Flexibility-conditioned protein structure design with flow matching
Authors: Vsevolod Viliuga
Leif Seute
Nicolas Wolf
Simon Wagner
Arne Elofsson
Jan Stühmer
Frauke Gräter
Venue: ICML 2025 poster (2025)
Keywords: Proteins
Protein design
Molecular Dynamics
Flow Matching
Conditional Flow Matching
Generative models
Equivariant models
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We propose a flow-based generative model for conditioning protein structure design on per-residue flexibility

Abstract: Recent advances in geometric deep learning and generative modeling have enabled the design of novel proteins with a wide range of desired properties. However, current state-of-the-art approaches are typically restricted to generating proteins with only static target properties, such as motifs and symmetries.

In this work, we take a step towards overcoming this limitation by proposing a framework to condition structure generation on flexibility, which is crucial for key functionalities such as catalysis or molecular recognition. We first introduce BackFlip, an equivariant neural network for predicting per-residue flexibility from an input backbone structure. Relying on BackFlip, we propose FliPS, an SE(3)-equivariant conditional flow matching model that solves the inverse problem, that is, generating backbones that display a target flexibility profile. In our experiments, we show that FliPS is able to generate novel and diverse protein backbones with the desired flexibility, verified by Molecular Dynamics (MD) simulations. FliPS and BackFlip are available at https://github.com/graeter-group/flips.

Flexibility-conditionel protein structure design with flow matching
Vsevolod Viliuga 13 Leif Seute 134 Nicolas Wolf Simon Wagner
Arne Elofsson? 2
Jan Stühmer
Frauke Gräter
Abstract
the way for more controllable and precise design of protein structures with specific functional properties. For example, proteins with on-demand chemical reactivities or binding propensities are of particular interest for biotechnological applications (Lovelock et al., 2022), therapeutics develop- ment (Bennett al., 2024) and even sustainability problems, such as plastic degradation (Chen et al., 2022).
Recent advances in geometric deep learning and generative modeling have enabled the design of novel proteins with wide range of desired proper- ties. However, current state-of of-the-artr art approaches are typically restricted to generating proteins with only static target properties, such as motifs and symmetries. In this work, we take a step towards overcoming this limitation by proposing a frame- work condition structure generation on flexibil- ity, which is crucial for key functionalities such as catalysis or molecular recognition. We first in- troduce BackFlip, an equivariant neural network for predicting per-residue flexibility from an in- put backbone structure. Relying on BackFlip, we propose FliPS, an SE(3)-equivarian conditional flow matching model that solves the inverse prob- lem, that is, generating backbones that display a target flexibility profile. In our experiments, we show that FliPS is able to generate novel and di- verse protein backbones with the desired flexibil- ity, verified by Molecular Dynamics (MD) simula- tions. FliPS and BackFlip are available at https: //gi tiggtuuu..... com/ graeter prraeter-group//0 group/flips. flips.
Only recently, several deep generative models for protein structure design based on diffusion (Song et al., 2021) or flow matching (Lipman et al., 2023; Albergo Vanden- Eijnden, 2022) have successfully demonstrated high bio- physical consistency of generated samples (Watson et al., 2023; Bose et al., 2024; Yim et al., 2023a). The molecular context that such state-of-the-art models can be conditioned on is typically restricted to static properties like the pres- ence of a certain motif in the final structure, adherence to specific symmetry groups or binding to target proteins or lig- ands (Ingraham al., 2023; Krishna et al., 2024; Yim et al., 2024; Lin et al., 2024; Huguet et al., 2024). Protein struc- tures generated by deep learning methods tend to be highly thermostable, (Watson et al., 2023; Zambaldi et al., 2024), which suggests high degree of structural rigidity (Scandurra et al., 1998), and incorporating flexibility into specific parts of a backbone remains a great challenge. Proteins, however, must possess dynamic behavior order to encode func- tion such as chemical reactivities specific recognition of binding partners. For example, enzymes have complicated free-energy landscapes (Benkovic et al., 2008) and traverse from one conformation to another in their catalytic cycle, which includes loop opening and closure, concurrent do- main movements and local folding and unfolding events (Miller & Benkovic, 1998; Liao et al., 2018; Zinovjev et al., 2024). Also molecule-specife protein binders require local structural flexibility in order engage in a complex with DNA, RNA or ligand molecules (Glasscock et al., 2023; Pacesa et al., 2024; Guo et al.. 2024). Since protein struc- ture defines protein function, the limited structural diversity and high rigidity of proteins designed with state-of-the-art generative models impede the exploration a vast potential functional space.
1. Introduction
The past few years have arguably marked the golden age of de novo protein design. Rapid advancement of deep learning-bast approaches and their application to the pro- tein structure prediction problem have enabled an unprece- dented level of modeling accuracy (Jumper et al., 2021; Dauparas al., 2021: Lin al., 2022). This, in turn, paved
'qqual contribution 'Heidelberg Institute for Thcoretical Stud- ies, Heidelberg, Germany Deptt of Biochemistry and Biophysics at Stockholm University and Science for Life Laboratory, Stock- holm, Sweden MMa Planck Institute for Polymer Research, Mainz, Germany "IWR, Heidelberg University, Heidelberg, Germany 'IAR, Karlsruhe Institute of Technology, Karlsruhe, Germany.
In this work, propose frame- work for generating protein structures conditioned on struc- tural flexibility relying on two key innovations. We first
Main contributions
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Flexibilty- protcin structure design with flow matching
ABRA
FliPS
Flexibility
(1)
(2)
Flexibiity-condttioneedd structure design
Residue
(3)
Target profile
max
Molecular Dynamics
Flexibility
(5)
(4)
BackFlip
flex
Flexibility prediction
Residue
Candidate
Figure 1.
protein backbone design. Given a target flexibility profile (I), the conditional generative model FliPS used sample several candidate backbones (2). Flexibility profiles of backbones predicted with BackFlip (3) and the most promising candidate selected (4). The flexibility the protein can then validated Molecular Dynamics simulation (5).
equivariant neural network trained to predict per-residue flexibilities of a given pro- tein backbone entirely independent of sequence information. Utilizing BackFlip for large-scale flexibility annotation and an auxiliary loss, we propose (2) FliPS generative model for protein structure conditioned on per-residue flexibili- ties, defined as magnitude of local fluctuations in Molec- ular Dynamies (MD) simulation. FliPS is extension of the recent Geometric Algebra Flow Matching (GAFL), an SE(3)-equivariant flow matching model for protein struc- tures with diverse secondary structure composition of the generated backbones (Wagner et al., 2024). We combine the two proposed models and introduce flexibility condi- tioning pipeline, in which backbones are generated with the conditional model FliPS and ranked with BackFlip (Fig. 1).
introduce (1) BackFlip
design. (Komorowska al., 2024) propose conditioning pre-trained diffusion model on normal modes that approx- imate local harmonic movements around an equilibrium state. Instead of learning conditional model, gradients of an analytical loss on normal modes are used for guidance upon inference. (Kouba et al., 2024) introduce the back- bone flexibility prediction model FlexPert-3D that relies on embeddings from large protein language model (pLM), which we evaluate in Tab. A. They propose framework to condition sequence generation models to better align with target flexibility, leaving the structure invariant. We discuss relevant works in more details in Appendix A.I.
2. Background
2.1. Flow Matching for protein structure generation
In our experiments we demonstrate that the proposed texibility-condiiionn conditioning pipeline generates novel and diverse protein structures that display given flexibility profile, ver- ified by MD simulations on the timescale of ns.
flexibility-
Riemannian Flow Matching The goal of flow matching (Lipman et al., 2023; Tong et al., 2024) is to sample from an
unknown distribution Pl : -M - [0, 1] on the data domain M by learning a flow Φi : [0, 1] M + M that transforms simple prior distribution po to the target p1 via the pushfor- ward (01]-Po = P1- The flow parametrized by a learnable, time dependent vector field v(x, (): M [0, 1] T.M the tangent space T.M according to the flow ODE,
1.1. Related work
Generative models for protein structure have been condi- tioned on static structural features, such as symmetry groups or motifs (Watson et al., 2023; Yim al., 2024) and on pro- tein ligand targets (Watson et al., 2023; Ingraham al., 2023; Krishna al. 2024). Deep learning-basee tools have been successfully combined with classical modeling tools in the past to design dynamic protein structures, such as al- losterically switchable assemblies (Pillai et al., 2024) or pH- responsive filaments (Shen et al., 2024), without explicitly harnessing intrinsic flexibility, Only few works attempted incorporating dynamics into generative models for protein
d ).(() = (x)) 6o(x)= =x,
which is numerically integrated during inference. Lipman et al. (2023) have shown that the vector field u can be learned by defining trajectories between samples Pi and regressing on the tangent vectors. common choice for the trajectories are geodesics (Chen Lipman, 2024). For more details, we refer to Yim et al. (2023a).
Po and ø
2

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Flexibilty-cconditoned protcin structure design with flow matching
Protein backbone representation Many state-of-the-art models represent the spatial structure of protein backbone of N residues as a sequence of N rotations and translations, that is as rigid-bor bidiybody frames T E .M= = SE(3)* / (Jumper et al., 2021; Yim et al., 2023b). In diffusion and flow match- ing for protein structure, it is common practice to learn the flow vector field v (Eq. 1) by predicting a de-noised target structure Ti (Ti, t) from the intermediate structure Te with neural network and then caleulating a prediction for as the tangent vector of the geodesic between Tr and Ti. Yim et al. (2023a) have shown that, for the manifold SE(3) the tangent vectors to geodesics T' = (2) connecting Tp = (22,re) re) and T1= = (I1, 11) can be calculated as
PLDDT Recently, the confidence measure LDDT pre- dicted by AlphaFold2 (pLDD)T) has been suggested as proxy for the flexibility and disorder in proteins (Ruff & Pappu, 2021; Alderson et al., 2023). An advantage of PLDDT values over B-factors is that they can be obtained in silico, without requiring wet-lab experiments.
One of the most established metries for assess- ing per-residue flexibility in protein is the Root Mean Square Fluctuation (RMSF) derived from Molecular Dy- namics (MD) simulations or in-solution NMR experiments. RMSF measures positional deviation of each residue after aligning protein states to a reference conformation. How- ever, there is no wide consensus on the choice of the refer- ence conformation. Moreover, RMSF in its conventional definition relies on the global superposition of the entire protein, causing a non-locality that can lead to ambiguities and artifacts, we illustrate in Section A. and Fig. A.3.
RMSF
dr 21 Lt dr logr, (11) 1-t
dt 1 t
dt
where the frames T E SE(3) N are decomposed into transla- tional part E R3N and rotational part r E (SO3))
Local RMSF Tackling the above-mentionel issues of global RMSF, we propose a generalization in which the fluctuations of a residue are measured with respect to its lo- cal surrounding instead of the whole protein. We introduce fluctuation scale S that quantifies the number of neighbors used for alignment of each residue, such that S oo cor- responds to the conventional global RMSF. We define the local flexibility & of residue i in a set of conformations C and reference conformation g-(ref) as
Model architecture In order to learn the function Ti (Ti,t) t) as described above, Yim et al. (2023b) introduce a neural network architecture based on the structure block of AlphaFold2 (Jumper et al., 2021). The input features include the noised frames Tü, their pairwise spatial distances, posi- tional encodings of absolute and relative sequence positions, and the flow matching time f. These features and the frames defining the protein structure are updated consecutively in a series of six blocks that use Invariant Point Attention (IPA), introduced in Jumper et al. (2021), as a central component.
Aligo ) x (raf)
E,
=
(3)
|C|
Wagner et al. (2024) propose replacing the point-valued features of IPA with features embedded in the Projective Geometric Algebra (PGA), yielding a geometrically ex- pressive latent representation for protein structure. They integrate this extension, termed Clifford Frame Attention (CFA), into the flow matching framework FrameFlow (Yim al. (2023a)), and call the resulting model GAFL.
ZEC
where the transformation (-()) the window W =
locally aligns residues in {i - 5..,. 5
Align
(-() Align argmin
T TO W (ref) 112.
(4)
In our experiments, we use 5 = 12 neighboring residues in the sequence. In order to remove potential biases induced by the choice of reference conformation, we randomly choose Net 10 reference conformations from the MD trajectory. This results in Nier local RMSF values for each residue, across which we take the median. We visualize the differ- ence to global RMSF in Fig, A.4.
2.2. Metrics of protein flexibility
Proteins display dynamic behavior to perform functions, such as catalysis, association, or regulation (Teilum al., 2009). To quantify protein flexibility, several metrics are being used in the field, based experiments simulations.
