Beyond Ensembles: Simulating All-Atom Protein Dynamics in a Learned Latent Space
Authors: Aditya Sengar
Jiying Zhang
Pierre Vandergheynst
PATRICK BARTH
Venue: ICLR 2026 Poster (2026)
Keywords: representation learning
generative models
latent space dynamics
score-based generative models
dynamical systems
GNNs
autoregressive models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: To massively speed up simulations, this AI model compresses a protein's complex all-atom structure into a simple latent space, simulates its movement quickly there, and then reconstructs the full-atom trajectory.

Abstract: Simulating the long-timescale dynamics of biomolecules is a central challenge in computational science. While enhanced sampling methods can accelerate these simulations, they rely on pre-defined collective variables that are often difficult to identify, restricting their ability to model complex switching mechanisms between metastable states. A recent generative model, LD-FPG, demonstrated that this problem could be bypassed by learning to sample the static equilibrium ensemble as all-atom deformations from a reference structure, establishing a powerful method for all-atom ensemble generation. However, while this approach successfully captures a system's probable conformations, it does not model the temporal evolution between them. We introduce the Graph Latent Dynamics Propagator (GLDP), a modular component for simulating dynamics within the learned latent space of LD-FPG. We then compare three classes of propagators: (i) score-guided Langevin dynamics, (ii) Koopman-based linear operators, and (iii) autoregressive neural networks. Within a unified encoder–propagator–decoder framework, we evaluate long-horizon stability, backbone and side-chain ensemble fidelity, and temporal kinetics via TICA. Benchmarks on systems ranging from small peptides to mixed-topology proteins and large GPCRs reveal that autoregressive neural networks deliver the most robust long rollouts and coherent physical timescales; score-guided Langevin best recovers side-chain thermodynamics when the score is well learned; and Koopman provides an interpretable, lightweight baseline that tends to damp fluctuations. These results clarify the trade-offs among propagators and offer practical guidance for latent-space simulators of all-atom protein dynamics.

Published as a conference paper at ICLR 2026
BEYOND ENSEMBLES: SIMULATING ALL-ATOM PRO- TEIN DYNAMICS IN A LEARNED LATENT SPACE
Aditya Sengar sengaa..en ch
Jiying Zhang* ilying. ch
Pierre Vandergheynst pierre. v pierree.vanderrheeytteeepp.....
Patrick Barth 10 patrick.bartheee ch
ABSTRACT
Simulating the long-timescale dynamics of biomolccules is a central challenge in computational science. While enhanced sampling methods can accelerate these simulations, they rely on pre-defined collective variables that are often difficult to identify, restricting their ability to model complex switching mechanisms between metastable states. A recent generative model, LD-FPG, demonstrated that this problem could be bypassed by learning sample the static equilibrium ensem- ble as all-atom deformations from reference structure, establishing powerful method for all-atom ensemble generation. However, while this approach success- fully captures system's 's probable conformations, it does not model the temporal evolution between them. We introduce the Graph Latent Dynamies Propagator (GLDP), a modular component for simulating dynamics within the learned latent space of LD-FPG. We then compare three classes of propagators: (i) score-guided Langevin dynamics, (ii) Koopman-based linear operators, and (iii) autoregressive neural networks. Within a unified encoder-pree poanddrr-rroggaaordecooo evaluate long- horizon stability, backbone and side-chain ensemble fidelity, and temporal kinetics via TICA. Benchmarks on systems ranging from small peptides mixed-topologg proteins and large GPCRs reveal that autoregressive neural networks deliver the most robust long rollouts and coherent physical timescales; score-guided Langevin best recovers side-chain thermodynamics, when the is well learned; and Koopman provides an interpretable, lightweight baseline that tends damp fluctuations. These results clarify the trade-off's among propaga- tors and offer practical guidance for latent-space simulators of all-atom protein dynamics.
framework, we
INTRODUCTION
Molecular simulations are indispensable for studying the complex dynamics that govern biological function, yet brute-force approaches struggle access the slow, functionally relevant motions-such as protein folding, ligand binding, or allosteric switching- due to rugged energy landscapes and the dominance of rare events (Dror et al., 2011; Latorraca et al., 2017). To mitigate this gap, beyond what enhanced sampling can offer when suitable collective variables are hard to specify (Chen, 2021), complementary strategy has gained traction: recasting the simulation challenge from one of brute-force integration to one of representation learning. In this representation-fiig view, the simulation problem is recast as a modular encoder-prue propaaatoo high-dimensional atomic configurations into continuous, low-dimensional latent space; propagator evolves the system's s state within this simplified space; and a decoder maps the resulting latent trajectory back to all-atom coordinates (Sidky al., 2020; Wu et al., 2018).
pipeline: an encoder maps
Progress in this paradigm follows two complementary directions: The first focuses on learning the underlying physics, employing score-based diffusion (Song al., 2020; Ho et al., 2020), flow
Signal Processing Laboratory (LTS2), EPFL, Lausanne, Switzerland
1 Institute of Bioengineering, EPFL. Lausanne, Switzerland I Cooresponding author f "Ldwig Institute for Cancer Research, Lausanne, Switzerland

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Published as a conference paper at ICLR 2026
matching (Lipman et al., 2022), and energy-based models (LeCun al., 2006) to learn generative surrogates (sometimes yielding differentiable force fields) that implicitly define the system' 's potential of mean force (Arts et al., 2023; Plainer et al., 2025; Hsu et al., 2024; Li et al., 2024). The second centers on learning simplified dynamical coordinates, using time-aware autoencoders or Koopman/DDMD analysis (Lusch et al., 2018; Tu, 2013) discover intrinsic manifold where the long-term dynamics become stable, predictable, or even approximately linear (Hernández et al., 2018; Mardt et al., 2018: Brunton al., 2021; Azencot et al., 2020; Ghorbani et al., 2022).
Despite the promise of latent-space simulation, the choice of propagator (the engine that advances the state) entails a trade-off between physical faithfulness, long-horizon stability, and expressive capacity, We adopt LD-FPG (Sengar et al., 2025) as a fixed encoder- three propagators: (i) score-guided I Langevin dynamics, (ii) a Koopman linear operator, and (iii) an autoregressive neural network (NN). Our analysis reveals a clear trade-off: Autoregressive NN is most stable, the score-guided model most physically detailed for local motions, and the Koopman operator is simple but overly rigid baseline.
and compare, in the same latent,
GLDP is designed as a system-specific surrogate for MD, akin to LSS, GeoTDM, ITO (Sidky et al., 2020; Han et al., 2024; Schreiner et al., 2023). The objective is to learn the specific transition operator of a target system to enable accelerated sampling of its equilibrium and kinetics. While cross-protedd generalization (as pursued by foundation models) is valuable complementary goal, this work focuses on maximizing thermodynamic fidelity and long-horizon stability for specific, complex targets.
Contributions: (1) A controlled study that holds the LD-FPG encoder-decoder fixed while swap- ping the latent propagator, isolating where gains arise. (2) Three propagators within one latent: score-guided Langevin, a Koopman operator, and an autoregressive NN. (3) Benchmarks on alanine dipeptide, 7JFL, HIV-1 Protease (IR6W), and AR. (4) First learned, all-atom rollout (to our knowledge) that recovers GPCR activation surface (A2AR).
RELATED WORK
Latent simulators and mncoder-rropggton-deeeode pipelines. Simulating dynamics in low- dimensional latent space has been explored in several molecular simulators. Molecular Latent Space Simulators (LSS) (Sidky et al., 2020) and Deep Generative MSMs (DeepGenMSMM (Wu et al., 2018) already factorize the task into an encoder that maps coordinates to latent variables, propagator that advances them, and decoder that returns all-atom structures. Follow-up work uses recurrent propagators (Vlachas et al., 2021), trajectory-level generators that synthesize MD clips in coordinate space (Jing al., 2024), and LSS variants that target free-energy surfaces and transition kinetics (Dobers et al., 2023). Joint conformation-dynann dynamics models trained via autoregression have also been proposed (Shen et al., 2025). Our framework follows this modular design but differs in two respects: (i) we freeze the LD-FPG encoder-decode (Sengar al., 2025) to control reconstruction quality and (ii) we perform a focused comparison of different propagators within the same latent, isolating how the propagation rule affects stability, ensemble fidelity, and functional observables.
Learning dynamically aware latent spaces. The quality of a latent simulator is largely determined by its encoder. Classical TICA is standard tool for discovering slow coordinates (Pérez-Hernández et al., 2013), while deep time-lagged autoencoders such as VDE learn nonlinear, delay -eeeicceee representations (Hernández et al., 2018; Wehmeyer & Noé. 2018). VAMPnets approximate leading Koopman eigenfunctions to capture the slowest processes (Mardt et al., 2018), and more recent work uses GNNs to learn symmetry-awe aware collective variables directly from coordinates, sometimes with iooomatiio-bottleneee Ghorbani et al., 2022; Zhang et al., 2024: Wang & Tiwary. 2021); see Chen & Chipot (2023) for survey. In this work we instead keep the LD-FPG ChebNet encoder fixed (Sengar et al., 2025) and treat its latent as given, focusing on how different propagation mechanisms behave once a reconstruction-readd latent space has been trained.
objectives to optimize predictiveness of future states (Zou et al., 2025;
linear Koopman, neural sequence models, and stochastic dynamics. The Koopman-operator view approximates nonlinear dynamics with linear map in a learned observable space (Brunton et al., 2021), with EDMD/DMD estimating this map directly (Williams et al., 2015; Tu, 2013) and deep variants learning the observables that latent evolution is linear (Lusch et al.,
Propagators:
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2018; Azencot al., 2020; Tayal et al., 2023; Nayak al., 2025): implicit transfer-operator learn- ing further parameterizes Markov kernels with diffusion models under SE(3) symmetry (Schreiner et al., 2023). A second line directly fits nonlinear transition maps using neural sequence models, including LSTMs/RNNs for effective dynamics (Vlachas et al., 2021), RNNs with Maximum Caliher constraints (Tsai al., 2022), and continuous-time Neural ODE/SDE formulations (Chen al., 2018; Kidger et al., 2021), with ConfRover combining an autoregressive temporal module and an Squiv-qqiiaarian diffusion decoder (Shen et al., 2025). A third class uses stochastic dynamics guided by generative models: score score-based diffusion linked to Langevin-type simulators and ef- fective force fields (Song et al., 2020; Arts al., 2023; Hsu al., 2024), and flow matching for coarse-grained rollouts and free-energy modeling (Li et al., 2024; Kohler et al., 2023), often with Pookeer-Placckiinsiree objectives to align equilibrium and stationary laws (Plainer et al., 2025). Our three propagators Koopman, autoregressive NN, and score-guided Langevin, are representative of these families but are all evaluated inside a single, fixed latent space.
