Unified Biomolecular Trajectory Generation via Pretrained Variational Bridge
Authors: Ziyang Yu
Wenbing Huang
Yang Liu
Venue: ICLR 2026 Poster (2026)
Keywords: deep generative model
molecular dynamics
trajectory generation
variational autoencoder
augmented bridge matching
adjoint matching
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Molecular Dynamics (MD) simulations provide a fundamental tool for characterizing molecular behavior at full atomic resolution, but their applicability is severely constrained by the computational cost. To address this, a surge of deep generative models has recently emerged to learn dynamics at coarsened timesteps for efficient trajectory generation, yet they either generalize poorly across systems or, due to limited molecular diversity of trajectory data, fail to fully exploit structural information to improve generative fidelity. Here, we present the Pretrained Variational Bridge (PVB) in an encoder-decoder fashion, which maps the initial structure into a noised latent space and transports it toward stage-specific targets through augmented bridge matching. This unifies training on both single-structure and paired trajectory data, enabling consistent use of cross-domain structural knowledge across training stages. Moreover, for protein-ligand complexes, we further introduce a reinforcement learning-based optimization via adjoint matching that speeds progression toward the holo state, which supports efficient post-optimization of docking poses. Experiments on proteins and protein-ligand complexes demonstrate that PVB faithfully reproduces thermodynamic and kinetic observables from MD while delivering stable and efficient generative dynamics.

Published as a conference paper at ICLR 2026
UNIFIED BIOMOLECULAR TRAJECTORY GENERATION VIA PRETRAINED VARIATIONAL BRIDGE
Ziyang Yu! Wenbing Huang Yang Liu
DDpartmentt of Computer Science and Technology, Tsinghua University
Institute for AT Industry Research (ATR), Tsinghua University
& Gaoling School of Artificial intelligence, Renmin University of China
'Beijing Key Laboratory of Research on Large Models and Intelligent Governance
yu-zy zy2 4 emmalss..............maaammmllmm www.iisssssssnnnng tsinghua. cdu. en, bwenbing@ruc. edu. cn, Suuyaang211ttsinghuue..o. cdu.
ABSTRACT
Molecular Dynamics (MD) simulations provide a fundamental tool for characteriz- ing molecular behavior at full atomic resolution, but their applicability is severely constrained by the computational cost. To address this, a surge of deep generative models has recently emerged to learn dynamics at coarsened timesteps for efficient trajectory generation, yet they either generalize poorly across systems or, due limited molecular diversity of trajectory data, fail to fully exploit structural infor- mation to improve generative fidelity. Here, we present the Pretrained Variational Bridge (PVB) in an encoder-decoder fashion, which maps the initial structure into a noised latent space and transports toward stage-specific targets through augmented bridge matching. This unifies training on both single-structure and paired trajectory data, enabling consistent use of cross-domain structural knowl- edge across training stages. Moreover, for protein-ligane introduce a reinforcement learning- brase optimization via adjoint matching that speeds progression toward the holo state, which supports efficient post-optimization of docking poses. Experiments on proteins and protein-ligand complexes demon- strate that PVB faithfully reproduces thermodynamic and kinetic observables from MD while delivering stable and efficient generative dynamics.
complexes, further
INTRODUCTION
Molecular Dynamics (MD) simulations are indispensable ingredients in chemistry, materials, drug discovery, and other fields (Van Gunsteren & Berendsen 1990, Lindorff-Larsen et al. 2011
Hollingsworth & Dror 2018 Lau al.) 2018). Given the molecular system of interest within a particular chemical environment, the essence of MD is to obtain the conformational ensemble that follows laws of physics through sílico simulation, which provides accurate estimates of kinetic (e.g., dissociation rate constant) and thermodynamic (e.g.. free energy) observables (Shaw et al.) 2010 Deng & Roux 2009 Wang et al. 2023 Chipot 2023).
Under certain conditions, the equilibrium measure of molecular ensembles follows the Boltzmann distribution (Boltzmann] 1877). To obtain such equilibrium samples in practice, classical MD simu- lations widely rely on Langevin dynamics (Langevin) 1908), which models the temporal evolution of the system through Stochastic Differential Equation (SDE) and induces a Markov-chain sam- pling process whose stationary distribution corresponds to the Boltzmann distribution (Paquet & Viktor 2015). Although these methods have been firmly established in practice, the stability of numerical integration requires exceedingly small time steps At (around femtosecond), making the computational cost prohibitively high for long-timescale or high-throughput simulations (Schlick) 2001).
Recently, the rise of deep learning methods has opened up new possibilities for boosting MD simulations and ensemble generation. In contrast to directly fitting the Boltzmann distribution that ignores temporal correlations, we focus on time-coarsened dynamics, which aims to fit the
Cooreeponningg authors: Wenbing Huang, Yang Liu.

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Published as a conference paper at ICLR 2026
conditional distribution p [x)), defined the transition density of the underlying MD process that maps a current state 2, to its distribution after a lag T > t, thereby enabling time-dependent
trajectory generation on coarse- -rasiie temporal scale (Schreiner et al., 2023 Klein et al. 2023 Jing et al. 2024b; Yu et al.] 2025). This paradigm achieves substantial efficiency gains while preserving both the kinetic and thermodynamic properties of the system. However, several challenges remain in the literature. First, the stability of trajectory generation over long timescales cannot yet be reliably ensured. In addition, most existing methods are confined to one specific molecular domain (e.g., proteins) and therefore fail to model cross-domain systems. In particular, UniSim (Yu al., 2025) employs 3D molecular pretraining to obtain a unified atomic representation model, demonstrating strong generalization for cross-domain MD simulations. However, the inconsistency between pretraining on single structures æ: and finetuning on trajectory pairs (2r4, TT++) could result in suboptimal transfer and hinder the effective use of pretrained knowledge. A further limitation of most existing methods is their focus on single-molecule simulations, while multi-molecular systems such protein-ligan -igand complexes have been much less explored.
To address the aforementioned issues, we pro- pose the Pretrained Variational Bridge (PVB), a novel generative model that leverages an encoder-decoder architecture to provide a uni- fied framework for both pretraining and fine- tuning. Specifically, we first enhance the cross- domain generalization by pretraining on a large variety of high-resolution single-structure data x, and then finetune on the paired data (I',, T(+r) drawn from MD trajectories to fit the condi- tional density u(Ee+r 2e) with predefined time step t. The key challenge is to elegantly resolve the objective mismatch between uncon- ditional generation during pretraining and con- ditional generation during finetuning. To this end, we introduce two random variables, Xo and Y,, and define a conditional probability measure q(Y), Xo) to be learned across train- ing stages. For single-structure data x, this measure is degenerate, assigning all probabil- ity mass to the point Y, = given X.- x. For finetuning on paired MD data (I2, T(+r), qdd,, X, = x.)) is defined such that its con- ditional density matches the target transition density u(I++r xx) of the underlying MD process. Since the measure collapses to a point mass in the single-structure pretraining scenario, we fur- ther introduce a latent random variable Yu to decompose the measure and prevent the model from degenerating into this trivial case:
initial state
trajectory generation RL-accelerated transition
target state
folded state
unfolded state
Figure 1: Two variants of PVB. Blue arrows show coarse-grained sampling that traverses unfolded and folded states reproduce the Boltzmann dis- tribution, while purple arrows denote accelerated transition that drives rapid access from the initial state to the target folded state via reinforcement learning.
f
a( E = (a(Xx EA Yo)o Xo), for any measurable set A.
(I)
Here de and qu denote the transition kernels of the Markov chain X, - Yo Y,, which are respectively parametrized by an encoder and separate generative decoder using augmented bridge matching (De Bortoli et al. 2023). The process of mapping the initial state to the noised latent space and back-projecting it to target states enables the model to acquire cross-domain structural knowledge in pretraining, which, due to objective consistency, greatly facilitates subsequent finetuning.
Furthermore, to better capture protein-ligand dynamics, we incorporate finetuning procedure into the generative framework based on Reinforcement Learning (RL). By following adjoint match- ing (Domingo-Enriee Enrich al. 2024), provide memory- -fffccettficientt finetuning scheme with explicit reward functions. The RL optimization is designed to modulate the generative distribution, guiding it toward the holo state and thereby enabling rapid evolution from the apo state within short generated trajectories. Figure illustrates the two generative modes of PVB described above.
Overall, our contributions in this work summarized below:
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We present PVB, novel generative model that integrates the encoder-decoder architecture with augmented bridge matching, offering a unified training framework for both single- structure and paired trajectory data to leverage the rich knowledge from pretraining.
We propose an RL-based finetuning procedure, applying the stochastic optimal control paradigm to augmented bridge matching. For the protein-ligant flexible docking task, PVB learns to bypass inefficient local exploration and rapidly evolve toward the holo state within short simulations, making it an efficient tool for post-optimization of docking poses.
For both protein monomers and protein-ligand complexes, PVB achieves comparable re- sults to MD in terms of thermodynamic and kinetic metrics, while delivering substantial improvements over the baseline in generation stability.
BACKGROUND
Ensemble Generation Deep learning techniques have introduced methodological innovations to ensemble generation. One promising approach is to replace empirical force fields with accurate all-atom neural network potentials accelerate simulations of large biomolecular systems, with representative work exemplified by BM (Wang et al. 2024b). This approach, however, does not overcome the inherent limitations of sequential trajectory generation with small time steps. To move beyond the framework of Markov chain sampling, a surge of methods leverage generative models to fit the empirical distribution from MD trajectories (Jing et al. 2024a| Wang al. 2024a Lewis et al., 2025), while Boltzmann generators further reproduce the Boltzmann distribution from the surrogate model by importance sampling (Noé et al. 2019 Klein & Noć 2024 Tan et al. 2025). Such methods enable highly efficient parallel generation of i.i.d. samples, yet the absence of temporal dependencies blocks further estimation of kinetic observables.
