TEMPO: Temporal Multi-scale Autoregressive Generation of Protein Conformational Ensembles
Authors: Yaoyao Xu
Di Wang
Zihan Zhou
Tianshu Yu
Mingchen Chen
Venue: NeurIPS 2025 poster ()
Keywords: Protein
protein dynamic
AI4Science
Primary area: machine_learning_for_sciences

Abstract: Understanding the dynamic behavior of proteins is critical to elucidating their functional mechanisms, yet generating realistic, temporally coherent trajectories of protein ensembles remains a significant challenge. In this work, we introduce a novel hierarchical autoregressive framework for modeling protein dynamics that leverages the intrinsic multi-scale organization of molecular motions. Unlike existing methods that focus on generating static conformational ensembles or treat dynamic sampling as an independent process, our approach characterizes protein dynamics as a Markovian process. The framework employs a two-scale architecture: a low-resolution model captures slow, collective motions driving major conformational transitions, while a high-resolution model generates detailed local fluctuations conditioned on these large-scale movements. This hierarchical design ensures that the causal dependencies inherent in protein dynamics are preserved, enabling the generation of temporally coherent and physically realistic trajectories. By bridging high-level biophysical principles with state-of-the-art generative modeling, our approach provides an efficient framework for simulating protein dynamics that balances computational efficiency with physical accuracy.

TEMPO: Temporal Multi-scale Autoregressive Generation of Protein Conformational Ensembles
Yaoyao Xu 1, Di Wang Zihan Zhou², Tianshu Yu Mingchen Chen
à School of Data Science, The Chinese University of Hong Kong, Shenzhen
4 Changping Laboratory, Beijing
{yaoyaoxu, zihanzhoul]@linkk cuhk edu. en, yutianshu@cuhk, edu cn (lotus, mwwgccenccenneeeeee....ooocc acp..ch.cccccc..c.c.... ac.
Abstract
Understanding the dynamic behavior of proteins is critical to clucidating their functional mechanisms, yet generating realistic, temporally coherent trajectories of protein ensembles remains significant challenge. In this work, introduce a novel hierarchical autoregressive framework for modeling protein dynamics that leverages the intrinsic multi-scale organization of molecular motions. Unlike existing methods that focus on generating static conformational ensembles or treat dynamic sampling protein dynamics as a Markovian process. The framework employs a two-scale architecture: low-resolution model captures slow, collective motions driving major conformational transitions, while high-resolution model generates detailed local fluctuations conditioned on these large-scale movements. This hierarchical design ensures that the causal dependencies inherent in protein dynamics are preserved, enabling the generation of temporally coherent and physically realistic trajectories. By bridging high-level biophysical principles with state-of-the art generative modeling, our approach provides an efficient framework for simulating protein dynamics that balances computational efficiency with physical accuracy.
independent process, approach characterizes
Introduction
The intersection of artificial intelligence and protein science has revolutionized our understanding of biological systems. Recent breakthroughs in AI for protein research have transformed structure and
function prediction 33 37 46 [22] protein design 3 25 and interaction modeling (9] 14147 (3)]
However, while static structural understanding has advanced dramatically, accurately modeling protein dynamics remains an outstanding challenge the frontier of computational biology.
Protein dynamics are characterized by two fundamental propertics. First, they are inherently hierarchi- cal and multi-scale, with motions naturally separating into slow collective movements (nanoseconds to microseconds) that typically correspond to functionally relevant conformational changes, and fast local fluctuations (picoseconds to nanoseconds) that reflect atomic-level interactions [15] 17). This multi-scale organization forms the theoretical foundation for analytical methods like Principal Component Analysis and Normal Mode Analysis R demonstrating that protein dynamics can be effectively decomposed into essential subspaces operating at different time scales. Second, protein motions exhibit strong temporal correlations, where the continuous evolution of conformational states follows specific pathways critical for biological functions. These temporally correlated dynamic ensembles have proven essential in understanding enzyme catalysis mechanisms [48], characterizing binding pathways 7 and elucidating allosteric regulation [3]] For instance, recent studies
Corresponding authors
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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have demonstrated that analyzing dynamic ensembles can reveal cryptic binding sites that only become accessible through specific conformational transitions [5].
While molecular dynamics (MD) simulations can naturally capture both properties by solving New- ton's equations of motion at atomic resolution, their computational demands make them impractical for large-scale applications, Even with specialized hardware and enhanced sampling techniques, MD simulations are typically limited to microsecond timescales and small protein systems, making it chal- lenging to systematically study slow conformational changes or analyze large protein datasets [38]. Current generative approaches, particularly diffusion- based models [99 23 10, fundamentally fail to leverage these characteristics. These methods generate conformational ensembles by simultaneously producing and optimizing protein states independently, by learning an energy landscape rather than capturing the true sequential and multi-scale nature of protein dynamics. This approach cannot accurately represent the causal chain of events that governs protein conformational changes, limiting their ability to generate physically consistent trajectories.
Motivated by these challenges. we propose TEMPO multi-scale autoregressive framework that models and generates protein dynamics across different temporal scales. Our approach combines low-resolution model capturing essential conformational transitions with high-resolution model generating detailed local fluctuations, directly translating biophysical principles into a computational framework for generating physically realistic trajectories. Extensive experiments demonstrate that our method achieves significant improvements over existing approaches.
Our work makes several key contributions:
Algorithm Design. TEMPO introduces novel multi-scale framework that captures both protein collective motions and local fluctuations, enabling efficient trajectory generation orders of magnitude faster than MD simulations.
Performance and Metrics. Our method achieves state-of-the-art performance in both structural accuracy and computational efficiency in various metrics, outperforming existing methods in matching MD ground truth while requiring fewer computational resources.
Extensive Analysis. We demonstrate TEMPO's 's ability to capture biologically meaningful protein motions through comprehensive case studies and analyses.
2
Related Work
Protein Ensemble Generation. Recent advances in deep learning have revolutionized the generation of protein conformational ensembles. Traditional approaches rely on MSA subsampling with Al- phaFold2 (32. which provides limited control over conformational diversity. Modern deep learning methods have introduced more sophisticated techniques. AlphaFlow [19] fine-tunes single-state predictors under flow matching framework to generate protein conformational ensembles. ESM- Flow 19 extends this approach by leveraging protein language models. BioEMU [23] employs diffusion-based framework to generate thermodynamically accurate ensembles. ConfDiff 43 incorporates force-guides networks with diffusion models to enhance generation fidelity, while Str2Str (27) introduces
translation framework with roto- translation equivari- ance. However, these methods primarily focus on generating conformational ensembles that match equilibrium distributions, without explicitly modeling the temporal evolution of protein structures.
Learning Molecular Dynamics. Machine learning approaches have emerged as powerful tools for accelerating and enhancing molecular dynamics simulations. VAMPNet (2 pioneered the use of variational approaches for Markov processes in molecular kinetics. Recent works like DifIMD [45] employ diffusion models to estimate conformational density gradients, while DFF [4] establishes connections between score-based generative models and molecular force fields. The Distributional Graphformer (DiG) [50] predicts equilibrium distributions of molecular systems, enabling efficient conformational sampling. However, these methods often focus on general-purpose force field learning or small molecular systems, making them computationally intensive for large proteins.
Multi-scale Dynamics Modeling. The inherent multi-scale nature of protein dynamics has long been recognized in computational biology. Traditional MD analysis methods decompose protein motions into slow collective changes, which are crucial for biological function, and fast local fluctuations contribute to overall stability. Recent deep learning approaches have begun address

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ESTAVHPL
Resolutio Generation
f
Slow Scale
MLP
X
X1
X
Emb
Qoose
Attention
Daseet
Depacial
Fcate
x1
x
X
GRU
Resesporn
High Resolution Generation
Nea
Figure I: Overview of our multi-scale protein dynamics generation framework. a) The hierarchical free energy landscape protein dynamics, where slow motions (upper) guide fast local fluctuations (lower). b) Our two-stage generation process: the low-resolution model fă captures slow collective motions, while the high-resolution model f fills in detailed dynamics. e) The neural architecture that parameterizes both models features spatial-temporal encoding of protein conformations.
this multi-scale characteristic. EigenFold 20 models protein structures as systems of harmonic oscillators, naturally inducing cascading-resolltion generative process along system eigenmodes. FoldFlow [6] proposes family of flow-based generative models on SE(3), where the continuous-time dynamics naturally capture multi-scale structural variations- from global conformational changes to local refinements through different time scales of the flow evolution. ITO [36] learns transition density operators that allow conditioning arbitrary timesteps, focusing on coarse-grain grained Ca representations with exponential distribution sampling. However, most existing approaches focus on either ensemble generation short-timescale dynamics, without explicitly bridging the gap between different temporal scales in protein motion.
Autoregressive Models in Structural Biology. While diffusion models have recently dominated protein structure generation, auto-regressive approaches are gaining traction for their ability to model temporally coherent and physically consistent dynamics. In the domain of bio-molecules, arDCA 41 applics a simple yet effective auto-regressive framework to model protein sequence distributions, capturing co-evolutionary couplings while enabling efficient sequence sampling and fitness prediction. For protein structure modeling, Structure Language Models (26) employ latent- space auto-regression
efficiently generate diverse backbone conformations, while equivariant models such as EquiJump [12] build an -03uivequiaa transport model that bridges long time intervals of all-atom protein MD by stochastically interpolating between snapshots. These advances demonstrate the promise of regr-ssgrese models in capturing complex hio-molecular dynamics across multiple timescales.
3 Method
3.1
Preliminaries
Given an initial protein structure Xo of sequence length L, our goal is to learn generative model that produces protein backbone trajectories X = _ XT), where each X; represents the backbone conformation time step While full-atom protein structure representation has been widely adopted [32 23 19,, we focus on backbone dynamics as they capture the essential conformational changes that determine protein function. This choice is motivated by several key observations: (1) many important biological processes, such as protein folding and large-scale conformational changes, primarily determined by backbone movements (33) (2) side-chain motions typically occur faster timescales. They can be considered as local fluctuations around backbone configurations [16].
Consistent with standard representations in protein modeling (32), describe each residue's back- bone conformation using an SE(3) frame along with the torsion angles (ò, 1h, w):
x = ((R, t), (e) w)), x ((SE(3) x T D T
(1)
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where t denotes the time step and indicates the residue index. The SE(3) frame, representing the rigid body transformation, consists of a rotation component encoded as a umit quaternion from the positive real part Q+ C I and a translation vector in I yielding a 7-dimensional representation. Each torsion angle is encoded as 2D vector [sin(0), cos(O)] E S to avoid discontinuities at the periodic boundary. Combining these components, each residue at each time step represented by 13-dimensional vector:
x @ + R3
x (5 CR'
(2)
This representation captures the essential geometric features of protein backbone dynamics while maintaining computational efficiency, and the detailed data processing could be found in Appendix B
3.2 SDE-based Protein Dynamics Modeling
Protein dynamics in solution naturally follows Langevin dynamics, which describes the motion of particles under both conservative forces and random collisions with solvent molecules [49]. The Langevin equation captures two key aspects of protein motion: (1) Deterministic forces arising from inter-atomic interactions that drive conformational changes, (2) Random forces from thermal fluctuations that contribute to the stochastic nature of protein dynamics ...
Motivated by this physical principle, we model protein dynamics as a stochastic differential equation (SDE) process, which can be viewed as a continuous- time generalization of the Langevin dynamics. In our framework, the evolution of protein conformations follows combination of deterministic drift and stochastic diffusion. Specifically, for a protein conformation X, at time t, its temporal evolution can be described as:
dxx, = ((X,)dtt dt + odW,
where (x(x,) represents the drift term that captures the deterministic dynamics, is the diffusion coefficient, and W, denotes standard Brownian motion. The drift term μ(X,) ) is learned by our model, while the stochastic component simulated through Gaussian noise injection.
