MarS-FM: Generative Modeling of Molecular Dynamics via Markov State Models
Authors: Kacper Kapuśniak
Cristian Gabellini
Michael M. Bronstein
Prudencio Tossou
Francesco Di Giovanni
Venue: ICLR 2026 Poster (2026)
Keywords: molecular dynamics
generative models
proteins
flow matching
markov state models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Molecular Dynamics (MD) is a powerful computational microscope for probing protein functions. However, the need for fine-grained integration and the long timescales of biomolecular events make MD computationally expensive. To address this, several generative models have been proposed to generate surrogate trajectories at lower cost. Yet, these models typically learn a fixed-lag transition density, causing the training signal to be dominated by frequent but uninformative transitions. We introduce a new class of generative models, 
**MSM Emulators**, which instead learn to sample transitions across discrete states defined by an underlying Markov State Model (MSM). We instantiate this class with Markov Space Flow Matching (MarS-FM), whose sampling offers more than two orders of magnitude speedup compared to implicit- or explicit-solvent MD simulations. We benchmark Mars-FM ability to reproduce MD statistics through structural observables such as RMSD, radius of gyration, and secondary structure content. Our evaluation spans protein domains (up to 500 residues) with significant chemical and structural diversity, including unfolding events, and enforces strict sequence dissimilarity between training and test sets to assess generalization. Across all metrics, MarS-FM outperforms existing methods, often by a substantial margin.

Published as a conference paper at ICLR 2026
MARS-FM: GENERATIVE MODELING OF MOLECULAR DYNAMICS VIA MARKOV STATE MODELS
Kacper Kapušniak I..1, Cristian Gabellini2.5
Michael Bronstein 1.4, Prudencio Tossou Francesco Di Giovanni University of Oxford, Valence Labs, "Recursion, *AITHYRA
ABSTRACT
Molecular Dynamics (MD) is powerful computational microscope for probing protein function. However, the need for fine-grained integration and the long timescales of biomolecular events make MD computationally expensive. To address this, several generative models have been proposed to generate surrogate trajectories lower cost. Yet, these models typically learn fixed-lag transition density, causing the training signal to be dominated by frequent but uninformative transitions. We introduce new class of generative models, MSM Emulators, which instead learn to sample transitions across discrete states defined by an under- lying Markov State Model (MSM). We instantiate this class with MARKOV SPACE FLOW MATCHING (MARS-FM) whose sampling offers more than two orders magnitude speedup compared to implicit- or explicit-solvent MD simulations. We benchmark MARS- -M'ss ability to reproduce MD statistics through structural observables such as RMSD, radius of gyration, and secondary structure content. Our evaluation spans protein domains (up to 500 residues) with significant chemical and structural diversity, including unfolding events, and enforces strict sequence dissimilarity between training and test to assess generalization. Across all metrics, MARS- FM outperforms existing methods, often by a substantial margin.
INTRODUCTION
Deep Learning has unlocked fast and accurate prediction of proteins' 3D structures (Jumper et al., 2021; Back al., 2021: Krishna et al., 2024: Abramson et al., 2024). However. these methods do not capture dynamic behavior of proteins (Lewis et al., 2024), whose conformational ensembles are governed by the Boltzmann distribution. To study such dynamics, the most reliable computational tool is Molecular Dynamics (MD) (Alder & Wainwright, 1959; Rahman, 1964; McCammon et al., 1977; Risken & Risken, 1996), which simulates atomic motion by integrating Newton's 's Law. Long MD tra- jectories provide samples from the Boltzmann distribution and reveal the mechanisms of biomolecular interactions, a key asset in drug discovery (De Vivo et al., 2016). Nevertheless, MD is costly, as events in biology occur over timescales vastly longer than the simulation timestep. This challenge has spurred a range of enhanced sampling methods to accelerate dynamics, often through non- non-physical forces (Laio & Parrinello, 2002; Hamelherg et al., 2004; Jiang & Roux, 2010; Sabbadin & Moro, 2014).
Recent works have proposed to avoid the computational burden of MD by using generative flows to sample from the Boltzmann distribution (Noé et al., 2019; Zheng et al., 2024; Klein & Noe, 2024). A key subclass is the family of MD Emulators (MD-Emus), which learn to emulate MD-sampling by modeling the transition density associated with fixed lag time T (Klein et al., 2023; Schreiner et al., 2023; Nam et al., 2024; Diez et al., 2024; Costa et al., 2024). That is, MD-Emus are models that generate future conformations xxt + 7) conditioned on some input frame xxt). At inference, these methods are applied autoregressively to produce surrogate MD) trajectories. Jing et al. (2024b) advanced this approach by training model to generate multiple frames jointly, all separated by a fixed interval т. However, learning dynamics at a fixed lag time introduces key challenges. Short lag times limit the achievable speed-up, while long lag times can skip over important meta-stable states. More fundamentally, the training signal is dominated by frequent but uninformative transitions observed during the simulation, while bigh-barrier transitions that drive exploration remain underrepresented. As result, MD-Enn Emus may struggle to capture rare, large conformational changes (Figure 1, left).
Worr done as an intern at Valence Labs/Recursion Cooresponding author; kacper.kapusnia.ee Code is available https: / g thub. com/va va lence labs tala/marss /maa/maaaaaaaa/ fm
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MD-Emus
MSM-Emus
MSM-Emus C MD-Emus
Figure 1: Comparison between existing approaches (MD-Emus) and our proposed novel class (MSM- Emus). MDD-Emus learn transitions within state (i.e. energy minima) well but could fail to generate transitions across different states (minima) since they are constrained by the data imbalance intrinsic to MD. Conversely, our framework learns sample from the distribution induced by Markov State Model (MSM). This modeling shift means that generative models are decoupled from temporal dynamics and can better learn sample inter-state transitions. During sampling, MSM-Emus can generate conformations in parallel or be combined with existing MD-Emus to capture both large conformational changes well as local dynamics within states.
Limitations of MD-Emus stem from the data imbalance in MD, whose temporal dynamic contains irrlevant high-frequency information. A common strategy extract meaningful signal from MD) through coarse- coarse-grained representations, is offered by Markov State Models (MSMs) (Noé et al., 2009; Prinz et al., 2011; Bowman et al., 2013). MSMs cluster frames into discrete states and describe dynamics via a Markov chain matrix T. While compressing the dynamical content, MSMs guarantee estimation of long-time time statistics through the equilibrium distribution induced by T.
Main Contributions, We present MSM Emulators MISs), a new class of generative models that learn sample from the Markov-chain transition induced by an MSM constructed from MD) trajectories. In this view, the MSM provides a coarse-grained Markov model of long-timescale dynamies between metastable states, and MSM-Emus focus on learning transitions between metastable states rather than to the full high-freyuency MD time series. This modeling shift has important implications as illustrated in Figure
Since target transitions now depend on state connectivity rather than specific observed paths, MSM- Emus benefit from much higher training diversity for learning rare conformational changes across high-energy barriers. In fact, MSM-Emuu interpolate across discrete macroscopic states associated with large conformational changes, such as folding or unfolding, rather than among specific frames, which is more robust and generalizable signal across different proteins. During sampling, frames are more easily decorrelated and autoregressive calls are greatly reduced, mitigating compounding error effects. We showcase this class with MARKOV SPACE FLOW MATCHING (MARS-FM), a novel framework optimized using Flow Matching (Lipman et al., 2023).
Evaluation with chemical and structural diversity. MD-Emus have usually been evaluated on small peptides or datasets with limited chemical Lindorff-Larseee et al., 2011; Majewski et al., 2023) or structural (Vander Meersche et al., 2024) diversity. To address this gap. we evaluate MARS-FM on MD-CATH (Mirarchi et al., 2024), large-scale dataset of thousands of protein domains with particular, by leveraging the highest-temperatuure replica, we ability to capture large conformational changes, such as unfolding. To
sequence length up to 500 residues. are able to test MARS-FM. assess generalization, we adopt the protocol Lewis al. (2024) and use MMSEQS2 (Steinegger & Söding, 2017) with maximum sensitivity to ensure that test proteins share no more than 20% sequence similarity with any protein the training set. We benchmark MARS-FM using structural observables reported in Jing al. (2024b); Mirarchi et al. (2024), including RMSD, radius of gyration, and secondary structure content, and show that consistently outperforms MID-Emus-ffie by a large margin. Crucially, MARS-FM explores the target distribution much more efficiently than both MD and MD-Emus, generating conformations across different states even in low-sample regimes.
2 PRELIMINARIES AND SETTING
A molecular system of N atoms, in thermal equilibrium at temperature T. explores conformations ac- cording to the Boltzmann distribution μ, i.e. q ~ p(q) x (app-HH()/(kpppppp with q = {) e X' denoting the system's 's position and velocity, H the Hamiltonian, and kB the Boltzmann constant. We
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focus on proteins, whose contions-sulle local unfolding cryptic pocket formation arise from structural fluctuations. To identify these conformational changes and quantify their frequencies (free energy), we aim to sample structures from the Boltzmann distribution μ. This enables the measure- ment of observables & f defined on the phase space X'avia expectations [[]]]
Since direct sampling from j. is intractable, widespread alternative is to use Molecular Dynamies (MD). MD simulates a continuous- timme Markov process, designed to be ergodic w.r.t p. That is, for any observable, time averages over long trajectory approximate expectations under μ (Schütte et al., 2023). A common formulation of MD is via Langevin dynamic; given a system with potential energy U, position and velocity of an atom i with mass mi, are updated as follows
= Ei m&
- - - + oo,,
with y and a controlling friction and thermal noise. In practice, we discard the kinetic component and focus the trajectories of positions f - x=t).. While MD is powerful tool, its computational cost is a major bottleneck. Many biologically relevant events occur on millisecond timescales, while Langevin dynamics requires integration with time steps of the order of femtoseconds, making long sim- ulations prohibitively expensive. Crucially, meta-stable states of protein are often separated by high- energy barriers, leading to MD simulations wasting time trapped in deep local minima (Wales, 2005).
Problem Formulation. Given an initial protein conformation z(0), MD produces an ensemble of structures by integrating eq. (1) and saving frames at regular intervals. Since MD is expensive, we aim to use generative models to sample surrogate distributions at fraction of the cost. To enable generalization across proteins, a single model is trained using available MD trajectories from multiple sequences. At inference, we provide sequence and input conformation from an unseen protein, and the goal is to generate samples that are statistically indistinguishable from MD trajectories with respect to set of physical observables é-that is, the model should match the distribution over structural or ther- modynamic quantities that practitioners care about. Next, we review existing approaches that attempt to address this problem by sampling from fixed transition density and discuss their limitations.