Trajectory Generation The paradigm of trajectory generation, also referred to in the literature as time-coarsened dynamics, seeks to learn the conditional distribution p(Z147 xz) at a coarse
time step 5 >> At, enabling the recovery of observables of interest from relatively short generated trajectories (Schreiner et al. 2023 Klein et al. 2023 Jing et al. (0224b] Yu et al. 2025). Notably, UniSim YYu al., 2025) first extends the task to cross domainn biomolecules, showcasing strong generalizability with the pretrained atomic representation model and force guided finetuming tech- niques. However, due to the inconsistency training objectives, the pretraining stage produces only a unified atomic representation model, failing to explicitly capture cross-domain structural information. In contrast, our approach integrates the generative model into pretraining via a noised latent space, which provides unified framework for training on both single-structure data and paired MD trajectories, enabling the exploitation of rich structural knowledge across training stages.
force-guided
METHOD
We now discuss the methodology and rationale of PVB, with its overall workflow illustrated in Fig- ure 2 We first present the problem formulation with necessary notations in Section [.1] Next, in Section 3.2] we discuss the overall generative framework of our pretrained variational bridge, which provides unified training interface for single-structure data and paired trajectory data. More- over, in Section 3.3] we develop a memory-effiiien RL-based finetuning framework grounded in stochastic optimal control, which can be applied accelerate exploration for the holo state of protein-ligand complexes. Details of the model architecture are shown in Section
3.1 TASK FORMULATION
In this work, the conformation of one molecular system is represented in simplified triplet form (z, C,).). where N denotes the atomic numbers of N heavy atoms, C' E N2xC represents C covalent bonds among heavy atoms the molecular topology, and x E Nx3 corresponds to the []] We consider two types of molecular systems: (I) systems represented by one or a few independent conformations, denoted as (z, C, {=}); and (II) systems represented by time-dependent MD trajectories, denoted as (z, C, {I F2st; }), where At is
Fuclidean coordinates of all heavy a
I "owercase is used for coordinates to distinguish them from random variable notations.
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Pretrained Variational Bridge
pretrain
Encoder
Decoder
finetune
qe
&d
dYp Vll-pa(i, dd)
X
Yo
b. RL finetuning
frozen
initialize
a1
Vimsd X
Encoder
Decoder
Pe
4'
Xo
X
initial state
predicted next state
holo state
update
dā, = (t.
Figure 2: Schematic of the overall PVB workflow. The unified framework for pretraining on single-structure data and finetuning on paired trajectory data. The encoder fee maps the initial state Xn to the latent variable Yo. which the decoder ęa then propagates to the stage-specific target Y via Equation (6). b. RI. finetuning with frozen ye and ş initialized from the finetuned yps. aiming to accelerate exploration of pooeein-ligand holo states. Given the predicted next state Y generated by Se and y the lean adjoint state aj is first computed from the reward function (i.e., the root mean square error to the holo state X and then propagated backward to ã, by solving Equation (12) with diffusion time t, which is subsequently used to update yo according to Equation
the integration time step, which may vary high-resolution structural data to acquire cross-domain full-atom generative capability, and (ii) to finetune on type-II MD trajectory data learn dynamies for trajectory generation, with appropriate framework and training objectives to maximally leverage the pretrained structural knowledge.
datasets. Our tasks are: (i) to pretrain type-T
3.2 PRETRAINED VARIATIONAL. BRIDGE
First and foremost, our primary objective to endow the model with the cross-domain generalizability difficult achieve by training solely MD data due to two key limitations: the high cost of generating accurate MD trajectories and the poor accuracy/adaptabbliiy of data from empirical force fields across diverse molecular domains. Inspired by UniSim YYu et al.) 2025), we aim to first pretrain on extensive high-resolution biomolecular structures to fully exploit structural information within a generative framework, and then seamlessly transfer this knowledge to trajectory generation through carefully designed finetuning objectives.
of biomolecular trajectory generation. Breakthroughs
Markov chain Formally, model the generation process in both pretraining and finetuning
Xo + Yo + Y1. For single-structure data (type-I systems) used in pretraining, we set the initial and target states as (Xo, Y1) (x,, 2². For paired trajectory data (type-II systems) used in finetuning, we set (Xo. Y1) = (xz, .2+T). The latent variable Yo is introduced to prevent the conditional law qdy,, X,) from collapsing
Dirac measure for type-I systems.
According to Equation [T. the target conditional law q(dY, X,) is realized as the composition of two Markov kernels: an encoder kernel qe (dYY Xn) and a decoder kernel qaYdY, Yo). We employ an encoder-decoder architecture to parametrize these two distributions separately, as detailed below.
Variational Encoder Given the initial state Zp. the conditional probability measure of the latent variable Y, is manually specified by q. (dY, X, = In) := N Nx T) where a, the hyper- parameter of deviation. Choosing appropriate prior distribution is crucial for performance, it
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should preserve sufficient structural information through encoding, and also avoid collapse of the decoding process into trivial cases (e-g.. a Dirac measure). To this end, we use de v0.5A in our experiments, which considerably larger than those typically adopted 3D molccular pretraining methods Zaidi et al. 2023; Jiao et al. 2024: Yu ct al. 2025).
Next, a neural network encoder SPe is leveraged to minimize the distance between the learned distribu- tion Pe Yd Xu) and the prior 9. ( Xg) under the Bullbback-eeibiee (KL) divergence measure,
with its architecture discussed in Section C By applying the reparameterization trick (Kingma &
Welling 2013), the encoder will yield the logarithm of the predicted variance log V yp. (Xo) R
Then the K1. divergence loss is given by:
CKL Ex, [DKL (pe(
(o) Xo))] - - + log V 22og 0e
Augmented Bridge Decoder Given the latent variable Yo ~ Pe(. Xo). we utilize a separate
decoder Pa fit the transition kernel qa(dY, Yo). Since the decoder stage is critical to model performance, we adopt the augmented bridge matching framework (De Bortoli et al. 2023). which ensures preservation of the coupling between the latent variable Yu and the target state Y denoted by Io
Specifically, we consider the path measure Q : C(O,, 11, Wx33) + R. associated with the Brownian motion (o (BB))) (0.,1)) with R4 predefined noise schedule. Next, define the twisted path measure P = ll. Qj0,1 where the trajectory Yo.1 ~ P can be generated by first sampling (Yo, Yi) from the coupling I then sampling from the Brownian bridge Q (- Yo, Y1) pinned down on the two endpoints.
well-studied case in the literature, the temporal marginal of diffusion time and the associated forward SDE are given by:
Y, = =Y, + (1- (1-4) a //1(11 ()B,, (Yo, Y1) Ilo.1:
dY, Y, Y, di dBd (Yo, Y1) lo..1-
De Bortoli et al. (2023) show that one can preserve the coupling 1o by first training a parameterized vector field şa conditioned on i, Yo, and Y, to minimize the quadratic loss:
Y, Y1 Y, 2
CABM = E -.)
E Ex,-Qua.
palt. Yo. Y.)
(5)
1-t
where 4(0, 1) denotes the uniform distribution over [0, 1]. Afterwards, simulating the resulting non-Markovian SDE ensures that the coupling Iu, is preserved:
dY, (7((, Yo, Y.) di dB., Yo~ Il,
where Hl is the marginal probability measure of Yo. and yoj is the minimizer of Equation (5). Finally, Proposition T demonstrates that our proposed encoder-decoder architecture provides unbiased estimate of the target conditional law q(dY, Xo) (see Section B):
Proposition Denote by pt the minimizer of Equation ( Tp" the path measure associated with Equation 66, and p the probability density function of The conditional measure Pij0- Then the following equality holds:
q( E A1 pa(Y e Yo)o Xo), for any measurable set A.
(7)
Overall, the objective WKL LKL. + WABM LABM is employed for both pretraining and finetuning, where WKL. and WABM are hyperparameters that control the relative weights.
3.3 MEMORY-EITICIEE STOCHASTIC OPTIMAL CONTROL
In this section, we further explore the application of the generated trajectories to protein-ligand flexible docking. Such tasks focus on locating the global minimum-energy conformation, rather
For convenience, omit and C' from the neural network parameters, and the same applies hereafter.

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than reconstructing the full Boltzmann distribution. Notably, the protein- -igand interaction occur on the millisecond timescale or longer, making its simulation computationally prohibitive even for integrate an RL-based finetuning
models with coarsened timesteps. To address this limitation, procedure with an explicit reward function r(x), designed to adjust the generative distribution and thereby accelerate the evolution of trajectories toward the holo state.
Formally, we first consider the following SDE with a control vector field u to be optimized:
dY, = (tā Yo. Yi) + ou(t, Yo. Y.)) dt - a dBi, Yo Ho.