In discrete time steps, our model approximates this continuous SDE process as:
X = X, + fo(X,) + GVAL
where fe is our neural network model parameterized by 0 that learns the drift dynamics, At is the time step, and é ~ (N0 represents the Gaussian noise. This formulation allows our model capture both the deterministic conformational changes and the stochastic nature of protein dynamics. This approach is theoretically justified as we learn the conditional expectation E[X++1 X.] in the finite time-step regime, mirroring how numerical MD integrators operate with deterministic updates plus controlled stochastic components for temperature regulation [49
However, protein dynamics typically exhibits non non-Marte Markovian behavior at short time scales [18), meaning that future states depend on multiple previous states rather than just the current one. To account for this memory effect, extend our model to consider multiple timesteps:
X = fo(X e A
where X₃-1:t represents two consecutive frames at times - 1 and f, and the model predicts the next two frames X.+1:t+2. This design choice is supported by prior research in physical system modeling (51) and latent ODEs (34), where incorporating temporal memory has been shown to significantly improve the accuracy of dynamical predictions.
3.3 Multi-scale Dynamics Learning
Building upon the SDE-based framework, we decompose protein motion into a bio-physically motivated two-timescale formulation capture both slow collective motions and fast local fluctuations 13 17. This decomposition is realized through coupled stochastic differential equations:
x Ho (X, dtt + a.dW;
(6)
d uy
(x/ )dt + aydW!
(7)
where X and X/ represent protein conformations at slow and fast timescales, respectively, with corresponding drift terms μ4 and μuy, and independent Brownian motions W,,, W!".

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To learn these multi-scale dynamics. we employ a unified neural architecture that operates at both timescales. Specifically, the drift terms p, and 2j are parameterized by the same spatiotemporal encoder architecture f but trained separately to capture scale-specific features. The discrete time evolution each scale follows:
X = (1(;) GV At,
(8)
X/ - -X/
+ ff + + Vat,
(9)
where At, and At, represent the time steps for slow and fast dynamics, with neural networks fä and fo learning the respective drift dynamics.
During inference, we employ a hierarchical sampling strategy that mirrors the natural organization of protein dynamics. Starting from an imitial conformation X², we first generate sparse trajectory (Xi, X , sc,) using the slow-scale model fz. which captures collective motions like domain reorientations. These scow-scale conformations then serve anchoring states for the fast- scale model f/. which generates the complete fine-grainee trajectory {x/ E [0, At ensuring that fast local dynamics remain consistent with the broader conformational changes.
3.4 Spatiotemporal Protein Encoder
Here detail the neural architecture that parameterizes the drift dynamics both timescales. Our encoder design captures both spatial relationships between protein residues and temporal correlations in conformational dynamics. As illustrated Figure the network comprises three functional encoding, and conformational prediction.
components: imput representation,
The input representation module processes protein conformations X, E RL413 with added noise terms that model the stochastic nature of protein dynamics. Specifically, we sample noise scale T noise ~ U(abb) b) and generate Gaussian noise ~ (N I) to obtain noisy conformations X, = X, + noise In parallel, protein sequences are embedded into feature vectors s IR xxdaat through learnable embedding layer. These sequence features g: E SE(3)* _ and noise scale an Invariant Point Attention (IPA) [2) module to capture geometric relationships. The final latent representation is computed as:
combined with frame representations
hatent MLP(X,) + IPA ( Embed (mnoise)
(10)
where MLP: 1R + Rddhan projects the input features to hidden dimension daidden- and Embed maps the scalar noise intensity to miistee-dimesiinall vector. The latent representation then undergoes -eappal-temppral processing through:
Ispatil = MulttHHeadAAtenntionnnnn (Hasn))
(11)
hrepprrl GRU (hapatia)))
(12)
The spatial module captures inter-residue interactions while the temporal module encodes frame-to- frame dependencies. The output module generates conformational updates:
X X, + MLP (heeppora)))
(13)
where MLP I deaster + 13 maps the latent features back conformational space.
The training objectives for both timescales follow the same formulation, consisting of two terms: a re- construction loss measuring the mean squared error between predicted and ground truth conformations and physical constraint loss penalizing steric clashes between backbone atoms:
Listal = | 2 +
Σ ReLU(112 - ||r - -rrl-
(14)
where r; and ry are positions of backbone atoms from different residues, and the minimum distance
threshold of 1.2Á is chosen following the widely adopted steric criteria in Rosetta [1. This objective applied independently to train the slow and fast dynamics models, with appropriate time intervals At, and yty, respectively.
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4 Experiments
4.1 Experimental Settings
Datasets. We conduct experiments on two comprchensive molecular dynamics datasets: md- CATH (9 and ATLAS [42]. For mdCATH, we randomly sampled 1,000 proteins and their 320K temperature trajectories with three independent seeds for training. Each protein sequence cated to 240 residues, and trajectories were standardized to 400 frames at Ins intervals through periodic extension or truncation. We randomly selected 50 proteins for validation and 64 proteins for lesting, ensuring no overlap with the training set. For ATLAS, we follow the data split and processing protocol established by MDGen (1I.. We rigorously quantified sequence similarity using mmseqs2, finding an average of 18.93% sequence similarity between training and test sets for mdCATH and 18.3% for ATLAS, well below standard thresholds (40%) for sequence relatedness.
trun-
Baselines. To evaluate our method's capability in capturing both protein dynamics and ensemble properties, conduct comprchensive comparisons with four state-of-the-arr baselines: BioEMU 23], AlphaFlow and ESMFlow [9]] and MDGen (21]. While the first three methods primarily focus on generating conformational ensembles, MDGGn, though not explicitly modeling dynamics, captures temporal evolution through training on the ATLAS dataset [42].
Implementation details. Our multi-scale modeling approach captures protein dynamics at two temporal resolutions. The low-resolution model generates trajectories at 20ns intervals, characterizing major structural transitions, while the high-resolution model operates at 1ns resolution to capture local fluctuations. We empirically chose the 20ns/Im Ins hierarchy based on established biophysical principles where the 20ns interval effectively captures major conformational transitions between different states in the free energy surface as visualized in Figure Ta. while the Ins resolution represents the dataset's s finest temporal sampling. The generation process follows hierarchical strategy. The low-resolution model first produces a sequence of conformational states { 20 at At = 20ns. Specifically, each
high-resolution segment is initialized by the corresponding low-resolution state (X/4 = fo(X;)) )), followed by autoregressive sampling (Xi+RAe fo fo(X (((x))))))))) for k = 2, ..., 20 where At Ins). This hierarchical process generates complete trajectory while maintaining consistency across different temporal scales.
During training. both scale-models simulate the forward process of protein dynamics SDE through autoregressive sampling with noise scales uniformly sampled from [0.01, 0.05]. At inference time, while the low-resolution model maintains similar noise levels, we increase the noise scale to 5.0 for the high-resolution model. This elevated noise level enables diverse conformational sampling on the learned energy surface.
Evaluation Framework. Our comprehensive evaluation framework encompasses both ensemble properties and trajectory-specifie characteristics. Following AlphaFlow, we analyze conformational flexibility through several complementary measures: Dynamic Range (the average Ca-RMSD between pairs of conformations within each ensemble, quantifying the overall conformational space explored), Local Flexibility (assessed through root mean square fluctuation (RMSF) analysis of atomic positions), and Distribution Accuracy (quantified using the root mean Wasserstein distance (RMWD) between predicted and ground truth conformational distributions).
The trajectory accuracy measured by the backbone RMSD error between generated and ground truth trajectories relative to the native structure: Error |RMSD pred - RMSD This metric, averaged across all frames and test proteins, quantifies the model's 's ability to capture conformational change magnitudes accurately [24]. Further biological validation includes contact dynamics analysis, where contacts between Ca atoms (8Å threshold) are classified as weak (initially present but dissociate in 10% of the ensemble) or transient (initially absent but form in 10% of the ensemble), with accuracy evaluated using Jaccard similarity between predicted and ground truth contact sets. Computational efficiency assessed through the average inference time per protein in generating 400 snapshots. Additionally, we calculate the clash ratio, defined as the proportion of conformations containing steric clashes among the 400 generated snapshots for each protein. Detailed definitions of all metries are provided in Appendix D

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Table 1: Evaluation on mdCATH. Comparing predicted ensembles with MD ensembles across various metrics. For protein flexibility and RMSF, ground truth values are in parentheses. Median values across 64 test ensembles are reported. The rightmost column shows TEMPO's performance
on the up-sampling task. r: Pearson correlation; J: Jaccard similarity; W2: 2-Wassrsneiin distance.
Metries
TEMPO
BioEMU
AlphaFLow-MD
ESMFLOw-MD
MDGEN TEMPO(Up)
Pairwise RMSD(= 3.26)
2.78
13.82
2.00
2.32
1.11
Pairwisc RMSD 1
0.77
-0.02
0.41
0.26
0.71
0.99
All-atom RMSF(= 1.64)
1.60
10.08
0.99
1.18
0,56
1.64
Global RMSF
0.67
0.13
0.41
0.34
0,67
0.99
Root W/2
4.21
10.70
5.62
4.08
3.36
1.06
MD PCA W32
2.33
2.49
2.38
2.36
2.62
0.63
% PC-sim > 0.5
7.81
9.38
21.88
25.00
17.19
95.31
Weak cuntacts
0.43
0.38
0.42
0.51
0,41
0.83
Trans, contacts J
0.20
0.12
0.27
0.28
0,20
0.60
% Clash ratio
4.75
19.7
15.5
5.23
0.42
4.12
RMSD Error
1.78
3.76
0.60
Inference (hour)
0.006
0.25
4.5
4.7
0.008
4.2 Results and Analysis
We evaluate our framework on both mdCATH and ATLAS datasets. Table 1 presents results on mdCATH, while Table results ATLAS where all methods trained and tested following MDGen's protocol. TEMPO demonstrates consistent strong performance across both datasets, achieving state-of-the-art results key metries.
Table Evaluation on ATLAS. All methods trained and tested following MDGen's protocol.
Metrics
BioEMU
AlphaFlow-MD
MDGen
TEMPO
Pairwise RMSD r
-0.02
0.48
0.48
0.91
Global RMSF
0.09
0.60
0.50
0.89
Root mean W2
19.23
2.61
2.69
1.49
MD PCA W2
3.61
1.52
1.89
0.60
% PC-sim > 0.5 f
14
44
10
Weak contacts J
0.26
0.51
0.62
0.74
Trans. contacts J
0.06
0.29
0.41
0.38
RMSD Error +
3.20
1.83
Our multi-scale framework demonstrates distinct advantages across three critical dimensions of protein dynamics modeling. On ATLAS, TEMPO achieves substantial improvements over baselines, with Pearson correlation of 0.91 for pairwise RMSD and 0.89 for global RMSF, significantly outper- forming other methods. The model captures 76% of principal components with similarity greater than 0.5, compared to 44% for AlphaFlow and 10% for MDGen, demonstrating superior ability to preserve essential collective motions. On mdCATH, in struetural flexibility metries, TEMPO achieves the closest match to molecular dynamics (MD) ground truth, with pairwise Ca-RMSD and residue-leve RMSF closely matching MD values. The strong Pearson correlation (r 0.77) between predicted and ground truth RMSD values reveals our hierarchical SDE formulation preserves the intrinsic roughness of protein energy landscapes.
In distribution matching metrics, TEMPO achieves the lowest MD PCA Wasserstein distance (0.60 on ATLAS, 2.33 mdCATH) across both datasets, suggesting accurate preservation of principal motion patterns. While MDGen shows slight advantages in average conformational Wasserstein distance on mdCATH, our approach achieves better backbone RMSD error both datasets (1.83 ATLAS, 1.78 on mdCATH compared to MDGen's 3.20 and 3.76). Furthermore, our physically constrained learning reduces steric clashes compared to ESMFlow, BioEMU, and AlphaFlow, validating the biological plausibility of generated conformations.
In computational efficiency, TEMPO generates complete 400-frame trajectories in approximately 22 seconds, significantly faster than AlphaFlow and ESMFlow. Among all baselines, only MDGen specifically designed for trajectory generation, yet it requires prohibitive computational resources. In
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contrast, our multi-scale decomposition enables training on a single NVIDIA A100 GPU, with both scow-scale and fast-scale models operating within memory constraints.
