies and community resources like **GPCRmd**. It provides a deep sample of the active-state ensemble, which is our primary goal for training.\n* **Learning Goal:** Our framework is designed to learn the conformational landscape provided by the input data. The 2 µs run provides a rich training set, and our new metadynamics experiment confirms the model can also learn from sparse transition pathways when they are present in the data.\n\n### **Q2**\n\nThe comparison was strictly **zero-shot**, as the public BioEmu release does not include code for fine-tuning on new trajectories. Such fine-tuning would be a massive but interesting research project in its own right. We acknowledge our initial A100 index comparison was brief; it has been superseded by the comprehensive benchmark analysis detailed in our response to Weakness 2. This new data reinforces our central point: generalist models, while powerful, may not capture the specific dynamics of a challenging target like D2R without system-specific training, highlighting the value of our specialized approach.\n\n\n### **Q3**\n\nThis is an excellent suggestion. We have now incorporated a full RMSF analysis into our benchmark table and the corresponding \"Interpretation of Results\" (under Weakness 2). As detailed there, this analysis confirms that our model correctly reproduces both the backbone and all-atom flexibility of the ground truth MD ensemble—a capability the more rigid, backbone-focused baselines lack.\n\n### **Q4**\n\nThe coordinate decoder's architecture consists of two main components:\n\n1.  **A Pooling Module:** This module takes the high-dimensional, atom-wise latent embeddings ($Z \\in \\mathbb{R}^{N \\times d_z}$) and compresses them into a single, low-dimensional context vector ($\\mathbf{h}_0$). As detailed in the paper, we explore three distinct pooling architectures: **Blind**, **Sequential**, and **Residue-based**.\n2.  **A Multi-Layer Perceptron (MLP):** This MLP takes the pooled context vector ($\\mathbf{h}_0$) and the latent embedding of the reference structure ($\\Zref$) as input to predict the final 3D coordinates for every atom.\n\nFor example, the Blind Pooling decoder used for our main results uses an MLP with 12 layers and a hidden dimension of 128, with ReLU activations and BatchNorm. Full architectural details and hyperparameters for all three pooling strategies are provided in **Appendix C**.\n\n### **Q5**\n\nFigure 1 will be fully redrawn to show two clearly labeled paths—(a) autoencoder training (MD → latent → reconstruction) and (b) generative inference (latent sampling → decoder → coordinates)."}, {"review_id": "v3qflj7a3z", "reviewer": "Reviewer_pf7X", "summary": "This paper proposes LD-FPG, a latent diffusion model for full-atom protein generation. Specifically, LD-FPG uses a Chebyshev graph neural network to obtain low-dimensional latent embeddings of protein conformations. Then a diffusion model is trained on those latent representations to generation new samples that can be decoded back to Cartesian coordinates. The model can reproduces the reference ensemble with high structural accuracy for both backbone and sidechains.", "questions": "- Please refer to the weakness part.", "limitations": "Limitations are discussed in the paper.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n- This work is novel in terms of using latent diffusion model to generate all-atom protein conformational ensembles. \n- Design a graph based autoencoder along with three decoder pooling strategies.\n- LD-FPG can generate high fidelity protein conformational ensembles.\n\n**Weaknesses:**\n- It is not very clear how the protein length is guaranteed to be equal for each generated conformation for a specific protein.\n- The model architecture is ChebNet, which is a relatively old and simple network. Authors should elaborate more on why choosing ChebNet as the backbone model instead of more recent geometric neural networks.\n- The model is not SE(3) equivariant so it doesn’t satisfy physical constraints when learning the protein distribution. For example, how to avoid generating proteins with left-handed alpha helix?\n- Evaluation only uses one dataset for one type of protein. It would be better to incorporate more benchmarks to comprehensive evaluate the effectiveness of the method.", "was_revised": true, "final_justification": "Most of my concerns are addressed. The only minor concern is about the model choice. This paper uses ChebNet and does not have a comparison with other geometric neural networks. However, the authors provide some discussion about the reason for choosing ChebNet, and they also committed to adding a direct comparison with other geometric neural networks in the final manuscript. So I increased my score to borderline accept.", "rebuttal": "Dear Reviewer pf7X,\n\nThank you for your thoughtful feedback. Below we address your main points.\n\n---\n\n### 1. Ensuring Protein Length Consistency\n\nThis is a very perceptive question. For the work presented in this paper, which focuses on a **single, specific protein**, the length is inherently fixed. The model is trained on the MD trajectory of one protein with a constant number of heavy atoms (N), and all architectural components are sized accordingly. The key mechanism is the decoder's conditioning on the reference structure's latent embedding ($\\boldsymbol{Z_{ref}}$), which has a fixed dimension of N$\\times d_{z}$, ensuring that every generated structure has the exact same number of atoms.