NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Authors: Urszula Julia Komorowska
Francisco Vargas
Alessandro Rondina
Pietro Lio
Mateja Jamnik
Venue: ICML 2025 poster (2025)
Keywords: generative models
diffusion models
protein design
conditional protein generation
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Protein's backbone flexibility is a crucial property that heavily influences its functionality. Recent work in the field of protein diffusion probabilistic modelling has leveraged Normal Mode Analysis (NMA) and, for the first time, introduced information about large scale protein motion into the generative process. However, obtaining molecules with both the desired dynamics and designable quality has proven challenging. In this work, we present NMA-tune, a new method that introduces the dynamics information to the protein design stage. NMA-tune uses a trainable component to condition the backbone generation on the lowest normal mode of oscillation. We implement NMA-tune as a plug-and-play extension to RFdiffusion, show that the proportion of samples with high quality structure and the desired dynamics is improved as compared to other methods without the trainable component, and we show the presence of the targeted modes in the Molecular Dynamics simulations.

NMA-tune:
Generating Highly Designable and Dynamics Aware Protein Backbones
Urszula Julia Komorowska' Francisco Vargas
Alessandro Rondina" 2 Pietro Lio" Mateja Jamnik
Abstract
Motif scaffolding encompasses various other tasks ranging from designing novel antibodies (Correia et al., 2014) to scaffolding enzyme active sites (Watson et al., 2023).
Protein's backbone flexibility is a crucial property that heavily influences its functionalityy. Recent work in the field of protein diffusion probabilistic modelling has leveraged Normal Mode Analysis (NMA) and, for the first time, introduced informa- tion about large scale protein motion into the gen- crative process. However, obtaining molecules with both the desired dynamics and designable quality has proven challenging. In this work, we present NMA-tune, new method that intro- duces the dynamics information the protein design stage. NMA-tune uses a trainable com- ponent to condition the backbone generation on the lowest normal mode of oscillation. We imple- ment NMA-tune as a plug-and-play extension to RFdiffusion, show that the proportion of samples with high quality structure and the desired dy- namics is improved compared to other methods without the trainable component, and we show the presence of the targeted modes in the Molecular Dynamics simulations.
However, often the biological mechanisms behind protein functionality depend not only on the structure, but also on the dynamics (Berendsen & Hayward, 2000). Currently, the beginning stage of the design, more than validation and re- finements stages, is limited by the available tools to include dynamics constraints. Often Molecular Dynamics (MD) simulations are ran to understand the relevant motions, then one makes an informed change in the current design model and runs the MD simulation again (Childers & Daggett, 2017). Since it is not immediately known how the MD tra- jectory will change under specific structure modification, this step must be reiterated many times, which is inefficient and costly.
Recently, a new method emerged that overcomes this design-vassess -asssss--ssfiiin loop. Komorowska et al. (2024) introduce dynamics conditioning, a way to design a protein backbone whose chosen subset of C. carbons have user- specified displacements in the lowest non-trivial normal mode, which corresponds to the large scale collective motion. They capture the coarse-grained dynamics using the Normal Mode Analysis (NMA) (Bahar & Rader, 2005), a computational method to obtain the eigenvectors of motion under an assumed force-field. This extended to joint conditioning, which is structure and dynamics conditioning applied together. Here, for ease, we call the dynamics con- ditioning from Komorowska et al. (2024) NMA AMA--uidance.
1. Introduction
Generative AI has had a tremendous impact on the field of protein design in the recent years. Denoising diffusion probabilistic models (DDPMs), such as REdiffusion (Wat- son et al., 2023), Chroma (Ingraham FrfmeDiff (Yim et al., 2023b), have been applied to design novel protein backbones with given structural properties. A common task is to perform motif scaffolding, that is, to design a stable protein backbone that consists of a pre-defined motif and a scaffold that holds the motif in place.
al., 2023) or
NMA-guidame takes first steps towards dynamics -ynammcc--nformed design, but also suffers from a number of shortcomings. In DDPMs, the protein generation is driven by the uncondi- tional term (does the new sample resemble real protein?) and conditional term (does the sample meet the imposed conditions?). Previous work uses simple analytical func- tion to link the dynamics of the protein with the probability of sampling such protein under conditional real data distri- bution. This method, while beneficial from the perspective of the model design and generalisability, is sensitive to the user defined sampling parameters. Moreover, the genera- tion of high quality samples with both desired structural and dynamical properties has not yet been achieved. This
'Department of Computer Science and Technology, Univer- sity of Cambridge, Cambridge, United Kingdom *Department of Molecular and Translational Medicine, University of Brescia, Brescia, Italy. Correspondence to: Urszula Julia Komorowska <ujk21 @cam.ac.uk>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
could be attributed to the significant challenge in structure
2.1. Normal Mode Analysis
conditioning with motifs as implemented in Komorowska et al. (2024).
Normal Mode Analysis (Bahar & Rader, 2005; Bahar et al., 1997; 2010) is a computationally efficient technique to de- scribe functionally relevant protein motions without running more expensive Molecular Dynamics simulation. Protein residues are often represented as coarse-grained system of N Ca atoms, with positions given by flattened vector of co- ordinates æ IR3N, and strengths of interactions given by the matrix R NMA assumes the atoms reside near the energy minima and undergo harmonic motions about them. The equation of the harmonic motion for the entire system can be written as Mi -xx, where M E 33N x is mass matrix. Eigenvectors solving this equation describe the amplitudes and frequencies of oscillations, as well as the directions of displacements of the individual atoms. NMA has been empirically verified to correctly describe wide range of motions in proteins (Bauer et al., 2019). Impor- tantly, often the lowest non-trivial modes of oscillations capture the large-scale collective motions, and a subset of the lowest modes can be sufficient to explain proteins' dy- namics (Bauer et al., 2019; Tama & Sanejouand, 2001).
To address the difficulty of balancing conditional and un- conditional term (as in XMA-guidancee we develop new method for dynamics conditioning that creates designable scaffolds with given dynamical properties. Our approach retains the advantages of analytical approximation to the conditional probability and the usage of some pre- pre-trained unconditional model, but crucially, we add novel trainable component that significantly corrects the conditional term. We call this new methodology NMA-tune. In contrast to the previous work on the dynamics conditioning that relied on the weighted sum of the unconditional noise and the educated guess of the conditional noise, NMA-tune learns their best combination. The conditioner is small neural network (530K parameters in our experiments), requires little computational resources to train, allows for faster com- putation of the loss-guidance term and therefore decreases sampling time, and lastly offers significant improvement in joint conditioning efficiency.
Our contributions are as follows:
2.2. Diffusion probabilistic models
We introduce NMA-tune, a method for turning an uncon- ditional diffusion model into dynamics-conditional one.
DDPMs (Ho et al., 2020;
et al., 2015) are
a class of diffusion generative models able to transform a sample from a standard normal distribution øy ~ ~N(0 1) into a sample from ground truth data distribution Po- The key observation is that one can transform zo Po into Fr (N by gradually adding Gaussian noise, and then neural network can learn to reverse this process. The noise is firstly added according to the variance sched- ule { that specifies the noise magnitudes for each step t E [0, For each noising step Pt (1 (x6-1) = N Mar, VT- - 3,44-1, B,7), and the marginal probability (" (r)) -xxx vārzo, (1-ā,)4) where ā. = =II' aq and a; 1 3,- The noisy sample has neat form of 21 = Vā 1 âje, where ~ ~N 1). It was shown that the above noising can be reversed (Sohl-Dickstein et al., 2015), and the reverse transition densities are given by ( However, the above forward and reverse densities are tractable only if conditioned on Ip. while by definition, there is no information about To when sampling novel data. This problem can be circumvented with the appro- priate denoiser training objective. The denoiser co can be trained with the noise-matching loss (Ho et al., 2020) L Et,,,, ||e ea(Vazzo the added noise given a noisy version of xo. With this formulation, the denoiser indirectly learns the conditional Pt (2) In the generation process, it iteratively makes guesses about To given 2z, starting from the fully noised ær.
We provide the conditioner as ready to use plug-in to the open-source model RFdiffusion. Since RFdiffusion is already able to do motif scaffolding, this constitutes an extension to joint conditioning that can easily be used by the wider research community.
We evaluate the effectiveness of the conditioner and its impact the sample designability using 3 proteins cho- sen from the literature. We found that for the dynamics conditioned samples, there exist protein sequences that will fold to protein backbones with desired normal modes, and NMA-tune outperforms existing state-of-the-art.
We run MD simulations on selected samples and perform Principal Component Analysis (PCA) on their trajectories. We observed higher overlap between the targeted normal mode and largest principal component for the dynamics conditioned samples. This orthogonal validation of NMA trajectories indicates our method translates to success in downstream, biological tasks.
= .V 2
-_ c), 1- 1 3,I).
VI-ā,e]]l² ep predicts
2. Background and related work
Since we utilise NMA and diffusion based and score models in our methodology, we start by providing an overview of their underlying theory, and then discuss the specifics of the RFdiffusion model.
2