These results collectively demonstrate that TEMPO's physics-informed multi-scale design achieves superior performance in conformational accuracy, temporal coherence, and computational efficiency across different datasets. Our experimental validation incorporates four analyses to verify TEMPO's spatiotemporal modeling: (1) Collective motion analysis to evaluate the capture of functionally relevant slow motions, (2) Up-sampling evaluates high-resolution reconstruction of local fluctuations, (3) State transition tracking examines temporal pathway fidelity, and (4) Free energy surface analysis to compare conformational sampling strategies. We focus subsequent analysis and visualizations on mdCATH its free energy landscapes contain more distinct energy basins with clearer conformational transition pathways, better suited for analyzing dynamics modeling. Visual examples of generated protein trajectories are provided in Appendix [. offering an intuitive demonstration of our model's capability to capture protein dynamics. Additional ablation studies and detailed analyses are provided in the Appendix. All analyzed cases are identified by their PDB IDs.
PCS Comparisan
PCi Comparisa
Similrty Compartinns
Trato
MAAK
PCL Indection
300 aoc 200 zoe lee
350
300 100
1EMPO
Phease
Phaaeer
(a) 1b1ОA02
(b) SexeA02
(c) 3c0wA02
(d)
Figure 2: Comparison of PC projections between MD and generated conformations. (a-c) PC1 trajectories from MD (blue) and our slow-scale trajectories (red) for three representative proteins. (d) Box plot of cosine similarity scores comparing TEMPO and MDGen across the test set.
Slow Motion generation. Protein slow motions, occurring on microsecond to millisecond timescales, often correspond functionally relevant conformational changes such as domain movements [17. Capturing these collective motions is challenging due to their long timescales and coordinated nature. We evaluate our model's capability in capturing such motions through principal component "apabilit analysis (PCA). as the first few PCs typically describe the dominant collective motions in protein dynamics 22 For representative cases (Figure
we projected both MD trajectories and our generated slow-scale conformations onto the first principal component (PC1) of MD trajectories. The cosine similarity between these projections ranges from 0.67 to 0.77, indicating strong alignment of collective motions. Extending this analysis across all test proteins (Figure d)), evaluated the maximum cosine similarity between the first two PCs of generated ensembles and MD trajectories to assess the capture of collective motions. TEMPO achieves better performance than MDGen, with mean similarity of 0.41 compared to MDGen's 's 0.36, suggesting potential for capturing slow motions while highlighting the challenging nature of this task.
Up-sampling. We evaluated our high- reghutesotution model's capacity to capture detailed protein motions through up-sampling experiments, using ground truth low-resolution protein conformation as input. The conformational sampling quality was quantitatively assessed via free energy surface (FES) analysis in reduced dimensionality space, obtained through PCA of protein backbone coordinates using Prody. The free energy landscapes were constructed by projecting conformations onto the first two principal components. followed by kernel density estimation and Boltzmann inversion at 300K. Figure 3 demonstrates the FES contour plots for two representative test proteins, comparing the distributions between MD ensembles and our generated trajectories. The close correspondence in FES characteristics, particularly in the location and depth of energy minima, validates our model's ability. More evaluation metrics across our test set (Table I further confirm that our high-resolution model effectively captures local protein fluctuations consistent with MD simulations. Additional case studies with extended protein sets are presented in Appendix [
State Transition. After evaluating the two scale models separately, we further investigated TEMPO's ability to capture complete conformational transition pathways. Using four representative proteins
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Frem Energy Surface IPull POA
Free Energy Surfacs (Gan PCA)
Eeengy
Free Energy Surface (Gun PCAI
100
10
-25
-50
-35
(a) 2eyzA03 md
(b) 2eyzA03 generated
(c) 3gyxA02 md
(d) 3gyxA02 generated
Figure 3: FES comparison between MD trajectories (a,c) and generated ensembles (b.d) for proteins
2eyzA03 (a.b) and 3gyxA02 (c.d). Colors represent free energy values from low (blue) to high (red).
Macipl Compureet boapps
Prisgpa Eodysss
tue
Figure 4: Comparison of conformational transitions in PC space between TEMPO and MDGGn baseline (bottom). Ground truth MD trajectories are shown in blue, while generated trajectories are in green. The polynomial fitting curves highlight the temporal evolution of conformational changes (Protein from left to right: 2e9xB01, 1s79A00, 1b0OAO2, 3cx5B01).
from our test set, we analyzed how well our integrated framework reproduces the sequential nature of conformational changes. We first constructed the PCA space using MD trajectories reference, then projected both our generated trajectories and MD trajectories onto the first two principal components to visualize the conformational landscape. As shown in Figure 4 the upper row demonstrates TEMPO's ability to generate trajectories (green) that follow similar transition pathways as MD simulations (blue). In contrast, the MDGen tends generate clustered conformations in limited regions of the PC space, failing to capture the full range of transitions. This comparison highlights the advantage of our temporal modeling approach over the indeendddeee generation strategy, particularly in reproducing the sequential nature of conformational changes. The polynomial fitting curves further illustrate how model better tracks the temporal evolution of these state transitions. Additional transition pathway analyses for an extended set of test proteins are provided in Appendix [
Free Energy Surface Coverage. Our quantitative metrics effectively assess conformational stability and structural accuracy, but provide limited insight into the comprehensive exploration of confor- mational space. To address this, we established reference framework using PCA derived from MD ensembles of four randomly selected proteins, subsequently constructing the corresponding free energy surface (FES). Projection of generated conformations onto this FES revealed that ESMFlow achieves broader coverage of the conformational landscape (Figure 3 consistent with its design for independent sampling across the energy surface.
This divergence reflects fundamentally different modeling objectives. ESMFlow optimizes conforma- tional diversity through independent sampling, valuable for exploring thermodynamically accessible
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states. In contrast, TEMPO's focused sampling is a design feature that preserves temporal conre lations and physical constraints governing real protein motion. Real proteins must respect energy barriers and cannot instantaneously jump between distant conformations; biological processes depend on sequential, pathway-dependen conformational changes where kinetic accessibility differs from thermodynamic accessibility. Such trajectory- -waae modeling ultimately serves our primary objec- tive of elucidating the kinetic mechanisms underlying biological function, rather than maximizing configurational sampling (additional analyses in Appendix K
Everg Sarfere
R
Free Lawyy Sarro
mm Carg/
Srergy Serfene
Figure 5: FES comparison between TEMPO (top row) and ESMFlow (bottom row) on four randomly selected test proteins. TEMPO's dynamic modeling shows more focused exploration of confor- mational space, whereas ESMFlow's 's independent sampling achieves broader coverage across the PCA-derived free energy surface (Protein from left to right: 1s79A00, Ix4tA01, 2e9xB01, 5b58T02).
Conclusion
In conclusion, our proposed TEMPO framework represents significant advancement in the modeling and generation of protein dynamics by effectively addressing the inherent complexities hicrarchical and multi-scale behavior. Through the integration of multi-scale autoregressive approach with stochastic differential equations, TEMPO successfully captures both slow collective motions and fast local fluctuations that characterize protein dynamics. Our comprehensive experimental validation demonstrates that TEMPO achieves superior performance across multiple metrics, from structural flexibility to distribution matching, while maintaining computational efficiency compared to existing methods. As we move forward, TEMPO's innovative design and proven capabilities hold the potential to facilitate further research in protein dynamics, ultimately contributing to more comprehensive understanding of biological systems and their underlying mechanisms.
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Acknowledgment
This work was supported in part by the National Science and Technology Major Project of China under Grant 20227DD01 16408. This work is also supported in part by the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence (2023B121201000) our gratitude to Changping Laboratory for their research funding and support. The resources and collaborative environment provided have been instrumental in enabling us to achieve the objectives of this project.
We express
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A Limitation
While TEMPO demonstrates promising capabilities in protein dynamics generation, several aspects warrant further exploration. The current model shows limited generalization capability to unseen proteins, especially for cases involving large conformational changes. This could be addressed through expanded training data incorporating diverse protein architectures and enhanced model designs for capturing large-scale motions. Our two-scale temporal decomposition, though effective for the demonstrated trajectory lengths, might benefit from auto-regressive resolution mechanisms similar to those successful in image generation [40] when modeling longer timescale dynamics.
The framework's ' current limitation to directions. Incorporating side-chain reconstruction and multi-molecular interactions could enable modeling of protein-ligand binding dynamics and DNA recognition processes, critical downstream applications drug discovery and biological mechanism studies. Additionally, while existing metrics provide useful validation, developing more biologically-grounder evaluation protocols could better assess trajectory quality through direct correlation with experimental observables and functional outcomes. These directions collectively suggest rich potential for expanding both the scope and practical utility of deep learning-baas baaedd protein dynamics modeling.
backbone dynamics could be extended in several
B Protein Structure Tokenization
We represent protein structures using a combination of local reference frames and torsion angles, resulting in rotation and translation equivariant representation. For a protein with T. amino acids, our tokenization procedure is as follows:
SE(3) Frame Representation. For each residue i, construct local reference frame using the backbone atoms (N, Co, C) following the approach similar to AlphaFold [32 Specifically:
The origin O placed at the Ca atom position
The x-axis à is aligned with the normalized Ca-N bond vector
The temporary vector ii is the normalized Ca-C bond vector
The z-axis 2; is computed as 2;
x.xd,
x
The y-axis 0: completes the right-handed coordinate system: 9: 2 d;
Each SE(3) frame consists of rotation matrix R& SO(3) and translation vector f; E R:
R
[ Di, 2, E 3x3 0 g
(15)
To obtain a compact representation, we convert the rotation matrix to a unit quaternion da E S3
following (39;
R, 0. + R3 11
+ 2,
2
fur.
n. 2.1 R.
4:
áz,b
R. 0.2 R. 2.0
4g
(16)
qy.i
4g
9z.i
R. 1.0 R. [0.1]
4g
Combined with the translation 19 qs 9y.ix 4z.6r tw tg. for each residue, resulting tensor of shape [L, for the entire protein.
vector, this yields a 7-dimensional vector
Torsion Angle Representation. angles (d, t. w), which define the protein's conformation. For residue i, these angles are defined as:
We complement the SE(3) frames with the backbone torsion
Φ diheddra(CC-1,,N, Cos,, Ci) y/: dihedral(N,, Cas, C₁ N; dihedral(Ca,, C₁ No41, Co541)
(17)
(18)
(19)
14

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Rather than using the raw angles. we represent each angle as a 2D vector (sin(0). (on()) to avoid discontinuities at = as commonly done in protein structure prediction models 44 This results in 6-dimensional vector per residue:
U = [sin (0:), cos(o;), sin( (1). (os))),, sin(w:), cos(w;))
(20)
This yields a tensor of shape [L, 6] for the protein's 's torsion information.
Final Representation. We concatenate the SE(3) frame and torsion angle representations obtain comprehensive protein structure encoding of shape [L, 13), where a 13-dimensional vector represents each residue:
2 &e 92 Gy.áo I5,5s tz by i sin (d)), cos(e,), sin(ti)) (1:), sin(w;;, cos(w)) (21)
Rotation and Translation Equivariance. Our final protein representation comprisiin unit quater- nions and translation vectors for SE(3) frames and backbone torsion angles is equivariant to global rigid-body motions. Specifically, a global rotation Rg E SO(3) and translation fg E IR3 transform each local frame (R,, f4) as R, R,t, ty). When using unit quaternions 9: to represent R,, this corresponds to a left quaternion multiplication:
= 9g @ qi R tt
where 4g is the unit quaternion corresponding to Rg and @ denotes quaternion multiplication.
The torsion angles (ø, 1b w), represented as [sin(0), cos(0)) pairs, are internal degrees of freedom and remain invariant under global SE(3) transformations. Therefore, our 13-dimensional representation o is globally equivariant and captures both spatial orientation and internal conformation. This formulation ensures that the learned model respects 3D geometric symmetries, consistent with the principles of equivariant neural networks (35.
C Algorithm
Algorithm describes our multi-scale training procedure, which is applied to both low-resolution and high-resolution models. The algorithm implementation differs in how we prepare the training trajectories for each scale:
Low-resolution Training. For the slow-scale model (At, 20ns), we sample frames from the full trajectory at 20ns intervals. Starting from the native structure, the model learns to predict conformational changes over longer time scales, capturing major conformational transitions.