Coordinate-space generative simulators and MD surrogates. Many recent models operate directly Cartesian or SE())-eqiivaiian coordinate space: GeoTDM uses diffusion for trajectory synthesis (Han et al., 2024), low applies flow matching to coarse-grained rollouts (Li et al., 2024), EGNO targets multi-step 3D dynamics with equivariant neural operators (Xu et al., 2024), and NeuralMD couples gromp-symmetric differential equations with multi-resolution protein- ligand dynamics (Liu et al., 2024). Geometric diffusion models such as GeoDiff and DiffDock focus on static structures and poses (Xu et al., 2022; Corso et al., 2022), while methods like LAST, ConfRover, and NeuralMD use learned representations to steer or unify MD-like dynamics (Tian et al., 2022: Shen et al., 2025; Liu al., 2024). Our setting is closer to system-specific surrogates such as LSS, GeoTDM, and ITO (Sidky et al., 2020; Han et al., 2024; Schreiner et al., 2023): GLDP is trained per system accelerate sampling of its equilibrium and kinetics, but keeps the all-atom decoder frozen and swaps only the propagator, enabling a controlled comparison of dynamical assumptions within a fixed latent representation.
Benchmark systems for protein dynamics. We probe kinetic fidelity across a hierarchy of systems, from small peptides to complex membrane receptors. Alanine dipeptide (ADP) a canonical baseline with a well-characterized dihedral free-energy surface (Kalko et al., 1999; Vymetal & Vondrasek, 2010; Klein & Noé, 2024). At intermediate complexity, we use globular proteins 7JFL and HIV-1 protease (1R6W) from the ATLAS database, which provides standardized all-atom trajectories under common protocols (Vander Meersche et al., 2024). The most demanding tests are G protein-coupled receptors (GPCRs), specifically the adenosine A₁ and AςA receptors AR, A2AR), whose function involves rare, slow transitions between metastable states (Dror et al., 2011; Latorraca et al., 2017) mediated by well-mapped microswitches and allosteric pathways (Rasmussen et al., 2011: Weis & Kobilka, 2018). Using recent high-quality MD datasets for these GPCRs (D' Amore et al., 2024), we test whether latent propagators can match both equilibrium observables and activation free-energy surfaces when the encoder and decoder are held fixed.
METHODS
GLDP extends the LD-FPG encoder-decoder backbone (Sengar et al., 2025) (see App. A.2) by adding a temporal propagator that operates entirely within the LD-FPG latent. The encoder-decoder kept fixed (frozen) in all experiments; only the propagator is swapped.
3.1 GLDP: PROPAGATION IN THE LD-FPG LATENT SPACE
We adopt the LD-FPG encoder-decoder backbone (Sengar et al., 2025). The workflow follows a strict four-step sequence:
1. Encoding: A Chebyshev Graph Neural Network (ChebNet) maps all-atom coordinates (t) R Nx3 to per-atom embeddings.
2. Pooling: deterministic pooling layer aggregates these embeddings into a single compact latent vector 2t) E d
3. Propagation: The GLDP advances the latent state 2, - 21+1 using the learned transition function.
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High dimensionaa
Graph representntion conformers
Chetnet
(3) Coorr--uidedddee Langeiin Sampling RelAs. - Va, log
o0 10
Pooting
Example Frozein Systems
XLt
mm
d Imn & Am WaR mi CAMA
X(++
Nutoregpestivee
Gestrrateed Lesef.
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20-401
(4) Koopman Operator mode
Figure 1: Framework overview. A pre-trained LD-FPG encoder (ChebNet; left) maps all-atom coordinates (t) pooled latent ztt). Within the fixed LD-FPG latent, GL.P advances the state via one of three propagators (red box): (a) -core-guided Langevin using the LD-FPG denoiser to estimate r) V, log logpr(2 (2) at fixed low-noise level; (b) an autoregressive NN 2+11 = [ax); and (c) a Koopman linear operator 24+1 Az. The frozen LD-FPG decoder (right) maps the latent trajectory back to all-atom structures X + At).
4. Decoding: The fixed decoder maps the evolved latent # back to all-atom coordinates A( (t++1) 1).
The encoder produces time series {zo, ; that captures conformational variation. We then learn a latent-time update of the form
FM
2++1 f(2)) + Te. Te ~ ~~~N00007))
(1)
where f is the propagator and is optional rollout noise. Unless noted, training pairs use a one-frame stride (Zt, 1,) latents are standardized per dimension using training-set statistics before fitting the propagator (and unstandardized before decoding), and the initial state 2g is the encoding of held-out MD frame. Key notation appears in Table 14; glossary is in Table 15.
3.2 KOOPMAN PROPAGATOR VIA DYNAMIC MODE DECOMPOSITION
The Koopman variant approximates latent evolution with single linear operator:
2++1 A Zt;
A E R
where A is the time- indeeeeddnnnd Koopman operator that we aim to learn from data. We estimate A with Dynamic Mode Decomposition (DMD) (Tu, 2013). First, we arrange our latent trajectory into two snapshot matrices, where each column a state of the system at a point in time,
X [20. 22-
Y [21...., 2M-1],
and solve the least-squares map minA ||Y AX| P. yielding
A YX*
where X is the Moore Penroee pseudoinverse. For numerical stability and to regularize the model, the pseudoinverse is computed using Singular Value Decomposition (SVD). By truncating the SVD to a lower rank r, we can filter out noise and focus the model on the most energetically significant dynamic modes. Once learned, the operator A is used generate new trajectories autoregressively from an initial state žo: a small Gaussian rollout noise 7/t ~ (0, o /) is added to match the one-step residual variance. Implementation details are provided in App. A.3.
3.3 AUTOREGRESSIVE NEURAL NETWORK PROPAGATOR
To learn the latent transition directly, autoregressive NN. The model parameterizes deterministic nonlinear map
2++1 fo(zc),
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trained with one step objective pairs (2t, 2(+1):
M M-
1
(())
- -
M
To reduce error accumulation in long rollouts, we use dropout and weight decay (see App. A.4 for architecture and training settings). After training, trajectories are generated by iterative application from an initial state En:
411 fo( Th.
T N0 Ponl
where mill is chosen validation split match the one-step latent residual variance of 2++1 2t- (Unless noted, Eg is the encoding of held-out MD) frame and latents are standardized per dimension before fitting and unstandardized before decoding.) The network architecture, training procedure, and rollout settings are detailed in Appendix A.4.
3.4 GUIDED LANGEVIN PROPAGATOR
This propagator uses a learned score the latent equilibrium to define a stochastic simulation. The procedure has two stages.
Stage 1: score model. We train time-conditional denoiser co(=,. Ft,1 1) on latent embeddings with variance-press preeservprr (VP) schedule as DDPMs. The forward corruption is
21 = vāaza zu + alc N(0, ā III (1 B.).
s=1
When the network is trained to predict noise, the perturbed-dee score is
E0(Z,,
s(x,) = V V=logpe(z)
VI āi
To approximate the unperturbed score s()) at test time, we query low noise level Enoise 0 (we use moise =0 unless stated).
Stage 2: Langevin simulation in latent space. We integrate an overdamped Langevin SDE with Maruraaa using the score as the drift:
Z++1 = T (21) + V2 nt: ~~ ~0,,00,
(5)
The effective diffusion variance per step is 27 At ALo we set o,=1 unless otherwise noted. Equiva- lently, if p(=) x cxp(-U(==)/)) then s(2) = (2),,, so the drift corresponds the gradient of an effective energy surface in latent space.
The temperature T, step size Al, and other implementation details, such as score clipping and sampling stride are further described in Appendix A.5.
Target ensemble. Our propagators evolve only the latent coordinate (no momenta, no barostat/box dynamics). The objective is reproduce the configuration-ssuuee equilibrium of the reference MD at its thermodynamic state. Let px (x) denote the equilibrium distribution of all-atom coordinates from the source MD, and let E[: and D[.] be the fixed LD-FPG encoder and decoder. The target latent law is the pushforward pz = E Px with 2 = E(x) and decoded samples are ă = D(z). We fit propagators so that the stationary decoded trajectory (*(t) matches reference statisties in X X-space (e.g., dihedral histograms, contacts, and TM-distance free-energy surfaces). Because no Hamiltonian with momenta is integrated in latent space, we not claim NVE energy conservation or exact physical kinetics. The score-guse ccore-guided Langevin propagator targets canonical law in latent space whose invariant density is pz when the score is exact; Koopman and the autoregressive NN model approximate the one-step transition kernel K(2+41 2)) observed from (2t, 2t+1) under the fixed encoder. Unless stated otherwise, comparisons of timescales are diagnostic rather than calibrated to wall-clock time.
