Learning residue level protein dynamics with multiscale Gaussians
Authors: Mihir Bafna
Bowen Jing
Bonnie Berger
Venue: ICLR 2026 Poster (2026)
Keywords: protein dynamics
flexibility
ensembles
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: Lightweight SE(3) invariant model for learning protein dynamics Gaussians and fast ensemble generation

Abstract: Many methods have been developed to predict static protein structures, however understanding the dynamics of protein structure is essential for elucidating biological function. While molecular dynamics (MD) simulations remain the in silico gold standard, its high computational cost limits scalability. We present DynaProt, a lightweight, SE(3)-invariant framework that predicts rich descriptors of protein dynamics directly from static structures. By casting the problem through the lens of multivariate Gaussians, DynaProt estimates dynamics at two complementary scales: (1) per-residue marginal anisotropy as 
 covariance matrices capturing local flexibility, and (2) joint scalar covariances encoding pairwise dynamic coupling across residues. From these dynamics outputs, DynaProt achieves high accuracy in predicting residue-level flexibility (RMSF) and, remarkably, enables reasonable reconstruction of the full covariance matrix for fast ensemble generation. Notably, it does so using orders of magnitude fewer parameters than prior methods. Our results highlight the potential of direct protein dynamics prediction as a scalable alternative to existing methods.

Published as a conference paper at ICLR 2026
LEARNING RESIDUE LEVEL PROTEIN DYNAMICS WITH MULTISCALE GAUSSIANS
Mihir Bafna
Bowen Jing'
Bonnie Berger
1 CSAIL, Massachusetts Institute of Technology
Dept. of Mathematics, Massachusetts Institute of Technology
3 Liquid AI
(mihirb14, bjing, babbemiit.cm
ABSTRACT
Many methods have been developed to predict static protein structures, but under- standing structure dynamics is what is essential for elucidating biological function. While molecular dynamics (MD) simulations remain the silico gold standard, its high computational cost limits scalability. We present DYNAPROP a lightweight, SE(3)-invariant framework that predicts rich descriptors of protein dynamics di- rectly from static structures. By casting the problem through the lens of multi- variate Gaussians, DYNAPROT estimates dynamics at two complementary scales: (1) per-residue marginal anisotropy x 3 covariance matrices capturing local flexibility, and (2) joint scalar covariances encoding pairwise dynamic coupling across residues. From these dynamies outputs, DYNAPROT achieves high accuracy in predicting residue-leve flexibility (RMSF) and, remarkably, enables reasonable reconstruction the full covariance matrix for fast ensemble generation. Notably, it does so using orders of magnitude fewer parameters than prior methods. Our results highlight the potential of direct protein dynamics prediction as a scalable alternative existing methods.
INTRODUCTION
Proteins rarely exist in static conformations. Due to interactions with ligands, other biomolecules, and external factors such temperature and pH, protein structures continuously fluctuate. Many enzymes rely on loop motions or domain rearrangements to form catalytically active sites (Zinovjev et al., 2024), allostery often involves shifting backbone or side chain conformations propagating signals over long distances (Yu and Koshland Jr, 2001), and even membrane proteins such as GPCRs switch between inactive and active conformational states essential for signal transduction (Zhang et al., 2024). Clearly, protein structures are dynamic. Understanding these dynamics is central for mechanistic insight and, potentially, the design of functions (Guo al., 2025; McCammon, 1984).
Capturing this ensemble-level behavior computationally has long been the domain of molecular dynamics (MD). MD simulates the time evolution of atoms under a force field, generating high- resolution conformational trajectories from which fluctuations, covariances, and time-dependent observables can be derived. MD remains the in sílico gold standard for protein dynamics, offering fine-gree fine-grained, physically grounded insights (Shaw et al., 2010; Hollingsworth and Dror, 2018; Childers and Daggett, 2017). However, it comes with an enormous computational cost: simulating 100 ns of dynamics for single protein can take days or weeks on specialized hardware. This limits its scalability, especially for proteome-wide applications or tasks requiring real-time dynamics estimates.
Recent work has explored the use of deep learning to approximate and accelerate this process. Generative modeling-basee
methods like ALPHAFLOW (Jing et al., 2024) repurpose AlphaFold2 (Jumper et al., 2021) under a flow matching paradigm to sample protein conformations, Along this vein, Lewis et al. (2024) recently introduced BIOEMU as a large-scale diffusion model pretrained on PDB (Burley et al., 2017) and AFDB structures, and fine-tuned efficiently generate protein conformations. Other methods, like MSA subsampling, make inference- time adjustments to the MSA input of AlphaFold2, yielding the structural ensembles (Del Alamo
200 ms of MD data, to

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al., 2022; Wayment- Steele al., 2024; Stein and Mchaourab, 2022). Still, all of these approaches necessitate large scale PDB pretraining and suffer from inference- time computational overhead, requiring multiple stochastic forward passes to generate meaningful structural diversity. Moreover, while these ensembles can be used to approximate protein dynamics, generating them remains time intensive, and the full ensemble of diverse conformations may not always be necessary. In many practical settings, compact and interpretable representations of dynamics often suffice. This motivates the need for models that can explicitly predict such dynamies descriptors without relying on implicit dynamics learners like expensive ensemble generation methods.
Current explicit dynamics predictors, like FLEXPERT3D (Kouba al., 2024), resort to predicting simple collective variables like per-residue RMSF, a scalar quantifying each residue's positional fluctuation. RMSF is widely used due its simplicity and interpretability, but it fundamentally limited: it captures only the magnitude of local motion and discards directionality and residue-residue coupling. Similarly, Wayment-Sten Stteee et al. (2025) trained DYNA-1 1 to predict labels of jus-ms motion by cleverly exploiting missing chemical shift assignments as hidden observables in NMR ensembles, these predictions also remain scalar and lack directionality. A different example an explicit dynamics predictor is Normal Mode Analysis (NMA), classic technique that approximates dynamics by identifying low frequency eigenmodes to describe the largest movements (Cui and Bahar, 2005; Skjaerven al., 2009). NMA does not learn from data however, and instead operates solely on the input PDB structure. It can estimate the principal global directions of motion and offers insights into collective flexibility, but is sensitive input structure quality and fails adequately capture local anisotropy or conformational heterogeneity (Ma, 2005). This raises a natural question:
Can we design models that lie on the Pareto frontier of expressiveness and effi- ciency capturic rich dynamic behavior without incurring the cost of sampling simulation?
We introduce DYNAPROT, a lightweight, inter- pretable, and expressive framework for predicting protein dynamics through the lens of Gaussian dis- tributions over structure (Section 2). Specifically, DyNAPROT predicts: (1) per-residue marginal anisotropy as 3 x 3 covariance matrices captur- ing local dynamics while encompassing RMSF, and (2) joint scalar N x N covariances encoding pairwise dynamic coupling across residues. Re- markably, while DYNAPROT was not explicitly trained directly model the full 3/ 3N joint distribution, we find that its marginal and pair- wise outputs can be composed into a reasonable approximation (Section 3.4), enabling extremely fast ensemble generation in IR3N DYNAPROT is trained on only ~~ 1,000 MD-derived proteins, without large-scale pretraining on PDB structures, and improves upon Normal Mode Analysis (NMA) in both predictive accuracy and efficiency, while remaining dramatically smaller and faster than existing ensemble generation approaches.