MD EMULATORS AND THE CHALLENGES OF FIXED LAG TIME TRANSITIONS
key object describing MD sampling is the transition density y ~ p- (ylaa) associated with a lag time r, which represents the probability of state evolving to state y within time T. Concretely, this estimated from MD trajectories by examining future transitions xx( + +T) given an input frame (t). MD Emulators (MD-Emus) are class of conditional generative models trained to approximate p- (=((())t) from MD data (Fu et al., 2023; Klein et al., 2023; Schreiner et al., 2023; Nam et al., 2024; Diez et al., 2024; Costa al., 2024). Specifically, MD-Emus Iearn to generate future conformations y ~ pf (y) (t)) given (t). Common training approaches include Normalizing Flows (Rezende &
Mohamed, 2015; Chen et al., 2018), Flow Matching (Lipman al., 2023; Tong al., 2023; Albergo et al., 2023), or Score Matching (Ho et al., 2020; Song et al., 2021).
These models train neural network Go- condiiiiio sequence and an input conformation =0()-too match the empirical distribution of transitions observed in the MD data. Jing et al. (2024b) refined such an approach in MDGen, by generating the next K future conformations at once. Explicitly, MDGen learns to sample from the joint density y = (g/,... 8K) ~ Pr.k [y]z)) = II. Pr (1++1 (1)). conditioned on yo x. In the following, we assume we are given an MD-Emu that is trained approximate Pr.K.
3.1 LIMITATIONS OF MD-EMUS
Training. One of the key challenges in learning from MD trajectories is the data imbalance between frequent but uninformative transitions in an energy minima, and rare but key transitions across differ- ent minima (Wales, 2005). To make things more concrete, consider the case where a protein MDD trajec- tory visits two (or more) macroscopic states SA and SB, such as folded and unfolded conformations, separated by high-energy barriers. Transitions SA + 5B are rare but critical as they reveal conforma- tional flexibility and functional dynamics. However in any given trajectory interval of length Kr, the system is far more likely to remain in single state, than to cross an energy barrier. As a result, training batches for MD-Emus tions. This data imbalance limits the model's 's ability to learn conditional distributions describing mean- ingful state changes. For example, to learn how to transition from folded unfolded states, the model
typically dominated by high-frequency, L owwifformation intra-state transi-
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must observe actual transitions SA x(t) + x(t+ + KT) E Sg in the data-but such events rare. This sparsity constrains the diversity and efficiency of training data and affects generalization, as MD- Emus learn to replicate fine-grained temporal resolution rather than large conformational changes.
Inference. MD-Emus need to be used autoregressively to generate long trajectories. This may lead to error accumulation, causing samples progressively drift away from the data manifold. Nam et al. (2024) introduced a refinement step to address this, yet this comes at the expense of training second network. Besides, using a larger K does not fully resolve the problem, as the number of required samples often exceeds the maximum window feasible during training for larger proteins.
The challenges faced by MD-Emmus stem from fixed lag time transitions often being uninformative and failing to capture meaningful conformational changes across different energy minima. To overcome these limitations, we first need to apply a coarse-grained representation to MD data so as to discard high-freet freuuerryyenc information. To this aim, we rely on Markov State Models (MSMs) and introduce novel class of generative flows to learn transitions directly among states (rather than frames). The key advantage is that, once trajectories have been clustered into metastable states, we are no longer restricted to training only on frame pairs of the form 4t + Tt+-r obtained by slicing the time series at a fixed resolution. Instead, any pair of frames assigned to successive MSM states can be used for training, greatly increasing the number and diversity of informative transitions.
4 THE NEW CLASS OF MSM EMULATORS
Markov State Models (MSMs) provide a coarse-grained representation that removes figeunrryyency time signal while still capturing long-timm tingtiimeccale statistics (Noé et al., 2009; Pande et al., 2010; Prinz et al., 1; Husic & Pande, 2018). In MSMs, frames x(t) are assigned to discrete states S S'M and the dynamics is modeled Markov chain over these states. Given an interval 7, estimates a transition matrix T, where Tiy is the probability of transitioning from S; to Sj within time T:
Cjj
Tú
=== (x((t) E S x(1 (((++) E S,)
Explicitly, that means we consider a probability density pT (-|xr((1) satisfying
S pT (vlz()))d -TTy Va(() E St i (1, (1....,,M) M).
(3)
s,
In its simplest form, MSMs assumes a uniform density within each state. That is, given z(1) E S and y Sj. pr (y) depends only the identity of the clusters, but not the specific conforma- tions. This coarse-graining enables robust estimation of ann-timesccale dynamics, even from limited transition data. In principle, one could initialize short MD simulations (of length r) across discrete states {S.). ). and propagate their dynamics via T. However, sampling from the states of an underlying (unseen) MSM remains a challenge. We propose class of generative models that addresses this issue.
Definition 1 MSM Emulators (MSM-Emus) are generative models that are trained to sumple conformations from pr()(z(t))), where T is the Markov chain transition matrix of a given MSM.
For frame x(t) Sj. the MSM defines a categorical distribution over successor states + T Ti-- We interpret pT( (t)) as the resulting mixture distribution over MD frames: one first samples a destination state S, according to T. and then samples a conformation from the empirical ensemble of MD frames assigned to Sy. MSM-Emus are thus trained to draw conformations from this MSM- induced mixture, rather than to learn the exact frame-to- to-frame MD transition density. By matching the distribution PT( pT(-|ar(()) (a((()))thh) that depends only on the MSM Midies-MM--Emaa bypass many of the data scareity and imbalance issues that affect MD-FEmus $3.
Increased sample diversity. MSM-Emul decouple the learning objective from the specific tran- sitions observed in the simulation. Rather than learning frame-to-frame transitions æx() + xrt + T). they learn state-to-state transitions S, - S,, enabling broader generalization (as validated in $5). Unlike MD-Emus, which attempt to replicate the exact temporal dynamics at fixed lag time т, MSM-Emus focus capturing macroscopic transitions identified by the MSM. In the scenario described in $3, MSM-EMSMM -Euus learn to generate transitions SA e x + y E Sn from diverse starting points &: E SA beyond those explicitly visited by the MD simulation. Crucially, the kinetics defined
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by the underlying MSM remains intact: during training. transitions are sampled from the distribution T, ensuring that the global dynamical behavior is faithfully preserved.
Fast exploration. At inference. MSM-Emus can explore the energy landscape by generating samples y ~ Pr (0)), without being bound by fine-grained temporal dynamics. This significantly improves sampling efficiency (as visualized in Figure 1 and validated in $5). Additionally, as many conformations can now be generated in paraltel, MSM-Emus reduce compounding errors due to autoregressive calls. While MSM-Emus are designed to preserve the coarse-grained kinetics encoded by the MSM (e.g., metastable-state populations and transition probabilities), they are not intended to faithfully reproduce all fin-grrinn dynamical observables of the MD trajectories, at the same level of accuracy as fixed-lag MD-Emus. By construction, training on state- so-stte jump transitions prioritizes long-timescale behavior over the detailed intra-state time ordering. In this sense, MSM- Emus are complementary to MDD-Emus: the former emphasize efficient exploration and sin-timesccal thermodyyammics//inetic whereas the latter more directly track the local, frame-to- ta-m-to-framm dynamics (see Appendix C.2 for concrete hybrid instantiation and torsion-angle decorrelation analysis).
4.1 A REPRESENTATIVE FRAMEWORK: MARKOV SPACE FLOW MATCHING
We introduce MARKOV SPACE FL.W MATCHING (MARS-FM), a novel representative framework of the MSM-Emu class. First, we describe concrete MSM instantiations defined over MD) trajectories.
Constructing MSMs. Specific versions of MARS-FM are instantiated by first building an MSM. To this aim, we use standard tools (Scherer et al., 2015; Hoffmann et al., 2021). The states {St} are defined in a lower-dimensional space of collective variables. A common approach for dimensionality reduction is Time-lagged Indepen- dent Component Analysis (TICA) HPérez-Hernández et al., 2013; Wu et al., 2017), which identifies directions W, that maximize the autocorrelation of wJ [1) at lag time r. Alternatively, clustering can be applied directly to observables such as radius of gyration and fraction of secondary structures (Figure
MSM Badlus Gyration Practions of structures
Clutte Centre
14.
0.--
Strnnday weneeeemmmmm....mmm
2). Given states, the Markov chain transition matrix is estimated at a lag time T.
0:33
0:31
Two key observations are worth emphasizing. First, MSM construction is carried out once as an offline pre re-processing step: for each dataset we fix the MSM hyperparameters and then build a separate MSM for every training domain from its MD trajectories. These per-domain state spaces and transition matrices remain fixed throughout training, ensuring minimal computational overhead. Second, since MARS- FM learns to interpolate between MSM states beyond the transitions directly observed in MD, we can afford to select larger T than MD-Emus without sacrificing the data availability. The lag time still sets the coarse-grainet timescale of the underlying Markov chain, but the number and diversity of effective training transitions is no longer tied to how often frames are saved in the MD trajectories: any pair of frames assigned to successive MSM states can be used to define training pair.
1.5
3.5
4.0
Radlas gyeztion (ox))
Figure 2: Clustering of MD conformations of protein 3ma5A00 from MD-Cath. We report cluster centres and Markov chain transitions from one representative state. We note how MSM states capture large structural differ- ences (folded vs unfolded).
We emphasize that MSMs are defined as pre-processing of training data; importantly, no MSM information is provided during inference. Additional details on how construct MSMs are reported in Appendix B.
Representation. To generate protein conformations, MARS-FM adopts the same SE(3) represen- tation as (Jumper et al., 2021; Jing et al., 2024b). Each residue of frame ax()) is represented as
x (t) = (o' r' (t). (cos(f( (t)), sin (8% (t)) 1):
where (t) E R$ is unit quaternion describing rotation and (t) R is translation, while (t(()) represent the & torsion angles. Given an input z(t), the target conformation y is represented as an offset in aoo-translational space. To model vector field by transporting a source distribution the
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target distribution pr (-]x(t)), we build the MDGen architecture (Jing al., 2024b). Specifically, Vg leverages DiT-style blocks (Peebles & Xie, 2023) and is conditioned on protein sequence a and the current conformation xxt) via IPA layers (Jumper et al., 202100222222 see Appendix A for a complete description. For notational simplicity, we omit the explicit conditioning on the protein sequence a.
