Simultaneous Modeling of Protein Conformation and Dynamics via Autoregression
Authors: Yuning Shen
Lihao Wang
Huizhuo Yuan
Yan Wang
Bangji Yang
Quanquan Gu
Venue: NeurIPS 2025 poster ()
Keywords: Protein Conformation
Generative Modeling
Molecular Dynamics
Diffusion Models
Language Models
Primary area: machine_learning_for_sciences
TL;DR: An autoregressive model that simultaneously learns both the conformations and dynamics of proteins from molecular dynamics data

Abstract: Understanding protein dynamics is critical for elucidating their biological functions. 
The increasing availability of molecular dynamics (MD) data enables the training of deep generative models to efficiently explore the conformational space of proteins.
However, existing approaches either fail to explicitly capture the temporal dependencies between conformations or do not support direct generation of time-independent samples.
To address these limitations, we introduce *ConfRover*, an autoregressive model that simultaneously learns protein conformation and dynamics from MD trajectory data, supporting both time-dependent and time-independent sampling.
At the core of our model is a modular architecture comprising: (i) an *encoding layer*, adapted from protein folding models, that embeds protein-specific information and conformation at each time frame into a latent space; (ii) a *temporal module*, a sequence model that captures conformational dynamics across frames; and (iii) an SE(3) diffusion model as the *structure decoder*, generating conformations in continuous space.
Experiments on ATLAS, a large-scale protein MD dataset of diverse structures, demonstrate the effectiveness of our model in learning conformational dynamics and supporting a wide range of downstream tasks. 
*ConfRover* is the first model to sample both protein conformations and trajectories within a single framework, offering a novel and flexible approach for learning from protein MD data.

Simultaneous Modeling of Protein Conformation and Dynamics via Autoregression
Yuning Shen ', LihaO Wang
Huizhuo Yuan', Yan Wang"t,
Bangji Yang
Quanquan Gu
ByteDancc Seed
School of Mathematical Sciences, Tongji University
3 Department of Automation, Tsinghua University
[yuning. www.auuu...uuuuubaedaaatt.t...e. gul.@bytednnce.cooom com
Abstract
Understanding protein dynamics is critical for elucidating their biological func- tions. The increasing availability of molecular dynamics (MD) data enables the training of deep generative models to efficiently explore the conformational space proteins. However, existing approaches either fail to explicitly capture the temporal dependencies between conformations ation of -nme-independent samples. To address these limitations, we introduce CONEROVER, autoregressive model that simultancously learns protein con- formation and dynamics from MD trajectories, supporting both time-dependent and time-independent sampling. At the core of our model is modular archi- tecture comprising: (i) an encoding layer, adapted from protein folding models, that embeds protein-specific information and conformation at each time frame into a latent space; (ii) temporal module, a sequence model that captures con- formational dynamics across frames; and (iii) SE(3) diffusion model the structure decoder, generating conformations in continuous space. Experiments on ATLAS, large-scale protein MD dataset of diverse structures, demonstrate the effectiveness of our model in learning conformational dynamics and supporting wide range of downstream tasks. CONFROVER is the first model to sample both protein conformations and trajectories within single framework, offering a novel and flexible approach for learning from protein MD data. Project website: https://bytteaance-ee seed. github. io/ConfRover
do not support direct gener-
Introduction
Proteins flexible molecules that adopt multiple structures, called conformations. Their ability to transition between different conformations enables biological processes critical to life. Characterizing the behavior of a protein, including its (I) dynamic motions, (2) conformational distribution, and (3) transitions between different states, is crucial for understanding its function and guiding the design of novel proteins [5] 34). Molecular dynamics (MD) simulations are the "gold standard" for studying protein conformational changes 88 21 These simulations use physical models describe the energy of a protein conformation and the forces acting its atoms. By simulating atomic motion through classical mechanics and iteratively sampling conformations over time, MD enables researchers explore different conformations, approximating the conformational distribution at equilibrium, and gain mechanistic insights in protein functions. However, MD simulations are both computationally expensive and technically challenging due to long simulation times and the tendency to become trapped in local energy minima.
"Equal contribution. Wook done during their internship Byte *Corresponding Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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A) Conformation generation tasks
B) Autoregressive generation p(x' (x=)
Trajectory simulation
(x!
II) Time indeeeddeeeeedttt sampling
s(x e))
Enncade
Diffusion
Decaada
MI Conformetion Interpolation
by
hf
Figure I: Key ideas of CONFROVER. (A) Conformation generation tasks with various conditioning configurations. Each block denotes a frame and arrows indicates the sequential dependencies among frames from autoregressive formulation. Initial conditioning frames are outlined in black. In conformation interpolation, the last frame is repositioned and prepended to the first frame for proper sequential dependencies. (B) CONFROVER models each frame as conditional distribution given preceding frames. Sequential dependencies are captured through latent variables h, and conformations are sampled from a diffusion decoder, conditioned on the updated latent.
These challenges have motivated the use of deep generative models to study proteins, leveraging the rich conformational and dynamic information provided by large-scale MD datasets [8] 43
(1) Generating the dynamic motions of proteins is a direct analog to MD simulation. Pioneering works modeled this by learning transition probabilities of future conformations from the current However, MD trajectories are often Non-Markovian due to partially observed coordinates (e.g.. protein-only atoms) and environmental coupling (e.g.. using Langevin thermostats). To mitigate this, Cheng et al. (7] incorporated higher-order information using multiple context frames, though this requires fix context windows and limits flexibility. Jing et al. [19] instead modeled the joint distribution over the entire trajectory, capturing complex dependencies among frames. Due to training on fixed-length trajectories and the non-autoregressive design, their model has limited inference-time time flexibility that cannot generate variable-length trajectories. Li et al. (27) introduced an autoregressive approach for flexible trajectory extension, but its deterministic formulation cannot capture trajectory distributions or generate diverse samples.
state 99. 12 38
(2) Learning the conformational distribution enables sampling time-independent conformations. Several methods [8] [5 335 44 49) train diffusion- or flow-based generative models on conformation ensembles from MD simulation data, bypassing the need for sequential sampling. While effective for generating samples in parallel, they disregard temporal information in MD trajectories and therefore cannot simulate physical motion of proteins.
(3) Conformation interpolation generate transition pathways between different states. Recent works [1] 19 have extended generative modeling to conformation interpolation, where the goal is to generate plausible intermediate samples between known start and end conformational states. Jing et al. [19] framed interpolation as a conditional trajectory generation task, but it requires training task-specific model and has not been evaluated on large proteins.
As these generative problems all stem from the same underlying physical principles and involve sampling from a protein's 's conformational space, a natural question arises: can we develop a general framework to learn all of these objectives?
We present CONEROVER, framework for simultancous learning protein conformation distribution and dynamics from MD trajectory data (Figure Our key observation that, for an MD trajectory xI=L of length L, by adopting a general autoregressive formulation, p(x==2) = I p(x' |x ), where x denotes all preceding frames of x in the sequence, we can unify multiple generation objectives instances of frame generation: (I) generating future frame conditioned on all previous frames, suitable for simulating non-Markovian dynamics; (2) unconditional single-frame generation, p(x(2), corresponding to time-independent conformation sampling; (3) flexible frame sequence ordering redefines the dependency structure, enabling tasks such as conformation interpolation.
Our contributions are summarized as follows:
We introduce a simple yet general framework to learn both the conformational distribution and dynamics from MD data, supporting multiple generation tasks including trajectory simulation, indepedent conformation sampling. and conformation interpolation.
We design modular architecture that captures temporal dependencies in latent space using efficient causal transformers (i.e., Ilama (42). and directly models conformations in continuous SE(3) space using diffusion decoder, avoiding discretization structure into tokens.

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Experiments show strong capabilities of CONFROVER: it outperforms MDGEN 19 in trajectory simulation, matches the performance of ALPHAFLOW and CONFDIFF 44 in time-dependent generation, and can effectively sample conformations interpolating two endpoints.
2 Background
2.1 Data Generation from Molecular Dynamies
Molecular dynamics describes the motion of molecules through Newtonian mechanics Mx = (V)(),, where x denotes coordinates of atoms in the system, M is the atomic mass, U(x) is the potential energy of the configuration and (VU(x) represents the forces acting on atoms. In practice, stochastic and frictional forces are integrated model energy exchange with the environment and maintain temperature control of the system, converting the equations of motion to Langevin process:
MX -VU(x) MM /2M*kgTn()),
where 7 is the friction coefficient, kB is the Boltzmann constant, T is the temperature, and (t( is a Gaussian noise term delta-correlated in time (m:(4)n)(()) = (1 - 1²). Sampling from this stochastic process generates time evolution of system configurations. Over time, the samples converges to the Boltzmann distribution p(x) x
The trajectory of protein coordinates, x1:L (xptot= x prott ..... ) is extracted and saved at prescrihed simulation intervals, providing both distributional and kinetic information in the protein conformational dynamics. For simplicity, omit the subscript 'prot' and use x denote protein coordinates throughout the paper.
2.2 Protein Representations
Proteins chain-like molecules composed of amino acid residues, each selected from 20 standard amino acid types. We parameterize the coordinates of heavy atoms in a protein using the SE(3)- torsional convention [20]: the backbone atoms (N-Ca-C) of each residue define local coordinate via a Gram-Schmidt process, referred to as a rigid. The position and orientation of each rigid relative the global coordinate system are described by tramslation-rotation transformation in SE(3) space. The backbone conformation of a protein with N residues can then be represented as (T, R) E SE(3)", where T E IR Nx3 and R. E SO(3) are the translation and rotation components. The coordinates of the oxygen atom of the backbone and the side chain atoms can be determined with the addition of up to 7 torsional angles (ø, 15, w2, Xi..... Xa) describing the bond rotation. Therefore, the complete configuration of a protein structure parameterized in the space: (T. R, s), Xi..... Xa) E (SE(3) x T
2.3
Siff)-Diftusion Models for Protein Conformation Generation
Diffusion generative models are capable of learning complex data distributions. Training involves progressively corrupting data with noise and learning to reverse this process through denoising, thereby modeling the original data distribution [14 39]. Recently, diffusion models operating in SE(3) space have been proposed to model protein backbone structures 44 48). Below, we provide a brief overview of diffusion model and defer the detailed SE(3) formulation to Appendix
Given protein backbone conformation X (To, Ro) SE(3) N, and conditioned the protein identity (omitted in the equations for clarity), we aim to train a neural network to jointly estimate the score functions of the reverse-time marginal distributions at varying diffusion time t, s0(x,, t) x Vx, log (xx)). This model is trained using the denoising score matching (DSM) loss:
CISM Exo x [A(t) (0)))sss(x)) (x), - Vx, log Pojo 2
Here (x Xg)) is the forward transition kernel defined in the SE(3) space, X, (T,, R.) is the noisy data at time I, and X(() is time-dependent weight. During inference, DPM generates clean conformations from random noise by simulating the reverse diffusion process with the learned score network s9(x,,t). t).
3

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3 CONFROVER
3.1 Modeling MD Trajectories through Autoregression
Autoregressive generative models factorize the distribution of a sequence as series of conditional generations over frames. We cast this idea to MD trajectories, modeling sequence of I. frames
p(x 1: (P)
p(x' x P)) II t=1
where x is the preceding frames and P denotes the protein-specific conditioning input.
Despite its simplicity, this formulation naturally supports multiple learning objectives in protein conformation modeling. In its base form, it models temporal dependencies among frames by learning generate the trajectory. When L
removes the frame context and reduces single-frame distribution p(x(P). learning to direct sample üime-independers conformations. In addition. the sequential dependency Equation (I) can be extended to any desired cradiiccnd generation tasks. By prepending conditioning frames K. to the sequence, we train the model to learn any conditional generation p(x!: IK, P). including conformation interpolation by setting K. = {xx,x" x²). A similar idea was applied in text infilling tasks by shuffling the order of text contexts
After defining the main learning objectives for trajectory simulation, single-frame (time-independdnn)) and conformation interpolation, we describe how to effectively model autoregressive dependencies over protein conformations in Section 3.2 More critically, in Section 3.3] we explain how adapt sequence models, traditionally designed for discrete tokens, to the continuous space of protein conformations. Lastly, we introduce a specific choice of architectures CONFROVER in Section 3.4
3.2 Latent Causal Modeling
We propose modular design composed of an encoder, a latent sequence model, and a stochastic decoder. This design enables the use of modern causal transformers. such as in Llama 422. to efficiently capture sequential dependencies between frames in the latent space (Figure 2 During training, the input sequence is shifted by one frame and mask token "[M]" is prepended; the generation process also begins with the mask token and conditioning frames (c.g., x
To model p(x'|x=1, P), the context frames x are first encoded into intermediate latent states h = th.,.... h1-1) using a shared encoder network with protein-specific condition P:
[M]
X
X²
NW
h = f (x', P),
1,2....,
(2)
A temporal module is then used to capture the sequential dependen- cies between the frame latents. Causal attention is used to ensure the latent for frame h' is only updated by its preceding frames:
b Pudaated
_ (h" h² (n?,
Figure 2: Causal sequence
Since both the encoder and the temporal module are chosen be deterministic, p(x|xx*) reduces to a conditional generation over the updated latent, realized through probabilistic decoder:
model to generate trajectory (x² (x?....) from the mask to- ken "[M]" and the condition- ing frame x! Each frame only attend to its previous frames. Attention activations for x are highlighted in orange.
x P) p4 (x' uppdated)
(4)
Details on this latent modeling are provided in Appendix C As result, we train model E, by jointly optimizing the three modules fene promp, and pde..
This setup easily accommodates learning single-frame distribution: by replacing all input frames with mask token and using identity attentions, where each frame only attends itself, we effectively disable inter-frame information flow. This trains the model to directly sample conformations. A similar strategy has been used in image-video training 16 29
3.3 Training Autoregressive Model with SE(3) Diffusion Loss
The main challenge in applying autoregressive modeling to conformation trajectories lies in rep- resenting the continuous distribution of protein conformations within a framework typically used

