Learning conformational ensembles of proteins based on backbone geometry
Authors: Nicolas Wolf
Leif Seute
Vsevolod Viliuga
Simon Wagner
Jan Stühmer
Frauke Gräter
Venue: NeurIPS 2025 poster ()
Keywords: Proteins
Boltzmann
Flow Matching
Protein structure
Protein conformation
Molecular Dynamics
Generative models
Primary area: machine_learning_for_sciences
TL;DR: We introduce a structure-based generative model for sampling protein conformations efficiently and eliminate the need for pre-training and evolutionary information.

Abstract: Deep generative models have recently been proposed for sampling protein conformations from the Boltzmann distribution, as an alternative to often prohibitively expensive Molecular Dynamics simulations. However, current state-of-the-art approaches rely on fine-tuning pre-trained folding models and evolutionary sequence information, limiting their applicability and efficiency, and introducing potential biases. In this work, we propose a flow matching model for sampling protein conformations based solely on backbone geometry - BBFlow. We introduce a geometric encoding of the backbone equilibrium structure as input and propose to condition not only the flow but also the prior distribution on the respective equilibrium structure, eliminating the need for evolutionary information. The resulting model is orders of magnitudes faster than current state-of-the-art approaches at comparable accuracy, is transferable to multi-chain proteins, and can be trained from scratch in a few GPU days. In our experiments, we demonstrate that the proposed model achieves competitive performance with reduced inference time, across not only an established benchmark of naturally occurring proteins but also de novo proteins, for which evolutionary information is scarce or absent. BBFlow is available at https://github.com/graeter-group/bbflow.

Learning conformational ensembles of proteins based on backbone geometry
Nicolas Wolf 1,2.3 Leif Seute +2,1,3
Vsevolod Viliuga 4.1 Frauke Gräter 1,2,3
Simon Wagner
Jan Stühmer* 2,5
1 Max Planck Institute for Polymer Research, Mainz, Germany Heidelberg Institute for Theoretical Studies, Heidelberg, Germany SWR, Heidelberg University, Heidelberg, Germany SciLifeLab and DBB at Stockholm University, Stockholm, Sweden SIRR, Karlsruhe Institute of Technology, Karlsruhe, Germany
Abstract
Deep generative models have recently been proposed for sampling protein confor- mations from the Boltzmann distribution, as an alternative to often prohibitively expensive Molecular Dynamies simulations. However, current ttat-oof-hhe-nt art ap- proaches rely on fine-tuning pre-trained folding models and evolutionary sequence information, limiting their applicability and efficiency, and introducing potential biases. In this work, we propose a flow matching model for sampling protein conformations based solely on backbone geometry BBFlow. We introduce geometric encoding of the backbone equilibrium structure as input and propose to condition not only the flow but also the prior distribution the respective equilib- rium structure, eliminating the need for evolutionary information. The resulting model is orders of magnitudes faster than current cate-offthheart approaches comparable accuracy, is transferable to multi-chain proteins, and can be trained from scratch in a few GPU days. In our experiments, we demonstrate that the proposed model achieves competitive performance with reduced inference time, across not only an established benchmark of naturally occurring proteins but also de novo proteins, for which evolutionary information is scarce or absent. BBFlow is available htthps: thub. com/ wwam/grrrerr-re gratieerrgrooupbbbloogo
Introduction
In recent years, the field of protein structure prediction has been revolutionized by geometric deep learning [9 27]. Jumper et al. [19] introduced AlphaFold 2, which predicts protein's 's structure using patterns found in naturally occurring protein sequences, so-called evolutionary information, upon inference. On the other hand, advancements in generative modeling such as diffusion (39) and have propelled the field of protein design, where several approaches for the generation of novel protein structures have been proposed [46] 52 7 Plausible protein structures conditioned on symmetry or motif can be designed without requiring an input sequence [15] 53).
flow-matching 29
Both, protein structure prediction and design methods, generate a single equilibrium structure of a protein. In contrast, protein function depends structural dynamics [3 11 5) that is, the protein's conformational ensemble as given by the Boltzmann distribution, assuming thermal equilibrium. To sample from the Boltzmann distribution, Molecular Dynamics (MD) simulations
established
method in the field (2). However, covering the state space extensively with MD requires long
Equual contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Ensemble samples
Equilibrium structure
Distance embedding of
Prior Samples
SE(3) Flow Matching
2o(x, xx,
Conditional Prior
X
Po(x, Ix(x,1x))
X, - Po
Figure I: Schematic representation of BBFlow. The equilibrium backbone structure Teq of an input protein is used condition SE(3) Flow Matching model the generation of protein backbone conformations ø1- Already the prior Po of the flow matching process is conditioned on the input protein via partial geodesic interpolation between pure noise and the equilibrium backbone structure.
simulation times in order to satisfy ergodicity by overcoming local free energy minima, making conformational sampling often prohibitively expensive. Recently, generative models have been suggested for emulating the sampling of MD conformations, offering inference times that are orders
of magnitudes faster than MD 33
For proteins, current state-of-the-art approaches for such generative models rely on modifications of AlphaFold where noise introduced into the MSA 47, the pre-trained folding model is fine-tuned on ensemble data [16), or the structure block is replaced by diffusion model (24). While these approaches are capable of generating realistic conformational ensembles, their efficiency is limited since they depend on large pre-trained folding models or the sampling of each state requires to predict the overall fold of the protein from the sequence. Consequently, the models rely on processing of evolutionary information such as MSA or weights from protein language models like ESM (7). This renders the models not only expensive, but can in addition introduce biases for sequences where evolutionary information is scarce or, as for de novo proteins, absent [27]
Main contributions In this work, we introduce BBFlow, generative model for MD-derived protein conformational ensembles based on backbone geometry that is more than an order of magnitude faster than the current state-of-the-art model AlphaFlow [16], at similar accuracy. To the best of our knowledge, it is the first protein ensemble generation model shown to be applicable not only to monomeric but also to multi-chain proteins. BBFlow relies on two key innovations.
(1) We formulate conformational ensemble prediction as protein structure generation task, conditioned on a geometric encoding of the equilibrium structure and (2) propose a conditional prior distribution for flow matching based on geodesic interpolation (Fig, [T. Notably, our work shows that neither pre-trained weights from a folding model nor evolutionary sequence information are necessarily required to generate conformational ensembles as observed in 300 nanoseconds (ns) of MD.
For benchmarking BBFlow, we train and test the model on the ATLAS dataset 42 which contains curated set of 300 ns long Molecular Dynamics trajectories for 1390 proteins the same dataset used for training AlphaFlow. We also test BBFlow on MD trajectories of de novo proteins, where we find similar performance as for naturally occurring proteins while AlphaFlow fails if the equilibrium structure not provided as template. We show that BBFlow although trained only on monomeric proteins generalize multi-chain proteins, which are not covered by other basclines.
1.1 Related Work
Ensemble generation Previous deep learning approaches for sampling conformational ensembles that apply invertible neural networks [33] or equivariant flow matching [21] usually require training on the specific system of interest. For proteins, a transferable model, AlphaFlow, has been recently
proposed [16], relying on fine-tuning the pre-trained folding model AlphaFold 2. Li et al. [25]
propose to speed up AlphaFlow upon inference time by only calling the evoformer once, however, the model is not publicly available and only subset of metrics is reported.

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Generating standardized MD-emulating ensembles AlphaFlow is trained on the ATLAS dataset 42], which contains MD conformations obtained in standardized setting via three times 100 ns long simulations at the same temperature, force field and water model. Training models to generate such standardized ensembles, which we refer to as MD emulation, is important for evaluating their distributional accuracy quantitatively. This is because ensemble metrics strongly depend on the duration and temperature of the corresponding MD (see also Tab, A.3)
Next to AlphaFlow, also ConIDiff 45 diffusion model that relies on a pre-trained sequence representation of AlphaFold 2, and MDGen [7]. model that generates ensembles of molecules in all-atom representation with consistent time evolution, are trained on the ATLAS dataset.
Generating other ensembles There also interest in generating protein ensembles that do not strictly follow a certain distribution induced by fixed MD time and temperature but rather sampling general alternative states, such BioEmu 24 We discuss such models in App. |..4 and Tab, A.3
Background
2.1 Flow Matching for protein structure generation
Flow Matching In order to sample from a target distribution p1 :M - [0, 1] on the data domain M. Lipman et al. [29] have proposed flow matching as generalization of diffusion models (39). A learned flow o -M x [0, 1] + M is used to transform samples Ty ~ Pu from simple prior distribution Po samples (( 1) from the target distribution P1. The key idea is to learn time- dependent flow vector field
v(x, t) M x [0, - T&M, (x. t) + v(r,t). t),
where T&M is the tangent space at point r. The flow or = o(. t) is then defined by i via integration of the flow ODE,
dt (t))) = (x), Φo(x) = x.
The vector field ot can be learned by sampling zu ~ Po and Φ ~ Pl. connecting them by a particle- wise flow (200 21, t) and regressing the time derivative of + [29 On Riemannian manifolds, vi is usually chosen as geodesic (9.
Application to protein structure A protein backbone can be represented a sequence of Euclidean frames x = 2r E SE(3) [19. each of which consists of a rotation r E SO(3) and a translation E R A flow matching process for protein structure can thus be formulated on the Riemannian manifold .M= = SE(3)*. By choosing the metric on SE(3) as in [5T]. the geodesic paths can be split into independent rotation and translation parts for each residue. Typically, one parametrizes both the ground truth and predicted vector field by current structure 2, and final structure æ1. It can be shown [51 7 that the vector field components are then given by
U8o(3) (ro (/r, = log,, (r1)
(2,, (211 = 21 2
(3)
1 1-t
1-t
A common choice for the prior distribution is independent Gaussians for the translations ~ A o²) and uniform distributions for the rotations To ~ U{()(3)) [T].
2.2 Evolutionary sequence information
In order to determine the structure of a protein, the challenging task of mapping from a one- dimensional sequence representation to a three-dimensional backbone geometry needs to be solved. To achieve this, folding models like AlphaFold 2 rely on evolutionary information in the form of Multiple Sequence Alignment (MSA) - an algorithm that aligns the input sequence with related natu- rally occurring protein sequences from a database during inference and training to identify patterns that encode information on folding states such as pairwise contacts. The calculation of an MSA during inference is computationally expensive. A more efficient strategy is to encode evolutionary information by extracting weights from a protein language model (36), which can be seen as learned evolutionary information [27]. While evolutionary information has been shown to be beneficial
3

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for predicting ensembles or alternative folding states 47, there is rising interest for methods that
perform well also without relying on evolutionary information [12 [44, which is not available for de-novo proteins or the disordered proteome.
3
Method
In this work, we propose to decouple protein conformational ensemble generation from the structure based purely on backbone geometry that does not
prediction task and introduce generative model rely on evolutionary sequence information. We achieve this by conditioning both the flow and the prior on the equilibrium structure of the protein.
Conditional flow matching for ensemble generation Inspired by FrameFlow [1] flow match- ing model for protein structure design, we formulate MD emulation as structure generation task, conditioned on the equilibrium state of the respective protein. In particular, we express the Boltzmann distribution of given protein as probability distribution p( Rey) of conformations æ, conditioned on the equilibrium state Teeq of the respective protein. In order to sample from p(x| we learn flow vector field,
v( t, te)) M [0, x Mey + TTM,
that receives protein equilibrium states Zeg Mey additional input. This defines a conditional flow Φe by
d oe (1(((1)) (,,
dt
Crucially, by conditioning the generation not on the sequence but the equilibrium structure, we eliminate the need for evolutionary information and pre-trained folding model weights. We summarize the training procedure in Algorithm [I]
We note that assuming the availability of an equilibrium structure is reasonable because, as MD) emulator, the use-case of the model is offer alternative to MD simulation, which also requires an initial structure (see I..Ip. If only a sequence is available, both MD and BBFlow first require structure prediction with folding model. In Tab, [.b.A.]] show that BBFlow remains accurate and fast if used as sequence-to-ensembble model in combination with AlphaFold 2.
Model architecture In order to learn the conditional flow vector field et, we adapt the model architecture of the recent protein design model GAFL [44), which is an extension of the FifmeDiff architecture proposed by Yim et [52] The input features include the frames 2, at time their pairwise spatial distances, and the flow matching time f. Crucially, in contrast to common protein
structure architectures [52 51 19 16), we do not use the residue indices as input feature. The reasoning behind this choice is that the ordering of the residues in the chain is already encoded geometrically equilibrium structure. Removing the residue index as an input feature reduces memorization and enables transferability from monomers to multi-chain proteins as explained in Sec.
The neural network is an SE(3) equivariant graph neural network, which uses invariant point attention (IPA) 19 as core element. In GAFL, IPA is extended Clifford frame attention (CFA), where geometric features are represented in the projective geometric algebra and messages are constructed using the bilinear products of the algebra. Frames are consecutively updated along with node and edge features in series of 6 message passing blocks to predict the target frames 2 Compared to Alphafold [19), this architecture is more shallow and operates only on structural data, hence a sequence processsngg module like the Evoformer of AlphaFold 2 is not required.
Encoding of the equilibrium structure For conditioning the flow vector field as in Eq, we modify the architecture such that the equilibrium backbone structure of the protein can be used as input feature. Inspired by the interpretation evolutionary information as contact map (2 we propose to encode pairwise distances of the equilibrium state Teg as initial edge feature,
8 bin (Il=, 2,l12)
where we bin the distance uniformly between 0 and 20 with bin count 51 encode the equilibrium structure in a more direct, geometrically meaningful way. Inspired by
Additionally,