(8)
Denote by Pa the path measure associated with Equation problem with KL-regularized objective:
We aim solve the optimization
max Ey P P)x.:
DKL ( ooo
Po oo
Yo)II,
(9)
where 3 controls the regularization strength. Based on the Girsanov theorem, the KI. divergence term can be expressed by the control vector field u (see Section B
max Eoo (x
(ron)
Yo. f
8 2. [. 2
|u(t, Juft,Yo
YOIl de
(10)
Although this training objective is already applicable for model optimization, it requires gradient accumulation during simulation along the SDE, which leads prohibitive memory overhead. Hope- fully, by first considering the inner expectation in Equation (T0) given that Y, takes the value you the dependence Yo in u(t, Yo. Y,) can be treated as fixed, casting the optimization problem into the standard Stochastic Optimal Control (SOC) paradigm (Bellman 1966). Based on the conclusions of adjoint matching Domingo-Enrich et al.) 2024), state the following proposition (see Section B):
Proposition 2. Assume that the function class of is rich enough so that the value of u(t,, yo. can be chosen independently for each yo, then the unique minimizer u of the following objective the optimal control of Equation 10p
Ladj EYo,-oo Be (. [llu(t, Yo, Y.) oālt, Yo:))1]] a sg(u), (11)
where a(s, Yo.) = -(s,, Yoa) Vy, Yo. Y.), ā(1, Yoa) - Vv,r(()) (12)
ds
Here sg(-) denotes the stop-gradient operator, and the random variable ā is known in the literature as the lean adjoint state. Denote by y the neural network optimized during RL finetuning, with parameters initialized from yat In light of Equation (6) and Equation (8), the control vector field can
be reparameterized by u = (4p% *f)) This formulation avoids the need to introduce any additional networks. For clarity, we provide the pseudocode of RL finetuning in Algorithm
EXPERIMENT
In this section, we provide details on the application of PVB to trajectory generation and related tasks.
Dataset For pretraining, PCQM4My2 (Hu et al. 2021) and ANI-Ix (Smith et al. 2020) for small organic molecules,
variety of datasets covering different molecular domains:
PDB (Berman al. 2000) for protein monomers, and PDBBiin2020 (Wang et al.] 2004 2005
Liu al. for protein-ligand complexes. After pretraining, the model is finetuned on domain- specific datasets: ATLAS (Vander Meersche al. (2024) and mdCATH (Mirarchi al. 2024) for
protein monomers, and MISATO (Siebenmorgen et al. 2024) for protein-lige ligand complexes, enabling trajectory generation in these domains. Additionally, we augment the PDBB2nd2020 protein-ligant complex data with short MD simulations using OpenMM (Eastman et al. 2023), which are further employed for RI. finetuning to explore protein- -aggid holo states. Comprehensive details of the datasets and preprocessing procedures are provided in Section D
Baselines
The following deep learning-based models tailored for trajectory generation are selected as baselines: (I) ITO (Schreiner al. 2023), conditional diffusion model that fits the transition

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kernel of molecular dynamics with varying temporal scales. (II) MDGEN (Jing et al. 2024b),
generative model based on torus and local frame representations of proteins, which can be applied to diverse dynamics-relater tasks. (III) UniSim (Yu et al. 2025), pretrained generative model with force-guided finetuning modules, which exhibits cross- domain transferability. (IV) AlphaFlow (Jing et al.) (0224)), a modification of AlphaFold (Jumper et al. (02T) with the flow-matching objective for finetuning, enabling the generation of independent and identically distributed (i.i.d.) protein conformational ensembles, For fair comparison, we follow the standard training procedures of each baseline: ITO and MDGEN are trained from scratch, UniSim is initialized from its officially released pretrained representation model, and AlphaFlow is initialized from the publicly released AlphaFold pretrained weights, all subsequently finetuned on our dataset.
4.1 TRAJECTORY GENERATION
We now investigate the capability of the model to generate trajectories on the protein domain. Building on the pretrained PVB parameters, we finetune on the ATLAS dataset to capture dynamic patterns, with 790/14 systems for training and testing, respectively. Specifically, for each 100 trajectory in the training set, we randomly sample 500 data pairs (Iz, Ø(+r) from the first 80 ns for training and 100 data pairs from the last 20 ns for validation, with the coarsened time step 100 ps. Given the finetuned model, generate trajectories for the 14 test protein systems by simulating Equation (6) via numerical discretization, with chain length of 1,000.
Proteins on ATLAS
To comprehensively evaluate the generated trajectory, we apply the following metries: (I) the Jensen-Shannon Divergence (JSD) calculated on projected feature spaces, including the radius of gyration (Rg), the torsion angles and ya of protein backbones (Torus). the slowest two lagged Independent Components (TICO and TIC1) PérezzHernández et al. 2013), and the state occupancy estimated Markov state models (MSM). (II) The proportion of conformations without bond break or clash between Cox atoms, denoted as VAL-CA. (III) The root mean square error of contact maps between generated trajectories and reference MD trajectories, denoted as RMSE-CONTACT. (IV) The decorrelation fraction with respect to TICO observed in the generated trajectories, termed as Decorr- TICO. The reference statistics for each metric are derived from three replicate MD trajectories each test system in ATLAS. Further details how the metrics are computed can be found in Section [+]
To facilitate comparison, the first 10 ns of one selected MD trajectory was included in the analysis for each test system. As presented in Table 1 PVB achieves performance comparable to the baselines across most metrics, with clear superiority in the validity of generated ensembles. Moreover, com- pared to 10 ns MD,, PVB maintains equivalent physical realism and achieves superior distributional agreement in TIC and MSM feature spaces. This further evidenced by a higher decorrelation ratio, confirming its ability resolve slow dynamical modes inaccessible short MD simulations.
Table Results on the ATLAS test set, with each metric reported as mean/std across 14 test protein monomers. The best and second- best results for each metric are shown in bold and underlined, respectively. * denotes models that generate i.i.d. protein samples instead of trajectories.
JSD (
MODELS
VAL-CA
RMSE-CONTACT (L) Decon-TICO (1)
Rg
Torus
TIC
MSM
MD (10 ns)
0.379/0.115
0.148/1.016
0.684/0.072
0.596/0.17 179
0.926/0.093
0.040/0010
0.000
AlphaFlow*
0.385/0.143
0.439/0.045
0.570/0.110
0,394/0.116
0.485/0.386
0.107/0.068
ITO
0,792/0.022
0.560/0.118
0.400/0.081
0.469/0.145
0.001/0.000
0,948/0.016
0.714
MDGEN
0.493/0.115
0.321/0.031
0.400/0.085
0.463/0.1 150
0.098/0.217
0.158/0.020
0.857
UniSim
0.538/0.135
0.344/0.11
0.372/0.078
0.344/0.097
0.106/0.050
0.129/0.019
0.786
PVB (owrs)
0.457/0.087
0,336/0.030
0.371/0.087
0.333/0.105
0.975/0.023
0.143/0.031
0.929
Proteins on mdCATH We further evaluate the model on the mdCATH dataset, which encompasses Danaonn a wider range of protein domains. Following Janson et al. (2025), we use 5, 5,049/40 split for training and validation, with 20 domains selected from the original test set for evaluation. Training data pairs (Iz, x(++)) are sampled every ns, with the coarsened time step 7 = 1 ns. After finetuning, generate 500-step trajectories for each of the 20 test systems, matching the standard trajectory length in mdCATH. For evaluation, four of the five replicate MD trajectories per system are used to construct reference statistics, while the fifth serves the MD oracle for direct comparison with PVB.
The evaluation results on mdCATH are shown in Table Similarly, PVB achieves MD-level physical plausibility in the generated conformations and shows good agreement with the slow modes (TIC)
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and metastable states (MSM) observed in the MD reference ensembles. Moreover, PVB outperforms the baseline on most metrics and, compared to the ATLAS results, exhibits a greater disparity in distributional similarity for projected features, thereby demonstrating enhanced generalization broader-scope dataset.
Table 2: Results on the test set of mdCATH. Values of each metric are shown in mean/std of all 20 test protein domains. The best result for each metric shown in bold.
JSD (
MODELS
VAL-CA
RMSE-CONTACT
Decon-TICO
Rg
Torus
TIC
MSM
MD Oracle
0.264/0.085
0.182/0.033
0.316/0.71
0,282/0.069
0.942/0.154
0,067/0.039
UniSim
0,562/0.160
0.357/0.050
0.454/0.100
0.402/0.114
0.522/0.371
0.116/0.04
0.20
PVB (ours)
0,437/0.142
0.347/0.035
0.376/0.050
0.362/0.124
0.963/0.049
0.177/0.030
0.90
Figure presents visualizations for four representative test systems, two from ATLAS and two from mdCATH. It demonstrates that the generated trajectories faithfully recover the underlying free energy landscapes, while the metastable state occupancies estimated by the MSM show strong agreement with the reference values. Together, these findings highlight the reliability of PVB in reliably capturing key dynamical properties during the long- time evolution of large molecular systems.
Free onargylet
Free energyotg
PVB
Free enangyike"
Free enampyite!
Fine enargy/e!
Fane energyaal
PVB
Free eningyyle
Free energyig!
MD
T
T
Figure 3: Illustration of generated trajectories for PDB 2bjq (row 1), PDB 7mm7 (row 2). CATH domain 3er0A02 (row 3), and CATH domain 1pyaA00 (row 4). Left: Representative structures from the first 10 frames of the generated trajectories. Middle: Free energy surfaces projected onto TICO and TICI, respectively. Right: Probability differences between PVB and MD across the 10 metastable states estimated by MSM.
Protein-Ligand Complexes MISATO Next, investigate the application to protein-ligand complexes using the MISATO dataset, with 13,765/1,595/20 systems for training, validation and testing, respectively. For the 8 ns trajectory comprising 100 snapshots, all frames utilized for
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finetuning, with lag time of r 80 ps. For consistency with the MD data, we generate trajectories of length 100 for evaluation.