Dynamics Pareto Frontier
DynaProt
2. FlexPert3D
Dynamics Information Content
AlphaFlow
MD
Efficlency
Figure 1: Dynamics methods information con- tent vs. efficiency.
To our knowledge, DYNAPROT is the first model to explicitly learn both marginal and pairwise Gaussian representations of protein dynamics, and the first to predict the full 3NV x 3NV covariance dtuucture-ake to NMA-in in data-driven, learnable fashion, rather than relying solely analytical approximations or less informative per-residue fluctuations.
GAUSSIAN REPRESENTATION OF DYNAMICS
We propose a perspective for modeling protein dynamics through distributions over atomic coor- dinates, relying on tractable approximations such Gaussians. Formally, we model protein structure with residues as a random variable X E R3N where each residue contributes the Cartesian coordinates of its C. atom. While this coarse graiied representation omits side-chain ficxibility,, it enables scalable modeling of backbone dynamics, which is the scope of our work. We consider ensemble be T independent samples after RMSD alignment. The
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distribution over conformational states is then represented multivariate normal distribution:
X (u, Ejoint),
μ E MER
Djjint RaNx3N
Here, μ corresponds to the average (or equilibrium) structure-ttyically the min- imum energy conformation-ann Ejsi captures the full covariance across all C. positions, encoding both local fluc- tuations and long-range correlated tions. This joint covariance matrix theoret- ically encodes all second-order informa- tion about the protein's dynamics: from it, one can derive wide range of col- lective variables including principal com- ponents (PCs) of motion, residue-residue distance variances, and global flexibility metrics. The Gaussian formulation pro- vides principled way to decompose pro- tein dynamics across different levels of granularity (Figure 2 and Table 1), de- pending on the modeling objective. At the local level, the marginal distribution (Figure 2.2) for single residue i is obtained by integrating out all other residue coordinates: p(x,) = p(z1...... .n dæ-a, where dx-4 : [z dæy. This results 3D Gaussian distribu- tion over the C. coordinates of residue
Ensemble/ Traemble/Traeectory
4. Joint full covariance
1. Per-residue scalar
2. Per-rasidue full covariance Joint scalar covariance
Figure 2: Gaussian view of dynamics.
X; M(4, (1)
marginal) 14 E R
(E) marginal E 3x3
where (+)
marginal the 3 x diagonal block of juint- These marginals can be interpreted as Gaussian blobs encoding anisotropic local fluctuations i..., spatial variance of where each residue may reside.
Notably, this formulation allows for simple derivation of scalar flexibility metrics such the root-
mean-square fluctuation (RMSF) RMSF; v TT marginal) ) RMSF (Figure 2.1) represents simple notion of dynamics: a single sealar per residue quantifying positional fluctuation. However, discards directional and covariance structure captured by the full marginal.
To capture dynamics beyond residue-loca fluctuations, we also consider a covariance matrix C' N of scalar pairwise coupling (Figure 2.3). Each entry Cij summarizes the dynamical coupling between residues and j. typically computed as a scalar projection the corresponding x block in the full joint covariance: Ejuim 31 3i + 3, 3j 3j + 3]. We choose MeanPooling as the scalar projection to compute each Cij. This compact representation enables efficient modeling of residue-residue coupling.
Table 1: Taxonomy protein dynamics representations under a Gaussian view.
Level
Description
Notation
Space
Captures
Per residue scalar (i.. RMSF)
Tr (
R
Magnitude of fluctuation per residue
2 Per-residue full (Gaussion blob)
x (4)
margiall
R N XXxx3
Anisotmoic local covariance residue
3 Joint scalar (pairwise coupling)
C₁
R N
Scalar covariance across all residues
Joint full covariance
Eyen
R x
Full spatial covariance across all residues
DYNAPROT focuses on levels 2 and 3 of this hierarchy-explicee predicting both 3 marginal Gaussians per-residue and N x N matrix of residue-residue couplings. As noted before, from the 3 x 3 marginals, we can easily derive RMSF (level 1). Interestingly, utilizing both the marginals and the pairwise coupling,
retrieve reasonable approximation of the full joint 3.N x 3NN (level 4; Section 3.4). This design strikes a balance between local interpretability and global coordination, while avoiding the intractability of directly learning the full joint covariance.
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Pairwise Coupling
singla Pep.
Pair Attention
Cholesky Readout
Invariant Point
MLP
SEL5)
Margiall Residue @aussians
[PA)
Cholesky Readout
MLP
xB
Figure 3: DYNAPROT architecture.
METHOD
3.1 DYNAPROT OVERVIEW
DYNAPROT (Figure 3) consists of models, each taking as input protein structure but designed to explicitly capture different granularities of protein dynamics: (i) marginal Gaussian blobs per residue (Section 3.2), and (ii) pairwise covariance across residues (Section 3.3). DYNAPROT given the input structure as a set of local Ca residue frames. The frames are denoted {T) where each frame T; E SE(3) is parameterized by rotation matrix R& E SO(3) and a translation vector t; E R Simply put, T, (R; (;) captures the local orientation and position of residue i. Additionally, an initial embedding layer is included to encode the amino acid sequence E RN
Both models share common architectural backbone composed of eight Invariant Point Attention (IPA) blocks from the structure module of AlphaFold2 (Jumper et al., 2021). These blocks are designed to encode geometric relationships between residues while maintaining invariance to SE(3) transformations (global transformations do not affect the learned residue- representations). The IPA backbone processes the set of residue frames and the sequence representation, outputting learned representation for each residue h E Ig. DD No pair representation is given as input to the model. The two models differ only in their readout layers, which define in the proceeding sections.
3.2 LEARNING MARGINAL GAUSSIANS
After the input sequence representations and residue frames are processed through the IPA backbone, simple MLP readout is used for marginal prediction. Given the hidden representation h; for each marginal E 3g33 modeling the local position (xyz) covariance of residue i. These outputs trained to match empirical marginal distributions derived from the MD data. Note that the mean of each Gaussian is not learned. Instead, we take the input structure's 's C,, coordinate t; 3 as the fixed mean μ4; of the distribution: assumption is motivated by the fact that the input structure usually corresponds to the experimentally determined (or
residue, the marginal readout outputs > (8)
μ4: := t,, X; ~N Σ marginall This
minimum energy confornation, and thus serves as a natural estimator of the ensemble mean. Consequently, the marginal prediction task reduces to learning the covariance matrices (1)
marginal alone.