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Score models (Hyvärinen, 2005; Vincent, 2011; Song & Ermon, 2019; Song et al., 2021) operate on similar prin- ciples to transform a sample from po into noisy IT. In a diffusion based score model (Song et al., 2021), both noising and denoising are expressed as the continuous- time Stochastic Differential Equations (SDEs), with for- ward dr. = - (()rrd + VB([)dw and the reverse dr- - ( V( In (x))]] dt+ (Bt)dū, where du and dē are the Wiener process in forward and re- verse time. Score models and DDPMs can be unified into a single framework using the score-noise equivalence VA, In (2 (mo) = -4/ViI ãi (for proof, see for exam- ple, Appendix F in (Komorowska et al., 2024)), such that any model that is score predictor is noise predictor as well.
hence
Va, In p(ylor) ~ Vz, In Inp(y/
(4)
22. In 11yp-1,((0) = V (40)
This formulation of loss-guidance is crucial for our method of dynamics conditioning. where we guide the sampling with the custom loss function.
2.4. Related work and RFdiffusion
Protein DDPMs such as FrameDiff (Yim et al., 2023b), Genie (Lin & AlQuraishi, 2023), REdiffusion (Watson et al., 2023) or Chroma (Ingraham al., 2023) were found to produce designable protein backbones both in the uncondi- tional generation and in generation constrained with prop- erties such as symmetry or presence of a given substruc- ture. Didi et al. (2024) offer an extensive survey of struc- ture conditioning methods, and additionally propose to use Doob's h-transform in the structure-conditiona training (motif amortisation). Yim et al. (2023a) use Continuous Normalising Flow (Chen et al., 2018) trained with flow- matching objective (Lipman et al., 2023) instead of the DDPM. In the follow-up work, unconditional FrameFlow was complemented with motif- gooff-guidncce (oos-guiiancce and motif amortisation, and was shown to beat RFdiffusion for some motif-scaffolding targets. The focus of the aforemen- tioned works was structure conditioning, and neither of them has considered dynamical properties. Eigenfold (Jing et al., 2023) incorporates the physical constraints for oscil- lations into the diffusion SDE, but it has not been shown to influence downstream dynamical protein properties. Includ- ing coarse- grasned NMA information in the conditioning is the most recent direction firstly proposed in (Komorowska et al., 2024). This work uses loss-guidance formulation to condition on the lowest normal mode, but in contrast to our methodology, it does not have trainable component, and conditioning efficiency suffers from the approximations needed for the loss- gos--giddacee term calculation. Also, Ko- morowska et al. (2024) show how one can implement two loss- loss-guidance components at the same time, one for dynam- ics, and second for the motif scaffolding. Since RFdiffusion was trained perform motif scaffolding if the motif in- formation is provided, we have no need to include loss- guidance for structure at inference, and we avoid fine-tuning two guidance scales at the same time at inference.
2.3. Loss-guided diffusion
Often, the task is not simply take sample from po, but to sample from po subject to conditions y. The com- mon methods for the conditional sampling include classi- fier guidance (Dhariwal & Nichol, 2021), classifier-free guidance (Ho & Salimans, 2022) and loss-guidance (Song et al., 2023). In those frameworks, the reverse SDE re- quires conditional Va, In pe (2+ly), where y is the target quantity, and (x(elly) is the probability density of dit given that za is subject to y. Note that by Bayes' rule V., (xx v. In p(ylax;)) n., (x.).
Both classifier guidance and classifier-free guidance rely on the assumption that the denoiser is able to learn the connection between Iz and y. When this connection is difficult to learn, such as in the case when y the eigenvector of Hessian matrix arising from moter-molecular forces, loss- doss-guidance is more suitable. The first step in the loss-guided diffusion is to specify the model p( p(mo) (xx) and Z,
[ p(zo) exp]-t,(ao)]daas where (=0) is a custom loss function enforcing the condi- tion at f = O, and Zy is the normalisation constant. This additional loss term turns the unconditional sampling distri- bution into a conditional one. p(y(=o) is key in finding the loss-guidance term Vx, In p(yla).)). Consider the integral
2,
p(yl0,) = I p(y) mo ( mn
(1)
Although intractable, it can be approximated using a point estimate of the mean of the posterior p( done by Chung et al. (Chung al., 2022)
RFdiffusion (Watson et al., 2023) is one of the leading dif- fusion models for protein design, trained to perform both unconditional and structure conditioned generation, and we use it the dynamics unconditional base model in this work. It operates on [N, C.,, C) representation of the protein back- bone. Atom coordinates are derived from two components: the translations of C. atoms from the origin, and residues' orientation frames that position N and C atoms w.r.t. Ca- (3) Because of this division, the diffusion is decomposed into p(y(1z) ~ / p(u))0)(8($0) - zoldro.
to : E [220211]
=
(2s + (1- az) V. Inp(zz)), (2)
V V
3

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Two SDEs as well, one purely on the Euclidean space of Ca translations, and the second SO(3) space of frame rotations. The generation starts from fully noised 27, and at each denoising step RFdiffusion directly predicts the fully denoised sample co which makes it particularly well suited for the approximation in the Equation 3. In the task of motif scaffolding, the information about motif target is explicitly contained in the noisy sample 2;; motif Ca atoms and frames set to the ground truth and never noised.
noise correction from the dynamics conditioner. The final noise prediction is sum of the RFdiffusion noise and the conditioner correction
CRF + Cp.
(5)
where we freeze the parameters of eRF during training.
3.1. Loss-guidanee term calculation
With the ideal choice of by (x), backpropagating through (áo) should give the desired guidance term. How- ever, our goal is to find the ideal Vy with the help of the trainable conditioner eØ. For this reason, we can avoid computationally expensive backpropagation through the RFdiffusion to obtain Vz, (dto). Instead, we calculate cheaper V V V (ão): this quantity describes how the residues of do should reposition to minimise the loss. Note that
3. NMA-tune
We tackle the following joint conditioning problem: can we design a protein backbone consisting of heavy atoms that will have a pre-defined structural motif, and motif Ca atoms will move in the lowest non-trivial normal mode according to the pre-defined displacement vectors?
Methodology. NMA-tune relies on finding a good approx- imation to the guidance term v., In p(ylyz xt) in Equation 4 using the neural network. We still take advantage of the insight that a well-chosen loss function allows to draw a probabilistic connection between r and y in cases when a neural network will be unable to do so. Here, y is the eigen- vector of the Hessian of the potential function. As pointed out in (Komorowska et al., 2024), there are currently no neural network architectures able to calculate eigenvectors of a matrix and to broadly generalise to weakly restricted set of unseen matrices.' 1 Our key innovation is that we take this idea further and we improve the dynamics condi- tioning by: 1) replacing V.. Ly with V (5x)) which improves sampling speed; and 2) passing this term to train- able conditioner that finds better conditional term than by directly using the loss lossguuid The conditioner receives both VE by (zo) and the unconditional score, and learns the best correction to the unconditional term along the path specified by the loss.
8gg
V V2,4y(50) =
Vaoly (80); and
(6)
0x.
0gg 1
dE Zx.
Vot (1 vI
0x
Therefore, the expensive V2,l2(00) is equal to cheap V Va (fo) multiplied by the term dependent on the Jaco- bian de. dr, We empirically found that ignoring the Jacobian and passing V as input to the conditioner yields sat- isfactory results. Similar observations about neglecting the Jacobian term were made by Poole et al. (2023). As detailed in Equation 5, we keep the parameters of the RFdiffusion network fixed and only train a substantially smaller correc- tion network that takes V (ão) as an additional context.
3.2. Dynamics loss for guidance
The condition y enforces that the selected subset of residues should have user-specified relative displacement directions and amplitudes. Therefore, the loss function should be invariant the protein rotations and its length. Note that the presence of the motif introduces the frame of reference, hence the need to use modified version of NMA-loss rather than its first formulation from (Komorowska et al., 2024). Modified NMA-loss is:
Notation. We denote the 3D coordinates of [N, C., C] backbone consisting of the L residues by the matrix E R3Lx3, From I. residues, M residues constitue the motif enforced to have a given atomic placement, and the remain- ing residues are the scaffold such that r = IMs U xg. The function Ui calculates, for a given x, the C. displacement vectors of M residues in the lowest non-trivial mode of oscillation, and the output has the shape uM(x) E xx3 The target displacements of M residues are arranged in the matrix y E R Mx3 We denote the noise prediction cal- culated from REdiffusion output do as CRF, and is the
ly(x) = 1- cos (ū, EM (x))
(8)
where ÿ and ÜM (1) are matrices and u (x) flattened into ID vectors and normalised. If ğ and uMs (z) have identical entries (up to the eigenvector sign), the motif residues move with the target amplitudes and angles with respect to each other. var (ar) uses the in-built PyTorch function to calcu- late matrix cigenvectors. The Hessian matrix was calculated using the Biotite (Kunzmann & Hamacher, 2018) implemen- tation of the Hinsen (Hinsen & Kneller, 1999) force-field with cutoff radius 1 13Á, which is a coarse ggriine model of
'Spectre (Martinkus al., 2022) shows way to generate graph from Laplacian eigenvectors where the neural network must exhibit an understanding of the eigenvectors. However, eigenvee- tors of the Laplacian depend only on the graph edge weights, while the dynamics conditioning problem requires calculating the second order derivatives well.