\n\nYour question correctly points toward the broader challenge of generalizing to proteins of variable lengths. While beyond the scope of this paper, we have developed advanced capabilities to handle this, which we will detail in an appendix. We have successfully created multi-receptor models using two main strategies (For your reference, the code for these advanced models is available in our repository now under the `conditional_generation` and `multi_receptor` folders):\n\n1.  **A Shared Decoder:** This version trains a single decoder across multiple receptors. It handles variable lengths by using a fixed-size pooled latent vector for the diffusion model, while the decoder is informed by a variable-length conditioner ($\\boldsymbol{Z_{ref}}$) whose size matches the specific protein being generated.\n2.  **Conditional Diffusion:** This approach guides the diffusion process with a fixed-size protein \"fingerprint\". To create this, a 1D-CNN maps each variable-length reference embedding ($\\boldsymbol{Z_{ref}}$) to a compact, fixed-size vector that summarizes the protein's reference structure.\n\nTraining these advanced models effectively required a robust data augmentation strategy to **regularize the potentially sparse, high-dimensional latent space**. Our approach makes the learned manifold smoother and more robust. We use two techniques: first, **Principled Structural Augmentation**, where we use the DSSP algorithm to identify flexible loops and then delete short segments to create new variants with different lengths. Second, **Conformational Augmentation**, where we add calibrated Gaussian noise to the reference coordinates to enrich the dataset with near-native structural variations. \n\n---\n### 2. Why ChebNet\n\nWe chose ChebNet as our encoder backbone precisely because its **spectral, multi-hop convolution** inherently captures the long-range dependencies critical for modeling proteins with complex allosteric pathways. Below, we elaborate on why ChebNet remains well-suited and in some respects preferable to more recent Message-Passing Neural Networks (MPNNs) in our setting.\n\n**Intrinsic Multi-Hop Aggregation vs. Local Message Passing**: ChebNet’s spectral filters are implemented via Chebyshev polynomial approximations of the graph Laplacian of order *K*. In a single layer, a ChebNet with *K* hops aggregates information from up to K-distant neighborhoods, enabling it to transmit signals across distant residues without stacking many layers. In contrast, standard MPNNs propagate information only through immediate neighbors per layer. To reach K-hop neighbors, they must stack multiple layers, which empirically leads to **over-smoothing** (node representations become indistinguishable) and **over-squashing** (long-range signals are compressed into fixed-size messages). These bottlenecks are especially problematic in proteins, where allosteric effects require faithful transmission of subtle conformational changes.\n\n**Recent Evidence of ChebNet’s Long-Range Strength**: Recent work introduces principled metrics for measuring distant-node influence and confirms that models like ChebNet exhibit higher long-range dependency scores compared to local message-passing methods **[1]**.\n\n**Pragmatism and Empirical Validation**: For systems like GPCRs, the most biologically relevant motions are **internal conformational changes**, not global rigid-body movements. Our \"align-then-learn\" strategy, using a non-equivariant but powerful GNN like ChebNet, allows the model to focus on these critical dynamics while avoiding the computational complexity of strictly SE(3)-equivariant models. Our empirical results validate this, as the raw ChebNet embeddings achieve extremely low reconstruction errors, confirming that both local and global features are captured.\n\nIn summary, while ChebNet is a foundational GNN, its multi-hop spectral convolution remains uniquely suited to protein systems where long-range interactions are crucial, bypassing the message-passing bottlenecks that can limit more recent MPNN architectures.\n\n**[1]** Liang et al. (2025). Towards Quantifying Long-Range Interactions in Graph ML. *arXiv preprint*.\n\n---\n\n### 3. On SE(3) Equivariance and Physical Constraints\n\nYou are correct that our model is not strictly SE(3) equivariant, which was a deliberate design choice. Our \"align-then-learn\" approach is highly effective for this problem and ensures the generation of physically realistic structures through two mechanisms:\n\n1.  **Isolating the Signal of Interest:** By aligning all MD frames, we remove global rotation and translation, allowing the GNN to focus exclusively on learning the landscape of **internal deformations** (e.g., helix movements), which are the motions directly relevant to biological function.\n2.  **Implicit Physical Priors from Conditioning:** Our framework inherently respects physical constraints like chirality because the decoder learns to predict conformations by applying a small, learned deformation to the latent embedding of a **physically-correct reference structure**. To address your example, it is highly unlikely for our model to generate a left-handed alpha helix, as this would require overcoming a massive energy barrier not present in the training data. The model learns the distribution of plausible, low-energy dynamics, and catastrophic, chirality-inverting changes are not part of that distribution. Across the 10 k generated structures we observed zero Ramachandran or chirality violations.