Training objective. We train MARS-FM using Flow Matching, which learns time-dependent vector field ti to transport a source distribution Po = (N 1) to the target distribution pi (-(x(())) - (+)) Concretely, given an input frame xx()) E S,, we first draw destination state Sj according to the MSM transition probabilities j + Tij and then choose a target MD frame among those assigned Sy. practice we sample uniformly from the finite set of MD frames in Sy. We denote the resulting target frame by æ1.
Independently, we draw a noise sample Io ~> po and define interpolations [0, 1] 9 s + I. connecting To to 21 using the standard schedule of Flow Matching (see Appendix A.2 for the explicit formula for v, and its analytic velocity d. We optimize vy by minimizing the velocity mismatch along the conditional paths + P- 2o, æ1) induced by these interpolations (Lipman et al., 2023; Liu et al., 2022; Albergo & Sanden-Eijnden, 2022). That is, we consider the training loss
LMARS-FM (0) (M- (001)))0--...0000000000-0--- pooop (o)) (s,
fell?
(5)
where i is the velocity of the conditional path. Crucially, we first select each state S; uniformly (see Appendix A) before conditioning specific frame xtt) E S. This state-based sampling ensures that rare states are encountered more frequently during training. In contrast, standard MD-Emus draw frames uniformly from the trajectory, which often biases training toward common intra-state transitions. A similar strategy was used by Costa et al. (2024), though their model still approximates fixed lag time transition. Further implementation and batching details are provided in Appendix A.
Compared to MDGen, which always takes 21 to be a future frame (tt+ + KT) from the same trajectory, MARS-FM samples 21 via the MSM transition kernel T, that the same Flow Matching objective is applied to a more diverse, state-conditionel set of training pairs. In particular, the model is not trained solely on transitions of the form ztt + Kr): the MSM provides transitions between all pairs of frames assigned to successive metastable states. Because each metastable state aggregates frames from different replicas and from multiple visits the same basin within trajectory, this construction exposes MARS-FF FM to transitions that span replicas and multiple barrier-crossigg events, even if only few such events occur in any individual time series. The lag time used to define the MSM still determines the coarse- graiind timescale of these moves, but the effective training set is no longer restricted to the specific at(t) att + 7) pairs obtained at fixed frame-saving interval. While these transitions still implicitly correspond to characteristic timescale set by the MSM lag time, the number and diversity of transitions available for training are no longer constrained by the temporal resolution used to slice the trajectories.
Sampling. Following standard evaluation for MD-Emus, we assume that at inference, we have se- quence and input conformation z0) from unseen protein. We explore two sampling strategies. MARS-FM @ MD-Emu. 1. Tree Sampling. We apply MARS-FM following a hierarchical sampling
scheme (Figure 3). We first generate n frames (yI....,,.. 3n ~ PT pT(: (=))) in parallel. Next, we generate frames from pT( (-(9)) for each yi- We continue extending the depth of the tree-sampling scheme based on our sample budget. We combine MARS-FI with MDGen (Jing et al.. 2024b). We first sample conformations using MARS-FM and then use MDGen to generate shorter trajectories from each of these points, separately. This approach mirrors the MSM paradigm, where statisties are inferred by initiating short simulations from different states. We introduce such hybrid scheme, in certain workflows temporal fidelity may also be relevant (De Vivo et al., 2016).
Figure 3: Hierarchical sampling used for MARS-FM.
For both cases, most conformations can be generated in parallel hence reducing autoregressive sampling. Crucially, as samples of MARS-FM are decoupled from temporal dynamics, they can explore the target distribution more efficiently (see Figure 5).

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5 EXPERIMENTS
A key condition for generative model to replace MD-sampling, that measurements of observables under the generated samples match those under MD. For this reason, our experiments, we compare dis- tribution of observables computed along MD trajec- tories and those computed along conformations gen- erated by the underlying model. We rely observ- ables used in previous generative works (Jing et al., 2024a;b) well additional ones reported for an- alyzing the MD-Cath dataset (Mirarchi et al., 2024).
MD
MDG
MarS
MarS MDGen
ADVO
Variants of MARS-FM. We first construct per-domain MSMs with dataset-specific hyperparametters- and then train MARS-FM to sample from pT())x(t)). Yet no information about MSMs is provided during inference. As explained in $4, during sampling, we consider two variants: MARS-FM ( MDGen and MARS-FM.
Figure TICA plot for random peptides in the test set, comparing MD ground-truth, MDGen, MARS (ours) and MARS MDGen (ours). Our frameworks explore modes that are otherwise entirely ignored by MDGen. Similar plots are reported in Appendix C.
Baselines. In our experiments, we take MDGen (Jing et al., 2024b) as the main representative of the MD-Emu class. Primarily, this is due to MDGen being the most versatile variant of MD-Emus due to the choice of the window of consecutive transitions at resolution т. For larger systems, additionally compare against BioEmu (Lewis et al., 2024), noting specific caveats discussed in Appendix B.7. Finally, we point out that other methods are either not scalable for MD-Cath domain (Klein al., 2023; Schreiner et al., 2023) have only been validated within the training distribution (Costa et al., 2024) (see Appendix B.7 for further discussion).
5.1 PRELIMINARY INVESTIGATION, TETRAPEPTIDES
We first adopt the dataset of tetrapeptides that provided the main evaluation for MDGen (Jing et al., 2024b). This consists of ~3000 trajectories training, 100 in validation and 100 test. Each trajectory is simulated in explicit solvent for 100 ns and conformations are saved every 10 That is, for any tetrapeptide in the test set, we have a held-out MD distribution of 10 frames. Accordingly, we generate distributions of 10a frames using both MD-Fimu (MDGen) and MSM-Emus (MARS-FM)
When training MARS- FM, we need define MSMs over the MD trajectories. As this dataset contains exhaustive simulations relative the size of the systems, rely TICA Pérrzz-Hrnández et al., 2013), selecting the minimal number of TICA coordinates whose cumulative kinetic variance exceeds 95%. We then apply k-means clustering to these coordinates, obtaining 100 microstates, which we further group into 10 metastable states using the PCCA+ spectral clustering method (Röblitz & Weber, 2013). This procedure yields the final MSMs constructed with lag time of 100 ps. Further details on MSM construction are provided in Appendix B.
MSM construction.
Evaluation and metrics. During sampling, report the Jensen-Shannon Divergence (JSD) different observables. Namely, we follow the same metrics as in MDGen Jing et al. (2024b): (i) Torsional angles (of both backbone and sidechains); (ii) Projection onto the slowest modes as identified by TICA; (iii) Equilibrium distributions induced by MSMs built over the test peptides. Inspired by Lewis et al. (2024), we also report the Macrostate MAE (mMAE), which measures the mean absolute error between model-generated and ground-trute MD free energies across metastable
Table JSD J. between sampled and ground-trut truth distributions for Tetrapeptides. Results based on sampled trajectories of 1 conformations compared to ground-trui distributions.
Teesions (bb)
Torsions (sc)
Torsions (all)
TICA-0
TICA-0.1 joint
MSM staties
Macrostate MAE
MD
0.06
MDGen-IOOW 1000
0.13
0.09
0.11
0.23
0.32
0.23
1.13
MDGea-200
0.14
0.10
0.12
0.24
0.33
0.27
L.12
MARS-FM MIXGen-211 (ours)
0.09
0.23
MARS-FM (ours)
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Table 2: Pearson 1 for RMSD and RMSF, forward KL divergence 4, JSD of gyration radius, fraction of secondary structures, and MSMs distribution, and folding free energy MAE (kcal/mol) 4- Results based sampled trajectories of 500 conformations compared to ground-trult distributions and averaged over inference runs.
Pairwise RMSD
Global RMSP r
Per target RMSP
Gyration Radius
Gyration Radius
Secondary Structures
Secondary Structures
MSM JSD
AGaa
KL
JSD
KL
JSD
MAE
(Oracle)
0.012
.008
.00
a:
MDGGcs-100
0.42 a: om7
0.49 .088
0.64 +. 008
1.13 a. 005
0.15 a oce
1.17 a oss
0.18 006
0.280 005
1.21 4: 210
MDGen-30
0.40 1
0:38 1.013
0.38 013
0.80 a 2.000
0.20 ore
1.85
0.30 n00
0.43 000
1.44 1. 1.000
MDGaa-100 (in parallel)
0,12
MDGes-20 (in panallel)
0.43
2.000
0.77 +: 006
2:71 003
001
2.96 =
0.28 001
0.46 007
3,68 1:
BioEmu
0.25
0:41 .094
0.66 001
3.83 0.0
0.40 .01
4.17 015
0.41 0011
0.51 000
4.67 0044
MARS-FM MDCGen-20
0.88
MARS-FM
0.65 .011
001
0.55
0.10
0.93 000
0.13 001
1.05 1
states. The mMAE provides direct measure of the model's ' accuracy in reproducing metastable state distributions. Further computational details and definitions are presented Appendix B.
Results. For the baselines, we train MDGen with window size K 200 and K 1000, and use the former combined with MARS-FM as per our hyhrid-scheme described in $4. Results in Table illustrate that MARS-FM achieves same or better performance than baselines. In particular, errors are often comparable with those encountered by different replicas of MD trajectories (oracle performance). Noticeably, large improvement is achieved on the Macrostate MAE, highlighting that MARS-FM can better sample from rare meta-stable states of the underlying TICA space of unseen peptides. This is also confirmed in Figure 4 additional TICA plots reported in Appendix C. In general though, we expect MD-Emus and MSM-Emus to be comparable for this dataset, as have very limited chemical diversity and large domain motion due to the size of the systems.
5.2 MD-CATH: PROTEINS WITH SEQUENCE DISSIMILARITY AND LARGE DOMAIN MOTIONS
Our main evaluation is based on the MD-Cath dataset from Mirarchi et al. (2024). This includes 5398 domains up 500 residues (average number of residues 137), simulated at 5 different temperatures, spanning 5 random replicas per setting. The average length each trajectory is 464 ns and frames saved at 1 ns intervals. The dataset is already built to ensure protein diversity by considering non- homologous domains at the 520 (20%) homology level-further details are reported Appendix B.