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tensor-based equivariant networks 38 and their formulation in terms of local frames [30, we include equivariant pairwise directions between residues that are closer than 5 as unit vectors,
eij = Er
21 2,112
and express them in the coordinate frame x; (ri z;) of residue Through the transformation into the co-rotating coordinate frame, the feature components become invariant and can be used together with Sty as initial edge feature.
We use amino acid identities as additional node features by transforming a one-hot encoding via a linear layer to 128- 18-dimeensional embedding. The reasoning behind encoding the amino acid type is that it carries information about the local degrees of freedom of the backbone, however, in an ablation we find that also without the amino acid identity, the model performs remarkably well.
(Tab. 3
Conditional prior distribution Unlike diffusion models [39], where Gaussianity of the prior po is a strict theoretical requirement, flow matching, in principle, allows the of general prior distributions [29] Non-Gaussian, unconditional prior distributions for proteins have been proposed by Ingraham al. [5] and Jing al. [16]. We take this idea a step further and propose conditional prior distribution Pn(2 (z)((e)) for flow matching. Samples vo ~ Po (Tag) are generated by interpolating between samples from unconditional prior Puncond and the equilibrium structure Teg.
Fancond ~ Punconds
Ho (Funcondd Fegq <),
(8)
where 7 is the geodesic between Fancond and Teq.
(Funoood Teq: 0) = Tuncond: (Funcond Fieq 1) Togg
(9)
and is a hyperparameter between 0 and that quantifies how close the noise sampled from the prior is to the equilibrium structure (see Fig. A.4). In our experiments, we set = 0.2. For an ablation of see Sec. A.8 For the unconditional prior distribution Puncond- we use the normal distribution for
translations and the uniform distribution for rotations [ST]. We note that this approach of conditioning the prior can be seen as generalization of partial denoising from diffusion [1] to the flow matching framework.
Loss function As explained in Sec. 2 we represent protein backbone structure as a of frames (r, E SE)() and define the flow matching process on the data manifold JM = SE(3) We learn a conditional flow vector field 6(2,, t, Fee)) (Eq. 4 on the tangent space of the data domain, parametrized by Eq. 3 For regressing on this vector field, we calculate the ground truth v as tangent vector to the geodesic Trm between the prior sample zu and target sample T1, and apply mean squared error loss,
CPM
||u 6(xe, t, Tey)) I1 (SS(3)
(10)
where xt is a point along the geodesic 7PM. xt 7FM (wo, 31, t), and Zeg denotes the equilibrium structure used as condition. The expectation in Eq. 10 runs over
t t~0,0,,) 1), (I1, #og) ~ Pdata 2g ~ Po (-e))
(11)
and the metric is defined as in 52
|| | Tr (u,u?) / |0=11
(12)
with the Euclidean 2-norm ||- lla and the projection on rotational and translational subspaces v U.). As [5I, we also use the auxiliary loss proposed in [22
Experiments
Training In order to directly compare the proposed model to the current state-of-the-ar MD emulator for proteins, AlphaFlow [66,, we train BBFlow on the ATLAS dataset [42] with the same split into training, validation and test proteins. The ATLAS dataset consists of three trajectories of 100 ns long all-atom Molecular Dynamies (MD) simulations for 1390 structurally diverse proteins, of which Jing et al. [16] select 1265 for training, 39 for validation and 82 for testing. We train the
5

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Table 1: Performance of BBFlow and baselines (Sec. the ATLAS test set. For each protein, we evaluate the metrics described in Sec. 4 and report the median of all proteins. We also report RMSF medians over all residues and indicate the MD reference value in parentheses. Inference time is reported per generated conformation of the 302 residue protein 7c45A. All metrics except for correlations r and transient contact accuracy Ju are reported in A. Errors are estimated as described in Sec. 4 and are shown in parentheses if they above precision. Best values are bold, second best are underlined. Note that BioEmu cannot be compared to other baselines directly, as explained in the paragraph Further Baselines below.
RMSF
Pw-RMSD
DCCM
PCA
Jr
Time
Median (MD=1.48)
(1)
MAE (J)
MAE ())
(())
W2 (1)
% (*)
[s] (L)
BioEmu"
0.83
1.29 (0.01)
2.34
2.84 (0.01)
0.80
1.65 (0.04)
36
1.9
AlphaFlow
0.86
0.59 (0.01)
1.51
(0.01)
0.86
1.47 (0.03)
41
32.0
ConfDiff
0.88
0.62 (0.01)
2.00
(0.01)
0.86
(0.03)
39
20.2
AlphaFlow-T
0.92
0.41 (0.01)
1.17
(0.01)
0.89
1.28 (0.03)
47
32.6
ESMFlow-T
0.92
0.52 (0.01)
0.94
(0.01)
0.89
1.48 (0.03)
47
11.2
AlphaFlow-Tane
0.92
0.68 (0.01)
0.90
1.41 (0.01)
0.88
1,43 (0.03)
42
3.3
AlphaFlow- din
0.90
0.85 (0.01)
0.68
1.80 (0.01)
0.87
1.60 (0.04)
24
1.2
BBFlow
0.90
0.42 (0.01)
1.49
0.77 (0.01)
0.87
1.33 (0.03)
29
0.8
Not trained to generate ATLAS ensembles.
model, and variants where leave out key features for an ablation study, for 3 days two NVIDIA A100-40GB GPUs from scratch, i.c. without initial weights from a pre- trained folding model. For all experiments, use the same hyperparameters number of timesteps, which we set to 20. Also the respective feature dimensions are increased by 128 for embedding the amino acid identity node feature and by 22 or 25, respectively, for embedding the equilibrium structure encoding with or without direction as edge feature.
in FrameFlow [5]] and GAFL 44 except for the
Baselines We compare BBFlow with models from [16 that were fine-tuned on the training set of BBFlow, but rely on pre-trained weights from the folding models AlphaFold 2 and ESMFold [27] that were trained much larger datasets. Next to the original AlphaFlow-MD. model (referred to in this work as AlphaFlow), we also evaluate AlphaFlow-MD with templates (AlphaFlow- receives the equilibrium structure input, encoded as template AlphaFold. Jing et al. [16 also introduce a model that relies not directly on the expensive MSA but on the protein language model ESM (ESMFlow-T) Additionally, templates that are optimized for efficiency by distillation (AlphaFlow-Taas)) decreasing the timesteps required from 10 to and by reducing the number of layers (AlphaFlow- -loo-TT2..w.dss) We evaluate all models above using the conformations deposited in the AlphaFlow GitHub repository? include the diffusion model ConfDifT 45 in our comparison (see Sec.
which
compare BBFlow with models based on AlphaFlow-MD with
We also
Further Baselines For completeness, and because of their recent impact the field, we also evaluate models that were not trained on the ATLAS dataset but on static structures, NMR data or longer MD trajectories, such as BioEmu (24] and other baselines 31 32] in the appendix (Tab. AA.).. These models are not trained to sample from the probability distribution of states visited during three times 100ns of MD and perform unfavorable if quantitatively evaluated in a standardized setting, as on the ATLAS test set (see Sec. I.I. Further, we show in Tab, A.3 that BBFlow outperforms MDGen [17], an all-atom approach for generating time-consistent ensembles trained on ATLAS. For comparison with the MSA subsampling approach [47] and the classical normal mode analysis (NMA) 88, we refer to [66, where is shown that both perform worse than AlphaFlow and BBFlow.
Metrics We evaluate the performance of the compared models by reporting metrics that quantify how well statistical properties of the generated ensembles agree with those obtained by MD under standardized settings described in Sec. [.I We report the key metrics, measuring similarity of the ensemble properties, Root Mean Square Fluctuation (RMSF), pairwise RMSD, principal

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components (PCA) and the Dynamical Cross Correlation Matrix (DCCM) that are established in the field of protein dynamics. To this end, we calculate the residue-wise Pearson correlation r for RMSF and DCCM, the mean absolute error (MAE) for RMSF and pairwise RMSD and the Wasserstein-2 distance the ensembles, projected on the first two principal components obtained from MD. To assess the role of pairwise distances in the provided equilibrium structure, we also report the accuracy of predicted transient contacts Jn as the Jaccard similarity between the generated ensemble and the MD ensemble. As in [16. we define transient contact pair of residues that are separated in the equilibrium structure, but in proximity in 10% of the ensemble states, using a C. distance threshold of A. In addition to these, Jing et al. 16 include new metrics, which we also report an exhaustive Furthermore, we investigate Ramachandran dihedral distributions in Sec. A.9 More details on the metrics and their interpretation can be found in Sec. A.33 In all experiments, set of MD conformations 100 times order to estimate the error caused by sampling finitely many states. All metrics are calculated using the Ca atoms of the protein structures.
evaluation table in the appendix (Tab. A.9
generate 250 conformations per protein, in AlphaFlow, and bootstrap the
Inference time Inference time is, even orders of magnitude smaller than MD, a critical factor for applications of MD emulators such annotation of datasets screening of proteins for target motion. As in [14], we evaluate the inference time per generated conformation of the 302-residue protein 7c45A, and the entire ATLAS test set in Fig. 5 using NVIDIA A100-80GB GPU. Note that for models based on AlphaFold2, the inference time for all-atom structures is dominated by backbone generation (Tab. A.8
4.1 ATLAS benchmark
We report the performance of BBFlow and the baselines evaluated on the ATLAS test set from AlphaFlow [16] in Tab. We find that BBFlow generates high-quality conformational ensembles faster than all baselines. For proteins of length 300, it around 40 times faster than AlphaFlow with templates (AlphaFlow-T), at comparable accuracy. While AiphaFlow-T is slightly more accurate in terms of RMSF and principal components, BBFlow outperforms by pairwise RMSD and median RMSF. BBFlow outperforms AlphaFlow, ESMFlow-T, the two distilled models and ConfDiff in almost all metrics while, at the same time, generating the ensembles faster. Indicated by small median RMSF, AlphaFlow-T systematically over-stabilizes the proteins and samples too close to the equilibrium structure. Additionally, we investigate the performance for different protein lengths (Fig. Fig. (.6) and find that, while the trends described above hold true for all lengths considered, BBFlow performs favorably for larger proteins. At transient contact accuracy, BBFlow underperforms the baselines, indicating that for predicting rare events, evolutionary information might be required. For weak contacts (see A.3 Tab. A.9 BBFlow is more competitive.
capturing flexibility quantified
Table 2: Performance of BBFlow and baselines for de novo proteins. Evaluation settings as in Tab, Errors shown parentheses above precision. Best values are bold, second best are underlined.
RMSF
Pw-RMSD
DCCM
PCA
Ju
Time
Median (MD=0.91)
(t)
MAE (L)
MAE (L)
(()
W2 (L)
% (?)
[s] (L)
BioEmu*
0.60
4.24 (0.01)
7.56
8.29
0.64
1.53 (0.04)
23
1.9
AlphaFlow
0.47
4.76 (0.01)
7.09
7.40
0.58
1.64 (0.03)
17
32.0
ConIDiff
0.62
3.82 (0.01)
6.35
7.26
0.65
1.72 (0.02)
15
20.2
AlphaFlow-T
0.89
0.25 (0.01)
0.74
0.38
0.85
0.66 (0.01)
55
32.6
ESMFlow-T
0.89
0.28 (0.01)
0.68
0.43
0.86
0.63 (0.01)
56
11.2
AlphaFlow-Tau
0.88
0.46 (0.01)
0.51
0.77
0.84
0.69 (0.01)
3.3
AlphaFlow- Tra
0.87
0.58 (0.01)
0.41
0.97
0.83
0.75 (0.01)
1.2
BBFlow
0.84
0.26 (0.01)
0.87
0.32
0.83
0.67 (0.01)
32
0.8
Not trained to generate ASeemmee...
4.2 De novo proteins
Recently, designing dynamical properties into novel proteins has gained attention hypothesize that BBFlow's greatly reduced inference time for generating high-quality ensembles
43
22 We
7

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A
BBFlow
AlphaFlow-T
AlphaFlow-Tan disc
2.5
1w-RMSD [Á]
3.0
6.0
MMSF MAE (A)
2.0
PCA W2
4.0
Time [s]
1.0
2.0
0.5
Alghaflow-Taa
0.0
0-100
100 100-300 300+
0.0
0-100
100-300 300+
0.0
0-100
100-300
300+
200
400
600
Protein length
Protein length
Protein length
Protein length
B
2.5
MD
RMSF [A]
Berlow
RMSF [A]
BBFKOw e: 0.05
RMSF [A] 2
RMSF [A]
2.0
BOFTOw
MABI 0.26
MAD: 0:18
MAE 0.22
1.5
MAEL 0.12
1.0
MA
50
100
50
100
50
100
Residue Index
Residue index
Residue index
Residue index
Figure 2: (A) Performance of BBFlow, AlphaFlow-T and AlphaFlow-TizL.di for different protein lengths. We divide the protein lengths in three bins and calculate RMSF MAE, the absolute error of pairwise RMSD and PCA W22 of each protein (lower is better) with length in the respective bin. The boxes depict the 0.25 and 0.75 quantile, minimum, maximum and median of all test proteins. We also show inference time per generated conformation function of protein length in lcg-scale, spanning several orders of magnitude. We show structures and RMSF profiles predicted by BBFlow and MD of four selected proteins from the dataset of de novo proteins along with Pearson correlation and MAE as reported in Tab.
the ATLAS test
(B) RMSF profiles of de novo proteins.
MD
BBFlow
AF-T
AF
BioEmu
Figure Ensembles of two novo proteins predicted by BBFlow, AlphaFlow- (AF-T), AlphaFlow (AF) and BioEmu compared with the ground truth molecular dynamics (MD) simulation. The proteins were generated by RFDiffusion and ProteinMPNN, and are colored by residue index.
makes the method interesting for applications in protein design pipelines, where efficient MD emulation would allow to screen for dynamics. However, since de novo proteins often have no evolutionary information available, the applicability of models that rely such information is unclear. order to evaluate conformational ensembles of de novo proteins, we generate small dataset of 50 proteins sampled with the established models RFdiffusion [46] and FrameFlow (53). respectively, and perform three 100-ns-long MD simulations for each, similar to ATLAS (A.10).
In Tab. 2 we report the performance of the models considered in Sec. for de novo proteins. We find that AlphaFloo without templates and BioEmu, which both heavily rely on evolutionary information, experience strong decline of performance compared to naturally occurring proteins (Tab. M.. The relative differences between BBFlow and the other baselines are comparable to the performance natural proteins. Fig. 2 displays structures and predicted RMSF profiles of four de