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A) Encoding Layer
Trajectory
Updatdd here
Proten andedding
Faree If (Kad)
Cenporaa (pplake
FoldiagNodaie (hooin))
Ptrastural
Mayy rudd (RC Tr
Derciand dgie (, Pr)
shuuuuro
Prane inpet stucae
encendonn
Secoder
Ditu
Figure 3: Architecture overview. (A) Encoding Layer embeds protein sequence and input structure to each frame as a frame latent representation h', comprised of single and pair embeddings; (B) The Trajectory Module then updates frame latent h using interleaved structural and temporal update blocks; (C) A diffusion-based Structure Decoder learns to denoise noisy conformations conditioned on the updated frame latent h'; during inference, it samples conformations from the prior distribution. See Appendix D.I for details.
for discrete token sequences. While some studies have approached this by discretizing the protein structural space into discrete tokens 13 30 31,, such approaches inherently suffer from discretization error, which can lead suboptimal performance in modeling protein conformations.
Instead, we propose to directly model the continuous conformational space using diffusion proba- bilistic models and employ the DSM loss for autoregressive model training. similar to Li et al. (26 Specifically, we perform DSM loss training in SE(3) space.
Given clean frame x (T Ry), its latent embedding with temporal context h (omit subscript "update" for clarity), the forward transition kernels o (TT, ) and (R, [R]) for the translation and rotation component of SE(3), and a score network to jointly estimate the translation and rotation
scores sø( (T), h and (R, h' f), the loss is defined as
DSM = E ((t)lls h(, Tay log log
+ E (0(I) (R,, h., VR log Pejo (R(
(5)
where the expectation is taken over the diffusion time i. and noisy structure x = (T). R4) sampled from the forward process. Gradients with respect to h' are then backpropagated to update the weights in the temporal module ft and encoder fare. During inference, we decode each frame autoregressively by performing reverse sampling as in Equation (6). replacing the scores with estimated values from sT(),, h t) and (R, h tt).
3.4 Model Architecture
An overview of model architecture shown in Figure 3 with detailed illustrations of each module provided in Appendix D.I
Encoding Layer. A FoldingModule, parameterized by pretrained OPENFOLD model (2), extracts protein-specte embeddings P consisting of a single representation (s) and a pair representation (z), shared across frames. For each frame, FrameEncoder, adapted from the template module used in prior works [88 20), encodes pairwise distance of pseudo-Cs, atoms via triangular updates and merges this frame pair representations zfra with the protein pair representation z. The resulting frame latent embedding. h' js', 2'], ]. is invariant to global translation and rotation of the conformations, and are passed to the Trajectory Module. Following causal sequence modeling, a masked frame token "[M]" is introduced by zeroing out the pseudo-C, pairwise distances to remove structural information.
Trajectory Module. The Trajectory Module models structural and temporal dependencies across frames, updating each frame's 's embedding based on its preceding frames. consists of interleaved StructuralUpdate and TemporalUpdat layers that operate on the frame-wise latent embeddings. Structural aupddte incorporates Puirformer layers, a core architecture in protein structure modeling, to update single and pair embeddings through triangular operations (2). TemporalUpdate employs a Llama-based causal transformer layer for channel-wise self-attention over the sequence of frame embeddings. Each channel in the single and pair embeddings is updated independently. Frame indices
5

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are encoded using Rotary Position Embedding A1. This interleaved design enables efficient updates while maintaining flexibility in modeling sequential dependencies.
Structure Decoder.
The updated latent emheddings from Trajectory Module serve as conditioning signals for generating the conformation at each frame. For the SH(3) diffusion model described in Section 2.3 we adopt CONF-DIFF [44] as the Diffusion onDecoder to generate 3D conformations. DiffusionDecoder
CONFDIFF composes of layers of Invariant Point Attention and Transformer (on single embeddings) to collectively update the residue SE(3) rigids, as well as single and pair embeddings. Trained with denoising score matching Equation 5 the DiffusionDecoder learns to iteratively denoise noisy frame structures drawn from prior SE(3) distribution, conditioned on the frame latent embeddings, to generate accurate backbone conformations of the frame. To reconstruct full-atom geometry. we additionally predict the 7 torsional angles (ø, ab w. Xi..... . X4) using light-weight AngleResNet, for the coordinates of backbone oxygen atom and side-chain atoms.
Experiments
Dataset. We evaluate model performance on ATLAS 43 a large-scale protein MD dataset covering ~ 1300 proteins with diverse sizes and structures. For each protein, it contains triplicated 100 ns simulation trajectories. All models are trained on training trajectories and evaluated on test trajectories split by protein identity [18, 19 44 This presents challenging task for assessing the generalization to unseen protein structures and dynamics.
Model training. We use OPENFOLD (with frozen pretrained weights) as CONPROVER'S FoldingModule and initialized the weights of the DiffusionDecodee from CONEDIFF, while training the remaining parts of the network from scratch. During training, trajectories of length L with varying timesteps (strides), corresponding to ~ 1024 MD snapshots saved at 10 ps intervals, sampled to enable learning across multiple timescales. For the base CONFROVER, hybrid training strategy with 1:1 ratio between trajectory and single-frame training objectives. To further enable conformation interpolation, continue training the hase model with 1:1:1 ratio of trajectory, single-frame and interpolation objectives, denotes the model as CONFROVER- See Appendix D.2 for training details.
adopt
Baselines. We compare CONFROVER with state-of-the-ar deep learning models for each task: For
trajectory simulationn, we compare against MDGEN [19], flow-based non-autoreeressive trajectory
model trained on ATLAS; For fine-independen generation, we evaluate against ALPHAFLOW [18] and CONFDIFF [44, flow- and tiffusion-basse conformation generation models finetuned on ATLAS; For conformation interpolation, no existing baseline is available for large proteins, therefore, we focus on analyzing results of our model. See Appendix A for more details the availability of baseline models.
4.1 Trajectory Simulation
Since trajectories from both MD and models are stochastic samples, directly comparing them using frame-wise error, such as aet-mean--squar-deeeeeee appropriate. Therefore, we evaluate the model's ability to recover trajectory dynamics from two perspectives: (1) how well it captures the magnitude of conformation changes across varying start conformations and timescales; (2) how well it recovers the conformational states and principal dynamic modes observed in long-time MD simulations.
(RMSD) between atomic coordinates, is not
Evaluating conformation change accuracy on multi-start benchmark. We curated a test bench- mark consisting of short trajectories with L = 9 frames, extracted from 82 ATLAS test proteins. For each protein, we choose from varying starting frames (snapshot index 1000 ~ 7000) and strides (128 1024 snapshots), resulting in total of ~ 2, 700 generation conditions. For each trajectory, measure three aspects of conformational changes: Trajectory, the total changes over the entire se- ((xxxx++1) x1+1); Frame, the changes of each frame relative to the starting frame d(x', x-); A Frame, the changes between consecutive frames d(x', x Here, d(.,-) measures the distance between two conformations. We report both the L'-distance in projected 2D PCA space and the RMSD (in Å) of alpha-carbon (Ca) atoms. This benchmark captures diverse dynamics at both the trajectory and frame levels and enables comprehensive evaluation across varying conditions and timescales (see Appendix E. for details).
quence Σ d(x',

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MD REF.
MDGEN
CONFROVER
MD REF.
MDGEN
CONFROVER
7C4S-A
6021-A
6IN7-A
7AEX-A
SZNO-A
Figure 4: Visualization of six proteins from multi-start. Trajectory conformations are colored by their secondary structures and superposed to show the dynamic ensemble. MDGEN primarily exhibits local movements, whereas CONFROVER captures conformations changes similar to MD simulations.
ConfRover shows superior performance in recovering the magnitude of conformational changes. We report the Pearson correlation of measured conformation changes between model- generated and reference trajectories in Table [ with additional results in Appendix E. Com- pared with MDGEN, CONFROVER shows significant improvement in correlation scores, mean absolute error and structural quality (Ta- ble indicating its stronger ability to recover the magnitude of conformation changes across different starting conditions in the conforma- tional space. The greater difference observed in PCA highlights that CONFROVER more ac- curately captures conformational changes along the feature dimensions most relevant to the struc- tural variance observed in MD. Figure 4 visual-
Table I: Pearson correlations between conforma- tion changes in model-generated and reference tra- jectories under the multi-star setting. The mean and standard deviations are calculated from five independent runs. Models with higher correlations are highlighted in bold.
coordinates
Traj.
Frame
AFrame
MDGEN CONFROVER
0.562003
0.47=n03
0.41
0.75
0.63
0.53
PCA 2D
Traj.
Frame
AFrame
MDGEN CONFROVER 0.73
0. 0008
0.15=mm
0.10
0.50
1.43
izes ensembles of conformations in generated trajectories, with additional examples in Figure CONFROVER exhibit more notable conformation changes than MDGEN, and reflects the major movements in the structured and loop domains observed in the MD reference.
MDGEN a non-autoregresssvve model trained on trajectories of length L = 250. Adjusting its inference setup results in degraded conformation. To ensure fair comparison, inference setting (S 40, L = 250) and downsampled the trajectories for evaluation. To confirm that this post-processing step does not introduce artifacts, we also trained MDGEN models under the evaluation setups. The results are consistent with the downsampled version (see Appendix E.5
the original
Assessing long trajectory generation 100 sim- ulation. We further evaluate model's ability to recover conformational states and principal dynamics of pro- teins. For each of protein, we simulate a trajectory of L 80 frames at stride S 120, approximat- ing the 100 ns MD simulation ATLAS. To assess state recovery, jected into a reduced PCA space and compared with the reference trajectory. Specifically, discretize each principal component into 10 evenly sized "states" and measure the distribution similarity using Jensen- Shannon Distance (JSD). We also compute precision, recall, and Fl-score on whether sample conformations
Table 2: Recovery of conformational states in the 100 ns simulation experiment. The mean and standard deviation are computed over five independent runs, and the better results are highlighted in bold. MD 100Ns serves as the oracle and is excluded from the comparison.
conformations are pro-
JSD (J) Recall (t)
(t)
MD 10ONS
0.31
0.67
0.79
MDGEN
0.56
0.29
0.42+
CONFROVER
0.51+am
0.42=n.00
0.58
fall within these known states
32
49
To evaluate dynamic mode recovery, we perform time-
7

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A)
ND 100ns
MDGen
ConfRover
B)
MO 1021S
ContRoNee
PC2
52Nj
2.5
0.20
0.30
23
0.16
Narson
7NNOQ-A
61149-A
6TGK-C
0.34
25
0
0
pe3 30
Lagtime (frames)
pCt
Figure 5: Results from 100 simulation. (A) Correlations of principal dynamic modes between sample and reference trajectories, evaluated at varying lag time. The mean and standard deviation are shown as line and shadowed area, computed from five individual runs for MDGEN and CONFROVER (B) Examples trajectories illustrating the states explored by different methods (downsampled by frames for visualization). The blue background indicates the density of the ground-truth conformation distribution from MD reference.
lagged independent component analysis (IICA) at varying lag times on both reference and sample trajectories. We then compute Pearson correlations between the per-residue contribution to the leading components, based on the tICA coefficients. See Appendix E.2 for details. We include one of the triplicate MD trajectories- excluded from ground-truth evaluation an "oracle" reference, denoted as MD 100NS, representing the performance expected if the model were as accurate as an MD simulation run.
ConfRover recovers more conformational states than MDGen and accurately captures the principal dynamics. As shown in Table CONFROVER outperforms MDGEN state recovery. achieving lower JSD, higher recall and Fl scores, showing its improved ability to capture diverse conformations. Additionally, CONFROVER shows clear advantage capturing the principal dynamic modes across varying lag times, performing even comparably to the MD oracle (Figure5 5 This results suggest CONFROVER can learn and generalize dynamics to unseen proteins and still capture the most important dynamic modes. We visualize simulated trajectories in the PCA space in F and Figure 12 These examples again confirm that CONEROVER is more capable of sampling over the conformational space of the proteins and covering diverse conformations. Yet, we also observe some cases where MD 10ONS overcame the energy barrier and achieved more remote states while CONFROVER did not (e.g., 7NMQ-A in Figure 5B).
Summary: These experiments demonstrate that CONEROVER outperforms the current state-of-the-art model in trajectory simulation, effectively learning protein dynamics from MD) data and generalizing well to unseen proteins. While gap remains compared the oracle MD 1 100Ns, particularly in state recovery. the improvement narrows the gap between deep generative models and established simulation methods.
4.2
Time-independent Conformation Sampling
We evaluate the time-independent sampling performance of CONFROVER, following the benchmark setup in Ye et al. [4]] For each protein, 250 independent conformations sampled and compared the model generated ensembles with MD reference ensembles. We summarize the mean and standard deviations from five independent with key metrics are summarized in Table 3 with full results in Appendix
ConfRover matches the performance of state-of-the-art ensemble generation models. Compared with ALPHAFLOW and CONFDIFF, CONFROVER demonstrate overall comparable performance and outperforms at least one of the SOTA models in five evaluation criteria. This demonstrate that CONFROVER, despite being a general-purposs. purpose model capable of trajectory generation, also performs strongly in sampling independent conformations that approximate the equilibrium distribution from MD simulation. In contrast, MDGEN, which is trained solely for trajectory generation, shows suboptimal results with sequentially sampled conformations.
Effect of hybrid training. Without explicit single-frame training, the model primarily learns time- dependent generation, with only the first frame of each trajectory learning to generate conformation

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Table 3: Results from the time-indepente -nneeenddennd generation experiment. ALPHAFLOW and CONFDIFF are state-of-the-ar models for direct conformation ensemble generation. CONFROVER is the base model trained for both trajectory and -umeiineeenndett generation, where as CON:ROVER-TRAN and MDGEN are trained exclusively for trajectory generation. The mean and standard deviation are computed from five independent runs. The best scores are highlighted in bold, and the second-best scores are underlined.
Pairwise RMSD
Per target
MD PCA
Joint
Weak contacts J (t)
Transient contacts J (t)
Exposed
RMWD (4)
RMSF
PCA
residue J J()
r (1)
r (1)
W2 (4)
W2 (4)
ALoH
0,56:0006
0.85
2.62=003
1.522005
2.26::001
0.62:::00
0.41 00.0
0.69
CONFDIFF
0,54+0000
0.85
2.70+0.m
1,44
2.22+0044
0.64:0000
0.40+0000
0.67=am
MDGEN
0.47+004
0.72+0.z
2.78=0004
1.86+aas
2.44
0.51
0.28
0.57=001
CONFROVER-TRAU
0,48
0.84
2.85+0.02
1.43
2.30
0.53
0.36
0.58+0001
CON-ROVER
0.51
0.85
2.66:0002
1.47
2.23
0.621001
0.37
0.662001
unconditionally (i.e., from a masked token input). To test the importance of hybrid training, we ablated the hinle-framme objective and trained variant, CONFROVER- solely trajectory generation. As shown in Table 3 while this variant still outperforms time-dependent results from MDGEN, it shows decreased performance across several metries compared to CONFROVER. This highlights the importance of hybrid training in balancing the learning objectives and enhancing the model's for generating independent conformations.
4.3 Conformation Interpolation
To enable CONFROVER for conformation interpolation, we continue training CONFROVER with hybrid objective combining trajectory, single-frame and interpolation, referred CONFROVER- INTERP. We select 38 short trajectories from multi-start for evaluation where the reference MD trajectories for these cases exhibit significant conformation changes and clear state transitions, see Appendix E.4 for details. To condition on both start and end frames, we prepend the end frame to the start frame and autoregressively generate the remaining (intermediate) frames. To evaluate whether the model generate smooth transitions towards the target end state, we measure Ca-RMSD and 1 distance in the PCA space between each intermediate frame and the start/end frames.
Training on the interpolation objective enables smooth interpolation between conformations. As shown in Figure AA, the distance the start frame increases while the distance to the end frame decreases with frame index, indicating smooth and directed transitions. Without explicit interpolation training. the original CONFROVER (dashed lines in Figure (A) generates trajectory that do not progress towards the end state. Figure 6B visualizes intermediate structures and transition pathways in PCA space, showing that intermediate conformations from CONFROVER-INTERR closely resemble those in the MD reference. In contrast, as shown in Figure [4] the original CONFROVER can miss key transitions and fails to reach the end state. Additional results and visualizations are provided in Appendix E.4 These results highlight the effectiveness of our interpolation training strategy: by adjusting the dependency order in the sequence model, MOFROVER-INTEER learns to generate smooth transitions between two conformations.
B)
MD RSF.
CONFROVEL-NIEAP
MD Rea
CowEROVER-ITTRRR
and Foaree
Sonsro 28
2.0
Figure 6: Results from conformation interpolation. (A) Ca-RMSD distance of intermediate frames to the start and end frames, normalized by the distance between start and end frames. Reported values are averaged over 38 cases selected from the multi-start benchmark. (B) Example interpolations results. CONFROVER-INTERY generates smooth pathways between the start and end frames, capturing the dynamics observed with the MD reference. Start and end frames are shown as solid structures; intermediate conformations are shown in fading colors. Main motions are indicated by blue arrows.