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
interactions, and ignores the presence of other atoms. All operations described above allow for backpropagation when taking V Van
C.
makes the prediction of the conditional noise correction, which is then summed with enF-
The key part of training is the minimisation of the noise- matching loss on the Ca translations. In this way the con- ditioner learns the valid protein structure, and to assure it is also dynamics-aware, we minimise the NMA-loss. In order to avoid the undesired effects of conditioning pointed out in (Komorowska et al., 2024). such as unrealistic chain dis- tances or radii of gyration, we introduce auxiliary losses to enhance training and weight them appropriately. We denote the expected position at = 0, calculated using the corrected noise c, by co,corr. original sample by Zo. chain distances between Ca in 2: by de, (z), and RG(x) is the radius of gy- ration calculated using Ct of x.. We define the chain loss as
is extracted from protein that contains the motif substruc- ture, and the orientation of y w.r.t. the origin depends on the motif orientation. For each žo, we find the best rotation matrix R of the target motif C. coordinates M. using Kabsch alignment (Kabsch, 1976; 1978), and calculate the loss only after applying R to y- This ensures that the loss is invariant to protein rotations, and the displacements' ori- entations w.r.t. motif residues orientations should match the target as well.
4. NMA-tune conditioner implementation and training
MSE(dc, (fo, (f d. (2o))
Lehain
(9)
MSE(dc. (0o), dc, (2o))
4.1. Architecture and inputs
and the radius of gyration loss as
eg is a rotation-equivariant Geometric Vectron Perceptron- Graph Neural Network (GVP-GNN) with five layers, which operates on vector and scalar features of nodes and edges. Node vector features are normalised vectors V V (ēo) and normalised unconditional noises cnr- Node scalar features are the magnitudes of vectors V (fo), the magnitudes of ERF, residue index along the backbone, and timestep t. Edge features are the displacement vectors between Ca atoms. Note that while RFdiffusion operates on [N, Ca, C] representation, GVP operates on a fully-connected graph of Ca atoms only, and corrects the translational component of the RFdiffusion noise without modifying the rotational part. Workflow and architecture details presented in Fig- ure 6 in Appendix B. Whenever we say that the denoiser takes as input scalar or vector feattures of I, we mean Ca features selected from [N, Ca. C1 backbone features. GVP does not add correction to the noise added to MM in order to avoid counteracting structure conditioning.
Lig |RG (fo, RG
(10)
Lehain and Lig ensure that large scale properties of conditioned backbones do not deviate from RFdiffusion predictions. All losses are weighted as
= 0.05 Luoise +0.8 LNMA + +0.1 Lebain + 0.05 Log (1I)
where LNMA is the same as (x() in Equation 8. eo is trained for 10 epochs with Adam optimiser, which took R 15h on a single Nvidia A A100 80GB. The learning rate le-4 is decreased by 0.1 after 5000 gradient updates and batch size is 32. The Luoise term is the standard denoising loss in the NMA-tune parametrisation Lnpoise E2, E...y. ||c (ERF (Vara v1 āie) teel eolvarvo+VV /1 āre, v))
Dataset for training eo was based the SCOPe database (Fox et al., 2013; Chandonia et al., 2021). We use the data preprocessing from Genie (Lin & AlQuraishi, 2023) to remove proteins with multiple chains or missing atoms, and to assure no two domains share >40% sequence identity. Additionally, we filter out proteins shorter than 50 or longer than 256 residues. 7139 proteins remained and we used rrii:valiiatiinntes split 0.8.0.1:0.1.
4.2. Training
The corrector eg is trained with the noise-matching ob- jective and a number of auxiliary losses. In a single train step. [N, Ca. C] backbone is noised time with the default REdiffusion noise schedule and Tmaz 50. RFdiffusion makes a prediction of žo. from which we get the Synamics-unconditions noise with motif scaffolding. For each protein, we randomly choose a motif that is kept constant in 2, with length uniformly sampled between 10-40 residues. Motifs longer than 25 residues are made discon- tinuous, while the ones equal to 25 residues and shorter are made to have 50% chance to be discontinuous. Timesteps of motif residues are set to O for both REdiffusion and eg. Sim- ilarly, motif (i0o) and CRF vectors are set to 0 when passed to the conditioner. With these inputs, the conditioner
Structure and dynamics targets We performed litera- ture search for proteins with well investigated protein hinge motions and prioritised having a smaller set of high qual- ity targets over a larger set but less carefully verified. We chose: triglyceride lipase (Derewenda et al., 1992) (PDB id: 4tgl), calmodulin (Khade et al., 2021) (PDB id: Iexr), and HIV-1 protease in semi-open conformation (Hornak et al., 2006) (PDB id: lhhp). We extract four targets from those, which we call after the protein they come from, that is, 4tgl. lexr, 1hhp for a single asymmetric unit of HIV-1 protease, and assembly for protease biological assem- bly (more discussion of the targets in the Appendix D).
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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
We used PACKMAN software (Khade et al., 2020) to verify that the hinge motion described in the literature is found at our targets. halp_assembly target, which we derive from two flexible flaps of two symmetrically positioned chains, is the only discontinuous target that we use. We did not use targets from (Komorowska et al., 2024), since those were shown to have low designability scores even without dynamics conditioning, thus preventing the generation of proteins. We focus hinge motions since they are the flagship example of impor- tance of flexibility for function, and also NMA-guidane, was originally evaluated on hinge targets and we aim for a comparable benchmark. Note that NMA-tune can in theory be applied to any collective type of motion.
high structure quality to assess whether the targeted NMA corresponds to the oscillation mode in the MID trajectory. We give detailed description of this procedure in Section 5.2.
5. Results
5.1. Conditioning efficiency
Empirically, we found that upscaling eo term was neces- sary to observe the effect of conditioning (sampling details and ablations are in the Appendix A). In RFdiffusion, but also in other works in the field (Yim et al., 2023b; Lin & AlQuraishi, 2023; Trippe et al., 2023), the diversity vs qual- ity trade-off is controlled by downscaling the Gaussian noise term at inference time by a parameter 1) E [o. 1]. Smaller nj can potentially improve the sample quality but at the cost of diversity. Since v) was an important part of the origi- nal RFdiffusion evaluation, we ablate how NMA-tune be- haves for 3 rj values and benchmark against NMA-guidana We reimplement and fine-tune NMA-guidanee as a part of RFdiffusion, so any difference in performance between the two methods cannot be attributed to the base unconditional model. Table I shows the comparison between structure- only (column dynamics-cond. none) and jointly conditioned samples (columns dynamics-cond. tune and dynamics-com -ond:: guid.). Note that structure-only samples set the upper bound on designability achievable for joint conditioning. Targets lexr and 1hhp have high designability in both conditioning scenarios, but 4tgl and Ihhp_assembly were not possible to scaffold even with structure conditioning only. For the latter two targets, we changed the designability threshold of sc-motif- mi--oofff-MMDD to < 3A and < 2Å respectively. How- ever, we still find these results informative, as they show that: 1) dynamics conditioning can be successful even for difficult structural targets; and 2) dynamics conditioning can work well for future models with improved motif scaf- folding capabilities. For each target we sampled scaffold of different lengths. To ensure that differences in sc-cossim be- tween NMA-tune and -MAdaandance are not caused by the differences in scaffold lengths, we seeded generation of each batch of 110 samples, and repeated the procedure for three random seeds. Table shows the mean over three seeds, and the results for the individual runs are in the Appendix E.