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novo proteins. We also visualize ensembles of two randomly chosen de novo proteins predicted by BBFlow, AlphaFlow-I, AlphaFlow and BioEmu, Both AlphaFlow without templates and BioEmu fail to sample ensembles consistent with MD (Fig. 3), and instead tend to predict unstable conformations.
Multi-chain proteins
In contrast to folding architectures [99 52), BBFlow does not rely absolute residue chain indices as input features but rather on geometric biases imposed by the distance matrix of the equilibrium structure (see Sec. 3 It thus naturally extends protein complexes that are made up of several protein chains, which are not covered by the baselines. This makes BBFlow, to the best of our knowledge, the first ensemble generation model shown to be applicable multi-chain proteins.
We demonstrate in Tab. A.4 that BBFlow, indeed, captures ensemble properties of five well studied multi-chain systems (see App. A.55 and show that AlphaFlow, given multi-chain features as input, fails (App. A.3 and Fig. A.Ip. p. We visualize multi-chain ensembles and their DCCM computed with MD and BBFlow in Fig. 4oto illustrate that both intra-chain and inter-chain motions are captured by the model. Remarkably, it is able to do so without being trained on multimeric proteins but only on the single-chain proteins from the ATLAS dataset.
18G5
MU
HD
0.87
9.3
022
100
0.15
Residue index
50
8.5
0.50
9.4
200
DCCM
100
Residue
6.0
300
-02
100
400
-0
150
0.4
-044
-0 50
150
Residue index
Residue index
Residue index
Figure 4: BBlow is applicable to multi-chain proteins. Dynamic cross-correlation matrices (DCCM) of conformational ensembles computed either with MD (upper triangle) or with BBFlow (lower triangle) for three protein dimers. Chain boundaries are indicated by black lines within matrices. r: Pearson correlation between entries of the triangular matrices. We show RMSF profiles in Fig. A.5
Table 3: Ablation study for key components of BBFlow. Metrics reported in Tab. Errors calculated as described in Sec. and displayed in parentheses if above precision.
Cond.
Distance encoding
Direction encoding
Amino acid enc.
Index encoding
RMSF MAE (L)
Pw-RMSD
DCCM
Name
prior
MAE (L)
(()
BBFlow
0.42(0.01)
0.77 (0.01)
0.87
0.52 (0.01)
1.15 (0.01)
0.85
b
V
0.48 (0.01)
0.90 (0.01)
0.86
v
0.42 (0.01)
0.82 (0.01)
0.88
0.54 (0.01)
0.93 (0.01)
0.85
(0.01)
7.08 (0.01)
0.55
4.4 Ablation
For quantifying the contributions of key components proposed or discussed in this work, we perform an ablation study on the ATLAS dataset and report the results in Tab. 3 We find that using the proposed direction encoding (a), the novel conditional prior (b) and eliminating indeed benefits the performance of the model. Additionally, we train model that is entirely backbone structure-bace based, without any sequence information (d), and find that it on par with non-template AlphaFlow. We also demonstrate the need for the distance encoding of the equilibrium structure (c).
index encoding (c)

---

4.5 Discussion
The results show that BBFlow achieves state-of-the- per- formance in the trade-off between speed and quality of the generated ensembles (see also Fig.(5). At comparable accuracy, it is more than order of magnitude faster than the current state-of-the-art model AlphaFlow-T and also faster than the distilled model AlphaFlow-T id..dt. Crucially, BBFlow does not suffer from the over-stabilization observed in AlphaFlow-I. impeding the exploration of conformational space. This perfor- mance is due to the proposed conditional prior and geometric encoding of the equilibrium structure, as shown in our ablation.
0.9
Alphat www.lllwwluaa 132.
RMSF MAE [A]
0.8-
Alphaflow- Tow
ConfDiff
0.6-
ESMFIow-T
BBFlow-light
0.5
BBFlow
AlphaFIow-T
0.4
10
Inference time per conf. [s] Figure 5: Trade-off between accu- racy and speed of MD emulation.
Not using templates in AlphaFlow can avoid overstabilization, but causes AlphaFlow to fail for de novo proteins. As conse- quence, BBFlow is the only model considered that accurately captures overall flexibility for de novo proteins. We attribute these observations to BBFlow being based entirely on back- bone geometry instead of evolutionary information, which is scarce for non-natural proteins. We also find that BBFlow is transferable to multimers- a class of proteins uncovered by current ensemble generation models.
While other methods are either effi- cient accurate, BBFlow performs well both. The accuracy met- ric RMSF MAE and inference time are averaged over the ATLAS test set. More metrics can be found in Fig. A.3
BBFlow-light: App. A.6
Limitations As MD emulation model, BBFlow's scope is fundamentally limited to reproducing MD-derived distributions, thus, it cannot predict states that very distant from equilibrium, such as alternative folding states, without being trained on correspondingly long MD trajectories. For predict- ing alternative states, specialized models, which are in turn less accurate at MD emulation, exist (see App.[A.4] A..4. In terms of sampling transient contacts, BBFlow underperforms like AlphaFlow, indicating that evolutionary information is especially helpful for predicting rare events. Note that the requirement of an initial structure as input is not a limitation for MD emulators in practice and be overcome (see App.[A.T. Tab. A.T). Similar the other backbone generation models in Sec. [.] BBFlow does not sample sidechain conformational ensembles or protein-ligand interactions, which is subject of further work.
approaches
5
Conclusion
The generation of MD-derived ensembles of proteins is key task in many protein-relate fields. Inspired by generative models for protein design, we propose BBFlow, a method for emulating MD by sampling ensembles with state-of-the-art performance in the trade-off between accuracy and efficiency. At the same time, BBFlow also avoids problems with de novo proteins and over-stabilization observed in current state-of-the -he- models. Crucially, BBFlow is also applicable to multi-chain proteins, which are not covered by other ensemble generation models. We achieve this by introducing conditional prior distribution and a geometric encoding of the protein's 's equilibrium structure as condition in How-matching model that is based on backbone geometry. This eliminates the need for evolutionary information and enables to train the model from scratch, without requiring weights from folding model that is trained on much larger dataset. We expect that both of these ideas using a conditional prior in fow-matching and replacing evolutionary information by condttoo- can also be applied to other problems in generative modeling and structural biology.
AlphaFlow [66] is widely used by practitioners to replace expensive MD simulations of proteins. We also see BBFlow as highly relevant in practice, given its significantly increased efficiency and transferability to multimers, allowing accurate MD emulation on a much larger scale. In particular, BBFlow can be applied in pipelines for de novo protein design, where it could enable the screening of generated structures for desired dynamics- property that is challenging incorporate into designed proteins so far.
Acknowledgments This study received funding from the Klaus Tschira Stiftung gGmbH (HITS Lab). We also acknowledge support by the state of Baden- Württemberg through bwHPC and the German Research Foundation (DFG) through grant INST 35/1597-1 FUGG.
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A
Appendix
A.1
On the requirement of initial structure for MD emulation
Since BBFlow is an MD emulator, also inherits the dependency on the initial structure from MD. AlphaFlow (without templates) be applied sequence-to-strruuue model by relying on evolutionary information encoded form of a Multiple Sequence Alignment (MSA). Since MD requires initial structure, it can be assumed that an initial structure is available in all practical workflows that currently include expensive MD simulation. However, in a scenario where a sequence, but not the structure is available, BBFlow can be combined with folding model and used as scquence-to-structure model that remains accurate and fast.
We show that the combination of AlphaFold2 and BBFlow remains both more accurate and more tine-efficient at sequence-to-struetue prediction than AlphaFlow' ' almost all metrics considered (Tab. A.I For de nove proteins, the equilibrium structure that is passed to BBFlow and AlphaFlow-T was obtained by the folding model EMSFold during creation of the dataset (Sec. A. thus already evaluate the combination of ESMFold and BBFlow, which acts as to-sruccttttttuuuureeee pipeline. BBFlow outperforms the sequence-to -eaeencceto-stuuuuuee model from AlphaFlow (without templates) in this setting (Tab. 2 Note that the inference time of the pipeline of AlphaFold 2 and BBFlow is still around 30 times faster than that of AlphaFlow because AlphaFlow requires structure prediction for every conformation at every time step while the combination of AlphaFold 2 and BBFlow only requires a single structure prediction at the beginning.
model at
We further investigate BBFlow's sensitivity with respect good input equilibrium structures by evaluating its performance when passing distorted cquilibrium structures (Tab. (A.2) to the model. To this end, add Gaussian noise to the Euclidean backbone coordinates of the ATLAS test set proteins and run inference with BBFlow. We find that the performance remains strong and decreases only slightly.
Table A.I: Performance of BBFlow as sequence-fo-ssrruture model. We evaluate the pipeline of predicting structure from sequence with AlphaFold 2 and passing this structure as input BBFlow, compared to the direct sequence-to-structure model AlphaFlow. We report metrics as in Tab.
RMSF
Pw-RMSD
(f)
MAE
Median (1.48)
MAE (1)
DCCM (1)
PCA W22 (-)
Time [s]
AlphaFlow
0.86
0.59 (0.01)
1.51
(0.01)
1.47 (0.03)
32.0
AlphaFold 2 BBFlow
0.87
0.52 (0.01)
1.49
1.07 (0.01)
0.85
1.47 (0.03)
0.3+0.8
Table A.2: Performance of BBFlow with distorted cquilibrium structures as input. We add Gaussian noise with the respective standard deviation to the backbone atoms equilibrium structure, choose random MD conformations as equilibrium structure, and evaluate each model on the ATLAS test set. Units and settings in Tab.
RMSF
Pw-RMSD
DCCM
PCA
1 (t)
MAE (L)
Median (MD=1.48)
MAE (4)
r (f)
W3 (L)
BBFlow
0.90
0.42
1.49
0.77
0.87
1.33
BBFlow-O.1A
0.90
0.48
1.53
0.77
0.87
1.33
BBFlow-0.2A
0.90
0.48
1.62
0.79
0.87
1.32
BBFlow-random conf.
0.90
0.47
1.56
0.83
0.85
1.56
A.2
The role of wewnsseeoon-timesclee MD simulations of proteins
In the biophysics community, Molecular Dynamics (MD) is one of the most established method to study protein function and folding by analyzing conformational landscapes [20] Most commonly, functional protein movements leading to molecular recognition, ligand binding, catalysis and protein
folding span the timescale between microseconds to seconds [23 48 Sampling rare, large-scale conformational transitions with MD on the biologically relevant timescale, however, often com- putationally prohibitive. Mhort-timescal. simulations, while typically insufficient to observe rare
15

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events, nonetheless provide valuable insight into local dynamics and mechanisms relevant to protein function.
Among a myriad of applications, short-time scale MDs in the range of hundreds of nanoseconds, emulated by BBFlow and AlphaFlow, were employed develop antibodies against NMDA receptors [40], to study allosteric regulation in enzymes [50] 37 6 6 to optimize biocatalysts for thermostability [49], and to shed light on the inhibition of SARS-CoVe NSP 13 helicase (35). Given that BBFlow is capable of accurately emulating MD ensembles of 300 ns while being orders of magnitudes faster than current baselines, we believe it can become useful tool in the hands of practitioners.
A.3 Metries for conformational ensembles
The Root Mean Square Fluctuation (RMSF) of Ca atoms measures the magnitude of positional deviations of individual residues across the set of conformations. For a given residue, these fluctuations are calculated in reference frame that defined by aligning the entire protein the equilibrium structure and thus implicitly depend on the positions of all other residues. Consequently, RMSF can be interpreted as measure for flexibility, but also encodes global collective behaviour. As in AlphaFlow, we calculate the Pearson correlation between RMSF profiles (for an example see Fig. 3) obtained from MD and generated ensembles in order to quantify how well the shapes the profiles match. We also include the Mean Absolute Error (MAE) of per-residue RMSF to measure how well RMSF amplitudes are reproduced, and compare the median over all residues with the ground truth in order to quantify systematic over- inder-stabllizatin..
RMSF
Pairwise RMSD For each protein, we calculate the average Ca RMSD between any two conforma-
tions
Narie
pwRMSD
RMSD Zy).
(13)
N2
This quantifies the magnitude of conformational changes without requiring specified reference state. We report the MAE of pairwise RMSD across all proteins.
PCA A metric that explicitly accounts for conformational changes, and quantifies how well the re- spective conformations are captured, relies on the Principal Component Analysis of the Co positions across the sampled conformations. We project the generated states on the first two principal compo- nents obtained from MD, thus receiving a two-dimensional "CA-projection each conformation. We report the 2ebsserstein-2dditanre between the distributions of PCA-projections.
DCCM Another metric that accounts for directional and long-range degrees of freedom is the Dynamic Cross Correlation Matrix (DCCM), It measures for each pair of residues i, j whether they rather move in parallel, antiparallel or with uncorrelated relative direction. The entries are thus defined as
((T, (Tx)) (7) (m,)))
DCCM, =
(14)
( (T.)) ((E, (2,))))
where (...) denotes the ensemble average and T. denotes the Car atom position of residue For comparing the similarity of the DCCM matrices obtained with MD and with the generated ensemble, we report the Pearson correlation between all entries of both matrices.
A.3.1 Other metries for protein ensembles
While the above metrics are frequently used and well established in the field of protein dynamics, Jing et al. [6) introduce two new, less established metrics for conformational ensembles, which we report for completeness in exhaustive evaluation tables in App. A.13
The root mean Wasserstein distance (RMWD) introduced in 16 assumes that atom positions in an ensemble are distributed according to three dimensional Gaussian distributions. It reports the Wasserstein distance between Gaussian distributions fitted both the generated and the MD ensemble. However, the assumption that the atom positions are distributed according to
RMWD
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---