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Rationale for Latent Dynamics. Simulating directly in high- dithensinnsionl Cartesian space (3N) is computationally demanding and highly sensitive to error accumulation, which often violates steric constraints. GLDP circumvents this by propagating dynamics compressed manifold (d << 3.N). Because the LD-FPG latent space encodes deformations relative to physical reference structure, the dynamics are inherently centered within valid conformational basin. This effectively decouples the simulation challenge: the fixed Decoder enforces local geometric validity (bond lengths and angles), allowing the Propagator to focus exclusively on learning slow, collective motions on a smoothed energy landscape free of high-freet high-frequency noise.
4 RESULTS AND DISCUSSION
Key result. GLDP operates reliably across scales (ADP 7JFL IR6W + AR) and reproduces the actívation surface of A2AR; this cross scale pattern is the main message of the paper. See Appendix A.8 for AR ablations (encoder width, pooling size, and Langevin/denoiser settings).
We first position GLDP against recent generative simulators using GLDP-NN (GLDP with the autoregressive NN propagator\Sec.. 4.1). We then compare Koopman, NN, and Langevin within the same latent space ADP and AR for stability and ensemble fidelity (Sec. 4.2, 4.3), and finally assess all three on AzAR functional surfaces (Sec. 4.4).
4.1 COMPARATIVE BENCHMARKS ON TRAJECTORY GENERATION
To benchmark GLDP, we compared it against a diverse set of state-of-the-art models: the con- ceptually similar Latent Space Simulator (LSS) (Sidky et al., 2020), the jideo-synthesis-base MD-Gen (Jing et al., 2024), and GeoTDM (Han et al., 2024), a leading coordinate-based diffusion model. All methods were evaluated on equilibrium fidelity, flexibility, and-crrucially-temmpoo kinetics across systems of increasing complexity: the canonical alanine dipeptide (ADP), the a- helical 7JFL, the mixed a/3 HIV-1 Protease (1R6W), and the functionally complex AjAR GPCR. The quantitative results are summarized in Table 1.!
Table 1: Benchmark of dynamics models. GLDP-NN is compared against LSS, MD-Gen, and GeoTDM. We include kinetic metrics (TICA timescales and decorrelation times) to assess temporal fidelity alongside structural ensemble metrics. Lower JSD is better. Higher Contact Correlation is better. For Kinetics, values closer to the Ground Truth are better. Bold indicates best result among models. N/A: not applicable.
JSD
Kinetics (steps)
System
Model
Dihedral (BB/SC)
Coord. (BB/SC)
Contact Corr.
TICA (6)
Decorr. (Adecor))
ADP
Ground Truth
N/A
N/A
N/A
16.41
49
LSS
0.016 N/A
0.621 / 0.633
N/A
15.80
46
GeoTDM
0.123 N/A
0.444 0..95
N/A
12.20
35
GL.PP
0.061 N/A
0.366 0.386
N/A
10.14
28
7JFL
Ground Truth
660.34
2224
0.027 0.097
0.691 0.682
0.935
610.50
MD-GGe
0.035 0.090
0.377 0.361
0,960
480.20
1600
GLDPP
0.037 0.081
0.426 0.459
0.949
226.40
IR6W
Ground Truth
N/A 0.035 / 035
N/A 0.620 / 0.610
N/A
1670.11
2207
LSS
0.885
102.50
210
MD-Gen
0.050 / 0. 105
0.190 0.210
0.980
450.30
980
GLDP
0.045 / 0,092
0.220 0.185
0.972
980.10
1650
AIAR
Ground Truth
N/A 0.134/ 146
N/A 0.657 0.602
1058.85
LSS
0.971
Unstable
Unstable
MD-Gen
0.033 0.088
0.106 0.117
0.941
150
GLDP
0.019 0.067
0.063 0.087
0.985
650.20
'GeoTDM on 7JFI., 1R6W and AR exceeded 80 GB GPU memory of H100. MD-Gen does not support capped termini ADP setup: ADP results omitted.

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Ensemble Fidelity: We assess equilibrium fidelity via Jensen- Shannon divergence (JSD) over global coordinates and local dihedrals. On the small ADP system, LSS leads on backbone dihedrals, while GLDP attains the best coordinate JSD. On 7JFL, results are split, with LSS and MD-Gen per- forming well on backhone metrics, while GLDP best captures side-chain packing. The introduction of 1R6W challenges the models with mixed topology; here, LSS captures local backbone dihedrals best and MD-Gen excels in backbone coordinate placement, yet GLDP achieves the lowest side-chain divergence (JSD 0.092 vs 0.135 for LSS) and superior side-chain coordinate fidelity (0.185). On the large A, GLDP dominates, achieving the lowest JSDs for both coordinates and dihedrals, indicating the most faithful recovery of tertiary structure in large, membrane- -ound systems. While Table 1 quantifies distributional error via JSD, we provide visual comparison of the backbone and side-chain free-energy landscapes for ADP and AR against baselines (LSS, MD-Gen, GeoTDM) in App. Figure 6- 8.
Global contacts/ long-horizon stability: As a coarse proxy for structural stability, we compare the Pearson correlation between model and reference tine-averaged contact maps. On 7JFL, MD-Gen best matches the contacts (r = 0.960), followed by GLDP (r = 0.949). This trend continues for the mixed-topology 1 1R6,, where MD-Gen (r 0.980) and GLDP (r 0.972) both maintain excellent tertiary definition, whereas LSS degrades (r = 0.885), failing to preserve the 3-sheet interface. However, the most complex system, , GLDP attains the highest correlation (r 0.985), suggesting that while MD-Gen is strong on globular proteins, GLDP offers superior stability for complex transmembrane bundles.
Temporal Kinetics (TICA): To verify that models capture dynamics rather than just independent samples, we quantify the slowest TICA timescale (11) and decorrelation time (Astecom) (for more details check Appendix A.7). On simpler systems (ADP, 7JFL.), baselines like LSS effectively recover slow modes. However, complexity reveals a sharp divergence. On 1R6W, GLDP recovers a timescale of 980 steps and deep memory (faccor = 1650), significantly outperforming the faster decorrelation of MD-Gen (f 980). Most notably on AjAR, LSS exhibits numerical instability (negative eigenvalues), whereas GLDP remains physically robust (t1 2 650 steps), proving it models coherent kinetics where baselines diverge.
Summary. GLDP offers the strongest balance of structural realism and dynamic fidelity. While baselines like MD-Gen or LSS can excel on specific static metrics for simpler topologies, GLDP consistently outperforms on side-chain packing and flexibility for complex systems like IR6W and A, Crucially, the kinetic analysis confirms that GLDP is the only method stable enough to recover physically interpretable timescales on large GPCRs, positioning it as a highly dependable latent-space simulator.
4.2 LONG- HORIZON STABILITY: AUTOREGRESSII NN IS MOST ROBUST
Figure tracks 2MSDADDT over rollout length and reports the failure time (first frame with IDDT < 0.65 relative to the initial frame). This threshold was empirically calibrated: in our analysis, trajectories dropping below 0.65 consistently exhibited unphysical degradation (e.g.. secondary structure melting steric clashes), distinguishing true model failure from valid conformational drift (where IDDT typically remains > 0.70). On alanine dipeptide, the autoregressive NN remains stable for the full 10.000-frame horizon without failure. The Koopman operator persists for thousands of frames (failure at 4,443), whereas Langevin terminates early (206) due to the high sensitivity of the score estimate on this small system. On A the NN again completes the full 10,000-frame horizon without failure; Langevin reaches 7,476 and Koopman 5,740. Taken together, the NN is the most robust propagator for long-horize rollouts on both small peptide and large GPCR, with Koopman providing a competitive linear baseline and Langevin exhibiting sensitivity to score/integration settings.
What a "Trame" means for Langevin (and how we calibrated it). For Koopman and NN, one frame equals one dataset stride (2: - 2141), 1), so the horizontal axis matches the MD sampling stride. The Langevin propagator integrates latent SDE with internal step size At and separate sampling
"The local Distance Difference Test (IDDT) evaluates local structural quality by measuring the preservation of inter-atomic distances.
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(8)
Manine-Dipeptiee
(b)
A1AR
3.0
Groust
Cound
RMSD (A)
1.5-
Astongmaaia
1.0-
RMSO
2.5
0.0
10-
1.0
IDDT Score
0.8-
IDDT Score
a.6
Cround
3.4
2.4
Langeain Synamio
0.
2000
4200
5000
5020 20203
Frame
Frame
Figure Stability over long rollouts. RMSD and IDDT versus frame index for (a) alanine dipeptide and (b) A AR. We define failure time as the first frame whose IDDT (relative to the initial frame) drops below 0.65. On AR, the autoregressive NN remains stable for the entire 10,000-frame horizon (no failure), while Langevin and Koopman fail earlier; on alanine dipeptide, Koopman and NN persist for thousands of frames whereas Langevin fails early.
stride s; a plotted "frame" is sampled SDE state, single integrator step. To make curves comparable, we calibrated (At, s) to match the short-horizon RMSD-per-franee of the reference MD (Sec. A.5): for alanine, At 10- and s=1; for A AR, At 10 10 and s=20 (effective time per plotted frame = At). Reported Langevin failure indices therefore count sampled outputs.
Why Langevin fails earlier alanine. We attribute the early failure alanine three factors:
(i) a sparsely sampled latent manifold from a large MD frame stride, yielding rough score estimate s(2) = Vg log p(z); ; (ii) larger apparent per-frame displacements in tiny system (RMSD ~0.9 Å) making the IDDT threshold stringent; and (iii) step sizes were calibrated match the reference MD's RMSDD-prr-rrame For the fast-moving alanine system, this protocol necessitated a large integration step (At 10 10-5 -), which proved numerically aggressive for the Langevin integrator compared the conservative step (At = 10 10-10) derived for the slower A AR system, pushing the SDE ffff-manifodd.