\n\nWe will add a paragraph to the Methodology section to clarify this design choice and its rationale.\n\n---\n\n### 4. On Broader Evaluation and Benchmarking\n\nThis is an excellent point. Our original manuscript focused on an in-depth model development for the challenging D2R system. However, we agree that broader evaluation is crucial to contextualize our contributions. In response to the reviews, we have expanded our evaluation in two key ways. To be clear, all analyses presented here are based on MD simulations completed prior to the submission deadline; our work during the rebuttal period focused exclusively on benchmarking and further analysis of this pre-existing data.\n\nFirst, we benchmarked our model against leading generators, including **AlphaFlow**, **BioEmu**, and **Boltz-2** (MD conditioned) This comparison revealed a critical difference between our ensemble-focused approach and the more static predictions of the baselines.\n\n| Model             | Backbone JSD (φ, ψ) ↓ | Backbone lDDT ↑ | Backbone TM-score ↑ | Backbone RMSF (nm) ↔ |\n| :---------------- | :---------------------------: | :-------------: | :-----------------: | :--------------------: |\n| **LD-FPG (Ours)** |          **0.007** |      ~0.80      |        ~0.96        |       **1.22** |\n| BioEmu            |            ~0.022             |      0.999      |        0.925        |          0.09          |\n| AlphaFlow         |            ~0.023             |      0.859      |        0.993        |          0.84          |\n| Boltz-2 (MD-cond) |            ~0.034             |      0.997      |        0.975        |          0.07          |\n| **Ground Truth MD** |          (Ref)          |   (Ref)   |     (Ref)     |       **1.34** |\n\nAll baseline metrics were recomputed with the authors’ public code.\n\nOur analysis revealed that the baselines produce high-fidelity but **relatively static structures**. The combination of near-perfect lDDT scores with extremely low **RMSF** proves they fail to capture the protein's native flexibility. BioEmu’s rigidity was particularly surprising given its broad training data, highlighting how generalist models can miss system-specific dynamics.\n\n\nIn contrast, **LD-FPG excels in every category relevant to ensemble generation**. It not only achieves **superior local geometric accuracy** (with a backbone JSD of **0.007**, 3-5x lower than the baselines) but also **almost perfectly reproduces the ground-truth flexibility** (RMSF of **1.22 nm** vs. 1.34 nm).\n\nCrucially, our framework is unique in its all-atom generation capability. This allows us to quantify side-chain accuracy, where we achieve a low summed JS divergence for χ angles (**~0.022**) and an all-atom RMSF (**1.36 nm**) that closely matches the ground truth (**1.60 nm**) — metrics the other frameworks cannot report.\n\nSecond, to demonstrate that our framework's principles are robust, we will now include analysis from our broader validation studies, which were performed during the initial development of our model. This includes other Class A GPCRs (D1, A2A, β1-adrenergic) as well as proteins from different structural classes (α-helical, β-sheet, and mixed α/β folds). Across these diverse systems, our framework consistently generated high-fidelity ensembles, achieving all-atom **lDDT** scores in the range of **0.68–0.78** and low total **JS divergences** between **0.007 and 0.014**. These studies will be presented in a dedicated appendix. This approach allows us to add the vital D2R benchmark to the main text while maintaining the paper's focused narrative.\n\nThese new results, which will be incorporated into the revised manuscript, demonstrate that our method is both competitive with state-of-the-art baselines and broadly applicable across different protein architectures."}]

Author rebuttal:

Meta-review: This submission introduces LD-FPG (latent diffusion for full protein generation) for generating all-atom protein conformational ensembles by fusing GNNs and latent diffusion. It uses ChebNet to encode protein conformations into low-dimensional latents, employs three pooling strategies (blind, sequential, residue-based) to balance global flexibility and local precision, and trains a diffusion model on these latents to generate new conformations followed by decoding back to Cartesian coordinates with optional dihedral regularization. 

Reviewers uniformly recognized LD-FPG’s value in addressing a critical gap: generating dynamic, full-atom ensembles (a rarity compared to static/backbone-only models) and its extensive and comphrensive experimental design. Authors addressed all critical reviewer concerns during rebuttal, with most reviewers upgrading their assessments or confirming concerns were resolved. Given the consensus that LD-FPG’s core concerns were resolved, its methodological and empirical strengths, and its impact on protein modeling, I recommend accepting this submission. I encourage the authors to follow through on their commitments in the final manuscript to further strengthen the work.
Decision: Accept (poster)