3D- AD-GGussians is a strong approximation for comparing real ensembles, the strong correlation between individual atoms should be taken into account 22
Weak contacts J Jing et al. [6 also report weak contacts as a metric defined as Co pairs that are contact (or not in contact) in the crystal structure but dissociate (or associate) in more than 10% of the ensemble structures, using an 8 Á distance cutoff. This metric informative for larger conformational changes, however, the cutoff values of 8 Å and 10% are arbitrary and might not apply to more heterogencous systems.
Table A.3: Evaluation of other ensemble generation methods (see App. A.4 as MD emulators on the ATLAS test set. Settings are the same as in Tab. | BBFlow-light is described in App. A.6
RMSF
Pw-RMSD
DCCM
PCA
Ju
Time
Median (MD=1.48)
(f)
MAE (.)
MAE (L)
r (t)
W2 (L)
% (1)
[s] (L)
MDGen
0.72
0.81 (0.01)
0.62
2.05 (0.01)
0.54
1.86 (0.03)
27
0.15
Str2Str
0.52
7.80 (0.01)
10.98
9.36 (0.01)
0.52
1.63 (0.03)
12
10.5
ESMDiff
0.69
(0.01)
3.00
4.95 (0.01)
0.75
1.84 (0.03)
27
0.39
BioEmu
0.83
1.29 (0.01)
2.34
2,84 (0.01)
0.80
1.65 (0.04)
36
1.9
BBFlow-light
0.89
0.48 (0.01)
L.38
0.86 (0.01)
0.86
1.32 (0.03)
31
0.14
BBFlow
0.90
0.42 (0.01)
1.49
0.77 (0.01)
0.87
1.33 (0.03)
29
0.77
A.4 Other ensemble generation models
For sampling alternative folding states and general ensembles that must not strictly follow the distribution of states obtain via MD simulation with fixed runtime and temperature, several models have been developed recently. Lewis al. [24] propose the generative model Bio-Emu, which is trained on large custom dataset containing MDs of greatly varying lengths with an architecture similar AlphaFold A diffusion module used for generating protein structures from the learned sequence representation. which relies on MSA. Another model trained on non-standardized MDs and also NMR data is ESMDiff (32), which relies on a Structure Language Model (SLM). In the proposed SLM, discrete variational autoencoder is used to encode structure into tokens, whose relationship to the protein sequence is modeled by a language model. By fine-tuning the SLM, protein conformations can be sampled from the sequence. It has also been proposed to generate ensembles using models that are only trained on static structures, such as Str2Str [31],, where noise added to the equilibrium structure and diffusion model is used to generate ensembles by partial denoising. Another example for such an approach is MSSA-ubbampliig [47), where AlphaFold 2 is applied with modified MSA in order to sample alternative folding states.
Since one would not expect such models to sample the distribution induced by standardized MD to high accuracy, we only benchmark them for illustrative purposes in the appendix (Tab.(..)) and focus on the more relevant baselines AlphaFlow and ConfDiff in the main part of this paper.
Also for ATLAS-likk ensembles, there related work alongside ConfDiff: Li et al. 25 propose modification of AlphaFlow, in which intermediate features of previous timesteps re-used, leading to improved efficiency. However, neither code or model weights are published and only limited subset of metries reported the paper, prohibiting direct comparison with BBFlow and other baselines.
A.5 Ensemble generation for multi-chain proteins
Multimeric systems order to assess BBFlow's 's performance for multimers, we simulate five multi-chain systems: a dimeric barnase- (PDB: IBGS), small homotrimer foldon (PDB: IRFO), homodimeric fructokinase in its apo state (PDB: 5EY7), heterodimeric nanobody-SARS. SSRS-CARS RBD complex (PDB: 7KGK) and heterodimeric nanobody-TNFRSFT
complex (PDB: 8HXR). We use simulation settings analogous to ATLAS (App. IO). Selected complexes span total residue count between 80 and 590. We report BBFlow's performance in Tab. A.4
17

---

Baselines for multi-chain proteins None of the baselines discussed in Sec. 1.1 or App. A.4
have been applied multi-chain proteins before. Nonetheless, we tested the most relevant baseline. AlphaFlow [16], for the multi-chain systems. By default, AlphaFlow does not allow to parse multi- chain proteins. We thus modified the parser accept multi-chain inputs.
More specifically, we took as an example a barnase-barstar complex (PDB: IBGS), computed an MSA (pair mode: umpaired_paires msa mode: mmseqs2_uniree www.uuiieueeee env) by querying the ColabFold mmseqs2 search server and used ColabFold's multimer parsing functions (unserialize e_msa(_))) followed by generate_inpae featurffeuureee (-)) to generate multimeric features that are compatible with AlphaFold's ss inference logics. We ran AlphaFlow inference with 10 timesteps by passing multimeric features as imput and observed that the model fails sample physical, protein-alike states despite good sequence coverage, as shown in Fig. A.1
Table A.4: BBFlow's performance for the five multi-chain proteins described in App.
A.5
evaluated
the settings Tab.
RMSF
Pw-RMSD
DCCM
PCA
(t)
MAE (L)
Med. (MD=1.2)
MAE (J)
(t)
W: (1)
BBFlow
0.82
0.31 (0.01)
1.40
0.41 (0.02)
0.85
0.71 (0.06)
A
B
1BGS
1.00
3000
Sequences
0.75
Coverage
2000
0.50
1000
0.25
0.00
0
50
100 150
Positions
Figure A.I: AlphaFlow predicts unphysical states for the multi-chain barnasc-barstar complex (PDB: 1BGS) if its parser is modified to accept multi-chain proteins. In (A). we show an example state sampled by AlphaFlow. (B) The parsed multimeric MSA features of the barnase-barsee complex show broad sequence coverage with many sequence hits for each of the chains.
A.6 BiFlot-light model
We also train BBFlow model with a set of hyperparameters different from those described in Sec. 4 chosen increase efficiency on the cost of accuracy. We call the model BBFlow-light and report its performance in Tab. A.3 We find that it is around 200 times faster than AlphaFlow and around 10 times faster than AlphaFlow-Tiz, dis while outperforming the distilled AlphaFlow models in most accuracy metrics.
In contrast to the vanilla BBFlow, BBFlow-light consists of 3 CFA message passing blocks instead of 6 and reduced node- and edge-feature dimensions of 96 and 48, respectively. BBFlow has around 18.2 M learnable parameters, while BBFlow-light contains only 2.5 M.
A.7 Training algorithm
In Algorithm
I
we summarize the training procedure described in Sec.
A.8 Role of the hyperparameter g
We study the effect of the hyperparameter that controls the interpolation strength between the equilibrium structure and the unconditional prior in Eq. & Smaller values correspond to noisier
18

---

Algorithm Training of BBFlow
(6) M Require: Dataset D ((eyy {ar Timll ), with one equilibrium structure Teey SE(3) and
several conformations ( E SE(3) per protein
Require: Model ø8
1: while training do
2: (Feq. 21)
Sample equilibrium structure and one conformation
3: To ~ pol: Tey)
Sample noise from conditional prior (Eq. 8)
4:
t~ U(0,1)
Sample flow matching time
5:
It = r( 11,
D Interpolated state It E SE(3) V 7 geodesic between æi and ø1 Calculate ground-truth flow vector T>, SE(3) from Eq. 3
6:
USE(3) (ate; (1)1)
7:
f to (2t, t, Feg)
> Calculate model output #1 E SE(33
8:
UsE(a) (zt, t(
D Calculate predicted flow vector T2, SE(3) from Eq.
9:
FFM ||v 8 a Laxx (1)
Flow matching loss
10: Update parameters using Ve
11: end while
initial states To, increasing ensemble diversity but also making training more challenging. Conversely, larger values bring 2g closer to the equilibrium structure, facilitating convergence but potentially reducing ensemble diversity.
Training- ablation. To quantify this tradeoff, we trained separate BBFlow models using different values of ξ. As shown in Tab. A.5 models trained with smaller & produce more diverse ensembles (higher RMSF), while those with larger & converge faster but exhibit reduced structural variability. We found that & = 0.2 provides good balance between accuracy and diversity, and use this value in all reported experiments.
Table A.5:
Ablation of models trained with different values of the hyperparameter &. Smaller increases ensemble diversity but slows convergence. Units and settings as in Tab, !
RMSF
RMSF MAB ())
RMSF (MD=1.48)
Pw-RMSD MAE (1)
DCCM r (1)
PCA Wa (l)
{
0.88
0.54
1.53
0.87
0.86
1.35
{ 0.4
0.89
0.44
1.39
0.89
0.85
1.33
Inference-time ablation. We further evaluated the model trained with & 0.2 using different values during inference. As seen in Tab. Tab. reducing & increases diversity (larger RMSF and PCA W2) but reduces agreement with molecular dynamics reference data. Increasing & has the opposite effect, leading to more constrained ensembles.
Table A.6: Inference-time ablation of & using the model trained with & = 0.2. Smaller & increases ensemble diversity but reduces agreement with MD reference structures. Units and settings in Tab.
3
RMSF r (t)
RMSF MAE (L)
RMSF (MD=1.48)
Pw-RMSD MAE (1)
DCCM r (t)
PCA W3 (L)
0.01
0.70
7,61
11.16
10.83
0.73
2.85
0.05
0.81
3.89
5.43
5.38
0.78
2.06
0.1
0.88
1.32
2.62
2.21
0.84
1.51
0.2
0.90
0.42
1.49
0.77
0.87
1.33
0,3
0.89
0.47
1.37
1.02
0.86
1.69
0.4
0.86
0.64
1.65
1.31
0.80
2.12
0.5
0.79
0.80
1.82
1.50
0.74
3.00
0.6
0.73
0.82
1.70
1.65
0.70
4.51
A.9
Backbone dihedral distributions
In addition the metrics above, we investigate the distribution backbone dihedral angles across ensembles, commonly visualized in the field by Ramachandran plots. We show the Ramachandran
19

---

AlphaFiow
100
Prequarr
Po
-180
10
-186
-180
10
AlphsFlow.T
AlphaFlpw-Tux.u L2
BloEmu
100
180
120
130
120
12
-30
-228
-10
-10
-120
-50
60
120
100
-30
-220
-00
60
120
100
-30
-120
-50
310
300
(a)
AlphaFlow
100
Requinccy
neep aetic
P2(
Red
-00
1.0
100
130
-30
RioFmil
12a
221
100
Iie
Freg
-100
12
-100
322
-180
(h)
Ph (1)
(hi
(b)
Figure A.2: Ramachandran plot, i.c. histogram of backbone dihedrals, for MD and generated ensembles of the representative proteins 6xxXA (a) and 6nl2A (b) from the ATLAS test set. We report Wasserstein distances of the distributions for the entire ATLAS test set in Tab. A.7
plot of generated ensembles for a representative protein in Fig. A.2 and report the median of in Tab. A.7 We find
the Wasserstein distance the MD distribution across the ATLAS that BBFlow samples dihedral angles that deviate slightly more from MD than ESMFlow-T and AlphaFlow-T, but is competitive with the distilled models and those without templates.
Table A.7: Similarity of Ramachandran dihedral distribution across generated ensembles. We calculate the Wasserstein-2 distance between the dihedral angle distribution induced by MD and the respective generated ensemble and report medians across the ATLAS test set.
Metrie
AlphaFlow
BioEmu
AlphaFlow-T
EsmFlow-T
AlphaFlow To
AlphaFlow-T
BBFlow
Rama. W/ (1)
0.53
0.66
0.48
0.47
0.51
0.55
0.52
20

---

A.10
De novo proteins dataset
Protein generation As described Sec. 4.2 we assess the performance of BBFlow on a set of de
proteins. We sample 20 protein backbones with FrameFlow [53] and RFdiffusion [46] for each
length L E [60, 65, ................... 512]. For each individual generated backbone, we earry out a colf-consisteney evaluation pipeline previously proposed [52] 26] by designing 8 sequence with ProteinMPNN [T] and refolding candidate sequences with ESMfold [27]. We then compute the length distribution of the ATLAS dataset and select 50 refolded backbones that have self-consistency RMSD (scRMSD)
of < 2.0 Á to the originally generated backbone that follow size distribution similar to the ATLAS
dataset each for FrameFlow and RFdiffusion.
MD setup MD simulations are performed using GROMACS v2023 M. utilizing the CHARMM27 all-atom force field. Proteins are embedded in periodic dodecahedron box, ensuring a minimum separation of nm from the box boundaries. The simulation system hydrated using the TIP3P water model (18), and the ionic strength is adjusted to a NaCI concentration of 150 mM. An initial energy minimization is carried out for 5000 steps. The system undergoes NVT equilibration for I ns with timestep of 2 fs, employing the leap-frog integrator. Temperature control is achieved at 300K using the Berendsen thermostat. This followed by NPT equilibration for 1 ns, where the pressure maintained at 1 bar using the Parrinello-Rahme extends over three 100 ns replicas. Throuugouut the simulations, covalent bonds involving hydrogen constrained using the LINCS algorithm (I3]. Long-range electrostatic interactions are treated using the Particle-Mesh Ewald (PME) method.
barostat. The production run of the simulation
A.11 ConfDiff inference setup
We evaluate ConfDiff using the ConDDiff-OF-r3MMD model, which is fine-tuned on the ATLAS dataset, available on GitHub We use the default hyperparameters for generating conformations.
A.12 Additional figures
As extension of Fig. we show the tradeoff between accuracy and speed with more accuracy metrics in Fig.
A.3
The construction of the conditional prior is visualized in Fig.
A.4
We show an extension of Fig. to all metrics from Tab.
D in Fig.
A.6
In Fig.
A.5
we show the RMSF profiles of the multimers displayed in Fig.
In Tab. A.8
report the inference time of AlphaFlow's backbone and sidechain module, compared
with BBFlow.
Alpha
pw- RMSD MAE [
1.75
AlphaFlow Tix, del
1.6-
AlphaFicuu FlooFFowwhho &er
RMSF MAE [A]
0.8
0.7
AlphaFlowsT.ww
1.50
Aiphollow Tey
Confboff
PCA x>>
1.5
ESMFlow-T
0.6
Contiir
1.25
ESMFIOW-T
1.4
Alphaflow-
ConfDitt
ESMFlow.T
0.51
BBFlow-light
BBFlow-light
AlphaFlow-T
BDFlOw
BBFlow
Nohaf low-T
1.3-
BBFlow-light
Alphaflow
0.4
0.75
10
10a
10
10
10"
10'
Inference time per conf. [s]
Inference time conf. [s]
Inference time per conf. [s]
Figure A.3: Trade-off between accuracy and speed of MD emulation. While other methods are either efficient accurate, BBFlow performs well at both. As extension of Fig show the accuracy metrics RMSF MAE, pairwise RMSD MAE and PCA W2 (all favorable if smaller). Both the accuracy metries and inference time are averaged over the ATLAS test set. The other metrics (Pearson correlations and Median PW-RMSD) do not show such clear trends in terms of correlation with inference time and can be found in Tab.
commit
21