3. BackFlip: Backbone flexibility predictor
B-factor Typically, structures of proteins resolved in ex- periments contain data on the deviation of atoms from their mean position, referred to as a B-factors (Sun al., 2019). Since B-factors strongly depend on the physical conditions under which the experiment was conducted, such as tem- perature and the radiation source (Eyal et al., 2005), com- paring B-factors of two independently determined protein structures is challenging and requires a series of normaliza- tions DDjinovicCCaruge & Carugo, 2015).
In this section, we introduce BackFlip (Backbone Flexibility Predictor), an SE(()-equivariant neural network capable of predicting per-residue protein flexibility from the backbone structure alone. Consequently, BackFlip does not rely on the protein's 's sequence and, in particular, does not use any evolutionary information and no pre-trained protein lan- guage model for its prediction, making it especially useful for de novo protein design applications.
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Flexdbilty-connditoonddd protcin structure design with flow matching
3.1. Architecture and training
In Tab. we report the correlation and mean absolute error (MAE) of BackFlip's 's predicted local RMSF compared with the value obtained from two of the three MD trajectories per protein in ATLAS. In order to quantify the noise present in the ground truth, which is due to the trajectories only covering finite time and thus not sampling the free energy landscape extensively, we hold out the third MD trajectory and report its correlation and MAE to the two other tra- jectories. We find that BackFlip outperforms any baseline considered and, indeed, reflects the shape of the local RMSF profiles with Pearson correlation coefficient of 0.8 on the 129 unseen proteins from ATLAS. Not only the shapes but also the amplitudes of the profiles are matched, indicated by an MAE of 17 A. BackFlip's 's remaining error can be in part attributed to the noise present in the ground truth since MD only performs slightly better, with a correlation of 0.84 and MAE of 0.14 A. We find that pLDDT only moderately displays trends of local RMSF and experimental B-factors correlate poorly with MD-derived local flexibility. For both pLDDT and B-factors, a direct conversion to local or global RMSF is not available, thus MAES are not reported.
BackFlip's architecture is based on the GAFL architecture, an -qqiivaaaa transformer with Clifford Frame Atten- tion (CFA) introduced in (Wagner et al., 2024) as extension of AlphaFold's invariant point attention (IPA) (Jumper et al., 2021). We represent a protein backbone as an element of SB(3),, that is as set of Euclidean frames, as in (Jumper et al., 2021). Node features are expressed in the local coordi- nate systems given by the frame of the respective residue and transformed into the target frame during message passing. This not only ensures -quivvaauv but also encodes geometric information about relative displacement and ori- entation of the residues. While in GAFL and AlphaFold2 the backbone structure is updated after each message passing block, the regression task performed by BackFlip does not require structural updates. Instead, we extract intermediate node features after neta blocks of CFA, feed them through node-wise Multilayer Perceptron (MLP) and map the output to the range [0, Smax] using a scaled sigmoid,
c = Smax o (MLP
(5)
In addition to the ATLAS test set proteins, we sample pro- tein backbones that match the length distribution of the ATLAS dataset with FrameFlow (Yim et al., 2024) and RF- diffusion (Watson et al., 2023), respectively, and investigate BackFlip's performance on a total of 100 designable de- novo proteins (Appendix A.6). We conduct three 100 ns long MD simulations per protein (Appendix A.8) and com- pute ground truth local RMSF profiles as for the ATLAS dataset. Also in this setting, BackFlip demonstrates strong performance in terms of correlation and MAE, indicating generalization beyond naturally occurring proteins (Tab. 1),
where the scalar value & represents the flexibility of residue i. For our experiments, we choose Neta 4, a node and edge embedding size of 64 and 32, respectively, and a maxi- mum flexibility of Emux= = 5 Å. With these hyperparameters, BackFlip has 0.68 M parameters, compared to 16.7 M pa- rameters of GAFL, which has six blocks of CFA and around four times higher embedding dimensions. A schematic rep- resentation of the architecture is depicted in Fig. A.I.
We train BackFlip with mean squared eтror loss on per-residue flexibilities of backbones from the ATLAS dataset (Vander Meersche et al., 2024). ATLAS is comprised of MD simulations of 1390 structurally non-redundant pro- teins conducted as three replicas for a total length of 300 nanoseconds (ns), from which we calculate ground truth per-residue flexibilities as local RMSF (see Sections 2.2 and A.) We filter ATLAS for proteins up to the length of 512 residues, resulting in a total of 1294 proteins, which we split in 1035 backbones for training, 130 for validation and 129 for testing (further details Appendix A.2).
We also compare BackFlip's performance with FlexPert-3D) (Kouba et al., 2024), a recent backbone flexibility prediction model trained on the ATLAS dataset, and find that Back- Flip is either on-par with or outperforms FlexPert, which relies on a large pre-trained protein language model, both natural and de novo proteins (Appendix A.5).
In Fig. A.2, we show that BackFlip's 's predictions are con- sistent for proteins of different sizes and visualize three selected backbones with color-coded flexibility profiles (Fig. 2A, B). We find that loops and turns are identified as the most flexible regions (Fig. 2B), which is expected since these structural elements typically have less stabilizing non- covalent interactions and thus can exhibit higher conforma- tional heterogeneity. The model also identifies rigid regions, such as a-helices and 3-sheets residing in the tightly packed core, as the least flexible regions. Notably, the model also reliably captures less obvious alterations in local flexibility, such as shorter secondary structure elements with interme- diate mobility (e.g. residues 80- 100 in 4zi3D, right panel).
3.2. Performance on unseen proteins
In order to test BackFlip's performance on unseen proteins, we evaluate on held-out test proteins of the ATLAS dataset. For that, we pass equilibrium experimental structures as input BackFlip and predict their local RMSF profiles. Since AlphaFold2's confidence measure PLDDT (Jumper et al., 2021) has been suggested as measure for disorder and flexibility before (Ruff& & Pappu, 2021; Akdel et al., 2022; Alderson et al., 2023), we compare BackFlip predictions with PLDDT. We also investigate how well experimental B-factors reflect local RMSF.
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Flexdbilty-conddtionndd protcin structure design with flow matching
Table 1. Performance of BackFlip for the local RMSF derived from two 100 ns long MD simulations on unseen proteins from ATL.S and 100 de novo proteins. We also evaluate the flexibility proxies B-factor and negative pLDDT. We report the Pearson correlation coefficient across all residues (Global r), the mean absolute error (MAE) and inference time for the 300-residue protein 7c45A on single NVIDIA A 100 without batching. Errors are computed by bootstrapping the 2N 20 ground truth RMSF profiles 100 times (see Section A.11).
ATLAS test set
De novo proteins
Global (1)
MAE [Å] (L)
Global r (t)
MAE [Å] (1) Time [s] (1)
MD (Ground Truth)
0.84 (0.00)
0.14 (0.00)
0.80 (0.01)
0.10 (0.00)
(
B-factor*
0.16 (0.01)
Negative PLDDT
0.54 (0.00)
0.48 (0.00)
118
BackFlip
0.80 (0.00)
0.17 (0.00)
0.73 (0.00)
0.11 (0.01)
0.6
*Computed for proteins where available.
A
native to training a conditional model, we also propose a training-free BackFlip- gertii-ggunncee (BG) strategy for flexibility conditioning that can be integrated into existing frame-based unconditional models like RFdiffusion (Watson et al., 2023), We also introduce a screening procedure based on BackFlip to find naturally occurring or generated protein backbones that display the desired flexibility profile.
Local RMSF (A)
4.0
Sacefip (ni0, 94)
5a67A
Local RMSF (A)
4.0
BackFlip (r.0.89)
3
Local RMSF (4)
4.0
Bacefip (nw0,90) 96)
4213D
3.0
MD
3.0
MD
3.0
MD
2.0
2.0
1.0
1.0
1.0
100
0.0g
Residue
Residue
Residue
B
4.1. Method
In order to sample from the conditional probability distribu- tion p (T()) of protein structures T E SE(3) with given per-residue flexibility profile & E RR.. we train a conditional flow matching model by approximating the conditional flow vector field v(z, () defined analogously to Eq. 1.
local RMSF
Figure 2. BackFlip accurately predicts protein backbone flexibil- ity. (A) Flexibility profiles three selected proteins from the test set, derived from Molecular Dynamics (ground truth) and predicted by BackFlip along with their Pearson correlation. (B) Vi- sualization of the corresponding proteins and their locally aligned MD-ensembles, colored by the BackFlip-predictee local RMSF.
d
or (TTE) v (o.,,,) dt
(o(()) T.
(6)
Training objective During training, we sample points in time t ~ u((0, 1)), prior structures To ~ Po and target struc- tures with flexibility profiles (T1, E1) ~ P1 from the training set. We randomly mask parts of {1, or the whole flexibil- ity profile, in order prevent memorization and retain the capability of unconditional structure generation (Appendix A.9). We calculate the intermediate structure T, along the geodesic connecting To and Ti and predict the vector field è (Te, Ei) with the model, using the parametrization in Eq. 2. We regress on the ground truth vector field v.
4. FliPS: Flexiblity-coonditioned protein structure generation
In this section we introduce flow-based generative model FliPS Ffexibility-cooddiioneeed Protein Structure generation.) capable of generating protein structures that display given target flexibility profile. The model is based on Geometric Algebra Flow Matching (GAFL) (Wagner et al., 2024) an unconditional generative model for protein backbone de- sign with strong performance in designability and secondary structure diversity of sampled backbones. In order condi- tion on structural flexibility, we propose modification of GAFL by introducing flexibility embedding, a flexibility auxiliary loss, and a flexibilitymmaskigg procedure during training. We term the resulting conditional model FliPS and demonstrate that conditional sampling yields protein back- bones that display flexibility profiles closely resembling the target profile while being structurally diverse. As alter-
L To [||c
0 (Ti, (Ti, 1| + Laux] (7)
using the norm and heuristic auxiliary loss function Ianx proposed in (Yim et al., 2023b). We extend 4aux by a novel flexibility loss as described below. The key difference to unconditional training is that we pass the flexibility profile ςi of the target structure to the model é, implicitly learning the relationship between flcxibility and structure.
Flexibility embedding In order to pass the fiexibility pro- file & to the model, we embed the per-residue flexibility
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Flexibilty-ccondtioned protcin structure design with flow matching
as additional node input feature. For this, we divide the flexibility into 8 bins with maximum value of Émax A.
similarity score by combining the two metrics in order to
evaluate both the shape and the amplitude of flexibility,
s(E, Eiet) = Woom r(c. Eiet)
Wmac MAE(C. Eset), (10)
Flexibility auxiliary loss We propose an additional auxil- iary loss term that explicitly penalizes the generation of structures with deviating flexibility profile during train- ing. To this end, we use the predicted target structure Ti (Ti, t, E1), obtained via Eq. 2, to retrieve predicted flex- ibilities for all residues (Ar(Ti) by applying BackFlip to the predicted target structure. Importantly, as the predictor {BP of BackFlip is differentiable, we can construct a dif- ferentiable auxiliary loss term that penalizes a deviation of predicted and target flexibility profile via the mean square error loss and add this term to the auxiliary loss in Eq. 7,
where we choose the weights W'corr = and Wmae
= 2.
4.2. Experiments
Training We train FliPS on the PDB dataset (Berman et al., 2000) introduced in FrameDiff (Yim et al., 2023b) annotated with BackFlip-predictter residue-level flexibili- ties', We filter the dataset for lengths between 60 and 512 residues and exclude all structures containing breaks, which results in a set of 22977 proteins. The hyperparameters are chosen as in GAFL (Wagner et al., 2024) and Atex is set to 100. We train the model for a total of 21 GPU days on eight NVIDIA A100 GPUs (details in Appendix A.9.)
Anex N
[1 ERF (Ti)|?²
laux Laxx, FD
(8)
where /mux, FD is the original auxiliary loss from FrameDiff (Yim et al., 2023b). Notably, this is only possible because BackFlip is differentiable with respect to the backbone struc- ture and does not rely on the protein sequence as input.
Evaluation setup To illustrate that FliPS generates plau- sible protein structures for range of different target flex- ibility profiles, we extensively evaluate it on set of 10 hand-drawn profiles and on 10 profiles of randomly cho- sen natural proteins. For each profile, we generate 100 samples per backbone length N E {60, 70, 120]. We further extend the evaluation to bigger proteins with length N E {200, 215, 2100....330,,, for which we pass another hand-drawn flexibility profiles as a condition. We apply BackFlip screening (Section 4.1) to choose the highest- ranked sample that best reflects the target profile and run three 100 ns long MD simulations to assess how well the target profile is recapitulated in practice.