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Table 4: Quality of model generated conformations. Conformations sampled from 38 trajectories are evaluated. Geometric metrics are reported as mean and standard deviations across 38 cases, and energy values are reported as mean with 95% percentiles. The best scores are highlighted in bold.
Ramachandran Rotamer
Clash
RMS
RMS
MolProbity soure (L)
MadraX
outliers % (L) outliers % (L)
score (L)
bonds (L)
angles ())
energy (L)
MD REFERENCE
0.38
1,02:0889
0.0410.16
0.0110000
18820005
0.72:0.18
(1793.0533)
MDGEN
0.93+2086
1.86
16.14=20.05
0.02+
2.13=0.30
2.24::0.0
-314.7 6.63.)
CONFROVER
0.58 6.63
1.98+144
7.81+66.22
0.01+
-.85
1.72+40.8
-522.2 (-185%.9, -5.4)
CON-ROVER -R-VERR
0.71
1.86
7.25
0.02:0011
1.91
1.61:10.51
-469.7 (-1712.3, -2.88
4.4 Conformation Quality
To ensure that CONFROVER generates physically plausible conformations, we further evaluate the quality using geometric assessments from MolProbity package [46] and energy profiles using a coarse-grained force field MadraX [36). We compared conformations across 38 trajectories shared
between the forward simulation and interpolation experiments, including results from MDGEN, CONFROVER, and an oracle MD REFERENCE. All structures are relaxed using the refinement to enable energy comparison.
pipeline in OpenFold [2
As shown in Table conformations generated by MD simulation exhibit the hightest overall quality, as expected. As a generative model, CONFROVER also produces high-quality conformations with fewer backbone (Ramachandran) and side-chain (rotamer) outliers, more accurate covalent lengths and angles, and achieving energy levels comparable to those of MD REFERENCE; outperforming MDGEN across all metrics. Furthermore, we compare conformations generated from forward simulation and interpolation tasks, where the latter includes additional constraints on terminal conformations. CONNRROVER-INEERE shows similar geometric and energetic metries, indicating that intermediate conformations maintains physical plausible when CONFROVER tries to interpolate between two conformational states.
Conclusions and Limitations
We introduce CONFROVER, general framework for learning protein conformational dynamics from MD trajectory data. Through autoregressive factorization, CONFROVER supports three tasks in a unified manner: trajectory simulation, uneeinndeenndent sampling, and conformation interpolation. This formulation reflects the temporal nature of MD while naturally encompassing conditional and unconditional frame-level generation. Extensive experiments and analyses highlight several empiri- cal advantages: (1) CONFROVER outperforms the current state-of-the-art in trajectory simulation, accurately capturing dynamic magnitude, state recovery, and principal motions; (2) Despite being a multi-purpose model, it achieves competitive performance in time-independent sampling compared to specialized methods; (3) With simple sequence reordering, CONFROVER effectively learns to interpolate between conformations.
Nevertheless, several limitations still remain: (1) Trajectory simulation and interpolation are emerging tasks with few available baseline models. We hope that this work, together with future developments in the field, will contribute to establishing more comprehensive benchmarks; (2) The dataset and evaluation metrics used in this study are limited and preliminary. The ATLAS dataset contains 100 ns simulations of single-chain proteins, which may not capture large conformational changes or the dynamics of protein complexes. Although we curated interpolation cases from ATLAS for demonstration purposes, future benchmarks reflecting realistic functional state transitions would be more meaningful. (3) Although CONEROVER narrows the gap with classical MD, it still falls short in fully capturing the conformational space with high structural fidelity. Future gains may come from scaling training data, using more efficient architectures, and leveraging additional information from MD, such as energy information. (4) Finally, while the triangular updates in the structural modules ensure high accuracy, their computational cost limits scalability to larger proteins and longer trajectories. Despite these challenges, CONFROVER demonstrates the promise of autoregressive models in molecular simulation, offering unified, efficient, and extensible approach to modeling protein dynamics.
6 Acknowledgments
We would like thank Dr. Hang Li for his invaluable support of this project. We also thank Zaixiang Zheng for insightful discussions, and Wesley Hsich, Yi Zhou, Nima Shoghi, Yuxuan Liu, Xiaolu Shen, Jing Yuan, Yilai Li, Fei Ye, and Wei Qu for their valuable feedback.
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A Additional Background
A.1 Related Work
Deep Generative Models for MD Trajectories. Recent works have explored generating protein trajectories as surrogate for MD simulations. Models such as TIMEWARP (22], ITO [38] and EQUIJUMP [9] learn stochastic transport functions to sample future conformations time (longer than MD intervals), reducing the computational costs of long- long-timescale simulations. However, these methods assume Markovian dynamics by relying solely on the current state for prediction, which may not be suitable for non- Mark-ovaiinnn dynamics common in protein MD data. To capture higher-order dependencies between the frames, ALPHAFOLDING [ incorporates history frames via "motion nodes". but it requires a fixed context window. MDGEN [99 takes a different approach by directly modeling the joint distributions of frames in trajectory and learning frame dependencies through "masked frame modeling", similar to masked language modeling. However, its key-frame parameterization requires separate models for different tasks, and its non-autoregressive paradigm limits flexible generation (e.g., not compatible for generating trajectories with variable lengths). GST 27 applies autoregression for future frame prediction, enabling variable-length conditioning context and prediction horizons. While the autoregressive approach is conceptually similar to CONFROVER, their work differs in several key aspects: performs deterministic prediction rather than generative trajectory sampling; it employs graph-based architecture with fixed structural priors from an adjacency graph, instead of full attention across all residues; it is trained and evaluated on a single protein instead of diverse proteins from ATLAS under transferable setting.
lagged
Beyond forward trajectory simulation, generative models have also been applied conformation interpolation, that is, generating the intermediate trajectories between two conditioned states. The non- autoregressive framework of MDGEN 19] can be extended sample transition pathways between such states; however, its key-frame parameterization requires training a separate model for this task, and it has not been tested large proteins like those in ATLAS. Du et [I]] proposed a simulation- free objective for transition cansition-pathway sampling based on Doob's 's h--rranffrmm but their approach has only been validated on numerical models and the small protein Chignolin. Its generalizability to larger, more diverse proteins remains unassessed.
Notably, the above models focus learning temporal dependencies between frames and do not support direct, lime-independent conformation sampling from the learned distribution. In contrast, CONFROVER is general framework that learns both the trajectory generation tasks well as direct sampling of independent protein conformations.
Deep Learning Models for Conformation Ensemble Generation. Another line of work focuses on direct sampling of conformations in a time-independent manner. Early efforts include perturbing the input to folding models (e.g., AlphaFold) [00 45 40) or perturbing the conformation using a structural diffusion model (32). However, these models trained solely static PDB structures and do not explicitly model the conformational distribution. Recent works have shifted to deep generative
paradigms that directly learn protein-specific conformational distributions 11 49 18
44
31 25).
Several models in this category, including ALPHAFLOW, CONFDIFF, and BIOEMULATOR, fine-tune pretrained structure models on large-scale MD datasets, enhancing their ability to capture conforma- tional distributions. A related approach trains normalizing flow models to approximate the Boltzmann distribution [35] 23 but their invertibility constraints limit scalability and transferability beyond small molecules and peptides. While these methods can generate time- -ndeeendenn conformations, they overlook temporal relationships and do not capture the kinetic aspects of protein dynamics.
Deep Learning Enhanced Molecular Dynamies. Another direction integrates deep learning with molecular conformation modeling through machine learning force field (MLFF) models [24, 33]. These models aim to incorporate higher-level accuracy (c.g., from ab initio calculations) into classical molecular dynamics simulations, improving fidelity while maintaining scalability. However, they still rely on sequential MD sampling with small integration steps and can be more computationally expensive than conventional MD. Two-for- One B does not directly learn an MLFF but instead trains a diffusion model for protein conformations and uses the resulting score function as an approximate coarse grained force field for MD simulation. Although it still depends on sequential MD sampling, this work provides an interesting perspective that connects conformation distribution modeling and trajectory generation. In contrast, different from these approaches, CONEROVER explicitly models
22

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and generates both individual conformation and trajectories, without relying on force field-based MD simulations.
Image-Videe Video Generation. The challenge of modeling protein dynamics conceptually parallels tasks in image and video generation, requiring both data distribution learning and temporal modeling. Recent advances in video generation offer valuable insights in addressing these challenges. Given limited video data, extending image generative models to video has proven effective. Several achieve this by incorporating temporal attention layers, enabling frame-to-frame communication. Disabling temporal attention reverts them to image models, allowing flexible training across both modalities. These approaches efficiently model time correlations without explicitly tracking offsets between frames. Meanwhile, the extension of autoregressive language models image and video domains has shown strong potential for sequential generation different data modalities. Li et al. [26] integrates language models' sequential modeling with diffusion models' ability to model continuous distributions, showing that discrete tokens are not essential for autoregressive models. MARDINI [29] extends the concept to video generation with efficient llama- style temporal planning and high-resolution video generation via diffusion decoder. By applying masked "frame" modeling. MARDINI allows the model to learn flexible temporal relationships and enables diverse tasks such as frame interpolation. CONFROVER differs from these works from video models in several aspects: it employs SE(3) diffusion for 3D structure generation; by using an autoregressive paradigm, it explicitly decouples the diffusion generation from temporal modeling, unlike the spatiotemporal denoising process in MARDINI; in addition, the causal autoregressive framework enables more flexible trajectory generation with variable lengths.
works []6 6 15]
A.2 Baseline Limitation
Modeling MD trajectories using deep generative models is still an emerging research area, few models currently support learning protein dynamics in transferable settings. Existing approaches based on forward transport operators have been primarily trained and evaluated on small peptides (e.g.. Timewarp [2 or small fast-folding proteins (e.g.. EquiJump (9). Although AlphaFolding 77] was also trained using ATLAS, the model weights are not publicly available at the time of this work. (We attempted internal reproduction and included its result in Appendix (E.6) Due to these limitations, we use MDGEN as the only available model for the main experiment. For conformation interpolation, neither sampling-basee based method []] nor video-like method MDGEN 19] have been trained and evaluated on large proteins. Therefore, we focus on demonstrating the interpolation results of CONFROV
23