Marginal dynamics module. Recall that covariance matrices are required to be symmetric and positive definite (SPD). Predicting all 9 elements of a 3 x 3 matrix would be overparameterized and does not guarantee SPD structure. Naively, one might consider symmetrizing an arbitrary matrix after predicting the independent elements, but this only guarantees symmetry. Instead, we leverage the fact that any SPD matrix can be uniquely defined by its Cholesky factorization. Thus, we enforce SPD constraints directly by parameterizing the covariance via its Cholesky factor. Specifically, the model predicts the entries {a,); of lower triangular matrix L, E _3x3 enforces positivity along the diagonal with the Softplus activation function (Glorot et al., 2011), and recovers the covariance:
softplus(a1)
Σ ( marginal
L.D., where L,
a2
softplus(as)
0
a4
a5
softplus(a6))
This factorization ensures that the predicted covariance matrix SPD by construction. Since SPD matrices lie Riemannian manifold with onnFuucidean geometry, using loss functions that respect
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this structure is critical for meaningful comparison. Standard Euclidean distances (e.g., MSE Frobenius norm) ignore the curvature of this space and can lead to unstable or distorted gradients (see ablations in Appendix A.3.2). We instead employ the log- oog-Euclidean (or log- Frobennu))) distance (Vemulapalli and Jacobs, 2015; Huang et al., 2015) that reflects the intrinsic geometry of the SPD manifold. The Bures- innnest (Bhatia al., 2019) distance also be used, but find the log-Frobenius distance to be more stable.
(4) As Riemannian manifolds are "locally Euclidean", this loss applies the matrix logarithm mapping the SPD matrix to its tangent space where Euclidean metric (canonical Frobenius norm) can be utilized.
LogFrob = || ( log E where log(E))
Q Qog((A)
3.3
LEARNING PAIRWISE DYNAMICS
Using the output representations h from the IPA backbone, the pairwise dynamics module produces scalar-valued N x N covariance matrix C, where each entry Cıý captures the dynamical coupling between residue pairs. These scalar couplings are derived from the full joint covariance matrix via averaging per block and trained to reproduce MD-derived pairwise fluctuations.
Pairwise Dynamics Module. To predict the global pairwise covariance structure, we first construct pairwise features for all residue pairs. For each pair (i,j), we concatenate their residue-level embed- dings [h; hy] E R2d and project them into lower-dimensional space: /1g W pmoj [h hy] Rd".
We pass these features through a stack of AlphaFold-style pairwise attention blocks based on the Evoformer architecture (Jumper et al., 2021), which include triangle updates and residue-wise message passing. These operations dependencies across residue pairs, and have been shown to be highly effective in structure-aware tasks: m PairwiseAttentionnBlon ( E R.d". The output f serves "learned basis""" over the space of residue-residue covariance structure. These basis features are then mapped to scalars through an MLP head, yielding covariance for each pair of residues: 2ij (LP((fa)..) 24j R. for 2 j. Following the same procedure as Section 3.2, we enforce SPD constraints on this covariance matrix by populating the lower-triangle entries of L E NNN with the values of 2; and applying the Softplus activation when i = j. Finally, the pairwise covariance matrix is reconstructed via Cholesky composition C LL and again equation is used for optimization.
designed model transitive and higher-order geometric
3.4 LEARNING THE FULL JOINT FOR ENSEMBLE SAMPLING
Joint reconstruction heuristic. Given a predicted scalar coupling matrix E N and a set of per-residue marginal covariances {Σ (t)
marginal E 33331 Ii=1, propose heuristic to reconstruct
approximate full joint covariance matrix Ejoint E 3NN3N
(() Each marginal covariance marriinal is SPD by construction, and thus admits Cholesky factorization (6)
manginal L; L, where L; 3x3 We then define block-diagonal matrix Lmaaginal E 3 x3N Imrrginal + = L, By construction, Lmarginal is lower triangular with positive diagonal entries,
since each L, satisfies these properties.
Drawing from the univariate identity Cov(i.)) Corr(i,j) o,Oy, define the multivariate cross- covariance block between residues i and j as (t.j)
= L.C.LLJ Here, the Cholesky factor L; marginal analogous to standard deviation in the univariate case. And, Č is correlation matrix found by standardizing C. Using the Kronecker product, we denote this heuristic cleanly follows,
joint
serves as a matrix square root of the covariance (1)
Σ Jjiiat
Imarginal (Ö 2 Is) L I
J and correlation matrix Č E N to be symmetric and positive definite, then the reconstructed joint covariance Ejoiut = Lemarginat @ L is also symmetric and positive definite.
Proposition 3.1 (SPD Closure). Given marginal covariances marginat E 13331N
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We refer the reader to Appendix A.I for the proof. This approximation combines local anisotropic uncertainty with global correlation structure. While not exact, we find it reconstructs the joint covariance reasonable degree and serves as a useful tool for downstream ensemble generation.
Ensemble sampling. Given the reconstructed joint covariance Ejoint and our assumption that the mean ll corresponds to the coordinates of the input structure (e.g., the PDB), have now retrieved our Gaussian distribution over conformations N (p, Eooma). To sample from this distribution, we apply a multivariate generalization of the reparameterization trick used in univariate Gaussian sampling.
Property 3.1 (Multivariate Gaussian Sampling). Given (u, E), where E E Rdxd is SPD and Σ LL" is its Cholesky decomposition. Then,
x x= μ Le, Io)
x N~N(H,E)
Note that this sampling relies directly the Cholesky factor (similar to matrix square root), mirroring the scalar case (Appendix A.1). Utilizing DYNAPROT predictions and this heuristic, ensemble sampling becomes extremely fast with minimal computational overhead.
4 EXPERIMENTS
Preprocessing. We construct ground-truth dynamics labels from the ATLAS molecular dynamics dataset, which comprises 1 1,390 proteins selected based on structural diversity using the ECOD domain classification (Vander Meersche et al., 2024). Following ALPHAFLOW (Jing et al., 2024) for preprocessing consistency, we concatenate each of the three replicate simulations of 100 ns per protein and extract the C. coordinates. From each ensemble, we compute the empirical full joint covariance matrix over time and extract the relevant dynamics labels (3 x 3 marginals per residue and x N residue coupling) as described Section We evaluate under two train/val/test split regimes. The primary matches ALPHA FLOW's (1265/39/88 while comparisons to FLEXPERT3D use DYNAPROT trained on their topology-buse based split (1112/1B) 139/ 139). For naming, we refer to DYNAPROT-M M for the model trained for marginals, DYNAPROT-J for the coupling predictions, and DYNAPROT for both.
Baselines. For faithful comparison, we mainly choose baseline methods that take protein structure as imput and predict dynamics deseriptors either implicitly (AFMD+TEMPLATES, or explicitly (FLEXPERT3D NMA). For NMA, utilize the ProDy package (Zhang et al., 2021), 1), specifically the Anisotropic Network Model instantiation. For a broader set of baselines, we also compare against some sequence based methods in Appendix A.3.3.