designable synamics-conditionn
Evaluation metries We evaluate the samples with respect to the dynamics and designability. We measure the success of the dynamics conditioning in terms of cosine similar- ity (cossim) of vectors ÿ and Ēs (z). Designability metric answers the question: for a given backbone structure, is there a sequence that will fold into it? The metric is com- puted using the standard pipeline of inverse-folding and folding again, commonly used in other works (Yim al., 2023b; Lin & AlQuraishi, 2023; Yim et al., 2023a), and explain it here again for consistency. We follow the REdiffusion steps to compute the oxygen positions in the RFdiffusion [N, Ca, C) generated backbone. For each such backbone, we generate eight amino-acid sequences using ProteinMPNN (Dauparas et al., 2022) at sampling temper- ature 0.1, and we use the version of the model that uses all four atoms instead of C. only. The target motif amino- acids are known and also passed to ProteinMPNN. Those sequences are then folded with ESMFold (Lin et al., 2022) to create the "ESMFold designs". For each sample we choose one design that has the lowest RMSD) to the original back- bone, which is called the self-consistency RMSD (scRMSD). Additionally, we compute the self- consistency RMSD of motif residues only, which is more informative about the success of the motif placement than the global scRMSD. Samples with scRMSD < 2 and sc-motif-RMSD < 1, and whose chosen designs have PLDDT > 0.7 and PAE < 10, are deemed designable. For calculations of both scRMSD and sc- -mmoif-RMSDD we use C. representation. In the end, we compute cossims between target displacements and displacements in: 1) REdiffussin C. backbone out- put (orig-cossim), and 2) ESMFold designed C. backbone crff-cooniistenc-cossiii or sc-cossim). The latter amounts to performing NMA again on the ESM design using the same force-field parametrisation as utilised in the condition- ing. Since ESMFold designs are a more faithful represen- tation of the protein that will be synthesised than the direct REdiffusion outputs, sc-cossim quantifies whether the con- ditioning effects can be observed in the real-world proteins. And finally, as the last evaluation step, we select samples of
Joint conditioning improves sc-cossim score across dif- ferent targets, and our method consistently outperforms NMA-guidance. Since dynamics conditioning is a very novel direction the generative protein modelling, there are no other methods to compare ours other than NMA-guidance. When report the fraction of sam- ples with sc-cossim above some threshold, we mean the fraction of all samples that are designable and have de- sired sc-cossim. Both NMA-guidance and NMA-tune sac- rifice designability to achieve conditioning. However, eval- uation of simple structural metrics such as C, backbone distance, radius of gyration and secondary structure compo-
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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Table Success rates for NMA-tune (tune) and NMA-guidance (guid.) evaluated targets and ablated with respect noise scale т)- For each datapoint we take 110 samples. We average the results over 3 random seeds. Even though NMA-guuddnne produces more designable samples overall, NMA-tune beats NMA-guidance terms successful structure and dynamics conditioned designable samples. *Designability for 4tgl and Ihhp.assemine is calculated according to the changed sc-motif-RMSD criterion. stands for Ihhp_assembly
17 7= 0.0
Target
1hhp guid.
lexr guid.
1hhp_a'
4tgl
Dynamics-cond
tune
none
tune
none
tune
guid.
none
tune
guid.
none
% designable*
58.2
67.3
72.7
53.3
68.8
81.8
20.3
47.7
51.8
11.5
13.2
13.0
% w. sC-cossim > 0.9
12.4
7.6
1.2
9.4
6,4
1.2
1.2
0.6
0.0
0.0
0.0
0.0
% se-cossim > 0.8
20.3
15.5
7.3
26.1
20.9
4.8
7.0
2.1
0.6
0.9
0.0
0.0
% sc-cossim 0.7
29.1
22.7
11.8
33.0
30.0
11.5
9.1
5.2
1.2
2.4
1.5
0.3
0.1
Target Dynamics-cond
lhhp guid.
lexr guid.
1hhp a²
4tgl
tune
none
tune
none
tune
guid.
none
tune
guid.
none
% designable*
53.6
63.0
70.3
53.9
67.6
81.2
17.9
49.5
54.5
10.9
12.7
13.9
% w. SC-cossim >
10.9
6.7
3.0
10.0
0.6
1.5
0.0
0.0
0.0
0.0
% w. sc-cossim > 0.8
20.3
13.3
10.0
22.4
21.5
7.0
4.2
1.5
1.5
0.0
0.3
0.0
% W. sc-cossim > 0.7
26.7
17.9
15.2
32.1
30,9
11.8
8.2
5.5
2.4
0.6
0.3
0.3
1.0 Target Dynamics-cond
Ihhp
lexr
Ihhp_a' guid.
4tgl
tune
guid.
tune
guid.
tune
none
tune
guid.
% designable*
42.7
58.5
57.3
46.4
59.7
68.2
15.8
36.7
37.6
8.2
13.9
13.3
% sc-cossim 0.9
6.1
3.0
0.9
4.8
0.9
0.3
0.3
0.0
0.3
0.0
0.0
0.0
% sc-cossim 0.8
13.9
11.5
6.1
17.3
9.7
1.8
2.4
0.9
0.3
0.3
0.3
0.0
% w. sc-cossim > 0.7
20.9
16.1
10.9
25.8
19.7
6.4
5.8
2.7
2.7
1.2
0.9
0.6
sition does not reveal significant differences between them. We show structural sanity checks for sets of all samples and for samples filtered for designability in Appendix B. Still, NMA-tune wins in terms of designability and dynamies- conditioning at the same time, which is the only subset of samples that can be counted as successful designs. It is not enough to obtain a high quality sample- a sample must be both designable and meet the dynamics conditions, and NMA-tune offers best designablity/conditionnng strength trade-off.
conditioning has negative impact on the designability. De- spite this undesired effect, the percentage of samples that are designable and exhibit target dynamies is higher when conditioning jointly, which means more efficient conditional sampling overall. Figure 7 in Appendix B shows the distri- butional shift of sc-cossims to higher values for designable, dynamics-conditioned samples.
Additionally, our method not only increases the proportion of effectively conditioned samples as compared to NMA- guidance, but also offers significant improvements of sampling speed. For example, 80GB, our method gives much faster sampling of a single protein (note that official implementation of RFdiffusion is compatible only with batch size 1). NMA-tune increases the sampling time only by 8% as compared to sampling without dynamics conditioning, while NMA-guidame causes the increase by about 75%.
Since force-field Hessian depends on the pair- pair-wise distances, modes of oscillation will be related to the structure, and structure conditioned samples will match the dynamics tar- gets to some degree. For example, a small fraction of struc- ture conditioned samples for lexr target has high sc-cossim 0.9. Still, jointly conditioned samples show improved match the dynamics target as compared structure con- ditioned only, as is in the case of lexr and 17 0.0 where the fraction of samples with best sc-cossim increases about 8 times with NMA-tune, and 5 times with NMA- -uuidacce And for some cases, as for lexr and = 1.0, generating the highest sc-cossim samples is possible only with the dy- namies conditioning present. As expected, the designability can go down as 17 increases for some targets, and dynamics
single Nvidia A100
To gain intuitive understanding of good and bad match to the targeted normal mode, we show in Figure 1 the visualisations of two samples, one with high sc-cossim and another with low sc-cossim.