High-resolution Training. For the high-resolution model (St, = Ins), we randomly sample continuous trajectory segments of 20ns length with Ins intervals. Each segment is an independent training sequence, allowing the model to learn local fluctuations and fast conformational changes.
Notations:
9e E SE(3) The frame sequence at time f, where each frame consists of rotation and translation components (R, in SE(3) space
X, E L2133 Protein conformation at time f, where each residue includes 13 features as described Section B
T': Number of timesteps used for both input and prediction windows
K: Total number of frames in the training trajectory (differs between scales)
Noise: Noise scale in our SDE formulation
Algorithm 2 details our hierarchical inference strategy for generating complete protein dynamics trajectories. During inference, we first use the low-resolution model f to predict conformations at coarse timesteps (At, 20ns), which captures slow collective motions. These coarse predictions then serve as anchoring points for the high-resolution model fá, which fills in the intermediate frames fine timesteps (Ai, = 1ns) to capture local fluctuations. This hierarchical approach ensures that the generated trajectories maintain consistency between collective motions and fast local dynamics.
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Algorithm 1: Autoregressive Training of Protein Dynamics Model with Multiple Timesteps
Input: ground truth frame sequences {9:) K-1 E (SE(3))) ground truth trajectories [X+) E (IR4113JK
amino acid identities A ......,0)4.
teaching force probability Ptf.
number of timesteps T (for both input and prediction)
Output: trained model parameters 0
for each training iteration do
// Initialize input buffers with first T frames Initialize Xbuffer + {Xi) Initialize fbuffer - {9, )=
Liotal 6- 0:
for 1 to K/T] do
Onoise ~ ((0.01, 0.05): No, I):
Motent f- W proj (Xtuf + Onoiset); 8 Embed(A):
10
for i to Pipa do //rrme Invariant Poinaatttiiitttttiiiiiiin (bu/ fers Inoise):
// spatial processing
12
hopatsal f- Matent + hfframe + (mmeed(onoise)
for = to natt do
14
Aspatial &- MulliHeadAAteeniionn/poaaa
// temporal processing Nemporol f- ispptiill for k = to NGRU do
15
16
17
hsemprrll -RUU(heempraaa)
18
X Xpred + W "roj (hremporal); Ipred SrggiddTaassoooDeooodoe prred):
19
20
Emoe t pred X (1
Leaash +- CoomputeClashSoore(XXp...
22
Lrotal + Lrotel + Lmse + ALelush
// Update buffers
23
r 1)0,1);
24
Pif then
25
Xbufser +- Xpred'
26
goaffer t- Spred ;
27
else
28
Xoufser + ((++)))((2)))
29
Jouffer + (
30
Update model parameters 0 using VoLrotal
D Evaluation Metrics
We employ a comprehensive set of metrics evaluate both structural accuracy and dynamic properties of generated protein ensembles. Our evaluation framework follows established protocols in protein ensemble generation [19], adapted trajectory-based assessment.
D.1 Structural Flexibility Metrics
Pairwise RMSD. For each ensemble, we quantify overall conformational diversity the average Ca- RMSD between all pairs conformations. This metric captures the range conformational space explored by the ensemble. We report both the absolute values and Pearson correlation coefficient
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Algorithm 2: Multi-scale Inference for Protein Dynamics Generation
Input:
initial frames {X; [x;]==0 T-1 {9 )=
// frames as initial buffer
coarse timestep At, 20ns,
fine timestep A Ins,
total simulation time Trotols trained coarse-resolution model sū.
trained fine-resolution model f
Output: complete trajectory X (X, (x,] == buuffer {= T-1 7 Pousser fer 5- ==0 X 1.
Testat11
for t 1000001 n2 do xireat Fpred x (X( Pouffer)
10
Xluffer t - predi Houf fer -Ppredi
13
Append X xrred X:
14
for & t1 y do
X pred Spread t fg (x buf for // Model forward pass
15
Append X pred to X:
16
Update x / Poaffer with prediction;
17
18
Update forrferr Pouffer with prediction;
19
return
between predicted and ground truth pairwise RMSD distributions assess whether our model captures the relative flexibility patterns across different proteins.
Root Mean Square Fluctuation (RMSF). To assess local flexibility, we compute the RMSF for each residue, measuring the standard deviation of atomic positions across the ensemble after optimal alignment. The Pearson correlation between predicted and ground truth RMSF profiles indicates how well the model captures residue-level flexibility patterns.
D.2 Distribution Accuracy Metrics
Root Mean Wasserstein Distance (RMWD). To generalize all-atom RMSD to ensemble comparison, we define the root mean Wasserstein distance between ensembles and as:
N
1
RMWD(X,))
Σ
W3 (WTx)),NDA))
(22)
where [[X; denotes 3D Gaussian fitted to the positional distribution of the ith atom in ensemble X. This metric reduces to standard RMSD for single structures and provides a distributional measure of positional accuracy.
Principal Component Analysis. To evaluate collective motions, we project the joint distribution
of Ca positions onto principal components computed from the MD ensemble. We measure: (1) the 2- Wasserrstein distance between predicted and ground truth ensembles in the PC space (MD PCA W2), and (2) the cosine similarity between the dominant principal components. A similarity > 0.5 indicates successful capture of the dominant collective motion. We report the percentage of test proteins achieving this threshold (% PC-sim > 0.5).
D.3 Dynamic Property Metrics
Contact Dynamies. We analyze intermittent contacts assess if the model captures thermal fluctuations. For each ensemble, we identify: (1) weak contacts Ca pairs (<(8A) in the native
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Protein 1b10A02
Protein 5exeA02
Time steps-
Figure 6: Visualization of protein conformational changes time for two representative proteins (Ib10A02 and SexeA02). Each row shows the temporal evolution of one protein. MD ground truth trajectories are shown in blue, while TEMPO-generated trajectories shown in pink, demonstrating the close structural alignment between generated and reference structures throughout the trajectory.
structure that dissociate in > 10% of ensemble structures. and (2) transient contacts pairs not in contact in the native structure but associate 10% of ensemble structures. We compute Jaccard similarity between predicted and ground truth contact sets.
Trajectory Accuracy. For trajectory generation methods, measure the backbone RMSD error: Error = IRMSD RMSDg1l where RMSD is computed relative to the native structure. This metric quantifies the model's ability to accurately capture the magnitude of conformational changes over time.
Clash Ratio. We compute the proportion of generated conformations containing steric clashes, defined as backbone atom pairs from different residues with distance < 1.2Å. This metric validates the physical plausibility of generated structures.
All metrics are computed on backbone atoms (N, Co,, C, O) after optimal rigid-body alignment to the native structure, unless otherwise specified. We report median values across the test ensure robustness to outliers.
E Trajectory Visualization
To provide intuitive visualization of our generation results, present trajectory comparisons for two representative test proteins. Figure(Jssoww snapshots of the trajectories at 80ns intervals, where the ground truth MD conformations are shown in pink. These visualizations demonstrate the close structural alignment between our generated conformations and the MD reference states.
shown in blue and "TEMPO-generatte conformations
F Thermodynamic Accuracy Analysis
Following BioEMU's established evaluation methodology we provide quantitative assessment of thermodynamic accuracy through free energy difference analysis. This analysis validates that our generated trajectories not only capture structural features but also preserve the underlying thermodynamic properties of protein dynamics.
Free Energy Difference Computation. We compute free energy differences by extracting reaction coordinates, specifically the fraction of native contacts, to calculate folding probabilities (Piot). The free energy differences are then calculated as:
Ptold
AG = KT log
(23)
Plold
where l is the Boltzmann constant and the temperature. We measure the free energy error as
AAG AG grounnd muth AGpeddiced
Results. TEMPO achieves an average AAG of 0.67 kcal/mol on the mdCATH test set, demonstrating good agreement with reference MD simulations within the acceptable range for biological applications
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(typically <1-2 kcal/mol). This result indicates that our method not only generates structurally accurate conformations but also preserves the thermodynamic properties of protein folding and unfolding processes.
Free Energy Profiles. Following BioEMU's evaluation framework, analyze generated trajectories through multiple perspectives:
ID Free Energy Profiles: We construct free energy profiles using three key reaction coordinates: RMSD from native structure, radius of gyration, and fraction of native contacts. Our generated trajectories exhibit high similarity ground truth MD simulations across all three coordinates.
2D Free Energy Surfaces: Two-dimensional free energy surface plots constructed from combinations of reaction coordinates show that TEMPO captures the essential features of the conformational landscape, including energy minima locations and barrier heights.
Time Series Analysis: The fraction of native contacts time series from our generated trajec- tories closely matches the temporal evolution patterns observed in ground truth simulations, validating model's ability to capture dynamic processes.
These quantitative thermodynamic evaluations complement our structural metrics and demonstrate that TEMPO generates trajectories that not only geometrically accurate but also thermodynami- cally consistent with reference MD simulations (see Figure 7 and Figure & for visualizations).
G Ablation Study: Multi-scale vs Single-scale Generation
While autoregressive methods typically suffer from error accumulation during sequential generation, hierarchical multi-scale design mitigates this issue by anchoring fine-scale dynamics coarse- scale predictions. To demonstrate the effectiveness of our multi-scale architecture, we compare TEMPO's full hierarchical framework against single-scale baseline that directly generates 400 frames without multi-scale guidance on mdCATH.
Table 3 shows that the multi-scale approach significantly outperforms single-scale generation across all metrics. The multi-scale model achieves substantially lower RMSD error (1.78À vs 8.62À) after 400 frames, demonstrating controlled error propagation that maintains trajectory stability over extended generation periods. Furthermore, the multi-scale design preserves conformational diversity (pairwise RMSD of 2.78Å vs 7.46Å compared to ground truth 3.26Å) and structural flexibility patterns (Pearson correlation of 0.77 vs O. 14 for RMSD, 0.67 0.15 for RMSF), while the single- scale approach shows severe degradation in capturing protein dynamics. These results validate that our hierarchical decomposition crucial for generating physically realistic long protein trajectories.
Table 3: Ablation study comparing multi-scale TEMPO with single-scale baseline on mdCATH. Ground truth values are shown in parentheses where applicable. The single-scale model directly generates 400 frames without hierarchical guidance.
Metries
TEMPO (Multi-scale)
TEMPO (Single-scale)
Pairwise RMSD (= 3.26)
2.78
7.46
Pairwise RMSD
0.77
0.14
All-atom RMSF (= 1.64)
1.60
4.27
Global RMSF 1
0.67
0.15
Root mean W2 |
4.21
8.27
MD PCA W2 J
2.33
2.53
% PC-sim > 0.5 f
7.81
3.12
Weak contacts
0.43
0.23
Trans. contacts
0.20
0.09
RMSD Error
1.78
8.62
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MMD Baskragy FoTE
Nyt kaug/ted
Pasken
Factel
Erod 05000
Oround Trats 125
F.CTime Sorfen Sapp cestinct
345
210
211
PRODOC
Slavi Sub
Gread
00021
Figure 7: Thermodynamic accuracy analysis for two representative proteins (7p46 A and 7asg_At A) selected from the ATLAS test set. For each protein, we show: 1D free energy profiles along three reaction coordinates (RMSD, radius of gyration, and fraction of native contacts), 2D free energy surfaces, and time series of fraction of native contacts.
H Training vs Test Performance Analysis
We acknowledge that capturing all distribution modes represents a fundamental challenge in protein dynamics modeling. While our primary objective focuses generating realistic conformational transitions rather than perfect mode coverage, we recognize the importance understanding the performance gap between training and test scenarios.
