Flexibility-conditioned protein structure design with flow matching

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.

• 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').

• 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').

i. BackFlip can predict per-residue flexibility without sequence information

General comment on the scope of BackFlip as MD-flexibility-emulator

We 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.

Scope of the comparison of BackFlip is limited to B-factors and pLDDT

Crystallographic 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.

We 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.

Comparison of BackFlip to NMR-derived flexibility

We 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.

We 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).

Table R5: Performance of BackFlip on RMSF prediction of 500 randomly selected NMR ensembles.

ii. Comparison of GAFL-Flex with SOTA structure generative models

We 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.

We 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').

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References

[1] Kouba, Petr, et al. "Learning to engineer protein flexibility." arXiv:2412.18275 (2024).

[2] Hoch, Jeffrey C., et al. "Biological magnetic resonance data bank." Nucleic acids research 51.D1 (2023): D368-D376.

[3] Bonomi, Massimiliano, et al. "Principles of protein structural ensemble determination." Current opinion in structural biology 42 (2017).

[4] Wayment-Steele, Hannah K., et al. "Learning millisecond protein dynamics from what is missing in NMR spectra." bioRxiv (2025).

[5] Huguet, Guillaume, et al. "Sequence-augmented se (3)-flow matching for conditional protein backbone generation." arXiv:2405.20313 (2024).

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.

• 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').

• 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').

i. Dynamics-Informed Protein Design with Structure Conditioning

We thank the reviewer for pointing us toward this work [1]. We have added a discussion in the related work section:

(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.'

While the approach may appear similar to the flexibility-conditioning proposed in our paper, both the task and the method are fundamentally different:

1. 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.

2. 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.

3. 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.

Since 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.

Q: Could this per-residue dynamics reflect specific patterns of protein motifs?

Since 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.

ii. More advanced conditioning techniques for steering dynamics

We 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'.

iii. Further elaborations on flexibility-conditioned design

We 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].

Note: We also evaluated another recent flexibility prediction model and observe that BackFlip outperforms it (see answer VDWk, Table R3, R4, Section i).

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References
[1] U. Komorowska et al. Dynamics-Informed Protein Design with Structure Conditioning. ICLR 2024

[2] Ho & Salimans, 2022 – Classifier-Free Guidance

[3] Khmelinskaia, Alena, et al. "Local structural flexibility drives oligomorphism in computationally designed protein assemblies." Nature Structural & Molecular Biology (2025): 1-11.

[4] Matsuo, Takashi, et al. "Global structural flexibility of metalloproteins regulates reactivity..." Chemistry–A European Journal 24.11 (2018): 2767-2775.

[5] Craveur, Pierrick, et al. "Protein flexibility in the light of structural alphabets." Frontiers in molecular biosciences 2 (2015): 20.

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.

• 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').

• 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').

i. Backbone flexibility prediction in absence of sequence information

We 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.

Table 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.

We 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.

These 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].

ii. On MD simulations of generated de novo backbones

We apologize for any lack of the clarity on MD procedure. We will clarify this in the final version.

We 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.

Indeed, 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.

Table R3: Performance of BackFlip retrained with or without sequence embedding on the global RMSF metric on ATLAS dataset split of FlexPert [1].

• No sequence embedding † One-hot sequence embedding ‡ pLM sequence embedding

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.

• No sequence embedding † One-hot sequence embedding ‡ pLM sequence embedding

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References

[1] Kouba, Petr, et al. "Learning to engineer protein flexibility." arXiv preprint arXiv:2412.18275 (2024).

[2] Radivojac, Predrag, et al. "Protein flexibility and intrinsic disorder." Protein Science 13.1 (2004): 71-80.

We thank the reviewer for the time they invested in reading the paper and their helpful suggestions!

i. General response

We 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.

Evaluation of the model trained on PDB

We 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).

Table R1: Performance of GAFL-Flex trained on the PDB (GAFL-Flex*) at the benchmark reported in Table 2 of the original submission.

ii. Experiments on longer proteins

We 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].

Table 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.

Remarkably, BackFlip-guidance (GAFL-BG*) achieves the same performance as the conditional model (GAFL-Flex*). GAFL-BG is about 20% slower.

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.

iii. Designability of flexibility-conditioned generated backbones

We 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.

If 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.

iv. Distinction between flexibilities in structured regions

Due 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.

Note: We also evaluated another recent flexibility prediction model and observe that BackFlip outperforms it (see answer VDWk, Table R3, R4, Section i).