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B Diffusion Models on SE(3) Space
Diffusion Probabilistic Models (DPM) model complex distributions through iterative denoising. In the context of protein conformations, DPMs defined over (3) translation-rotation space have been applied for protein backbone structural generation [48 44 Following Section 2.2 Xo = (To. Ro) E
SE(3)* denotes the translations and rotations of backbone rigids in data. The diffusion processes defined in the translation and rotation subspace add noise to corrupt the data:
dT, 2 P.PT, dt + B,P.,
dt of dw; 50(3)
dR, V
where t E [0, 1] is the diffusion time, 3, and o, are predefined time-dependent noise schedules and is projection operator removing the center mass. We and w; (,0(3)) are the standard Wiener processes in (0, @N and (3))(33) @N respectively.
The transition kernel of 'T satisfies Pejo (Tr (T) (TT Va, (1-04) I), where az = S3 8,ds The transition kernel of R satisfies Prjo [R, (Ro) IGSO, (R,; Ro. t), where IGS03 is the
isotropic Gaussian distribution SO(3) 48).
The associated reverse-time stochastic differential equation (SDE) follows:
1 T.T. Be Vo (T,) dt v B,Paw
dT, P
2
RR.
v (R.)d + V dt oidw, S0(3)
(6)
dt
80(3) denote standard Wiener processes in the reverse time. where w, and W
The reverse process can be approximated by neural network through the denoising score matching loss for translation and rotation:
()) (0) + (0)
=E (o VT, log (T TT) [|
E (R,) VR, log Pr(0 RR [Ro]
where X(() and (1) are time-dependent weights, so (Te, t) and sp (R,, are the score networks commonly parameterized with shared weights. The expectations are taken over diffusion time I ~ [1mmm 1], and over noisy and clean data pairs from the forward process (To, T.) and (Ra, R.).
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C Derivation of Equations in Section 3.2
Here we provide a more rigorous derivation of equations in Section 3.2 For clarity, we omit the conditioning variable P in the intermediate steps.
As defined in Equation T our goal is to model the frame-level conditional distribution p(x| x (x)) To
achieve this, we encode the previous frames (x {x] into sequence of latent embeddings {h") 1h1) and model inter-frame dependencies in this latent space.
By applying the Bayes' rule, we can factor the joint distribution over the current conformation x' and intermediate latent embeddings h (h', h as:
p(x' h hi
p(x' h h p(h' h x 5l p(h <l x
Integrating both sides over the latent variables h yields:
p(x' x=1) = / p(x' h h! x < p()('' hht x < p(h si xx
dh.
(8)
In our approach, both p(h' h x=1) and p(h <l x=1) are modeled using deterministic neural networks: an encoder (x') and an autoregressive temporal module _ (h**) respectively.
These mappings reduce both p(h' h- x**) and p(h-! xx1) to Dirac delta functions, and the conditional dependencies can be simplified
p(h h x²1) = p(hh((hh px'(h', h xx) p(x
Substituting into Equation ( gives:
p(x x p(x' p(x'(h') Ih))
p(h'(n=t) p(h "/h+11>" 1
(9)
Again, due to the deterministic nature of the encoder and temporal module, there is no marginalization involved in Equation yielding:
pxx1 = p(x
where
h = fyy (x'P), i= 1,2,......
and h' = St plemp (h) h2 h'-)).
Finally, we approximate (h)) with parameterized model p0 (x" hh).
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D Method Details
D.1 Detailed Module Architectures
Encoding Layer. The protein-specific single and pair representations are obtained from the Evo- former stack of pretrained OpenFold model (with frozen weights), after three recycle iterations. In addition, we encode residue-level sequence information by combining sinusoidal positional em- beddings of residue indices with learnable embeddings for the 20 standard amino acid types. These features are concatenated with the single representation from the FoldingModule.
To encode the structural information of each conformation frame, we introduced the FrameEncoder, a aseudo-beta-carbee (Cg) coordinate encoder similar to the InputEmbedding module from AL- PHAFLOW [8] (without diffusion time embedding). Specifically, this module first compute the pairwise distances between residues using C3 coordinates. These distances are then binned, embed- ded into latent embedding, and further refined through triangular update blocks including triangle attention and multiplication updates [20]. See Algorithm [ for the specifics. The resulting per-frama C3 pair embedding Z Frame concatenated with the pair representation from FoldingModule.
Both single and pair embeddings are projected into the same dimension of for simplicity, forming the latent embedding h' js' xz) for each frame. See detailed illustration in Figure 7
Encoding Layer
embesding bpsi
Sequence NAALLYLFVVBG.....
Frome Ippent
It (Fiaa)
FoldingNodule (frozen)
crcee
Go par pareaie distenc
lnput dructure
paik erowddin
Denagram embedding
InputPatrStack Alle
Traequaar Ya spliccaaion))
FraneEncoder
Figure 7: Architecture details of the Encoding Layer. A frozen FoldingModule encodes the protein- specific information from its sequence, containing prior knowledge on its chemical environment and folding structures. The single representation is further concatenated with additional amino acid embeddings and projected to a hidden dimension of size d: The pair representation concatenated with frame conformation information, encoded in C3 pair embedding, and projected to hidden dimension of size d. Both frame-level single and pair embeddings form the frame-level latent for downstream modules.
Algorithm 1 FRAMEENCODER
Input: Pseudo beta carbon (C3) coordinates x NN3, time E [0. 1]
Output: Input pair embedding z e Nx66
Zij | x,|
Zij + Bin(Zig, min 3.25 A, max 50.75 A, Nbins = 39)
Zij < Linear(OneeHl) (nnaaro(zz)o))
for 1 <- 1 to Nblacks = do
{z] += ThangeeAtteniionnSttriggggloeei = 64, Nucad = 4)
[z] + TraangeeAttenmiinEEnddngggeeeee (Zi,, 64, Ninead 4))
[z] += TiiageeAnutippiectttoooorgggggg (Zij, C = 64)
[z] + TringeeAuutipiictiinnnnnnnn Zij. C 64)
(z)r += Pair ThinTraaniiionn 2)
end for
Zij = LayerNorm(z..)
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Trajectory Module. In the Trajectory Module, we interleave layers of StructuralUpdate and TemporalUpdace iteratively update the latent zz', enabling the temporal reasoning across frames and structural refinement within each frame.
For the StructuralUpdate, updates the single and pair embeddings of the current frame through structural reasoning. After the StructuralUpdate the pair embedding is flattened from N, di [N x NV dJ and concatenated with the single embedding before being passed into the TemporalUpdate.
adopt Pairformer block from AlphaFold 3 2 which jointly
To model temporal dependencies between frames, we lightweight Llama architecture [42 We transpose the input such that the temporal dimension is treated as the sequence axis for channel-war self-attention across time. Rotary positional encoding 41 is applied to encode the temporal position for each frame. A causal attention mask is applied to restrict each frame to only attend to previous frames. After the temporal update, the latent embedding reshaped and split back into single and pair embeddings.
Figure 8 and Table provide the detailed module architecture and hyperparameter configurations, respectively. StructuralUpdate block included for every two TemporalUpdate layers.
Trajectory Module
Strscturealpdatee
In stchtee
Figure 8: Architecture details of the Trajectory Module. Trajectory Module contains interleav- ing blocks of StructuralUpdate and TemporalUpdate (only one block of each is shown). StructuralUpdate leverages the Pairformer architecture from Abramson et al. M. updating the pair embeddings with triangular updates and the single embeddings using with pair bias from the updated pair. The updated pair embeddings are flatten and concatenate with single embedding for channel-wise temporal update. The attention applied along the temporal dimension and update each single and pair embedding channels independently. The embeddings from TemporalUpdate are split and reshape back into single and pair embeddings.
Table 5: Hyperparameter choices of Trajectory Module
Hyperparameters
Values
Lightweight Llama (TemporalUpdate)
Number of layers
Dimension of the MLP embeddings
256
Dimension of hidden embeddings Number attention heads
128
Pairformer (StructuralUpdate))
Dimension of single embeddings Dimension pair embeddings
128
128
Dimension triangle multiplication hidden emheddings Number triangle attention heads
128
Dimension of pair attention embeddings
32
Transition layer expanding factor
4
Pair attention dropout rate
0.25
Structure Decoder. Conformation generation, conditioned on the temporal signals from the Trajec- tory Module, is performed using the model architecture from CONFDIFF [44), As shown in Figure 9 following CONFDIFF, the inputs to the denoising model include diffusion time I, pairwise distance between residue rigids, and residue indices (not shown). They are encoded and concatenated with the single and pair embeddings from FoldingModule. In addition, the single and pair embeddings from
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the Trajectory Module are projected back to the latent dimension de and d2, respectively, to modify the single and pair embeddings used by CONFDIFF.
The core of the Structure Decoder consists of multiple IPA-transformer blocks, which update single and pair embeddings well as the SE(3) rigids of noisy conformations. In the final block, torsional angles are predicted by TorsionPred and, together with denoised rigids, to reconstruct the atomic structure of generated conformation. The corresponding hyperparameter settings summarized in Table
Nes Neshiiilinglaaeee
Treaaiiee
Figure 9: Architecture details of the Structure Decoder. Single and pair embeddings from Trajectory Module is used to update the original embeddings from FoldingModule, The resulting embeddings are fed into blocks of IPA-transformer to update single, pair embeddings and denoise rigids, SE(3) representation of protein backbone conformations. Denoised rigids together with torsion angles predicted by TorsionPred recovers the atomic structure of protein conformation at this frame.
Table 6: Hyperparameter choices of the Structure Decoder
Hyperparameters
Values
Neural network
Number of IPA blocks
4
Dimension of single embedding (dg)
256
Dimension of pair embedding (dg)
128
Dimension IPA hidden embedding
256
Number of IPA attention heads
4
Number IPA query points
8
Number of IPA value points
12
Number transformer attention heads
4
Number of transformer layers
2
SE(3)-diffusion SDE
Number diffusion steps Translation scheduler
200
Linear
Translation 8min
0.1
Translation 8max
20
Rotation scheduler
Logarithmic
Rotation amin
0.1
Rotation Tmax
1.5
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D.2 Training and Inference Details
We train all CONFROVER models on the trajectories from the ATLAS training set, following the train-validation-ess split of previous works [18 44 19. Specifically, we exclude the training proteins longer than 384 amino acid residues, leading to 1080 training proteins.
Most components of CONFROVER models were trained from scratch, except for the FoldingModule where we used frozen weights from OpenFold extract the single and pair representations from three recycling iterations, and the Diffusi oorfuuieeeeeddrrr which was initialized from SonfDiff-OFFr3--. checkpoint provided by the authors
During each training epoch, we randomly sample stride length from 2 to 2 to extract sub- trajectories of length I. = 9 at varying time scales. With the of causal transformers, input frames were shifted forward by one frame with a [MASK] token padded at the beginning of the trajectory. Combined with the use of causal mask temporal attention, the design ensures that each frame is trained to sample conditioned only on previous frames and the first frame is generated unconditionally using only the [MASK] token as input.
We trained main CONFROVER model 180 epochs (~ 37 hrs) and SANFROVER-INTERR model for additional 220 epochs (~ 45 hrs). Additional training hyperparameters be found in Table All model training and sampling were carried out using NVIDIA H100 GPUs.
Table 7: Training hyperparameters
Hyperparameters
Values
Batch Size
Frames Num
8
Gradient Clip
1.0
Learning Rate
1x10-4
Optimizer
Adam (weight decay = 0.)
D.3 Training and Inference Cost
CONFROVER models in this work contains 19.6 M trainable parameters. All model training and sampling were carried out using 8 NVIDIA H100 GPUs with Distributed Data Parallel. We trained the main CONFROVER model for 180 epochs (~~ 37 hrs) and CONFROVER INTERP -NNTER model for
additional 220 epochs (~~ 45 hrs).
Inference varies with protein size and benchmark setups. To measure the potential speedups from using CONFROVER compared with classic MD simulation, we measured the wall-clock time required to generate 100 trajectories (80 frames) for ATLAS proteins of varying sizes and report the average inference time per size bucket. For comparison, we selected representative protein from each bucket and estimated the time required simulate 100 ns using OpenMM with implicit solvent. Both are performed on single NVIDIA H100-80G GPU, As shown in the Table 8 CONTROVER provides significant speedup for 100 ns simulation, with even more pronounced acceleration for larger proteins.
Table 8: Runtime comparison (in minutes) across different protein sequence lengths. Speedup is computed as MD runtime divided by CONFROVER runtime
Sequence length
(0,150)
(150,300)
(300,450)
[450,600)
[600,724]
ConfRover
6.99
7.53
10.92
15.88
20.83
MD
104.54
207.92
386.69
651.29
1099.13
Speedup
14.95
27.61
35.41
41.01
52.77x
For our large-scale experimental benchmarks, multi-starl trajectory simulation (2,700+ trajectories) took 8 hours and 30 minutes, while time-independent sampling (250 conformations per protein) took 3 hours and 20 minutes.
hhunss/gghhbbboomyyyyddnCCCCCCDt
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E Additional Experimental Results
E.1 Trajectory Simulation: multi-start
Benchmark Curation. In mulli-start, we sample short trajectories from varying starting point while ensuring the generation within the scope of the reference trajectory. For example, select frame index 1000, 3000, 5000, and 7000 as starting frames for stride 5 = 128/ /266, resulting in 12 test trajectories from triplicates; frame index 1000, 3000, 5000 starting frames for stride S 512, and frame index 1000 for stride S = 1024 to avoid exceeding total of 10000 frames. This provided us 2,706 different starting conditions from 82 proteins from the ATLAS test for evaluation.
PCA Projection. Following previous works [8 44], project the Ca coordinates of proteins into reduced PCA space to focus on the principal dimensions that best capture the structural variations observed in MD simulations. Briefly, for each protein, conformations from triplicate MD simulations in ATLAS are all aligned to the reference conformation (input structure for simulations). The coordinates of each Ca atoms are then flattened and used to fit a per-protein PCA model. For all subsequent analyses, sampled conformation are aligned to the reference structure before computing their PCA projections.
Additional Results. Here we also include the scatterplot of Pearson correlation in Figure and additional metrics from the multi-start experiments different strides in Table ( experimental included). Across different strides, CONFROVER models consistently outperforms MDGEN in recovering the correct level of dynamics.
Table Additional metrics from the multi-start benchmark. Results for different strides are shown in separate blocks. The better score in each block is highlighted in bold (excluding diversity). One inference run per model is used for this comparison.
Diversity
MAE on PCA-2D (L)
MAE on CA coordinates (4)
Quality
Pairwise RMSD
Frame Next
Frame Next
PepBond Break % (1)
Trajectory
Frame
Trajectory
Frame
Stride=128
MDGEN
1.26
6.53 3.10
1.28
0.91
5.02
0.96
0.71
27.9
CONFROVER
1.63
1.10
0.74
3.83
0.81
0.64
17.3
Stride=256
MDGEN
1.34
7.66
1.06
1.14
27.9
CONFROVER
1.78
3.91
1.28
0.87
4.60
0.91
0.75
16.6
Stride=512
MDGEN
9.12
1.94
1.25
7.27
1.41
28.0
CONFROVER
1.89
4.84
1.53
1.01
5.66
1.07
0.89
16.7
Stride=1024
MDGEN
1.51
11.48
1.55
1.80
28.1
CONFROVER
2.04
6.75
1.25
7.39
1.26
1.09
16.7
30

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CA coordinates
PCA-2D
NDGmn
Con ioofoovvr
MDGee
ConfRever
Saepecti
Trajectory
20
63
80
23
42
93
60
20
10
62
10
MDGGH
Canfilover
MoDan
Contilouer
Frame
Steglo
tra)
MD 10)
MD 001
NDG
Confflver
MDGen
Confflever
23
Somplene
AFrame
15
KD tmu)
HD
KD
MD tru
Figure 10: Scatterplots of conformation changes in sample trajectories versus those the reference trajectories, measured by trajectory-level conformation changes, frame-level conformation changes, and next-frame difference (AFrame). measured by the RMSD of alpha carbons (unit: Á) or L distance in the projected PCA space. MDGEN tends underestimate the magnitude of conformation changes while CONFROVER generate samples at similar level as the MD reference. The exact match of measured conformation changes is not possible due stochastic sampling both MD simulation and generative models.
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Additional Visualization of Trajectory. We additional unfiltered examples (randomly selected) for visual comparison of conformations generated by different models the multi-start experiments. as shown in Figure
MD REF.
MDGEN
CONEROVER
MD REF.
MDGEN
CONFROVER
7RM7-A
6GUS-A
6H86-A
7P46-A
6Y2X-A
6XDS-A
6TGK-C
6Q]0-A
6LUS-A
6IN7-A
Figure 11: Visualization of 10 trajectories randomly selected from the Multi-start benchmark. Trajectory conformations are colored by their secondary structures and superposed to show the dynamic ensemble. MDGEN exhibits primarily local motions while CONEROVER better reflects the motions observed in MD reference.
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E.2
Trajectory Simulation: 100 ns Long Trajectory Simulation on ATLAS
Details on Evaluation Metries. Conformational state recovery evaluated by comparing the distribution of model-generated and reference conformations in a PCA space. Same as in the multi-start benchmark, each conformation projected into the PCA space parameterized by the 3D coordinates of Ca atoms. To compare distributions, each principal component is discretized into 10 evenly spaced bins. After projecting the conformations into this space, we count their occurrences Jensen-Shannon
in each bin and compute the distribution similarity using Jensen Shannon Distance (JSD). We
also binarize the occupancy counts to compute precision, recall, and Fcore evaluating whether sampled conformations fall within known states, following prior work 32 44 49].
Dynamic mode recovery assessed using time-lagged independent component analysis (IICA) applied separately to reference and generated trajectories across varying lag times. After fitting, we extract tCA coefficients for each Ca atom and compute Pearson correlations between the per-residue contributions to the leading components, evaluating alignment of dynamic modes.
Additional Visualizations.
We additionally provide unfiltered (randomly selected) PCA plots in
Figure 12
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-10
-10
-00 20 -22 10
20
10
Figure 12: Visualization the 10LAAS-100ns trajectories from 20 randomly selected cases. The blue background indicates the density of the ground-truth conformation distribution from MD reference. CONEROVER shows improved conformation state recovery in several cases (c.g., 7JRQ-A, 6YHU-B, 7AQX-A, 6LAL-A, 60ZI-A, etc), sampling more diverse conformations. Yet, the gap between the oracle MD 100NS and deep learning models is evident in some cases (e.g.. 7JFL-C, 6TLY-A, 6JWH-A, etc)
34