Performance Gap Analysis. Table Aable4come compares TEMPO's performance on training and test sets of mdCATH. where the training set results are evaluated on 60 randomly selected proteins from the
20

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Iooksaye PoTts
hoa Mragg Polia
FMCHRA Phale
135
Stard
V/
Prediten: 20 PC5
Crund 22.725
DNC Tree Soree 1A00-AU2 m900
112
122
168
199022
175
17.5
O Stat
11:2
275
Grund Thsto
1002
Figure 8: Thermodynamic accuracy analysis for two representative proteins (2ijd101 and lr10A02) selected from the mdCATH test set.
training data. The training set demonstrates significantly better mode coverage accuracy, with 46.7% of principal components achieving similarity greater than 0.5 compared to 7.81% % the test set. Similarly, the training set shows improved performance across most metries, including lower MD PCA Wasserstein distance (1.31 vs 2.33) and better contact prediction accuracy (0.53 vs 0.43 for weak contacts, 0.37 vs 0.20 for transient contacts).
This performance gap reflects the inherent complexity of generalizing to novel protein architectures in mdCATH, where most proteins contain distinct energy basins and unique conformational landscapes. The challenge of mode coverage is not unique to our method even extensively pretrained methods like BioEMU show significant free energy surface (FES) deviations as shown Appendix Figure TT] The difficulty stems from the fundamental nature of protein dynamics: each protein has its own
21

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characteristic energy landscape shaped by its unique sequence and structure, making it challenging for models perfectly capture the full conformational distribution of unseen proteins.
Despite the performance gap, our test set results still demonstrate strong capability in capturing essential protein dynamics, as evidenced by high structural flexibility correlations (Pearson 0.77 for pairwise RMSD, r 0.67 for RMSF) and accurate trajectory generation (RMSD error of 1.78A). The stronger training set performance validates our model's 's capacity to learn complex protein dynamies when sufficient data is available, suggesting that expanded training data incorporating diverse protein architectures could further improve generalization.
Table 4: Comparison of TEMPO performance on training set versus test set. Ground truth values are shown in parentheses. The training set demonstrates significantly better mode coverage.
Metries
Train Sample
Test
Pairwise RMSD
3,06 (2.51)
2.78 (3.26)
Pairwise RMSD
0.79
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I Additional Up-sampling Analysis
Additional up-sampling experiments across multiple test proteins further validate our high-resolution model's ability generate full protein dynamics from ground truth low-resolution protein conforma- tions. Figure shows the FES contour plots for four randomly selected proteins (Ir10A02, 1x4tA01, 1zpdA02, and 2ndpA00) computed from their backbone conformations.
J Additional State Transition Analysis
To further validate our model's capability in capturing protein conformational transitions, we present additional state transition analyses on representative proteins from our test set. Figure each subplot compares the conformational trajectories generated by TEMPO and MDGen with MD simulations in the space of the first two principal components. Consistent with our observations in the main text, these additional examples demonstrate TEMPO's s robust ability to generate physically meaningful transition pathways. MDGen typically exhibits more clustered sampling patterns, fails to capture the continuous nature of conformational transitions. These results further support our framework's s advantage modeling the temporal dependencies inherent in protein dynamics.
K Additional Free Energy Surface Analysis
To complement the FES analysis, we randomly selected four additional proteins from our test set for detailed comparison. As shown in Figure Based on the FES analysis across four randomly selected test proteins, observe several distinctive patterns in conformational sampling strategies: TEMPO demonstrates precise conformational sampling that closely aligns with the ground truth distributions. The generated conformations (orange points) are concentrated within physically meaningful energy basins, suggesting coherent trajectory generation. MDGen shows similar sampling quality, though with slightly more dispersed distributions in some cases.
In contrast, AlphaFlow exhibits broader but less focused sampling, often deviating from the main energy wells. BioEMU shows the most scattered sampling patterns, frequently generating physically implausible conformations that outside the main energy basins. This comparison highlights the advantage of our trajectory aware approach over independent sampling methods while methods like
-warr
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Figure 9: FES plots of four randomly selected test proteins. The FES is computed from: (top) trajectories generated by TEMPO performing up-sampling tasks, (middle) ground truth molecular dynamics simulations, and (bottom) trajectories produced by the baseline model MDGen, performing simulation tasks. Each column represents a distinct test protein (from left to right: Ir10A02, 1x4tA01, 1zpdA02, and 2ndpA00).
AlphaFlow and BioEMU may achieve wider conformational coverage, they often do so at the cost of physical realism.
These observations consistently demonstrate that TEMPO's multi-scale framework effectively bal- ances conformational exploration with physical constraints, producing trajectories that maintain both continuity and thermodynamic plausibility. The results validate our design choice of incorpo- rating temporal dependencies, which proves crucial for generating biologically meaningful protein dynamics.
L Per-Residue RMSF.
The RMSF analysis reveals our model's ability to capture local protein dynamics across different scales. Figure 12 iillustrates a representative case study using protein 4impA02, where the generated trajectories closely mirror the MD simulation's 's Ca fluctuation patterns. In this example, both profiles exhibit characteristic mobility signatures, with enhanced fluctuations the N-terminus (residues 0-20) and C-terminus (residues 190-210), while maintaining relatively stable conformations (RMSF < 2Å) in the central regions. Across our entire test set, the generated trajectories maintain a reasonable correlation (average Pearson 0.667) with MD simulations in terms of RMSF profiles, suggesting that our model consistently reproduces biologically relevant flexibility patterns. This global performance indicates that the model has learned meaningful protein dynamics patterns rather than generating arbitrary motions. Additional case studies are provided in Figure T3
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Priseed Anutes
Figure 10: Comparison of conformational transitions in PC space between TEMPO and MDGen baseline (bottom). Ground truth MD trajectories shown in blue, while generated trajectories in green. The polynomial fitting curves highlight the temporal evolution of conformational changes.
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shange Sarffce
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Figure 11: Comparison of FES for four randomly selected test proteins (from left to right: 3cj8B02, 3cx5E01, 303,A03, 4fomA03). Results are shown for TEMPO, MDGen, AlphaFlow, and BioEMU (from top to bottom).
MD (r = 0.927)
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Figure 12: Comparison of Root Mean Square Fluctuation (RMSF) between MD simulation trajectory and generated trajectory for protein 4impA02. The RMSF values reflect the Ca fluctuations protein residues during the simulation. The Pearson (r) between the two RMSFs is 0.93.
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Figure 13: RMSF profiles comparison between MD simulations and TEMPO generated trajectories for 4 randomly selected test proteins. Each plot demonstrates the Ca atomic fluctuations along the protein sequence (Protein from top to bottom: 1y4mA00, 3b8xA02,3bjdA01, 33y23330101133
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
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Reviews:
[{"review_id": "vEQCrT4iz8", "reviewer": "Reviewer_ezhj", "summary": "The authors propose a method for generating protein conformational ensembles, TEMPO, which aims at generating protein dynamics across different temporal scales. The two temporal scales include slow collective motions/changes and fast local fluctuations, quantified as “low resolution generation” and “high resolution generation.” Protein dynamics are represented by the standard SDE process, and decomposed into two coupled SDEs representing protein motion at both slow and fast timescales. The authors use the same architecture to learn both sets of dynamics, but the architecture is trained separately to parameterize the drift terms. The authors train and evaluate on the mdCATH dataset, where low resolution operates at 20 ns intervals, and high resolution at 1 ns: they sample low-resolution states and then do autoregressive sampling, with both scales simulating protein dynamics. The authors evaluate on different metrics including comparing the conformational distributions, comparison to MD, and steric clashes and compare to other protein conformational ensemble generation methods.", "questions": "- What is the relationship with such an approach and coarse-graining methods? There seems to be a lot of similarity, but coarse-graining is not mentioned at all.\n\n- What are the sizes of the proteins that are being trained/tested on, in terms of number of atoms?\n\n- What does the error propagation look like during autoregressive sampling? \n\n- Can you provide more details on the mdCATH dataset? What force field was used to generate the data and is it fully unbiased simulations? What does it mean “trajectories were standardized to 400 frames at 1 ns intervals through periodic extension or truncation”?\n\n- Can the authors quantify the overlap in similarity between the training and test sets?", "limitations": "Limitations are included in the appendix.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- I found the paper well-written and easy to follow in describing the motivation and the approach.\n\n- The idea of splitting the generation process into different scales intuitively makes sense in modeling the different scales of protein dynamics.\n\nWeaknesses:\n\n- The authors say that the training and test sets were chosen randomly, which means there’s likely a high probability of a lot of overlap in the sequences. This likely means that the evaluation framework is not truly testing on new systems. Rigorously quantifying this is important. \n\n- TEMPO is trained on MDcath but the baselines are trained on different datasets (for example, MDGen is trained on ATLAS). If the models are all evaluated on MDcath trajectories, this doesn’t seem like a fair comparison: what does evaluation look like on other systems (such as systems the other models were trained on), or when systems are all trained on the same data?\n\n- I also the evaluation framework to not be ideal from a scientific point of view, as many of these are hacky ML metrics. For example, what do macroscopic observables look like over autoregressive generation? From the dynamic trajectories, can one compute a free energy? I understand that for some of these systems it may be difficult because they are large (but the underlying MD trajectory does exist).", "was_revised": true, "final_justification": "I appreciate the authors responses to these questions. I am borderline on this paper (raising my score from 3 to 4). I think that there are some reasonable ideas in this paper, but I agree with the other reviewers that the results don't seem fully convincing over baselines (some of this is also related to it being hard to have clear evaluations in this field).", "rebuttal": "We sincerely thank the reviewer for their valuable feedback on our evaluation framework, dataset fairness concerns, and methodological rigor. Below are our detailed responses addressing each point.\n\n**W1&Q5**\n\nWe appreciate the reviewer's concern. The mdCATH dataset is composed of non-homologous protein domains at the CATH S20 level (sequence similarity <20%), ensuring low sequence similarity between any two domains, and we rigorously quantified sequence similarity using mmseqs2, finding averaged **18.93%** sequence similarity between training and test sets, which is well below standard thresholds (40%) for sequence relatedness and ensures our evaluation truly tests generalization to novel protein systems, with the ATLAS dataset showing similar results at 18.3% average sequence similarity.\n\n**W2**\n\nWe appreciate the reviewer's concern. We chose mdCATH over ATLAS because its free energy landscapes contain more distinct energy basins with clearer conformational transition pathways better suited for our dynamics modeling task. At the same time, we evaluated baselines using their provided checkpoints since BioEMU has already been pretrained on diverse MD data and PDB structures; ESMFlow and AlphaFlow were not designed for MD generation. MDGEN encounters severe memory limitations on mdCATH due to its full-trajectory attention mechanism (requiring batch size of 1 and still experiencing out-of-memory errors on 8 A100 GPUs when training 400-frame conformations), but to address fairness concerns, we also trained TEMPO on ATLAS following MDGen's splits and results in table below demonstrate our method's strong performance across different datasets, achieving **state-of-the-art** results on ATLAS, with FES visualizations of generated ensembles corroborating our quantitative results, comprehensive visualizations will be provided in the revision due to current supplementary figure limitations.