4.1 PREDICTING RESIDUE FLEXIBILITY
Since DYNAPROT-M is trained to predict marginal Gaussians per residue, it inherently captures residue-level flexibility, as RMSF is defined as the square root of the trace of each marginal covariance (see Section 2). To evaluate DYNAPROT: M's ability to recover this, we compare against what is, to our knowledge, the only method that explicitly predicts residue flexibility: FLEXPERT-3D. For fair comparison, we train and evaluate DYNAPROT-M M under the same ATLAS train/val/test split defined in Kouba et al. (2024). Despite solving the more challenging task of predicting marginal anisotropy rather than scalar fluctuations alone, DYNAPROT-M achieves sub- stantially higher Pearson correlation with MD-derived RMSF (median r = 0.865, 75th percentile 0.930) than FLEXPERT-3D 3D (Table 2), while using three orders of magnitude fewer parameters (955K vs. 1.2B) and without NMA as input. This allows DYNAPROT-N. M to generalize better while being more parameter efficient. See Appendix A.4 for DYNA PROT-M additional RMSF plots.
Table 2: RMSF Pearson correla-
tion (r) against ATLAS MD-derived RMSF (FlexPert test split). Median and 75th percentile reported.
Method
RMSF
Params
DYNAPROT-M
0.865 / 0.930
955 K
FLEXPERI-3D
0,830/ 0,899
1.2B
NMA (ANM)
0.697 0.697/0.784

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A
MD (SRWT_A) Time: OT2)
DynaProt
B
RMWDDD Var Centr Contributiio
RAMAD Cone foution
Concribution
Time
RMND 0.74
oas
DyraPros
(mOOEE)
DynaPPot Time: -0022
AFMD-Semmpaiee
Normal Mode Arwgett yeis
Normal Aroignk
Time: onzi hrs
Accuracy of N x pairevinn coupling
DynaProx
Pearson
[D (80 900
NMO
KL: C.9
Sequence discance ()1-1l is
Figure 4: DYNAPROT marginal Gaussian and residue coupling analysis. A. Renderings of predicted marginal Gaussians compared to ATLAS MD constructed Gaussians (mean symmetric KL divergence and RMWD are reported). B. Joint distribution (within 75th percentile) of DYNAPROT performance vs. (AFMD+T, NMA). C. Band-wise Pearson correlation between predicted and ground-truth residuc-residue coupling matrices function of sequence distance.
4.2 PREDICTING RESIDUE FULL ANISOTROPY
To assess the faithfulness of DYNAPROT-M'S predicted marginals, we compare against both physics- based and learned ensemble methods. In practice, only NMA (ANM) is a feasible baseline, as ALPHAFI.OW is prohibitively slow: A single 271 271-Iength protein (71ao A) requires 7000 com- pared ~0.02 s for DYNAPROT-M (Table 3). Note that DYNAPROT-M predicts this directly, but for ALPHAFLOW, first sample 250 structures per protein and then caleulate the empirical covariance to define the marginal Gaussians. With NMA, we retrieve the full joint via the normal modes and extract the marginal block diagonals.
To quantify the accuracy, we compute the variance contribution of the symmetric KL divergence (see Appendix A.2) and the root mean 2- Wasserstein distance (RMWD) as described in Jing et al. (2024), compared to the ground truth marginal Gaussians computed from the ATLAS test set (AFMD split). Despite being orders of magnitude faster and smaller (955k vs. 95M parame- ters), DYNAPROT-M achieves compet- itive accuracy. DYNAPROT-M attains a median RMWD of 1.18 and symmet- ric KL divergence of 0.91, both sub- stantially better than NMA (1.45 and 4.56, respectively), and comparable to 1.10 and 0.60. Moreover, rather than relying solely summary statistics, we also visualize the distributions (75th percentile) of RMWD and mean symmet- ric KL variance contributions across test proteins (Figure 4B). These plots compare DYNAPROT-M to both AFMD+T and NMA on a per-protein basis. Points below the diagonal (highlighted in pink) indicate that DYNAPROT-N. outperformed the method in question on that particular protein. From this, we see that DYNAPROT-A M achieves comparable performance to AFMD+T. Notably, within the 75th percentile it often outperforms AFMD+T (examples visualized in Figure 4A.B) on RMWD variance contribution. Moreover, DYNAPROT- M significantly outperforms NMA across both RMWD and symmetric KL. This further corroborates DYNAPROT-M. ability to capture local anisotropic structure well despite being much smaller and faster than other methods.
Table 3: Comparison of methods on anisotropic blob pre- diction (ATLAS test split). Runtime for length 271 1 pro- tein (7la0 A). 25th %ile / Median reported (4 is better).
Method
RMWD Var
Sym.
Params
Time
DYNAPROT-M
0,84/ 1.18
0.53/ 0.91
955 K
00.02
AFMD+T
0,87 1.10
0.37/ 0.60
95 M
~7000
AFMD+TEMPLATEE
NMA (ANM)
1.147 1.45
3,03/ 4.56
~5.37s
4.2.1 ZERO-SHOT CRYPTIC POCKET DISCOVERY OF ADENYLOSUUCIIATTE SYNTHETASE
Beyond accuracy, DYNAPROT-M's marginals can also provide functional insight. Many pro- teins are considered to be undruggable as their apo form may not display clear binding pocket. However, the druggable pocket may only become apparent after the drug bound (holo form)- so called "cryptic pocket." " Identification of cryptic pockets is therefore an important task in drug discovery (Mou et al., 2025; Hollingsworth et al., 2019; Comitani and Gervasio, 2018).
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As case study, sought investigate DYNAPROT's ' ability in cryptic pocket iden- tification for the enzyme adenylosuccinate synthetase, as it is known to exhibit a cryp- tie pocket (Meller et al.) and both the apo and holo forms are available in the PDB (IADE / 1CIB).
Apo (IADE)
Holo (ICIB)
We applied DYNAPROT-M to zero shot pre- dict the marginal Gaussians on the apo form. When we look at the predictions, we notice two features (Figure 5). First, the residues with the largest variance are exactly those that encompass the binding pocket. Second, more than just the magni- tude of the Gaussians, the directionality consistent with a potential pocket opening motion (when compared with the holo form 1 ICIB). These early results suggest the potential of DYNAPROTT utility in cryptic pocket discovery, but a systematic exploration left for future work.
Figure 5: DYNAPROT-M predicted residue Gaussians (ellipsoids) overlaid the apo form.
4.3 PREDICTING SCALAR COUPLING
To evaluate DYNAPROT-J in modeling residue-residue couplings, we compare its predicted N N scalar covariance matrices against those derived from NMA, classical method for capturing global structural dynamics. While DYNAPROT-J directly predicts these scalar coupling matrices, we construct comparable matrices for NMA by computing full 3N x 3N anisotropic network models (using ProDy) from each test protein, and then projecting them into N x N scalar covariances as described in Section 2. We obtain per-residue correlation matrices by normalizing the entries to be unit diagonal and constrained to the range [-1, 1].