Considering the characteristics of protein-ligand complexes, adopt more specialized evaluation metrics: (I) the Wasserstein-1 distance (Villani ct al. 2008) of the ligand RMSD relative to the MD initial state after superposition onto the pocket backbone (EMD--igand), between the center of mass of the pocket and the ligand (EMD-CoM). All distances are reported in nanometers. (II) The root mean square error of the contact occupancy between heavy atoms of the ligand and the pocket, also denoted as RMSE-CONTACT. Further details can be found in Section]
well as the distances
The results in Table show that, PVB achieves the closest match in lig- and poses and pocket-ligand center-of- mass distances compared to MD tra- jectories, while also reaching compa- rable performance in heavy-atom con- tacts. Notably, the EMD corresponds to errors on the order of atomic resolu- tion, demonstrating the accurate char- acterization of protein-ligand complex interactions.
Table 3: Results on the test set MISATO, Values of each metric are shown in mean/std of all 20 test protein-ligand complexes. The best result for each metric is shown bold.
MODELS
EMD-ligand (L)
EMD-CoM (L)
RMSE-CONIACI (J)
ITO UniSim
0.494/0.240
0,479/0.258
0.987/0.001
0.196/0.137 137
0.128/0.1 111
0.049/0.009
PVB (ours)
0.133/0.072
0,089/0.078
0.055/0.008
Meanwhile, Figure 4 presents visualizations of ligand RMSD and pocket-ligand CoM distance for two representative cases. It can be observed that the trajectories generated by PVB closely match MD results in both distribution and temporal evolution, showing that the model accurately captures the dynamical behaviors of the molecular system.
RMSO UA,
Col (A)
Time (nc)
Time (ris)
RMSD (
CoM (A)
Time (ns)
Time (ns)
Figure 4: Illustration of ligand RMSI) and CoM along the trajectories for PDB 2ww0 (top row) and PDB 5n2f (bottom row). Left: Visualization of the complex, with the receptor shown in blue and the ligand in green. Middle: Ligand RMSD over time. Right: Pocket-ligand CoM distance over time.
4.2 EXPLORATION FOR PROTEIN-LIGAND HOLO STATE
We further evaluate the model-generated trajectories for exploring the holo state of protein-ligand complexes. Generating trajectories directly from the apo the holo state is computationally pro- hibitive, for which we employ two complementary strategies. First, we reformulate the task as post-optimization of docking poses: an existing docking model used to generate coarse docking pose, which then initializes trajectory generation for refinement. Second, we employ the RL finetun- ing procedure in Section 3.3 guide the generation process toward the holo state. Specifically, for a protein-ligamd system with holo state X the reward is defined r(X)= -rmsd(X. X,et).
The training data are constructed from the PDBBind2020 database, with 23 systems reserved for testing, detailed in Section D. With the data prepared, perform RL finetuning using the algorithm in Algorithm]] [] initializing the model with weights from the version trained on MISATO. For evaluation, begin with the initial state constructed by docking the ligand into the apo protein

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Published as a conference paper at ICLR 2026
structure via AutoDock Vina (Eberhardt et al. 2021), ), and generate trajectories of length 100 (8 ns). The predicted holo state is identified as the conformation with the highest binding affinity, which is estimated by EPT (Jiao et al. 2024). Following Zhang al. (2025), report the root mean square emorr of the ligand (Ligand RMSD) and the pocket (Pocket RMSDD as evaluation metrics.
The evaluation results the test of PDBBind is shown in Table 4 Firstly, the holo state predicted by PVB without RL finetuning deviates further from the reference than the initial state, likely due insufficient sampling from the vast conformational space within the short simulation window. Meanwhile, RL finetuning delivers substantial gains over the non-RL model across all metrics and consistently surpasses Vina-docked poses in ligand placement, suggesting that learns bypass local free-energy minima and progress more efficiently toward the holo state through RL.
Table 4: Results on the test set of PDBBind.
Ligand RMSD
Pocket RMSD
MODELS
25%
50%
75%
mean
<5A
25%
50%
75%
mean
<2
<4À
Autk Vina
4.771
6.268
8.479
6,368
26.1%
0.707
1.021
1.380
1.211
91.3%
95.7%
PVB w/o RL
4.797
6.613
9.515
6,904
26.1%
2.106
2.683
3.121
2.718
26.1%
87.0%
PVB w/ RL
4.178
5.745
7,500
5.918
43.5%
1.515
1.962
2.281
1.965
56.5%
95.7%
In addition, the predicted and co-crystal structures of PDB 6d15 and PDB 6j0g are displayed in Figure [5] two representative protein-ligan ligand complex systems. Regardless of whether the initial Vina-docked pose is close to the ground truth (e.g., PDB 6d15) or far from it (e.g., PDB 6j0g). PVB with RL finetuning consistently evolves toward the correct binding pose within short simulation window, demonstrating the capability to perceive global interactions rather than local ones. Therefore, we claim that PVB can serve as an efficient tool for post-optimization of docking poses.
holo ligand structure
holo ligand structure
AutoDock Vina: 5.115 A
AutoDock Vina: 6.133 A PVB w/o 6,427
PVB w/o RL: 3.081A A
PVB w/ RL: 2.667
PV8 4.721 A
a. PDB 6d15
b. PDB 6j0g
Figure 5: Comparison of the predicted holo state with the co-crystal structure for a. PDB 6d15 and b. PDB 6j0g. The holo protein structure is shown in blue, and the displayed ligand pose is presented after aligning the predicted protein to the holo structure, with the ligand transformed accordingly.
CONCLUSION AND FUTURE WORKS
In this work, we introduce the Pretrained Variational Bridge (PVB). novel generative model for trajectory generation for cross-domain biomolecular systems. By leveraging the encoder-decoder architecture, PVB provides a unified framework for both pretraining on single-structure data and finetuning on MD trajectories, thereby enabling seamless transfer of rich structural knowledge across stages. Extensive experiments show that, with the unified training objective, the model reproduces thermodynamic and kinetic observables consistent with classical MD and achieves substantial gains in generative fidelity over basclines. Furthermore, to facilitate efficient exploration of -lgaid co-crystal structures, we integrate RL finetuning via adjoint matching, employing a reward function to reweigh the generative distribution in favor of the holo state. Benchmarking on PDBBind suggests PVB as an effective tool for post-optimization of docking poses.
Looking ahead, sequential trajectory generation remains a limitation despite the coarsened timestep. and developing parallelized methods that maintain temporal correlations is an important avenue for future work.
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ETHICS STATEMENT
This study introduces a generative model for the simulation of cross-domain biomolecular systems. The work primarily methodological and computational, with an emphasis on assessing the model's capacity to accurately reproduce physical observables and efficiently explore metastable states of the molecular system. All experiments were conducted in silico using publicly available databases or author-curated datasets. As the study does not involve human or animal subjects, clinical data, or primary biological experiments, no specific ethical approval was required.
ACKNOWLEDGMENTS
This work is jointly supported by the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (No. JYB2025XDXM101) the National Natural Science Foundation of China (No. 62236011, No. 62376276), and Beijing Nova Program (2023048427)).
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APPENDIX
A REPRODUCIBILITY
Our code is available at https:// com/yaledeus//VD) which provides detailed instructions for training and evaluation, along with released model weights for both pretraining and downstream tasks.
B PROOF OF PROPOSITION
PROPOSITION
Given the initial state To. the unique optimal solution to Equation (2) satisfies P- X, = zo) = ge(- Xo .o) almost surely. Further, according De Bortoli al. 2023). the path measure P associated with Equation (6) preserves the coupling, i.e., lo, Po, Po Therefore, we can derive the following equalitres:
(dyo) = / pd ao) pxo (dro) = f q=(dvo) 2o) px, (dxo) Po(d)o).
(13)
hence for PPo-almost every yo.
Pijo yo) Pý (dyo, dwi)
Ho, (dyo, dy1) Pio(dy) 10).
(14)
(dyo)
Po(dyo)
By combining Equation
we complete the proof.
B.2 PROPOSITION 2
We first provide the derivation of Equation (TO) from Equation (9). Given the same initial state yo- we consider the following two SDEs, assumed to admit unique strong solutions on [0, 1]:
dY; Pt,t, Yo. Y,) df + + dB, Yo yo-
(15)
dY, = (vi(1, Yo. Y,) ou(1, Yo, Y,)) di a dB, Yo = yo-
(16)
Let 1P% and Py be the probability measures of the solutions to Equation 15) and Equation (16).
respectively. Since the drift terms are both adapted to filtration of B,, we can derive the Radon- Nikodym derivative by applying the Girsanov theorem:
log a (Y..) = - =-f u(t, Yo. dB 2 [.
dlp
|lu(t. Yo. Y.) dt, Yo. P
(17)
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Accordingly, the KL divergence be simplified by:
dlg 10
DKL (PP (P11)P)T)) EYonPP (log
(Yu1) Yo = yol
(18)
dl
jo dip
--Eoo blog dlp
Yo1) Yo = =yol
(19)
=Yoor L u(l, vo, Y,) dB, + 2 I ||((,, vo. Y.)II
1
(20)
-YoorpP ja 1 / ||u(t, yo, x
(21)
The last equality follows from the property that stochastic integrals are martingales. By substituting this equality into Equation ( we obtain Equation (IO).