\n\n\n| Metrics | BioEMU | AlphaFlow-MD | MDGen | TEMPO |\n|---------|-------|-------------|----------|---------|\n| Pairwise RMSD r ↑ | -0.02 | 0.48 | 0.48 | **0.91** |\n| Global RMSF r | 0.09 | 0.60 | 0.50 | **0.89** |\n| Root mean W2 ↓ | 19.23 | 2.61 | 2.69 | **1.49** |\n| MD PCA W2 ↓ | 3.61 | 1.52 | 1.89 | **0.60** |\n| % PC-sim > 0.5 ↑ | 14 | 44 | 10 | **76** |\n| Weak contacts J ↑ | 0.26 | 0.51 | 0.62 | **0.74** |\n| Trans. contacts J ↑ | 0.06 | 0.29 | **0.41** | 0.38 |\n| RMSD Error ↓ | - | - | 3.20 | **1.83** |\n\n\n**W3** \n\nWe appreciate the reviewer's comment regarding evaluation rigor, and following BioEMU's established methodology, we quantitatively compute free energy differences by extracting reaction coordinates (fraction of native contacts) to compute the folding probabilities ($p_{\\text{fold}}$), calculating free energy differences as $\\Delta G = -kT \\cdot \\log(p_{\\text{fold}}/(1-p_{\\text{fold}}))$, and measuring $\\Delta\\Delta G = \\Delta G_{\\text{ground truth}} - \\Delta G_{\\text{predicted}}$, with TEMPO achieving an averaged $\\Delta\\Delta G$ of 0.67 kcal/mol on the mdCATH test set, which demonstrates good agreement with reference MD simulations within the acceptable range for biological applications. BioEMU's evaluation framework further reveals that our generated trajectories exhibit high similarity to ground truth in 1D free energy profiles constructed using RMSD, radius of gyration, and fraction of native contacts, 2D free energy surface plots, and fraction of native contacts time series analyses, with comprehensive visualizations to be provided in the revision due to current supplementary figure limitations. As protein dynamics generation is an emerging field where standardized benchmarks are still under development, we agree that advancing more sophisticated biophysical evaluation frameworks would benefit the entire community.\n\n\n**Q1**\n\nWe thank the reviewer for this question and clarify that while our approach shares conceptual similarities with coarse-graining methods, we employ temporal coarse-graining (sampling at 20ns vs 1ns intervals) combined with backbone-only spatial representation rather than traditional spatial coarse-graining of molecular interactions.\n\n**Q2**\n\nmdCATH proteins range from 50-482 residues with sequences truncated to a maximum of 240 residues during training for computational efficiency, but inference is performed at the original protein lengths, and since we generate 4 backbone atoms per amino acid residue, the average protein in our dataset contains approximately 548 atoms (137 average sequence length).\n\n**Q3** \n\nWhile autoregressive methods typically suffer from error accumulation during sequential generation, our hierarchical multi-scale design mitigates this issue by anchoring fine-scale dynamics to coarse-scale predictions, resulting in significantly lower RMSD error (1.78Å) after ~400 frames and demonstrating controlled error propagation that maintains trajectory stability over extended generation periods. We provide additional ablation analysis below showing error propagation when directly generating 400 frames without multi-scale guidance.\n\n| Metrics | TEMPO (Multi-scale) | TEMPO (Single-scale) |\n|---------|---------------|-----------|\n| Pairwise RMSD(=3.26) | **2.78** | 7.46 |\n| Pairwise RMSD r ↑ | **0.77** | 0.14 |\n| All-atom RMSF | **1.60** | 4.27 |\n| Global RMSF r | **0.67** | 0.15 |\n| Root mean W2 ↓ | **4.21** | 8.27 |\n| MD PCA W2 ↓ | **2.33** | 2.53 |\n| % PC-sim > 0.5 ↑ | **7.81** | 3.12|\n| Weak contacts J ↑ | **0.43** | 0.23 |\n| Trans. contacts J ↑ | **0.20** | 0.09 |\n| RMSD Error ↓ | **1.78** | 8.62 |\n\n\n**Q4** \n\nThe mdCATH dataset contains 5,398 protein domains with 134,950 fully unbiased molecular dynamics trajectories simulated using CHARMM22 force field and TIP3P water model in the NVT ensemble with Langevin thermostat, neutralized and ionized with Na+/Cl− at 0.150 M concentration, and \"standardized to 400 frames at 1ns intervals\" means trajectories longer than 400ns were truncated while shorter ones were cyclically repeated to reach uniform 400-frame length for batched training, with 1ns representing the dataset's finest temporal resolution as detailed in the original mdCATH paper."}, {"review_id": "4rkjSlgnJZ", "reviewer": "Reviewer_geUV", "summary": "This paper introduces an autoregressive method that models protein conformational dynamics at two time-scales, slow (low resolution) and fast (high resolution). In contrast to ensemble/steady-state prediction methods, the authors focus on the problem of generating whole trajectories to infer time dynamics. The slow time-scale model captures collective motions while the fast one captures local fluctuations conditioned on the collective movements. By having a hierarchical structure, this method aims at preserving the temporal consistency and realism of generated conformations. The authors train and evaluate on a subset of mdCATH, comparing primarily with AlphaFlow and BioEMU, which are steady-state ensemble predictors, based on the ability to accurately predict conformational trajectories of heldout proteins.", "questions": "* How are the Wasserstein distances in table 1 calculated?\n* If the fast timescale predictions are made for each timestep of the slow timescale predictions, then can the trajectory of the fast predictions starting from one particular slow timescale prediction diverge too much from the next slow timescale prediction? If so, there will be discontinuity between each segment of fast predictions. How is this addressed to ensure continuity between slow timescale steps? \n* Higher noise levels are used for the high-res model. Do the predicted conformations still remain in the data distribution when the noise levels are high?\n* In table 1, is TEMPO only using low-res (slow timescale) and TEMPO(up) also using high-res (fast timescale)?\n* In fig 3, is there a way to compute FES difference quantitatively?\n* How well to the MD trajectories themselves replicate, i.e. what would the errors be in table 1 for an MD trajectory replicate?\n* How different are the train/val/test proteins?\n* How important are the different components of the proposed model? I.e. are the slow and fast timescale components actually necessary for performance? What about the components of the structure representation?\n* How are sequence embeddings learned or are they derived from a pre-trained model of some kind?\n* How sensitive are the results to the initial frame buffer? This seems to be providing (potentially) a significant amount of additional information to the model at inference time that BioEMU and AlphaFlow etc are not being provided. What if zero or only one initial frame is provided? I don't actually see this number mentioned anywhere in the experiments.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths \n* Dynamics have not been the focus of previous methods, but are clearly critical for understanding protein mechanistics\n* The hierarchical and the autoregressive modeling of the protein conformational dynamics makes sense for short and long time scale processes\n* Comparisons are done with several state-of-the-art methods with various metrics\n* The time and memory efficiency of the method seems to be an advantage over other methods\n* The paper is well written\n\nWeaknesses\n* It would make the paper stronger if it is compared to the methods that are also designed to learn explicitly the protein conformation's temporal evolution. There are also no simple baselines or ablation studies of the model itself\n* Not clear how reproducible the MD trajectories are themselves, therefore how close should we expect a perfect predictor to be to any given \"ground truth\" trajectory?\n* Many metrics in Table 1 are not explained or it's not clear why they are good measures of model performance (e.g., Pairwise RMDS indicates whether the conformational ensemble has a similar diversity to the MD trajectories, but this does not suggest the structures themselves are correct)\n* Not clear how well the model generalizes given lack of information regarding sequence diversity\n* In fig 2, the plots of the predictions look very smoothed out compared to the GT. Why is this the case and how to improve this? Also, while the proposed method gives a better cosine similarity than MDGen, it seems still quite low (0.41)\n* Requires seeding the trajectory with initial frames", "was_revised": true, "final_justification": "Overall, I think this is a reasonable paper where there remain sufficient weaknesses to warrant my recommendation of weak accept. It's a paper that, with the revisions and clarifications provide in the rebuttal, is appropriate for the conference, but I wouldn't advocate strongly against rejection depending on the consensus of the other reviewers.", "rebuttal": "We sincerely thank the reviewer for their insightful questions regarding our method's design rationale, evaluation metrics, generalization capability, and ablation studies. Below are our detailed responses addressing each concern.\n\n**W1**\n\nWe thank the reviewer for this question and note that MDGen represents the current sota open-source method designed for protein dynamic generation, while other temporal methods were either not publicly available or published after our submission, and we have conducted ablation studies comparing our hierarchical TEMPO framework against single-scale baselines where the high-resolution model directly generates all frames without slow-scale guidance as shown in table below, demonstrating the importance of our multi-scale design.\n\n| Metrics | TEMPO (Multi-scale) | TEMPO (Single-scale) |\n|---------|---------------|-----------|\n| Pairwise RMSD(=3.26) | **2.78** | 7.46 |\n| Pairwise RMSD r ↑ | **0.77** | 0.14 |\n| All-atom RMSF | **1.60** | 4.27 |\n| Global RMSF r | **0.67** | 0.15 |\n| Root mean W2 ↓ | **4.21** | 8.27 |\n| MD PCA W2 ↓ | **2.33** | 2.53 |\n| % PC-sim > 0.5 ↑ | **7.81** | 3.12|\n| Weak contacts J ↑ | **0.43** | 0.23 |\n| Trans. contacts J ↑ | **0.20** | 0.09 |\n| RMSD Error ↓ | **1.78** | 8.62 |\n\n**W2** \n\nWe appreciate this question, and our evaluation framework appropriately focuses on statistical and thermodynamic consistency rather than exact trajectory reproduction by measuring free energy surface fidelity, dynamic consistency, and physically plausible conformational transition patterns in PCA space, recognizing that even independent MD simulations from identical starting conditions would show different specific trajectories while maintaining the same underlying statistical distributions and exploring similar regions of conformational space.\n\n**W3** \n\nWe apologize for the insufficient description and clarify that these metrics, adopted from AlphaFlow's evaluation framework, collectively assess different aspects of protein dynamics: pairwise RMSD measures conformational diversity, RMSF correlation captures local flexibility patterns, Wasserstein distances quantify distribution similarity in conformational space, contact dynamics evaluates biologically relevant inter-residue interactions, RMSD error measures structural deviation accumulation during trajectory evolution, and clash ratio assesses physical plausibility. We will provide more detailed descriptions of their biological significance in our revision.\n\n**W4&Q7**\n\nOur training set of 1,000 proteins spans diverse structural classes from the CATH database representing different folds, topologies, and homologous superfamilies, with the mdCATH dataset inherently ensuring structural diversity through non-homologous protein domains, and we rigorously quantified sequence similarity using mmseqs2, finding averaged **18.93%** sequence similarity between training and test sets, which is well below standard thresholds(40%) for sequence relatedness.\n\n**W5** \n\nThe smoothed curve in Figure 2 occurs because we compare our coarse-resolution trajectories (20ns intervals) with full MD ground truth (1ns intervals), as our coarse model is designed to capture slow collective motions rather than high-frequency fluctuations. While our cosine similarity of 0.41 improves over MDGen (0.36), we acknowledge this reflects the fundamental challenge of capturing large-scale conformational changes and believe future improvements through enhanced model generalization, physical priors, and pretraining strategies could further advance this metric.\n\n**W6** \n\nThe requirement for initial frames is inherent to the nature of molecular dynamics simulation and trajectory generation, as our method aims to generate temporally coherent MD trajectories that capture physically realistic transitions from a given starting conformation, and importantly, our model's performance is not limited by the source of the initial structure, allowing it to be coupled with any structure prediction method to generate dynamics from sequence-predicted conformations.\n\n**Q1**\n\nFollowing AlphaFlow's evaluation framework, Root mean W₂ is computed as the average Wasserstein distance between 3D-Gaussians fitted to positional distributions of each atom across ensembles, while MD PCA W₂ measures the Wasserstein distance in the space of the first two principal components derived from PCA of joint Cα position distributions, where PCA is computed from the MD ensemble and the distance quantifies distributional similarity in this reduced conformational space.\n\n**Q2** \n\nWe appreciate this question regarding trajectory continuity, and while potential discontinuities between fast-scale segments could theoretically occur in our hierarchical approach, our evaluation demonstrates that such instances are infrequent and remain within acceptable error ranges as evidenced by our low RMSD error metrics (Table 1) and smooth trajectory coherence (Figure 4 and Figure 8 in Appendix), and developing more sophisticated continuity constraints represents an important direction for future work.\n\n **Q3**\n \nSince fast fluctuations are more stochastic, we increase noise levels to encourage diverse local conformational sampling. We empirically found that a higher noise scale in the high-res model provides an optimal balance between conformational diversity and trajectory fidelity, maintaining conformations within reasonable proximity to the data distribution while enabling sufficient local exploration, as validated by our low clash ratios and strong correlation metrics in Table 1.\n\n **Q4** \n\nTEMPO results use our complete hierarchical pipeline where the low-resolution model generates coarse trajectories that are then fed into the high-resolution model to produce full trajectories, while TEMPO(up) represents an up-sampling task using ground truth coarse-resolution conformations as anchors with only the high-resolution model filling intermediate frames, demonstrating that our fast-scale dynamics modeling component can effectively reconstruct complete trajectories when given accurate anchor points.