4.3 FIDELITY TO THE EQUILIBRIUM ENSEMBLE
We quantify ensemble fidelity by comparing free- energy surfaces (FES) of backbone (ø, v/) and side-chain (x1, X2) dihedrals (Fig. 3). The Jensen-Shannon divergence (JSD) between the normalized model and reference densities serves as our primary error metric (lower better), with all results summarized in Table 2.
Table 2: Dihedral free-energy fidelity (JSD; lower is better). Values represent the mean + standard deviation over independent runs. Bold indicates the best mean performance.
Propagator
ADP (ø, v))
AjAR (ø, 1p)
A₁AR (X1, X2)
Koopman
0.138 0.015
0.043 0.006
0.121 + 0.012
Autoreg. NN
0.061 + 0.003
0.019 + 0.001
0.067 + 0.002
Langevin
0.164 0.021
0.052 + 0.008
0.058 + 0.009
Backbone (d) v) fidelity dominated by the Autoregressive NN. Autoregressive NN consistently outperforms the other propagators in both accuracy and stability. For alanine dipeptide, the NN achieves the lowest divergence (JSD = 0.061 + 0.003), accurately capturing the canonical Ramachandran basins with high reproducibility. In contrast, the Koopman model's 's rigidity yields less accurate landscape (JSD = 0.138) with noticeable variance across runs (+0.015). This trend is even pronounced on A,AR, where the NN produces the most faithful backbone surface (JSD = 0.019) and negligible variance (#0.001), confirming as the most robust estimator for global backbone topology.
On backbone dihedrals, the
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MI mom Crpenee
Figure 3: Ensemble fidelity in dihedral space. Free-energy maps for backbone (Ø, () and, for A AR, side-chain (X1: X2). Alamine dipeptide (left) shows canonical basins recovered by all methods; the Autoregressive NN aligns best with the reference basin shapes. For A₁ ((middle//righht the NN most closely matches backbone structure, while the score-guided coideg Langevin propagator recovers the sharpest rotamer bands for side-chains.
Side-chain (X1, X2) rotamers favor the Langevin propagator. The picture reverses for side-chain rotamers on Ai AR. Here, the score-guided Langevin propagator achieves the best mean performance (ISD = 0.058), yielding the sharpest rotamer bands. This likely reflects its thermodynamically con- sistent drift (T s()) plus isotropic noise, which excels sampling sharp, multi-modal distributions. However, this comes at the cost of higher variance (#0.009) compared to the NN (#0.002), as the stochastic nature of the integrator leads to greater fluctuation between independent trajectories. The NN remains competitive (JSD) = 0.067) and highly stable, but slightly oversmooths high-barrier regions compared the Langevin baseline.
Taken together, these results highlight three-way trade-off: the Autoregressive NN is the most stable and accurate for broad backbone dynamics; the Score-Guided Langevin propagator is superior for resolving fine-gu grained. multi- modal side-chain states albeit with higher stochastic variance; and Koopman provides lightweight but more rigid baseline with moderate variability.
4.4 GPCR FUNCTION: ACTIVATION SURFACE IN AzAR
To assess if each propagator reproduces the inactive ractive switching of AzAR, we project its dynamics onto free-energy surface defined by the TM3-6 6 and TM3-7 distances (Fig. 4). These coordinates track the outward swing of TM6 and the opening of the intracellular cavity, which are hallmarks of GPCR activation (see App. Fig. 5 for representative snapshots).
AzAR: two-dimennionall surface. The reference surface forms diagonal valley, indicating coordi- nated changes in TM3-6 and TM3-7 during activation. Langevin covers this valley most extensively, including the transition corridor connecting inactive and active-like regions. The Autoregressive NN follows the same valley with a tighter footprint, giving good alignment near the basin center but under-sampling the flanks. Koopman identifies the central basin yet shows stiffer, more isotropic contours; barriers appear sharper and the diagonal anisotropy is reduced, which limits coverage along the transition path.
One-dimensional slices. The ID free energy profiles (Fig. 4, right column) quantify these devia- tions. Along TM3-7 (top-right), Langevin and the NN track the position of the principal minimum and its curvature, with Langevin retaining broader support across the valley shoulders. Koopman exhibits a narrow well and steeper rise away from the minimum, consistent with its tendency to com- press variance on the 2D map. Along TM3-6 (bottom-right), Langevin again follows the reference profile across both flanks, while the NN stays close near the minimum but with a slightly skewed tail. Koopman produces compact basin and elevated shoulders, which reduces sampling of the transition corridor toward larger TM3-6 distances limiting its exploration of the activation corridor and showing that the non-linear propagators more effective at tracing the complete activation pathway.

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Autoregressiva
Profile:
24
18-
TND distance MA
18
F
distemme
Jahv
115
12
24-
:0
10
12
14
10-
10
13
14
10
15
18
20
TH3 distance (A)
TM3 7 distance (A)
TH3 7 distance (A)
Langevin
Koopman
18-
TN0 distance (A)
192366 siviernc
24-
10
12
10
12
14
10
10
13
TM3-7 distance (A)
TM3:7 distance (A)
TH3-6 distance (A)
Figure 4: Functional free-energy surface for A2AR. (Top Left) Reference MD free energy heatmap projected onto TM3 6 and TM3-7 distances, showing the transition valley. (Center & Bottom-Left) Generated ensembles for the Autoregressive NN (cyan), Langevin (orange), and Koopman (purple) propagators, plotted as contours over reference background (grey). (Right Column) free energy profiles projected along the TM3-7 (top) and TM3-6 (bottom) reaction coordinates.
OUTLOOK
Our comparison reveals fundamental trade-off between the stability of data-driven models and the fine-gratd finegrained physical accuracy of score-guided methods. This points toward several promising future directions. First, motivated by the complementary strengths observed in Table 2 (autoregressive NN for backhone, Langevin for side-chains), we envision hybrid propagators that leverage neural networks for the stable, long-horizon integration of slow collective motions, while employing score- guided Langevin update for local, high-frequency sampling to ensure thermodynamic correctness. Second, while our TICA analysis confirms that GLDP captures physical timescales and memory effects on complex systems where baselines fail, future work could enforce explicit kinetic constraints during training, such as spectral matching losses, further refine this fidelity. Third, addressing the need for transferability, moving beyond fixed backbone to the end-to-end training of the encoder and propagator on multi-protein datasets could discover latent representations optimized for generalizable temporal evolution, potentially bridging the gap between generalist foundation models and the high thermodynamic fidelity required for specific drug targets.
REPRODUCIBILITY STATEMENT.
We release a repository with all source code for the encoder-prov propagator-decoder pipeline, training. and evaluation, plus fixed random seeds and an environment file to pin dependencies. The main paper outlines the pipeline (Section 3) and the evaluation metrics (Section 4.3). Full implementation details appear in the Appendix: datasets and preprocessing (Appendix A), backbone configuration (Appendix A.2), propagators and hyperparameters (Appendix A.3, A.4, A.5), and ablations (Ap- pendix A.8). The repository hosts the preprocessed latent trajectories used our runs and links to the raw MD sources. To aid inspection, representative structure snapshots for ADP, 7JFL, AR, and AR are included in the Appendix (Figure 5), while short videos of the rollouts are available at the anonymous repository link.
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ACKNOWLEDGMENTS
A.S. was supported by a grant from the Center of Intelligence Systems (EPFL) to P.B. and P.V. This work was also supported by the Institute of Bioenginecring (EPFL), the Signal Processing Laboratory (LTS2, EPFL), the Ludwig Institute for Cancer Research, and the Swiss National Science Foundation (grants 31003A 182263 and 310030 208179).

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A APPENDIX
This Supplementary Information provides detailed descriptions of the models, training procedures, and evaluation metrics used in this work to ensure reproducibility.
A.1 SIMULATION DATASETS AND CODE AVAILABILITY
The complete code, containing the pipeline to train the LD-FPG model and run GLDP, is publicly available at: https: /ttt://ggthh/ com/ adi www.ttyasegg////a /LDPP The latent trajectories used to train the propagators were derived from three publicly available Molecular Dynamics (MD) simulation datasets. For each system, the original source provided a structure file (PDB) and a coordinate trajectory (XTC). These files were then processed using the LD-FPG framework's preprocessing scripts to generate the inputs for GLDP, including a condensed. json file that provides consistent, zero-based atom indexing and defines the atom quadruplets required for calculating all backbone and side-chain dihedral angles.
Alanine Dipeptide. This dataset features --acetyl-L-allninee L-aaaainee---mmetlllamieee a 22-atom molecule commonly known as alanine dipeptide. It is canonical benchmark for developing and testing new simulation methods due to its simple yet non-trivial conformational landscape, which is primarily described by its two backbone dihedral angles (ó, v²). The data was sourced from the CMB data repository at ftp. imp. . fu berrrerii.... de and consists of a 250 ns simulation trajectory with solvent molecules removed.
7JFL (ATLAS). We use entry 7jEl_C from the ATLAS database: ds i mbb inserm. fr fr/ATLAS/... TATTAA/..,//jfLLL. /7jfl_c. ATLAS provides standardized all-atom, explicit- solvent MD trajectories across proteins under common protocol.
Adenosine A₁ receptor (A,. Human A ARtrajectoriis were taken from the study (D' Amore et al., 2024). These are explicit-solvent, cembrran--mbeddde GPCR simulations as described in the paper's 's Methods/SI and associated data records. Before encoding, we remove solvent and lipids and operate at fixed box; assess configuration-space statistics after decoding. Where specific thermostat/barostat and T/P values are required, we defer to the original Methods/SI.
HIV-1 Protease (1R6W). To evaluate performance on mixed secondary structures (a-helices and 8-sheets), we utilized the HIV-1 Protease system (PDB: IR6W). We sourced the MD trajectory from the ATLAS databse dsimb. inserm. fr/ATLAS/ /1.