BackFlip guidance (BG) As alternative training flow vector field v(T,, (, directly, as described above, we also pro- pose BackFlip guidance (BG) - a training-free approach based on BackFlip for guiding an unconditional model upon inference. Inspired by mussifiierguiidanee (Dhariwal & Nichol, 2021), we add the gradient of the deviation between Backelip-predicted flexibility {RF and target flexibility& & of the intermediate structure Te at flow matching time t,
model to approximate the
Baselines Since, to the best of our knowledge, FliPS is the first model for the generation of flexibilit--condiionn protein structures, we can not directly compare it to exist- ing approaches. However, we apply BackFlip screening to find backbones with the desired flexibility properties sam- pled with the established unconditional models FoldFlow2 (Huguet et al., 2024) and REdiffusion (Watson et al., 2023), For that, we calculate a flexibility similarity score for all generated backbones and pick the highest-scored structures, as described in Section 4.1. We also apply the same screen- ing procedure to 3673 natural proteins from the SCOPe dataset (Fox et al., 2014; Chandonia et al., 2022) comprised of protein lengths between 60 and 128 residues (see Ap- pendix A.9). Additionally, we perform an extensive ablation of BackFlip guidance (Section 4.1) and find that its perfor- mance is inferior to the conditional FliPS sampling. Results are reported in Appendix A.13 and Tab. A.4.
Doodd (Tet,, = t) V 11E-EBF (TO)I
unconditional
cond. guidance term
where the BG scale 17 is a hyperparameter. In an abla- tion study (Tab. A.4), we find that BackFlip guidance with FrameFlow as unconditional model underperforms the con- ditional model FliPS, which we describe in more details in Appendix A.13.
BackFlip screening (BFS) After (ffeiiblity-conddtioneee) backbone generation, we identify protein backbones that best display the target flexibility profile Exe using a flexi- bility screening strategy relying on BackFlip. We note that eexibility-screening of de novo backbones benefits from an efficient model for flexibility prediction that independent of the sequence in order to apply the screening before the expensive refolding pipeline (see e.g. (Yim et al., 2023b)).
Metrics
We evaluate the similarity of flexibility profiles by calculating the Pearson correlation coefficient and the
For a set of backbones, we predict the flexibility profiles with BackFlip and compute (i) the Pearson correlation r and (ii) the mean absolute error (MAE) between the predicted profile & and the profile of interest Exe. We assign a profile
Source code, model weights and the fiexibilityaannttace dataset are available at
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Flexibilty-conddtioned protcin structure design with flow matching
mean absolute error (MAB) for each profile, measuring how well shape and amplitudes of the profile are recapitulated. We also report established metrics for protein structure gen- eration that measure structural similarity across samples (diversity), similarity to the naturally existing structures (novelty) and consistency with folding and inverse folding models (scRMSD) (Yim et al., 2023b; Bose et al., 2024). For diversity, novelty and scRMSD, we use the same defini- tion GAFL (Wagner et al., 2024) (see Appendix A.7).
per bound of around "MD R 0.8, set by the fundamental uncertainty between independent (unconverged) MD runs as observed in Tab. 1. Flexibility-conddioonnng substantially outperforms the flexibilit--ccreeinng of unconditionally gen- erated de novo and natural proteins in similarity to the target profile. We also find that, in terms of novelty, backbones generated with FliPS are on par with backbones sampled by FoldFlow2 and RFdiffusion, which are trained on larger datasets. Remarkably, with only 700 generated samples, FliPS outperforms screening of over 3,500 natural proteins from the SCOPe dataset in terms of capturing flexibility profiles, showcasing the potential of the proposed flexibil- ity conditioning method (Tab. 2). In Fig. 4 we visualize proteins generated with FliPS and their MD-dderived flexi- bility profiles. The samples are composed of both a-helices and 3-strands and display local RMSF profiles that mainly follow the shape and magnitudes of the target profiles. Es- pecially for bigger proteins, peak positions and magnitudes are recapitulated by the designs. All other target profiles can be found in Fig. A.5.
Notably, since scRMSD depends on the superposition of a generated backbonn with the refolded structure, and flexibil- ity induces a fundamental uncertainty in the structure of a protein (Halle, 2002). it can be expected that more flexible proteins have worse scRMSD scores. Indeed, we establish relation between scRMSD and flexibility on natural pro- teins from the SCOPe dataset (Fig. 3), which are designable by definition, questioning the role of scRMSD if generat- ing flexible backbones is intended. We observe similar relation between scRMSD and novelty of FrameFlow and REdiffusion samples (Fig. 3).
Conditioning increases similarity to the target flexibility To separate the effect of BackFlip screening from the condi- tioning in FliPS, we also report the same metrics for all 700 generated samples per profile, that is, without screening for top samples. Since evaluating the flexibility of all structures in MD simulations is prohibitively expensive, we estimate the flexibility with BackFlip and report average values over all profiles (Tab. 2, A.3; BackFlip on all samples). We find that for both existing and hand-drawn profiles, protein backbones sampled conditionally with FliPS, on average, reflect the target flexibility profiles substantially better. The strongly improved performance of FliPS over unconditional sampling can be attributed to flexibility conditioning, which is especially prominent for bigger proteins where uncondi- tional samples, on average, do not resemble the flexibility profile at all (Tab. A.3).
SCRMSD [A]
8.0
ScRMSD [Å]
2.0-
2.0
0.5
0.5
0.0-0.5
0.5-1.0
1.0+
0.0-0.50.500.6 0.6-0.7
Binned Avg. Flex
Binned Novelty
Figure 3. Relationship between scRMSD and average local RMSF
predicted by BackFlip for proteins from the SCOPe dataset (left)
and between scRMSD and novelty for the novo proteins (right) sampled with RFdiffusion and FrameFlow described in Sec- tion 3.2. Vertical axis is log-scaled. Boxes represent 0.25 and 0.75
quantiles. Means are shown as grey dashed lines.
4.3. Results
FliPS generates diverse structures with various see- ondary structure compositions Additionally, we investi- gate how structurally diverse the generated natural protein backbones are on set of small proteins from Tab. 2. For that, we pick the 10 top-ranked backbones for each tar- get profile identified with BackFlip screening and report diversity and novelty along with the flexibility correlation in Tab. 3. Note that this is in contrast to Tab. 2, where we apply expensive MD simulation only for one sample per profile. FliPS achieves a maximum pairwise TM score smaller than 0.5 and thus generates structurally distinct backbones, being better or on par with the unconditional baselines. SCOPe proteins display the highest diversity, which can be expected, as this dataset was filtered to contain structurally non-redundant proteins. The experiment also shows that not only one but several backbones with high
Tab. 2 summarizes the results of the experiment described above on domain-sized proteins and Tab. A.3 on bigger pro- teins. For each profile, we obtain a total of 700 samples and single top-ranked sample selected by BackFlip sereening. We conduct all-atom MD simulations (see Appendix A.8) of the top-ranked, refolded samples and report novelty of the structure and similarity of the MD-derived flexibility profile to the target (MD of top samples).
FliPS generates structures that recapitulate target flexi- bility profiles in MD We find that FliPS generates back- bones whose MD-derived flexibility, indeed, reflects the target profile (Tab. 2 and Fig. 4). For the target profiles considered, the average Pearson correlations r of 0.70, 0.78 for smaller and 0.79 for bigger proteins approach the up-
7

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Flexibilty-conddtionndd protcin structure design with flow matching
Table Flexibility conditioned backhone generation for 10 existing and 10 hand-drawn target profiles. For each of the profiles, we generate backbones spanning lengths .N {60, .o.... ] using FliPS by conditioning on the flexibility profiles. We apply BackFlip screening (BFS) to obtain best-ranked candidate sample each profile. For each of the baselines, we sample the same number structures unconditionally and also apply BFS screen the entire SCOPe dataset. For quantifying the effect of conditioning, we also report average values across all sampled backbones without ranking (BackFlip on all samples). correlation (r) and the mean absolute error (MAE) to the target profiles, and TM-similarity to the PDB (Novelty). For the best-ranked candidates for each of the profile (MD top samples), we calculate the profile of the generated protein with MD. We report standard deviations obtained by bootstrapping MD profiles as in Tab. and by bootstrapping the set of all generated backbones, respectively.
report the per-target Pearson
MD of top samples
BackFlip on all samples
r (t)
MAE [Å] ())
Novelty (L)
r (t)
MAE [Å] (4)
Novelty (L)
10 existing profiles
FliPS
0.70 (0.02)
0.16 (0.00)
0.62 (-)
0.52 (0.00)
0.15 (0.00)
0.63 (0.01)
RFdiffusion-BFS
0.58 (0.03)
0.16 (0.00)
0.60 (-)
0.28 (0.00)
0.19 (0.00)
0.62 (0.02)
coldFlow2-BFS
0,45 (0.04)
0.16 (0.00)
0.57 (-)
0,27 (0.00)
0.18 (0.00)
0.57 (0.01)
ACFSe-BFS
0.52 (0.02)
0.19 (0.00)
1.0 (+)
0.19 (0.00)
0.25 (0.00)
1-)
10 hand-drawn profiles
FliPS
0.78 (0.01)
0.31 (0.01)
0.55 (-)
0.56 (0.00)
0.43 (0.00)
0.64 (0.03)
REdiffusion-BB
0.56 (0.02)
0.44 (0.00)
0.60 (-)
0.13 (0.00)
(0.00)
0.62 (0.02)
FoldFlow2-BFS
0.50 (0.03)
0.45 (0.01)
0.57 (-)
0.08 (0.00)
0,50 (0.01)
0.57 (0.01)
SCOPe-BFS
0.61 (0.02)
0.35 (0.01)
(-0
0.09 (0.01)
0.48 (0.00)
1-0
MD
Target
3.0
3.0
3.0
3.0
0.8.4
2.337
0.84
Local RMSF
2.5
0.20
Local RMSF
2.5
MAE:
Local RMSF
2.5
MAE:
Local RMSF
2.5
2.0
2.0
2.0
2.0
1.5
1.5
1.5
1.5
1.0
0.5
0.5
0.5
0.5
0.0
60
100
0.0
20
0.0
0.0
20
40
60
80
100
Residue
Residue
Residue
Residue
1:0093
0:7.
0.8.8
r: 0:78
Local RMSF
2.5
MAE: 0.18
Local RMMF
2.5:
MAE: 0.22
Local RMSF
2.5
MAE: 0.17
Local RMSF
2.5-
MAE: 0.24
2.0
2.0
2.0
2.0
1.5
1.5
1.5
1.5
1.0-
1.0-
1.0
1.0-
0.0
50
100
150
200
250
0.0
50
100
150
200
250
0.0
50
100
150
200
0.0
50
100
150
200
Residue
Residue
Residue
Residue
Figure 4. FliSS samples for two selected existing and hand-drawn target profiles from Tab. 2 (top panel) and for all hand-drawn profiles from Tab. A.3. The generated protein's flexibility computed from MD simulation. Corresponding protein structure is depicted with sequential coloring (N-terminus blue, C-terminus red).
8

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Flexibilty-condittonndd protcin structure design with flow matching
Table 3. Diversity and novelty of fexibblity-condiiioned backbones from Tab. Metries are reported as averages over the 10 top-ranked samples for all profiles, selected with BackFlip screening (BFS). We report diversity and novelty based on pairwise TM similarity, scRMSD is computed by aligning the refolded with the generated backbone (see Appendix A.7). We report Pearson correlation r and mean average error (MAE) between the BackFlip predicted and target flexibility profiles. Standard deviations are obtained by bootstrapping all
generated samples 10 times.
scRMSD
Diversity (1)
Novelty (1)
(t)
MAE [Å] (L)
Helix pet.
Strand pet.