---

X uncond ~ P Purconal
Xo ~ Po xx)
For given equilibrium structure as condition Teq. we sample noise Funcond from the unconditional prior Puncond and interpolate along the geodesic between Funcomd and Jeq (Eq. 9 to obtain sample zo from the proposed conditional prior po(-(2ey)) In the experiments, we choose the hyperparameter g 0.2; in the figure, we show a state sampled with & = 0.5 for better visualization.
Figure A.4: Construction of the conditional prior (Eq. 8
1BGS
8HXR
5EY7
NO BERTOM
RMSF [Á]
BeH
0-82
RMSF [A]
RMSF [Å]
MAE 0.21
MAG:
WAAA Masan wam
w
0
100
200
50
100
150
200
400
600
Residue index
Residue index
Residue index
Figure A.5:
RMSF profiles of three dimeric proteins whose DCCM matrices are depicted Fig. computed either with MD or BBFlow. Chain boundaries are indicated by black vertical lines.
Table A.8: Inference time by module of a single AlphaFlow and BBFlow forward pass on a 300 residue protein. The sidechain module of AlphaFlow is entirely separate from the backbone module and only takes a fraction the backbone module's 's runtime. Inference time is therefore dominated by the backbone prediction task, for which BBFlow achieves speedup.
Module
Backbone [s] (L)
Sidechain [s] (L)
AlphaFlow-T BBFlow
32.5
0.12
0.8
A.13
Exhaustive evaluation tables
We report the performance of BBFlow and baselines including the new metrics introduced in [6
(see Sec. A.3) in Tab. A.9 and Tab. A.1I
A.14 Societal impact
We consider the societal impact of this work mostly positive since understanding protein dynamics is essential for the development of new drugs, therapies and even materials, which outweighs potential risks.
22

---

MD
BBFlow
AlphaFlow-T
AlphaFlow-Te 122,
TW-RRMDD AE (A]
RMSF MAE [A]
2.0
PCA W2
2.0
1.0
2.0
0.0
0.0
0-100
100
300
300+
0 100
100 300 300+
0-100
100
300
300+
Protein length
DW-RMSD [Å]
10.0
RMSF [A]
7.5
RMSF
2.0
0-100
100-300
0-100
100-300
0-100 100-300 300+
Protein length
Figure A.6: Additional metrics for the performance of BBFlow, AlphaFlow-T and AlphaFlow-T TZL.dis on the ATLAS test set for different protein lengths. We divide the protein lengths in three bins and calculate per-residue RMSF, RMSF MAE, RMSF correlation r, per-protein RMSD, absolute error of pairwise RMSD and PCA W2 of each protein with length in the respective bin. The boxes depict minimum, maximum, median, and the 0.25 and 0.75 quantile.
MD
BBFlow
AlphaFlow-T
NiphaFlow-T1x.0n
DW-RMSD AE [Á]
RMSF MAE [A]
2.0
0.B
1.5
1.0
0.5
1.0
0.5
0.2
0.5
0.0
0.0
0.0
0-100
100-300
300+
0-100
100-300
300+
0-100
100-300
300+
Protein length
1.5
pw- SD [A]
1.0
RMSF [A]
2.0
RMSF
0.5
1.0
D-100
100-300
0-100
100-300
300+
100-300
300
Protein length
Figure A.7: Performance of BBFlow, AlphaFlow-T and AlphaFlow-TizL.d on the de novo protein dataset for different protein lengths. We divide the protein lengths in three bins and calculate per- residue RMSF, RMSF MAE, RMSF correlation r. per-protein RMSD, the absolute error of pairwise RMSD and PCA W2 of each protein with length in the respective bin. The boxes depict minimum, maximum, median, and the 0.25 and 0.75 quantile.
23

---

Table A.9: Evaluation the ATLAS dataset including the metrics introduced in [6 RMSF and RMWD are calculated from Ca atoms.
(see Sec.
A.3)
AlphalFlow
BioEmu
ConfDifl'
AlphaFlow-T
BsmFlow-T
AphhFllowTiz...c
BBFlow
Pairwise RMSD (=2.9)
2.89
4.29
3.43
2.18
2.0
1.4
3.09
Pairwise RMSD y
0.48
0.62
0.94
0.82
Pairwise RMSD MAE
1.35
2.84
1.45
0.91
1.22
0.77
C. RMSF (=1.48)
1.51
2.34
2.0
1.17
0.94
0.68
1.49
Global RMSF y
0.58
0.46
0.7
0.91
01.4
0.74
0.84
target RMSF
0.86
0.88
0.92
0.92
0.9
target RMSF MAE
0.59
1.29
0.62
0.41
0.52
0.85
0.42
Global D
0.8
0.73
0.85
0.88
0.87
0.85
0.84
Per target DCCM
0.86
0,80
0.86
0.89
0.89
0.87
0.87
target DCCM MAE
0.15
0.19
0.14
0.12
0.12
0.15
Root mean Wy-distance
2.38
2.56
1.72
1.91
Traassation contrih.
2.17
2,51
2.02
1.47
1.52
1.73
1.65
Variance contrib.
1.18
1.99
1.22
0.82
0.92
1.2
0.93
PCA Wa-distance
1.47
1.41
1.28
Joint PCA Wa-distance
2.26
2.90
2.19
1,58
1.76
1.93
1.72
36 PC-sim 0.5
43.85
24.49
37.76
44.6
47.94
39.12
40.1
Weak contacts
0.62
0.46
0.63
0.62
0.59
0.56
0.57
Transient contacts
0.41
0.39
0.47
0.47
0.29
[s]
32,0
20.2
32.6
11.2
Table A.10: Evaluation the de novo dataset including the metrics introduced in 16 RMSF and RMWD are calculated from Ca: atoms.
(see Sec.
A.3
AlphaFlow
BioEmu
ConfiDiff
AlphaFlow-T
FsmFlow-T
Alphahlow-T
BBFlow
Pairwise RMSD (=1.59)
8.08
7.9
7.27
1.25
1.2
0.68
1.47
Pairwise RMSD y
0.2
0.24
0.86
0.7
Pairwise RMSD MAE
7.4
8.29
7.26
0.38
0.43
0.97
0.32
C. RMSF (=0.91)
7.09
7.56
6.35
(1.74
0.68
0.41
0.87
Global RMSF r
0.27
0.25
0.28
0.86
0.86
0.83
0.77
target RMSF
0.47
0.62
0.89
0.84
target RMSF MAE
4.76
4.24
3.82
0.25
0.28
0.58
0.26
Global DOCM
0.52
0.53
0.54
(.779
0.8
0.77
0.77
Per target DCCM
0.58
0.64
0.65
0.85
0.86
0.83
0.83
target DCCM
0.22
0.21
0.11
0.1
0.14
Root mean Wy-distance
10.17
7.32
1.02
Translation contrib.
8.01
4.25
6.91
0.9
0.85
1.0
1.06
Variance contrib.
5.5
5.2
4,29
0.45
0.47
0.73
0.54
PCA Wa-distance
1.64
1.72
0.66
0.67
Joint PCA Wa-distanc,
9.08
5.23
7.32
0..5
0.89
1.13
1.09
& PC-sim 0.5
9.86
7.46
16.09
48.05
$0.19
40.99
39.37
Weak contacts
0.38
0.39
0.44
0.61
0.61
0.57
0.57
Transient contacts
0.17
0.15
0.55
[s]
32.6

---

Table A.11: Evaluation of ensemble generation methods from App. A.4 1.45
as MD emulators on the
ATLAS dataset including the metrics introduced in [16 (see Sec. A.3 RMSF and RMWD are calculated from Ca atoms.
MDGen
Str2Str
ESMDiff
BioEmu
BBFlow-light
BBFlow
Pairwise RMSD (=2.9)
1.34
13.29
5.36
4.29
2.6
3.09
Pairwise RMSD
0.48
0.15
0.22
0.23
0.81
0.82
Pairwise RMSD MAE
2.05
9.36
4.95
2.84
0.86
0.77
C.. RMSF (=1.48)
0.62
10.98
3.0
2.34
1.38
1,49
Global RMSF
0.49
0.28
0.31
0.46
0.82
0.84
Per target RMSF
0.72
0.52
0.69
0.83
0.89
0.9
Per target RMSF MAE
0.81
7.8
1.57
1.29
0.48
0.42
Global DCCM r
0.35
0.46
0.71
0.73
0.83
0.84
Per target DCCM r
0.54
0.52
0.75
0.80
0.86
0.87
target DCCM MAE
0.23
0.26
0.19
0.19
0.16
0.15
Root mean W2-distance
2.59
9.58
4.84
3,36
2.04
1.93
Translation contrib.
2.15
4.74
3,48
2.51
1.75
1,65
Variance contrib.
1.32
8.11
2.63
1.99
0.96
0.93
MD PCA Wy-distance
1.86
1.63
1.84
1.65
1.32
1.33
Joint PCA Wa-distance
2.47
6.49
3.94
2.90
1.81
1.72
PC-sim > 0.5
13.74
2.5
21.1
24.49
37.43
40.1
Weak contacts J
0.5
0.3
0.49
0.46
0.5
0.57
Transient contacts
0.27
0.12
0.36
0.36
0.31
0.29
Time [s]
0.15
10.5
0.39
1.9
0.14
0.77
Table A.12: Ablation study including the metrics introduced in [16].
Extension of Tab.
BBFlow
d
Pairwise RMSD (=2.9)
3.09
2.58
2.35
2.62
2.4
11.12
Pairwise RMSD
0.82
0.74
0.81
0.83
0.82
0.04
Pairwise RMSD MAE
0.77
1.15
0.9
0.82
0.93
7.08
C. RMSF (=1.48)
1.49
1.55
1.3
1.25
1.34
8.26
Global RMSF r
0.84
0.72
0.81
0.85
0.81
0.19
Per target RMSF
0.9
0.88
0.9
0.9
0.87
0.44
Per target RMSF MAE
0.42
0.52
0.48
0.42
0.54
5.88
Global DCCM r
0.84
0.82
0.85
0.85
0.84
0.48
Per target DCCM
0.87
0.85
0.86
0.88
0.85
0.55
Per target DCCM MAE
0.15
0.16
0.16
0.14
0.16
0.23
Root mean Wi-distance
1.93
2.09
1.95
1.97
2.19
7.81
Translation contrib.
1.65
1.75
1.67
1.82
4.22
Variance contrib.
0.93
1.04
0.98
0.94
1.03
6.3
MD PCA Wy-distance
1.33
1.46
1.44
1.32
1.47
1.32
Joint PCA Waddistance
1.72
1.93
1.8
1.82
1.96
4.53
% PC-sim 0.5
40.1
37.39
46.77
41.82
38.4
9.2
Weak contacts
0.57
0.52
0.45
0.55
0.52
0.31
Transient contacts J
0.29
0.3
0.32
0.31
0.26
0.1
25