We observe that in the ground truth N N correlations, magnitude of entries diminish rapidly with distance from main diagonal, indicating weak long-range coupling. This is indicated by the dotted gray line Figure 4C, which shows the mean absolute value of the entries from the principal diagonal up to the kth diagonal band. To focus on meaningful and prominent interactions, we define a diagonal band of width
50 residues (i - j| S 50), which captures local and medium-range interactions. This essentially measures residue-residue coupling as function of sequence distance (how distal are and j along the backbone). We compute the Pearson correlation between predicted and ground- truth ceidue-residue correlation matrices for the entries along each diagonal band k, by iteratively extracting the upper-triangula entries satisfying I - jl I & for k = 1 to 50. This is repeated for each of the 82 test set proteins and the median Pearson correlation is reported (each point in Figure 4C). This band-wise analysis enables us to compare the accuracy of coupling signals at increasing residue distances, and we find that DYNA (peak correlation of 0.71) strongly outperforms NMA (peak correlation of r 0.59) particularly at short to mid-range coupling distances, where the coupling the strongest.
4.4 ENSEMBLE GENERATION
As described in Section 3.4, given the output 3 x 3 marginal covariances and N N residue coupling from DYNAPROT-M. M and DYNAPROT-J respectively, we reconstruct a full joint covariance using the heuristic defined in Eq. 5. This direct access the joint distribution enables extremely fast sampling of diverse structures. For evaluation, we sample 250 structures with DYNAPROT, AFMD+ and NMA to form ensembles for each of the 82 test set proteins in the AFMD split. Following Jing et al. (2024), we assess these ensembles in their flexibility accuracy, distributional similarity, and the ability to reproduce complex observables. For flexibility accuracy, measure the pairwise RMSD to ground truth MD and RMSF correlation at the global and per- target level. For distributional coverage, we measure the 2- Wasserstein distance after projecting the ensembles onto the first two principle components derived from the MD trajectory (MD PCA W2) the combined (MD+sampled) trajectory (Joint PCA W₂). Table 4 summarizes the ensemble evaluation results across AFMD+TEMPLA. DYNA PROT, and NMA. PROT achieves performance
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MD (7
AlphaFlow-Templatte
DynaProt 0.923
0999
X
12 hours
2-3 hours
sec
Figure 6: Comparison of DYNAPROP generated ensemble vs. AFMD+T to ATLAS MD simulation (PDB 7q)u_A) overlaid on reference. RMSF Pearson correlation and sample time reported.
comparable to AFMD+TEMPLATI on key flexibility metrics such pairwise RMSD and per-target RMSF correlation, while lagging slightly behind on distributional similarity and observable recovery. Some examples of where DYNAPROT out- performs AFMD+TEMPLATER on ensem-
Table 4: Comparison of C. ensemble evaluation metrics on ATLAS MD Dataset between AFMD+Templates, DYNAPROT, and NMA.
ble flexibility correlation visualized in Figure 6 and Appendix A.4. For the vi- suals. all atom reconstruction is enabled by PULCHRA (Rotkiewicz and Skolnick, 2008). Moreover, DYNAPROT consistently outperforms NMA across nearly all eval-
Metric
AFMD+T
DYNAPROT
NMA
Pairwise RMSD (=2.89)
2.18
2.17
0.91
RMSF (=1.48)
1.17
1.10
0.52
Global r (1)
extions-except for transient contact pre-
Per-target RMSF (t)
0.92
0.86
0.76
diction-particular excelling in measures of local flexibility and pairwise distance preservation. Notably, DYNAPROT quires only 2.86 million parameters (vs. 95 million for samples ensembles over 70,000 faster on average (~0..14 vs. ~ ~~10,000s) s), all while being trained only to predict marginal and scalar covariances. This efficiency advantage is maintained when compared against sequence-baig baseed methods (i.e. ConfDiff, BioEmu, ESMDiff, see Appendix A.3). DYNAPROT even outperforms them on modeling flexibility and anisotropy.
MD PCA W2 (L)
1.25
1.74
1.86
W/ (L)
2.39
Weak contacts J (t)
0.62
0,51
0.43
Transient contacts J (1)
0.47
0.29
0.33
and
Parameters (4)
95M
2.86 M
Ensemble sampling time (4)
000
Finally, assess DYNAPROT's generalization longer timescale dynamics, by comparing its zero-shot ensemble of BPTI to the Ims trajectory from Shaw et al. (2010). Even with these larger conformational changes. DYNAPROT performs reasonably well. It achieves RMSF correlation of 0.88 (c.f. 86 on ATLAS), anisotropy with RMWD of 0.52 f(c.f. 1.18 Á on ATLAS), and strong recovery of transient contacts J 0.54, c.f. 0.29 on ATLAS). See appendix A.3.1 for more.
5
CONCLUSION
Protein dynamics is critical for understanding biological function. Existing approaches to modeling dynamics often rely on complex generative models with large-scale PDB pretraining and expensive ensemble generation. In this work, we introduce DYNAPROT, lightweight and data-driven alter- native akin Normal Mode Analysis (NMA), but designed directly predict structured dynamics descriptors in the form of per-residue and pairwise Gaussian representations. This formulation enables extreme parameter efficiency while outperforming traditional baselines on key metrics, including flexibility estimation, marginal anisotropy, and residuc-residue coupling. Remarkably, DYNAPROT'S outputs also support ultra-fast ensemble sampling with reasonable structural fidelity offering a compelling alternative conventional ensemble generation methods. While further scaling may be needed to match the full capabilities of state-of- the-art generative methods, our approach highlights a promising alternative grounded in explicitly learning structured representations of dynamics.

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ACKNOWLEDGEMENTS
We would like to thank Alexander Amini, Jeffrey Li, Yuseong (Nick) Oh, Ryan Mei, Manoj Niverthi, Renzo Soatto, and Kabir Doshi for their useful ideas and discussions.
This work was supported by the National Institute of General Medical Sciences of the National Institutes of Health under award number IR35GM141861-0)
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A APPENDIX
A.1 METHOD DETAILS
SPD closure of joint reconstruction heuristic (restating Proposition 3.1). Given marginal
covariances (>6) (Sigiinal E [3333] Ji=1 and a correlation matrix C E NxN that symmetric and
marginal
positive definite, then the reconstructed joint covariance Ejoint marginal symmetric and positive definite.
(č 1g L
marginal is also
Proof. Let marginal RR3Nx3N be the block-diagonal matrix defined
marginal
Li
where each L; E Tx333 the Cholesky factor (or any valid matrix square root) of the positive definite matrix (6)
marginal- Since each [1) iarginna > O. it follows that each L, is full rank, lower triangular, and has positive diagonal entries. Consequently, marginal is full rank and lower triangular with positive diagonal blocks.