MSM construction. MSMs are constructed independently for each domain using k-means clustering directly applied to normalized observables identified as critical by the original dataset analysis (Mirar- chi et al., 2024). Specifically, residues in a- a-helical and 5-sheet secondary structures additional details are reported in Appendix B. We cluster these normalized features into 10 metastable states, providing physically meaningful par- titioning of the conformational space. This yields the final MSMs constructed with lag time of 50 ns.
consider features: (i) the radius of gyration and (ii) the fraction of
Starting Pruze NEDGE Tremee Dranse
MDGon
Mars
2.5
Figure 5: First samples generated by MDGen and MARS-FM for the domain 2ynmD03 in the test set. As MARS-FM interpolates among states independently of temporal dynamics, it can explore the energy landscape more efficiently. In fact, the secondary structure content varies significantly among these samples (note that there is no ordering as they are generated in parallel). Conversely, MDGen samples all belong to the same energy minimum which reduced sampling efficiency and exploration.
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Table 3: Pearson 1 for RMSD and RMSF, forward KL. divergence + and JSD of gyration radius, fraction of secondary structures, MSMs distribution, and folding free energy MAE (kcal/mol) 4. Results based on sampled trajectories of 100 / 1000 conformations compared to ground-truth distribu- tions and averaged over inference runs.
Pairwise RMSD
Global RMSF
Per target
Gyration Radius
Gyration RaJius
Secondary Structures
Secondary Structures JSD
MSM
A MAR
RMSF
KL
ISD
KL
JSD
MD (Oracle)
0
0.77/0.92
2.39/
0.08/0055
2.75/0.26
0.49 0.22/0.05
/.80
MIXGcen-100 100
0.34/00 /..28
0.46/
0.60/ 0.46
3.66 /.77
/D.14
3.601449 1.49
0.29/0.26
0.51 70.27
2.58/ 1.52
MDGGen-20
0.57 7 10.28
0.62. 0.223
0.61 /0.20
2.4//133
0.22/0.33
2.08/2.34
0.19/0.39
0.48/0.55 70.51
1.52/2.01
MDG parallet)
0.62/0.71 0.56
3.62/188
0:31/0.21
3.51/1554
0.28/0.10
0.52/0.38
2.68/2.12
MDXGen-20 (in porallel)
0,41 70.43
0.53 0.55
0.67 0.71
3.56 2.50
0.30/
3.77
0.31/0.27
0.53
3/771
BioEmu
0,40 00.26
0.42
0.64 0.66
4.75 43.55
5.08/3.91 0.33/0.39
0,44/0.41
0..4/
4.82
MARS-FM MIXCeen-U
0.59 (166
0.71
0.79/0.84
1.99 /0.7
0.20/
1,85 0.66
6.18/0.11
0.43
1,38/1022
MARS-FM
0066
0.68 0.71
0.84/ 0.90
1.74 (1.42
0.18 0.09
1.9//.95
0.18/0.14
0.41/0.17
1,25 1.20
Large domain motions. To validate the ability of MARS-FM to generate large conformational changes, our main evaluation is performed over the Seehetttemperature replica (450 K), in which proteins exhibit unfolding. In fact, proteins are quite stable at lower temperatures over the simulated timescales, meaning that often no significant conformational change has occurred- as reported in Mirarchi et al. (2024)which would result in an casier generative task. Nonetheless, for a more complete evaluation, we also train MARS-FN on lowest-eemperature replica and sample conformations in this regime-results are reported in Appendix C.
Assessing generalization. First, we consider random traiiin--validttoon the total number of domains according to 10%-00%10%% We then follow the strategy in Lewis et al. (2024) and use MMSEQS2 (Steinegger & Söding, 2017) with highest sensitivity setting to filter out any test protein sharing more than 20% sequence similarity with any training protein (note that Lewis et al. (2024) use more forgiving 40% threshold). As result, our test set consists of 495 domains that are meaningfully diverse from those whose trajectories have been used during training.
split partitioning
Metries and Observables. For evaluation, we focus on observables studied in the original dataset analysis of Mirarchi et al. (2024), namely the radius of gyration and fractions of residues in secondary structures (a-helices and (3-sheets). We measure the forward KL divergence to quantify the model's exploration capabilities and the LemsenSShanoop Divergence (ISD) for general distribution alignment. Inspired by Jing et al. (2024a), we also quantify ensemble flexibility using the Pearson correlation (r) computed on pairwise backbone Root Mean Square Deviation (RMSD), global Root Mean Square Fluctuation (RMSF), and per-target RMSF. We additionally evaluate folding free energies via the mean absolute error (AGr MAE). Lastly, we calculate the JSD between reconstructed MSMs and reference MSMs, providing a direct measure of the ability to explore states of domains dissimilar from those observed in training. Additional details on metrics are provided in Appendix B.
Extended baselines. We benchmark different variants of MDGen to highlight how advantages of our framework cannot be simply replicated by changing the lag time altering the sampling approach. As such, we train MDGen with window size K = 20 and K = 100, corresponding to total lag time of 20 ns and 100 ns, respectively. Additionally, to demonstrate that the improvements of MARS-FM cannot be simply ascribed to reduction of autoregressive calls, we also evaluate MDGen in parallel, meaning that we sample conformations only conditioned on the input frame. Finally, we also report the performance of one MD replica against 4 held-out ones (and average over all possible 5 combinations) to be treated as oracle performance.
Results. We report our evaluation in Table 2. Both hybrid sampling (MARS-FM + MDGen) and hierarchical sampling (MARS-FM) significantly improve over all extended MDGen baselines, often by a large margin. This confirms that our framework can extrapolate over unseen large conformational changes better than MD-Emuus Crucially, MARS-FM can even match the oracle performance in the reconstruction of the unseen underlying MSM. This validates how the framework is capable to generalize across unseen MSMs and points to such a signal being easier and more robust than the one derived from fixed-lag time transitions. As all test proteins share no more than 20 % sequence similarity to any training protein, we again emphasize that the results in Table 2 provide a stringent test for the generative models to sample conformations over meaningfully different domains.
Changing sample budgets. Finally, we report results across different sample budget in Table 3. Namely, we consider the same metrics as above but compare distributions obtained by gener-
Ablation:

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Published as a conference paper at ICLR 2026
ating 100 and 1000 samples, respectively. In the low-sample regime, MARS-FN, also surpasses the oracle performance as given by the first 100 frames (at ns resolution) sampled by MD. This con- firms that by decoupling the training objective from temporal dynamics, MARS-FM can sample large conformational changes much more efficiently as the model interpolates directly across states rather than across frames separated by time interval. We illustrate this phenomenon in Figure 5. Our frame- work also offers optimal performances at higher sapling regimes, as it significantly reduces autore- gressive calls hence mitigating compounding error effects. Overall, the results in Table confirm that MARS-FM improves upon the baselines across different sample budgets due to (i) Its ability to sample large-domain transitions independent of temporal ordering; (ii) Reduced autoregressive sampling.
Sampling speed. A key requirement for generative models to serve as practi- substantial computational speed-ups.
cal replacements for MD is that they offer Table 4 summarizes the wall-clock time required to gen- erate the equivalent of 500 ns of trajectory (500 confor- mations) for a 159-residue protein (domain 4dhkB00) an NVIDIA H100. For a fair comparison, we report MD in implicit solvent, which is substantially more efficient than explicit- solvent simulations. For such a system, im- plicit MD runs at 2400 ns/day the same hardware for a 1 159-residue protein (OpenMM Team, 2025). Under these conditions, MARS-FM provides computational speed-up of 600x.
Table 4:
Wall-clock time (in seconds, 4) to generate the equivalent of 500 ns trajectory (500 conformations) for 159- residue protein.
Method
Time (s)
MD (implicit solvent)
Rs 18,000 (- 5h)
MDGen-100
31.70 0.21
MDGea-20
100.87 : 2.21
MARS- FM è MDGen-20
19.79 1. 0.04
MARS-FM
30.34 0.08
6 CONCLUSIONS
Related works Beyond MD-Emus, our work is related to the class of Boltzmann Generators (BGs), normalizing flows that are trained via the potential energy (Noé et al., 2019; Wirnsberger et al., 2020; Köhler et al., 2021; Rizzi et al., 2021; Garcia Satorras et al., 2021; Rizzi et al., 2023; Midgley et al., 2023; Klein & Noe, 2024; Tan et al., 2025). However, BGs have shown limited scalability beyond small peptides. Alternatively, Jing et al. (2024a); Lewis et al. (2024) proposed requence-fo-struccue generative flows trained on MD data, after ignoring any temporal ordering. These methods though suffer from the same data inbalance issues affecting MD-Emus. Learning to sample diverse protein conformations was also investigated in (Jing et al., 2023; Zheng et al., 2024; Lu et al., 2024) and is related to works that perturbed AlphaFold at the MSA level (Del Alamo et al., 2022; Wayment-Steele et al., 2024) or refined its predictions via experiments (Maddipatla et al., 2025). However, these works may fail to attain quantitative distributional matching. Finally, MSMs have been used, for single peptide dataset, to reweigh samples when fine-tuning a generative model in Lewis et al. (2024).
L Limitations and Future Works. In general, public MD data of protein dynamics is limited, particularly for long unbiased simulations. This generally hinders the development of generative models-a key reason as to why to assess large domain motions with sufficient chemical diversity we had to focus on higher-temperature simulations. In terms of scope, in this work we have focused on protein representation. Natural next steps would entail extending MSM-Emus to complexes, for example protein-ligane ones, and inorganic systems. MARS- FM requires an input 3D structure at inference. To further accelerate sampling and increase applicability of the framework, we will study how to leverage recent sequence-to-stee to-suuccuree models to be able to generate protein conformations starting from sequence only. Finally, would be interesting to explore whether MARS- can be combined with MD simulations, for example by sampling different input conformations across an underlying MSM, and then initializing shorter MD simulations in parallel.
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ACKNOWLEDGMENTS
The authors would like to thank Daniel Cutting, Berton Earnshaw, Nikhil Shenoy, and Stephan Thaler for their helpful discussions, input, and feedback on this work. The authors also thank Lars Holdijk for his assistance with high-performance computing resources and support.
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OUTLINE OF APPENDIX
In Section A we describe the MARS-FF -M architecture, and our MSM-informed procedure for drawing training pairs; Section B reports dataset details, supplementary MSM construction choices, evaluation metrics, nyper-parameters, sampling settings, and compute resources; Section C gathers supple- mentary figures and tables-including tetrapeptide TICA plots, MD-Cath samples plots, expanded MD-Cath flexibility metrics, and the full 320K evaluation that complement the main text; Section D summarizes usage of LLMs; finally, Section reflects on the socictal benefits and potential misuse risks of generative models such as MARS-FM FM and Section F contains reproducibility statement.