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Reviews:
[{"review_id": "VsIfhfjtmj", "reviewer": "Reviewer_aKex", "summary": "In this paper, the authors present a method called \"BBflow\" for sampling protein structures from MD-derived steady state ensembles using a flow matching framework. The key idea is to condition on a template structure (the \"equilibrium structure\") in order to avoid the problem of requiring the neural network to be able to predict the general structure from sequence before learning ensemble conformations. As a result, the author's model can be trained from scratch rather than building on a pre-trained structure predictor and does not require multiple sequence alignments as input.\n\nBBflow is trained on the Atlas dataset and is primarily compared with AlphaFlow, where BBflow shows similar to favorable metrics. The authors also consider de novo designed proteins and multi chain proteins which are challenging or impossible for current methods.", "questions": "* How sensitive is the model to the quality of the equilibrium structure and can samples escape the energy well around that structure?\n* Similarly, for conditional prior distribution, does the change of xi affect the training time, e.g., if we have a prior closer to the equilibrium structure, the model converges faster? But then, will this change the diversity of the predicted conformations maybe?\n* In principle, evolutionary information should encode protein dynamics if those dynamics are important for function/fitness. Are there situations where excluding this information is harmful?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n* The paper is well written, the method and results are clearly described, and the method is well motivated\n* The core idea is elegant (condition on the equilibrium structure and let the model just learn the physics rather than needing to also fold the structure from scratch) and its merit bears out in the experiments\n* Experiments are comprehensive\n\nWeaknesses\n* This is not specific to this work, but the metrics used (e.g. in Table 1) are not intuitive. It's hard to understand how exactly these correspond to, e.g., whether the densities of the learned distribution match the ground truth and what good baselines would be. For example, pairwise RMSD is influenced by the actual diversity of the MD samples (if the MD samples are high diversity, pairwise RMSD should be worse) but we don't have an MD replicate baseline to understand how far these methods are from \"perfect\"\n* Not clear how sensitive the method is to poorly estimated equilibrium structures and the method seems pretty reliant on these being accurate. Can it effectively escape the well of the equilibrium structure to sample other low energy conformations or does it primarily stay around the equilibrium structure? Looking at a protein with known large conformational shifts would be interesting\n* Some more details on the implementation/algorithms for the components used from existing works (although cited) would make the method easier to follow as a stand alone work", "was_revised": true, "final_justification": "The rebuttal mostly addressed my questions, but I think this is a useful contribution anyway and is high enough quality for acceptance.", "rebuttal": "We cordially thank the reviewer for their feedback on our paper. We are happy to read the reviewer finds our paper well-written, the core idea elegant and the experiments comprehensive. Below, we address these points raised in the review and present corresponding additional experiments that we ran during the rebuttal period.\n\n&nbsp;\n\n*A1.1 - Addressing W2/Q1: Sensitivity of the model to the quality of the equilibrium structure*\n\n**Short answer:**\n\nIf the quality of the equilibrium structure is reduced, BBFlow's performance only decreases slightly, indicating robustness under noisy equilibrium structures.\n\n**Long answer:**\n\nWe agree with the reviewer that robustness under changes of the equilibrium structure is a good consistency check and also relevant in practice. In Table A.1 of the original submission, we reported the performance of BBFlow if AlphaFold2-predicted structures, instead of the equilibrium structures from the ATLAS dataset, are passed to BBFlow. In this setting, we find only a slight decrease in performance compared to the equilibrium structures used in ATLAS, which can be expected since the MD simulations are not long enough (not converged to statistical equilibrium) for assuming independence from the starting structure. Importantly, BBFlow is still more accurate than the other baselines that have no access to the equilibrium structure from ATLAS, such as AlphaFlow without templates. \n\nWhile we regard AlphaFold2-predicted structures as the most relevant class of equilibrium structures in practice (AlphaFold2 is heavily used by practitioners and AlpaFold2-predicted structures are available for millions of naturally occurring proteins), we also ran an additional experiment during the rebuttal period, where we distort the equilibrium structures by adding noise.\nTo this end, we added Gaussian noise to the Euclidean backbone coordinates of the ATLAS test set proteins and reran inference with BBFlow. We find that the performance remains strong and decreases only slightly. We report the results in Table R1 below.\n\n||RMSF r|RMSF MAE|RMSF (MD=1.48)|pw-RMSD (MD=2.9)|pw-RMSD MAE|DCCM r|MD PCA W2|\n|:-|-:|-:|-:|-:|-:|-:|-:|\n|BBFlow|0.9|0.41|1.49|3.09|0.77|0.87|1.33|\n|BBFlow-0.1A|0.9|0.48|1.53|3.17|0.77 |0.87 |1.33|\n|BBFlow-0.2A|0.9 |0.48 |1.62 |3.3  |0.79 |0.87|1.32|\n|BBFlow-random conf as input |0.9 |0.47 |1.56|2.99|0.83|0.85|1.56|\n\n*Table R1. Performance of BBFlow on distorted equilibrium structures. We add Gaussian noise with the respective standard deviation to the backbone atoms equilibrium structure, or choose random MD conformations as equilibrium structure, and evaluate each model on the ATLAS test set. Units and settings as in Table 1.*\n\n&nbsp;\n\n*A1.2 - Addressing W2/Q1: Escaping the energy well around the eq. structure*\n\nWe can understand the concern of the reviewer that BBFlow might be biased towards predicting states that are close to the equilibrium structure since our method heavily relies on that structure. However, already from the median RMSF column reported in Tables 1 and 2 from the original submission, it can be seen that BBFlow predicts off-equilibrium states to a similar extent as MD, while AlphaFlow-T samples too close to equilibrium, as indicated by the smaller median RMSF. We will stress this important point more in the final version.\n\nTo make the observation more explicit, we introduced an additional analysis during the rebuttal period. We calculated the average RMSD of the generated samples from the equilibrium structure for each protein and report the results in Table R2 below. We find the same trend as in the original submission: BBFlow samples states that are on average as far away from the equilibrium structure as those explored by MD, while AlphaFlow-T samples states that are closer to the equilibrium structure.\n\n\n| Method       | Mean RMSD to Eq. Structure [Å] |\n|--------------|--------------------------|\n| MD           | 3.0                      |\n| BBFlow       | **2.5**                  |\n| AlphaFlow-T  | 2.1                      |\n| BioEMU       | 4.0                      |\n| AlphaFlow    | **3.5**                  |\n\n*Table R2. Deviation from the equilibrium structure on the ATLAS test set. Closer to MD is better.*\n\n&nbsp;\n\n*A1.3 - Addressing Q2: Effect of hyperparameter xi on training time and diversity of ensembles*\n\nWe thank the reviewer for suggesting this experiment, demonstrating the effect of xi, which we performed during the rebuttal period and report the results in Table R3. We indeed observe that diversity increases with smaller xi, in the sense that pairwise RMSD and median RMSF of the model trained with xi=0.4 is smaller than that of the xi=0.2 and xi=0.1 models. For the inference-xi-ablation (Table R4), we observe a tradeoff between accuracy (RMSF r, RMSF MAE, PCA W2) and diversity.\n\nAs anticipated by the reviewer, we also observe that training converges faster with larger xi, but we did not perform extensive analysis on this effect since it requires training several models per xi and we regard the importance of speeding up training as not too high, compared to speeding up inference.\n\n||RMSF r|RMSF MAE|RMSF (MD=1.48)|pw-RMSD (MD=2.9)|pw-RMSD MAE|DCCM r|MD PCA W2|\n|:-|--:|-:|-:|--:|-:|-:|-:|\n| xi=0.1 |0.88 |0.54 |1.53 |2.74 |0.87 |0.86 |1.35|\n| xi=0.4 |0.89 |0.44 |1.39 |2.64 |0.89 |0.85 |1.33|\n\n*Table R3. Ablation of models trained with different choice for the hyperparameter xi. In the paper, we chose xi=0.2 based on manual optimization on the validation set. Units and settings as in Table 1.*\n\n| xi   | RMSF r | RMSF MAE | RMSF (MD=1.48) | pw-RMSD (MD=2.9) | pw-RMSD MAE | DCCM r | MD PCA W2 |\n|-----:|-------:|---------:|---------------:|-----------------:|------------:|-------:|----------:|\n| 0.01 |   0.7  |     7.61 |          11.16 |            14.15 |       10.83 |   0.73 |      2.85 |\n| 0.05 |   0.81 |     3.89 |           5.43 |             8.19 |        5.38 |   0.78 |      2.06 |\n| 0.1  |   0.88 |     1.32 |           2.62 |             5.2  |        2.21 |   0.84 |      1.51 |\n| 0.2  |   0.9  |     0.41 |           1.49 |             3.09 |        0.77 |   0.87 |      1.33 |\n| 0.3  |   0.89 |     0.47 |           1.37 |             2.49 |        1.02 |   0.86 |      1.69 |\n| 0.4  |   0.86 |     0.64 |           1.65 |             2.5  |        1.31 |   0.8  |      2.12 |\n| 0.5  |   0.79 |     0.8  |           1.82 |             2.6  |        1.5  |   0.74 |      3    |\n| 0.6  |   0.73 |     0.82 |           1.7  |             2.33 |        1.65 |   0.7  |      4.51 |\n\n*Table R4. Ablation of the model from the paper, trained with xi=0.2, for inference with different values of xi. Units and settings as in Table 1.*\n\n&nbsp;\n\n*A1.4 - Addressing Q3: Evolutionary information encoding dynamics via fitness*\n\nIndeed, evolutionary information (e.g. MSA) can encode information about functionally or structurally important sites in proteins and their dynamics. \n\nAs we demonstrate in our paper, evolutionary information is not strictly required for emulation of 300ns MD simulation if abundant training data (such as the ATLAS dataset) is present. We expect that evolutionary information might become a necessity for predicting conformational changes occurring at longer timescales, where the task becomes increasingly more challenging, especially given the lack of large datasets covering micro- and millisecond protein dynamics. This is, however, out of scope of the paper as we describe in sections 1.1, A.3 and 'limitations' of the original submission. We added the above discussion on the role of evolutionary information to section A.3.\n\n&nbsp;\n\n*A1.5 - Addressing W3: More detailed explanations on models and algorithms cited from other papers*\n\nWe thank the reviewer for making us aware of this. We will add an overview on the respective resources to the appendix of the final version."}, {"review_id": "tKudVdqzks", "reviewer": "Reviewer_QtLV", "summary": "This work proposed BBFlow, a flow matching framework to generate conformational ensembles of protein. With conditional prior, the model is capable of generating high-fidelity conformational ensembles of protein backbones with short running time. The model can also generate conformational ensembles of multimers, which is valuable.", "questions": "1. A comment about the necessity of weights of a folding model or evolutionary sequence information: The structure itself contains sequence information implicitly. As BBFlow uses a prior that’s derived from the equilibrium structure, I believe it’s reasonable to expect the folding model weights or MSA is not critically needed. It is nice that the authors validated that through this paper. \n1. It is not very fair to claim significant speedup over AlphaFlow because the BBFlow only generates the backbone ensemble instead of the all-atom ensemble. The computation burden of all-atom vs backbone generation is not comparable. This is my largest concern regarding this work. \n1. Index encoding: Do authors have any intuition on why index encoding can hurt the performance of the model as shown in ablation? I understand the explanation in line 136-139, but that does not give intuition on the negative effect.\n1. Ablation study: I am very curious about the ablation of the hyperparameter $\\xi$ in the conditional prior. It would be nice if the author can show that. \n1. Given conditional prior being close to the equilibrium state (or one of the many equilibrium states), I would imagine the model can better sample conformations that are close to the equilibrium state (geometrically or on the energy landscape). How does the model perform in terms of sample diversity? AlphaFlow offers some analysis on this front, and it would be nice if the authors can provide some analysis on sample diversity or energy landscape coverage. \n1. Line 108: typo “benefitial”.\n\nI'll be happy to raise my score if my concerns and questions are addressed.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strength**\n1. Interesting idea of backbone ensemble generation and well-executed experiments.\n1. The proposed model BBFlow is capable of generating the ensemble of multimers, which is highly valuable.\n\n**Weakness**\n1. Inference time benchmark is not an apple-to-apple comparison. Baseline models such as AlphaFlow and ESMFlow generate protein conformation ensembles with all-atom resolution, while BBFlow only generate backbone ensembles. \n1. Although the value of generating the backbone ensemble should be recognized, lack of side-chain details prohibited some important result analysis such as ramachandran plots and ensemble energy landscape.", "was_revised": true, "final_justification": "Nice paper for protein backbone conformation ensemble generation that balances efficiency and accuracy. Despite the absence of side-chain related features and experiments, the work is valuable and should be accepted.", "rebuttal": "We cordially thank the reviewer for the time they invested in reading the paper and their constructive critique. We are happy that the reviewer finds our core ideas interesting, the experiments well-executed and highlights the valuable capability of BBFlow to sample ensembles of multimers!\nIn the following, we address all points raised in the review. We will include the respective additional experiments and explanations in the final version of the paper.\n\n\n&nbsp;\n\n*A2.1 - Addressing W1/Q2: Inference time comparison is not fair*\n\n**Short answer:**\n\nThis point is technically true, however, changing the settings to a strictly fair comparison only changes the values marginally. In the paragraph below, we provide a table with the respective values and remarks on the fairness of the comparison, which we also include in the final version of the paper. We thank the reviewer for bringing this potential misunderstanding to our attention, which allows us to clarify this in the final version as below.\n\n**Long answer:**\n\nAlphaFlow and ESMFlow first predict the backbone in the SE(3) frame representation, similar to BBFlow. As in AlphaFold2, the sidechain positions are then predicted in a post-processing step with a separate small neural network that only takes a fraction of the backbone-prediction inference time to run. Without considering sidechain conformations, AlphaFlow-T's inference time (per conformation on the protein 7c45A from Tab. 1 in original submission) is reduced from 32.6s to 32.5s while BBFlow only requires 0.8s. Thus, also in the strictly fair setting of backbone-only generation, BBFlow outperforms AlphaFlow-T in terms of efficiency, with virtually the same relative margin as reported in the original submission. In Table R5, we report the runtime of the individual modules of AlphaFlow and compare with BBFlow.\n\n| Module                               | Backbone | Sidechain |\n|--------------------------------------|----------|-----------|\n| AlphaFlow-T                          | 32.5 s    | 0.12 s     |\n| BBFlow                               | 0.8 s     | --        |\n\n*Table R5: Inference time by module of a single AlphaFlow and BBFlow forward pass on a 300 residue protein. The sidechain module is entirely separate from the backbone module and only takes a fraction of the time.