---

NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Table 2. Values of PCA-cossim at 20-100 interval for NMA- tune
and dynamics-uncondiiee unconditional sam- ples. PCA-cossing shifted to higher values for the dynamics- conditioned samples.
dynamics-cond.
dynamics-unconee
sc-cossim
0.92
0.97
0.16
0.32
(a)
(b)
PCA-cossim
0.86
0.58
0.23
0.34
Figure 1. Examples of (a) a design that matches the targeted nor- mal mode (sc-cossim- 0.96), and (b) one that shows poorer match (sc-cossim= 0.68). Both visualisations show the ESMFold de- signs for the REdiffusion backbones. Green arrows show the displacements of the targeted C. atoms in the ESMFold designs, and purple show the target displacement (arrows are upscaled for clarity). Greater se-cossim assures that the relative amplitudes and angles of the green displacement vectors reflect amplitudes and angles between purple displacement vectors.
Table 3. Values of PCA-cossim computed using the tailored time interval (durations in Table in Appendix C).
dynamics-cond.
dynamics-
se-cossim
0.92
0.32 0.97
0.16
PCA-cossim (interval)
0.50
0.48
0.51
0.80
pendix C. Before running the simulations, the ESMFold designs were minimised and equilibrated, with C. RMSDs to their initial structures remaining below 0.5 Á. A detailed description of the simulation protocol and the subsequent evaluations can be found in Appendix C.
5.2. MD evaluation
As final step in our evaluation, we checked whether the targeted normal mode is observed in the MD simulation of the successfully conditioned sample. To this end, we chose 4 ESMFold designs that correspond to designable samples, two with high and two with low sc-cossim, all for 1hhp target. Since the way in which protein explores available conformations is inherently stochastic, pendent 500 ns simulations (replicas) per ESMFold design. The following simulations show that short-term motions of dynamics-conditioned samples are consistent with the targeted mode. In the next paragraphs, we explain the evaluation protocol and relevant metries.
Initially, PCA-cossim values were analysed during the early simulation period (20-100 ns), once the proteins had equi- librated but had not significantly deviated from their start- ing structures. Table 2 shows the results: the first two columns are for dynamics--ondittiones samples and the last two columns are for dynamics-uncondition samples. The PCA-cossim values are higher for dynamics-come conditioned samples, indicating a higher overlap between the PCA mode and the targeted mode. While improvements in PCA-cossim due dynamics conditioning are not as pronounced im- provements in sc-cossim, they remain significant. Dynamies conditioning remains effective in the short-term simulated motions, which is promising for the potential downstream applications.
run three inde-
We perform PCA on the generated trajectories of C, atoms and extract the largest principal component. Similarly, with sc-cossim evaluated on the ESMFold design, we com- pute the cosine similarity between the displacement vec- tors of motif residues in the PCA mode and in the targeted mode - we call this metric PCA-cossim. Note that due to the probabilistic nature of MD simulations. proteins may ex- plore less likely motions, even if the targeted normal mode corresponds to the most probable collective motion. Since there is never guarantee that the most likely domain mo- tion will occur in given single replica, we report the best PCA-cossim value from the three replicas for each sample.
Finally, we analysed the presence of the targeted mode dur- ing the later stages of the simulation, once the proteins had enough time sample alternative conformations. To this end, we find the stable temporal regions by plotting Ca RMSD vs time (Figure 8 in Appendix C), and perform PCA on the trajectories from stable time intervals. As expected, PCA-cossim is lower as compared to initial time intervals (Table 3). While it unsurprising that the direction of the lowest oscillation mode differs between alternative con- formations, the ideal scenario would involve the targeted normal mode persisting in the new stable states reached during the simulation. Together with the early PCA-cossim results, these findings suggest that the initial motions of mnnaicc--conitioons samples better align with the targeted mode, indicating successful conditioning of the starting con- formations.
Two designs were chosen from successfully dynamics- conditioned NMA-tune samples, and the other two from dynamics-uncoondiiooal samples. We selected designs with low MolProbity score (as assessed by SwissModel ser- vice (Waterhouse et al., 2024)): dynamics-conditioned with sc-cossim 0.92 and 0.97 and MolProbity (Williams et al., 2018) scores 1.68 and 1.91, and dynamics-undn unnpiiiionnooniiionnlllll with sc-cossim 0.16 and 0.32 and MolProbity scores 1.52 and 1 1.69. For intuitive explanation of MolProbity see Ap-
8

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
5.3. Limitations
Impact Statement
There is room for improvement terms of stability of the designed proteins. The targeted mode does prevail for alternative states in the simulation in a consistent way, but it still might occur spontaneously (sample sc-cossim-0.32).
This paper presents work whose goal is to advance the gen- eral methodology of generative protein design. The potential consequences of our work for medicine and chemistry are very broad and hard to quantify, and there is nothing specific we are able to highlight here.
Like NMA-guidance, NMA-tune takes the displacements' directions into account, which does not allow for a direct control of the absolute flexibility in the conditioned region. Important direction for further work is to extend condition- ing the absolute values of displacement vectors well.
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12

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Appendices
A. Sampling details
We used 50 sampling steps, which is the default in REdiffusion. We observed it was necessary to upscale the conditional term by some guidance scale to obtain samples both designable and with improved dynamics. We obtained best results if introduced the sinusoidal time dependence for the guidance scale gs, such that gs @Sconst sin t T,, where in the sampling the normalised time flows from to O. Firstly, we set / 0.0 and then searched over several gSoonst. We did not add dynamics conditioning correction in the last 3 generation steps. Then, the best guidance scales for each target were used for experiments with 0.1 and 1.0 (the best guidance scale in a sense of the largest proportion of designable samples with sccosim >0.9). We searched over @Sconst E 20, 40, 60, 80, 100 and additionally 120 for dubp.sssemlly and we chose gSeonst = 80 for all targets expect Ihhp_ aassseeblly for which gSronst 100. Additionally, for targets 1hhp and lexr we ablate the time scaling function in Table 4.
Table NMA-tune ablation with respect the guidance strength time scaling in the sampling. Linear corresponds gs gSoonst t, where time flows from to 0, and constant is simply unscaled gSeonst. Linear and constant scalings perform similar to sinusoidal scaling from the main text, and still outperform NMA-guidance. We also searched for gSconst with finer interval. Values of gSconet chosen for each linear scaling and target were 70 for 1hhp and 30 for lexr, and for constant scaling were 70 for lhhp and 20 for lexr.
1
0.0
f 0.1
1.0
Target
1hhp Constant
lexr Constant
1hhp
lcxr Constant
1hhp
lexr
Time scaling
Linear
Lincar
Linear
Constant
Lincar
Lincar
Constant
Línear
Constant
% designable
59.39
38.18
63.64
64.55
55.76
31.52
62.42
68.79
39.70
27.58
61.21
57.88
% w. sc-cossim 0.9
11.21
9.09
11.52
11.82
7.88
10.00
7.58
12.42
5.15
5.76
6.06
7.27
% w. sc-cossim 0.8
19.39
16.67
29.39
32.12
17.27
14.85
25.76
33.64
13.33
11.21
18.48
18.79
% sc-cossim 0.7
27.58
21.21
38.18
40.91
23.94
19.09
37.58
47.27
17.58
14.24
29.39
29.39
In every single protein generation, we uniformly sampled length of the scaffold before (prefix) and after (suffix) the motif, between min and max values dependent on the target. For the discontinuous motif 1hhp assembly we also sampled gap length between 20-50 residues.
We followed the steps in (Komorowska al., 2024) to implement the benchmark method and also performed the gSconst search. We evaluated the following values of g8const in 1000 intervals: 2000 to 9000 for lexr and 1hhp; 2000 to 5000 for lhhp_assembly; 2000 to 4000 for 4tgl. We chose 8000 for lexr, 7000 for 1hhp, 4000 for Ihhp.assembly, 2000 for 4tgl. Similarly as in our method, we do not add the conditioning update in the last 3 steps.
For both methods we rescaled the corrected noise (which the sum of the unconditional and conditional noise) such that it has the same variance as the unconditional noise, in order avoid too large denoising steps and improve designability, However upon further investigation found this effect was not affecting designability in consistent way across different time scalings and guidance strengths. In the end, fine-tuning the guidance strength was the crucial factor that determined designability.
Table 5. Prefix and suffix ranges used in sampling.
Prefix min
Prefix max
Suffix min
Suffix max
4tgl
30
70
20
70
lexr
30
70
20
70
lhhp
30
50
30
50
30
50
30
50
Sampling was done on single NVIDIA A100 80GB and the compute resources are comparable to only running RFdiffusion sampling.
13