\n\n**Q5** \n\nWe quantitatively measure FES differences using the MD PCA W₂ metric reported in Table 1, which computes the Wasserstein distance in the space of the first two principal components derived from PCA of joint Cα position distributions, where our FES visualizations in Figure 3 are constructed using inter-residue Cα distances as features projected onto these same first two principal components. The TEMPO(up) results in Table 1 provide the quantitative performance evaluation corresponding to the upsampling task demonstrated in Figure 3, achieving an MD PCA W₂ distance of 0.60, which demonstrates strong distributional similarity in the reduced conformational space.\n\n**Q6**\n\nWe thank the reviewer for this question. Our evaluation framework appropriately focuses on statistical and thermodynamic consistency rather than exact trajectory reproduction by measuring free energy surface fidelity, dynamic consistency, and physically plausible conformational transition patterns in PCA space, recognizing that even independent MD simulations from identical starting conditions would show different specific trajectories while maintaining the same underlying statistical distributions and exploring similar regions of conformational space, which is why we evaluate against ensemble properties rather than deterministic trajectory matching.\n\n**Q8**\n\nWe have added the ablation study with our multi-scale decomposition, showing significant performance improvements over single-scale baselines, and the table below compares error propagation when directly generating 400 frames without multi-scale guidance. During model training, we found that both the spatial encoder (multi-head attention) and temporal encoder (GRU) components are essential for effective training, as removing either component leads to poor training convergence and substantially reduced model fitting capability on the training data.\n\n| Metrics | TEMPO (Multi-scale) | TEMPO (Single-scale) |\n|---------|---------------|-----------|\n| Pairwise RMSD(=3.26) | **2.78** | 7.46 |\n| Pairwise RMSD r ↑ | **0.77** | 0.14 |\n| All-atom RMSF(=1.64) | **1.60** | 4.27 |\n| Global RMSF r | **0.67** | 0.15 |\n| Root mean W2 ↓ | **4.21** | 8.27 |\n| MD PCA W2 ↓ | **2.33** | 2.53 |\n| % PC-sim > 0.5 ↑ | **7.81** | 3.12|\n| Weak contacts J ↑ | **0.43** | 0.23 |\n| Trans. contacts J ↑ | **0.20** | 0.09 |\n| RMSD Error ↓ | **1.78** | 8.62 |\n\n**Q9** \n\nThe sequence embeddings are learned through a standard nn.Embedding layer during training.\n\n**Q10** \n\nPlease refer to our response to W6. Our method aims to generate temporally coherent MD trajectories that capture physically realistic transitions from a given starting conformation, and we use two initial frames as the optimal hyperparameter (with slight performance degradation observed when using only one frame), as these initial conditions enable the temporal dependencies that are essential for dynamics modeling, which differ from the equilibrium ensemble generation approach targeted by BioEMU and AlphaFlow."}, {"review_id": "8QruCD2DSo", "reviewer": "Reviewer_tNDp", "summary": "This paper introduces TEMPO, a generative model for simulating protein dynamics through a multi-scale autoregressive architecture. It models both slow global motions and fast local fluctuations in protein conformations using SDEs and a two-level temporal model. They show the model can produce realistic, physically plausible, and temporally coherent trajectories.\n\nThe main contributions include:\n1) They propose a multi-resolution temporal autoregressive framework for protein conformation generation.\n2) They develop a spatiotemporal encoder that captures inter-residue geometry and temporal continuity.\n3) The proposed method achieves strong experimental results and the sampling is efficient.", "questions": "1. The model seems to be trained from scratch. Have explored finetuning a casual-aware model from a pretrained structure prediction method?\n2. How difficult would it be to extend TEMPO to model full-atom structures?\n3. How do you choose the resolution hierarchy (variable $\\Delta $ t) empirically? \n4. Have you explored training a single resolution (say 1ns) model on full MD trajectories (in this case, 400 steps)?", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n1. The decomposition of dynamics into slow and fast components looks interesting, which may reflect the biophysical principles.\n2. Unlike many ensemble-focused methods, TEMPO ensures causal consistency across time via autoregressive sampling.\n3. The proposed method achieve strong empirical results, while requiring fewer computational resources.\n4. The paper is well written and they provide implementation details.\n\n## Weakness\n1. The two-scale time hierarchy is fixed to 20ns and 1ns, which might be suboptimal for arbitrary MD trajectories. Making the resolution learnable could improve the robustness of the algo.\n2. Each high resolution trajectory fragment is initialized with the corresponding low resolution state, which might lead to inconsistent trajectories. The conformation may \"jump\" from one state to another during the fragment transition.\n3. Currently the model only support backbone-only modeling, while sidechains are crucial for function and binding.\n4. Unlinke  ensemble-based methods, the algorithms depend heavily on the quality and diversity of the training data (mdCATH dataset). It can not be extended to other cases (such as multimeric proteins) if we lack the data. \n5. The algorithm takes the initial structure as input to generate the MD simulation and can not handle cases where initial structures are unavailable.", "was_revised": true, "final_justification": "Although some shortcomings remain, such as the fixed time hierarchy and potential inconsistencies in the generated trajectories, the proposed method provides a nice foundational framework for modeling protein conformational trajectories. The experiments are conducted on mdcath, a large-scale MD dataset that also deserves greater attention in the field. Overall, the strengths of the approach outweigh its limitations, and the combination of methodology and experimental validation makes me inclined to recommend acceptance of this paper.", "rebuttal": "We sincerely thank the reviewer for their thoughtful suggestions on adaptive timescale learning, trajectory continuity concerns, and potential extensions to full-atom modeling. Below are our detailed responses addressing each point.\n\n**W1**\n\nWe appreciate this constructive suggestion and acknowledge that while our current fixed two-scale decomposition reflects the established biophysical separation of protein dynamics into slow collective motions and fast local fluctuations, our hierarchical SDE formulation can naturally be extended to learnable or adaptive timescales,  and we view this as a promising direction for future work that could enhance the framework's robustness across diverse protein systems with varying characteristic timescales.\n\n**W2**\n \nWe appreciate this question regarding trajectory continuity, and while potential discontinuities could theoretically occur in our hierarchical approach, our evaluation demonstrates that such instances are infrequent and remain within acceptable error ranges as evidenced by our low RMSD error metrics (Table 1) and smooth trajectory coherence (Figure 4 and Figure 8 in Appendix), and developing more sophisticated continuity constraints represents an important direction for future work.\n\n**W3**\n\nWe thank the reviewer for this valuable suggestion and clarify that, as an initial attempt, we focused on validating backbone trajectory generation within the TEMPO framework since backbone atoms determine the primary conformational changes of proteins, but we fully agree that sidechain generation is equally important and valuable. Extending to full-atom representation requires significantly larger model capacity and more physical constraints, making the incorporation of sidechain dynamics generation one of our key objectives for future model improvements.\n\n**W4** \n\nWe appreciate this observation regarding data dependency, and note that while the current scarcity of comprehensive MD benchmarks and large-scale pretraining datasets for protein dynamics generation constrains our evaluation to mdCATH, our framework is designed to leverage temporal dependencies that become increasingly valuable as larger and more diverse MD datasets emerge, and we believe our approach will demonstrate advantages as the field develops richer benchmarks for protein dynamics modeling.\n\n**W5** \n\nThe requirement for initial frames is inherent to the nature of molecular dynamics simulation and trajectory generation, as our method aims to generate temporally coherent MD trajectories that capture physically realistic transitions from a given starting conformation, and importantly, our model's performance is not limited by the source of the initial structure, allowing it to be coupled with any structure prediction method to generate dynamics from sequence-predicted conformations.\n\n**Q1** \n\nWe appreciate this valuable suggestion and acknowledge that leveraging pretrained structure prediction models could potentially enhance our framework's performance, though we have not yet explored this direction due to computational constraints, representing an important avenue for future investigation.\n\n**Q2**\n \nExtending TEMPO to full-atom structures is conceptually straightforward, as our framework can accommodate higher-dimensional representations, though computational constraints currently limit our focus to backbone dynamics, which capture the essential conformational changes for most biological processes, and full-atom extension represents a key objective for our future model improvements.\n\n**Q3**\n \nWe empirically chose the 20ns/1ns hierarchy based on established biophysical principles where the 20ns interval effectively captures major conformational transitions between different states in the free energy surface as visualized in Figure 1a, while the 1ns resolution represents the dataset's finest temporal sampling, and as a model hyperparameter, different datasets may require empirical exploration of various resolutions to optimize performance since protein dynamics timescales can vary significantly across different systems and functional contexts.\n\n**Q4**\n \nWe have explored training single-resolution models at 1ns intervals and found that autoregressive error accumulation becomes problematic for extended trajectories, with single-scale generation beyond approximately 400 frames resulting in structural deviations exceeding 8Å RMSD relative to the native structure compared to ground truth. We also conducted ablation studies comparing our hierarchical TEMPO framework against single-scale baselines where the high-resolution model directly generates all frames without slow-scale guidance, as shown in the table below, demonstrating the importance of our multi-scale design.\n\n| Metrics | TEMPO (Multi-scale) | TEMPO (Single-scale) |\n|---------|---------------|-----------|\n| Pairwise RMSD(=3.26) | **2.78** | 7.46 |\n| Pairwise RMSD r ↑ | **0.77** | 0.14 |\n| All-atom RMSF(=1.64) | **1.60** | 4.27 |\n| Global RMSF r | **0.67** | 0.15 |\n| Root mean W2 ↓ | **4.21** | 8.27 |\n| MD PCA W2 ↓ | **2.33** | 2.53 |\n| % PC-sim > 0.5 ↑ | **7.81** | 3.12|\n| Weak contacts J ↑ | **0.43** | 0.23 |\n| Trans. contacts J ↑ | **0.20** | 0.09 |\n| RMSD Error ↓ | **1.78** | 8.62 |"}, {"review_id": "jtjAhc49Sm", "reviewer": "Reviewer_HVNL", "summary": "In this paper, the authors present TEMPO, a method to generate trajectories of protein dynamics by learning a multi-scale dynamical propagator parameterized by a neural network. Specifically, capitalizing on the multi-scale nature of protein dynamics, the authors propose to learn a coarse propagator which evolves the system at a large timestep (e.g. 20 ns) to capture large-scale conformational motions, as well as a fine propagator which refines the coarse prediction at a smaller timestep (e.g. 1 ns). Results are demonstrated on proteins from the mdCATH dataset, showing that TEMPO can recover conformational ensembles and transition pathways of proteins well relative to baseline methods.", "questions": "1. Is there a reason the authors chose to frame the prediction task as a deterministic mapping from $X_t$ to $X_{t+\\Delta t}$ and adding Langevin noise after the prediction, as opposed to treating the problem in a generative modeling/distribution matching/stochastic interpolant framework (i.e sample from the distribution $p(X_{t+\\Delta t} | X_t)$). It seems that that would make more sense, because the ground truth $(X_t$ to $X_{t+\\Delta t})$ tuples that the method is being trained on are presumably the result of stochastic dynamics, so it would be incorrect to treat this $X_{t+\\Delta t}$ as the unique ground truth value. Additionally, this way it would not be necessary to set/sample an empirical noise scale during sampling.\n\n2. Does switching from the fine to the coarse dynamics every e.g. 20 steps introduce boundary effects/discontinuities in the sampled dynamics?\n\n3. During training, through how many autoregressive steps are you able to optimize the loss function without exhausting memory/getting unstable gradient norms? Within this limit, does reducing the teacher forcing probability and backpropagating though more autoregressive steps lead to performance improvements?\n\n4. Just to clarify, were all baselines retrained on the identical training split of mdCATH?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is generally well-written and easy to follow.\n\n2. The proposed method of learning a fast and slow dynamical propagator is conceptually simple and easy to implement. It allows for the possibility of simply generating coarse dynamical trajectories without resolving the finer/faster dynamics, which is often sufficient for many practical applications.