*\n\nLastly, we would like to note that AlphaFlow, during post-processing of the predicted backbone structure, uses the same sidechain module as AlphaFold2 without introducing methodological changes or ablating performance of the sidechain module. In AlphaFlow, as in our paper, the focus is on the backbone structure, which encodes larger conformational changes and the overall structure of the protein. If practitioners are interested in sidechain configurations of the generated backbone ensembles, it is, in principle, possible to combine the sidechain module from AlphaFold2 also with BBFlow, as done in AlphaFlow. However, this requires re-training and is out of scope for the paper at hand.\n\n\n&nbsp;\n\n*A2.2 - Addressing W2: Lack of Ramachandran analyses and sidechain energy landscapes*\n\nWe added Ramachandran histograms of backbone atom dihedrals to the paper, comparing BBFlow, AlphaFlow-T and MD for four randomly sampled proteins from ATLAS.\nWe find that the Ramachandran histograms of both BBFlow and AlphaFlow-T indeed show overlap with MD (we cannot upload the plots during rebuttal since the global response with PDF is not supported this year and links are not allowed).\nThe Ramachandran plots are possible because backbone dihedrals can be calculated from BBFlow's predicted Euclidean coordinates for the backbone atoms. For energy landscapes, however, the sidechain conformation is required in order to apply a force field.\nWhile further sidechain analyses would be interesting indeed, they are out of scope for both our paper and also AlphaFlow. Instead, the main contribution of both papers is predicting ensembles of backbone structures. The backbone structure captures overall conformational changes and dominates computational cost of state-of-the-art models like AlphaFold2 or Alphaflow, as shown above. Thus, also the authors of AlphaFlow (the most relevant baseline) do not perform analyses on the recovery of the sidechain conformations, Ramachandran plots or energy landscapes, possibly because the sidechain prediction method is not novel but done as post-processing step with the same network as in AlphaFold2, as described above.\n\n\n&nbsp;\n\n*A2.3 - Addressing Q4: Ablation study on the xi hyperparameter*\n\nWe thank the reviewer for suggesting this experiment, which we performed during the rebuttal period and report the results in Table R6. We observe that diversity, as measured by median RMSF and pairwise RMSD, increases with smaller xi, in the sense that xi=0.1 and xi=0.2 gives better diversity than xi=0.4. For the inference-xi-ablation (Table R7), we observe a tradeoff between accuracy (RMSF r, RMSF MAE, PCA W2) and diversity.\n\nWe also observe that training converges faster with larger xi, but we did not perform extensive analysis on this effect since it requires training several models per xi and we regard the importance of speeding up training as not too high, compared to speeding up inference.\n\n\n||RMSF r|RMSF MAE|RMSF (MD=1.48)|pw-RMSD (MD=2.9)|pw-RMSD MAE|DCCM r|MD PCA W2|\n|:-|--:|-:|-:|--:|-:|-:|-:|\n| xi=0.1 |0.88 |0.54 |1.53 |2.74 |0.87 |0.86 |1.35|\n| xi=0.4 |0.89 |0.44 |1.39 |2.64 |0.89 |0.85 |1.33|\n\n*Table R6. Ablation of models trained with different choice for the hyperparameter xi. In the paper, we chose xi=0.2 based on manual optimization on the validation set. Units and settings as in Table 1.*\n\n\n| xi   | RMSF r | RMSF MAE | RMSF (MD=1.48) | pw-RMSD (MD=2.9) | pw-RMSD MAE | DCCM r | MD PCA W2 |\n|-----:|-------:|---------:|---------------:|-----------------:|------------:|-------:|----------:|\n| 0.01 |   0.7  |     7.61 |          11.16 |            14.15 |       10.83 |   0.73 |      2.85 |\n| 0.05 |   0.81 |     3.89 |           5.43 |             8.19 |        5.38 |   0.78 |      2.06 |\n| 0.1  |   0.88 |     1.32 |           2.62 |             5.2  |        2.21 |   0.84 |      1.51 |\n| 0.2  |   0.9  |     0.41 |           1.49 |             3.09 |        0.77 |   0.87 |      1.33 |\n| 0.3  |   0.89 |     0.47 |           1.37 |             2.49 |        1.02 |   0.86 |      1.69 |\n| 0.4  |   0.86 |     0.64 |           1.65 |             2.5  |        1.31 |   0.8  |      2.12 |\n| 0.5  |   0.79 |     0.8  |           1.82 |             2.6  |        1.5  |   0.74 |      3    |\n| 0.6  |   0.73 |     0.82 |           1.7  |             2.33 |        1.65 |   0.7  |      4.51 |\n\n*Table R7. Ablation of the model from the paper, trained with xi=0.2, for inference with different values of xi. Units and settings as in Table 1.*\n\n\n&nbsp;\n\n*A2.4 - Addressing Q5: Analysis on sample diversity*\n\nWe can understand the concern of the reviewer that BBFlow might be biased towards predicting states that are close to the equilibrium structure since our method heavily relies on that structure. However, already from the median RMSF column reported in Tables 1 and 2 from the original submission, it can be seen that BBFlow predicts off-equilibrium states to a similar extent as MD, while AlphaFlow-T samples too close to equilibrium, as indicated by the smaller median RMSF. We will stress this important point more in the final version.\n\nTo make the observation more explicit, we introduced an additional analysis during the rebuttal period. We calculated the average RMSD of the generated samples from the equilibrium structure for each protein and report the results in Table R8 below. We find the same trend as in the original submission: BBFlow samples states that are on average as far away from the equilibrium structure as those explored by MD, while AlphaFlow-T samples states that are too close to the equilibrium structure.\n\n| Method       | Mean RMSD to Eq. Structure [Å] |\n|--------------|--------------------------|\n| MD           | 3.0                      |\n| BBFlow       | **2.5**                  |\n| AlphaFlow-T  | 2.1                      |\n| BioEMU       | 4.0                      |\n| AlphaFlow    | **3.5**                  |\n\n*Table R8. Deviation from the equilibrium structure on the ATLAS test set. Closer to MD is better.*\n\n\n&nbsp;\n\n*A2.5 - Addressing Q3: Intuition for removal of index encoding*\n\nIntuitively, removing the index encoding reduces the potential for overfitting and incentivizes the model to generalize better. If the index of a certain (challenging) structural motif in the training set is available, the model might use the combination of motif and index in order to memorize the dynamics in the training set. The index should have no physical meaning but only structural context and amino acid identity, removing it as input can thus be interpreted as physical inductive bias. We will add a comment on this to the final version and thank the reviewer for making us aware of the need for clarification.\n\n\n&nbsp;\n\n*A2.6 - Addressing Q1: Comment about the necessity of weights of a folding model or evolutionary information*\n\nWe agree with the reviewer and identify this as one of the core ideas that underlie our method. We would like to note that also all other reviewers find the idea 'elegant', 'technically compelling' or 'conceptually reasonable'.\n\n&nbsp;\n\n*A2.7 - Addressing Q6: typo*\n\nWe corrected the typo and thank the reviewer for pointing this out!"}, {"review_id": "dUuPTrvvSM", "reviewer": "Reviewer_EssD", "summary": "The paper presents BBFlow, a deep generative model designed to learn conformational ensembles of proteins based solely on backbone geometry. By reframing the problem as a conditional generation task conditioned on equilibrium structural information, the authors aim to eliminate reliance on evolutionary data such as MSAs and large pre-trained protein folding models. The key contributions include a conditional encoding scheme that uses only equilibrium backbone geometry and a novel conditional prior integrated into the flow matching process.", "questions": "1.\tHow is the hyperparameter $\\xi$ in the conditional prior chosen? Could the authors provide insights into how sensitive model performance is to this parameter?\n\n2.\tIn Table A.3, BBFlow performs significantly better on multi-chain systems than on the ATLAS test set (Table 1). Given the increased complexity and out-of-distribution nature of multi-chain systems, this result is unintuitive. Could the authors offer an explanation?", "limitations": "The main limitations of the work are restricted empirical validation and marginal empirical gains over existing baselines like AlphaFlow-T.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Pros:\n\n•\tThe motivation to replace evolutionary features with pure structural priors is well-articulated and technically compelling.\n\n•\tBBFlow demonstrates state-of-the-art performance in terms of the speed–accuracy trade-off on a widely accepted benchmark.\n\n•\tThe model architecture—particularly the omission of residue indices—makes BBFlow naturally extendable to multi-chain systems, broadening its applicability.\n\nCons:\n\nDespite the promising framework, several issues limit the practical impact and technical novelty of the work:\n\n•\tGiven that BBFlow is conditioned on equilibrium structure, its most relevant baseline is AlphaFlow-T. The performance improvements over AlphaFlow-T are modest. In addition, as shown in the ablation study (Table 3), removing the conditional prior results in performance comparable to AlphaFlow-T, suggesting that the empirical impact of equilibrium structure encoding is limited.\n\n•\tSince BBFlow depends heavily on equilibrium structures for conditioning, the quality and accuracy of these structures could critically influence performance. The paper does not discuss robustness to noisy or approximate inputs, which would be important in real-world applications.\n\n•\tThe model is primarily evaluated on short-timescale MD simulations, where conformational variability is relatively small. The paper would benefit from additional results on proteins with complex, multi-state dynamics, such as those in BioEmu [2], to demonstrate real-world utility.\n\n•\tThere is no direct discussion or comparison with AlphaFlow-Lit [1], which also uses feature-conditioned generative modeling for efficient protein dynamics and serves as a relevant baseline.\n\n\n[1] Li S, Li M, Wang Y, et al. Improving AlphaFlow for efficient protein ensembles generation[J]. arXiv preprint arXiv:2407.12053, 2024.\n\n[2] Lewis, Sarah, et al. \"Scalable emulation of protein equilibrium ensembles with generative deep learning.\" bioRxiv (2024): 2024-12.", "was_revised": true, "final_justification": "The rebuttal has resolved most of my concerns. Thus, I will raise the score.", "rebuttal": "We thank the reviewer for the time they invested in reading the paper and their helpful feedback. We are happy that the reviewer finds our method technically compelling, sees it as state-of-the-art on a widely accepted benchmark and recognizes its applicability to multi-chain systems.\nFor resolving the concerns raised in the review, we conducted additional experiments, which we will include in the final version of the paper. In the following, we address all points individually.\n\n&nbsp;\n\n*A3.1 - Addressing W1: Performance improvements over AlphaFlow-T are modest*\n\nWe would like to note that performance is not only assessed solely via accuracy but also efficiency has to be considered. In this combination, the tradeoff between accuracy and efficiency, BBFlow's improvement over AlphaFlow-T is large: BBFlow is 40 times faster, at comparable accuracy. Additionally, it covers multi-chain systems, an important class of proteins for which AlphaFlow is not applicable. Both of these points are also recognized by other reviewers.\n\n&nbsp;\n\n*A3.2 - Addressing W1: The empirical impact of equilibrium structure encoding is limited*\n\nAs the reviewer states, removing the conditional prior indeed reduces performance (Table 3, line b). But removing the equilibrium structure encoding and keeping the conditional prior (line e) reduces performance even stronger. Thus, both of these innovations are needed to achieve state-of-the-art performance. Of course, line b suggests that the conditional prior applied within AlphaFlow-T (i.e. without eq. structure encoding but with MSA and Evoformer) might lead to the same performance in terms of accuracy, however, efficiency would be greatly reduced. Thus, also the equilibrium structure encoding can be regarded as crucial, especially for efficiency.\n\n&nbsp;\n\n*A3.3 - Addressing W2: Robustness under noisy or bad quality equilibrium structures*\n\n**Short answer:**\n\nIf the quality of the equilibrium structure is reduced, BBFlow's performance only decreases slightly, indicating robustness under noisy equilibrium structures.\n\n**Long answer:**\n\nWe agree with the reviewer that robustness under changes of the equilibrium structure is a good consistency check and also relevant in practice. In Table A.1 of the original submission, we reported the performance of BBFlow if AlphaFold2-predicted structures, instead of the equilibrium structures from the ATLAS dataset, are passed to BBFlow. In this setting, we find only a slight decrease in performance compared to the equilibrium structures used in ATLAS, which can be expected since the MD simulations are not long enough (not converged to statistical equilibrium) for assuming independence of the starting structure. Importantly, BBFlow is still more accurate than the other baselines that have no access to the equilibrium structure from ATLAS, such as AlphaFlow without templates. \n\nWhile we regard AlphaFold2-predicted structures as the most relevant class of equilibrium structures in practice (AlphaFold2 is heavily used by practitioners and AlphaFold2-predicted structures are available for millions of naturally occurring proteins), we also ran an additional experiment during the rebuttal period, where we distort the equilibrium structures by adding noise.\nTo this end, we added Gaussian noise to the Euclidean backbone coordinates of the ATLAS test set proteins and reran inference with BBFlow. We find that the performance remains strong and decreases only slightly. We report the results in Table R9 below.\n\n||RMSF r|RMSF MAE|RMSF (MD=1.48)|pw-RMSD (MD=2.9)|pw-RMSD MAE|DCCM r|MD PCA W2|\n|:-|-:|-:|-:|-:|-:|-:|-:|\n|BBFlow|0.9|0.41|1.49|3.09|0.77|0.87|1.33|\n|BBFlow-0.1A|0.9|0.48|1.53|3.17|0.77 |0.87 |1.33|\n|BBFlow-0.2A|0.9 |0.48 |1.62 |3.3  |0.79 |0.87|1.32|\n|BBFlow-random conf as input |0.9 |0.47 |1.56|2.99|0.83|0.85|1.56|\n\n*Table R9. Performance of BBFlow on distorted equilibrium structures. We add Gaussian noise with the respective standard deviation to the backbone atoms equilibrium structure, or choose random MD conformations as equilibrium structure, and evaluate each model on the ATLAS test set. Units and settings as in Table 1.*\n\n\n&nbsp;\n\n*A3.4 - Addressing W3: Evaluation on short-timescale MD simulations, not on more complex dynamics*\n\nIt is correct that also longer MD timescales and larger conformational changes are interesting. However, as we state in section 1.1, A.3, 'limitations' and 'Further baselines' in the original submission, this task is different from the one solved by AlphaFlow and BBFlow, in which MD is emulated under fixed, standardized settings. The ground truth ensembles strongly depend on these settings (temperature, timescale, forcefield). Thus, evaluating distributional accuracy quantitatively (and not rather qualitatively as in BioEmu) is only sensible if these settings are fixed. This becomes apparent in the poor performance of BioEmu on ATLAS as done in Table A.2 of the original submission and on de novo proteins in Table R10. This poor performance is expected because BioEmu does not claim to solve this specific task, conversely it can be expected that BBFlow and AlphaFlow would perform poorly if evaluated on BioEmu's task.\n\nLastly, we would like to note that not only the prediction of far alternative states but also emulation of short-timescale MD under standardized setting has a lot of applications. Shese kind of MD simulations are employed on a regular basis by practitioners all around the world. To state a few concrete examples: MD on the 300 ns timescale can be used for assessing opening-closing movements, hinge movements or flexibility of binding pockets [1, 2, 3].