Next, consider the optimization problem of the inner expectation conditioned on yo at t 0
in Equation 10
3. E 2 // [lue, Yo. x.) Yo yo]-
max Ey (
(22)
Since Yo = is fixed, the conditioning of the control vector field u on Yo is absorbed, yielding Uyo (1, Y.) u(4,yo yo, Y.). Therefore, the optimization problem of Equation (22) can be interpreted as Stochastic Optimal Control (SOC) problem, with its standard form given by:
minEl (-' 2 Ill (1, Y2) di gYY)),
(23)
s.t. dY: (b(1, Y,) + yo (1, Y.)) di + t dB, Yo 890
(24)
where b(t, Y.) = (t, Yo. Y.). g(Y = -
our case, and 8g represents the Dirac delta measure centered on æ. Based on the theorem proposed by [Domingo-Enrcch al. (2024), the optimal control u of the SOC problem defined in Equation (23) is the unique critical point of the following objective:
minEy..... Yon ~P [/2 L/" 11 (t, Y,) + oā(t, 1f dt Yo -yol û = sg(u),
1
(25)
d a(t, Yo.1) = -ā(t Yua) VV, Y.), ā(1, Yoa) = 9Y,)(
s.t. dt
(26)
Assume that the function class for u is sufficienlly expressive. Then there exists a function w" (t, yo. Y,) such that, for every yo supp( the map Y, ト> u" (t, yo, Y,) coincides with the conditional minimizer "no (4, Yz). We further prove that u" is the unique minimizer of Equa- tion (IT), which can be reformulated as:
Ladj (u) Ex, Ex (upo)) f Ryo Il
(27)
s.t. Ryo (u) FE-U((.1) Yoa Pa [[lugo (1, Y,) + oā(l, You) Yo = vol, ū = =s(().
(28)
By definition of u for every E supp(Ilo) and every u, we have Rego (u S Ryo (u). Integrating over yo with respect to Tl. we have:
/ Ryo (dyo).
Ludj (a")(")) - Ryo (u (odg) s
(29)
for every u. Thus un is the global minimizer of Ladi- Let u' for any global minimizer of Ladj- Consider the set:
(vo Ryo (u') Rys (u
(30)
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If (A) 0 then:
/ (Ryo (u') Ryyo (u (
Lads(u') (uad(("))
(31)
/ (Rzo (u') Ryo (u ( 1) (yy)
> 0.
(32)
This contradicts minimality of u'. Hence (A) O. Therefore for Ho-almost every yo we have Ryo (u') Ryo ( ). Recall that " is the unique minimizer of Equation (5 thus u' u" almost surely w.r.t. Hy. This completes the proof.
C MODEL ARCHITECTURE
C.I BLOCK-LEVEL REPRESENTATION
While atomic numbers offer basic representation of molecular systems, different classes of molecules are often characterized by fomain-speciff building blocks (e.g., amino acids in proteins). To enhance the model's ability to generalize across molecular domains, we aim represent molecular systems in a hierarchical manner based on these basic units. Inspired by Kong et al. (022)), We constructed vocabulary of building blocks that includes all atomic types with atomic numbers from to 118, as well as the 20 natural amino acids. Based on this vocabulary, we provide the block-level representation E NV for the molecular systems, where b[i] (0 S i < N) is defined as:
the atom type of the i-th atom, if it is part of small molecule; the amino acid type containing the i-th atom, if is part of a protein.
The atomic type and the block representation b will be projected into the same latent space dimension H through two separate embedding layers. The two embeddings are summed to form the initial atomic feature representation H R. NXH which is then passed the subsequent network.
C.2 GRAPH-LEVEL REPRESENTATION
The molecular system is further represented as graph g = (V,E), where V includes the invariant features H and SO(3) (3)--errvvariant Euclidean coordinates of N nodes (i.e., heavy atoms), and E comprises the edge index and associated edge features.
Specifically, for each molecule the system, we construct intra-molecular edges using the kNN algorithm (Cover& & Hart 1967), linking each node to its k nearest atoms in the molecule. In multi-molecule systems such as protein-ligand complexes, we further build inter-molecular edges by connecting each ligand atom to its k nearest protein atoms. The constructed edge index is denoted by E E x where the number of edges. Moreover, from the covalent bond index C, we derive the edge feature {0,1) 1] E which explicitly encodes the molecular topological structure:
1, if the i-th edge belongs to C, OSi<E
clt
(33)
0, otherwise.
C.3 NEURAL NETWORK BACKBONE
With the molecular graph prepared, the well-known TorchMD-NET (Pelaez et al. 2024) is leveraged as the backbone model to preserve equivaarnre We propose a simple modification in which the topology edge feature c: concatenated to the default edge attributes in the network after embedding.
In particular, to allow the decoder şa to condition on both Y, and Y,, we first construct the graph from Y.. Next, the equivariant coordinates of Yo. Y, are converted into invariant interatomic distances as edge features, and processed by lightweight equivariant graph neural network to obtain invariant node features of dimension H. These features are concatenated, fused through an MLP with single hidden layer, and then fed into the subsequent attention layers. Finally, the Geometric Vector Perceptron (GVP) used in (Yu et al. 2025) is applied as the output layer to yield invariant node features and SO(3) -equivariant vectors.
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D DATASET DETAILS
Firstly, the following datasets are used for pretraining:
PCQM4MY2 (Hu et al. 2021). A quantum chemistry dataset comprising approximately 3M small molecules, with DFT-calculated 3D structures.
ANI-1x (Smith et al. 2020). A dataset with 5M small organic molecules, consisting of multiple QM properties from density functional theory calculations.
PDB (Berman al. 2000), A large collection of experimentally determined 3D structures of biological macromolecules, where we use a subset processed by Jiao et al. (002)). comprising 58,576 protein monomers for training and 1,477 for validation.
PDBBind2020 (Wang et 2004 2005] Liu et al. 2015). A curated collection of 19,443 crystal protein-ligand complex structures with experimentally measured binding affinities. In this work, we adopt the data split proposed by (Lu et al. 2024). which consists of 12,826 complexes for training, 734 for validation, and 334 for testing.
For datasets without a pre-defined split, we randomly partition them into training and validation sets with 4:1 ratio.
Next, we benchmark the trajectory generation task on proteins and protein-ligante complexes. We use the following three datasets for the trajectory generation task:
* ATLAS (Vander Meersche et al. 2024). The ATLAS dataset provides long- tong-timescale
trajectory data for thousands of proteins generated from 100 ns all-atom MD simulations performed with GROMACS (Abraham et al. (2015) in explicit TIP3P water (Jorgensen et al., 1983). Following Yu et al. (2025), we use the curated subset comprising 790 proteins for training and 14 for evaluation, where the split is constructed by clustering sequences with MMseqs2 Steinegger & Söding, 2017) under a 30% sequence-identity threshold to ensure non-redundancy between the traiinn and test sets.
mdCATH (Mirarchi et al. 2024). The dataset contains all-atom molecular systems for 5,398 CATH domains, each simulated using a modern classical force field. For every domain, five independent MD trajectories are generated at each of five temperatures ranging from 320 K to 450 K, with the frame spacing of ns. We follow the data split proposed by [Janson al. (2025). comprising 5,049/40/90 protein domains for the training, validation, and test sets, respectively. To reduce the computational cost, we use only the trajectories simulated at 320 K for training and evaluation. Training data pairs (zie, (et+) are sampled every 5 ns from MD trajectories, with the coarsened time step T = 1 ns, yielding approximately 1.5M training pairs and 18K validation pairs. The first 20 domains of the original test set are used as our evaluation subset.
MISATO (Siehenmorgen et al. 2024). The MISATO dataset consists of over 10,000 pro- tiindl complexes, each accompanied by a 10 ns explicit-solvent molecular dynamics trajectory. Following the original protocol, the first 2 ns of each trajectory are discarded, and the remaining ns are provided as equilibrated conformational ensembles. In our exper- iments, we adopt the original dataset split of 13,765 13,765/1,595 1,595/1,612 for vaaaaing/vadidaiion/tssst respectively, while restricting the evaluation to the first 20 test complexes due to the substan- tial computational cost of trajectory generation on large systems.
To explore the protein-ligand holo state, we benchmark on the PDBBind2020 database (Wang et al., 2004; 20055 Liu al.) 2015). To prepare data for RL finetuning and ensure the availability of apo protein and unbound ligand structures for evaluation, we apply the following pre-processing pipeline to the training and test sets, respectively.
PDBBind-train Given the sensitivity of protein-ligand interactions to structural accuracy, we use only the refined set of PDBBind2020, which contains 5,316 diverse protein-ligand complexes with high-resolution crystal structures. To prevent sequence-leve overlap, we first cluster complexes using MMSeqs2 (Steinegger & Söding 2017) based on their bind- ing pocket residue sequences with a 60% sequence identity threshold. retaining a single representative from each cluster. Next, we adopt the time-based split used in previous
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studies (Corso et al. 2022 Stärk et al. 2022 Lu al. 2024), where structures deposited before 2019 are randomly partitioned into training and validation sets in a 4:1 ratio, and those deposited in 2019 are held out for testing. Furthermore, since RL finetuning requires not only crystal structures but also binding intermediates of protein-ligand complexes, we select small subset from the training and validation sets for additional MD simulations, yielding a 265/70 split. For each selected system, a short 20 ps MD simulation is performed using OpenMM (Eastman et al. 2023) in the NVT ensemble 300 K. The AMBER 14 force field (Maier et 2015) is applied, with explicit TIP3P water (Jorgensen et al. 1983) and 1.0 nm solvent padding. Long-range electrostatics are treated with PME, and all hydrogen bond lengths are constrained. The system is integrated using Langevi nMi ddl e eIntegrate with a friction coefficient of 0.5 ps and 2 timestep. Finally, 10 snapshots are extracted from each trajectory at 2 ps spacing for RI. finetuning.
PDBBind-test. For the test set, we first remove complexes that contain non- canonical amino acids, and then exclude those overlapping with the training and validation sets of MISATO, leaving a total of 33 test cases. For each test case, the unbound ligand and apo protein structures generated using MMFF94 (Halgren) 1996) and Boltz-1 (Wohlwend al. 2025), respectively. These structures are rigidly aligned to their corresponding parts of the crystal structure via the Kabsch algorithm (Kabsch, 1976). The aligned structures then serve input for AutoDock Vina (Eberhardt et al. (021), which predicts docking pose of the ligand while keeping the apo protein structure fixed, using an exhaustiveness parameter of 32. After removing the cases where Vina-docked ligands fail r rákit (rdk) sanitization, 23 test systems are retained, where the Vina-docked complex structures will be used as initial states for trajectory generation.