Now consider the matrix Č I E R3NN3N Since Č and I3 > 0, the Kronecker product Č & I3 > as well (Kronecker product of two SPD matrices also SPD). Finally, the product
E joint marginal (7 2 13 L
a congruence transformation the SPD matrix Ć x I3 by the full-rank matrix L marginal- Since congruence preserves positive definiteness, we conclude:
Dpint > 0
Moreover, Ejsi is symmetric because is the form ABA
Multivariate Gaussian Sampling. Let (O, I) be standard multivariate normal in Rd. and let μ E Rdd Σ E Raxd be a symmetric positive definite matrix. Suppose E Rdxd satisfies E LL (e.g., via Cholesky decomposition matrix square root). Then the transformation x μ + Le yields a random variable æ (N, E),
Proof. Since Gaussian distributions are fully characterized by their first two cumulants (mean and covariance), it suffices to show that the transformed variable has the desired mean and covariance.
Mean of w:
E [ Elu + LE) L Elej μ
Covariance of æ::
Cov(æ:
E [(e u)( T
= E [LC)(Le
E LEE L
L E.EE T| TT
L I Ig. LT
LL =
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A.2 EVALUATION METRICS
RMWD Variance Contribution. To evaluate the efficacy of the marginal Gaussian predictions, adopt distributional similarity metrics used in Jing et al. (2024). The first of these is the root mean 2-Wasserstein distance (RMWD), specifically its variance contribution term. The 2-Wasserstein distance between two multivariate Gaussians has closed-form expression follows.
Let No N (wo, Σo) and Mi N (pr, E1) be two d-dimensional Gaussian distributions. The squared Wasserstein distance between them is given by:
m1ll + Tr Eo E1 -2 -2 2 ) 1/2
W (Nv, =1
(6)
This expression consists of two additive components: mean contribution and covariance (variance) contribution. This metric is also referred to as the Bures-Wasserstein distance (Bhatia et al., 2019).
Since method predicts only the covariances, we isolate and evaluate only the second term. We define the RMWD variance contribution across residues as follows:
Tr Eo, + Σi,- - 2 Eo. 1/2) 1/2)
RaWDyaar N1)
Symmetric KI. Divergence Variance Contribution. Alongside the Wasserstein metric, we also evaluate the discrepancy between predicted and ground-truth marginal distributions using the symmetric Bellbackk-eibler (KL) divergence, defined as the mean of the two directed KL divergences mentioned in (Kullback and Leibler, 1951; Jeffreys, 1998):
KLyym (No [[) = [KL[NG (I + KL(NG |[
For two d-dimensional Gaussian distributions No = NN Eo) and N₁ = -N(, E1), the KL. divergence from No to M₁ given by:
(m (m 10) d log det ES
KTNNN ([) = 2 Tr(E + (m) - pa) E
(8)
det Eg
This expression consists of both contribution, comprising the trace and log-determinant terms. Since our method predicts only covariances (and uses the input structure coordinates means), we isolate the variance terms by omitting (μy = Po) the mean term:
contribution-the Mahalanobis term-and variance
KNver(N (IA)) = Ir(E, Eo) -oo d + log det Σ1
(9)
det Eo
To symmetrize the variance contribution of the divergence, we define the symmetric variance KL as:
1
(No.
2 [KLvar (|A + KL (M (|.Vo)
(10)
1
(Tr(E) Σn) + Tr Σn) - 2d)
(11)
4
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A.3 ADDITIONAL EXPERIMENTS
A.3.1 DYNAPROT ZERO-SHOT ENSEMBLE GENERATION OF BPTI
We note that DYNAPROT was trained on the ATLAS MD dataset comprising 100 ns per replicate trajectories. In contrast, D.E. Shaw Research performed simulations of BPTI (PDB: 5PTI) millisecond-scale revealing structurally distinct conformational states (Shaw et al., 2010). Thus, in an effort to understand DYNA ability to generalize to long-timescale dynamics, we applied it to BPTI and compared to the DESRES trajectory.
Listed Table 7b, we compute the ensemble evaluation metrics from Jing et al. (2024) and observe that DYNAPROT performs remarkably well: e.g., RMSF correlation of 0.88 (c.f. 86 on ATLAS), local anisotropy with RMWD of 0.52 Å (c.f. 1.18 Á on ATLAS), and strong recovery of transient contacts (Jaccard similarity 0.54, c.f. 0.29 ATLAS). These metrics emphasize that DYNAPROT is able to model larger conformational changes at high fidelity.
MD (5PTI)
DynaProt
Metric
DYNAPROT
Pairwise RMSD (=1.57)
1.36
RMSF(=0.84)
0.86
Per-target RMSF r (f)
0.88
RMWD Var Contrib (1)
0.52
MD PCA W2 (L)
0.49
Joint PCA W2 (J)
0.81
Weak contacts J (1)
0.54
Transient contacts / (t)
0.54
# Parameters (L)
2.86M
Ensemble sampling time (L)
~0.55s
(b) DYNAPROT zero shot ensemble generation of
BPTT (PDB: 5PT1), compared DESRES MD tra- jectory (Shaw al., 2010). Note that the global RMSF Pearson correlation is omitted as there only one protein so global = per-target.
(a) Visualization of DYNAPROT zero shot BPTI (PDB:
SPTI) ensemble generation.
A.3.2 DYNAPROT ABLATIONS
To test both the importance of Dy-
Table 5: DYNAPROT-M ablations of the log Frobenius loss loss and SE(3) invariance.
NAPROT's Riemannian aware loss
(log Frobenius norm) and the SE(3) invariance from the IPA layers, we have performed the following ab- Unsur-
No LogFrob Loss
No SE(3) Invariance
Metric
DynaProt
lations listed in Table prisingly, replacing the log Frobe- nius norm objective with standard Mean Squared Error loss signifi- cantly degrades performance as the optimization is over the space of positive definite covariance matrices, which lies well-studied Riemannian manifold. Replacing the IPA blocks with standard MLPs also degrades performance, suggesting that SE(3) invariance is crucial in this low-data, low-parameter regime.
RMWD Var (L)
1.18
2.70
1.92
Sym KL Var (L)
0.91
9.26
4.46
RMSF r (t)
0.87
0.38
0.48
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A.3.3 SEQUENCE INPUT BASELINES
There are many methods worth noting that aim predict ensembles dynamics descriptors from sequence itself: standard ALPUAFFLW (AFMD), MSA-SUBSAMPLING FLEXPERT-SEO, ESMDIFF (Lu et al., 2024), CONFDIFF (Wang al., 2024), SEQDANCE (Hou and Shen, 2024), and SEAMOON (Lombard et al., 2024).
Though DYNAPROT's true comparison NMA as is a data driven and learnable alternative, we still is we compare against some of these sequence based methods in ensemble generation. DYNAPRot outperforms these methods on local RMSF correlation and marginal anisotropy prediction and comparable with other distributional metrics. Moreover, the efficiency advantage is clear with DYNAPROT's sub-second sample time.
Table 6: Comparison of DYNAPROT generated with ensemble generation methods that take in sequence imput. ESMDiff, ESMB entries reported from Lu et al. (2024).