10 existing profiles
FliPS
1.52 (0.01)
0.36 (0.02)
0.57 (0.01)
0.86 (0.01)
0.09 (0.00)
0.53 (0.01)
0.19 (0.01)
RFdiffusion-BFS
0.72 (0.00)
0.40 (0.02)
0.61 (0.01)
0.68 (0.01)
0.15 (0.02)
0.78 (0.01)
0.09 (0.01)
FoldfFlow-BFS
0.72 (0.01)
0.44 (0.02)
0.57 (0.01)
0.67 (0.02)
0.16 (0.01)
0.83 (0.01)
0.02 (0.00)
SCOPc-BFS
0.79 (0.01)
0.33 (0.02)
(-)
0.72 (0.02)
0.12 (0.00)
0.29 (0.01)
0.33 (0.01)
10 hand-drawn profiles FliPS
2.01 (0.03)
0.47 (0.02)
0.60 (0.01)
0.86 (0.01)
0.28 (0.00)
(0.01)
0.15 (0.01)
RFdiffusion-BF BFS FoldfFlow-BFS
0.74 (0.01)
0.44 (0.02)
0.59 (0.01)
(0.01)
0.39 (0.01)
(0.01)
0.11 (0.01)
0.72 (0.01)
0.45 (0.02)
0.59 (0.01)
0.58 (0.02)
0.43 (0.02)
0.84 (0.01)
0.02 (0.00)
SCOPe-BI'
1.96 (0.02)
0.32 (0.02)
(-)
0.76 (0.02)
0.28 (0.02)
0.31 (0.01)
0.24 (0.01)
BackFlip-predicter simillrtyy can be found, as indicated by the Pearson correlations of over 0.8 between Backelip-predicted and target profile. Remarkably, while better reflecting flexibility profiles of interest, FliS back- bones also contain higher amount of 3-strands than both RFdiffusion and FoldFlow2, more closely resembling the secondary structure composition of SCOPe proteins.
guidance, proteins sampled conditionally with FliPS best reflect the desired flexibility, while being structurally non- redundant, novel, and contain both a-helices and B-strands. This finding suggests that FliPS learns a relation between structure and flexibility, exploring a structural space beyond natural proteins in order to find distinct backbones that dis- play the desired flexibility profile.
Biophysical consistency of protein structures depends on the target profile To assess consistency of the generated backbones, we calculate scRMSD for the top 10 samples for each profile for the same set of small-sized proteins as de- scribed in the previous subsection (Tab 3). We find that, for most profiles, FliPS generates designable backbones with scRMSD below 2 Å. However, backbones generated with FliPS have higher scRMSD values than unconditionally gen- crated samples. We observe that scRMSD strongly depends on the target profile (Fig. A.6), which can be expected since scRMSD is correlated with flexibility and novelty (Fig. 3), both of which can be influenced by the target profile. In this context, we find that even natural proteins from SCOPe display elevated scRMSD values for samples screened for agreement with the hand-drawn profiles.
5. Conclusion
We introduce a generative model for protein structure de- sign conditioned on per-residue flexibilities. The proposed criibllty-conditioonn framework relies on the structure- based flexibility prediction model BackFlip, enabling large scale flexibility annotation of proteins, a flexibility auxiliary loss and flexibility screening procedure to find protein backbones that best display flexibility profile of interest. Our experiments demonstrate that leads to the generation of diverse and novel backbones that indeed display the respective target flexibility profile in Molecular Dynamics simulations. Thus, the proposed frame- work enables overcome the current restriction of deep sarring-bbssel protein design to static protein structures. While functional protein motions often involve timescales beyond the 300 ns of MD used as flexibility definition in this work, this limitation might be overcome as soon as better ground truth data for fiexibility on longer timescales becomes available. Flexibility conditioning can be straight- forwardly combined with motif or symmetry conditioning as used to design binding sites or protein assemblies, for which flexibility is required. Since flexibility is crucial for wide range of protein functionalities, we believe that the proposed method for controlling the flexibility of designed proteins on a per-residue level paves the way for exploring a yet uncharted but highly relevant space of protein structures.
4.4. Discussion
In our experiments we demonstrate that the proposed framework is capable of delivering protein backbones of varying small and bigger lengths that display a range of different realistic and custom hand-drawn flexibility profiles. We note that there is fundamental uncertainty in MD-derived flexibility due to deviations be- tween simulation runs, as observed in Tab. 1, making the generation of backbones that recapitulate the MD-derived flexibility especially challenging. Importantly, compared to unconditional models augmented by flexibility screening or

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Flexibilty-conddttonndd protcin structure design with flow matching
Acknowledgments
Alford, R. F., Leaver-Pay, A., Jeliazkov, J. R., O'Meara, M.J., DiMaio, F. P., Park, H., Shapovalov, M. V., Renfrew,
This study received funding from the Klaus Tschira Stiftung gGmbH (HITS Lab). We acknowledge the National Academic Infrastructure for Supercomputing in Sweden (NAISS), partially funded by the Swedish Research Coun- cil through grant agreement no. 2021-29 for awarding this project access to the Berzelius resource provided by the Knut and Alice Wallenberg Foundation at the National Su- percomputer Centre. The authors acknowledge support by the state of Baden-Wüine Württemberg through bwHPC and the German Research Foundation (DFG) through grant INST 35/1597-1 FUGG.
P. D., Mulligan, V. K., Kappel. K., Labonte, J. W., Pacella, M. S., Bonneau, R., Bradley, P., Dumbrack, R. L. J., Das, R., Baker, D., Kuhlman, B., Kortemme, T., and Gray, J. J. The rosetta all-atom energy function for macromolecular modeling and design. Journal of chemical theory and computation. 13(6):3031-3044 2017.
Baker, D. and Church, G. Protein design meets biosecurity. Science, 383(668) 313499349, 2024.
Benkovic, S. J., Hammes, G. G., and Hammes-Schifted S. Free-energy landscape of enzyme catalysis. Biochemistry, 47(11):33173322) 2008.
Impact statement
Bennett, N. R., Watson, J. L., Ragotte, R. J., Borst, A. J., Sce, D. L., Weidle, C., Biswas, R., Shrock, E. L.,, Leung, P. Y., Huang, B., Goreshnik, I., Ault, R., Carr, K. D., Singer, B., Criswell, C., Vafeados, D., Garcia Sanchez, M., Kim, H. M., Vázquez Torres, S., Chan, S., and Baker, D. Atomically accurate de novo design of single-domain antibodies. bioRxiv, 2024.
The societal impact of this work is considered mostly posi- tive, as de novo protein design holds the promise to develop drugs against diseases and personalized therapies, which out-weights potential risks (Baker& & Church, 2024). While of course security concerns remain, the presented method does not directly enable the design of proteins with a certain biological function, so risk of misuse is low.
Berman, H. M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T. N., Weissig, H., Shindyalov, I. N., and Bourne, P. The protein data bank. Nucleic acids research, 28(1): 235-242, 2000.
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Flexibilty-conddttonndd protcin structure design with flow matching
A. Appendix
A.I. Previous work on flexibility fexiibilitycoonditionedd protein design
In the past, it was possible to introduce flexibility into the desired protein region with classical modeling tools, such as Rosetta MotifGraft (Alford et al., 2017) along with Rosetta BackRub (Lauck et al., 2010) sampling, Modeller (Sali & Blundell, 1993) LoopGrafter (Planas-Iglesias et al., 2022). However, these tools require structure as input to specify the context into which loop or any given motif should be incorporated. This limits the applicability of these tools to designing scaffolds, but not the novel protein structures, where the starting structure best suited to encode the desired dynamical properties is not known. Another common downside of the classical tools is their relatively high computational cost due to an extensive iterative sampling relying on aapiriically-arammeeeiieee energy functions. Nonetheless, by combining classical tools and deep learning-bass approaches, allosterically switchable proteins (Pillai et al., 2024), pH-dependent filamentous self-assembling protein complexes (Shen et al., 2024) and fold-switching proteins (Guo et al., 2024) have been successfully cases, deep learning, however, is only used
designed without explicitly conditioning on flexibility. In most for protein structure prediction of designed sequences, not for generating target structures. We see potential for improvement by using generative deep learning with the flexibility conditioning framework proposed in the main text of the paper in design studies described above.
In their work (Komorowska et al., 2024) condition a pre-trained diffusion model on the lowest non-trivial normal modes computed with the Normal Mode Analysis (NMA). Lowest normal modes are obtained from the eigenvectors of the Hessians of the potential energy relying on force field that assumes the harmonicity of the system, which inherently limits sampling protein structure states to the movements around an equilibrium state. Conditioning is achieved by the gradient guidance whilst gradients computed from an analytical normal mode loss. Different their work, we propose training a conditional model akin to classifier-free guidance, whereby the flexibility is derived directly from Molecular Dynamics (MD) simulations. Thus, our approach is not limited to analytical conditioning but handles non-analytical conditions. Since we train our model to approximate a conditional flow, we avoid potential conflicts between unconditional and conditional gradient terms, which might arise in the gradient guidance relying on the analytically computed gradients. We implemented a classifier-gun -uaffiir--g strategy, which we term BaockFlip-uuidnne (BG), as an alternative to the conditional FliPS model. However, we found that BG performed worse than conditional sampling and often violated the physicality of the generated backbones, as discussed in A.13.
(Kouba et al., 2024) propose pipeline to condition protein sequence design on MD-derived flexibility. They introduce flexibility prediction model termed FlexPert-3D, which they use as scoring model to fine-tune the sequence design model ProteinMPNN (Dauparas et al., 2022) on generation of sequences that better capture the target flexibility observed in MD simulations. FlcxPent-3D is a regression model that relies on the pre-trained weights from a protein Language Model and has trainable CNN head to combine pLM embeddings with Hessian matrices obtained from the Anisotropic Network Model (ANM) computations The approach operates exclusively in sequence space and requires both an input structure and evolutionary information encoded by the pL. embeddings, rendering it fundamentally different from our work. In our experiments we show that BackFlip outperforms FlexPert on both natural proteins from ATLAS and de novo proteins in A.5,
A.2. Training details of BackFlip
In order to train the BackFlip specifically local flexibility, for each protein in the ATLAS dataset, we first compute local RMSF values as described in Sections 2.2 and A. We split the dataset into train, valid, and test set according to the splitting scheme 80:10:10 10 and limit protein size to 512 residues. An epoch is defined as a single pass through the whole dataset, whereby one of the Nrey conformations is randomly selected for each protein. We batch proteins together during training, where each batch contains at most 32 proteins. We train the model on a single NVIDIA A100 GPU and use the value of RMSE between the ground truth and predicted local flexibility scores on the validation set as the early stopping criterion. We then select the best checkpoint based on the RMSE on the validation set and evaluate the model on the test set.
A.3. BackFlip's architecture
We illustrate a schematic, high-level overview architecture of BackFlip in Fig. A.I.
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Flexbbilty-conddtioned protcin structure design with flow matching
4x Feature Update
Readout Block
h
CiffordFrameettentioo
Sigmoid
& R
Edge Update
4
Figure A.I. High-level overview of the architecture of BackFlip. Current node h1 and edge = embeddings at layer updated by the Clifford Frame Attention (CFA) block 4 times. Updated features are input to the Readout Block and processed. The output is a tensor Ç N
where denotes the length a protein. Blocks are indicated with dashed lines.
A.4. Additional results on BackFlip performance on held-out test set proteins of ATLAS
We find that BackFlip consistently predicts local RMSF profiles for proteins from the ATLAS test set of varying lengths (Fig. A.2A). Also we visualize the distribution of the predicted and ground truth per-residue local RMSF values for the same set of proteins in Fig. A.2B).
A
64
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Frequency
48
40
32
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24
0.0
300
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Local RMSF [Á]
Figure A.2. (A) Pearson correlation coefficients of BackFlip predictions on the held-out test set proteins from the ATLAS dataset to the simulations as a function protein length. (B) Distribution the BackFlip- local
ground truth local RMSF computed RMSF values compared with the distribution of the ground-truth local RSMF for the same of proteins in (A).
A.5. Comparison of BackFlip with FlexPert on the ATLAS and de novo datasets
We compared the performance of BackFlip to another recently released backbone flexibility predictor FlexPert-3D (Kouba al., 2024). As discussed in Section 1.1 and Appendix A. FlexPert relies on the protein Language Model (pLM) embeddings and has more complicated architecture with trainable CNN head. We retrained BackFlip the global RMSF metric and the dataset split used in FlexPert, both without and with one-hot encoded sequence embeddings, to also assess their effect. Without any sequence embedding, BackFlip outperforms FlexPert on both global and per-target Pearson correlation (as reported the FlexPert paper), while performing slightly worse in terms of MAE (Tab. A. On the the de novo protein set, BackFlip significantly outperforms FlexPert on all metrics (Tab. A.2). We hypothesize that the reason for this might be that pLM embeddings are not informative for these proteins, as there no evolutionary information available.
Table I. Comparison of performance of BackFlip FlexPert on the ATLAS test set. BackFlip was retrained on the global RMSF metries and the ATLAS dataset split used in FlexPert
Model
Global r
MAE [Å] 4
Per-target r
Per-target MAE [Å]
BackFlip" 0.78
0.61
0.88
0.73
BackFlip
0.81
0.56
0.88
0.72
FlexPent!
0.74
0.44
0.83
0.47
"No sequence embedding Onnehot sequence embedding embeddings "LLM embeddings
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Flexxbiltt- protcin structure design with flow matching
Table A.2. Comparison of performance of BackFlip PlexPent the of 100 MD simulations of de novo proteins from Tab. For inference with BackFlip, we use the same model checkpoint as in the Tab. A.1.