E TRAINING DETAILS
E.I POSITION PERTURBATION
Recall that our objective is to generate trajectories from the initial state of molecular system, process prone to error accumulation due the inherent instability generative models. Consequently, if the model is trained solely on reasonable conformations such crystal structures snapshots from MD trajectories, even minor deviations during generation may cause the model to fall out of distribution. To enhance the robustness of trajectory generation, introduce small perturbation to the initial state Xg before feeding into the encoder during both pretraining and finctuning. Formally,
X X + O2e, ((O,
(34)
where o is the hyperparameter of perturbation strength.
F.2 DATA AUGMENTATION FOR RI.
Since the training data for reinforcement learning are limited and sampled only from short MD trajectories, we apply data augmentation during RI. finetuning. Specifically, for each training iteration, we randomly sample a rotation angle ~ U1 T 3: 3 and unit rotation axis, which can random rotation matrix Q E SO(3) using Rodrigues' rotation formula. Next,
be transformed we sample translation vector t from the standard Gaussian distribution. The (ee(3))tranffommatin (Q, t) is then applied to the ligand pose the initial state Xo. which served as the new input.
E.3 ALGORITHM FOR RL FINETUNING
We present the RL finetuning algorithm for protein-ligand complexes in Algorithm [ where the operator data augmentation discussed in the previous section.
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Algorithm 1 RL finetuning for protein-ligand complexes with PVB
1: Input: protein-ligand complex system with position Zo, the corresponding holo state Tref, the reference models SPe, iPa, the regularization strength 3, discretized SDE steps
2: freeze parameters of Pe and
3: Pd < Pd
4: A --
5: for training iterations do
6: Zo f- (uataaaaunmeentiio(()))))
7:
~~ [((0 A, 2A, 1))
8:
y/ ~ ( Xn = zn) (reparameterization)
9:
Yo.1
Yo, Ya, "..,..., Yi]
pāc Yo yo) (Equation g
10:
â1
Yy,rmdd (rmms((11220) 1,
11:
s+1
12:
while 8 > f do
13:
0g-44 A ú, + a₂ Vr, Yo, Y,) A
14:
8 8+3
15: end while
16:
Ladj f (t, Yo, Y,) - palt, Yo. Y,) + 0²ā,
min Cadj
18: end for
E.4 HYPERPARAMETERS
The hyperparameters used for training and inference with PVB are summarized in Table
Table 5: Hyperparameters of PVB.
Hyperparameters
Values
Hidden dimension H
256
FFN dimension
512
RBF dimension
32
RBF cutoff
10 Á
Attention heads
8
# Encoder layers
2
Decoder layers
8
Neighbors k
32
Variational noise scale de
0.. A
SDE noise scale
0.2 A
Perturbation Strength Op
0,2 A
KL loss weight wkr.
0.8
ABM loss weight WARM
1.0
Regularization strength
4c-4
Learning rate (pretrain)
lc-4
Learning rate (finetune)
5e-5
Optimizer
Adam
Warm up steps
1,000
Warm up scheduler
LambdaLR
Training scheduler
ReduceL Pawaaooooalaaaattrrrrasaao patience=5, min_Ir=le-7)
Discretized SDE steps T
[8.10]
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F EVALUATION DETAILS
F.1 FEATURIZATION OF PROTEINS
To featurize trajectories of proteins in ATLAS, we select as collective variables the sine and cosine of backbone & and torsions and all side-chain torsions, and the Euclidean distances of contacting heavy-atom pairs, which are defined as pairs of C, N, or S atoms separated by more than two bonds in the molecular topology graph and lying within 4.5 Á in the initial conformation reference trajectory (Yuan et al. 2012). In particular, since each trajectory in mdCATH contains relatively few frames, we instead use only the backbone ó and & torsions as TICA features for this dataset avoid ill-conditioning. The extracted features are then projected onto the slowest two time-lagged independent components (TICO and TIC1) Pérenz-errnndez et al. (013) using deept ime (Hoffmann et al. 2021). with a lag time of 100 ps for ATLAS and ns for mdCATH.
the MD
F.2
DIVERGENCE
We compute the Jensen-Shannon Divergence (ISD) on various feature spaces using scipy (Jones) al. 2000T. For 1-dimensional JSD, we discretize the range spanning the minimum and maximum values from MD trajectories into 50 bins. For multi-dimensional features, we first compute the JSD separately along each dimension and then report the mean value across all dimensions.
F.3 DECORRELATION
Following (Jing et al. 2024b), we report the decorrelation percentage for TICO to characterize the kinetic properties of the generated trajectories. Firstly, we define the autocorrelation as follows:
E[( - () - -1]]/o²
(35)
where and a are the mean and standard values of TICO from the MD reference trajectory, re- spectively. By computing the autocorrelation using sta smodels (Seabold & Perktold 2010), the generated trajectory is defined as decorrelated if its autocorrelation of TICO starts above and falls below 0.5 within 1,000 frames (100 ns). Since all MD reference trajectories have been veri- fied to decorrelate within 100 ns, the decorrelation fraction observed in the generated trajectories (Decorr- TICO) serves as a quantitative indicator the ability to capture dominant slow modes during simulations.
F.4 MARKOV STATE MODEL
We employ Markov state models to assess the fidelity of metastable state reproduction. We first cluster the two-dimensional features (TICO and TICI) from the MD reference trajectory into 100 microstates using the KMeans++ algorithm (Arthur & Vassilvitskii 2006). Next, we construct the Markov state model based on the microstates with a lag time of 100 ps, and apply the PCCA+ algorithm (Röblitz & Weber 2013) to coarse-grain the dynamics into 10 metastable states. Specifically, after the PCCA+ the nearest active cluster,
analysis, we reassign clusters corresponding to inactive microstates thereby ensuring connectivity among all microstates. Finally. the clustering and PCCA models are employed assign metastable states for each frame of both the MD reference trajectory and the generated trajectory, and JSD is computed between their probability distributions over the 10 metastable states.
F.5 ALIGNMENT OF PROTEIN-LANE LIGAND COMPLEXES
To assess the -rigad complexes, the frames should be superimposed onto reference con- formation at first. Specifically. for the PDBBind dataset, the holo state is used as the reference conformation, whereas for the MISATO dataset, the initial state of the MD reference trajectory is selected. Given the reference conformation, the pocket of the complex is defined as the set of residues containing least one heavy atom within 20 Å of the ligand center. Each frame in the trajectory then aligned to the reference conformation by applying an (3(3)-transformatio based on the backbone Co, C, and N backbone atoms of residues in the pocket.
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F.6 CONTACT OCCUPANCY
Following Janson et al. (2023), for protein with R residues, we define the contact occupancy r(i,j) of the residue pair J as;
T Σ
r(i,j) T
(101) isisjs R, ==1
(36)
where T denotes the number of frames in the trajectory, and de (i,j) denotes the Fuclidean distance between Ca atoms of residues and j in the t-th frame. Then the root mean square error of the contact occupancy is computed by:
RMSE-CONTACT
2 R(R -
Σ (r(i.)) - (ee(())) ISiCiSR
(37)
|
where ji is calculated from the MD reference trajectory.
Similarly, for protein-ligand complexes, we select the heavy-atom pairs between the ligand and the pocket to calculate contact occupancy, and correspondingly compute the RMSE relative to the MD reference trajectory.
F.7 VALIDITY
For the molecular system of proteins, we calculate the validity of generated conformations assess the stability in long timescale simulations. Following Wang al. (2024a), conformation is defined as valid if: (I) the Euclidean distance between Ca atoms of residue pairs is no less than 3.0 and (II) the Euclidean distance between Ca atoms of adjacent residue pairs is more than 4.19 A. The proportion of valid conformations in the trajectory is reported, denoted as VAL-CA.
G ADDITIONAL EXPERIMENTAL RESULTS
G.1 FAST-FOLDING PROTEINS
In this section, we assess whether PVB can explore essential biological processes in long-timescale MD simulations. As a case study, we focus on the well-established fast-folding proteins, which capture protein folding and unfolding dynamics on the microsecond timescale (Lindorff-Laasen cct al. 2011). Among twelve fast-folders, we select NTL9 (an a/3 protein), Protein B (a three-helix bundle protein). Villin (an actin-associated mini-protein), the WW domain (a 8-sheet protein), and X- repressor (a melix-turn-held DNA -innding protein) as five representative test systems, with reference trajectories obtained from MD simulations ranging from 100 to 3000 jıs, saved at frame spacing of 200 ps.
To better characterize folding dynamics, we employ global structural descriptors TICA collective variables: the sine and cosine of backbone (c) 1h) and side-chain torsions, along with Ca-Ca distances for residue pairs separated by least residues. The lag time for TICA is taken as 20 ns.
Using the PVB model trained on ATLAS, we generate trajectories of length 100,000 for the five test systems in a zero-shot manner, each corresponding to 10 μs in wall-clock time. The results visualized in Figure 6 Here, contact occupancy is defined as the fraction of residue pairs separated by least two residues, whose Ca atoms are within 12 A (Kukic et al., 2014).. We observe that for NTL9, PVB fluctuates mainly around the initial conformation during the 10 simulations without reaching the folded state, which is likely due to the limited simulation timescale and the system being partly out of distribution for the model. For Protein B, although the limited simulation time precludes a full recreation MD findings, PVB nonetheless provides an initial characterization of metastable states consistent with MD. Moreover, for Villin, the WW domain, and X-repressor, despite minor artifacts in phase space regions undersampled by MD (e.g., for X-repressor), the model nevertheless recovers the essential free energy landscape and captures the rapid transition between metastable states, highlighting its capacity to generalize to reliable long-timescale molecular dynamics of fast-folding proteins spanning diverse secondary structures.