ConfDiff OF-r3-MD
AlphaFlow -MD
ESM3
ESMDiff (I))
Metric
DynaProt
BioEmu
(ID)
Pairwise RMSD (=2.89)
2.17
3.43
2.89
3.57
RMSF (=1.48)
1.10
2.21
1.68
2.47
Global RMSF r (1)
0.71
0.67
0.60
0.63
0.19
0.49
Per-target RMSF 7 (1)
0.86
0.85
0.85
0.77
0.67
0.68
RMWD Var Contrib ())
1.18
1.40
1.30
2.04
4.35
3.37
MD PCA W2 (L)
1.74
1.44
1.52
2.05
2.06
2.29
Joint PCA W2 (L)
2.39
2.25
2.25
4.22
5.97
6.32
Weak contacts J (1)
0.51
0.59
0.62
0.33
0.45
0.52
Transient contacts J (t)
0.29
0.36
0.41
0.19
0.26
0.26
# Parameters (L)
2.86M
12.64M
95M
31M
1.4B
1,4B
Sampling time (L)
0..
~ 5700s
10,000s
240s
70s
70s
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A.4
SUPPLEMENTARY FIGURES
6pxz_ B: 0.945
6ndw B: = 0.896
HD Gound Truth DynsProt
ND Ground Tuth DyraProt
ROMSF (A:
75 Residue Index
175
Residue Index
6Frv A: 0.971
6q10 A: 0.976
15
HD dound Truss DynaFro:
10-
ND Ground Tuth DynaPot
(A)
RMSSI (A)
10
20
120
120
140
7jrq 0.983
0.923
ND Gound Truth DynaFro:
ND Ground Tuth DynoProt
RMST t/c
5.0
5-
2.5
20
40
80
100
120
50
100
150
200
250
300
350
Residue Index
Residue
6kty A: 0.931
6rwt
0.904
15
RMSF (A)
10:
50
75
100
125
150
175
200
25
50
75
100
125
150
175
Residue Index
Residue Index
7n0 E: r= 0.905
6ovk, R: r 0.945
Ground
Ground Truth DynaPrt
MMST (A)
BMST
5
Residue Index
Residue Index
Figure 8: DYNAPROT-M predicted RMSF correlations. Visualized test examples of predicted RMSF per residue (derived from the predicted marginal Gaussians) compared to ground truth RMSF derived from MD trajectories. Pearson correlation coefficient (r) between predicted and ground truth RMSF is reported.
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computed and compared the Published as a conference paper at ICLR 2026
Mean alignment to apo -> holo direction for cryptic pocket residues
Manginal gaussian pcl alignment (mean)
0.8
0.6
0.4
0.2
0.0
P79345
074933
POAEXO
P08581
P08037
P60338
rapeea
P12823
P45568
P26281
POA7D4
P02787
P64080
P0AG16
Q58L87
PS6817
Q27686
P12758
002763
QaWSFa
P00766
P61586
P69441
POA7F3
POASV6
P19491
Q29495
PE2593
P24182
P08709
016658
P44542
Figure 9: Evaluation of DynaProt-M zero shot prediction for BioEmu cryptic pocket proteins.
For each reference cryptic pocket protein from Lewis et al. (2024), the apo structure was provided to DynaProt-M, predicting marginal Gaussians for every residue. For each annotated cryptic-pocket residue, a displacement vector from the apo to the holo structure principal eigenvector of the respective predicted Gaussian. Cosine similarity between the two vectors quantifies how well DynaProt anticipates the direction of motion associated with pocket formation.
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7lao_A A
613r E
7e2s A
6q10_A A
6odd_ B
6uof A
6o6y_A
7nmq_A
7n0j
Figure 10: Sample ATLAS free energy landscapes. Free energy landscapes obtained by projecting ATLAS MD trajectories onto their top two principal components computed per protein. 2D density estimates transformed into free energy reveal dominant conformational basins explored during simulations.
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Closed conformation
Open conformation
Closed conformation
Open conformation
ATLAS PDB (6)Po_A)
AILAS PDB (6y2x A)
Figure 11: Sample ATLAS proteins with opening motions. Visualized are two ATLAS proteins (6q9c_A and 6y2x_A) that display significant opening motions over the course of their trajectories.
20


Reviews:
[{"review_id": "KZnt4BH7ku", "reviewer": "Reviewer_6FkX", "summary": "DynaProt is a Gaussian-based framework for ultra-fast protein conformation ensemble generation. Unlike traditional generative models that learn protein structure distributions via sampling, DynaProt learns a hierarchical Gaussian parameterization composed of residue-level anisotropic Gaussians and a protein-level isotropic residue–residue covariance matrix that captures coupling between residues.\nThis formulation enables DynaProt to analytically compute dynamics-related quantities such as RMSF without sampling; reconstruct the full 3N×3N covariance matrix using simple heuristics to support conformation sampling, generate large conformational ensembles extremely efficiently by sampling from reconstructed Gaussian distribution.\nCompared with classical normal mode analysis, which also relies on Gaussian approximations, DynaProt learns variances and couplings directly from data rather than through simplified physical potentials, resulting in more accurate ensemble statistics.\nExperimental results show that DynaProt achieves competitive performance on some ensemble-level metrics compared to diffusion generative models while being orders of magnitude faster. Although some accuracy is sacrificed due to the inherent Gaussian assumptions, the method offers a compelling balance between speed, accuracy, and physical interpretability, making it a strong and distinctive solution for conformation sampling and protein dynamics analysis.", "questions": "### 1. Relationship between DynaProt-M, DynaProt-J, and DynaProt\n\nIn the Experiments “preprocessing” section, DynaProt-M, DynaProt-J, and the IPA backbone appear to be trained as three separate models. However, in Section 4.4, the final DynaProt results are described as being composed from the output of DynaProt-M and  DynaProt-J. Could the authors clarify which is the case? And if separate models are trained, why is it necessary (vs. a single model with two prediction heads)?\n    \n### 2. Cryptic pocket case study and sampling\nIn Section 4.2.1, the authors show that predicted marginal variances can indicate ligand-binding dynamics, such as cryptic pocket formation. Have the authors sampled from the full reconstructed Gaussian distribution for this protein, and verified whether both open and closed conformations can actually be generated?\n    \nAdditionally, BioEmu (https://www.science.org/doi/10.1126/science.adv9817) includes several benchmarks with cryptic pocket motions. Evaluating DynaProt on those systems and reporting pocket coverage metrics would further strengthen the results.\n    \n### 3. Extending to atomic-level or protein-ligand complexes\n    \nDynaProt demonstrates impressive speed and potential scalability for residue-level modeling. Is it feasible to extend this approach to atomic-level resolution or to explicitly model protein-ligand complexes? Are there technical challenges such as dimensionality of the covariance matrix or parameterizing atom-level couplings?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The manuscript is well organized, clearly written, and easy to follow, with clean and informative figures that effectively support the main ideas.\n- The model addresses a unique and practical perspective in conformational modeling, where not only accuracy but also the ability to efficiently generate large numbers of samples is essential. DynaProt improves upon classical normal mode analysis while being orders of magnitude faster than modern generative models.\n- The method produces versatile outputs (e.g., RMSF, residue-level anisotropic Gaussian profiles, residue-residue coupling) and achieves competitive accuracy on some ensemble-level metrics compared to current state-of-the-art models.", "weaknesses": "- The core limitation lies in the Gaussian approximation and reliance on input structures, which restricts expressiveness compared to more flexible generative models - but this is aligned with the paper’s goal of trading some accuracy for major gains in speed and efficiency.\n- Given this design philosophy, clearly demonstrating the cases where DynaProt is useful becomes important. Cryptic pocket modeling is a relevant application, but only one case study is provided.