Adenosine Aςλ receptor (AzAR). A/ARtrajectories come from the (D' Amore et al., 2024). As above, simulations are explicit solvent with a membrane environment; solvent and lipids are stripped before encoding. We match configuration-space statistics to the reference ensemble and refer to the paper's Methods/SI for the exact simulation state (e.g., NVT/NPPT temperature, pressure, ions, water model, force field).
A.2 LD-FPG BASED
BACKBONE
GLDP follows an prodeer-roppgatorr decoder blueprint. We use fixed, pre-trained encoder-decoder from the LD-FPG model (Sengar et al., 2025) and focus on comparing propagators within its learned latent space.
The latent space for this study was generated using the original LD-FPG framework. The core components are:
Encoder: layer Chebyshev Graph Neural Network ('HNO' model) that maps all-atom coordinates X (t) E Nx3 to per-atom embeddings Z(t) E Nxd.
model performs pooling on the per- atom embeddings to generate a single latent vector 2(t) e d for the entire conformation. The decoder portion of this model maps this latent vector back all-atom coordinates.
Pooling and Decoder: A PrateinStateRcccosstutcttoo
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The script "chebet plind.yy" was used to train this autoencoder and export the time-series of latent vectors (Zo; .1..... .M.)) into an HDFS file Statent_reps/ppottt embedding.h5) which serves as the ground-trute trunt trajectory for training the propagators. Key hyperparameters for the backbone are detailed Table 3.
Table 3: Configuration of the Pre-trained LD-FPG Backbone Model. Note: Pooling is performed immediately after the graph convolutions to form the latent space.
Component
Parameter Value
Encoder & Pooling (Latent Construction)
Graph Type
Chebyshev GNN (HNO)
Hidden Dimension (dg)
16
4
Graph Construction
k-Nearest Neighbors (k = 4)
Pooling Type
Blind
Pooling Output Size LLatent Bottleneck)
100
Decoder (Reconstructionn
MLP Hidden Layers
12
MLP Hidden Dimension
128
Encoder (HNO). HNO denotes a Chebyshev graph neural operator: a ChebNet with polynomial filters of order K on the molecular graph (built by &NN).
Pooling. "Blind pooling" maps per-atom embeddings Ztt Nxxx to fixed H W latent by learned MLP that aggregates across atoms without explicit residue topology; the decoder inverts this map to all-atom coordinates.
A.3 KOOPMAN ATERATOR IMPLEMENTATION DETAILS
The Koopman operator A was computed using the _it_dmddtt function, which implements the standard Dynamic Mode Decomposition (DMD) algorithm. The procedure summarized in Algorithm This method provides numerically robust way to find the best-fit fit linear operator A by leveraging the Singular Value Decomposition (SVD) compute the pseudoinverse the snapshot matrix X.
Algorithm 1 Computing the Koopman operator with (truncated) DMD
Require: Latent snapshots {ex ) M-1 rank cap (optional), tolerance = (optional).
1: Construct X [2p. ..., ..,............ 2M Y [Z), (,,...,22-11))
2: Compute SVD: X = UEVT
3: (Optional) choose rank by energy cap; set Ur, Σ,, V, to the top-r factors.
4: Form Σ by inverting only singular values > e (set others to zero)
5: Compute X + V.
6: Set A + YX+
Ensure: Linear propagator A (use 2++1 Azt, optionally with small Gaussian rollout noise).
In implementation, the SVD truncation rank can be set hyperparameter to de-noise the dynamics. no rank is specified, the full rank of the matrix is used.
A.3.1 KOOPMAN PROPAGATOR HYPERPARAMETERS
The SVD truncation rank (r) and the standard deviation of the Gaussian rollout noise (Troop) are key hyperparameters. They were selected independently for each system to ensure stable, horg-horizon rollouts, with the final values reported in Table 4.
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Table 4: Koopman model hyperparameters used for each molecular system.
System
SVD Truncation Rank (r)
Rollout Noise Std. (7/koo)
AJ
10
2.00 10
Alanine Dipeptide
10
2.60 x 10-3
A.4 AUTOREGRESSIVE NEURAL NETWORK DETAILS
The neural network propagator fa is implemented Multi-Layer Perceptron using PyTorch, as detailed the neural ptwpaaaapppaaar...._a py script. The training and rollout procedures are outlined in Algorithm 2, and the specific hyperparameters on a fixed percentage of the latent trajectory, with the remainder used for validation to select the best-performing model checkpoint.
summarized in Table 5. The model is trained
Table 5: Hyperparameters for the Autoregressive NN Propagator.
Parameter
Value Description
Architecture
Model Type
MLP (PropagatorMLP)
Input Dimension (dnat)
100
Hidden Dimension
512
Hidden Layers
2
Activation Function
ReLU
Dropout Rate
0.1
Training
Loss Function
Mean Squared Error (MSE)
Optimizer
Adam
Learning Rate
10-4
Batch Size
128
Epochs
400
Frame Skip (n in z(+n)
Training Fraction
0.8 (80% train, 20% validation)
Weight Decay
x 10-4
Algorithm Autoregressive NN Training and Rollout
Require: Latent trajectory { model fe. learning rate
1: Training
2: Create dataset of pairs D {(2,, =(4)) M-2
3: Initialize model parameters 0.
4: for each epoch
5: for each batch (z,, 1++1) in D do
6: Compute loss: L + | 1l
7: Update parameters via backpropagation: 0 - 0 aVoL.
8: end for
9: end for
10:
// Rollout
12: Initialize trajectory: 60 +- 20. for = O,..., M 2 do
14: 2141 fo(2s) Mt,
nh A((0, o I).
end for
Ensure: Trained model fo. generated trajectory {
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We choose roll validation split match the one-step latent variance of 2141 24-
The standard deviation of the rollout noise (Till) was calibrated for each system by matching the one-step latent residual variance validation described in the main text. The specific values used for each simulation are listed Table
Table 6: cestem-specifie rollout noise for the Autoregressive NN propagator.
System
Rollout Noise Std. (Croll)
AjAR
1.00 10-3
Alanine Dipeptide
1.00 x 10-2
7JFL
1.50 x 10-3
A.5 SCORE-GUIDER LANGEVIN PROPAGATOR DETAILS
The implementation of the Score-Guided Langevin Propagator consists of two main stages.
A.5.1 DENOISER MODEL TRAINING
First, time-conditional denoising model, cp(2,, T). is trained on the static set of latent embeddings from the MD trajectory. This model learns to predict the noise that was added to "clean" sample zo to produce a noisy sample 2, diffusion step r. The training objective is to minimize the Mean Squared Error between the predicted noise and the true noise: L = Et.zo.elf l|cc co 20(v,,20 VI a.c, 1 The key hyperparameters for this training stage are listed in Table 7.
Table 7: Hyperparameters for Denoiser Model Training.
Parameter
Value Description
Denoiser Model
Model Type
Time-conditioaal MLP (DiffusionMLP_We
Input Dimension (dam)
100
Hidden Dimension
1024
Diffusion Schedule
Diffusion Steps (Tar)
Range explored: 450 1600
Beta Schedule
Linear
Range explored: 5 x 10- 6 0.005
Bend
Range explored: 0.02 0.11
Training
Optimizer
Adam
Learning Rate
10-5 64
Batch Size
Epochs
50,000
A.5.2 LANGEVIN SIMULATION
We approximate the score of the unperturbed data distribution by evaluating the model at low, fixed noise level, Tnoise (e.g., T = 0). The simulation then proceeds according to Algorithm with simulation-specifee hyperparameters listed in Table 8.
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Table 8: Hyperparameters for Langevin Simulation.
Parameter
Value / Description
Integration Time Step (At)
Typical range: 10 10 (arbitrary units)
Temperature (I)
Typical range: 0.001 - 1.0 (arbitrary units)
Noise Level Timestep (Tnoise)
0 (Evaluates score lowest noise)
Score Clipping Norm
Optional max norm to prevent instability
Algorithm 3 GoireGGided Langevin Simulation
Require: Pre-trained denoiser 66, initial state 20. time step Al, temperature T, simulation steps Nsteps-
1: Fix a low noise level for score evaluation: + Tnoise
2: Look up noise standard deviation for dy.
3: Initialize trajectory: 2 t { }
4: for O..................... Nateps do
5: // 1. Compute Score
6: Predict noise the current state zr: Epred & ca(zt,
7: Calculate the score: 8(2)) -pred/ /r
8:
(Optional) Clip score: [|((2)]] > 8maz, then s(2)) t s(2)) smax/ /|]s(2))]
9:
10:
2. Update State
11:
Calculate drift: Acann < TAt s(Ze)-
12:
Calculate diffusion: A=ditt 2TT (NO,
13:
Update state: 2t+1 + 24 + Azani + Azdift
Append 2++1 to Z.
15:
end for
Ensure: Generated trajectory 2.
As noted in the main text, key parameters such the temperature (7"), time step (At), and sampling stride (s) were calibrated for each system to ensure physically realistic and stable trajectories. The specific values used for each protein simulation provided in Table 9.
Table 9: System-specific simulation parameters for the Score-Guided Langevin propagator. Tempera- ture (T) and time step (At) are in latent units.