A IMPLEMENTATION DETAILS
A.1 ARCHITECTURE
Inspired by MDGEN (Jing al., 2024b), the velocity network eg (R²))A (R21)1, with being the overall number of residues, employs modified DiT blocks (Jing et al., 2024b: Peebles & Xie, 2023) and Invariant Point Attention (IPA) layers (Jumper et al., 2021). A pseudocode is given in
Algorithm
Algorithm 1 Velocity Network
Require: conditioning tokens x E R211, target tokens x!" R2!!, conditioning roto ottotrranslations
g00) (3) amino acid identities a, flow matching time ~ 2((0) Ensure: velocity U E 22IL
+ TIMEEMBED(s)
2: I <- AMINOACIDEMBED(O)
to T4PA do 4: L +- ILW.RIANPPONTAATEEIIONNNN (0)
3: for
5: I + r' + LINEAR(X + LINEAR(X!")
6: for l to RIDTA do
7: + TAPFUSOOTTAASSOOMEEEEETTTTTTTTT
8: return WWTTUSOONTTTANSFFREEEEEELLLLLEE
At flow-matching time [0, 1]. we construct the noisy representation linear combination of the ground -tutt frame and i.i.d. Gaussian noise E:
( ((())++((s)x+
a(s) = sin(
(s)) cos( 3)
The derivatives d(s) = I co and a(s) = sin ( enter the target velocity (1) during
training.
A.2 DRAWING TRAINING PAIRS
Drawing Procedure. Given per-protein Markov State Model (MSM), for each training example we draw pair (Zo. 21), as follows,
Select source states. We always include the state containing the first frame of the trajectory. The remaining source states are sampled uniformly (with replacement) from all Markov states.
Draw destination states. For each source state S,, we sample fixed number of destination states S, independently using the MSM transition probabilities T.₃ categorical distribution.
Select frames. For each source-destination pair (S,, S,), we uniformly sample one frame =(t) S; as the conditioning frame and one frame Iı Sj as the target frame. These form the training pair 21). This step can be repeated a few times
This strategy balances sampling between and frequent metastable states while preserving the transition structure encoded in the MSM. Our sampling method allows to sample across different replicas.
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B EXPERIMENTAL DETAILS
B.I DATASETS
Tetrapeptides We use the tetrapeptides dataset introduced in Jing et al. (2024b), where trajectories are trained at a resolution of 10 ps over a total duration of 100 ns. We adopt the exact same split as introduced by the authors, comprising 3,109 tetrapeptides for training, 100 for validation, and 100 for testing.
MD-Cath We use the MD-Cath dataset introduced in Mirarchi et al. (2024), which provides molecular dynamics simulations for 5,398 protein domains. Frames are trained at a resolution of ns, with average trajectory length of 464 ns per replica and the majority of trajectories being 500 ns long. Each domain (ranging from 50 to 500 residues, average 137) is simulated at five temperatures from 320 K to 450 K, with five replicas per temperature, yielding 25 trajectories per domain. For the train/validationnesst split, we follow the similar strategy to Lewis et al. (2024) (but more strict) and MMSEQS2 (Steinegger & Söding, 2017) with default parameters and maximum sensitivity to exclude from the test set any domain that shares more than 20% sequence identity with any training or validation domain. This results in final split of 4,304 domains for training, 538 for validation. and 495 for testing. We primarily evaluate our models on the highest-lemperature subset (450 K) of MD-Cath, where most proteins undergo (partial) unfolding within 100-500 ns. In contrast, lower temperature simulations (e.g., at 320 K) often remain near the native structure and exhibit limited conformational diversity- for quantitative comparison see Appendix C. We use all available replicas for the given temperature.
B.2 ADDITIONAL DETAILS ON MARKOV STATE MODEL CONSTRUCTION
We follow the MSM construction described in Section 5. All MSMs are implemented using the deeptime library (Hoffmann et al., 2021). The resulting transition probability matrices are symmetrized post hoc as P (P + PPPPTT )/2.
For clustering of MD-Cath dataset we use standardized Radius of Gyration and Secondary Structures Fractions defined in the Appendix B.3.
B.3 EVALUATIONS AND METRICS
B.3.1 TETRAPEPTIDES
Evaluation metrics. We follow the evaluation protocol of Jing et al. (2024b) and compute the Jensen-Shannol Divergence (JSD) between monel-generated and reference MD trajectories across several observables: (i) distributions torsional angles, (ii) TICAs, and (iii) equilibrium distributions derived from Markov State Models.
Macrostate Mean Absolute Error (mMAE). To complement the evaluation and JSD-based observables, we additionally report the mMAE. The mMAE measures the discrepancy in metastable- state populations between the generated and reference ensembles, after both are coarse- grained into macrostates using the MSM calculated as described in Section 5. is computed as the mean absolute difference in free energy across macrostates,
mMAE
Gmmulel G where G-=kk n,
Here, G; denotes the free energy of macrostate kg is the Boltzmann constant, the temperature, and is the stationary distribution (normalized histogram count) of macrostate i. We use small floor value (c.g., 10 10-4) to avoid numerical instability in the logarithm. Since we do not have access to additional replicas, we did not report the mMAR of the Oracle.
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B.3.2 MD-CATH
Predicting Flexibility. We follow the exact procedure of Jing et al. (2024a) compute pairwise Root Mean Square Deviation (RMSD), global Root Mean Square Fluctuation (RMSF), and per-target RMSF. These metrics capture the flexibility and structural variability of conformational ensembles.
Distribution Alignment Metries. To assess the similarity between model-generale and reference ensembles, we compute both the forward Seelbacck-eibbed (KI. divergence and the Jensen- Jensen-Shannon -hannno Divergence (JSD). Both distributions are estimated by binning values into histograms with 100 bins and normalizing the counts to obtain discrete probability distributions. To avoid numerical issues, we apply small floor value (e = 10- to the model probabilities before computing KL. Specifically:
The forward KL divergence, DKL ( Q), computed with smoothed model probabilities: Q = max(Q,c).
The
Divergence is computed as the squared JSD distance - scipy's jensenshannon function (Virtanen et al., 2020).
Radius of Gyration. We compute the radius of gyration for each frame in generated trajectory and compare its distribution against that of generated trajectories across all replicas given temperature. It is defined as,
E m; Xx Xeenter
Rg
(6)
Σ m
where m. and x, denote the mass and position of atom i, and Xcemer is the center of mass of the structure. Higher Ry values typically correspond more extended unfolded conformations.
Secondary Structure Fractions. We compute the fraction of residues in secondary structure per frame based on DSSP assignments. Specifically, we include canonical a-helix and 3-strand states, using r''s compute dssp function (McGibbon et al., 2015),
Jss T. Σ [s, e (H. G.I. B)),
where L. is the number of residues in frame, 8, is the DSSP-assigned secondary structure of residue j. and ll is the indicator function.
Markov State Model Recovery. We build MSMs using the same procedure for training (Sec- tion 5), and then we compare the resulting stationary distributions via Sensen-Shamme Divergence.
Folding Free Energies. To estimate folding free energies, we follow the BioEmu protocol (Lewis et al., 2024). For each protein, we first compute a native-contadd (FNC) score using all heavy- pairs with sequence separation | - jl >> 3 and reference distance diff < 10 A. For a given trajectory frame at time each pair contributes
(t) = [1 + exp(13
(8)
with 3 = 5 and X 1.2. The overall contact score is then defined Q(t) = (t)) averaged over all native contacts. To separate folded and unfolded ensembles, we determine a midpoint threshold Q1/2 from the kernel density estimate the 320 K MD reference distribution. Specifically, we locate the deepest minimum in the range 0.45-0.90; if no minimum is found, we set Q1/2 0.70. This same value then reused to evaluate both 450 K MD and all generative trajectories.
Given this threshold, we compute the folded probability of each frame as
10, and report its ensemble average Plots Finally, the folding free energy at temperature T is defined as
(t) [ + exp( [Q(t) Q1/22))
Prold
AG = -k3 In
(10)
Prold
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MD Oracle performance To quantify MD Oracle performance for 100 and 500 samples, we hold out one replica and compare it against a reference set containing the other four replicas. For 1000 samples, we hold out two replicas and use the remaining three the reference.
B.4 TRAINING DETAILS
We follow the apper-paammetrrs of Jing et al. (2024b) as closely as possible. Throughout all experiments use an exponential moving average with decay 0.999, and AdamW (Loshchilov & Hutter, 2017) optimizer with a learning rate of x 10-1 We use an architecture described in Appendix A. I with 5 transformer layers, 384-dimensional token embeddings. 16-head multi-head attention, and an IPA layers with 4 heads, 32-dimensional head size, and 8 query-key as well as 8 value points.
For the tetrapeptide dataset we recreate the setup of Jing et al. (2024b), sampling MDGEN-IO01 from the released checkpoint and training our MARS-FM model for 1000 epochs and MDGEN-200 for 2500 epochs, both with a batch size of 8.
On MD-Cath dataset, MARS-FM is run for 1000 epochs with batch size 8 while sampling training pairs from two source clusters (zo) and two corresponding destination clusters (x1) with 12 sequences per cluster; MDGen- 100 is trained for the same epoch budget with batch size 1, and MDGen-20 for 2500 epochs with batch size 4.
B.5 SAMPLING DETAILS
All trajectories are sampled from the first MD frame in the test set; for the MD-Cath data, we use the first frame of the first replica. MDGen is then applied autoregressively, where the last of generated frames at each rollout serves as the conditional input for the next rollout. For MDGen in parallel mode, all calls are conditioned the initial MD frame; we simply call it multiple times independently.
For MARS-FM @ MDGen, we first draw a set of total number of frames frames using MARS-FM in MDGen window size parallel. Each output of MARS-FM is then used as the conditional frame for independent MDGen rollout.
For hierarchical MARS-FM sampling, we begin from the initial frame and sample 200 first-layer children in parallel. Each of these nodes is then expanded once, and the process continues recursively until the required number of frames is reached.
B.6 COMPUTE RESOURCES
All preprocessing tasks (including TICA projection, Radius of Gyration calculation, Secondary Structures Fractions calculation, clustering), and evaluation metric computation (such as divergence measures, RMSD, and RMSF), are performed on CPU nodes of the compute cluster.
Training and sampling are conducted on a single NVIDIA H100 GPU. Training on the tetrapeptide dataset takes approximately 1-2 days, while training on the MD-Cath dataset requires around 4 6 days. For sampling, generating 10,000 conformations for the full tetrapeptides test set takes approximately 1.5 hours; for MD-CATH, generating 500 conformations for all test domains takes around 5 hours.