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
B. Extra figures
Figures 2 and show mean C. backbone distance, radius of gyration and secondary structure composition for samples 77 = without filtering for sc-RMSD, For both NMA-tune and NMMA-guiancee the Rg distribution has longer tail for the dynamics-conditioned samples as compared "ynamics-unconditione however it remains in the physical range. Both methods do disturb secondary structure distribution, however the ratio helix:sheet:coil is preserved.
Uncord. NKMA tune
Uncond
100
04
MM4--ume
67.9
88.0
100
0.0
Uncond Hellx
MMA-tune
5.75
3.76
3.77
3.78
15.0
17.5
20.0
22.5
Backbooe
Radiun gyration
Sheet
Coll
0.6
18.6
Uncord.
NMMuuus
25.1
27.0
0.
56.2
51.0
Uncond
NMA-tuue
31.75
3.27
3.78
Backbonn moan Co distance (N]
gyration Fi, [A]
Hulkx
Show:
Coll
Figure 2. Structure metrics NMA-tune Top row: target. Bottom row: Ihhp.assembly target.
0.4
Uncand.
Uncond. NMAeggitance
guidance
0.1-
0.0
Unrond
NMAA-qudanra
3.76
3.77
Backbore mean C_-distance
Radion of gyration (A)
18.6
19
Uhcord.
Uncord guidane
25.1
24.7
0.6
56.2
50.1
0/
00
Uncond
guidacce
1..75
3.75
3.77
3.78
135
200
C. distance
gyration
Hallx
Show:
Cail
Figure Structure metrics for NMA -MMinniiance. Top row: lexr target. Bottom row: Ihhp_assembly target.
Figures 4 and show the same statistics as Figures and 3, but for samples filtered for designability across seeds. Statistics follow similar trend when calculated over samples without filtering.
In the end, perform sanity check whether the differences in designability come from sampling To this end, to evaluate novelty and diversity for targets lexr and 1hhp assembly. For novelty, we compute the TM-score to AFDB and PDB 100 databases available at Foldseck (van Kempen et al., 2024) server, and for each sample retain the max score. Table shows mean values of those max TM-scores (the lower, the more novel samples). While it seems that NMA-tune outperforms NMA- guiddacce by a narrow margin, novelty of both methods remains in the range comparable to other generative models. We calculate diversity using MaxCluster (Herbert Sternberg, 2008) with hierarchical clustering
novel structures.
14

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
15.0
Uncond: NK
Unsond NMA--ume
Denety
09.0
50-
0.2
Uncond
NMA-tune
Backboore mcan C2 distance (A]
Radios of gyratioon Ri, [A]
Sheet
Coll
19.3
23.4
Uncond NMA
Uncond. NMMA-uue
25.4
00-
Uensty
L.0
55.3
20-
42.0
0.0
Uncond
NMA
3.74
3.70
3.76
3.77
3.77
14
20
Backbone
Gi-diittnce
Radius of Ryration
Helx
Sheet
Coll
Figure 4. Structure metrics for NMA-tune (designable samples). Top row: lexr target. Bottom row: Ihhp.assembly target.
4P
H2
Uncand.
0.4
Uncond NM gultane
100
NiMi-uuidarce
0.2
0.0
Uncond
NMA guidance
3.715
1,770
33777
7.00
3.785
Rackbore
Rodiion of gyrntion (A)
Uhcond.
0.8
Uncond
19.3
15.7
MMMeguidaacee
NM Auiiane
0.4
20
55.3
00
Uncond
3.74
3.75
3.76
3.77
3.78
14
26
155
20
Cu-diutance
Eeration Hr
Hellx
Showe
Cail
Figure 5, Structure metries for NMA-guidance (designable samples). Top lexr target. Bottom row: Ihhp.assembly target.
lexs, 9=0.0
lexr, 1j=10
1hhp a. =0.0
1hhp 1j=1.0
NMA-tune
0.68
0.64
0.52
0.53
NMA-guid.
0.71
0.68
0.57
0.55
Table 6. Mean TM-score across different targets and noise levels.
(single linkage method), in sequence independent mode, with TM-score threshold 0.6. From the set of all 110 samples generated per target per noise scale for one seed, we take samples for r 0.0 and 1.0 together, and discard the non-designable samples. Results for the remaining designable samples are as follows (clusters / num of designable samples):
NMA-tune: lexr target: 13/118; 1hhp aassemlly target: 32/34
UMA-guidanee lexr target: 10/137; 1hhp aasemblly target: 52/91
As expected, diversity depends on the scaffolding target. Neither of the methods collapses to sample from a single cluster.
15

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Tunis 0.8 INNI +
Letain
0.05 Iig
naisy dalapoint
GRP +
REdillusion
Eu
CRF
scalar features
Tuple norm GVP projecton
|
tuple
GVP feed forward
x5
tuple
vector features
GVP convolution
NMA-1088
V (
tuple (s.v)
(80)
Figure 6. Conditioning framework diagram. The conditioner is GVP based graph neural network, which operates scalar (s) and vector (v) inputs. Firstly the tuple (s,v) are projected onto node and edge features. Then, inside the conditioner there are stacks of graph convolution where the message passing function is the gated GVP, followed by the normalisation layer that normalises scalar and vector features. feedforward layer with a single gated GVP and another normalisation layer, which sums up to 530K parameters. In training step, RFdiffusion takes noised [N,C,C, C., coordinates of the protein with pre-selected structural motif and makes the dyyamics-uncondition prediction of fully denoised sample do. The dyyaamic--ncondiife
noise is derived from Ji. The lowest oscillation mode in is used calculate the gradient of the NMA-loss. Next, the features passed the conditioner which performs message- paaggegg on the graph of Ca atoms, and it outputs its correction.
1.0
0.8
cossim
0.6
SC cossim
0.4
0.4
0.2
0.2
0.4
0.8
0.0
0.6
0.8
orig_com cossim
orig_cossim cossim
Figure 7. The plot of sc-cossim vs orig-cosssn cossco for conditional (left) and unconditional (right) samples for the lexr target at ns=0.0. Conditional samples exhibit both higher orig-cossim and sc-cossim. Backbones with scRMSD=0 should on the line y
16

---

NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
C. Extended MD discussion
C.1. Scoring sample quality
MolProbity a validation tool for macromolccular 3D structures, such as proteins and nuclcic acids. Its overall score is derived from combination of geometric and stereochemical parameters (e.g., clash score, percentage of Ramachandran outliers, sidechain rotamers) that assess the quality of an atomic model. lower MolProbity score indicates fewer geometric inconsistencies and generally denotes higher quality structure, ensuring the accuracy of atomic coordinates. SwissModel is homology modeling server that generates 3D protein models based on known structural templates and provides quantitative assessments of model quality.
C.2. Simulation protocol
Molecular dynamics (MD) simulations were performed using GROMACS 2024 version and NVIDIA A100 80GB GPU, Simulations were carried out with Amber94 forcefield, in explicit solvent using TIP3P water models, with the system solvated in an octahedral box. Counter ions were added to neutralise the system, with a salt concentration of 150 mM. Periodic boundary conditions were applied to all replicas. The initial energy minimisation was performed using the steepest descent algorithm, with a maximum force tolerance of 10 kJ/mol/nm. maximum 50,000 steps was set to ensure system stabilisation. Following energy minimisation, the system was equilibrated in the NVT ensemble for 100 ps using leap-frog integrator. The temperature in the NPT conditions for 100 ps under similar conditions: the pressure was maintained at 1 bar using Parrinello-Rahman barostat. The simulations were performed in NPT conditions for total of 500 ns each, with time step of fs. Coordinates and velocities were recorded at 10 ps intervals. All simulations were carried out with cut-off distance of 1.0 nm for short-range electrostatic and van der Waals interactions. Long-range electrostatic interactions were treated with the Particle Mesh Ewald method. When selecting the appropriate samples for simulations, we verified the motif residues in the ESMfold are indeed flexible, and the L2 norm of concatenated displacements vectors of the motif residues were 0.31, 0.56, 0.90, 0.60 for samples with sc-cossim 0.97, 0.92, 0.32, 0. respectively. Figure shows how Cg RMSD w.r.t the starting structure
maintained at 300 K using a velocity-rescale thermostat, Next, the system was equilibrated
RMISD of 50 cossim-0 sample
RMGG cossimm 97 comple
SAID wwwsim-2..0000 cample
RM/SD consim-m.. sample
reclis?
Fanal
Final ic
0.20
300
020
300
430 350
700
325
300
19an
Figure 8. C. RMSD vs. time during the simulations. Best-of-three replica, whose PCA-cossim is reported Table 2, is plotted in red.
changes in time. Different replicas reach stable states in different time intervals, therefore for each one of them we choose the tailored time interval where RMSD flattens (Table 7). Even if given replica does not seem to converge to a stable region for longer time, we still try find the best interval, and perform PCA for consistency.
Table 7. Time intervals of trajectories used to compute PCA- cossim in Table 3.
sc-cossim
0.92
0.16
0.97
0.32
replica
3
1
3
1
1
2
interval start, end (ns)
300, 500
100, 300
80,
500
225, 500
225, 400
225,
500
150, 300
1500, 400
100, 225
100,
200
100, 250
100, 250
Computing PCA-cossim all cases requires finding the rotation matrix (as in the NMA-loss in the Equation 8). R aligns the frame of reference of the target displacements with the frame of the motif in the simulated sample. Since the motif residues are moving in the simulation, there no one constant frame of reference. For this reason we align the target motif
17