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Regarding contact occupancy. PVB demonstrates robust stability for NTL9, Protein B, Villin, and the WW domain throughout the 10 jıs simulations, with trajectories displaying average values and fluctuations broadly consistent with the MD benchmarks. In contrast, the X-repressor a declining trend, signaling simulation failure and subsequent structural collapse. These findings underscore PVB's capacity maintain long-timescale fidelity for small- medium-sized proteins, with potential challenges in scaling to larger systems.
reveals
Grepp
MM
VD
7/B
Time
Time
10
Time tol
1823
The
Figure 6: Visualization of the generated trajectories of NTL9 (Row 1), Protein B (Row 2). Villin (Row 3), the WW domain (Row 4), and X-repressor (Row 5). Column 1: Experimental determined structures from Protein Data Bank (PDB). Column 2-3: Comparison of free energy surfaces projected onto the slowest two time-lagged independent components (i.e., TICO and TICI) for the entire MD reference and PVB-generated trajectories. Column 4-5: Comparison of the contact occupancy during the first 10 of simulation for the MD reference and PVB- PVB-generated trajectories.
G.2 ABLATION STUDY
To validate the effectiveness of the proposed pretrained variational bridge in leveraging structural prior knowledge, we perform the following ablation experiments on the ATLAS dataset under three different configurations:
PVB w/o finetuning. This setting corresponds to directly using the pretrained model to generate trajectories the test systems, without any finetuning on the training data.
PVB w/o pretraining. This setting corresponds to training the model from scratch on the ATLAS training set, without using any pretrained weights.
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PVB-ITO. This setting replaces our bridge- mateee-mmtcchhtiiieee based decoder with the conditional diffusion model used in ITO (Schreiner et al. 2023). while keeping the model backbone unchanged.
The evaluation results are presented in Table First, they show that PVB with pretraining achieves substantial improvements across all metrics, with particularly pronounced gains in validity. This un- derscores the critical role of the rich, cross-domain structural knowledge acquired during pretraining.
Second, directly evaluating the pretrained PVB model ATLAS without finetuning reveals that the fidelity of generated conformations largely stem from pretraining. While finetuning slightly reduces residue contacts, it consistently improves distributional similarity, as reflected by lower JSD values. Together, these findings provide strong evidence that PVB functions effectively as a unified framework, benefiting from both pretraining and finetuning.
Finally, we also tested the scenario where the PVB decoder is entirely replaced with the conditional diffusion model used in ITO, while fully replicating the PVB pretraining and finetuning scheme. The experimental results show that the modified model performs poorly overall and fails to generate physically plausible protein conformations. This suggests, to some extent, the limitations and incompatibilityy of ITO's generative framework in this setting. We also emphasize that, although bridge matching is not necessarily irreplaceable, it does perform remarkably well under the current experimental setup.
Table 6: Ablation results on the ATLAS test set, with each metric reported as mean/std across 14 test protein monomers. The best result for each metric is shown in bold.
ISD (1)
MODELS
VAL-CA (1)
RMSE-CONTACT (1)
Rg
Torus
TIC
MSM
PVB
0.457/0.087
0.336/0.030
0.371/0.087
0.333/0.105
0.975/0.023
0,143/0.031
PVB w/o finetuning
0.561/0.
0.310/0.018
0.463/0.094
0.407/0.155
0.968/0.039
0,097/0.014
PVB w/o pretraining
0,710/0.063
0,481/0.011
0,408/0.072
0.581/0.127
0.022/0.007
0.223/0.033
PVB-ITO
0.676/0.082
0,577/0.014
0,487/0.072
0,460/0.158
0.000/0.000
0.269/0.042
G.3 PHYSICALITY OF GENERATED PROTEIN-LIGAND COMPLEXES
In the docking pose post-optimization task for protein-ligand complexes (Section we relied solely on RMSD as the evaluation metric, which may obscure the extent to which the generated conformations are physically plausible. In this section, we introduce more fine-graines evaluation criteria to analyze the physicality of both protein and ligand conformations produced along the generated trajectories, and further compare how it changes before and after reinforcement learning.
Specifically, to evaluate the physicality of the generated complexes, we introduce the following metrics:
VAL-BL. A ligand conformation bond-length valid if every covalent bond length deviates by more than 20% from the Distance Geometry (DG) reference bounds defined in PoseBusters (Buttenschoen et al. 2024).
VAL-BA. A ligand conformation is bond-angle valid if every bond angle stays within 20% of the DG reference angular bounds defined PoseBusters.
SCF. A ligand conformation is steric clash-free if all non-covalent inter-atomic distances are at least 80% of their DG lower bounds defined in PoseBusters.
SI. A ligand structure satisfies stereochemical integrity only if (i) all topologically stere- ogenic atoms possess well-defined CIP configurations, and (ii) all stereogenic double bonds exhibit valid E/Z annotations without ambiguity. The analyses are carried out using rákit (rdk).
VAL-CA. To assess the validity of protein conformations, we adopt the same metric defined in Section[F] considering a protein structure valid only if there are no broken bonds between adjacent residues and no clashes between any residue pairs based Ca distances.
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Note that for all the above metrics, first compute each metric over the 100-frame generated trajectories for each test system and take the average. We then report the mean and standard deviation of these averages across all 23 test systems, which are shown in Table 7 Based on the comparison of the experimental results, we draw the following observations.
First, the model trained MISATO (i.e., PVB w/o RL) preserves high degree of physical plausibility in the generated all-atom conformations of both ligands and proteins. This further demonstrates that large-scale pretraining on diverse single structures, followed by light MD finetuning, effectively imparts domain knowledge applicable across molecular families.
Second, after reinforcement learning, the model exhibits slight improvements on ligand-related met- rics and a notable enhancement in the validity of generated protein conformations. This indicates that reinforcement learning preserves, and may even enhance, the model's ability generate physically plausible trajectories supported by additional data.
Finally, we find that the two models perform identically on the stereochemical integrity metric. Upon inspection, we determined that in of the 23 test systems, the initial ligand conformations from the original database already failed this test. Since the input specifies molecular topology, the generated trajectories do not alter stereochemistry of the ligand, leading all ligand conformations in the trajectories fail the check as well. This accounts for the unusual consistency observed for this metric.
Table Physicality of the generated protein-ligand complex trajectories the test of PDBBind. The metrics are reported in mean/std over 23 test systems.
MODELS
VAL-BL (t)
VAL-BA (1)
VAL-CA (f)
SCF (t)
SI (1)
PVB w/o RL PVB RL
0.874/0.200
0.827/0.212
0.615/0.217
0.962/0.051
0.870/0.344 0.870/0.344
0.895/0.204
0.852/0.220
0.813/0.224
0.963/0.070
G.4 INFERENCE EFFICIENCY
We assess the inference efficiency for different models by computing the inference time per step on the same GPU device. Results are shown in Figure 7 where PVB delivers approximately 5-10x faster inference than the second-best model (MDGEN), while also exhibiting the lowest variance across all baselines.
Inference Time per Step (s)
8-
6-
5
3
PVB
UniSim
MDGEN
ITO
Model
Figure 7: Comparison of per-step inference time the ATLAS test set for deep learning models, evaluated on a single identical GPU.
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H COMPUTING INFRASTRUCTURE
All experiments were performed on NVIDIA GeForce RTX 3090 GPUs. During the pretraining stage, used five GPUs for approximately five days. To fully utilize GPU memory across molecular systems of varying sizes, the mini-batch size was dynamically adjusted, maintaining an average of about 2.500 atoms per batch, In addition, balance the contribution datasets with different sizes, each training epoch randomly samples up to 60,000 datasets, all available samples are traversed within the epoch.
from each dataset. For smaller
Following pretraining, finetuning was conducted on up to five GPUs, with training durations ranging from one day one week depending on the specific task and dataset size.
Finally, all model evaluations, including both baseline methods and our proposed approach, were performed on single GPU to ensure a fair and consistent comparison.
I DISCLOSURE OF LARGE LANGUAGE MODEL USAGE
Large language models (LLMs) were used solely general-purpott writing assistants in the prepara- tion of this manuscript. Specifically, LLMs were employed to polish wording, improve clarity, and correct grammar. They were not used for research ideation, methodological design, experimental plan- ning, data analysis, or any scientific decision-makin All technical content, results, and conclusions were developed and verified by the authors, who take full responsibility for the manuscript.