Model
Global r
MAE [Á]
rer-target
Per-target MAE [Å]
BackFlip" 0.63
0.49
0.85
0.48
FlexPent
0.51
0.62
0.63
0.60
No sequence embedding pLM embedklngg
A.6. Generation of the de novo protein dataset
We generate 20 protein backbones for each specified length N [60, 65, 512] using both FrameFlow (Yim al., 2024) and REdiffusion (Watson et al., 2023). Each generated backbone undergoes self-consistency pipeline as introduced in prior work (Trippe et al., 2023; Watson et al., 2023; Yim et al., 2023b). This process involves designing eight sequences per backbone with ProteinMPNN (Dauparas et al., 2022) and refolding these sequences using ESMfold (Lin et al.. 2023). We next analyze the length distribution of the ATLAS dataset (Vander Meersche et al., 2024), and select 50 backbones from each method (FrameFlow and RFdiffusion) that exhibit a self-consistency RMSD (scRMSD) of s 2.0 Å, and closely match the size distribution observed in the ATLAS dataset.
A.7. Designability, novelty and diversity of protein backbones
In Tab. I we select 50 designable backbones sampled each with FrameFlow and REdiffusion and in Tab. 2 we report novelty for backbones sampled for certain flexibility profile. In order find designable backbones, we follow pipeline of self-consistency (Trippe et al., 2023; Watson et al., 2023) well-established in the protein design field. For each sampled backbone, we design 8 candidate sequences with ProteinMPNN (Dauparas et al., 2022), and predict their 3D structures with ESMfold (Lin et al., 2023), and define scRMSD as the smallest RMSD between our generated backbone and the 8 refolded, aligned candidates. Similar to (Watson et al., 2023; Yim et al., 2023b), we call backbone designable if the generated backbone has scRMSD < 2
We compute novelty analogously to (Wagner et al., 2024) by comparing the structures of protein backbones to the ones found in the PDB database using FoldSeek (Van Kempen al., 2024) in TMalign mode (-alignment-type 1), and report novelty as the average of the highest TM score calculated for given backbone, averaged over all proteins considered. Diversity from Tab. 3 is computed pairwise TM-similarity score for set of protein backbones also as defined in (Wagner et al., 2024).
A.8. Settings of Molecular Dynamies (MD) simulations
We conduct MD simulation using GROMACS v2023 (Abraham al., 2015) with the all-atom force field CHARMM27. Since FliPS generates only protein backbones, we provide input to MD simulation the structure of the best-scoring sequence refolded with ESMfold in terms of scRMSD. Proteins are placed in the periodic dodecahedron box with at least 1 nm distance from the box edge. The system is solvated with the TIP3P water model (Jorgensen et al., 1983) and NaCl concentration is adjusted to 150 mM. The system is energy-mining minimized for 5000 steps. NVT equilibration is performed for 1 ns with a 2 fs timestep using the leap-frog integrator with the temperature of 300K maintained with the Berendsen thermostat. Next, the system is NPT cquilibrated for another 1 ns with the pressure set at I bar and maintained with the RarhnnnlloRRahan barostat. The production run is performed starting from the last frame of the NPT equilibration as three replicas of 100 resulting in the total simulation time of 300 ns. Covalent bonds involving hydrogen atoms were constrained using the LINCS (Hess, 2008) algorithm in all the simulations. Long range electrostatic interactions were accounted for using the Particle-Mesh Ewald (PME) method.
A.9. Details on FliPS training
As mentioned in Section 4.2, we train FliPS on a subset of the PDB dataset. We observed that first pre-training smaller SCOPe dataset (Chandonia et al., 2022) annotated with JackFlip-predicter flexibilities helps stabilize training on the bigger PDB dataset and results in better performance. SCOPe can be considered as a benchmark dataset in the field of protein design, as it has been manually curated and used for training several generative models for protein backbones (Yim et al., 2023a; Lin AlQuraishi, 2023). We filter SCOPe dataset clustered by 40 % sequence identitiy for lengths between 60 and
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Flexdbilty-coondttonee protcin structure design with flow matching
128 residues, which results in of 3673 structures. The hyperparameters are chosen in GAFL (Wagner al., 2024) and Amex is set to 100. We train FliPS for total of four GPU days two NVIDIA A100 GPUs. Then, starting from this model checkpoint, we continue training on the filtered PDB dataset, as introduced in Section 4.2 for another 17 GPU days on 8 NVIDIA A100 GPUs, totalling 21 GPU days.
During training, 10% of the time we fully mask one-hot encoded per-residue flexibility and perform training of FliPS as an unconditional model with the original loss function introduced in (Yim et al., 2023a; Wagner et al., 2024). For remaining 90%, we randomly unmask flexibility as window of size [{022, 0.4]) and with the window center positions M[1,NN where N stands for the total protein length. We do not use padding during the masking procedure.
A.10. Sampling different protein lengths for a flexibility profile of interest with FliPS
In Section 4.2, we demonstrate that FliPS samples protein backbones of different lengths that display a flexibility profile given input. For this, linearly scale the profile such that it has the same length the generated sample.
A.11. Definition of a local measure for flexibility
Figure A.3. A globally-aligned helix-turn-helix motif. Since the smaller helix has lower weight in the global alignment, it is assigned large (global) RMSF values although it locally stiff. dotted lines are the positions the residues in the reference conformation.
The Root Mean Square Fluctuation (RMSF) commonly obtained either from conformational ensembles C of NMR- determined or MD-simulated protein structures is computed as
RMSF
Topt o x4 =.rel
(11)
C
WEC
with the reference conformation rf and where denotes the residue index. The transformation Topt is defined as
Top argmin TESE(3)
(T oTo 2107) 2
(12)
where is the number of residues.
In simpler formulation, the RMSF shows the extent of positional deviation for given residue from its reference state, all expressed in relation to the whole protein. Since the superposition the definition of Topt (Eq. 12) a global quantity that depends on the whole protein, it is apparent that RMSF is non-local in the sense that the value of given residue depends on the position of all other residues, even if their spatial distance is large. This suggests that the global RMSF cannot be the only measure for per-residue flcxibility, as it often simply captures the movement of an entire, locally stiff subdomain with respect to the rest of the protein. This not only contradicts the notion of per-residue flexibility (as opposed to flexibility of subdomains or collective degrees of freedom), the non-locality can also lead to discontinuities. An example for such a case is illustrated in Fig. A.3A, where, due to the higher weight in the alignment, the longer helix in the helix-turn-le
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Flexibilty-condiiionne protcin structure design with flow matching
motif is rendered as stiff by the global RMSF and the other is rendered flexible. If the shorter helix would be slightly longer, the global alignment would render it stiff and the other helix flexible instead. For and the same movement, the RMSF profile thus shows a large change upon a slightly change in the protein's topology. We not that this is no discontinuity in the strict mathematical sense since continuity cannot be defined for functions that depend on the length of individual subdomains because the length is discrete.
Addressing this issue, we propose a new, local definition of flexibility that avoids global superposition of the whole protein the reference state in Section 2.2.
To illustrate the difference between local and global RMSF, aligned 25 conformations of 324 residues long protein (PDB: IRM6) available from the ATLAS dataset according to the definition of local flexibility with sequence-window size 2K + 1 = 13 and compare with standard global alignment (Eq. 12). Fig. A.4 demonstrates that in the case of local alignment, the secondary structure elements, such as a-helices remain stable and well resolved, whereas in the case of global alignment, the helices experience significant positional changes and fluctuate more. As mentioned above, this will be manifested the computation of RMSF, which will be significantly higher for the helical regions experiencing fluctuations in the global alignment.
A
B
Local RMSF
Global RMSF
Figure A.4. Comparison of 25 conformations of a 324 residues long protein (PDB: 1RM6) aligned either locally, according to the definition local flexibility in Eq. (A) or globally (B). Bold circles highlight secondary structural elements where difference between local and global alignment most pronounced.
A.12. Conditional FliPS sampling for proteins of bigger lengths
Table A.3. Flexibility conditioned backbone generation for 4 hand-drawn target profiles illustrated in the lower panel of Fig. 4. For each the profiles, we generate backbones spanning lengths {200, 215, .... 3000 using FliPS by conditioning on flexibility profiles. See caption to Tab. 2 for other details.
MD of top samples
BackFlip all samples
r (t)
MAE (Ā] (4)
Novelty (4)
(t)
MAE [Â] (4)
Novelty (L)
4 hand-drawn profiles FliPS RFdiffusion-BBl BFS
0.79 (0.01)
0.21 (0.00)
0.61 (-)
0.76 (0.00)
0.27 (0.00)
0.57 (0.01)
0.41 (0.02)
0.31 (0.00)
0.57 (-)
-0.00 (0.00)
0.32 (0.00)
0.58 (0.01)
FoldFlow2 -FF
0.35 (0.02)
0.34 (0.00)
0.48 (-)
-0.00 (0.00)
0.33 (0.00)
0.48 (0.00)
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Fleddbilty-condittonndd protcin structure design with flow matching
A
MD
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1uoy
2pne
2qsk
3a39
4kfd
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0:0.64
0.71
0.7.7
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Local RMSF
2.5
MAE: 0.14
Local Local RMSF
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MAS: 0:13
Local RMSF
2.5
0.20
Local RMSF
2.5
MAE: 0.13
Local RMSF
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5075
5vdc
6dcm
6p9g
8a55
3.0
3.0
4: 0.45
coo MAE 0.30
0.84 MAB: 0.20
0./5
Local AMSF
2.5
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Local RMSF
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Figure A.5. Target profiles for benchmarking FliPS as in Tab. 2 and Tab. 3. (A) 10 realistic profiles (BackFlip-predicttonn of natural proteins outside the training set), (B) 10 hand-drawn profiles. Local RMSF computed from MD simulation of the generated samples is depicted in purple, and the target profile in green.
Realistic Profiles
Hand Drawn Profiles
5
Samples
Samples
4-
FliPs
FliPS
scope
scope
RMSD [A]
3
3-
RMSD
luoy
2pne
2ask
3a39
afkd
5075
Sudc
6dcm
6p9g
8a55
3
Profile
Profile
Figure A.6. scRMSD values of FliPS samples or naturally existing SCOPe proteins for given set of flexibility profiles. Means and errors
are computed as in Tab. by bootstrapping the samples and selceting top 10-best ranked backbones 100 times.
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Flexdbilty-conddtionedd protcin structure design with flow matching
A.13. BackFlip guidance strategy and its ablation
We train FliPS model akin classifier-free guidance approach (Ho & Salimans, 2022), which requires the presence of interesting conditioning information at training time. We achieve this by BackFlip (Section 4.1) and passing flexibility profiles of the respective proteins to the model as a feature. The model then learns to approximate a conditional flow vector field (Eq. 6). A more general, training-free sampling strategy is classifier guidance (Dhariwal & Nichol, 2021), which combines the unconditional model score with a gradient term computed from a pre-trained classifier at test time. This allows to approximate the same conditional flow vector field as described above. Multiple studies have shown that classifier guidance-derived gradients might conflict with the unconditional generative direction (Dinh et al., 2023), which leads to con-convereence (Lou & Ermon, 2023) and quality- sought investigate how classifier-guidance would compare to the FliPS model that is trained to directly approximate conditional flow vector field.
the PDB dataset with
trade-off. We
As described in Section 4.1, we implement updating the prediction of the flow vector field made by unconditional model with a gradient computed by BackFlip w.r.t to intermediate structure T3 at inference time (Eq. 9). We select the flexibility profile in the lower right corner in Fig. 4 as target and sample 100 backbones for each length N E {200, 215,... 300} using FliPS in unconditional model setting (see A.9) but with BG. For comparison, we also sample backbones unconditionally and using FliPS conditional model (BG)
-uielaace (BG) for condiiioy--odiiioneee backbone generation by
without BG. We assess the impact of various BackFlip-guidane 10), a linear scheduling scheme, and length-dependent scaling on performance across flexibility metrics (Section 4.2), secondary structure composition, and scRMSD of the generated backbones. In addition, we implemented BG within another unconditional model FrameFlow (Yim et al., 2024) and evaluated its performance with the same settings as described above.
absolute BG scales (5 and
Tab. A.4 reports results of the experiment. Indeed, application of BG significantly outperforms the unconditional sampling on flexibility metrics, but does underperform compared to FliPS sampling with conditional flow (FliPS*) regardless of the hyperparameters tested. We observe that the BG approach with high absolute scale can significantly compromise backbone physical validity in favor of flexibility metries, as indicated by high scRMSD values and illustrated in Fig. A.7. Although favorable flexibility metrics are achieved, there is clear trend of over-representation of ax-helices when BG is applied. The optimal performance regards to both target flcxibility, physical validity and secondary structure composition is achieved when the the effective BG scale is scaled with protein length. For FrameFlow-BG, we report inference results using the best- pert-rerfonnngg length-scaling schedule, and find similar performance when applying the same settings within the GAFL architecture. We also find that BG is approximately 20% slower than FliPS sampling with conditional flow.