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E.3 Time-independent Conformation Sampling
We follow the evaluation protocol of Ye al. [47 assess time-independent conformation sampling on the ATLAS test set. For each protein, 250 independent samples are generated. Since MDGEN does not support time-independent sampling, we approximate its performance using samples from its 100- is trajectory, serving sequential-samplinn baseline. The performance of state-of-the-art models, ALPHAFFLW and CONFDIFF, is taken from Table of Ye et al. [47], using their best-performing variants: ALPHAFLOW-ME and CONEDIFF-OPEN-ME -ID. Full results are shown in Table I0
Table 10: Performance on time-independent conformation generation on ATLAS. A total of 250 conformations sampled for each protein, and the and standard deviation of metries computed from five independent runs. The best performance is highlighted in bold, and the second- best is underlined. CONFROVER-TRAI and MDGEN are trained exclusive generate trajectories. MDGEN does not support time-independent sampling and the metrics are evaluated on sequential sampling result.
Diversity
Flexibility: Pearson r on
Distributional accuracy
Pairwise RMSD
Pairwise RMSI>
Glohal
Per target
MD PCA
Joint
sim
RMSF
RMWD
f
RMSF
Wa J
PCA Wal
>
%
ALPHAFLOW
2.85
1.632am
0.5
0.66:884
0.85
2.62:4as
1.52
2.262am
39.52
CONFDIFF
3.59+am2
2.18+0m2
0.54-000
0.65+000
0.85+am
2.70+001
1.44+am
2.22+004
41.00=1.
MDGEN
1,34
0.77
0.47=004
0.50
0.72
2.78
1.86
2.44
10.24
CONFROVER-TRAU
3.19
1.74:0m0
0.48=000
0.62+001
0.84+0m
2.85+
2.30+nol 1.43aam
37.08+2283
CONEROVER
3.68
2.25
0.51
0.64#001
0.85
2.66+0002
1.47
2.23
38.28 +4.4
Ensemble observables
Quality
Weak contacts
Transient
Exposed //f residue
Exposed MI
CA clash
PepBond
matrix p
break %
ALPHAFLOW
0.62:am0
0.41:00
0.69
0.35-
0.00
10:00
CONFDIFF
0.64:0.0
0.40=am
0.67=000
0.33+000
0.50=om:
6.20
MDGEN
innt
0.28
0.57
0.26
0.24
30.8 10.84
CONEROVER-TRO.
0.53
0.36220
0.58_om
0.27
0.32100:
11. 11.781024
CONFROVER
0.62gmm
0.37
0.66
0.32
0.502am
19. 18
35

---

E.4 Conformation Interpolation
In conformation interpolation experiment, we selected trajectories from multi-start. These trajectories exhibit sufficient conformation changes (e.g.. RMSD between the start and end frames Å) and clear interpolation path in the PCA space. The list of selected cases are in Table 11
The L distance of generated intermediate frames the start and end frames in the PCA spaces reported in Figure [3 Distances are normalized by the distance between start and end frames. Similar to the results measured by Ca-RMSD, CONFROVER-INTERR shows smooth interpolation between the start and end frames while CONFROVER does not. This result shows that by continue training the model on interpolation objective, CONFROVER can learn generated interpolating trajectories conditioned on the end state.
Table 11: List of 38 selected cases from multi-init for interpolation test. Naming conventions: [ep__[Chai__ID__R..
repeat)F([tarting index S "
S2NI-A-R2FI00S512
FLHL--ARI/F000S2566
6TGK-C-RI F-5000 7JFL-C-R2F30002556
RFF70-A-3333000512
AURD-A-RIFF100SS0022 6TT.Y-A-R210008226
TIRQ-A-RIFF1000SI
SOUSS--R2270002220
12FS-A-R2F30001128
2XDDS-A-R
7LAG-A-RIFI0W0S51
3
GH49-A-R2FI000S 1024
6OVK-R-R22H R2F70005128
FLP1-A-R2F500SS256
6H86-A R2F1000S 1024
RE11-A-RFF00005112
6Y2X-A R2F30005512
DPL-DD-R233005226
GIN7-A-R3F50005126
APSH-A-RIFI000SII
OP46-A-33550005256
6
GJ56-A-RIF30005256
6P5X-B- DP5X-B-RIF3000S23 RIF3005226
FAQX-AAR2F3000S511
7RM7-A-R3F30005
6JPT-A-R2T5000S512
409C-A-R31300005
7ASG-A-RIIF70001128
7586-A-R3750005556
8
GJV8-A-R3F10088256
6QJ0-A-RIF1000S22
EBWF-B-R3330001.1
BRRV-A-RIFIO00S22
FC45-A-R1150000552
vs. start frame
ConfRaver-interp
vs. start frame
ConfRoven-interp
Vs. end frame
ConfRover
vs. end frame
MD reference
1.0
Normalized Distance on PCA
0.9
Normalized Distance
0.6
0.6
0.2
0.2
0.0
0
1
2
3 4 5
6
8
Frame index
Frame index
Figure 13: Normalized PCA distance of intermediate frames to the start and end frames, averaged over 38 cases selected from the multi-start benchmark. [Left] a comparison between CONFROVER-INTEDS and CONEROVER, where CONEROVER-IN -NTERP generates smooth pathways while CONFROVER does not; [Right] a comparison between CONFROVER INTERP and reference trajectories.
36

---

Additional Visualizations.
Here we include additional visualization on interpolation results.
MD REF.
CONFROVER
CONFROVER-
QS6 RIF30005236
HD Fat Goffovereter
G-566- 100051024
R1F10105256 S0j0
5.0
-50
TEMF-2-0131 50005128
1.5
Cart
JFL R2F10005256
Figure 14: Example interpolations results. CONFROVER-INTERR generates smooth pathways between the start and end frames, capturing the dynamics observed with the MD reference while CONFROVER does not show the correct intermediate conformations. Start and end frames are shown as solid structures; intermediate conformations are shown in fading colors. Main motions indicated by blue dashed arrows. These examples highlight the difference between the original CONFROVER and CONFROVER- INTERP that further trained the interpolation objective. The original CONFROVER can miss key motions of the transition while CONFROVER INTERP correctly capture these motions.
37

---

GETE A/333220552
GOUD A-RZ ARSSTOOO000 TOOCOADO
CI40 AO4D-ATTTER AFOFLCSOOLOGA
OFIDO MOFFCCOOLOOO
G147 (47-4-20 53P5005128
05GAR1FH00525
QPTA-ROPS0005112
4CTVAR3F10005100
5.0
CLAL REF30005225
SLADD RUF00000222
SLUS A R2-33005122
GOVE R2F70025126
6OZE A ARIFI0005122
Gar
3-7
23
SOXCA-IS5 50305229
0200412F3000225
27
-20
STGBC-RL750005220
0TISA-RQF 12005256
12260-2271000551
1BWF-B-0030000120
35
5.0
-25
23
Carthond Vilep
to net.
NO
Figure 15: PCA plot of 38 selected interpolation cases. MD reference trajectories and results from CONFROVER-IN INTERP are shown in each plot.
38

---

E,5
Retraining MDGen for Experiment-Specifie Setups
Due to its non-autoregressive design and training on fixed-length trajectories, MDGEN cannot directly generate sequences of varying lengths. Therefore, in evaluation, we generate trajectories using the original settings (named HDGGE-S((trides (((enth))( and subsample them to match the evaluation setup. However, this post-processing may introduce artifacts. To address this concem, retrain MDGEN under the evaluation settings and compare the results on the Multi-start benchmark (stride 256) and 100 long trajectory simulations, shown below in Table Table 13 Table 14 and Figure T6 Experimental results show no significant difference of performance for key metrics observed comparing MDGEN with post-processed results and models specifically trained at the evaluation settings, suggesting no evident decrease of trajectory quality from the subsampling post- process. We use one inference run per model for this experiment.
Table 12: Compare MDGEN- MDGEN-S256F9 with MDGEN from subsampling post-pres process. Here is the table summarizing the Pearson correlations conformation changes between sampled and reference trajectories in multi-stari. ADGEN-S256F9 is trained and sampled with stride of 256 MD snapshots and length of 9 frames. The best scores are highlighted in bold.
Ca coordinates
Trajectory
Frame
AFrame
MDGEN
0.57
0.46
0.41
MDGEN-S2556
0.56
0.45
0.38
CONFROVER
0.77
0.62
0.53
PCA 2D
Trajectory
Frame
AFrame
MDGEN
0.18
0.13
0.11
HDDGE-SS26FP
0.21
0.19
0.11
CONFROVER
0.75
0.5
0.44
Table 13: Compare HDDGEN-S2662 with MDGEN from subsampling post-process. Here is the table summarizing additional metrics multi-star start benchmark. MDGEN-S256F9 trained and sampled with stride of 256 MD snapshots and length of 9 frames. The best scores highlighted in bold.
Diversity
MAE on PCA-2D (1)
MAE on CA coordinates (J)
Quality
Pairwise RMSD
PepBond Break %()
Traj.
Frame
FFrame
Traj.
Frame
A Frame
MDGEN
1.34
7.66
1.54
1.06
6.00
1.14
0.85
27.9
MDGEN-S256F9
1.59
6.90
1.47
1.03
5.33
1.13
0.82
16.2
CONFROVER
1.78
3.91
1.28
0.87
4.60
0.91
0.75
16.6
Table 14: Compare MDGEN- 92 with MDGEN from subsampling post-process. Here the
table summarizing the state recovery performance in 100 long trajectory simulation. MDGEN- 1120880 is trained and sampled with stride of 120 MD snapshots and length of 80 frames. The best scores are highlighted in bold. MD 100NS is included as the oracle.
JSD (L)
Recall (t)
F1 (t)
MD 1OONS
0.31
0.67
0.79
MDGEN
0.56
0.30
0.44
MDGEN-S120F80
0.57
0.29
0.42
CONFROVER
0.51
0.42
0.58
39

---

MD 100ns
MDGen
MDGen 51200-
PC
Canfilover
0.20
Pearsnn Conetation
0.18
0.16
0.14
0.98
TICA Lagtime (frames)
Figure 16: Compare MDGEN-S256F7 with MDGEN from subsampling post-process. This figure shows correlations of main dynamic modes between sampled trajectory and reference trajectory. MDGEN-S120FN trained and sampled with stride of 120 MD snapshots and length of 80 frames.
E.6 Reproducing AlphaFolding
Given the limited available baseline models for protein trajectory generation, attempted to reproduce AlphaFolding 77). a diffusion-based model that generate short trajectories (i.e., blocks) and can be extended to longer through iterative generation. We followed the authors' official implementation D. setting the motion token count to and the generation horizon (block length) to 16 frames. To improve generation efficiency, increased the stride from the default to 40, matching the setup in MDGEN. Training ALPHAFOLDING on the full ATLAS dataset resulted in out- of-memory
NVIDIA A100-80GB GPU. Therefore, we adopted the authors' filtering criterion of a maximum sequence length of 256 residues. Similarly, we encountered the out-of-memory error for the five largest proteins during inference. The model was trained for 65K steps where we saw convergence. For evaluation, we iteratively extended the 16-frame outputs to generate 256 frames and retained the first 250 frames to match the 100 simulation setting.
We compared ALPHAFOLDING with MDGEN and CONFROVER on the ATLAS 100 ns simula- tion task, using inference run per model. As shown in Table [5] both ALPHAFOLDING and CONFROVER outperform MDGEN capturing the principal coordinates of the dominant dynamics, although ALPHAFOLDING slightly lags behind CONFROVER. However, shown in Table [6] ALPHAFOLDING produces lower-quality conformations, exhibiting inflated backbone and side-chain rotamer outliers well higher clash rates. We found
evident degradations from error accumulation (Table [7]. likely due to its iterative block-wise extension, which conditions only on the last frame of the preceding block. In contrast, MDGEN employs non-autoregressive attention across all frames, while CONFROVER maintains full attention history via KV cache. Owing to the noisy nature of its generated trajectories, FOLDING also shows inflated recall in conformational state recovery (Table 18). as the increased structural noise likely leads to artificially higher coverage in state space.
Although our implementation is a reproduction of 7 and may not faithfully reflect the authors' original model, this analysis provides valuable insights into the performance and potential limitations of block-extension-ball trajectory models such as ALPHA FOL.DIGG.
40

---

Table 15: Pearson correlations of principal dynamic modes (PC1 and PC2) between sampled and reference trajectories, evaluated at varying lag times (At. in frames). The best scores are highlighted bold.
Al 1
Al =
At = 10
Al 20
PC I
MDGEN
0.11
0.12
0.10
0.12
PC 1
ALPHAFOLDING
0.15
0.15
0.16
0.18
PC
CONFROVER
0.19
0.17
0.19
0.19
PC 2
MDGEN
0.12
0.10
0.10
0.11
PC 2
ALPHAFOLDING
0.18
0.15
0.16
0.20
PC
CONFROVER
0.19
0.17
0.18
0.17
Table 16: Conformation geometric quality evaluated using MolProbity. Unrelaxed conformations used for evaluation. The best scores are highlighted in bold.
Ramachandran outliers (4)
Rotamer outliers % (4)
Clash score (L)
RMS bonds (L)
RMS angles (J)
MolProbity
score (L)
ALPHAFOLDING
2.91
15.37
151.35
0.07
4.76
3.94
MDGEN CONFROVER 1.75
1.87
2.85
128.08
0.04
3.14
3.28
3.30
76.89
0.05
3.69
3.00
Table 17: Average Ramachandran outliers across trajectory frame ranges. Compared with MDGEN and CON:ROVER, ALPHAFOLDING exhibits an increased level of backbone outliers when simulating longer trajectories.
Frame range
(0,15)
(15,31)
(31,47)
(47,63)
(63,79)
ALPHAFOLDING
1.77
2.78
3.21
3.34
3.57
MDGEN
1.59
1.85
1.96
1.95
2.05
CONFROVER 1.34
1.61
1.77
1.93
2.08
Table 18: Recovery of conformational states in the ATLAS 100 ns simulation experiment. Although ALPHAFOLDING shows the highest coverage, this may due lower sample quality that artificially inflates diversity.
JS- 4
Recall
ALPHAFOLDING
0.47
0.51
0.65
MDGEN
0.55
0.29
CONFROVER 0.51
0.42
0.58
E.7 Broader Impacts
This work aims to advance machine learning for protein modeling, with broad applications in biology and drug discovery. By enabling efficient simulation of protein dynamics and conformational changes, our method has the potential accelerate research in structural biology and therapeutic development. Beyond proteins, the approach may also be adapted to other domains of computer-aided design, including small molecule design, materials science, and chip design. While the potential benefits are significant, it is important to ensure responsible use and prevent misuse of the technology. Developing appropriate safeguards and regulatory frameworks will be essential mitigate any potential negative impacts.
41