\n\nWeaknesses:\n1. It seems that the training procedure requires pre-specification of the integration timestep during training, and that this cannot be changed during inference. In contrast, previous works directly [1] learn the transition density operator, which allows for conditioning on an arbitrary timestep (and recovering i.i.d sampling as $\\Delta t \\rightarrow \\infty $). In addition, the hierarchy is fixed to be 2-levels. This seems to limit the flexibility of the approach - there may be systems for which less or more temporal resolution is required.\n\n2. The strength of the results appears inconsistent and unconvincing. \n\na) In Table 1, apart from the RMSD metrics which loosely capture whether the correct amount of conformational diversity is being captured, performance from TEMPO appears mixed/weak relative to baselines. In particular, the % clash ratio metric (which presumably measures the frequency of steric clashes in the produced dynamics) is considerably higher than MDGen, making it unclear whether the proposed steric clash loss is having the intended effect. \n\nb) Additionally, it appears from Figure 2 that the dynamics learned by TEMPO are highly smoothed compared to the reference. Is this because of the 1 ns fine timescale? \n\nc) In Figures 4 and 5, it is not at all clear to me that TEMPO-produced trajectories are superior to those of the baselines. In most cases, the sampled points do not even cover all the modes of the distribution, suggesting that large-scale conformational transitions are not being captured accurately. Is this perhaps a generalization issue (i.e are results better on the training proteins but fail to translate to the test set)? \n\nd) The claim that TEMPO produces more \"focused exploration\" of the conformational space than the baselines is at odds with the claim that it enables sampling large-scale conformational transitions. \n\n[1] Schreiner, et. al \"Implicit Transfer Operator Learning: Multiple Time-Resolution Surrogates for Molecular Dynamics\"", "was_revised": true, "final_justification": "I thank the authors for addressing some of my concerns during the rebuttal period, and have raised my score from 3 to 4.\n\nIn general, this paper proposes an interesting approach for generating temporal trajectories of protein dynamics by coupling fast and slow timescales. While this type of approach is not entirely novel and has a history in the molecular dynamics literature, to my knowledge it has not been combined with generative models in this way, and appears to offer promising speedups over baselines (although one point that was unclear to me after going over the inference times reported in Table 1 is whether the reported time for 400 snapshots is comparing correlated snapshots in the case of TEMPO/MDGen to uncorrelated snapshots for ESM/AlphaFlow - if this is the case, then this would be an unfair comparison, as the latter is sampling more statistically independent samples in the given time. I would request the authors to clarify this in their final version). \n\nOverall, I recommend weak acceptance due to the fact that the ideas contained in the paper are interesting, but I still don't find the results/claims of superior sampling to be very convincing (e.g Figures 4, 8, 9). I think the paper would be strengthened more with clearer examples of how the generated samples are substantively more useful/accurate than the baselines - the visual results in the aforementioned figures are not convincing on their own.", "rebuttal": "We sincerely thank the reviewer for their detailed feedback on our modeling framework choices, evaluation methodology, and concerns about result consistency and generalization performance. Below are our detailed responses addressing each point.\n\n**W1** \n\nWe appreciate this comment. While our current implementation uses pre-specified timesteps, this design choice reflects the established biophysical principle that protein dynamics naturally separate into slow collective motions and fast local fluctuations. The framework can be naturally extended to multiple scales by adding coupled SDE terms, similar to adaptive resolution schemes in image generation.\nUnlike ITO, which focuses on coarse-grained Cα representations and uses exponential distribution sampling, our approach models full backbone dynamics while maintaining temporal coherence and biological interpretability. ITO shows challenges in long-time scale prediction (as evidenced by negative VAMP-2 gaps that increase with lag time). In contrast, our hierarchical approach decomposes complex long-timescale dynamics into more tractable multi-step processes. Additionally, ITO was only evaluated on 12 folding proteins, making it difficult to assess its model capacity and generalization ability on larger protein datasets, whereas our model is trained on the comprehensive mdCATH dataset and tested on novel proteins to demonstrate robust generalization capabilities. We will provide the discussion of these methodological differences in the revision.\n\n\n**W2.a** \n\nWe appreciate this observation and note that unlike ensemble methods that perform independent sampling, TEMPO generates temporally coherent MD trajectories while maintaining thermodynamically consistent free energy surfaces, demonstrating comparable ensemble quality and superior trajectory fidelity through lower RMSD error compared to MDGEN, and while our clash ratio is higher than MDGEN, it remains significantly lower than ensemble-based baselines like BioEMU and AlphaFlow, with visualizations revealing that MDGEN‘'s lower clash ratio comes at the cost of severely restricted sampling confined to very small conformational regions on the mdCATH dataset, and our clash loss has already substantially improved inference stability with future incorporation of enhanced physical constraints expected to further reduce the clash ratio.\n\n**W2.b**\n \nThe smoothed curve in Figure 2 occurs because we compare our coarse-resolution trajectories (20ns intervals) with full MD ground truth (1ns intervals), as our coarse model is designed to capture slow collective motions rather than high-frequency fluctuations.\n\n**W2.c** \n\nWe thank the reviewer for raising this concern about mode coverage challenges and acknowledge that capturing all distribution modes represents a fundamental difficulty in protein dynamics modeling, as even extensively pretrained methods like BioEMU show significant FES deviations (Appendix Figure 9). While our primary objective focuses on generating realistic conformational transitions rather than perfect mode coverage (Figure 4 and Appendix Figure 8), we recognize the performance gap between training and test scenarios reflects the complexity of generalizing to novel protein architectures in mdCATH where most proteins contain distinct energy basins. We have added TEMPO's performance on the training set in the table below, evaluated on 60 randomly selected proteins, where the training set demonstrates significantly better mode coverage accuracy. Due to conference policy restrictions on supplementary figures, we will provide comprehensive training set visualizations in the revision to illustrate this performance.\n\n| Metrics | TEMPO (train sample) | TEMPO (test) |\n|---------|---------------------|----------------------|\n| Pairwise RMSD | 3.06(2.51) | 2.78(3.26) |\n| Pairwise RMSD r ↑ | **0.79** | 0.77 |\n| All-atom RMSF | 1.16(1.31) | 1.60(1.64) |\n| Global RMSF r | **0.78** | 0.67 |\n| Root mean W2 ↓ | **3.42** | 4.21 |\n| MD PCA W2 ↓ | **1.31** | 2.33 |\n| % PC-sim > 0.5 ↑ | **46.7** | 7.81 |\n| Weak contacts J ↑ | **0.53** | 0.43 |\n| Trans. contacts J ↑ | **0.37** | 0.20 |\n| RMSD Error ↓ | **1.51** | 1.78 |\n\n**W2.d**\n\nWe clarify that \"focused exploration\" refers to TEMPO's adaptive sampling around physically relevant conformational regions rather than uniform dispersion, which enables more efficient capture of large-scale transitions when they occur (Figure 4 and Figure 8 in the Appendix). This contrasts with baseline methods that show more dispersed but less physically meaningful patterns, particularly evident in Figure 9 in the Appendix, where our trajectory-aware approach maintains tighter distributions around energetically favorable states while still capturing the essential conformational transitions that define protein dynamics.\n\nWe appreciate the reviewer's feedback and respectfully note that protein dynamics generation represents an emerging and challenging field where our work addresses the more complex task of capturing temporal conformational evolution rather than just equilibrium ensemble generation, and our comprehensive evaluation demonstrates consistent improvements over baselines across multiple metrics on the demanding mdCATH dataset, with additional experiments on the ATLAS dataset(kindly refer to our respose to Q4) showing our model's substantial performance advantages over baselines through both quantitative metrics and visualizations, confirming our model's advanced ability to capture conformational transition pathways, particularly for proteins with complex free energy landscapes featuring multiple basins, while our multi-scale SDE formulation significantly reduces computational demands during training and maintains high efficiency during inference compared to direct trajectory decoding approaches used in prior work.\n\n**Q1**\n\nWe thank the reviewer for this question about our modeling framework choice. Our deterministic prediction approach is theoretically justified because we are essentially learning the conditional expectation E[X_{t+1}|X_t] in the finite time-step regime, which mirrors how numerical MD integrators operate with deterministic updates plus controlled stochastic components for temperature regulation[1,2]. Additionally, our approach parallels transformer-based language models, where deterministic training (cross-entropy loss) still enables diverse generation during inference - similarly, we preserve stochastic diversity via Langevin noise while learning conditional expectations through MSE loss. At the 1ns sampling interval used in mdCATH, consecutive frames exhibit strong deterministic correlations due to physical inertia, making our model appropriate for capturing dominant drift dynamics, and our SDE formulation dX_t = μ(X_t)dt + σdW_t directly mirrors actual MD implementation, where deterministic force calculations drive conformational changes and stochastic thermostats introduce thermal effects.\n\n[1] Effective stochastic dynamics on a protein folding energy landscape.\n\n[2] Memory effects in irreversible thermodynamics.\n\n**Q2** \n\nWe appreciate this question regarding trajectory continuity, and while potential discontinuities could theoretically occur in our hierarchical approach, our evaluation demonstrates that such instances are infrequent and remain within acceptable error ranges as evidenced by our low RMSD error metrics (Table 1) and smooth trajectory coherence (Figure 4 and Figure 8 in the Appendix), and developing more sophisticated continuity constraints represents an important direction for future work.\n\n**Q3**\n \nDuring training, we can optimize through approximately 20 autoregressive steps corresponding to our trajectory length under the multi-scale setting without encountering memory or gradient stability issues, and while we have not systematically explored reducing teacher forcing probability or extending backpropagation through longer sequences, this represents an important direction for future investigation that could potentially improve long-term trajectory coherence.\n\n**Q4**\n\nWe appreciate the reviewer's question. We chose mdCATH over the widely used MD dataset ATLAS because its free energy landscapes contain more distinct energy basins with clearer conformational transition pathways better suited for our dynamics modeling task. At the same time, we evaluated baselines using their provided checkpoints since BioEMU has already been pretrained on diverse MD data and PDB structures; ESMFlow and AlphaFlow were not designed for MD generation. MDGEN encounters severe memory limitations on mdCATH due to its full-trajectory attention mechanism (requiring batch size of 1 and still experiencing out-of-memory errors on 8 A100 GPUs when training 400-frame conformations), but to address fairness concerns, we also trained TEMPO on ATLAS following MDGen's splits and results in table below demonstrate our method's strong performance across different datasets, achieving **state-of-the-art** results on ATLAS, with FES visualizations of generated ensembles corroborating our quantitative results, comprehensive visualizations will be provided in the revision due to current supplementary figure limitations.\n\n\n| Metrics | BioEMU | AlphaFlow-MD | MDGEN | TEMPO |\n|---------|-------|-------------|----------|---------|\n| Pairwise RMSD r ↑ | -0.02 | 0.48 | 0.48 | **0.91** |\n| Global RMSF r | 0.09 | 0.60 | 0.50 | **0.89** |\n| Root mean W2 ↓ | 19.23 | 2.61 | 2.69 | **1.49** |\n| MD PCA W2 ↓ | 3.61 | 1.52 | 1.89 | **0.60** |\n| % PC-sim > 0.5 ↑ | 14 | 44 | 10 | **76** |\n| Weak contacts J ↑ | 0.26 | 0.51 | 0.62 | **0.74** |\n| Trans. contacts J ↑ | 0.06 | 0.29 | **0.41** | 0.38 |\n| RMSD Error ↓ | - | - | 3.20 | **1.83** |"}]

Author rebuttal:

Meta-review: This submission introduces TEMPO, a hierarchical autoregressive framework designed to generate temporally coherent and physically realistic protein conformational trajectories by modeling the multi-scale nature of protein dynamics. Its core design decomposes dynamics into two coupled scales: a low-resolution component (20 ns intervals) that captures slow, collective conformational transitions, and a high-resolution component (1 ns intervals) that generates fast local fluctuations conditioned on the large-scale movements. 

Reviewers consistently highlighted TEMPO’s strengths, including its novel integration of biophysical principles into autoregressive modeling as well as comprehensive ablation studies. During the rebuttal period, the authors addressed reviewer concerns, including training TEMPO on ATLAS to confirm cross-dataset SOTA results for dataset fairness; providing results for validating component importance and trajectory continuity concerns; and clarifying about the sampling behavior. Although limitations, especially rigor evaluation and comparison and more convincing visualized evidence remain, given its strengths alongside reviewers’ positive feedback, I recommend accepting this submission, and encourage the authors to further improve the work in the final version in line with reviewers’ comments and discussion.
Decision: Accept (poster)