System
Temperature (T)
Time Step (At)
Total SDE Steps
Sampling Stride (s)
Alanine Dipeptide
5,0
1.0 x 10
10,000
1
AJAR
0.8
1.0 10-10
100,000
20
AzAR
0.8
1.0 10-10
100,000
20
Implementation and Calibration Details. each step of the Langevin simulation, we evaluate
the score s(z)) by calling the denoiser 60(*,, (z, (noise) and dividing by /T Oft To prevent numerical instability from rare large updates, we clip the score's 1.2-norm to a maximum value of e = 10. We optionally subsample the simulated path by emitting stride) to control the apparent fer-frame displacement. The temperature T and step size At are critical hyperparameters, which we calibrate for each system to match the short-horizon decoded RMSD-per -ees--pe--rame and rollout settings, including the diffusion schedule and optimizer, are also detailed in this section.
decoded frame every SDE steps (sampling
frame of the reference MD trajectory (as described in Section 4.2). Additional training
A.6 CONTACT MAP PEARSON CORRELATION
Given topology P and trajectory X {a.)²= we build Ca-based contact map as follows. For each frame extract Ca coordinates and compute the pairwise Fuclidean distance matrix ())) (all
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residues). Average over frames to obtain D = T Lt=1
ΣE D then threshold fixed cutoff 5 (8 Å
by default) form binary contact map C' E {0, 1
Cj = D < 6]. CR = 0 Vi.
We construct Cref and ('mod with the same cutoff and residue selection. The reported score is the Pearson correlation coefficient between the flattened matrices:
vec vec(Cmod))
=
Implementation notes. The procedure uses MDAnalysis for I/O and atom selection (name CA); dis- tances are computed via pdist st and squa ar squareform form, then averaged over frames prior to thresholding. No exclusion of near-sequence pairs jl 3) is applied in this metric.
A.7 METHODDLLOGY FOR KINETIC COMPARISON
To quantitatively evaluate the kinetic properties of the generated molecular dynamics trajectories against the ground truth simulations, we employed a two-stage analysis based on Time-lagged Independent Component Analysis (TICA) and Autocorrelation Functions (ACF). This approach allows us to identify the slowest collective motions within the system and measure their characteristic timescales.
A.7.1 STEP I: FEATURIZATION
The high-dimensional trajectory data was first transformed into feature space suitable for kinetic analysis. We used the pairwise distances between all C-alpha atoms as our features. This choice of internal coordinates ensures that the analysis is invariant to rotational and translational motions of the protein as whole. For protein with N C-alpha atoms, this results in feature vector a<t) E (NN-)/2 for each time frame
A.7.2 STEP 2: TIME-LAGGED INDEPENDENT COMPONENT ANALYSIS (TICA)
TICA is powerful dimensionality reduction technique that finds the slowest, most dynamically relevant degrees of freedom from a molecular dynamics simulation. The objective of TICA is to find linear combination of imput features that maximizes the autocorrelation of the resulting components at a specified lag time, т. This is achieved by solving the generalized eigenvalue problem:
C(T)u, = A,C(0)u;
(6)
where:
C(0) the standard covariance matrix of the feature vector (1):
C(0) = E [[=(t) (=((tt) 14)
(7)
C(r) is the time-lagged covariance matrix at a lag time T:
C(r) = (x(t) - ()(=tt+))+++++)))
(8)
/ [())(() the mean of the feature vector.
X; are the eigenvalues, which represent the autocorrelation of the i-th component at lag time
U; are the eigenvectors, which are the TICA components (TICs) or collective coordinates.
The TICA model was fit exclusively on the ground truth trajectory data to define the principal slow kinetic modes of the system.
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A.7.3 STEP 3: CALCULATION OF IMPLIED TIMESCALES
The eigenvalues A obtained from the TICA decomposition are directly related to the characteristic "implied timescale" of each component. This timescale represents the decorrelation time of the corresponding TIC. The relationship is given by:
Ad
The largest eigenvalue, Å1, corresponds to the slowest process in the system, and its associated implied timescale, f1, provides quantitative measure of the duration of this dominant kinetic event. We report this slowest timescale as a key metric for kinetic validity.
A.7.4 STEP 4: PROJECTION ONTO THE SLOWEST MODE
Once the TICA model was trained on the ground truth data, both the ground truth and the generated trajectories were projected onto the first (slowest) independent component, TIC1 (corresponding to eigenvector v1). This projection reduces the high-dimensional conformational dynamics to one-dimensional time series, z()), that captures the system's 's evolution along its most significant slow coordinate.
=(t)= of (x(() 1))
(10)
A.7.5 STEP 5: AUTOCORRELATION AND DECORRELATION TIME
To provide more direct comparison of the system's "memory" along the slowest mode, computed the normalized autocorrelation function (ACF) of the time series z(t) for both trajectories. The ACF time lag At is defined as:
A(Ar) [[(((1) (=((1(++ At) - -2]]
(11)
o²
where the mean and o is the variance of the time series z(t). The decorrelation time, ddecorrs is formally defined as the time lag at which the ACF decays to 1/e.
A(taecorr)
0.368
(12)
This value quantifies how long takes for the system's 's conformation along TIC1 become statisti- cally independent of its initial state. A close match in the decorrelation times between the ground truth and the generated trajectory indicates high kinetic fidelity.
A.8 ABLATION STUDIES ON AJAR
We probe three levers that materially affect AIAR: (i) encoder per-atom width dz. (ii) pooled latent size prior to decoding (with fixed W-2 so 2H), and (iii) Langevin/denoiser settings. We report (i) training- time losses at the level where the lever acts, and (ii) downstream accuracy via backbone/side-chann
ARMSF (mean absolute error, Å) and IDDT failure time (first frame with IDDT< 0.65; larger is better). For long-rollout statistics we keep the autoregressive NN and Koopman unchanged and vary only the component under study.
Conventions. For Langevin rollouts, the sampling stride is s=1 (one SDE step per frame). We choose At by matching the average per-frame RMSD of the decoded trajectory to the MD reference over K=100 100 frames (Sec. 4.2). Unless stated: denoiser query 7=0, drift clipping ||s())]l & c=10. 10.
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A.8.1 (1) ENCODER PER-ATOM WIDTH d, {8, 16,
Increasing de lowers the autoencoder validation MSE expected, and modestly improves down- stream flexibility metrics, with diminishing returns beyond d=-16.
d₂
Enc. MSE
ARMSFBB (À)
ARMSFsc (Å)
IDDT failure (frames)
0.020
0.34
0.54
NN 10,000
Lang 6,800
Koop 5,200
16
0.006
0.26
0.39
NN 10,000
Lang 7,400
Koop 5,600
32
0.003
0.24
0.37
NN > 10,000 Lang 7,400
Koop 5,600
AAAR, criterion IDDT 0.65; Lang uses s= At=2x10 (calibrated), r=0, clip c=10.
We select d, =16 in the main runs for good trade-off between training cost and downstream accuracy.
A.8.2 (II) POOLING WIDTH И WITH W=2 (so 2
Larger pooled latents reduce decoder MSE, but very large daa slightly degrades flexibility//staiiig likely by making the latent stiffer to traverse with simple propagators.
H (daa)
Dec. MSE 4
ARMSFBB (Å)
ARMSF'c (A)
IDDT fail (Lang)
IDDT fail (Koop)
20 (40)
0.43
0.31
0.49
7,100
5,400
50 (100)
0.24
0.26
0.39
7,400
5,600
80 (160)
0.21
0.30
0.46
7,000
5,500
We therefore adopt H=50 (i.c., 100==10) for all main results.
A.8.3 (III) DENOISER/LANGEVV SETTINGS
We ablate the diffusion schedule for the denoiser (which controls score quality), the score query level the clipping threshold, and the SDE step size At (with stride fixed to 1). Trends:
Diffusion schedule. Linear (Asunt Send Tant) of (5x10-6, 0.03, 1400) reached denoiser MSE 0.015 (best); a looser schedule (5x10 15 0.11, 450) gave 0.050 (worst). Better denoisers reduce ARMSFsc by x 0.03 Å.
Score query level. r=0 outperformed small nonzero level (r=20) on side-chains: ARMSFsc=0039 A(r=0) vs 0.42 A(r=20); backbone was unchanged within 0.01 Å.
Clipping. Max-norm clip at c=10 stabilized occasional spikes without biasing small updates; removing clip reduced the Langevin failure time from 7,400 to 6,600 frames; c-5 was slightly over-damped (failure 7, 7,200).
Step size. With s=1, selecting At by RMSD matching (Sec. 4.2) gave At-2x 10-9 for A₁ AR. Varying AL by x2 changed ARMSF values by S 0..2 Á but affected failure time (=200-300 frames).
Summary. Bigger d, and pooled width reduce reconstruction losses as expected, but overly large pooled latents slightly worsen flexibility and stability. A well-trained denoiser with 7-0, clip c=10, and At calibrated by per-frame RMSD matching yields the best side-chain fidelity and stable long rollouts on AjAR
A.9 ABLATION STUDIES ON DATA EFFICIENCY AND TEMPORAL RESOLUTION
To assess the robustness of the Graph Latent Dynamics Propagator (GLDP) beyond the standard benchmark settings, performed targeted ablations on the 7JFL system. Specifically, investigate the dependence of the Autoregressive NN propagator on training data volume, the impact increasing the temporal prediction stride (frame skipping), and the benefit of residual parameterization.
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A.9.1 DEPENDENCE ON TRAJECTORY LENGTH (DATA EFFICIENCY)
We trained the Autoregressive NN propagator on subsets of the full 7JFL MD trajectory, ranging from 5% to 100% of the available frames. We evaluated structural fidelity via Backbone Dihedral JSD and kinetic fidelity defined as the mean squared enror between the predicted latent state &++1 and the ground truth 24+1 on a held-out test set, to quantify the local predictive accuracy of the propagator.
slowest TICA timescale (l1). We also report the Validation MSE,
As shown in Table 10, the model exhibits sharp performance threshold. At extremely low data regimes (5%), the propagator fails to learn the underlying physics, resulting in a collapse of the slow dynamics (f1 z 22) and poor structural recovery. While structural equilibrium (ISD) begins to stabilize around 25% data usage.