Reviews:
[{"review_id": "P7MVtqjgMQ", "reviewer": "Reviewer_NHvs", "summary": "In this paper, the authors propose a new model for the simulation of protein conformations that can perform multiple tasks: time-independent sampling, trajectory simulation and interpolation between two conformations. This is achieved by defining all tasks as conditioned generation, where the conditioning is generated by an encoding layer based on sequence only in the case of sampling or on one or multiple frames for the other tasks. The support for multiple frames is ensured by a specialized trajectory module working on a sequence of frame embeddings with a causal mask. The model is trained on a dataset with a large amount of proteins but with limited MD time sampling, ATLAS, and shown to achieve better results with respect to the open-source state of the art on each of the tasks.", "questions": "* It is not very clear to me how the “MD timestep” is inserted in the model. Is it an explicit input to the model, or is the model left to infer it from the previous frames? And in this second case: are a few frames sufficient to univocally infer the desired timeframe? In either case, it would be interesting to see the trajectories generated for different timesteps and compare their statistics.\n* What is the exact setting of the multi-frames task? I.e. how many frames are presented to the model and how many frames are generated?\n* This may be a miss on my part in appendix but from the main description of the model, it does not seem to be equivariant under rotations of the protein. Can the authors comment on this point? Is any augmentation used on the training dataset?\n* For the interpolation task, the trajectories are shown to interpolate “smoothly” between start and end point. However, from the PCA plots, several trajectories seem to pass through potentially unphysical paths. Would it be possible to perform some rough energetic analysis to verify that the intermediate states retain some physical plausibility? \n* While some information on model cost is presented in appendix, it would be nice to see this section slightly expanded, especially covering a more direct comparison for specific proteins on the same hardware, since the scaling of this model to larger systems is a concern.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths: \n* The model is based on validated components in its modular design, e.g. openfold2, confdiff. These models have been shown to work well for their respective tasks.\n* The model can generate structures in multiple settings, unifying conformation generation and molecular dynamics trajectory generation, altho it is unclear how important it is to unify these.\n* The model performs well on the ATLAS dataset, appearing to simulate convincing MD trajectories for some of the proteins outside of the training set according to visual inspection, however numerical validation is incomplete.\n* Introduction of a modern causal transformer for capturing the dependency between frames is a significant improvement for the practitioners in the field. \n\nWeaknesses: \n* Although the modular design is great, the modules are computationally very expensive: encoding layer of the model is based on a pre-trained OpenFold model, which may be limiting for large protein or multimer use case. The trajectory module is also based on the pairformer architecture, based on triangle operations, which are notoriously expensive. This leads to a rather expensive model, making it potentially unfeasible for the simulation of larger proteins. Although there are some math libraries coming up for such operations, the cost still should be discussed. \n* While the results shown indicate the ability of the model to produce more realistic trajectories than a similar approach, a more rigorous analysis of the trajectories would be needed to convince the reader of the statistical validity of the sampled states. Authors may follow analysis from the papers they cite, e.g. TimeWarp or EquiJump. Altho I 100% agree with the issue of not having open source baselines limiting the reproduction of former work and 1-to-1 comparison, there needs to be more analysis on the statistical validity of the MD trajectories. The unfiltered results in Figure 12 in Appendix appear somewhat lacking alignment with MD. \n* Not a weakness but a comment that authors may want to address in review stage for the general authorship that might share the same opinion as this reviewer - If I understand correctly, there is a strong remark about the unification of non-time aligned vs time aligned sampling, ie conformer generation vs trajectory generation, being an important contribution, and this being the first model to do so. I am not sure this is as significant as the paper puts it to be. Non autoregressive trajectory generation models cited in the paper can be easily modified to be conformer generation models as well, since they are based on simple conditioning using an interpolant scheme. If they are trained with interpolation from noise to data frame, rather than previous frame to next frame, these models would function as conformer generators. Repeating that I don’t believe a unified framework is a weakness, but just may not be the strength it is emphasised to be, especially if the conformer generation model becomes too computationally heavy in order to use the same framework of the trajectory generation model. \n\nMinor: \n* Line 187: updateing -> updating\n* Line 188: StructualUpdate -> StructuralUpdate\n* Line 189: StructralUpdate -> StructuralUpdate\n* Line 306: trajecotry -> trajectory", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for acknowledging the significance of our overall design improvements, the strong performance across tasks in transferable settings, and the significant value of introducing the causal transformer for researchers in the field. Here we include our point-to-point response to the questions raised:\n\n### Q1: Encoding of MD timesteps.\nThe MD timestep is encoded in the Llama layer of the TemporalModule using rotary position encoding (RoPE) [1]. Specifically, we represent time as discrete snapshot indices (e.g., 128, 256, 384, ...) corresponding to frame indices (0, 1, 2, ...). RoPE allows the model to learn dynamics across multiple temporal scales and, by encoding relative rather than absolute time, ensures invariance to global time shifts. In our multi-start experiments, we observe that the diversity of generated trajectories increases with larger stride values (see below), highlighting the model's ability to capture scale-dependent dynamics:\n|   Stride  | Pairwise RMSD (A) |\n|:----------|:--------------|\n| 128 | 1.63 |\n| 256     | 1.78 | \n| 512      | 1.89 |\n|1024 |2.04\n### Q2: Frame settings for training and evaluation.\nWe train models on sub-trajectories of 8 frames sampled from original MD. For evaluation of different tasks: 9 frames for multi-start and conformation interpolation, 80 frames for Atlas-100ns simulation, and one frame for time-independent sampling\n### Q3: Model equivariance.\nWe did not require data augmentation thanks to the use of invariant sequence and structure embeddings, SE(3)-diffusion, and equivariant denoising models. Specifically, the encoding layer is invariant to global translations and rotations: it encodes input structures via pairwise distances between pseudo-beta atoms, and the resulting single and pair embeddings are invariant, enabling invariant reasoning in the TemporalModule. For the decoder, we adopt Invariant Point Attention from AlphaFold, which predicts residue rigid-body updates relative to the input structure, making the denoising network equivariant to global rotations.\n### Q4: Additional quality metrics.\nThank you for the suggestion. We have added detailed structural quality and energy evaluation of generated intermediate states. Specifically, we used MolProbity [2] to assess geometric accuracy and MadraX [3] for heavy-atom-only energy evaluation. To verify whether interpolation tasks still generate physically plausible conformations, we evaluated the intermediate conformations (excluding the first and last frames), labeled ConfRover-interp, and compared them to conformations from forward simulations (without terminal constraints, labeled ConfRover-fwd) and reference MD simulations (oracle). All structures were processed using constrained local relaxation and evaluated with the default settings of MolProbity and MadraX. We found such relaxation necessary to obtain accurate energy evaluations, even for MD references. Results are summarized in the following table (all values reported as mean ± std; energy values as 95% percentile ranges):\n  - Comparing ConfRover-interp to ConfRover-fwd, we observe similar structural quality, suggesting that interpolated conformations are physically realistic. Slightly higher energy in interpolated samples may reflect the presence of regions with energy barriers.\n  - Other comparisons show ConfRover generates conformation with high quality. ConfRover achieves comparable performance to MD references in backbone dihedral (Ramachandran), bond geometry, and overall energy. The largest discrepancy lies in clash scores, which depend on ad hoc hydrogen addition and may bias evaluation of heavy-atom-only models. ConfRover consistently outperforms MDGen across all metrics.\n| exp               | Ramachandran outliers % | Ramachandran favored % | Rotamer outliers | Clashscore | RMS(bonds) | RMS(angles) | MolProbity score | MadraX Energy          |\n|-------------------|--------------------------|--------------------------|-------------------|-------------|-------------|---------------|-------------------|----------------------------------------|\n| MD Reference       | 0.38±0.49                | 97.41±1.49               | 1.02±0.89         | 0.04±0.16   | 0.01±0.00   | 1.88±0.05     | 0.72±0.18         | -519.3 (-1793.0, -53.4)                |\n| MDGen              | 0.93±0.86                | 94.98±2.04               | 2.86±1.59         | 16.14±20.05 | 0.02±0.02   | 2.13±0.30     | 2.24±0.40         | -314.7 (-1483.8, 263.6)                |\n| ConfRover-fwd      | 0.58±0.63                | 96.93±1.45               | 1.98±1.48         | 7.81±6.52   | 0.01±0.01   | 1.88±0.25     | 1.72±0.38         | -522.2 (-1858.9, -53.4)                |\n| ConfRover-interp   | 0.71±0.94                | 96.90±1.95               | 1.86±1.46         | 7.25±8.74   | 0.02±0.01   | 1.91±0.32     | 1.61±0.51         | -469.7 (-1712.3, -42.8)                |\n### Q5: inference time across different protein sizes.\nWe measured the wall-clock time (minutes) required to generate 100 ns ATLAS trajectories (80 frames) for proteins of varying sizes and report the average inference time per size bucket. For comparison, we also selected a representative protein from each bucket and estimated the time required to simulate 100 ns using OpenMM with implicit solvent. As shown on the table below, ConfRover can sample at least 10x speedup over implicit solvent MD simulations. The acceleration becomes more pronounced as protein sizes increases, suggesting ConfRover might be more advantageous for larger proteins. Notably, the cost of generating additional frames remains nearly constant due to our autoregressive generation with KV-caching, which eliminates redundant computation of attention activations.\n|  seqlen|      (0,150) |     [150,300) |    [300,450) |   [450,600) |   [600, 724] |\n|----------|-------|--------|--------|-------|--------|\n| ConfRover | 6.99 | 7.53 | 10.92 | 15.88 | 20.83 |\n| MD  |104.54 | 207.92 | 386.69 | 651.29 | 1099.13 |\n| Speedup | 14.95 | 27.61 | 35.41 | 41.01 | 52.77 |\n\n### Weakness 1: Heavy architecture.\nThe detailed computational cost of applying ConfRover to larger proteins is summarized in our response to Q5. We agree that standard architectures in the field, such as Pairformer, are computationally intensive. However, designing efficient neural architectures for protein structure modeling remains a broad challenge in the field. In this work, our focus is on proposing a framework for learning protein dynamics, rather than advancing neural architectural innovations. Therefore, we adopt proven architectures to build ConfRover, which demonstrates competitive performance, scalability to proteins with over 700 amino acids, and reasonable acceleration compared to MD. As the reviewer noted, recent acceleration libraries such as cuEquivariance [4] could be beneficial for further speeding up our model, and we will consider integrating them in future work.\n\n### Weakness 2: Lack statistical analysis of dynamics.\nUnlike Timewarp or EquiJump, which focus on small systems with well-equilibrated dynamics, our work specifically targets modeling dynamics for large proteins in transferable settings.  As noted in MDGen, trajectories in ATLAS are not fully equilibrated, making it difficult to take the Markov state models (MSMs) approach for evaluating long-term dynamics. \n\nDue to this limitation, we additionally designed experiments and metrics to assess whether the model captures dynamic changes across different time scales and initial conditions (multi-start benchmark) and whether major dynamic modes are correctly reflected (as in the TICA analysis in Figure 5). While these metrics are not perfect, we believe they provide statistical insights into the dynamic behavior of the models and are effective in revealing some limitations of current state-of-the-art approaches. \n\n### Weakness 3: Significance of a unified framework.\nA unified framework for protein conformational dynamics is appealing for several reasons. First, tasks like forward simulation, interpolation, and time-independent sampling share similarities, naturally to address in a single framework. Second, both time-dependent and independent conformations stem from the same underlying distribution, as also noted in prior work [5], supporting joint learning. Third, our framework’s ability to leverage both structures and trajectories enables flexible training and paves the way for foundation models in protein conformations. \n\nWhile alternative approaches like stochastic interpolation are possible, our early experiments with them were unsuccessful. In contrast, our autoregressive LLM + diffusion model performs robustly across tasks and offers a strong basis for learning conformational dynamics.\n\nFinally, we clarify that ConfRover adds minimal overhead for structure generation: the temporal module is lightweight, and for time-independent tasks, it reduces to a simple MLP. This design enables dynamic modeling with little additional cost over the conformation diffusion model.\n\n------\n\n[1] Su, Jianlin, et al. \"Roformer: Enhanced transformer with rotary position embedding.\" Neurocomputing 568 (2024): 127063.\n\n[2] Williams CJ, Headd JJ, Moriarty NW, et al. MolProbity: More and better reference data for improved all-atom structure validation. Protein Sci. 2018;27(1):293-315. \n\n[3] Orlando G, Serrano L, Schymkowitz J, Rousseau F. Integrating physics in deep learning algorithms: a force field as a PyTorch module. Bioinformatics. 2024;40(4):btae160. \n\n[4] cuEquivariance: https://docs.nvidia.com/cuda/cuequivariance/\n\n[5] Arts M, Satorras VG, Huang CW, et al. Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics. http://arxiv.org/abs/2302.00600"}, {"review_id": "i7tURFcYUP", "reviewer": "Reviewer_tPch", "summary": "The authors introduces Confrover, an autoregressive framework that couples a transformer “trajectory module” with an SE(3) diffusion decoder so that one model can (i) roll out protein trajectories, (ii) draw single, time-independent conformations, and (iii) interpolate between two end states. Training is performed on the ATLAS MD data set with a hybrid loss that mixes trajectory, single-frame and (optionally) interpolation objectives. On held-out proteins the authors report better dynamical metrics than MDGen (but worse than MD) and competitive equilibrium-ensemble metrics with AlphaFlow and ConfDiff.", "questions": "1. Given deterministic encoder & time modules, how diverse are the multiple samples from identical conditioning frames?\n2. Have the authors checked energy conservation or compared implied timescales against MD?\n3. How is CA clash/peptide bond break calculated exactly? How about other clashes/bond length distribution?\n3. There are many things for which ML based MD sampling like AlphaFlow does not work, e.g. https://www.biorxiv.org/content/10.1101/2024.08.31.610605v1, does the Confrover work better?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The motivation and objective for the paper are clear, the task is important, and the modular design of the architecture is sensible. The observation that frame autoregression subsumes the three target tasks is very sensible. \n2. The joint training for the different tasks makes sense and improves equilibrium sampling.\n3. Empirically, the paper demonstrates reasonable (albeit still not excellent) performances.\n4. Overall, the paper can be easily followed.\n\nWeakness:\n1. This paper is primarily empirical, with a few previous studies exploring similar ideas (e.g., autoregressive + diffusion has been explored in MarDini for image/video, and AlphaFolding). The architecture also reuses existing components and fuses them. That said, this should not result in the rejection of the paper.