B.7 BASELINE CAVEATS
BioEmu For larger systems, we also benchmark against BioEmu (Lewis al., 2024). Several caveats should be noted when interpreting these results. First, BioEmu uses a backbone-only protein representation, whereas MARS-FM explicitly models both backbone and side-chain torsion angles. Second, BioEmu trained on substantially more data, including proprietary MD simulations, which makes direct comparisons less controlled. Third, BioEmu currently provides only an inference pipeline, with publicly available training code. Despite these limitations, we performed inference evaluations on MD-Cath using the pre-trained BioEmu model, generating 500 conformations for comparison with reference MD simulations.
Other Methods Several other works have aimed, similar to MDGen (Jing et al., 2024b), at speeding up molecular dynamics, but they exhibit important limitations, The Implicit Transfer Operator (ITO)
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(Schreiner et al., 2023) relies coarse- grained Ca representation and assumes fully connected graph with ((M²) scaling, making application to systems with thousands of atoms infeasible. Time Warp (Klein et al., 2023), based RealNVP and Transformer stack, incurs prohibitive training costs even for short peptides and has only been demonstrated on tetrapeptides. EquiJump (Costa al., 2024) does not provide publicly available code and has far only reported performance on the same fast-folding proteins used for training, without evidence of generalization.
C ADDITIONAL RESULTS
C.1 SUPPLEMENTARY TETRAPEPTIDES TICA PLOTS
MD
MIXGen
MarS
MarS MDGen
MD
MDGen
Marš
Marš MDGen
RTVD
EHEN
INEEC
LENY
EaiC
KAEP
INIYY
Figure 6: TICA plot for 10 random peptides in the test set, comparing MD ground-truth, MDGen, MARS (ours) and MARS MDGen (ours). Our frameworks explore modes that are otherwise entirely ignored by MDGen.
C.2 ADDITIONAL. DYNAMICAL. OBSERVABLES FOR TETRAPEPTIDES
To complement the thermodynamic metrics in the main text, report additional dynamical observ- ables on the 4AA tetrapeptide benchmark. Following MDGen (Jing et al., 2024b), we consider the normalized torsion autocorrelation
Co(At) (cos (0: - ((+2()))
and define relaxation time To as the lag at which (a(At) decays below 1/e (up to 10.s).
Figure 8 compares backbone and sidechain decorrelation curves for representative test peptides (ALDA. IWHF, KSIY, PGKM. SSNN) across MDGen-1000, MARS-FM. and the hybrid MARS- FM + MDGen-200 scheme. Figure 7 correlates model and MD relaxation times for all backbone and sidechain torsions in the test set.
Across all three models, ADGen-1000 unsurprisingly provides the closest match to MD torsion relaxation times, as explicitly trained to emulate fixed-lag MD transitions at 10 ps. con- trast, MARS-FM alone shows systematically shorter relaxation times and faster decay of Ca (At), reflecting its design as an MSM-EN MSM-Emu: it is trained to sample across MSM states rather than to reproduce fine- -nnignd torsional kinetics. The hybrid MARS-FM @MDGen-20 interpolates between these behaviors, recovering near-MD agreement Tr while retaining the improved exploration of metastable states and macrostate free energies highlighted in Tables and 2.
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Buckban
15
ND taricca insuatioon
MD sasioal vibuddic
MDGen-1000
MARS-FM
MARS FM #MDGen- 200
Figure 7: Torsion relaxation times for 4AA tetrapeptides. Scatter plots of model versus MD relaxation times 76 for backbone (blue) and sidechain (orange) torsions. Axes are logarithmic; the dashed line indicates x. Pearson correlations r for backbone and sidechain torsions are reported in the legend of each panel.
MDGen-IO0
MARS-FM
MARS-FF FM 4MDGen-200
Figure 8: Backbone and sidechain torsion decorrelation for 4AA tetrapeptides. Normalized autocorrelation Co(At) as function of lag time for backbone (left panels each image) and sidechain (right panels) torsions in five representative test peptides (rows: ALDA, IWHF, KSIY, PGKM. SSNN) and three models (columns: 10DGen-1000, MARS-FM, MARS-FM @MDGen-200), Solid lines denote MD reference trajectories, dashed lines the corresponding model trajectories.
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C.3 SUPPLEMENTARY PLOTS OF GENERATED SAMPLES FROM MD-CATH DOMAINS
20yqA07
MD Tegieben Hi0ee Finces
NDDOO
Mars
1.5
2qfyA00
Cropetanio VEDiien
MDG
Mark
035
2-5
4c23B01
MOOI
Mars
4egwA02
Trajeceraas
MDGee
m
Mars
0.0
4lmoA00
Strting Premn Murti Tremms
AUAGES
Mars
Figure 9: First 4 samples generated by MDGen and MARS-FM for 5 random domains in the MD- Cath test set.
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C.4 ABLATION ON HIERARCHICAL TREE SAMPLING
MARS-FM samples MD-Cath trajectories using hierarchical tree sampling ($5.2), where each node spawns N children parallel before the tree is further expanded. To study the impact of this branching factor, we fix the total sample budget to 500 conformations per domain on the 450 K MD- Cath test set ($5.2) and vary N E {10, 20, 50, 100, 500). We then evaluate the same structural and thermodynamic metrics as in Table 2.
Results are reported in Table 5. Very small trees (N 10) require more expansion depth to reach the target sample budget, increasing the number of autoregressive steps and leading to noticeable degradation in structural flexibility and folding free energies. Increasing N to intermediate values (N x 100-250) consistently improves RMSD/RMSF correlations and reduces divergences in radius of gyration and
distributions. However, pushing N too high (NV = 500) makes the tree very shallow: most samples are generated close to the initial frame, which harms state-space exploration and MSM reconstruction, as reflected by the higher MSM JSD and weaker RMSF correlations. Overall, intermediate branching factors around N 100-250 provide the best trade-off between limiting temporal error accumulation and accurately recovering the MSM and long- -igecile observables; our default choice of N 200 used in the main experiments lies in this regime.
Table 5: Effect of branching factor N in hierarchical MARS-FN tree sampling on the MD-Cath 450 test set. We report Pearson r 1 for RMSD and RMSF, forward KL. divergence 4. JSD 4 of gyration radius, secondary-structur fractions, and MSM distributions, and folding free energy MAE (kcal/mol) L. Results are based on 500 generated conformations per domain and averaged over inference runs.
Gysation Radius K1.
Gymitio Radius ISD
Secondary Structure KL
Secondary Stracture JSD
Pairwise RMSD
Global RMSF
Per target RMSE
MSM ISD
AGra
N 10
0.54 a .007
0.64 + 005
0.89 0002
0156 .001
0.13 001
2.12 .uss
0.24 s: .001
0.17 + 052
2:58 .001
N
0.62
0.69
0:90
0.51
0.11
1.51
f.18
0.18 +
1.73
0.65
0.71
0.90 0000
000
1.31
0.17
0.17 a. 003
2.005
N 100
0.65 + -002
0.71 +: 002
0.90 2001
0.52 a00
0,10 001
1.13 .007
f.15 + -001
0.15 : 002
1.25 =.008
N 250
0.65 a
0.71 1 002
0.89
000
0.57 1.400
0.10 00
0.87
0.12 1 .001
0.19
1 055
1.03 000
N
500
0.61
0.69
0.84
0.87
0.12
0.80
0.12
0.25
1.20
C.5 ABLATION ON MSM STATE CONSTRUCTION FOR MD-CATH 450 K
In the main MD-Cath experiments ($5.2), we construct per-domain MSMs by k-means clustering of normalized radius of gyration and eeoondayy-structur fractions into c 10 metastable states with lag time = 50 ns. To assess the sensitivity of MARS-FM to this choice, we perform an ablation over the MSM discretization on the 450 K test set, varying both the number of clusters e and the MSM lag time while keeping all other settings fixed.
Specifically, we retrain MARS-FM and the hybrid MARS-FM + MDGen-20 using MSMs built with (c, E {110, 50). (10, 100), (10, 20), (20, 50) (5, 50)}. where e denotes the number of k-means clusters and the lag time (in ns) used to estimate the transition matrix. For each configuration we generate 500 conformations per test domain. Results are reported in Table 6.
Overall, we observe that MARS-FM is robust to reasonable changes in the MSM discretization. For both MARS-FA and MARS-FM I MDGen-20, all (c, () settings yield very similar RMSD/RMSF correlations and divergences on radius of gyration and secondary- -ttuucueee distributions. Increasing the MSM lag time from 50 ns to 100 ns slightly improves MSM JSD and AGrola MAE, while shortening the lag to 20 ns degrades the gyration-radius KL and MSM JSD for both models. Varying the number of clusters at fixed lag time has only modest effects: coarse discretizations with 5 tend to worsen secondary sroonday-sttructure structure KL/JSD and MSM JSD, whereas e = 20 slightly improves MSM recovery without changing the overall conclusions. Taken together, these results indicate that the default MSM choice (c = 10, = 50 ns) used in the main text offers a good trade-off across metrics, and that our improvements over MD>-mmus do not rely on a finely tuned state construction.
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Table 6: Ablation on MSM discretization for MD-Cath 450 K. Pearson 1 for RMSD and RMSF; forward KL divergence & and JSD + of gyration radius, fraction of secondary structures, MSMs distribution, and folding free energy MAE (kcal/mol) 4. Results based on sampled trajectories of 500 conformations compared to ground-truth distributions. MSMs are labeled by the number of clusters c and lag time (e.g.. 10, lag 50 ns).