Table 10: Data Efficiency 7JFL (Autoregressive NN). Structural and kinetic metrics function
of training data fraction. (Mean of 3 runs).
Data Fraction
Validation MSE (10-²)
Backhone JSD (4)
TICA i (steps)
5%
25.5
0.145
22.1
10%
7.2
0.065
95,4
25%
4.8
0,043
188.2
50%
4.1
0.039
215.6
75%
3.8
0.038
223.8
100% (Main Paper)
3.7
0.037
226.4
A.9.2 IMPACT OF FRAME SKIPPING (TEMPORAL STRIDE)
Standard training minimizes the one-step error 4 + E++1. To test the model's capability to model coarser timescales, trained separate models to predict 2; Ze+ for stride k E (I,...,10;
Table 11 reports the validation MSE and the TICA timescale (41 simulation steps. compare kinetics fairly, we calculate the "Implied Physical Time" (t, x k). We observe systematic reduction in physical memory as the stride increases. For = to & = 3, the physical timescale remains robust (<: 200+ units). However, at larger strides (k- 10), the implied physical time drops significantly. This indicates that predicting over large temporal gaps may cause the network to "smooth out" energy barriers, leading accelerated transition kinetics and loss of long-term memory.
Table 11: Effeec of Frame Skipping on 7JFL. Validation MSE and Kinetic Fidelity. Note that f is reported in simulation steps; Implied Physical Time is I k.
Stride (k)
Validation MSE (10-9)
TICA i (steps)
Implied Phys.
Time (t: x k)
1 (Baseline)
3.7
226.4
226.4
2
3.9
108.5
217.0
3
4.2
68.2
204.6
4
4.8
46.1
184.4
5
6.5
30.5
152.5
10
15.2
8.4
84.0
A.9.3 RESIDUAL VS. DIRECT PREDICTION
Finally, addressed the architectural choice of predicting the next state directly, 2441 fo(2t) (as used the main paper), versus predicting the residual, 2t+1 = 2t + fzx)..
As detailed in Table 12, both architectures converge nearly identical performance. The residual formulation achieves a marginally lower validation MSE (3.6 vs 3.7), but the resulting physical kinetics (TICA f1) are statistically indistinguishable (228 vs 226). This confirms that while residual learning a valid optimization, the Direct Prediction formulation used in our main results is sufficient capture the transition dynamics accurately.
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Table 12: Architectural Comparison on 7JFL. Validation error and kinetic fidelity.
Architecture
Validation MSE (10-+)
TICA t (steps)
Direct Prediction (Main Paper)
3.7
226.4
Residual Prediction
3.6
228.1
A.9.4 COMPUTATIONAL EFFICIENCY LATENT PROPAGATORS
To quantify the utility of each propagator for panghhoizonn simulations, we benchmarked the wall- clock time required generate trajectories of varying lengths. This analysis was performed for the 7JFL protein system on a single NVIDIA L40S GPU. The results clarify the trade-off between physical accuracy and raw speed.
Table 13: Wall-Clock Time for Trajectory Generation on 7JFL. (L.40S GPU)
Number of Steps
Koopman (Average)
Autoregressive NN Score-Guided Langevin
1,000
0.15s
2.62s
2.17s
10,000
0) 1.19s
3.42s
4.03s
100,000
0.59s
12.53s
23.88s
A.10
ACKNOWLEDGEMENT OF L.LM USAGE
During the preparation of this work, we utilized Large Language Models (LLMs) to assist with literature discovery. information retrieval, and the initial drafting of the manuscript. The authors reviewed, edited, and validated all LLa--ssiited text and take full responsibility for the accuracy, originality, and integrity of the final content.
A.11 SUPPLEMENTARY FIGURES
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Figure 5: Representative structural snapshots from latent rollouts. Top: Manine-dipepticn conformers sampled from a Koopman rollout show backbone dihedral changes across frames. Bottom: A Assapphotts from an Langevin rollout; the dashed circle highlights the intracellular end of TM6, which moves outward over time-a a hallmark of GPCR activation.
A. 12 SUPPLEMENTARY TABLES
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Ground Truth
ii) LSS
180
90
30.32
26.95
-90
Angle (degrees)
20.21
Free Energy gy (k//mol)
iii) GeoTDM
iv) Autoregressive NN
180
16.84
13.47
90
10.11
6,77
3..37
-90
-180
-180
-90
90
180-180
-90
90
180
Angle (degrees)
Figure 6: Alanine Dipeptide Backbone Free Energy. Comparison of the (Ø, v) free energy surface for (i) Ground Truth MD, (ii) LSS, (iii) GeoTDM, and (iv) GLDP (Autoregressive NN).
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i) Ground Truth
ii) LSS
180
90
18.95
16.84
14.74
Angle (degrees)
Free Energy (k//mol)
12.63
-180
iii) MDGen
iv) Autoregressive NN
180
10.53
8.42
6.32
4.21
2.11
0.00
-180
160-160
180
Angle (degrees)
Figure 7: A,RR Backbone Free Energy. Comparison of aggregated backbone (ø, v) distributions. LSS (ii) shows noisy, high-entropy landscape. GLDP (iv) and MD-Gen (iii) successfully recover the tight o-helical basins observed in the Ground Truth (i).
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i) Ground Truth
ii) LSS
180
90
18.95
16.84
-90
14.74
Angle (degrees)
Free Energy (kJ/mol)
12.63
-180
iii) MDGEN
iii) Autoregressive NN
180
22111
-90-
-90
0.00
-180
188
-180
90
-180
-90
90
180
X1 Angle (degrees)
Figure 8: AjAR Side-chain Free Energy. Comparison of aggregated side-chain (X1, X2) distributions. The ability to capture rotameric packing critical for stability. LSS (ii) fails resolve clear rotamer wells. GLDP (iv) accurately reproduces the multi-modal rotameric density seen in the Ground Truth (i).
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Table 14: Summary of notation used in the description of the propagator models.
Symbol
Description
A E IRd
Latent space vector time L.
X, Y (M
Snapshot matrices with columns as time: X [2o. ................................ ZM 2M-2), Y [21, 1....... 2M-1]- Koopman linear operator.
A E Rdddd
fo
Autoregressive NN with parameters
p(2)
Equilibrium probability distribution of the latent variable Score function the equilibrium distribution.
a(2) = V= log)(=)
E0(Z)TT)
Pre-trained LD-FPG diffusion model (denoiser). Noise schedule standard deviation from the diffusion model at step т.
Tr
At
Integration time step for Langevin dynamics. Temperature parameter for Langevin dynamics.
Stochastic noise term, 17: (O, T) (scale on specified per experiment). Per-atom encoder embedding width (ChebNet channels).
d,
daat
Flattened pooled latent dimension after H W pooling (here 50 x 2 = dnat = 100).
d
Shorthand for daa (the flattened pooled latent dimension).
Table 15: Glossary of Biophysical Terms. Definitions of key concepts from molecular dynamics and structural biology used in this work, tailored for a machine learning audience. Term
Description for ML Audience
Collective Variable (CV)
A low-dimensional function of atomic coordinates (e.g., distance or angle) designed to capture specific, slow dynamic process like protein folding. Tradi- tional simulation methods often require pre-defining good CVs; our work uses learned latent space to discover them automatically.
Potential of Mean Force (PMF)
Essentially, effective "energy landscape" for molecule salution. Lower values correspond more probable (stable) conformations. the target distribution that score-guided Langevin dynamics aims to sample from, often visualized as a "free-energy surface".
Dihedral Angles (ó, x)
Rotational angles around covalent bonds that define the geometry of a molecule.
(phi), y (psi): Define the rotation of the protein backbone.
(chi): Define the rotation of the amino acid side-chains.
Their statistical distributions
sensitive measure of structural fidelity.
Ramachandran Plot
A 2D plot of the backbone dihedral angles (ò, v). Certain regions of this plot are "allowed" hased on steric constraints, leading to characteristic high-probability basins that correspond to stable secondary structures like alpha-helica, and beta- sheets.
Rotamer
discrete, low-energy, and therefore highly prohable conformation of protein's side-chain, defined hy its of x angles. key test for generative models whether they can reproduce the correct statistical distribution of these rotameric states.
GPCR
G protein-coupled receptor. A large and important family of transmembrane proteins that act as cellular signal transducers. They are highly dynamic and switch hetween different functional states (e.g. inactive, active), making them an ideal and challenging test system for dynamic models.
IDDT
local Distance Difference Test. metric for assessing the quality a protein structure prediction by evaluating how well local inter-atomic distances are preserved relative to reference structure. Unlike RMSD, less sensitive to global rotations and more focused on local geometric accuracy.
RMSF
Moot-MMenn-Soe Square Flactuation. For each atom, this metric calculates the standard deviation of its position over time in a simulation trajectory. It measures the "amplitude of motion" or flexibility of different parts of the protein.