\n2. The evaluation is primarily based on RMSD, PC-L2, JSD, etc. These statistical metrics have no physical meaning. In my opinion, kinetics (e.g. implied timescale)/energies/Rama plots should be evaluated to show that the model is physically grounded. Specifically, since the model breaks peptide bonds, I suspect it is not very good at distinguishing which dihedrals are more rigid and which ones are more flexible. The interpolation task should also have physical validation (e.g. minimum free energy paths; especially since there are no baselines). \n2a.  I note the empirical performance of the paper, while better than MDGen (unclear how good it is compared to AlphaFolding or other similar methods due to closed-source code), it is still very far from MD itself. The model often breaks peptide bonds (which should never happen) and has some C-alpha steric clashes. I am open to raising my score if evaluations comparisons can be made with AlphaFolding or more SOTA methods.\n3. There's a potential data leakage problem. Presumably, the frozen OpenFold has been trained on (some) PDB structures of the ATLAS test proteins.", "was_revised": true, "final_justification": "The paper has shown an interesting technique that allows can draw single conformations but also generate protein dynamics trajectories. While the empirical results are yet to match MD, this work does mark a step forward in modelling protein dynamic trajectories using generative methods.", "rebuttal": "We appreciate the reviewer's recognition that our work addresses important tasks and the proposed autoregressive framework for multiple tasks sensible. We also thank the reviewer for their valuable input and constructive feedback. We organize our response as follows:\n\n\n### Weakness 1: originality and relation to similar ideas\nThe key contribution of this work is the introduction of a typical autoregressive framework $p(x|x^{<l})$ into the modeling of MD trajectories. We demonstrate that this formulation naturally aligns with several important tasks in protein conformation sampling by managing sequential dependencies across frames. By leveraging modern causal transformers, ConfRover offers efficient training and can be readily extended to generate longer trajectories.\n- **vs MarDini**: While ConfRover shares the general idea of combining autoregressive modeling with diffusion, there are key differences. First, the two methods target fundamentally different application domains. MarDini focuses on video data, whereas ConfRover is designed for molecular dynamics simulations. Second, ConfRover emphasizes a broader multi-task setting, including time-independent sampling and interpolation tasks, which were not explored in MarDini.\n- **vs AlphaFolding**: While AlphaFolding extends MD trajectories by iteratively generating fixed-size blocks conditioned on motion and reference frames, our causal autoregressive model differs in key ways: (1) ConfRover generates long trajectories frame-by-frame using KV-cache, unlike AlphaFolding’s block-wise extension that discards historical context. (2) ConfRover operates at the frame level with clear causal dependencies, whereas AlphaFolding’s block-level autoregression lacks explicitly defined intra-block causality. (3) ConfRover supports forward simulation, time-independent sampling, and interpolation - capabilities not directly supported by AlphaFolding.\n- **Reuse existing components**: This work focuses on proposing a general autoregressive framework. To demonstrate this idea, we intentionally use well-established architectural components. Architectural improvements are beyond the scope of this paper but represent a promising direction for future work.\n\n### Weakness 2: conformation evaluation and baseline comparison\n**1. Physically grounded evaluation.** thank you for the suggestion. We agree that demonstrating the physical plausibility of generated conformations is important. To this end, we introduced standard structural and energy evaluation pipelines using MolProbity [1] for geometric quality and MadraX [2] for heavy-atom-based energy assessment. We compared conformations from reference MD, MDGen, and ConfRover (in both forward simulation and interpolation settings).\nAll structures were evaluated using the default settings of MolProbity and MadraX. We found that energy and structural metrics can be sensitive to minor deviations and it is necessary to apply local relaxations even for MD reference conformations. We applied the standard protocol from OpenFold to all conformations to ensure fair comparison and reliable energy estimates. Results are summarized in the following table (values are reported as mean ± std; energy values as 95% percentile ranges):\n| exp               | Ramachandran outliers % | Ramachandran favored % | Rotamer outliers | Clashscore | RMS(bonds) | RMS(angles) | MolProbity score | MadraX Energy          |\n|-------------------|--------------------------|--------------------------|-------------------|-------------|-------------|---------------|-------------------|----------------------------------------|\n| MD Reference       | 0.38±0.49                | 97.41±1.49               | 1.02±0.89         | 0.04±0.16   | 0.01±0.00   | 1.88±0.05     | 0.72±0.18         | -519.3 (-1793.0, -53.4)                |\n| MDGen              | 0.93±0.86                | 94.98±2.04               | 2.86±1.59         | 16.14±20.05 | 0.02±0.02   | 2.13±0.30     | 2.24±0.40         | -314.7 (-1483.8, 263.6)                |\n| ConfRover-fwd      | 0.58±0.63                | 96.93±1.45               | 1.98±1.48         | 7.81±6.52   | 0.01±0.01   | 1.88±0.25     | 1.72±0.38         | -522.2 (-1858.9, -53.4)                |\n| ConfRover-interp   | 0.71±0.94                | 96.90±1.95               | 1.86±1.46         | 7.25±8.74   | 0.02±0.01   | 1.91±0.32     | 1.61±0.51         | -469.7 (-1712.3, -42.8)                |\n\nThe results show that ConfRover generates high-quality conformations, matching MD references in backbone dihedrals, bond geometry, and energy, while outperforming MDGen across all metrics. **Interpolation quality**: when comparing ConfRover-interp with ConfRover-fwd, they shows similar structural quality, indicating the conformation generated from interpolation maintain physically plausible.\n\n**2. Dihedral flexibility analysis.** While prior results suggest the conformations are physically plausible, we additionally performed dihedral flexibility analysis as suggested by the reviewer. We extracted dihedral angles from the 100 ns simulations and computed their circular variance. Pearson correlation with the MD reference was used to assess how well the model captures ground-truth flexibility. The result shown below shows reasonable agreement between ConfRover generated sample and MD reference.\n|           | phi           | psi           |\n|:----------|:--------------|:--------------|\n| ConfRover | 0.480 ± 0.158 | 0.455 ± 0.163 |\n\n\n**3. Comparison with other models like AlphaFolding.** The lack of baseline models is a common challenge in this field. We contacted the authors of AlphaFolding to request their model for comparison, but the weights are not available at this time. However, we identified code from a related open source project and are currently working on retraining their model for evaluation on the ATLAS benchmark. We are happy to update our results if preliminary comparisons become available during the discussion phase.\n\n### Weakness 3: potential data leakage from OpenFold model\nOur objective is to model the distribution and dynamics of protein conformations, not to predict a single folded structure. Accordingly, our setup and evaluation metrics focus on structural variation and temporal behavior rather than alignment to a fixed reference. Since OpenFold is trained on static PDB structures and lacks such dynamic information, we do not believe data leakage is a concern under the current setting.\n\n### Questions:\n\n1. Data diversity conditioned on the same conditioning frames.\nWe are currently investigating the diversity of frame-conditioned generation and will share updated results during the discussion phase.\n\n2. Regarding energy conservation or implied timescales against MD.\nImplied timescales are commonly used in Markov State Models (MSMs) to characterize the dynamics of equilibrated systems. However, the MD trajectories in ATLAS are not well equilibrated, making standard implied timescale analysis unreliable in this context. To address this limitation, we instead analyze the dominant tICA component across varying lag times (Figure 5A) as a proxy.\nWe are currently investigating energy conservation during trajectory generation using the newly integrated MadraX and will update the relevant results during the discussion phase.\n\n3. CA clash peptide bonds quality metrics.\nWe calculated the CA clash/break rate and peptide bond break rate following the evaluation code of ConfDiff [3]. Specifically, CA clash is determined when the distance of two alpha carbon atoms are within 2 * 1.7 - 0.4 = 3.0 Å, and CA break is determined if CA of two consecutive residues are greater than 3.8 + 0.4 = 4.2 Å. For peptide bond, we consider a broken peptide bond with have bond length greater than 1.4 Å.\nAs discussed in the response to Weakness 2, we have include a standard set of protein conformation quality evaluation using MolProbity.\n\n4. Conformation sampling for proteins with stable alternative locations.\nWe thank the reviewer for bringing this work to our attention. The referenced paper presents challenging protein cases where conformational changes are difficult to sample using state-of-the-art generative models. As the benchmark includes experimental data that differs from the MD-based benchmark we use, we are currently investigating these cases and are willing to share results if obtained during the discussion period.\n\n------\n\n[1] Williams CJ, Headd JJ, Moriarty NW, et al. MolProbity: More and better reference data for improved all-atom structure validation. Protein Sci. 2018;27(1):293-315.\n[2] Orlando G, Serrano L, Schymkowitz J, Rousseau F. Integrating physics in deep learning algorithms: a force field as a PyTorch module. Bioinformatics. 2024;40(4):btae160.\n[3] https://github.com/bytedance/ConfDiff"}, {"review_id": "taQF9HNaiU", "reviewer": "Reviewer_JNBc", "summary": "The paper presents a method that holistically takes into account both time-independent and time-sequence protein conformations.\nThis is done by a representation where the encoder is at the level of individual frames, and the sequence are modeled using masked auto-regression. The same model is used for conformation dynamics, time-independent conformations and conformation interpolation.", "questions": "Q1: From the perspective of molecular dynamics, the concept of (machine learning based) coarse grained dynamics is related to your work, I suspect. Could you please comment on this? See e.g.:\n- Arts et al., \"Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics\".\n\nQ2: You start the paper with Langevin dynamics, which is a second order differential equation, thus the full state would not only be the positions of the atoms, but also their velocities. As far as I understand, you don't model velocities, and don't take them into account in any way. Do you have any comments or thoughts on this?\n\nQ3: You explicitly do not take any temporal information into account in the FrameEncoder, because you want to model both time-independent frames, and frame sequences. Specifically for the task of frame sequences, do you think this causes a decrease in performance?\n\nMinor remarks/suggestions:\n- Fig. 1: iii): the second block has a black outline. Should this not be the last block, corresponding to $x^L$?\n- line 89: insights **in** the\n- line 129: **A** similar'\n- line 140: sequence\n- line 148: Since both the encoder ..\n- line 151: Details .. are\n- line 183: **is** invariant .. and **is** ..\n- line 219 Baselines. Can you please provide citations here for MDGen, AlphaFlow and ConfDiff?\n- line 248: compare**d**", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strenghts:\n\n- The paper is very well written and pleasant to read (though I advise thoroughly proof-reading during rebuttal for typo's, see my minor suggestions below). The authors have carefully thought about which concepts to explain, and I appreciate the space given to practical insights and discussions, and motivations for the experimental setup, while leaving prior (relatively well-known) concepts, such as the SE(3) diffusion model, for the appendix or relevant literature.\n- Experimental validation yields convincing results and the setup seems reasonable and is well motivated.\n- I appreciate the clear and thorough Limitations section.\n\nWeaknesses:\n\n- The method is relatively simple, from a theoretical perspective, though this is not necessarily a weakness.\n- Validation is a bit tailored to the problem, but this seems common in this relatively new research area, where standard benchmarks are not yet widely established (as the authors point out themselves). I've only recently switched to this application domain, so other reviewer might be better positioned to gauge this.", "was_revised": true, "final_justification": "Thank you for the extra evaluations and the clear answers to my questions. I retain my score", "rebuttal": "We thank the reviewer for affirming our manuscript as well-written, well-organized, and pleasant to read, and for recognizing the value of our insights and discussions, as well as the soundness of our results.\n\nBelow are our responses to the questions and weaknesses raised:\n\n### Q1: Relations and differences with two-for-one\nWe appreciate the reviewer for bringing the \"Two-for-One\" paper to our attention. While both Two-for-One and ConfRover explore the relationship between conformational distributions and dynamics using diffusion-based generative models, their focuses and methodologies differ: Two-for-One does not explicitly learn a temporal generative process $p(x|x^{<l})$; instead, it uses a score function learned via diffusion as an approximate coarse-grained force field and requires simulating dynamics through MD; In contrast, our model directly captures both the static distribution $p(x)$ and temporal dynamics $p(x|x^{<l})$ within the generative framework itself, allowing us to generate both time-correlated and uncorrelated samples via diffusion.\n\nNonetheless, we find the Two-for-One paper insightful in unifying distribution learning and dynamics generation from a physics perspective, and we will include it appropriately in the revised manuscript.\n\n### Q2: ConfRover does not explicitly consider velocity\nIt is correct that velocity is not explicitly encoded in ConfRover, as it is absent in ATLAS. As a result, the model relies solely on temporal relationships between coordinate frames.\nHowever, we expect the attention-based temporal module to infer higher-order motion information, akin to velocity, by reasoning over coordinate differences and time intervals between frames. Unlike velocity, which captures local changes over short time scales, frame-to-frame differences across multiple scales may provide richer dynamic information.\nYet, explicit velocity data could offer additional guidance on motion direction, and we would be happy to explore its potential benefits when such data becomes available in future datasets.\n\n### Q3: FrameEncoder does not contain temporal information\nIn ConfRover, the FrameEncoder encodes only structural information, while temporal information is incorporated separately in the temporal module using rotary position encoding (RoPE) [1]. Specifically, we encode time as discrete snapshot indices (e.g., 128, 256, 384, ...) corresponding to each frame (0, 1, 2, ...). We use RoPE for temporal encoding based on two key considerations: (1) It enables the model to learn dynamics at real temporal scales, matching the snapshot interval resolution (e.g., 10 ps in ATLAS). (2) Unlike hard-coding time into frame-level embeddings, RoPE encodes relative time rather than absolute time, maintaining invariance under global time shifts.\n\nWhile we intentionally avoid injecting absolute time into the FrameEncoder, it may be beneficial to encode the trajectory \"stride\" to enhance multi-scale learning. We leave this direction for future investigation.\n\n### Weakness 1: The method is relatively simple\nWe consider the simplicity of our model a merit rather than a weakness. Despite its simple form, the core idea of learning different perspectives of the protein conformational landscape via conditional probability is carefully designed so that we can use a modern causal language model architecture enables a unified and flexible approach across multiple protein conformation tasks.\n\n### Weakness 2: the evaluation are not standard \nThis is an inherent challenge in developing models for protein dynamics, and we see it as a potential contribution of our work. \nStandardized evaluation protocols are lacking for this problem, especially for large proteins where the MD trajectory are not fully equilibrated. We designed metrics to capture aspects of dynamic behavior as comprehensively as we can. Our evaluation assesses how models respond to different initial conformations and time scales and, for long-time simulation, whether they capture main dynamics and recover conformational states.\n\nDuring the rebuttal, we also added structural quality and energy-based evaluations using standard pipelines to complement our analysis, particularly on whether generated conformations are plausible. Specifically, we used MolProbity [2] to assess geometric accuracy and MadraX [3] for heavy-atom energy evaluation. \nWe evaluated 38 cases tested in forward simulation and conformation interpolation tasks, and compare across different models.