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Reviews:
[{"review_id": "rqowkUu439", "reviewer": "Reviewer_i298", "summary": "The paper proposes a strategy for developing next-step MD emulators by first pretraining on a bridge that maps between distributions of static structures. This allows models trained on vast amounts of rich static data to be easily tuned on dynamics prediction. The authors show that models pretrained on the PDB and PDBBind can be fine-tuned on ATLAS (protein simulations) and MISATO (protein-ligand simulations) to replicate observables. The authors also develop a RL training strategy for the bridge to steer rollouts towards the holo state of a protein-ligand complex.", "questions": "What is the state matrix in Figure 3, right? The caption says \"Probability differences between PVB\nand MD across the 10 metastable states estimated by MSM.\" --- this shouldn't be a matrix, then.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The idea of pretraining a bridge to recapitulate the initial state, and then fine-tuning it to produce the evolved state, is quite interesting. The work also touches upon simulation of protein-ligand simulations, which have been somewhat neglected in the ML for MD literature, despite their significant practical importance.", "weaknesses": "**Method**\n* The RL formulation of the holo complex finetuning task seems gratuitous. In particular, if the reward is the RMSD to the holo state, why can't the holo state be used in a supervised fine-tuning fashion? If would seem that if the reward is simply the similarity to an explicit, known state, that is the setting of supervised learning, not reinforcement learning.\n\n**Experiments**\n* There are missing controls that make the value of the pretraining bridge hard to interpret. What if we pretrain without a bridge, such as AlphaFlow (with templates)? What if we don't pretrain at all, but use the same architecture? (I assume the retrained ITO baseline is using the ITO architecture).\n* Although I am willing to judge these as not the focus of the paper, the protein-ligand docking evaluations are extremely sparse - the single baseline is AutoDock Vina, despite vast amounts of recent literature.", "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": "dZDRlXy6Uh", "reviewer": "Reviewer_JWxB", "summary": "This paper introduces pretrained variational bridge (PVB). The core contribution is a unified framework that first pretrains an encoder-decoder model on a large and diverse dataset of single, static molecular structures to learn generalizable structural features. This pretrained model is then finetuned on paired molecular dynamics (MD) trajectory data to learn system-specific dynamics. Furthermore, the paper presents RL finetuning procedure using adjoint matching, which efficiently optimizes the model to guide trajectories toward specific target states, such as the holo conformation in protein-ligand docking.", "questions": "- Have the authors checked the physicality of the ligands (and proteins as well)? Not only the bond break or clashes, but also stereochemical errors. Does that get better or worse with RL finetuning?\n- The paper claims cross-domain generalization, but it is not specified whether the train/test splits of datasets were performed based on sequence similarity or other metrics to prevent data leakage and rigorously test generalization to unseen protein folds.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The paper proposes a novel methodology to include pretraining on datasets with static structures but diverse chemical space, and then finetuning on dynamical data. This enables the model to achieve better chemical transferability despite the limited chemical space coverage of the dynamical data.\n- The integration of RL with adjoint matching for pose-optimization in docking is a novel application. And the authors have shown the improvement in the ligand pose after the finetuning.\n- The model's performance is thoroughly benchmarked across multiple demanding tasks, including protein dynamics, protein-ligand complex dynamics, and holo state exploration. The comparison against several relevant baselines on different datasets demonstrates the effectiveness of the method\n- PVB shows outperformance over baselines across most metrics.\n- Ablation studies have been performed to show the benefit of pretraining and finetuning procedure", "weaknesses": "- While the paper evaluates against other trajectory-based models, it assesses performance on free energy landscapes. While most of the metrics compared in the paper are actually thermodynamic properties, they can be evaluated with i.i.d. (time-agnostic) sampling models. It will be helpful to benchmark against those methods as well.\n- In the meanwhile, although the time-dependent model describes dynamics, it's not obvious from the benchmarks and applications shown in the paper why the time-dependence is needed, what is its advantage over i.i.d. sampling model. It will help to justify the motivation if the authors can clarify that (time-dependence makes the model to describe thermodynamics better than i.i.d. sampling model) or show some cases when kinetics/dynamics are of practical interests in applications.\n- The rationale behind using two separately finetuned models for the protein (ATLAS) and protein-ligand (MISATO) tasks is not explained. One might expect that a single model finetuned on both could offer better transferability, especially for the protein component of the 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": "LdsUVLUqyf", "reviewer": "Reviewer_WXYF", "summary": "The paper proposes a method for coarse-grained molecular dynamics simulation, using Brownian bridges. More specifically, they first go to a latent state, and then to the target state. This allows generalization over both single-structure and paired trajectory data, especially in terms of pretraining. Method is evaluated on relevant datasets.", "questions": "- Fig. 1. This can be a very informative figure, but the elements are quite small (the arrows and black dot). Consider indicating on the figure what the meaning is of the three small modes on the left and the big one on the right.\n- Why do you use the rmsd, and not the (Gaussian) log likelihood?\n- At the start of 3. Method, Z and C are defined, but are not used in the remaining sections. Do you also model these? If so, how exactly?\n- Why exactly do you use \\mu vs. p? They both indicate probability measures I assume?\n- Please define more clearly what \\mu, and X are on line 41. In general, please take some time to rethink where you define the different mathematical concepts and objects. Now it's a bit all over the place and does not seem to follow a structured explanation, or logical build up in your story.\n- eq. 11, why is there a gradient stop on u when sampling Y_0:1?\n- footnote on line 225. Please don't put this in a footnote, it is very confusing! The bridge is from \\tilde X to X_1, correct? Why not call it Y from the start, please take some time to think about this, I'm sure there's ways to make this paper much more readable if you decide on certain notations from the start and don't start changing/adding things in the middle of your explanation.\n- Prop. 3.2: You are solving an optimal control problem analytically, correct? Is solving the ODE for \\tilde a related to solving the HJB equation? Can you please compare your approach to [1], where this is done for the same kind of ELBO, for linear SDEs?\n\n[1] https://arxiv.org/abs/2505.17150\n\n\nMinor suggestions:\n- line 92. RL is not defined.\n- line 102: 'applying'\n- I would not use (so much) abbreviations in the abstract, it does not improve readability, and RL is not defined. The abstract is quite wordy which makes it also harder to read (e.g. 'inefficiency' -> cost, 'nevertheless', 'remarkable', ..)\n- please use larger brackets and ||, e.g. \\left[ \\right], see eqs. 5, 7, ..\n- line 57: 'domain' (singular)\n- line 234: 'prove', this sentence is also not clear?\n- line 278: Y=Y, very confusing. I suppose one is a stochastic variable and the other one is a realization. You could e.g. use small letters for the realizations.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper works on a relevant problem.\n- Optimal control methods are leveraged for more efficient training.\n- The method is evaluated on relevant benchmarks.", "weaknesses": "- The explanation of the theory and the notation is quite confusing. I sympathize that this is not trivial, especially with having in a mind a relatively broad target audience from diverse research backgrounds. But considerable effort should be made to improve the writing. I try to make some concrete suggestions below. I'm certainly willing to raise my score if readability is improved!", "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": "CmeWJba9vt", "reviewer": "Reviewer_BUEn", "summary": "This paper introduces the Pretrained Variational Bridge (PVB), a unified generative framework for biomolecular trajectory generation that leverages pretraining on static 3D structures and finetuning on coarse-grained MD trajectory data.\nThe central idea is to bridge the gap between single-structure pretraining and trajectory-conditioned finetuning with a unified objective:  $\\mu(X_1 \\mid X_0)$. \n\nDuring pretraining, the model learns from diverse molecular structures to capture cross-domain structural knowledge.\nDuring finetuning, the model is finetuned on paired transition data $(X_t, X_{t + \\Delta t})$ from dynamic datasets (e.g., ATLAS and MISATO).\n\nAn additional innovation is the RL-based adjoint finetuning using stochastic optimal control, enabling direct optimization for holo state generation in protein–ligand systems.\n\nEmpirical results show that PVB reproduces thermodynamic and kinetic observables (Rg, torsion, TIC projections, MSM occupancy) with stability comparable to MD and substantial improvement over baselines (ITO, MDGen, UniSim) in validity (VAL-CA = 0.97) and decorrelation metrics.\nIn protein–ligand docking, PVB with RL finetuning outperforms AutoDock Vina and non-RL variants.", "questions": "1. I am particularly interested in the experimental results on the large-scale MDCath dataset, as well as the runtime analysis of the proposed method. Could the authors provide more details or quantitative comparisons to illustrate the computational efficiency and scalability of PVB?\n\n2. Is it possible to extend the proposed framework for parallel trajectory generation, rather than sequential sampling? This could further improve scalability, especially for long biomolecular simulations.\n\n3. Lastly, could the authors elaborate on the role of the latent variable X0? Specifically, in the statement “The latent variable X0 is introduced to avoid the collapse of the conditional probability µ(X1|X0) from degenerating into a Dirac delta measure,\" it would be helpful to clarify why this latent variable is necessary and what would happen if one directly generated X1 conditioned on X0 without it.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* PVB elegantly integrates structural pretraining and trajectory learning through a shared encoder–decoder bridge, aligning objectives across domains. Pretraining on heterogeneous biomolecular structures allows transfer to proteins, small molecules, and protein–ligand complexes without retraining.\n\n* The adjoint-based stochastic control formulation enables memory-efficient fine-tuning toward functional states (e.g., holo forms) without additional networks.\n\n* PVB consistently achieves better or comparable results to classical MD and generative baselines in reproducing both kinetic and thermodynamic observables.\n\n* The RL variant shows meaningful progress toward real drug-design applications, improving ligand placement beyond traditional docking and static generative methods, which can serve as an alternative method for docking.", "weaknesses": "* While conceptually elegant, the experimental scope is somewhat narrow and the demonstrated benefits are modest under realistic scales. The datasets (ATLAS, MISATO) are relatively small, and the observed improvements, though consistent, are incremental, especially given that baseline MDGen and UniSim already yield physically valid trajectories. Including results on the recently released MDCATH dataset will be make the manuscript stronger.\n\n* As mentioned by the authors, the generation remains sequential, limiting scalability to long timescales or high-throughput ensemble generation. The paper does not analyze the runtime of the PVB for trajectory generation, which is also a concerning fact. While coarse timesteps improve efficiency conceptually, inference speed, wall-clock cost, and scaling to larger systems (e.g., >10⁴ atoms) remain unreported.\n\n* The claimed cross-domain transferability is supported only by protein and protein–ligand tasks; other molecular domains (RNA, materials, polymers) are underexplored.", "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)