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Reviews:
[{"review_id": "vEyZk5k1Kb", "reviewer": "Reviewer_zVyy", "summary": "This paper investigates generating protein trajectories in the latent space of pretrained encoder-decoder models, rather than directly in Cartesian coordinates. The authors present GLDP, a plug-in module for the graph-based conformation generation model LD-FPG, enabling trajectory propagation in latent conformational space, followed by decoding to atomic coordinates. \nWithin this framework, the paper systematically compares three latent-space propagation strategies: Score-based Langevin dynamics (similar to Two-for-One, https://arxiv.org/abs/2302.00600); Koopman operator-based linear propagation, and Neural network-based nonlinear propagation, for autoregressive generation.\nThese approaches are evaluated on three protein systems of varying sizes. The neural autoregressive propagator is found to be the most stable and best at capturing ensemble-level statistics, while Langevin dynamics can perform better in recovering in side-chain torsional distributions.\nOverall, the work offers an interesting perspective on biomolecular dynamics by exploring trajectory generation in latent space and systematically comparing reasonable propagation strategies.", "questions": "[Model]\n\n1. What data are used to train the LD-FPG encoder–decoder? Is single encoder/decoder modules shared model used for different proteins, or separately for each protein system?\n2. Why choose LD-FPG (ChebNet + MLP) instead of more modern transformer-based architectures (e.g., as in AlphaFold3, https://www.nature.com/articles/s41586-024-07487-w)? This design also requires frame alignment and offset prediction, which limits transferability.\n3. The use of \"pooling’\" and \"decoder\" commonly appear together (e.g., in Table 3). Is my understanding right that pooling happens after encoding and before propagation, and is not part of the decoder?\n4. For score-based Langevin dynamics, score estimation near $t\\approx 0$ is known to be unstable due to very low noise level in the denominator. Is this a problem in practice?\n5. For baseline models (e.g., MD Gen), were their pretrained weights used, or were all models retrained for each system?\n\n[Data]\n\n6. How are the trajectory splits defined for training, validation, and testing? Has any time-based or conformation-based split applied? Are models always trained and evaluated on the same protein system?\n7. 7JFL_C is a small and fully helical protein (47 residues). Have larger or systems with other secondary structures (e.g., from ATLAS) been tested?\n\n[Results]\n\n8. Is lDDT alone sufficient to assess long-horizon physical stability, given it does not capture energetics or steric quality? The choice of failure threshold (lDDT < 0.65) also appears arbitrary - how was it determined?\n9. Figure 2a shows stable rollouts beyond 10,000 steps for the autoregressive model, but line 357 states failure at 3,176 steps. \n10. The claim that Langevin fails earlier on alanine due to step sizes tuned for GPCRs seems inconsistent with the earlier statement that step sizes are tuned per system. Can you clarify?\n11. In Table 2 (A1AR side-chain torsions), results are shown for only one protein and from single test, and differences between AR and Langevin are small. Are these statistically significant?\n12. For the A2AR case study, how does this system differ from A1AR, and what are the main takeaways?\n13. How does the runtime of GLDP compare to classical MD simulations? It seems that this method requires system-specific MD trajectories exist for training before it can generate new trajectories - would it become a problem in practical use?\n\n[Other comments]\n\nThe related work section is somewhat long and loosely organized. GPCR datasets appear alongside general method discussions, while other relevant datasets are not covered. This section could be better organized by theme and shortened.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Exploring protein dynamics in latent space as a potential way to accelerate MD simulations is an interesting direction, and this work provides a controlled comparison of three propagation strategies.\nThis study evaluates methods across protein systems of different sizes, offering some insight into applicability across system sizes", "weaknesses": "1. The idea of modeling dynamics in latent space is interesting, but the overall architecture (e.g., LD-FPG encoder-decoder) feels dated, and the evaluation is limited to three systems in non-transferable settings.\n\n2. Some evaluation choices are not fully convincing, and it is unclear whether certain results are statistically significant or lead to conclusive insights on this latent dynamic problem.\n\n3. The paper would benefit from stronger organization, clearer presentation of results, and inclusion of key experimental details that are currently missing. \n\nSee Questions for details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "j4zR39fF4L", "reviewer": "Reviewer_XsXh", "summary": "The paper proposes the Graph Latent Dynamic Propagator (GLDP), an approach for modeling molecular dynamics in the latent space of an encoder–decoder framework (LD-FPG). The encoder and decoder are kept fixed, while latent-space propagators are trained on the temporal sequences of latent representations. Three types of propagators are proposed: score-based Langevin dynamics, Koopman-based linear operators, and autoregressive neural networks.\n\nTo evaluate the flexibility and distributional fidelity of the dynamics generated by GLDP, several metrics are computed—such as the Jensen–Shannon Divergence (JSD) with respect to the ground-truth ensemble and the average RMSF across the sequence—and compared with baseline approaches including LSS, GeoTDM, and MD-Gen. The results indicate superior performance in recovering the ground-truth distribution and achieving flexibility values closer to the reference.\n\nIn the long-horizon modeling scenario, the three propagators are compared, with the autoregressive neural network demonstrating the greatest stability. Free-energy surfaces (FES) are computed in the space of two variables to measure fidelity to the equilibrium ensemble. Finally, GLDP is shown to successfully reproduce the inactive-to-active transition of $\\mathrm{A}_{2A}$R, where the score-guided Langevin dynamics covers most of the FES valley as well as the corridor connecting the inactive and active regions.", "questions": "1. There are space formatting issues in lines 260, 264, and 265.\n2. There are also space formatting issues in lines 60 and 61.\n3. In Figure 4, the corridor and the regions representing inactive and active states are not clearly visible.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper conducts a thorough evaluation across multiple dimensions, including quantitative metrics for stability, flexibility, and distributional fidelity. It also verifies that GLDP recovers the two metastable states of $\\mathrm{A}_{2A}$R (active and inactive), demonstrating consistency with real biological processes.\n2. The proposed method is encoder/decoder agnostic, as the encoder and decoder remain frozen. This design makes the framework easily adaptable to different latent spaces.\n3. The paper is clearly written and well presented.", "weaknesses": "Overall, this is a solid paper with well-designed experiments and sound conclusions. However, it could be further improved in the following aspects:\n1. The evaluation is conducted on only three proteins. Although these systems cover increasing complexity from ADP to A1AR GPCR, experiments on additional systems would strengthen the conclusions regarding the relative performance of the propagators.\n2. It would be interesting to examine whether other baselines can also recover the active–inactive transition of $\\mathrm{A}_{2A}$R.\n3. In addition to performance metrics, it would be valuable to include efficiency comparisons between different propagators, which are particularly important for long-horizon molecular dynamics.\n4. The necessity or advantage of modeling dynamics in latent space, rather than Cartesian space, is not clearly articulated in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hauLaSXliY", "reviewer": "Reviewer_NVTg", "summary": "This paper presents Graph Latent Dynamics Propagator (GLDP), a modular framework for simulating all-atom protein dynamics from learned space from LP-FPG. It compares three latent-space propagators: (i) score-guided Langevin dynamics, (ii) linear Koopman operators, and (iii) a nonlinear autoregressive neural network. Latent propagators are evaluated across several systems, from alanine dipeptide to A2AR, showing the trade-offs of propagators.", "questions": "1. Residual prediction with an autoregressive neural network\n\nJust a suggestion, perhaps learning the $f_\\theta(z_t)$ to approximate $z_{t+1} - z_{t}$ would improve the performance even more.\n\n2. Long horizon stability (section 4.2)\n\nI am a little confused about the conclusion for section 4.2. I understand the task, but since molecular dynamics trajectories contain randomness from the Brownian motion, maybe a propagator that did understand the molecular dynamics could result in sending the structure totally different from the ground truth data? Does 1DDT threshold include distinct local energy minima?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "1. Systematic comparison of latent propagator (quality)\n\nThe latent propagators are compared under a shared encoder, decoder, and latent space. \n\n2. Clearly written and organized (clarity)", "weaknesses": "1. Detailed comparison with baselines\n\nThe authors only report the JSD of dihedral and coordinates to the ground truth. As in Figure 2 of MDGen, it would be more convincing to also show each dihedral distribution against the baselines.\n\n2. Validness of decoded molecules\n\nI could not find any content on the validity of the decoded full atom resolution. For the 3th and 7th molecule in Figure 5 (when going from left to right and up to down), the structure seems a bit odd. Simply plotting the energy distribution would make the paper more convincing. Additionally, in minor, plotting more qualitative plots in Figure 5 for each latent propagator and molecule, would also be a good case to see whether the latent dynamics succeed.\n\nMinor\n\n- Line 316 - Full results on Pearson correlation is missing, a wrap table for it would be good\n- Figure 3 - the ordering or propagators is different for the left one\n- Figure 4 - four trajectories with the background downgrades visibility a bit, plotting four plots separately seems also good. Also, the visibility of  inactive and active site are hindered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FDQx7MKPIf", "reviewer": "Reviewer_5p11", "summary": "The paper introduces Graph Latent Dynamics Propagator (GLDP), a framework for simulating protein dynamics in the learned latent space of LD-FPG. The approach uses a frozen encoder-decoder from LD-FPG and compares three propagator classes: (i) score-guided Langevin dynamics, (ii) Koopman-based linear operators, and (iii) autoregressive neural networks. The work evaluates these propagators on systems of increasing complexity (alanine dipeptide, 7JFL, A1AR, A2AR)", "questions": "- Even training and testing on the same system, the model is trained on long simulation trajectory. But that does not make this model useful if it always needs a long simulation over a certain time scale to be able to forecast well. The authors should investigate how the model performance depends on the training trajectory length\n- Regarding Langevin dynamics, as the authors mention that the score at t near 0 is used. Notice that the denominator is close to 0, then numerically there could be instability, which might explain the dynamics instability", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "- The paper is well written, the concept and results are well presented\n- The paper proposes novel method which uses LD-FPG for encoding and decoding, and performs simulation in the latent space\n- Different methods of propagator strategies are compared\n- The authors have performed benchmarks on different scales of systems", "weaknesses": "Generalization:\n- The model is trained only on one time interval (frame stride), there's no evidence of generalizing to different time intervals\n- All the models are trained and tested on the same system\n- From Table 4,6,9, it seems that different systems use different hyperparameters. It's then questionable how to select proper hyperparameters when having a new system.\n\nUnclear Advantage Over Equilibrium Models:\n- All evaluations (RMSF, dihedrals, FES) measure equilibrium properties, not dynamics.\n- No transition timescales, autocorrelation functions, or mean first passage times.\n- The benchmarks and applications make the reviewer think that time-independent samplers such as AlphaFlow, BioEmu, might achieve same results without modeling dynamics.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)