---

NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
residues onto the average positions of the motif residues in the sample, and the positions are averaged over any given time interval specified in text.
18

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
D. Details of the hinge targets
4tgl (Figure 9) has been found exhibit hinge motion of the lid residues 82-96. We found that choosing residues 70-89 as the motif resulted in protein close-to-deseee -eddeiggabb.... Additionally we used PACKMAN (Khade et al., 2020), tool for finding protein hinges in the query structure, and we found presence the hinge residues 75-90.
Khade et al. (2021) found hinge motions at residues 62-87 of lexr (Figure 10). Choosing residues 64-81 as the motif resulted in the designable samples.
1hhp and 1hhp assembly (Figures 11 and 12) are targets are extracted from flexible flaps of HIV-1 protease. Protease undergoes transformation between closed and open conformations with a pronounced movement of flaps (residues 43-58). We obtained two targets, one extracted from a single chain the asymmetric unit, the other from a biological assembly with two chains. Residues for Ihhp target were 46-54, and for Ihhp assembly target they were 46-50 and 145-149. For 1hhp assembly, the exact hinge motion was found by PACKMAN not exactly at the flap, but at the spatially close residues that are placed near the flap in the gap between the two units in the assembly. We further confirmed that the target residues in 1hhp assembly are indeed active in the lowest frequency motion using the elNemo software (Suhre & Sanejouand, 2004).
Figure 9, 4tgl target
Figure 10. 1exr target
19

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
Figure 11, 1hhp target
Figure 12. 1hhp assembly target
20

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NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
E. Individual results for 3 sampling runs
E.1. seed 1997
0.0
Target Dynamics-cond
1hhp guid.
lexr
1hhp_a guid.
4tgl
tune
no
tune
guid.
no
tune
no
NMA-t
guid.
no
% designable
60.9
63.6
65.5
53.6
66.4
85.5
20.9
46.4
50.0
10.0
16.4
16.4
% w. sc-cossim 0.9
11.8
4.5
0.9
10.0
3.6
0.9
0.0
0.9
0.0
0.0
0.0
0.0
% w. sc-cossim 0.8
20.9
15.5
8.2
30.9
18.2
5.5
4.5
3.6
0.0
0.0
0.0
0.0
% W. SC-cossim 0.7
27.3
19.1
12.7
40.0
27.3
13.6
6.4
4.5
0.0
1.8
0.9
0.0
" 0.1
Target
1hhp
lexr
1hhp guid.
4tgl
Dynamics-cond
tune
guid.
no
tune
guid.
no
tune
no
NMA-t
guid.
no
% designable
50.9
61.8
64.5
51.8
73.6
83.6
16.4
47.3
58.2
13.6
11.8
16.4
% w. sc-cossim > 0.9
16.4
8.2
3.6
11.8
6,4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
% sc-cossim > 0.8
22.7
12.7
10.0
23.6
23.6
8.2
4.5
1.8
1.8
0.0
0.0
0.0
% Sc-cossim > 0.7
27.3
17.3
12.7
32.7
37.3
12.7
9.1
7.3
2.7
0.0
0.0
0.0
1.0
Target Dynamics-cond
1hhp guid.
lexr guid.
1hhp a guid.
4tgl
tune
no
tune
no
tune
no
NMA-t
guid.
no
% designable
41.8
57.3
60.0
53.6
58.2
69.1
10.0
36.4
36.4
10.0
15.5
13.6
% w. sc-cossim
0.9
5.5
0.9
0.0
8.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
% w. sc-cossim 0.8
12.7
12.7
3.6
20.9
6.4
0.9
0.9
0.9
0.0
0.0
0.0
0.0
% sc-cossim 0.7
19.1
17.3
11.8
30.0
19.1
1.8
4.5
2.7
2.7
0.0
0.9
1.8
21

---

NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
E.2. seed 1999
-
Target Dynamics-cond
lhhp guid.
lexr
1hhp_a
4tgl
tune
no
tune
guid.
no
tune
guid.
no
tune
guid.
no
% designable
55.5
69.1
73.6
55.5
68.2
80.9
16.4
45.5
52.7
16.4
10.0
13.6
% w. sc-cossim > 0.9
13.6
9.1
1.8
10.9
10.0
1.8
0.9
0.9
0.0
0.0
0.0
0.0
% w. sc-cossim > 0.8
21.8
16.4
7.3
24.5
23.6
7.3
5.5
0.9
0.9
2.7
0.0
0.0
% w. sc-cossim > 0.7
36.4
24.5
9.1
31.8
29.1
12.7
8.2
3.6
1.8
5.5
1.8
0.0
0.1 Target Dynamics-cond
lhhp guid.
lexr guid.
1hhp_a guid.
4tgl
tune
no
tune
tune
no
tune
guid.
no
% designable
56.4
60.9
77.3
57.3
70.0
82.7
19.1
46.4
48.2
11.8
8.2
10.9
% w. sc-cossim > 0.9
8.2
6.4
4.5
9.1
3.6
0.9
2.7
0.9
0.0
0.0
0.0
0.0
% sc-cossim 0.8
21.8
12.7
10.9
21.8
18.2
9.1
3.6
2.7
2.7
0.0
0.0
0.0
% w. sc-cossim 0.7
28.2
19.1
18.2
32.7
27.3
13.6
7.3
6.4
3.6
1.8
0.0
0.0
11 1.0
Target Dynamics-cond
Ihhp guid.
lexr
1hhp_a a
4tgl
tune
no
tune
guid.
guid.
no
tune
guid.
% designable
44.5
56.4
56.4
47.3
60.0
69.1
19.1
39.1
42.7
7.3
11.8
13.6
% w. sc-cossim > 0.9
6.4
3.6
1.8
4.5
0.9
0.0
0.9
0.0
0.9
0.0
0.0
0.0
% w. sc-cossim > 0.8
16.4
9.1
5.5
19.1
10.9
1.8
4.5
1.8
0.9
0.9
0.0
0.0
% w. sc-cossim > 0.7
26.4
14.5
10.0
29.1
19.1
9.1
6.4
4.5
3.6
1.8
0.9
0.0
22

---

NMA-tune: Generating Highly Designable and Dynamics Aware Protein Backbones
E.3. seed 2007
-7=0.0
Target Dynamics-cond
lhhp guid.
lexr
lhhp_a .a
4tgl
tune
no
tune
guid.
no
tune
guid.
no
tune
guid.
no
% designable
58.2
69.1
79.1
50,9
71.8
79.1
23.6
50.0
52.7
8.2
16.4
9.1
% w. sc-cossim > 0.9
11.8
9.1
0.9
7.3
5.5
0.9
2.7
0.0
0.0
0.0
0.0
0.0
% w. sc-cossim > 0.8
18.2
14.5
6.4
22.7
20.9
1.8
10.9
1.8
0.9
0.0
0.0
0.0
% w. sc-cossim 0.7
23.6
24.5
13.6
27.3
33.6
8.2
12.7
7.3
1.8
0.0
1.8
0.9
0.1
Target Dynamics-cond
lhhp guid.
lexr guid.
1hhp -a
4tgl
tune
no
tune
tune
guid.
no
tune
guid.
no
% designable
53.6
66.4
69.1
52.7
59.1
77.3
18.2
52.7
57.3
7.3
17.3
14.5
% sc-cossim 0.9
8.2
5.5
0.9
9.1
7.3
0.9
1.8
0.0
0.0
0.0
0.0
0.0
% sc-cossim 0.8
16.4
14.5
9.1
21.8
22.7
3.6
4.5
0.0
0.0
0.0
0.9
0.0
% w. sc-cossim
0.7
24.5
17.3
14.5
30.9
28.2
9.1
8.2
2.7
0.9
0.0
0.9
0.9
71 1.0
Target Dynamics-cond
Ihhp guid.
lexr guid.
Ihhp_a aa
4tgl
tune
no
tune
tune
guid.
no
tune
guid.
% designable
41.8
61.8
55.5
38.2
60.9
66.4
18.2
34.5
33.6
7.3
14.5
12.7
% w. sc-cossim > 0.9
6.4
4.5
0.9
1.8
1.8
0.9
0.0
0.0
0.0
0.0
0.0
0.0
% w. sc-cossim > 0.8
12.7
12.7
9.1
11.8
11.8
2.7
1.8
0.0
0.0
0.0
0.9
0.0
% w. sc-cossim > 0.7
17.3
16.4
10.9
18.2
20.9
8.2
6.4
0.9
1.8
1.8
0.9
0.0
23