Table A.4. Ablation of the BackFlip-Guidanss (BG) strategy implemented either within FliPS or FrameFlow code. As the target we defined the flexibility profile in the lower right corner in Fig. 4 and sampled 100 backhones for cach length N E [200, 215, 300). BG scale in linear schedule is computed as scale(t) scalemax (t/tmax) with scale set as 5. We report metrics averaged over 10 top-ranked samples per profile. Standard deviations are computed by bootstrapping all generated samples 10 times before ranking them as described above. Unconditional flow means that the flexibility profile is not passed as condition during inference. For more details see caption to Tab. 3.
Med. scRMSD
(t)
MAE (L)
Helix pet.
Strand pet.
Coil pet.
Static schedule'
2.78 (0.13)
0.73 (0.01)
0.30 (0.01)
0.74 (0.13)
0.05 (0.07)
0.22 (0.02)
Static ?
7.58 (0.53)
0.78 (0.00)
0.26 (0.01)
0.65 (0.11)
0.02 (0.03)
0.33 (0.03)
Linear schedule
1.60 (0.07)
0.71 (0.01)
0.31 (0.00)
0.70 (0.15)
0.07 (0.07)
0.22 (0.02)
Length scaling
1.80 (0.11)
0.76 (0.00)
0.25 (0.01)
0.52 (0.20)
0.07 (0.08)
0.40 (0.04)
Flone-lwww--ettt scaling
3.23 (0.14)
0.78 (0.01)
0.25 (0.01)
0.46 (0.09)
0.08 (0.06)
0.46 (0.02)
FliPS FliPS*
1.15 (0.04)
0.47 (0.05)
0.33 (0.01)
0.51 (0.24)
0.21 (0.17)
0.28 (0.03)
2.78 (0.12)
0.89 (0.00)
0.24 (0.00)
0.49 (0.08)
0.19 (0.05)
0.32 (0.00)
"BG = 5
|BG
10
UUncodittioaal flow
*Conditional flow
20

---

Flexibilty protcin structure design with flow matching
A
B
Figure A.7. (A) Example a backbone generated with BackFlip guidance with static schedule BackFlip guidance scale set to 10 from Tab. A.4 with distorted, unphysical structure (sRRMSD 15.1 Å). Especially prominent are improper torsional angles of the helix in green and discontinuous helix in yellow. (B) Undesignable sample generated with FliPS using conditional flow with no BackFlip guidance (scRSMD 4.1 Å). The backbone does not display obvious physical violations or clashes, in contrast to (A).
21


Reviews:
[{"review_id": "NuArLp4NaB", "reviewer": "Reviewer_3jah", "summary": "This paper proposed a framework for conditional structure generation conditioning on desired flexibility, a key characteristic in catalytic interactions and molecular recognition. They develop BackFlip, a backbone flexibility predicter that can be used for large-scale flexibility annotation, and combine it with a Geometric Algebra Flow Matching model to achieve flexibility conditioned generation.\nThey show that GAFL-Flex can generate novel protein backbones with the desired flexibility, verified by Molecular Dynamics (MD) simulations.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time they invested in reading the paper and their helpful suggestions!\n\ni. **General response** \n\nWe retrained GAFL-Flex on the BackFlip-annotated PDB dataset of 22977 monomeric protein structures filtered by the (i) length between 60 and 512 residues and (ii) absence of breaks in the structure, and conducted a series of new experiments based on the review. The resulting model generates protein backbones that better match the desired flexibility profiles, demonstrates improved novelty, and succeeds at designing larger proteins for challenging flexibility profiles. We developed a BackFlip-guidance (BG) approach similar to classifier guidance that performs well but is slower than the original conditional model. Due to time constraints, we applied it with our unconditional model but will extend it to RFdiffusion in the final submission. We discuss new experiments below.\n\n**Evaluation of the model trained on PDB**\n\nWe re-ran the experiment from the main text of our submission reported in the Table 2 and found that GAFL-Flex trained on PDB performs better than GAFL-Flex trained on SCOPe in terms of correlation and yields more novel backbones (Table R1).\n\n**Table R1: Performance of GAFL-Flex trained on the PDB (GAFL-Flex\\*) at the benchmark reported in Table 2 of the original submission.**\n\n||r (↑)|MAE [Å] (↓)|Novelty (↓)|\n|-|-|-|-|\n|**10 existing profiles**|||||\n|GAFL-Flex\\*|**0.52 (0.00)**|0.20 (0.00)|**0.64 (0.02)**|\n|GAFL-Flex|0.45 (0.00)|**0.17 (0.00)**|0.68 (0.02)|\n|GAFL-uncond.|0.20 (0.00)|0.20 (0.00)|0.73 (0.02)|\n|SCOPe proteins|0.19 (0.00)|0.25 (0.00)|1.0 (-)|\n|**10 arbitrary profiles**|||||\n|GAFL-Flex\\*|**0.56 (0.00)**|0.44 (0.00)|**0.64 (0.02)**|\n|GAFL-Flex|0.47 (0.00)|**0.43 (0.00)**|0.68 (0.03)|\n|GAFL-uncond.|0.09 (0.00)|0.48 (0.00)|0.72 (0.02)|\n|SCOPe proteins|0.09 (0.01)|0.48 (0.00)|1.0 (-)|\n\nii. **Experiments on longer proteins**\n\nWe defined 4 new target flexibility profiles suitable for longer proteins (given by 3 to 5 rectangular peaks with widths of 10 to 20 residues and amplitudes of 1 to 2.5 Å). We sampled 100 protein backbones for each length in [200, 250, 300]. We also included the SOTA unconditional structure generative models RFdiffusion and FoldFlow2 [1].\n\nTable R2 reports the results of the experiment. Conditional generation yields protein backbones that closely follow the respective target profiles. Unconditional sampling, on the contrary, produces samples that do not reflect desired profiles. Similar to the experiment from our submission, we observe that conditioning improves novelty of sampled backbones. Average flexibility is elevated compared to unconditional generation.\n\nRemarkably, BackFlip-guidance (GAFL-BG\\*) achieves the same performance as the conditional model (GAFL-Flex\\*). GAFL-BG is about 20% slower.\n\n**Table R2: Performance of GAFL-Flex for longer proteins. We evaluated 4 new target flexibility profiles. Sampled lengths L ∈ [200, 250, 300], each 100 backbones. Metrics are evaluated using BackFlip on all samples.**\n\n| Method| r (↑)| MAE [Å] (↓)| Novelty (↓)| Avg. Flex [Å]|\n|-|-|-|-|-|\n| **4 arbitrary profiles**|||||\n| GAFL-Flex\\*|**0.757**|**0.268**|0.569| 0.730|\n| GAFL-BG\\*|**0.757**|**0.267**|0.566|0.730|\n| GAFL-uncond.\\*|-0.03|0.33| 0.67|0.56|\n| FoldFlow2|0.00|0.33|**0.48**|0.46|\n| RFdiffusion|-0.01|0.32| 0.58|0.55|\n\niii. **Designability of flexibility-conditioned generated backbones**\n\nWe observe that the more novel and flexible even natural protein backbones are, the higher the self-consistency RMSD (scRMSD) computed in the refolding pipeline becomes (Figure 4 in the main text of the submission). Designability depends on the target flexibility profile (Figure A.8), which is not surprising as flexibility introduces fundamental uncertainty to scRMSD. We regard our finding as an important contribution to rethinking the designability definition for flexible protein design. We think approaches for alternative definitions could include making the refolding-RMSD threshold dependent on novelty and flexibility or to only consider stiff parts for calculating scRMSD.\n\nIf one would rely on the well-established cut-offs for designability (e.g. scRMSD < 2.0, pLDDT > 70), one would inevitably introduce a bias towards selecting rigid proteins. It is important to note that these cutoff values were conceived having static protein representation in mind.\n\niv. **Distinction between flexibilities in structured regions**\n\nDue to time constraints during the rebuttal phase, we were not able to make an experiment in this regard. However, we believe this is a great suggestion and we think can answer this question by computing the Pearson correlation and MAE to the ground-truth MD flexibility by masking loops during the computation of metrics. We will include this analysis in the final version of the paper.\n\n**Note: We also evaluated another recent flexibility prediction model and observe that BackFlip outperforms it (see answer VDWk, Table  R3, R4, Section i).**", "claims_and_evidence": "The claim that BackFlip can accurately predict flexibility profile (as measured by MD RMSF) is supported by experimental results onthe  ATLAS test dataset. Although the true flexibility might be different from the MD simulation. \nThe flexibility-conditioned generation performance is supported by some evidence from small proteins, but its capability to generalize to bigger proteins (that are likely to contain more flexible regions) is undetermined.", "methods_and_evaluation_criteria": "The authors proposed to use local RMSF instead of global RMSF to quantify flexibility and generate 10 conformations from MD to produce ground-truth training data. The conditional flow matching model is trained using similar techniques in classifier-free guidance that balances conditional and unconditional training and uses auxiliary loss that penalizes the deviating (predicted) flexibility of the generated structure.  The methodology overall makes sense, although the training data seems to focus only on short proteins with len 60-128 (3673 structures) and might be too small to reliably learn complex geometries of the protein backbones.\n\nSince the BackFlip score is differentiable, maybe another baseline they should compare is directly using BackFlip to do test time guidance of pre-trained large-scale backbone generative models such as RFDiffusion.", "theoretical_claims": "The paper is mostly method development and empirical evaluation and therefore does not have theoretical results to assess.", "experimental_designs_or_analyses": "As mentioned in the Methods and Evaluation criteria, experimental designs are valid. It would be nice if a comparison with strong unconditional models, as well as classifier-guidance results, could be added (e.g., using DPS with BackFlip prediction as objective). \n\nIn addition, experiments on longer proteins (>128 aa) are needed to prove that the model can be scaled to more complex protein backbones (and likely to contain more coils).", "relation_to_broader_scientific_literature": "The paper is one of the first method that this reviewer know on flexibility conditioned design. The task can be generally related to molecular dynamics generation of proteins, such as MD trajectory learning, conformation sampling, and structure prediction.", "essential_references_not_discussed": "The references are covered quite comprehensively.", "other_strengths_and_weaknesses": "It seems comparison on pLDDT and RMSD is omitted in the evaluation as the authors thought that more flexible protein will lead to worth RMSD/pLDDT. However, since both metrics are still important in current de novo design, how should we measure the designability/foldability of flexibility-conditioned designs if these two metrics lose their meanings?", "other_comments_or_suggestions": "No other comments.", "questions_for_authors": "Does the model scale to larger data and longer proteins?\nDo we know if the model is not solely recognizing alpha-helix and beta sheet? Are there quantitative measures on how well the model differentiate flexibilities in non-loop regions?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0tJlXDjYAd", "reviewer": "Reviewer_VDWk", "summary": "This paper introduces a novel framework for de novo protein design that explicitly incorporates residue-level flexibility—a dynamic property critical for biological function—into the generative process. Current methods prioritize static structural features (e.g., motifs, symmetry), limiting their ability to engineer proteins for dynamic processes like catalysis. The authors address this gap with two key innovations: BackFlip, an SE(3)-equivariant network predicting flexibility from backbone structures, and GAFL-Flex, a conditional flow model that inversely generates backbones conditioned on target flexibility profiles. Extensive experiments show that BackFlip can accurately predict the flexibility with a high Pearson correlation coefficient on unseen data and GAFL-Flex can generate plausible proteins conditioned on target flexibility profile. The paper is overall well-written.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We cordially thank the reviewer for their time invested in reading the paper and for their constructive review. We discuss questions and concerns below line by line.\n\n- We note that we retrained GAFL-Flex on the larger PDB dataset and observe enhanced performance at the original benchmark (answer to 3jah, section 'Evaluation of the model trained on the PDB').\n\n- We also note that we introduce a novel BackFlip-guidance approach for conditional generation that we evaluate on longer proteins (answer to 3jah, section 'General response' and 'Experiments on longer proteins').\n\ni. **Backbone flexibility prediction in absence of sequence information**\n\nWe compared the performance of BackFlip to FlexPert [1], another backbone flexibility predictor that combines embeddings from a large protein language model (pLM) with structural features. Similar to BackFlip, FlexPert is trained on the ATLAS dataset. We retrained BackFlip on the global RMSF metric and dataset split used in FlexPert, both without and with one-hot encoded sequence embeddings, to assess their effect.\n\nTable R3 reports inference results on the ATLAS test set. Without any sequence embedding, BackFlip outperforms FlexPert on both global and per-target Pearson correlation (as reported in the FlexPert paper), while performing slightly worse in terms of MAE. Indeed, one-hot encoded sequence improves the performance of BackFlip, but it is clearly possible to estimate the flexibility already from the structure alone. Since we utilize BackFlip in the auxillary loss during training of GAFL-Flex and screening of de novo generated backbones, where the sequence is not defined, we find it advantageous that BackFlip demonstrates such a strong performance without requiring any sequence as input. We regard this as an important contribution since it contrasts the paradigm of relying solely on evolutionary or sequence information for predicting dynamical properties.\n\nWe also compared BackFlip and FlexPert on a set of de novo proteins (from Table 1 in the main text of the submission), with results shown in Table R4. BackFlip significantly outperforms FlexPert on all metrics. We think the reason for this might be that pLM embeddings are not informative for these proteins, as there is no evolutionary information available.