\nResults are summarized below (all values reported as mean ± std; energy values as 95% percentile ranges). It shows both ConfRover (forward simulation) and ConfRover-interp have similar structural quality, comparable to MD references in backbone dihedral (Ramachandran), bond geometry, and overall energy. While there are more clashes detected than MD references, it also depends on ad hoc hydrogen addition and may bias evaluation of heavy-atom-only models. Overall, ConfRover consistently outperforms MDGen across all metrics.\n\n| exp               | Ramachandran outliers % | Ramachandran favored % | Rotamer outliers | Clashscore | RMS(bonds) | RMS(angles) | MolProbity score | MadraX Energy          |\n|-------------------|--------------------------|--------------------------|-------------------|-------------|-------------|---------------|-------------------|----------------------------------------|\n| MD Reference       | 0.38±0.49                | 97.41±1.49               | 1.02±0.89         | 0.04±0.16   | 0.01±0.00   | 1.88±0.05     | 0.72±0.18         | -519.3 (-1793.0, -53.4)                |\n| MDGen              | 0.93±0.86                | 94.98±2.04               | 2.86±1.59         | 16.14±20.05 | 0.02±0.02   | 2.13±0.30     | 2.24±0.40         | -314.7 (-1483.8, 263.6)                |\n| ConfRover      | 0.58±0.63                | 96.93±1.45               | 1.98±1.48         | 7.81±6.52   | 0.01±0.01   | 1.88±0.25     | 1.72±0.38         | -522.2 (-1858.9, -53.4)                |\n| ConfRover-interp   | 0.71±0.94                | 96.90±1.95               | 1.86±1.46         | 7.25±8.74   | 0.02±0.01   | 1.91±0.32     | 1.61±0.51         | -469.7 (-1712.3, -42.8)                |\n\n\nWe hope these efforts provide a comprehensive understanding of ConfRover and inform future benchmarks.\n\n\n### Minor points:\n  - Clarification of Figure 1A (iii): This figure illustrates how we condition on both the first and last frames in the causal sequence model. The second box represents frame 0, the original first frame, and the first box represents the last frame (arrows indicate moving it to the front of the sequence). Both boxes have black outlines to mark them as conditioning frames. We will clarify this further in the revised manuscript.\n  - We thank the reviewer for pointing out typos and will correct them in the revision.\n\n------\n[1] Su, Jianlin, et al. \"Roformer: Enhanced transformer with rotary position embedding.\" Neurocomputing 568 (2024): 127063.\n[2] Williams CJ, Headd JJ, Moriarty NW, et al. MolProbity: More and better reference data for improved all-atom structure validation. Protein Sci. 2018;27(1):293-315. \n[3] Orlando G, Serrano L, Schymkowitz J, Rousseau F. Integrating physics in deep learning algorithms: a force field as a PyTorch module. Bioinformatics. 2024;40(4):btae160."}, {"review_id": "pPvZnMSu0D", "reviewer": "Reviewer_A8Hu", "summary": "The paper introduces a new method for generative modelling of protein MD trajectories. An MD trajectory is considered as a sequence of 3D structures that are temporally related. Thus, the proposed model autoregressively generates/samples a trajectory frame-by-frame. The model is benchmarked for 3 sampling problems compared to other existing models for the same tasks, and tends to perform favorably.", "questions": "See questions within weaknesses.", "limitations": "Yes, I think some limitation are discussed.\n\nIts worth asking: Why are proteins dynamic and undergo conformational changes? There is some amount of inherent dynamics of atomic systems involved, and this paper tackles the problem well. However, another major reason proteins undergo conformational changes is to interact with other partner molecules. It may be worth mentioning that the current method is limited in modelling interactions (or at least, can only implicitly model interactions at present).\n\nIt may also be interesting to discuss relative trade-offs vs. emerging methods in machine learning interatomic potentials for organic systems including proteins. Examples include MACE OFF (https://pubs.acs.org/doi/10.1021/jacs.4c07099) and the Orb family of models, which were applied to a simulating the dynamics of a couple enzyme proteins.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- **Novel idea:** The idea of framing generative modelling of molecular dynamics trajectories as autoregression is exciting and new, to the best of my knowledge.\n\n- **Good results on standard benchmarks:** Empirical results look strong relative to baselines, overall. The experimental setup largely follows past published papers based on my understanding (except S.4.3 highlighted below). I will rely on other reviewers' assessment on the rigour of the experiments.\n\n- Figure 1 does a good job showing the pipeline and main contributions.\n\nWeaknesses:\n\n- **No standard deviations:** For almost all the tables, there are no standard deviations or variations reported. Only an averaged metric is reported. This is esp. important for several of the claims of outperforming other models when the results are very close (eg. Table 3). It may be the case that several models are within an error range of one-another, making it harder to make an outright claim of SOTA performance.\n     - I realise that it may be conventional to train only one big model and report the best mean metric. However, it can still be useful to readers to additionally report some appropriate measure of variance in performance over your test set.\n     - I will adjust my assessment and increase my score if the authors can address this concern in the rebuttal.\n\n- **Missing inference time metrics:** I would be interested to see some metrics on the inference time for the proposed model as well as the previous models being compared to, as well as how that compares to actually running an MD simulation. I think this is important, b/c the goal of this line of work broadly is to accelerate and augment MD simulations. So are current models actually faster than running MD?\n    - As an example, in the conformation interpolation task: Say I am a user and I know the two conformational states of my protein. Now, I can either run an MD simulation or I can use your AI method (which is slightly less accurate than MD). Thus, I will only ever run your AI method if it is much faster than MD. Would you agree with this assessment?\n\n- (Minor) **Conformation interpolation results and associated claim (S.4.3)** The model requires finetuning to perform this task. However, when I read the Abstract/Intro/Conclusion, I got the impression that the model can perform this task out of the box (e.g. statements like \"CONFROVER supports three tasks in a unified manner\"). I think this claim needs better support/more nuance.\n\n---\n\nSubjective note on presentation:  As a reader who works in closely related fields, but not deeply familiar with all aspects of this paper's field, I found that the paper's presentation of the technical content (method, experimental setup) was a bit hard to grasp. Esp. the methods were presented in what I felt was a very dense/condensed form, and required jumping back and forth to references in the appendix. I would suggest trying to improve the presentation of the methods.\n\nThe Intro/Conclusion/etc. reads very well.", "was_revised": true, "final_justification": "The response addresses my concerns satisfactorily. I also read the other reviews' discussions, and have updated my assessment as a result.\n\nI think that, if possible, authors should try to run repeats of AlphaFlow, too, for completing the revised tables with replicates.", "rebuttal": "We thank the reviewer for acknowledging this work contains exciting novel ideas with strong empirical results, and the general evaluation follows standard practice. And we appreciate the constructive feedback on the current manuscripts and we provide responses as follows:\n\n### Weakness 1: Missing variance estimate on model performance\n\nWe thank the reviewer for the suggestion and agree that providing variance estimation on metrics would enhance the interpretability of the results beyond the standard mean metrics. We repeated the inference for our model and baseline models five times with different random seeds. From which, we report the mean and standard deviations for each experiment as follows:\n\n- Pearson correlations of conformation changes between sample and reference on multi-start (Table 1):\n|    Type   |      Model     | Traj.     | Frame     | ∆Frame   |\n|----------|-----------|----------|----------|----------|\n| CA | MDGen      | 0.56±0.03 | 0.47±0.03 | 0.41±0.02 |\n| CA | ConfRover  | 0.75±0.01 | 0.63±0.01 | 0.53±0.01 |\n| PCA | MDGen     | 0.18±0.01 | 0.15±0.01 | 0.10±0.01 |\n| PCA | ConfRover | 0.73±0.01 | 0.50±0.01 | 0.43±0.00 |\n\n\n- State recovery in 100ns simulation (Table 2). *MD reference does not contain repeats.*\n| exp       | Avg JS-Dist   | Avg Recall   | Avg F1    |\n|:----------|:--------------|:-------------|:----------|\n| MDref     | 0.31      | 0.67     | 0.79  |\n| MDGen     | 0.56±0.01     | 0.29±0.01    | 0.42±0.01 |\n| ConfRover | 0.51±0.01     | 0.42±0.00    | 0.58±0.00 |\n\n- Pearson correlation on main dynamic mods between sample and reference trajectory. (Figure 5A shown as table.) *MD reference does not contain repeats.*\n|  Components    |     Model     | lag=1         | lag=5         | lag=10        | lag=20        |\n|------------------|----------------|--------------|-------------|------------|----------|\n| PC 1 | MD 100ns     | 0.13  | 0.17  | 0.18  | 0.22  |\n| PC 1 | MDGen     | 0.10±0.01 | 0.11±0.02 | 0.12±0.01 | 0.13±0.01 |\n| PC 1 | ConfRover | 0.16±0.02 | 0.18±0.01 | 0.19±0.01 | 0.20±0.01 |\n| PC 2 | MD 100ns     | 0.17  | 0.16  | 0.18  | 0.18  |\n| PC 2 | MDGen     | 0.11±0.01 | 0.11±0.01 | 0.11±0.01 | 0.12±0.01 |\n| PC 2 | ConfRover | 0.18±0.01 | 0.17±0.00 | 0.18±0.01 | 0.19±0.01 |\n\n- Time-independent conformation sampling (Table 3). *We were not able to get repeated results from AlphaFlow due to long inference time and limited resource.*\n| exp       | Pairwise RMSD r   | Per target RMSF r   | RMWD      | MD PCA W2   | Joint PCA W2   | Weak contacts J   | Transient contacts J   | Exposed residue J   |\n|:----------|:------------------|:--------------------|:----------|:------------|:---------------|:------------------|:-----------------------|:--------------------|\n| AlphaFlow | 0.53              | 0.85                | 2.64      | 1.55        | 2.29           | 0.62              | 0.41                   | 0.69                |        \n| ConfDiff  | 0.54±0.00         | 0.85±0.00           | 2.70±0.01 | 1.44±0.00   | 2.22±0.04      | 0.64±0.00         | 0.40±0.00              | 0.67±0.00           |\n| ConfRover | 0.51±0.01         | 0.85±0.00           | 2.66±0.02 | 1.47±0.03   | 2.23±0.04      | 0.62±0.01         | 0.37±0.01              | 0.66±0.01           |\n| MDGen     | 0.47±0.04         | 0.72±0.02           | 2.78±0.04 | 1.86±0.03   | 2.44±0.04      | 0.51±0.01         | 0.28±0.01              | 0.57±0.01           |\n\n\nWith the updated results, we confirm the following:\n(1) In forward simulation, ConfRover outperforms MDGen in predicting dynamic levels, capturing major motions, and recovering conformational states.\n(2) For time-independent conformation generation, ConfRover performs on par with specialized models such as AlphaFlow and ConfDiff. Its performance on metrics including 'per-target RMSF r', 'RMWD', 'MD PCA W2', 'Joint PCA W2', 'Weak contacts J', and 'Exposed residues J' is generally within one standard deviation or better, compared to ConfDiff and AlphaFlow.\n\n### Weakness 2: Inference time metrics comparisons with MD simulation.\n\nWhile we briefly discussed inference time in Appendix A.2, we recognize the importance of providing more detailed comparisons with typical MD simulation runs. Below, we include a detailed runtime analysis comparing ConfRover and MD simulations, using the same hardware (NVIDIA H100-80G). Specifically, we measured the wall-clock time required to generate 100 ns ATLAS trajectories (80 frames) for proteins of varying sizes and report the average inference time per size bucket. For comparison, we also selected a representative protein from each bucket and estimated the time required to simulate 100 ns using OpenMM with implicit solvent.\n\n|     seqlen|      (0,150) |     [150,300) |    [300,450) |   [450,600) |   [600, 724] |\n|:----------|-------:|--------:|--------:|-------:|--------:|\n| ConfRover | 6.99 | 7.53 | 10.92 | 15.88 | 20.83 |\n| MD  |104.54 | 207.92 | 386.69 | 651.29 | 1099.13 |\n| Speedup | 14.95 | 27.61 | 35.41 | 41.01 | 52.77 |\n\nAs shown in the table, ConfRover provides clear speedup for 100ns simulation, with even more pronounced acceleration for larger proteins.\nBeyond faster simulation with larger strides, ConfRover also supports time-independent sampling for generating independent conformations in parallel, and interpolation between terminal conformations, capabilities that are not easily achievable with classical unbiased MD simulations.\nWhile targeted MD simulations could in principle be used for transition pathway sampling, implementing and benchmarking such setups is beyond the scope of this short rebuttal period. Nonetheless, we believe the reported unbiased MD simulation speeds offer a concrete and fair basis for comparing inference efficiency, and they highlight the practical advantages of ConfRover.\n\n### Weakness 3: unification of tasks\nThank you for bringing this up. By \"unification\" we refer to a shared modeling framework that can learn from multiple tasks, rather than a single model that performs all tasks equally well out of the box.\nWhile training a single model for all three tasks is possible, different training strategies often yield models with varying strengths. In our early experiments, we trained a single model jointly on three tasks, but observed slightly degraded performance on time-independent sampling (e.g., on contact prediction). As a result, we adopted a two-stage training strategy, continuing training for interpolation to better balance task performance.\n\n### Other comments\n\n1. **Methods part being too dense.** We are sorry that the reviewer found the Methods section difficult to follow. Due to page limitations, we had to condense some technical content and move certain model details (e.g., SE(3) diffusion) to the supplementary materials. With additional space available in the final version, we aimed to provide smooth reading experience for readers from diverse backgrounds. We would greatly appreciate any specific suggestions to help us strengthen the revised manuscript.\n\n2. **Limitations on modeling complexes and interactions.**\nWe appreciate the reviewer for pointing out this important limitation. While the ConfRover framework is architecture agnostic and can, in principle, incorporate more advanced components such as AlphaFold3-style encoder-decoders for modeling complex interactions, the current implementation does not explicitly model such interactions. Additionally, a key challenge in this direction is the scarcity of high-quality datasets capturing interaction dynamics. We have identified promising datasets such as MISATO [1], which include protein-ligand interaction trajectories. We will acknowledge this limitation in the revised manuscript and highlight it as an important avenue for future work.\n\n\n3. **Comparison with MLFF.**\nThanks for the suggestion. ConfRover differs from machine learning force fields (MLFFs) in terms of the trade-off between accuracy and efficiency, as well as the timescales targeted. MLFFs, such as MACE-OFF and Orb, are designed to integrate high-level quantum accuracy into classical molecular dynamics (MD) simulations, aiming to improve accuracy while retaining scalability. However, these models still rely on MD-like sequential sampling with small integration steps (fs level) and are often more computationally expensive than classical MD, hindering their ability to efficiently capture long-timescale conformational changes.\nIn contrast, dynamic generative models like ConfRover are tailored for capturing slower dynamics on longer timescales, such as ps to ns. In addition, ConfRover enables non-sequential sampling strategies, including time-independent conformation generation and interpolation between terminal states, making it well-suited for exploring broad conformational landscapes. We will add this to discussion.\n\n\n-------\n[1] Siebenmorgen T, Menezes F, Benassou S, et al. MISATO: machine learning dataset of protein–ligand complexes for structure-based drug discovery. Nat Comput Sci. 2024;4(5):367-378."}]

Author rebuttal:

Meta-review: (5,5,5,5) This paper introduces ConfRover, a generative model for protein conformations learned from MD data. The model explicitly captures temporal dependencies via an autoregressive framework that unifies time-independent sampling, forward simulation, and conformational interpolation. All reviewers were positive, noting strong performance on the ATLAS benchmark (100 ns MD simulations), where ConfRover outperforms other learning-based approaches and achieves substantial speedups over classical MD, though still falling short of full physical fidelity. The rebuttal added thorough structural/energy evaluations, runtime analysis, and new baselines.
Decision: Accept (poster)