\n- The current formulation models only coarse-grained residue-level dynamics. It would strengthen the paper if the authors could demonstrate potential extensions to atomic-resolution modeling or protein-ligand complexes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iEmhuarTaM", "reviewer": "Reviewer_HMqK", "summary": "The paper presents DYNAPROT, an innovative and lightweight SE(3)-invariant model that predicts protein dynamical features—marginal anisotropy and residue-level couplings—directly from static structures. It achieves accuracy comparable to leading generative models without large-scale PDB pretraining, while being more efficient. By constructing covariance matrices and introducing scalar covariances, the model can capture some key features, which are useful for fast ensemble generation.", "questions": "The same as weaknesses. In multi-conformation generation tasks, more attention is paid to the changes in system transitions, (rather than the local fluctuation). If the author is willing to explain this point, I am willing to increase the score", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well organized and clearly stated.\n2. The authors propose a protein dynamics framework based on a Gaussian hierarchical representation, which incorporates per-residue covariance tensors and the global coupling matrix into a Gaussian hierarchy.\n3. Compared with a single scalar RMSF, the covariance matrix is richer: it captures both the shape and direction of fluctuations.", "weaknesses": "1. The core issue is model misspecification. A Gaussian hierarchy simplifies learning and sampling, and it can not reflect the thermodynamics of MD data. Molecular energy landscapes are multi-well with distinct basins. Gaussian models are locally valid only near a single basin (approximately harmonic vibrations) and cannot capture complex, anharmonic shapes or multimodality across multiple minima. This defect was masked in the atlas dataset. Proteins in Atlas are relatively large, so the existing MD trajectories predominantly capture local fluctuations rather than informative transitions. Could the authors validate the method on simpler systems—e.g., a 2D double-well potential and fast-folding proteins—to demonstrate effectiveness of model?\n2. Although Eq. 5 and Proposition 3.1 guarantee positive semidefiniteness, the construction in Eq. 5 lacks clear physical motivation. The marginal covariance $\\Sigma_{marginal}$ rightly emphasizes per-residue anisotropy, which is a strength. However, the coupling matrix $C$ enforces isotropic interactions—i.e., identical influence along all three spatial directions—between residues. This design oversimplifies residue–residue coupling and risks weakening key directional effects that are critical in complex biomolecular systems. Could the authors elaborate on the motivation and theoretical basis for this design choice?\n3. When constructing $C$, the use of **mean pooling** as a scalar projection to compute each $C_{ij}$ appears casual and overly simplistic for modeling residue–residue coupling. A more completed ablation is needed to justify this design choice.\n4. What is the $\\mu$ of the BPTI results in Appendix A.3.1? Are the two main conformational changes caused by $\\mu$ or by $\\Sigma$?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8g1XX3jibj", "reviewer": "Reviewer_LoLM", "summary": "This paper introduces DYNAPROT, a novel, lightweight, SE(3)-invariant deep learning framework for directly predicting protein dynamics from static protein structures. Unlike molecular dynamics (MD) simulations, which demand substantial computational resources, or generative models requiring extensive pre-training and costly sampling procedures, DYNAPROT is an efficient alternative.", "questions": "The model was trained on 100ns of data. Beyond the single example of BPTI, have the authors tested the model on additional proteins exhibiting known long-timescale (e.g. > 1us) dynamic characteristics? Is there a possibility that the model might overfit to short-timescale fluctuations, thereby underestimating genuine large-scale motions?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Compared to models such as DYNAPROT (with 2.86 million parameters in total) and AlphaFlow (95 million parameters), the parameter count is reduced by several orders of magnitude. Its inference and sampling speeds far exceed those of other methods, rendering it exceptionally practical. Moreover, the model performs well and is trained solely on molecular dynamics data from approximately 1,000 proteins.", "weaknesses": "The model was trained on the ATLAS dataset comprising 100 ns simulations. For numerous biologically significant processes—such as large-scale domain motions and allosteric transitions—100 ns is wholly inadequate, as these typically unfold over timescales ranging from microseconds to milliseconds or longer. Consequently, the model may have predominantly learned local, transient thermal fluctuations while exhibiting limited capacity to capture large-scale conformational changes. Although the paper's zero-shot testing results for BPTI appear favourable, this represents a single protein case study. One cannot conclude from this solitary example that the model generalises well to all long-timescale dynamics.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jdEifuOG6z", "reviewer": "Reviewer_ad6B", "summary": "This paper proposes a pipeline for training a model to predict protein dynamics.  The authors model protein dynamics as sampling conformations from a normal distribution with a low-energy mean and a non-identity covariance matrix.  The overall covariance is represented as a combination of a local residue-wise 3×3 covariance matrix and a global residue-wise N×N covariance matrix, matching the dynamics learned from MD simulations.", "questions": "1. When sampling from the normal distribution of protein structures predicted by the model, how often do unphysical protein structures appear?  What prevents this approach from producing impossible torsion angles or bond lengths?  \n\n2. How does the model handle proteins with pronounced conformational changes, such as large domain rotations, extensive motion cascades, or highly distinct alternate conformations like those observed in 4OLE?\n\n3. How does the model behave when the provided protein is not in its lowest-energy state?  How would the model perform as part of a dynamical system, where a protein conformation is iteratively updated along the principal directions of each residue? Would the structure return toward the mean state, or would it break down entirely?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "This paper addresses protein dynamics prediction not implicitly through ensemble generation, but directly by modeling the protein’s residue C$\\alpha$ covariance matrix.  This approach enables direct analysis of protein dynamics.  \nModeling the full covariance matrix using a heuristic that combines the cross-residue covariance matrix with the intra-residue position matrix provides an efficient and meaningful representation of dynamics.  \nThe experiments presented are diverse and interesting, demonstrating the versatility of the proposed approach.", "weaknesses": "1. Throughout the paper, the examples presented focus on local dynamics, where the protein does not deviate significantly from the mean. Proteins that undergo large conformational changes may not be well represented by modeling their dynamics as a normal distribution with a single covariance matrix. For proteins that open and close or contain highly flexible regions, the full dynamic range might collapse into a single blurred representation under this parameterization. Nevertheless, this work represents an important step toward explicit modeling of protein dynamics.  \n\n2. The authors assume that the provided protein structure is in a low-energy state, but there is no discussion of how the model behaves when the input structure is not in such a state.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)