\n\nThese results support our hypothesis that the geometry of a backbone and secondary structure composition of a well-folded, globular protein are sufficient to infer short-range nanosecond backbone flexibility without sequence information. However, we agree with the reviewer that sequence information is more important when it comes to long-range protein flexibility. We expect that BackFlip will generalize worse to highly dynamical systems, such as intrinsically disordered proteins, where dynamics is in the range of micro or milliseconds and is dictated to a large extent by the sequence [2].\n\nii. **On MD simulations of generated de novo backbones**\n\nWe apologize for any lack of the clarity on MD procedure. We will clarify this in the final version.\n\nWe conduct all MD simulations following the protocol published in the ATLAS paper, that is, all-atom simulations for 300 ns conducted as 3 replicas with explicit TIP3P water as solvent. The input structure to the MD simulation is the ESMfold-refolded protein (sequence is designed by ProteinMPNN, see A.7 section in the paper) with the lowest scRMSD to the backbone generated by GAFL-Flex. We report all metrics (r and MAE) on Cα RMSF profiles, but these are extracted from the all-atom trajectories.\n\nIndeed, BackFlip is trained to predict Cα RMSF from the ATLAS dataset structures, which are all-atom. Accordingly, all evaluations report Cα RMSF metrics. BackFlip only sees [N, CA, C] backbone atoms during training, thus it only implicitly predicts effects of side chain atom interactions - like AlphaFold2.\n\n**Table R3: Performance of BackFlip retrained with or without sequence embedding on the global RMSF metric on ATLAS dataset split of FlexPert [1].**\n\n| Model|Global *r* (↑)|MAE [Å] (↓)|Per-target *r* (↑)|Per-target MAE (↓)|\n|-|-|-|-|-|\n|BackFlip *|0.78|0.61|0.88|0.73|\n|BackFlip †|0.81|0.56|0.88|0.72|\n|FlexPert ‡|0.74|0.44|0.83|0.47|\n\n* No sequence embedding † One-hot sequence embedding ‡ pLM sequence embedding\n\n**Table R4: Performance of BackFlip without sequence embedding retrained on the global RMSF metric on ATLAS dataset split of FlexPert [1] on MD simulations of 100 de novo proteins sampled with RFdiffusion or FrameFlow.**\n\n|Model| Global *r* (↑)|MAE [Å] (↓)|Per-target *r* (↑)|Per-target MAE [Å] (↓)|\n|-|-|-|-|-|\n|BackFlip *|0.63|0.49|0.85|0.48|\n|FlexPert ‡|0.51|0.62|0.63|0.60|\n\n* No sequence embedding † One-hot sequence embedding ‡ pLM sequence embedding\n\n---\n**References**\n\n[1] Kouba, Petr, et al. \"Learning to engineer protein flexibility.\" arXiv preprint arXiv:2412.18275 (2024). \n\n[2] Radivojac, Predrag, et al. \"Protein flexibility and intrinsic disorder.\" Protein Science 13.1 (2004): 71-80.", "claims_and_evidence": "This paper claims that \"Back-Flip is a backbone flexibility predictor which is entirely independent of sequence information\". However, flexibility is intrinsically coupled with sequence as side-chain type and conformation are keys to protein thermostability and flexibility. Therefore, my concern is that flexibility can not be accurately predicted without any sequence information as an input. I kindly suggest authors can elaborate on this point.", "methods_and_evaluation_criteria": "The proposed method of training a backbone flexibility predictor followed by conditioned generation model makes sense to me.", "theoretical_claims": "This paper does not claim theoretical contributions, so there are no theoretical claims. I have checked the formulas used in this paper. They are correct and understandable.", "experimental_designs_or_analyses": "The experimental design raises concerns regarding the role of side chains in evaluating protein dynamics. While protein flexibility depends on both backbone and side-chain interactions, the proposed approach relies solely on backbone MD simulations to validate the results in Table 2o, which introduces uncertainty. A direct comparison of flexibility between backbone-only and full-atom MD (e.g., using the Atlas dataset) is necessary to assess whether backbone-only simulations sufficiently capture dynamic behavior or if side-chain contributions are critical. Additionally, there is a potential inconsistency in dataset usage—the Back-Flip model is trained on the Atlas dataset, which includes side-chain information, yet the generative module is validated using backbone-only MD simulations. If Back-Flip is trained properly, it predicts all-atom flexibility, which should not be directly compared to de novo protein backbone flexibility. Addressing these discrepancies through additional controlled experiments would strengthen the validity of the approach.", "relation_to_broader_scientific_literature": "Traditional protein modeling tools such as Rosetta MotifGraft (Alford et al., 2017) with BackRub sampling (Lauck et al., 2010), Modeller (Šali & Blundell, 1993), and LoopGrafter (Planas-Iglesias et al., 2022) enable flexibility engineering in specific regions through structure-guided iterative sampling using empirical energy functions, but their applicability is constrained by reliance on predefined input structures and high computational costs. While hybrid approaches combining classical tools and deep learning have successfully designed allosteric proteins (Pillai et al., 2024), pH-responsive complexes (Shen et al., 2024), and fold-switching systems (Guo et al., 2024), current deep learning methods primarily focus on structure prediction rather than generative target structure design and lack explicit flexibility conditioning. In concurrent work, Kouba et al. (2024) propose FlexPert-3D, a sequence-based pipeline using molecular dynamics-derived flexibility to fine-tune ProteinMPNN (Dauparas et al., 2022) via evolutionary priors from protein language models. However, their framework operates solely in sequence space and depends on evolutionary information, fundamentally contrasting with our structure-centric generative approach that directly encodes flexibility into structural design.", "essential_references_not_discussed": "I did not find.", "other_strengths_and_weaknesses": "No other strengths and weaknesses.", "other_comments_or_suggestions": "I have no further questions for authors. I will keep a positive rating if my concern can be well addressed.", "questions_for_authors": "## Update after rebuttal\nI thank authors for thier detailed responses and I would like to keep my positive rating to accept this paper.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "c1Wkq8hS3n", "reviewer": "Reviewer_7Wxe", "summary": "This paper takes a step towards overcoming this limitation by proposing a framework to condition structure generation on flexibility, which is crucial for key functionalities such as catalysis or molecular recognition. The authors first introduce BackFlip, an equivariant neural network for predicting per-residue flexibility from an input backbone structure. Relying on BackFlip, we propose GAFL-Flex, an SE(3)-equivariant conditional flow matching model that solves the inverse problem, that is, generating backbones that display a target flexibility profile.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time invested in reading the paper and for their constructive feedback. We are happy that the reviewer agrees with our claims and evidence and appreciates the methods and evaluation criteria and our experimental design. Below we will discuss the comments line by line.\n\n- We note that we retrained GAFL-Flex on the larger PDB dataset and observe enhanced performance at the original benchmark (answer to 3jah, section 'Evaluation of the model trained on the PDB').\n\n- We also note that we introduce a novel BackFlip-guidance approach for conditional generation that we evaluate on longer proteins (answer to 3jah, section 'General response' and 'Experiments on longer proteins').\n\ni. **Dynamics-Informed Protein Design with Structure Conditioning**  \n\nWe thank the reviewer for pointing us toward this work [1]. We have added a discussion in the related work section:\n\n(U. Komorowska et al., 2024) propose conditioning a pre-trained diffusion model on normal modes—i.e., Hessians of the potential energy predicted by a force field that approximate local movements around an equilibrium state. Conditioning is achieved via inference-time guidance, using gradients computed from an analytical normal mode loss.'\n\nWhile the approach may appear similar to the flexibility-conditioning proposed in our paper, both the task and the method are fundamentally different:\n\n1. We propose training a *conditional* model: akin to classifier-free guidance [2], we pass the condition as input and learn a conditional flow. In contrast, [1] uses an *unconditional* diffusion model guided during inference by gradients from an analytical scoring function, following a classifier guidance scheme.\n\n2. The conditioned quantity differs: we use flexibility derived from Molecular Dynamics (MD) simulations, while [1] relies on Normal Mode Analysis (NMA)—a simulation-free approximation that assumes a harmonic potential and is only valid near equilibrium.\n\n3. The guidance approach in [1] requires an *analytical* condition to compute gradients. In contrast, our flexibility-conditioning approach can handle *non-analytical* conditions (e.g., MD-derived flexibility) because we train the model to approximate a conditional flow, rather than apply analytical gradient guidance.\n\nSince the models condition on different quantities (analytical NMA vs. MD-derived flexibility), they solve different tasks and are not directly comparable. While GAFL-Flex can accept any flexibility profile (e.g., derived via NMA), making it a general method for dynamics-informed design, the approach in [1] is limited by its reliance on the harmonic assumption in NMA, which may not always hold.\n\n**Q: Could this per-residue dynamics reflect specific patterns of protein motifs?**\n\nSince lowest non-trivial modes in [1] are computed for the entire protein structure, and motifs are subsequently sampled as sub-regions of these structures, it can indeed be expected that MD-derived flexibilities will correlate with the amplitudes of lowest non-trivial modes of oscillations from NMA.\n\nii. **More advanced conditioning techniques for steering dynamics**\n\nWe retrained GAFL-Flex on BackFlip-annotated PDB of monomeric structures (22977) and developed BackFlip guidance in analogy to classifier guidance. With BackFlip-guidance, we can achieve the same flexibility-similarity as with the conditional model on the four new flexibility profiles for longer proteins, however, it is around 20 percent slower and requires more memory. Training on the larger dataset improves the conditioning performance and yields more novel backbones. For more details we refer to the answer to 3jah, sections 'General response' and 'Experiments on longer proteins'.\n\niii. **Further elaborations on flexibility-conditioned design**\n\nWe also regard this as promising direction and plan to use the framework introduced in this paper in more application-related work in the future. For instance, structural flexibility is believed to be important for the functionality of protein assemblies [3], for enzymatic catalysis [4] and for binding events of flexible receptor and respective ligand-proteins [5].\n\n**Note: We also evaluated another recent flexibility prediction model and observe that BackFlip outperforms it (see answer VDWk, Table  R3, R4, Section i).**\n\n---\n**References**  \n[1] U. Komorowska et al. Dynamics-Informed Protein Design with Structure Conditioning. ICLR 2024\n\n[2] Ho & Salimans, 2022 – Classifier-Free Guidance\n\n[3] Khmelinskaia, Alena, et al. \"Local structural flexibility drives oligomorphism in computationally designed protein assemblies.\" Nature Structural & Molecular Biology (2025): 1-11.\n\n[4] Matsuo, Takashi, et al. \"Global structural flexibility of metalloproteins regulates reactivity...\" Chemistry–A European Journal 24.11 (2018): 2767-2775.\n\n[5] Craveur, Pierrick, et al. \"Protein flexibility in the light of structural alphabets.\" Frontiers in molecular biosciences 2 (2015): 20.", "claims_and_evidence": "This paper introduces a generative model for protein structure design conditioned on per-residue flexibilities using flow matching.\n\nThe experiments demonstrate that flexibility-conditioning leads to the generation of diverse and novel backbones that indeed display the respective target flexibility profile in Molecular Dynamics simulations.", "methods_and_evaluation_criteria": "This paper introduces a generative model for protein structure design conditioned on per-residue flexibilities using flow matching. The proposed flexibility-conditioning framework relies on the structure-based flexibility prediction model BackFlip, enabling large scale flexibility annotation of proteins, a novel flexibility auxiliary loss and a flexibility screening procedure to find protein backbones that best display a flexibility profile of interest.\n\nThis paper also proposes a generalization of RMSF as Local RMSF, in which the fluctuations of a residue are measured with respect to its local surrounding instead of the whole protein. This paper evaluate the flexibility with MD.", "theoretical_claims": "There is no significant theoretical claims.", "experimental_designs_or_analyses": "The experiments demonstrate that flexibility-conditioning leads to the generation of diverse and novel backbones that indeed display the respective target flexibility profile in Molecular Dynamics simulations.", "relation_to_broader_scientific_literature": "This is a direct application of Riemannian Flow Matching in Protein design.", "essential_references_not_discussed": "The most related work, Dynamics-Informed Protein Design with Structure Conditioning (ICLR2024) needs a discussion, and comparison is appreciated.", "other_strengths_and_weaknesses": "Conditioning on protein dynamics is a promising direction, further elaborations (e.g., case study about the application) on this are appreciated.", "other_comments_or_suggestions": "More advanced conditioning techniques for steering (protein) dynamics are appreciated.", "questions_for_authors": "Could this per-residue dynamics reflex some specific patterns of protein motifs as in Dynamics-Informed Protein Design with Structure Conditioning (ICLR2024) ？", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "p4V8gMNm4h", "reviewer": "Reviewer_jPCN", "summary": "This paper introduces a novel framework for flexibility-conditioned protein structure design. The authors present BackFlip, an SE(3)-equivariant neural network that predicts per-residue flexibility from protein backbone structures. Using BackFlip, the authors propose GAFL-Flex, a flow matching-based generative model that conditions protein backbone generation on per-residue flexibility.