\n\n|| RMSF r | RMSF MAE | RMSF (MD=1.48) | pw-RMSD (MD=2.9) | pw-RMSD MAE | DCCM r | MD PCA W2 | Time/conf|\n|:--|--:|--:|--:|--:|--:|-:|--:|--:|\n|BioEmu|0.6|4.24 |7.56|7.9|8.29|0.64 |1.53 | 1.9|\n|BBFlow|**0.84**|**0.25**|**0.87**|**1.47**|**0.32**|**0.83**|**0.67**|**0.8**|\n\n*Table R10. Performance of BioEmu (and BBFlow) on the de novo proteins from Table 2 in the original submissions. Units and settings as in Table 2. Full set of metrics in Table R16.*\n\n--- \n\n[1] Raubenolt, Bryan A., et al. \"Molecular dynamics simulations of the flexibility and inhibition of SARS-CoV-2 NSP 13 helicase.\" Journal of Molecular Graphics and Modelling 112 (2022)\n\n[2] Kiss, Gert et al. \"Molecular dynamics simulations for the ranking, evaluation, and refinement of computationally designed proteins.\" Methods in Enzymology. Academic Press, 2013.\n\n[3] Wijma, Hein J. et al. \"A computational library design protocol for rapid improvement of protein stability: FRESCO.\" Protein engineering: Methods and protocols, 2017. 69-85.\n\n\n&nbsp;\n\n*A3.5 - Addressing W4: No direct discussion or comparison with AlphaFlow-Lit*\n    \nIn AlphaFlow-LIT inference time is sped up by calculating the evoformer output only once. We refrained from including AlphaFlow-LIT as baseline because, although published as workshop paper at ICML 2024, no code, model weights or generated samples are publicly available. These would be required for evaluating the model. This would be necessary since AlphaFlow-LIT [1] does not report all metrics from AlphaFlow [2] and since in Table 1 of [1], inconsistent values for AlphaFlow are reported (see Table R11), raising the question whether the same metrics are being used by [1].\n\nWe will include AlphaFlow-LIT as related work in the background section and explain, as above, why it cannot be evaluated.\n\n||AF|AF in [1]|BBFlow|AF-Lit|\n|:-|-:|-:|-:|-:|\n|Pairwise CA-RMSD (=2.90)|2.89|2.89|3.09|2.43|\n|Pairwise CA-RMSD r|0.48|0.49|0.82|0.58|\n|MD CA RMSF|1.48|1.94|1.48|1.94|\n|CA RMSF|1.51|1.88|1.49|1.65|\n|Per target RMSF r|0.86|0.75|0.90|0.77|\n*Table R11. Performance of AlphaFlow-LIT as reported in [1]. There is an inconsistency between AlphaFlow (AF) as reported in [1] and as reported in [2] and our paper, respectively.*\n\n[1] Li S et al. Improving AlphaFlow for efficient protein ensembles generation. arXiv preprint arXiv:2407.12053, 2024.\n[2] B. Jing et al. AlphaFold Meets Flow Matching for Generating Protein Ensembles. ICML 2024\n\n&nbsp;\n\n*A3.6 - Addressing Q1: How sensitive is the model with respect to the hyperparameter xi?*\n\nThe hyperparameter xi was chosen by brief, manual optimization on the validation set.\n\nWe thank the reviewer for suggesting an ablation study, which we performed during the rebuttal period and report the results in Table R12. We observe that diversity, as measured by median RMSF and pairwise RMSD, increases with smaller xi, in the sense that xi=0.1 and xi=0.2 gives better diversity than xi=0.4. For the inference-xi-ablation (Table R7 in answer to QtLV), we observe a tradeoff between accuracy (RMSF r, RMSF MAE, PCA W2) and diversity.\n\nWe also observe that training converges faster with larger xi, but we did not perform extensive analysis on this effect since it requires training several models per xi and we regard the importance of speeding up training as not too high, compared to speeding up inference.\n\n||RMSF r|RMSF MAE|RMSF (MD=1.48)|pw-RMSD (MD=2.9)|pw-RMSD MAE|DCCM r|MD PCA W2|\n|:-|--:|-:|-:|--:|-:|-:|-:|\n| xi=0.1 |0.88 |0.54 |1.53 |2.74 |0.87 |0.86 |1.35|\n| xi=0.4 |0.89 |0.44 |1.39 |2.64 |0.89 |0.85 |1.33|\n\n*Table R12. Ablation of models trained with different choice for the hyperparameter xi. In the paper, we chose xi=0.2 based on manual optimization on the validation set. Units and settings as in Table 1.*\n\n&nbsp;\n\n*A3.7 - Addressing Q2: Why are metrics better for multi-chain systems reported in A.3?*\n\nWe thank the reviewer for bringing this potential source of confusion to our attention. The reason for this effect is that some proteins are more challenging than others, thus the metrics between different sets of proteins cannot really be compared on an absolute scale, rather different baselines on the same set of proteins. Also, we would argue that there is no clear trend but it is rather mixed: RMSF r, RMSF median and DCCM r are worse than in Table 1 of the submission."}, {"review_id": "q4Usz5JnJs", "reviewer": "Reviewer_ndHw", "summary": "The paper presents a SE(3)-flow matching method for emulating MD ensembles, given access to a seed structure. Specifically, the model learns a flow from a prior centered around the seed structure to the MD ensemble using relatively lightweight (i.e., pairformer-free) architectures. When trained on the ATLAS dataset, it is shown to perform comparable to AlphaFlow while possessing significantly faster runtimes. Despite being trained only on naturally occuring protein monomers, the model is also shown to have nontrivial performance on de novo proteins and for dimers.", "questions": "See above.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n* The paper's architectural contribution is conceptually reasonable and a noteworthy improvement in terms of efficiency upon prior methods. \n* The paper's main claims are fully supported by the experiments and the paper is well written.\n* The paper provides the first demonstration of an MD ensemble emulator for dimers and de novo, a noteworthy step forward.\n\n\n**Weaknesses**\n* The main challenge with ensemble emulators is that practitioners are interested in sampling rare events, such as transitions to kinetically separated Markov states. It is very hard to assess whether these events are captured using the provided metrics. The transient and weak contacts metrics from AlphaFlow come closest to measuring these phenomena, but the presented method lags behind AlphaFlow by those metrics. The authors have also deferred these metrics to the appendix, which weakens the paper.\n* Why is BioEmu not included in the results in the main text?\n* Please provide the full set of metrics for the de novo designed proteins.", "was_revised": true, "final_justification": "I maintain my positive assessment of the work and its empirical results.", "rebuttal": "We thank the reviewer for their feedback and constructive critique. We are glad to read the reviewer finds our paper well-written, conceptually reasonable and the efficiency improvement noteworthy. Below we address weaknesses mentioned in the review.\n\n\n&nbsp;\n\n*A4.1 - Addressing W1: Rare events sampling*\n\n We agree with the reviewer that sampling conformational changes triggered by rare events is interesting for practitioners. However, these typically occur on MD timescales well beyond 300ns, which is the MD timescale of the ATLAS dataset. As we state in section 1.1, A.3, 'limitations' and 'Further baselines' in the original submission, sampling such large conformational changes and alternative states is a task that is different from the one solved by AlphaFlow and BBFlow.\n \n In AlphaFlow and BBFlow, MD is emulated under fixed, standardized settings.\n The ground truth ensembles strongly depend on these settings (temperature, timescale, forcefield). Thus, evaluating distributional accuracy quantitatively (and not rather qualitatively as in BioEmu) is only sensible if these settings are fixed. This becomes apparent in the poor performance of BioEmu in all metrics on ATLAS as done in Table A.2 of the original submission and on de novo proteins in Table R16. This poor performance is expected because BioEmu does not claim to solve this specific task, conversely it can be expected that BBFlow and AlphaFlow would perform poorly if evaluated on BioEmu's task.\n\n We would like to note that not only the prediction of rare events and far alternative states but also emulation of nanosecond-timescale MD has a lot of applications. These kinds of MD simulations are employed on a regular basis by practitioners for multiple different purposes. To state a few concrete examples: MD on the 300 ns timescale can be used for assessing opening-closing movements, hinge movements or flexibility of binding pockets and is useful for rational enzyme design. [1, 2, 3].\n\n--- \n\n[1] Raubenolt, Bryan A., et al. \"Molecular dynamics simulations of the flexibility and inhibition of SARS-CoV-2 NSP 13 helicase.\" Journal of Molecular Graphics and Modelling 112 (2022) \n\n[2] Kiss, Gert et al. \"Molecular dynamics simulations for the ranking, evaluation, and refinement of computationally designed proteins.\" Methods in Enzymology. Vol. 523. Academic Press, 2013. 145-170.\n\n[3] Wijma, Hein J. et al. \"A computational library design protocol for rapid improvement of protein stability: FRESCO.\" Protein engineering: Methods and protocols. New York, NY: Springer New York, 2017. 69-85.\n\n\n&nbsp;\n\n*A4.2 - Addressing W1: Contact metrics in appendix*\n\nAs we argue in section A.2.1, we focus on metrics that are established in the field of biomolecular dynamics and provide the contact metrics, which were only recently introduced in AlphaFlow, for completeness in the appendix. As stated in that section, the metrics contain arbitrary thresholds of 8 Å and 10\\%. BBFlow indeed performs slightly worse than AlphaFlow but still better than the distilled model (Table A.4). We regard the distilled model as relevant baseline since it is the only AlphaFlow model that comes close to BBFlow in terms of efficiency, demonstrating BBFlow's favourable performance in the tradeoff between efficiency and accuracy. Notably, BBFlow also performs better in the contact metric than BioEmu (Table A.4), which is commonly considered as good model for off-equilibrium sampling. However, we included BBFlow's slightly worse performance in the contact metrics as limitation in the 'limitations' section in the main text.\n\nWe would like to note that we ran an additional experiment during the rebuttal phase that shows that there is no clear trend, which model is generally better at sampling off-equilibrium states. In addition to the contact metrics, we introduce another metric that measures the difference between equlibrium structure and generated samples. Instead of using contacts as difference measure between states, we explicitly calculate the RMSD between samples and equilibrium structure for each protein and report the values in Table R13.\nWe find the reversed trend - BBFlow being better than AlphaFlow-T at capturing off-equilibrium states.\nIn terms of RMSD, BBFlow samples states that are further away from the equilibrium structure, while AlphaFlow-T samples closer to the equilibrium structure. We believe this demonstrates that slight performance advantages may depend on the metric definition and that there is no clear candidate that is generally better at sampling larger conformational changes.\n\n| Method       | Mean RMSD to Eq. Structure [Å] |\n|--------------|--------------------------|\n| MD           | 3.0                      |\n| BBFlow       | **2.5**                  |\n| AlphaFlow-T  | 2.1                      |\n| BioEMU       | 4.0                      |\n| AlphaFlow    | **3.5**                  |\n\n*Table R13. Deviation from the equilibrium structure on the ATLAS test set. Closer to MD is better.*\n\n\n&nbsp;\n\n*A4.3 - Addressing W2: Omission of BioEmu from the main text*\n\nAs discussed in answer A4.1, we find reporting BioEmu’s performance on the ATLAS test set would place it in an unfavourable evaluation setting, as BioEmu is not designed to emulate 300ns of MD with distributional accuracy. It is rather designed to sample off-equilibrium states by training on longer MD timescales and fine-tuning on folding transitions with experimental data. We think this makes a comparison to BBFlow and other models trained on ATLAS dataset (300 ns) unfair (for BioEmu) and misleading and should not be emphasized in the main text. We stress this in Section 4 of the original submission under Further Baselines.\n\nAppendix Tables A.2 and A.5 of the original submission illustrate that BioEmu, indeed, performs poorly across nearly all metrics on the ATLAS test set in standardized settings. We also predicted ensembles for de novo proteins with BioEmu during the rebuttal period and report its performance in Table R14. Probably due to shallow evolutionary information for de novo sequences, BioEmu performs even worse for those proteins. Conversely, it can be expected that BBFlow and AlphaFlow would perform poorly if evaluated on BioEmu's task.\n\n|       | RMSF r | RMSF MAE | RMSF (MD=1.48) | pw-RMSD (MD=2.9) | pw-RMSD MAE | DCCM r | MD PCA W2 | Time/conf|\n|:------|-------:|---------:|---------------:|-----------------:|------------:|-------:|----------:|--:|\n| BioEmu |   0.6  |     4.24 |           7.56 |             7.9  |        8.29 |   0.64 |      1.53 | 1.9 |\n| BBFlow |   0.84 |     0.26 |           0.87 |             1.47 |        0.32 |   0.83 |      0.67 | 0.8 |\n\n*Table R14. Performance of BioEmu (and BBFlow) on the de novo proteins from Table 2 in the original submissions. Units and settings as in Table 2. Full set of metrics in Table R15.*\n\n\n&nbsp;\n\n*A4.4 - Addressing W3: Provide full set of metrics for de novo proteins*\n\nWe thank the reviewer for their suggestion. We report the full set of metrics for de novo proteins in Table R15.\n\n| metric                  |    AF | ConfDiff |  AF-T |  EF-T | AF-T-D | AF-T-12L-D | BioEmu | BBFlow |\n|:------------------------|------:|---------:|------:| -----:|-------:|-----------:|-------:|-------:|\n| Pairwise RMSD (=2.9)    |  8.08 |     7.27 |  1.25 |  1.2  |   0.87 |       0.68 |   7.9  |   1.47 |\n| Pairwise RMSD r         |  0.2  |     0.24 |  0.86 |  0.86 |   0.82 |       0.83 |   0.13 |   0.7  |\n| Pairwise RMSD MAE       |  7.4  |     7.26 |  0.38 |  0.43 |   0.77 |       0.97 |   8.29 |   0.32 |\n| CA RMSF (=1.48)         |  7.09 |     6.35 |  0.74 |  0.68 |   0.51 |       0.41 |   7.56 |   0.87 |\n| Global RMSF r           |  0.27 |     0.28 |  0.86 |  0.86 |   0.84 |       0.83 |   0.25 |   0.77 |\n| Per target RMSF r       |  0.47 |     0.62 |  0.89 |  0.89 |   0.88 |       0.87 |   0.6  |   0.84 |\n| Per target RMSF MAE     |  4.76 |     3.82 |  0.25 |  0.28 |   0.46 |       0.58 |   4.24 |   0.25 |\n| Global DCCM r           |  0.52 |     0.54 |  0.79 |  0.8  |   0.79 |       0.77 |   0.53 |   0.77 |\n| Per target DCCM r       |  0.58 |     0.65 |  0.85 |  0.86 |   0.84 |       0.83 |   0.64 |   0.83 |\n| Per target DCCM MAE     |  0.22 |     0.21 |  0.11 |  0.1  |   0.11 |       0.11 |   0.21 |   0.14 |\n| Root mean W2-distance   | 10.17 |     8.39 |  1.02 |  0.98 |   1.1  |       1.27 |   7.32 |   1.2  |\n| -- Translation contrib. |  8.01 |     6.91 |  0.9  |  0.85 |   0.88 |       1    |   4.25 |   1.06 |\n| -- Variance contrib.    |  5.5  |     4.29 |  0.45 |  0.47 |   0.6  |       0.73 |   5.2  |   0.54 |\n| MD PCA W2-distance      |  1.64 |     1.72 |  0.66 |  0.63 |   0.69 |       0.75 |   1.53 |   0.67 |\n| Joint PCA W2-distance   |  9.08 |     7.32 |  0.95 |  0.89 |   0.97 |       1.13 |   5.23 |   1.09 |\n| % PC-sim > 0.5          |  9.86 |    16.09 | 48.05 | 50.19 |  48.25 |      40.99 |   7.46 |  39.37 |\n| Weak contacts J         |  0.38 |     0.44 |  0.61 |  0.61 |   0.54 |       0.57 |   0.39 |   0.57 |\n| Transient contacts J    |  0.17 |     0.15 |  0.55 |  0.56 |   0.51 |       0.38 |   0.23 |   0.32 |\n\n*Table R15.: Evaluation on the de novo proteins from Table 2 including the metrics introduced in AlphaFlow. Units and settings as in Table A.2 and Table 1, respectively.*"}]

Author rebuttal:

Meta-review: This paper introduces BBFlow which generates samples from the Boltzmann distribution by conditioning on a starting equilibrium structure in the model as well as modifying the prior. All reviewers found the idea sensible, and the performance gains over existing approaches like AlphaFlow are appreciated. A few reviewers questioned the inference speedup over AlphaFlow, but the authors included additional details in their rebuttal responses to clarify this aspect. Overall, the proposed idea is empirically interesting and well executed, but does not provide a lot of technical novelty. As a result, I recommend acceptance as a poster.
Decision: Accept (poster)