MSM (clusters, lag)
Model
Painwise
Giohal
Per target RMSF
Gyration Radius
Gyration Radius
Sccundary Stractures
Secondary Structures
MSM
AGau
KL
JSD
KL
JSD
10, log 50
MARS-FM @ MIXGen-20
0.63
0.69
0.83
0.98
0.13
0.73
0.11
0.24
1422
MARS-FM
0.65
0.71
0.55
0.10
0.93
0.13
0.19
1.05
10, log 100 us
MARS-FM MDGGn-20
0.69
0.73
0.85
0.70
0.12
0.74
0.12
0.20
0.91
MARS-FM
0.69
0.75
0.90
0.50
0.11
1,05
0.15
0.15
1.21
10, lag 20
MARS-FM MDGen-20
0.56
0.66
0.82
1.07
0.14
0.75
0.12
0.25
0.97
MARS-FM
0.98
c= 20, lag. 50 ns
MARS-FM @ MDGGen-20
0.64
0.69
0.83
0.94
0.13
0.76
0.12
0.24
0.9
MARS-FM
0.59
0.67
0.89
0.64
0.11
1,10
0,14
0.20
0.96
lag
MARS-FM MARS-FM
0.95
0.57
0.65
0.89
0.57
0.10
1.58
0.18
0.20
1.23
C.6 JSD FOR PAIRWISE RMSD AND GLOBAL RMSF, AND PEARSON r FOR FOLDING FREE ENERGIES
To further quantify structural flexibility and thermodynamic accuracy, we report the mean JSD on the test set between reference MD samples and 500 samples generated by our models and baselines, along with the Pearson correlation (r) for folding free energies (A (Aor)))). The results are presented in Table 7. For each protein, trajectories are aligned with respect to the first MD frame of the first replica before calculating the metrics. Both hybrid sampling (MARS-FF FMM + MDGen) and hierarchical sampling (MARS-FM more effectively capture the structural distribution of the trajectories and recover folding free energy correlations than the extended MDGEN baselines, often by a large margin.
Table 7: JSD & for pairwise RMSD and global RMSF, and Pearson r t for Folding Free Energies (AGrola). Results based on sampled trajectories of 500 conformations compared ground-trute distributions and averaged over 5 inference runs.
Pairwise RMSD ISD
Global RMSF JSD
AGios
MD (Oracle)
0.16 1. onss
0.09 1. o00
0.90 J. .002
MLXGen-10 100
.00s
000
1005
MDGen 20
0.38 1. ons
0.52 1 Loto
0.85 1. ons
MDGen- parallel)
0.54 :: 081
0.58 :: 000
0.83 4: 001
MDGen 20 (in parallel)
0.73 on2
0.71 1. o00
0.86 1. o00
BioEinu
0.70 J. 0001
0.72 1. 0001
0.71 1. 0002
MARS- @ MDGGen
0.15
0.24
0.88
MARS. FM
0.18 000
0.28 o0
01.8 1. o0s
C.7 VALIDITY METRICS
We report validity metries that directly probe native-contact preservation, C. geometry, and backbone covalent structure the MD-Cath 450 K test set. Following the EquiJump (Costa et al., 2024), we compute ries for fraction of native contacts (FNC) and for the Global Distance Test-Total Score (GDT-TS). We additionally report backbone bond- and angle-RMSZ values.
divergences (JSDs) between MD and model genee--gneeratdd trajecto-
Fraction of native contacts. FNC quantifies how well a sampled conformation preserves the interatomic contact pattern of a reference structure (Best et al., 2013). Given the set of native contacts Cref defined on the reference structure (pairs of atoms closer than a cutoff deu)). the FNC of conformation is
FNC(z) #( E Cref : dd)(()) S dem)
(11)
#Cre!
and we report the JSD between the FNC distributions of generated and MD trajectories.
Global Distance Test-Total Score. GDT-TS measures C, -level similarity to a reference struc- ture (Zemla, 2003). For a conformation with C, coordinates {r=(x)){, and reference coordinates
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(r calculate,
#( {I..... LJ (((((() r d
GDT-TS(1)
Σ
(12)
We again compute the JSD between the GDT-TS distributions of generated and MD trajectories.
Backbone covalent-gene geometryy RMSZ. length and bond-angle mous-uean- Z-scores (RMSZ) (Chen et al., 2010). For each backbone bond k we form a Z-score
To assess bonding quality, we compute backbone bond-
E
gideal R
2k
(13)
Ok
where & is the observed bond length, and & and aR the Engh-Huber crystallographic target mean and standard deviation for that bond type (Engh & Huber, 1991). The bond RMSZ is then
Mandd
Bond RMSZ
2k
(14)
Nbond
with an analogous definition for the angle RMSZ over backbone bond angles. Values close the MD oracle indicate MD-like covalent geometry, whereas large RMSZ reflects systematic distortions.
Table summarizes these additional validity metries. Both MARS-FM and MARS-FM @ MDGen-20 achieve FNC- and GDD--TS-base JSDs that are closest to MD, indicating that they best reproduce MD-like distributions of native contacts and Co, distances. For covalent geometry, the MDGen-20/100 parallel variants obtain the lowest RMSZ by construction, around a single conditioning structure; however, they underperform strongly on dynamical diversity metrics (Table 2). In contrast, MARS-FM models explore configuration space substantially more, yet still markedly improve bond and angle RMSZ over MDGen and BioEmu.
they perform only minimal moves
Table 8: JSD J. for fraction of native contacts (FNC) and GDT-TS, and RMSZ 1 for backbone bond lengths and bond angles. Results based on sampled trajectories of 500 conformations compared to ground-truth MD distributions and averaged over inference runs.
FNC
GDT-TS JSD
Bond RMSZ
Angle RMSZ
MD (Oracle)
0.13 .00
0.11 1: .004
2.23 a .022
2.19 .c09
MDGen-100
0.20
0.17
76.58
5.27
MDGen-20
0.30 .008
0.22 + 002
107.82 6.807
7,00 Xa1
MDGen- parallel)
0.29 .011
:: 0012
:: .00
2.56
MDGen-20 (in parallel)
0.40 + .001
02 + 001
2.88 a 0011
2,32 .011
BioEmu
0.53 .001
0.19 + .011
15.25 + 8.151
2.90 .001
MARS-FM MDGen-20
0.18
0.16
8.34
2.68
MARS-FM
0.15 .001
0.12 + 001
12.62 + 144
3.06 .001
C.8 MD-CATH AT 320K
In contrast to the high-temperature replicas, trajectories at 320 K display no large-scale domain motions. Both the radius of gyration and secondary structure fractions exhibit nearly an order-of- magnitude lower variability (Table 9). Similarly, all-atom RMSF and secondary structure fluctuations remain tightly clustered around their native values. These statisties confirm that proteins largely remain stable at lower temperatures, with no significant conformational changes--onsissentt with the findings reported by Mirarchi al. (2024).
For methodological consistency, we built the Markov State Model (MSM) with the same hyper- parameters at 320 K as for the 450 regime (10 clusters and 50 ns lag time). However, MSM could be temperature-specific for instance, by shortening lag times changing the number of clusters to capture subtler motions. As shown in Tables 10 and Table 11, MARS- --M continues to outperform all baselines in reproducing structural flexibility, radius of gyration and MSM state reconstruction. All methods show only marginal differences on secondary-structurr KL/JSD which could potentially improved by hyper-tuning MSM settings.
25

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Published as a conference paper at ICLR 2026
Table 9: Structural variability at 320K versus 450K in the MD-CATH dataset: standard deviations of radius gyration and secondary-structure structure fractions, plus mean pairwise RMSD and mean all-atom RMSF
MD-Cath 320K
MD-Cath 450K
Radius of Cryration std (nm)
0,045 .05
0.265 4.164
Socondary Structures Fractions std
0.016 024
0.101 4.000
Pairwise RMSD (A)
2.00
44.71
All Atom RMSF (A)
2.81
10.34
Table 10: Pearson r 1 for RMSD and RMSF, forward KL divergence + and JSD 1. of gyration radius, fraction of secondary structures, MSMs distribution, and folding free energy MAE (kcal/mol) at 320K. Results based on sampled trajectories of 500 conformations compared to ground-truth distributions and averaged over 5 inference runs.
Global RMSF
target RMSF
Gyration Radius
Gyration Radius ISD
Secondary Strectures
Secondary JSD
AGuu
RMSD
KL.
KL.
JSD
MAE
MD (Oroclc)
0.90
0.88
0.91
0.82 A28
0.10
0.37 Am
0.05 002
0.13 +
0.52
MDGGen-1l
.0LL
0.78
A22
001
0.18
MDGen-20
0.79 s.o
0.75 # .i10
0.79 .012
1.30 .085
0.19 = .007
0.78 .tw
0.15 0011
0.19 a: 4002
1.10 s.013
MDGen- 100 (in parallet)
0.87 1.04
0.83 + .02
0.87 = 20003
1.60 4227
0.18 s
0.44 + 0os
0.07 = .000
0.14 + .012
0.64 #.012
MDGen-20 parallel)
0.81
0.85
0.21
0.71
0.18
BioEma
0.58 c:0
0.63 2:04
0.84 000
2.67 2.02
0.36 = o0
0.94 #:011
0.15 on
0.25 #:08
0.83 2.00
MARS-FM FM MDXka-20
0.91 001
0.87 001
0.89 0011
0.74
0.13 .00.
0.46 +
0.09 001
0.10
0.62
MARS-FM
0.87
0.90
0.72
0.14
0.68
0.12
0.14
0.58
Table 11: Pearson r 1 for RMSD and RMSF, forward KL divergence 4 and JSD 1. of gyration radius, fraction of secondary structures, MSMs distribution, and folding free energy MAE (kcal/mol) J at 320K. Results based on sampled trajectories 100 1000 conformations compared to ground-truth
distributions and averaged over 5 inference runs.
Gyration
Gyration Radias
Secondary Strictures
Secondary Strectures
Pairwise RMSD
Global RMSF
Per larget RMSF
Radias
MSM
AGi
MD (Oracke)
0.87 /0.93
0.85
/093
2,21 70.44
0,18 0.007 /0.07
0.96 /0.19
0.09
0.43/0.88
0.58/0.32
MDGen-100
0.82 00.75
0.77/0.73
0.81 10.77
3.4
0.29/0.20
1.23/1.00
0.12/0.20
0.20/0.22
0.70/ 1.15
MDGen-20
0.79/0.68
0.73
0.25 /0.22
0.11/0.22
0.18/0.24
0.62/1.0
MDGen-100 (in paraliel)
0.81 /0.88
0.77
0.888
3.52/1.24
0.30/0.16
1.19/0.33
0.11/0.06
02110.13
0.71
MDGen-20 20 (in parallel)
0.81 0083
0.78 0.88
0.83 0.86
3,26 1.86
0.27 /0220
1.28/0.62
0.12 0.09
0.10/0.18
1.10/ 1.09
BioEma
0.18/0.14 0.886
0.42/0.24
MARS-FM e MDGea-20
0
0.86 0,87
0.87 / 0.90
2.23 0.57
0.22 00.12
0.89 /0:40
0.12/0.09
0.14/0.11
0.60 /066
MARS-FX
0.89 0.90
0.86 0.57
0,89 / 0.90
1,99 /..61
0.21 /0.13
1.25 /0.60
0.15/
0.15/0
0.59 /0.58
D LLM USAGE
We used large language models (GPT-5) assist in improving the grammar of the manuscript, as well as facilitate writing code.