\n\nThe authors evaluate both BackFlip and GAFL-Flex using molecular dynamics (MD), demonstrating that BackFlip accurately predicts structural flexibility, while GAFL-Flex successfully generates protein structures that exhibit dynamic behavior in accordance with target flexibility profiles.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive and helpful review. We are happy the reviewer finds the problem of flexibility-conditioned design important. Due to character constraints, we try to focus on the most important concerns below.\n\n- We note that we retrained GAFL-Flex on the larger PDB dataset and observe enhanced performance at the original benchmark (answer to 3jah, section 'Evaluation of the model trained on the PDB').\n\n- We also note that we introduce a novel BackFlip-guidance approach for conditional generation that we evaluate on longer proteins (answer to 3jah, section 'General response' and 'Experiments on longer proteins').\n\ni. **BackFlip can predict per-residue flexibility without sequence information**\n\n**General comment on the scope of BackFlip as MD-flexibility-emulator**\n\nWe first want to emphasize that BackFlip serves as a speed-up for MD simulations, which are prohibitively expensive for dataset annotations, and can thus be seen as an MD-flexibility-emulator. Thus, by learning to predict flexibility derived from MD simulations, BackFlip inherits biases of and correlations between ground-truth MD-derived flexibility and any experiment-derived flexibility.\n\n**Scope of the comparison of BackFlip is limited to B-factors and pLDDT**\n\nCrystallographic B-factors and pLDDT are widely regarded as metrics of flexibility and we regard showing that they do not correlate well to MD as important.\n\nWe include another recent flexibility prediction model, FlexPert [1], which combines embeddings from a large protein language model (pLM) with structural features, as a baseline. Similar to BackFlip, FlexPert is trained on the ATLAS dataset. We retrained BackFlip on the global RMSF metric and the dataset split used in FlexPert. **For the results, we refer to the Tables R3, R4 in the response to the reviewer VDWk.** BackFlip without sequence embedding is better or on-par with the sequence-informed, larger model FlexPert on the ATLAS test set and outperforms it on the set of MD simulations of 100 de novo proteins. We conclude that BackFlip is the current SOTA model for predicting MD-derived flexibility.\n\n**Comparison of BackFlip to NMR-derived flexibility**\n\nWe thank the reviewer for their suggestion. As discussed above, BackFlip solves the task of predicting MD-derived flexibility and predicting NMR-derived flexibility is out of scope. We expect BackFlip to inherit the upsides and downsides of MD, i.e. also the correlation of MD- to NMR-derived flexibility. However, we did apply BackFlip to 500 randomly selected NMR ensembles from the BMRB database [2] and compared it to other baselines, where it achieves SOTA performance as well and outperforms FlexPert, in particular. Table R5 summarizes the results.\n\nWe find it is important to note, while NMR structures are commonly deposited as conformational ensembles, these often represent averages over heterogeneous states over broad time spans and cannot be seen as true statistical ensembles [3]. Even the (very) recent micro-milisecond dynamics predictor Dyna-1 [4], trained directly on spin relaxation times observed in NMR, correlates very poorly with the flexibility of PDB-deposited NMR ensembles (Table R5, Dyna-1).\n\n**Table R5: Performance of BackFlip on RMSF prediction of 500 randomly selected NMR ensembles.**\n\n|**Model**|**Global *r* (↑)**|**Global MAE (↓)**|\n|-|-|-|\n|Negative pLDDT|0.45| -|\n|FlexPert|0.43|2.1|\n|Dyna-1| 0.16| -|\n|**BackFlip**|**0.65**|**2.0**|\n\nii. **Comparison of GAFL-Flex with SOTA structure generative models**\n\nWe followed the reviewer's suggestion to compare GAFL-Flex with other baselines. We chose as baselines RFdiffusion and FoldFlow2 [5], as these two models demonstrate SOTA performance for unconditional generation. \n\nWe retrained GAFL-Flex on a BackFlip-annotated PDB dataset. We compare this new model (GAFL-Flex\\*) with the baselines and find that conditional sampling results in proteins that, on average, more closely follow the desired flexibility profile and that are significantly more flexible compared to proteins sampled unconditionally using the baselines. The results and more details on the experiments can be found in (answer to 3jah, section 'Experiments on longer proteins').\n\n---\n**References**\n\n[1] Kouba, Petr, et al. \"Learning to engineer protein flexibility.\" arXiv:2412.18275 (2024). \n\n[2] Hoch, Jeffrey C., et al. \"Biological magnetic resonance data bank.\" Nucleic acids research 51.D1 (2023): D368-D376.\n\n[3] Bonomi, Massimiliano, et al. \"Principles of protein structural ensemble determination.\" Current opinion in structural biology 42 (2017).\n\n[4] Wayment-Steele, Hannah K., et al. \"Learning millisecond protein dynamics from what is missing in NMR spectra.\" bioRxiv (2025).\n\n[5] Huguet, Guillaume, et al. \"Sequence-augmented se (3)-flow matching for conditional protein backbone generation.\" arXiv:2405.20313 (2024).", "claims_and_evidence": "1.BackFlip can predict per-residue flexibility from backbone structures without sequence information.\n\nWhile BackFlip demonstrates strong performance, its evaluation is limited in scope. The authors primarily compare it against pLDDT, a confidence score from AlphaFold, and B-factors, which measure atomic displacement in X-ray crystallography and reflect thermal motion rather than intrinsic flexibility. Although pLDDT and B-factors correlate with flexibility, they are not standard flexibility benchmarks, raising concerns about the completeness of the validation. A more comprehensive evaluation could include alternative methods, such as using ProteinMPNN to generate multiple sequences for a given backbone, refolding them, and analyzing structural variability to see if BackFlip correctly predicts flexibility across different sequence designs. Additionally, the authors could apply BackFlip to NMR datasets, where proteins naturally exist in multiple conformations, providing a real-world test of its ability to capture experimentally observed flexibility. \n\n\n2. GAFL-Flex can generate protein backbones that match desired flexibility profiles.\n\n While the results indicate that GAFL-Flex can generate structures with desired flexibility, the paper does not compare its performance against established baseline models in protein structure generation. RFdiffusion [1], FrameDiff, and other generative models have been shown to produce diverse and designable protein backbones, yet the paper does not examine whether GAFL-Flex actually generates more flexible structures than these existing approaches. A direct comparison with these models would provide stronger evidence that GAFL-Flex improves flexibility-aware design rather than simply being another generative method. The authors could conduct MD simulations on protein backbones generated by different models to assess whether GAFL-Flex produces statistically more flexible structures. Additionally, analyzing flexibility variations across multiple backbones from various generative models would further support the claim that GAFL-Flex introduces a meaningful improvement in flexibility control.\n\n\n[1]Watson, Joseph L., et al. \"De novo design of protein structure and function with RFdiffusion.\" Nature 620.7976 (2023): 1089-1100.\n[2]Yim, Jason, et al. \"SE (3) diffusion model with application to protein backbone generation.\" arXiv preprint arXiv:2302.02277 (2023).", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria do not fully align with the standard benchmarks used in protein flexibility prediction and generative protein structure modeling. While the authors demonstrate that BackFlip accurately predicts flexibility and that GAFL-Flex generates structures conditioned on flexibility, the evaluation remains incomplete due to the choice of baseline comparisons and the absence of key benchmark models.\n\nFor BackFlip, the authors validate its flexibility predictions using MD-derived local RMSF and compare it against AlphaFold’s pLDDT and B-factors from experimental data. However, while these metrics correlate with protein flexibility, they are not standard benchmarks for flexibility prediction. Although it is challenging to establish a proper baseline for a novel task—predicting flexibility solely from backbone structure—the authors should still include alternative baseline methods to provide a fair comparison. Additionally, a better validation would involve using NMR-derived ensembles, which capture experimentally observed conformational heterogeneity. NMR data provide multiple conformations of the same protein in solution, offering a real-world test of whether BackFlip can accurately predict backbone flexibility across different dynamic states.\n\nSimilarly, GAFL-Flex lacks a direct baseline comparison for protein structure generation. While the method is novel, structure generation is a well-established task, and comparing GAFL-Flex against existing models such as RFdiffusion and FrameFlow would provide stronger evidence of its effectiveness. The authors could also perform partial structure generation on NMR datasets, where they condition GAFL-Flex on rigid regions and generate flexible segments. This would allow them to test whether flexibility-conditioned generation outperforms naïve generative models in producing realistic flexible regions observed in experimental structures.", "theoretical_claims": "The paper does not focus on formal theoretical claims or proofs but builds on established principles from flow matching, geometric deep learning, and SE(3)-equivariant networks.", "experimental_designs_or_analyses": "As mentioned above, while the experiments are well-structured, they lack comprehensive baseline comparisons and real-world flexibility benchmarks. The evaluation of BackFlip is limited to comparisons with pLDDT and B-factors, which are correlated with flexibility but not standard benchmarks. A stronger validation would include NMR-derived ensembles to assess flexibility across experimentally observed conformations. Similarly, GAFL-Flex is not compared against existing generative models like RFdiffusion or FrameFlow, making it unclear whether flexibility conditioning provides a meaningful advantage. Expanding the evaluation with NMR datasets and alternative generative models would significantly strengthen the findings.", "relation_to_broader_scientific_literature": "This paper presents a well-motivated approach to rapidly evaluating protein dynamics and generating flexible protein structures, which addresses an important gap in the field of protein design. Current state-of-the-art protein design methods, such as those used for binder design, have achieved significant success when targeting static structures. However, for flexible targets, the success rate is notably lower, as existing generative models do not explicitly account for conformational dynamics. By introducing flexibility-aware protein structure generation, this work has the potential to impact areas such as antibody design, enzyme engineering, and intrinsically disordered protein modeling.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Strengths: The paper tackles an important and underexplored challenge in protein design—integrating flexibility into protein structure generation. This is crucial for designing proteins that target dynamic systems, such as antibodies and enzymes, where flexibility is essential for function.\n\nWeaknesses: The study lacks baseline comparisons, making it unclear how GAFL-Flex compares to existing structure generation models. BackFlip's flexibility predictions are only evaluated against pLDDT and B-factors, which are not standard flexibility benchmarks.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents a novel and well-motivated framework for protein structure generation conditioned on per-residue flexibility—an important but under-explored aspect of protein design. The authors introduce BackFlip, a backbone-only model for predicting flexibility from MD data, and GAFL-Flex, a conditional flow model that generates backbones to match target flexibility profiles. The approach is validated through comprehensive experiments using Molecular Dynamics simulations.

The paper tackles a timely challenge: designing proteins with dynamic, functional properties such as flexibility. The proposed models are thoughtfully designed, and the results demonstrate that GAFL-Flex can generate novel, flexible backbones aligned with specified profiles.

Reviewers raised concerns about generalization to longer proteins, comparisons to baselines like RFdiffusion, and whether flexibility can be predicted without sequence input. The authors responded with new experiments, retraining models on larger datasets, benchmarking against strong baselines, and evaluating on NMR data. These additions substantially strengthen the paper and clarify its contributions.

While not all reviewers updated their scores, the rebuttal clearly addressed the main concerns with compelling evidence. This work lays a strong foundation for future research in dynamics-aware protein design.

I recommend acceptance.
Decision: Accept (poster)