E BROADER IMPACTS
The MARS-FM model's ability to generate realistic protein conformations over two orders of mag- nitude faster than conventional MD accelerate drug discovery and other socially beneficial molecular design tasks, while also reducing the energy footprint of large-scale simulations. Con- versely, the same speed-ups and accessibility could facilitate malicious protein engineering or foster overconfidence when the model is applied outside its intended scope.
F REPRODUCIBILITY STATEMENT
All details necessary to reproduce the experiments are present in the paper. All code necessary to reproduce our results has been included in the supplementary material as part of the submission and will be released publicly upon acceptance.
26


Reviews:
[{"review_id": "EJEFdJoarH", "reviewer": "Reviewer_pGe2", "summary": "The paper proposes to use Markov state models to augment the training of generative models that sample molecular dynamics (MD-Emus) by targeting the distribution of transitions as defined by the MSM transition matrix. Such models are called MSM-Emus. The method is validated to yield comparable or improved results on tetrapeptide simulations and improved results on the mdCATH dataset of protein simulations.", "questions": "No specific questions.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "The paper poses an insightful question for training MD rollout emulators - how do we more effectively learn from MD trajectory datasets beyond simply training on observed frame pairs? The approach is sound and seems to lead to useful empirical results.", "weaknesses": "**Clarity**\n\nMy chief concern is the clarity of the writing. First, the authors write that they \"learn to sample from the Markov chain transition induced by an MSM, rather than emulating the noisy temporal dynamics induced by MD.\" This phrasing seems to suggest that the the MSM transitions somehow differ and improve upon, by design, the dynamics in the MD. In truth, the MSMs are themselves fit to the \"noisy temporal dynamics induced by MD\" by enforcing coarse-graining and Markovianity approximations. Second, MSMs are not usually taken to be representations of fine-grained dynamics, yet the present model still learns full structural dynamics. Third, the authors argue that the time lag is a limitation of MD-Emus, yet MSM-Emus also have a specific lag time - that used in the construction of the MSM.\n\nA better way to express the proposed idea is as follows: Instead of learning the transition density faithfully, MSM-Emus are trained to predict next frames as a mixture distribution corresponding to MSM states in proportion to the MSM transition probabilities. While this corrupts the underlying dynamics, it allows the model to focus on learning the state change probabilities rather than paying attention (and potentially overfitting) to the exact starting frame within the state and for rare starting states to be identified and upsampled during training. This ultimately and empirically leads to better emulation of long-timescale dynamics (in the absence of the MSM) at inference time.\n\n**Justification for score**\nI would support to accept this paper if the authors can convincingly tidy up the presentation.", "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": "edbAcmxwKM", "reviewer": "Reviewer_H61Z", "summary": "The authors developed a model called MarS-FM for combining molecular dynamics (MD) modeling using flow matching and a traditional method markov state model. The work is build upon MDGen, and using similar dataset to test the performance of the generated MD.", "questions": "1. Authors should test their models on long MD trajectories including DESRES fast folding proteins and dynamic proteins (including WW domain and BPTI). Using short trajectories from MDGen is not convincing for either MSM models and conventional simulations\n2. This model should tested on large proteins compared with models like AlphaFold subsampling and BioEmu. Only small protein is limited to toy datasets\n3. Authors should also report multiple quality metrics including the protein validity, the bonding, the interaction distances, the Ca-Ca distances, etc.\n4. MSM models are the best fit for generate protein folding trajectories and pathway between rare event. I suggest authors add several case studies on these examples.\n5. A large issue for the tICA representation of conformation generation models are that incorrect conformation and unfold conformation are mixed in the same place. Authors should give multiple examples of the generated conformations based on k-means clustering, to show the possible generated incorrect conformations.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 1, "contribution": 1, "strengths": "There are not many work in machine learning conference that combines the MSM with MD generation models. This MarS-FM model sets an good example for combining traditional model and recent developments.", "weaknesses": "The benchmark is only on short trajectories, which highly limit the ability of MSM model (which desiged for long trajectories) and the application of this model.", "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": "BK6nMtuSYE", "reviewer": "Reviewer_2MiT", "summary": "This paper introduces a novel strategy to improve modeling of molecular dynamics (MD) based on the introduction of a Markov State Model to better represent state transitions at training time. The paper introduces MarS-FM as a model in this space based on Flow-Matching. MarS-FM  sampled molecular trajectories are compared against samples from MDGen and traditional MD simulations to demonstrate that the proposed method can achieve superior fidelity, even when unfolding simulations of molecules not observed during training.", "questions": "1. Section 4.1 reports that the MSM construction is performed once per dataset. Does this mean that different molecules share the same state space or that only the TICA projection are performed on the whole dataset but the clustering and transition modeling are molecule-specific? In the first case, how can a small state space (e.g. 10 x 10 transitions) effectively capture the dynamics of multiple molecules?  \n \n\n2. Can the authors clarify the sampling strategy described in lines 248-251? The paper reports that $x_1$ is sampled uniformly withing the state $S_j$. In practice, the density $p(x_1|S_j)$ is far from uniform on its domain. Do the authors sample uniformly from the data-points that are assigned to $S_j$ in the training trajectory? \n\n 3. How are the intervals $\\tau$ picked in practice at training time? The parameter $\\tau$ is implicit in $p_T(x_1|x(t))$ but it seems quite a relevant detail to balance the focus on different temporal horizons. Furthermore, this parameter may affect  the accuracy of the MSM model since a limited number of states (e.g. 10) may not be sufficient to describe smaller lag times since the transitions. How does Mars-FM work for varying $\\tau$ and number of MSM states? \n\n 4. How does the transition distribution modeled by Mars-FM compare against MDGen or other models in literature that model a specific transition interval $\\tau$? Do the conditional statistic also improve or does Mars-FM improve only in terms of the statistics of the equilibrium distribution?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* The paper tackles a practical problem in molecular modeling, demonstrating the effectiveness of the proposed MarS-FM solution against a relevant method in literature. \n\n* The paper reports convincing results on a large set of peptides demonstrating the generalization capabilities of MarS-FM on a disjoint set of trajectories using a wide range of metrics. \n\n* The paper is overall well-motivated and effectively explains limitation of existing MD Emulators with clear illustrations.", "weaknesses": "* The key part of the method and its training objective is quite compressed in a few paragraphs in section 4.1. Without any additional pseudocode, illustrations, and references, the details are quite hard to follow. \n   * The notation for the noise sample $x_0$, initial $x(t)$, next $x_1$ and interpolated $x_s$ frames is quite confusing. The time index $t$ is dropped in $x_1$ (and $x_s$), which should depend on $t$ and $\\tau$.  \n\n   * The paper reports that $x_1$ is sampled uniformly withing the state $S_j$. In practice, I expect that the density $p(x_1|S_j)$ is far from uniform on its domain. Please clarify the distinction between the empirical and modeled density in this paragraph. \n   \n    * A methodology comparison with MDGen is mentioned but not fully detailed. The method and background sections describe MSM constructions and input representations but the notation and flow-matching objective are hardly introduced, which makes following the objective in equation 5 more challenging. \n\n* The paper reports extensive empirical comparisons with MDGen but limited or no ablation on the effect of the modeling choices such as the lag time $\\tau$ or number of MSM states. The reported metrics focus on statistics of the equilibrium distribution, but no comparison or ablation evaluates the fidelity of the transitions learned by MarS-FM.", "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": "DUyzAVAwCL", "reviewer": "Reviewer_3Wf3", "summary": "This work presents a new class of generative models called MSM Emulators, which is designed to overcome the issue in previous approaches where learning transition densities with fixed time steps may cause the training signal to be dominated by frequent but uninformative transitions. During training, the proposed model, dubbed as MarS-FM, leverages flow matching to learn state transitions defined by the discrete transition matrix of preconstructed Markov State Models (MSMs) rather than those based on fixed time steps, thereby enhancing its ability to cross energy barriers and access different macroscopic states. Experiments on the tetrapeptide and MD-CATH datasets demonstrate that MarS-FM outperforms existing methods across all metrics by a substantial margin, while exhibiting a large improvement of sampling from rare metastable states.", "questions": "1. Equation (5) presents the flow-matching-based training objective, but I would like to know more about certain training details. For two macroscopic states $i$ and $j$ with corresponding samples $x(t)$ and $x_1$, the conformational difference between them may be substantial, making the conditioning on $x(t)$ potentially uninformative for generating $x_1$. Would this increase the training difficulty, and does the conditioning network indeed provide effective guidance in such cases?\n2. Lines 264–278 describe two generation strategies built on MarS-FM. I would like to know whether, and under what conditions, each strategy can unbiasedly reproduce the Boltzmann distribution, which is crucial for MD simulations.\n3. The Tree sampling strategy mentions the number of samples $n$ generated in parallel at each iteration. I would like to understand how the choice of $n$ affects the resulting ensemble under a fixed sampling budget. Intuitively, a larger $n$ may concentrate generated conformations within local states and miss global diversity, whereas a smaller $n$ may explore macroscopic states more broadly but lack local detail. The authors are encouraged to provide an ablation study on $n$ and investigate whether there exists a hyperparameter setting that achieves an optimal trade-off between local accuracy and global coverage.\n4. As mentioned in the Weaknesses section, the authors could introduce dynamical metrics for further evaluation to verify whether MarS-FM can capture the true dynamical behavior of the MD process.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper presents clear motivation and well-structured logic. Aiming to address the limitation of existing methods that learn transition densities at fixed time steps, it proposes a generative model based on the Markov State Model (MSM) that directly learns state transitions from the probabilistic transition matrix, enabling the model to cross energy barriers and access different macroscopic states.\n- Extensive evaluations on the tetrapeptide and MD-CATH datasets have been conducted. The results demonstrate the model’s out-of-distribution (OOD) generalization to unseen biomolecular systems and its substantially superior ability to access diverse metastable states compared with existing methods.\n- The training objective is simple and effective, making the method easy to reproduce. Remarkably, even with a single objective, the model achieves impressive experimental results, demonstrating the effectiveness of the proposed approach.", "weaknesses": "- As a comparison to existing models that learn transition densities at fixed time steps, MarS-FM appears to lack an analysis of dynamical observables for the generated trajectories, such as the decorrelation time of torsion angles. Evaluating the model solely using thermodynamic metrics does not allow one to distinguish it from an unconditional generative model that targets the Boltzmann ensemble, and therefore fails to highlight the advantages of the conditional generation approach.", "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)