Reviews:
[{"review_id": "Fr27zsF31r", "reviewer": "Reviewer_FHJ7", "summary": "The paper aims to propose a solution to conditioning protein structure generation on structural dynamics. The authors define protein structure dynamics as the lowest normal modes of oscillations computed with Normal Mode Analysis (NMA) and propose an efficient strategy to incorporate this information into existing generative models for protein structure. In particular, the authors demonstrate that pretrained, unconditional diffusion-based generative models, such as RFdiffusion, can be turned into dynamics-conditional ones via loss guidance without requiring retraining or fine-tuning, and call their framework NMA-tune.\n\nThe authors extend the work of Komorowska et al. (2024) and introduce an SO(3)-equivariant conditioner, which learns the conditional loss guidance term in contrast to the analytical approximation proposed previously in Komorowska et al. (2024). The conditioner essentially learns the set of translations that need to be applied to the unconditional noise in order to sample from the conditional probability distribution. The authors also propose a training strategy along with an SE(3)-invariant NMA-loss for their conditioner to solve the problem of generating structures with certain functional motifs that encode a movement of interest. In a series of experiments, the authors demonstrate the efficiency of their method and outperform a baseline.\n\n## Update after rebuttal\n\nI will keep my score. I believe this work makes an interesting contribution and consistently achieves better performance on relevant metrics.", "questions": "", "limitations": "", "rating": 4, "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": "Thank you for reading our manuscript in great detail. We are glad to know that you find our work well-written and you appreciated the strength of our experimental evaluation. \n\nThank you for pointing out that the “conducted experiments convincingly demonstrate that the proposed method outperforms a previously published baseline”. We are grateful for your feedback on how to improve the clarity of the manuscript. Please note that during the rebuttal period there is no option to upload a revised version of the paper, but we certainly will implement those changes in the future version of the manuscript. Nevertheless, we are happy to report the extra evaluations you would like to see here.\n\n1. “Report Cα-Cα distances and secondary structure distributions without filtering for designability.”\n\nWe recalculated the results presented in Figures 2 and 3 in the Appendix B, but using only designable samples. The number of designable samples is much lower than the total number of samples, therefore we recomputed results using designable samples from all 3 seeds (for the Figures 2 and 3 we used 110 samples). The  $C_{\\alpha}$- $C_{\\alpha}$ distances and secondary structure distributions are now as follows:\n\nFor 1hhp assembly\n\n$C_{\\alpha}$ dist (A): NMA-guid.:  3.7679 ± 0.0075, NMA-tune: 3.7616 ± 0.0080, Uncond.: 3.7679 ± 0.0075\n\n### Secondary Structure Usage\n\n|        | NMA-guid.          | NMA-tune         | Uncond.          |\n|--------|--------------------|------------------|------------------|\n| Alpha  | 0.5338             | 0.4286           | 0.5532           |\n| Beta   | 0.2689             | 0.3378           | 0.2539           |\n| Coil   | 0.1974             | 0.2337           | 0.1929           |\n\nFor 1exr\n\n$C_{\\alpha}$ dist (A): NMA-guid.:  3.7767 ± 0.0032, NMA-tune: 3.7780 ± 0.0025, Uncond.: 3.7803 ± 0.0033 \n\n### Secondary Structure Usage\n\n|        | NMA-guid.          | NMA-tune         | Uncond.          |\n|--------|--------------------|------------------|------------------|\n| Alpha  | 0.8846             | 0.8783           | 0.8903           |\n| Beta   | 0.0027             | 0.0013           | 0.0015           |\n| Coil   | 0.1127             | 0.1204           | 0.1082           |\n\n\nChanges in the above statistic follow the same patterns as when calculated using all samples. $C_{\\alpha}$ distances remain in the realistic range both for NMA-tune and NMA-guidance, and again both methods slightly disturb the secondary structure statistics, which is expected since conditioning induces a distribution shift.\n\n2. “Absolute inference time for each sampling method would be informative.”\n\nWe calculated the running time of the sampling loop (with the default RFdiffusion number of time steps equal to 50) for NMA-tune and NMA-guidance. Mean loop running time for each target on our Nvidia A100 80GB machine, averaged over 110 samples per target, was about 36 seconds for NMA-tune, and about 63 seconds for NMA-guidance, **which gives about 75% speedup**.\n\n3. “It would be informative to include novelty and diversity metrics”\n\nAccording to your suggestion, we use Foldseek [1] and MaxCluster [2] to evaluate novelty and diversity for two targets: 1exr and 1hhp_assembly. For novelty,  we compute theTM-score to AFDB and PDB100 databases available at Foldseek server, and for each sample retain the max score. We then report the mean of those max TM-scores (the lower, the more novel samples). \n\n| Novelty (TM-scores)      | 1exr, $\\eta$=0.0 | 1exr, $\\eta$=1.0 | 1hhp_a, $\\eta$=0.0 | 1hhp_a, $\\eta$=1.0 |\n|--------------|----------------|----------------|---------------------|---------------------|\n| NMA-tune  | 0.68           | 0.64           | 0.52                | 0.53                |\n| NMA-guid. | 0.71           | 0.68           | 0.57                | 0.55                |\n\n\nWhile it seems that NMA-tune outperforms NMA-guidance by a narrow margin, novelty of both methods remains in the range comparable to other generative models (FrameDiff [3], FrameFlow [4]).\n\nWe calculate diversity using MaxCluster with hierarchical clustering (single linkage method), in sequence independent mode, with a TM-score threshold 0.6. From the set of all 110 samples generated per target per noise scale for one seed, we take samples for $\\eta=0.0$ and $\\eta=1.0$ together, and discard the non-designable samples. Results for the remaining designable samples are as follows (clusters / num of designable samples):\n\nNMA-tune: 1exr target: 13/118; 1hhp_assembly target: 32/34\n\nNMA-guidance: 1exr target: 10/137; 1hhp_assembly target: 52/91\n\nAs expected, diversity depends on the scaffolding target. Neither of the methods collapses to sample from a single cluster.\n\n\n[1] https://www.nature.com/articles/s41587-023-01773-0 \\\n[2] https://www.sbg.bio.ic.ac.uk/maxcluster/ \\\n[3] https://arxiv.org/pdf/2302.02277 \\\n[4] https://arxiv.org/pdf/2302.02277", "claims_and_evidence": "Thoroughly conducted experiments convincingly demonstrate that the proposed\nmethod outperforms a previously published baseline. The authors report informative\nmetrics on conditionally generated protein structures, such as secondary\nstructure composition, and assess the effect of conditioning on the well established\ndesignability metric. They justify the selection of the proteins used\nin experiments, conduct extensive Molecular Dynamics (MD) simulations to\ncross-validate recapitulation of the NMA-derived lowest modes passed as a condition,\nand clearly state their limitations.", "methods_and_evaluation_criteria": "Yes. Although NMA, while being computationally efficient, only captures types\nof harmonic motions about the equilibrium state, which might not be sufficient\nto condition on biologically relevant, functional directional movements [1],\nsuch as state-transitions performed during ligand binding [2] or catalysis [3].\nThe authors acknowledge this limitation and compare their results with MD\nsimulation-derived lowest modes in different time windows (Tables 2 and 3).\nTo the best of my knowledge, no precise and computationally efficient methods\nexist to generate and incorporate reliable mid- to long-range protein structure\ndynamics during training of the conditioner. Thus, the motivation to use NMA\nis clear.\n\n[1] Alexandrov, Vadim, et al. ”Normal modes for predicting protein motions: a\ncomprehensive database assessment and associated Web tool.” Protein science\n14.3 (2005): 633-643.\n[2] Deng, Hong, Nick Zhadin, and Robert Callender. ”Dynamics of protein ligand\nbinding on multiple time scales: NADH binding to lactate dehydrogenase.”\nBiochemistry 40.13 (2001): 3767-3773.\n[3] Schramm, Vern L. ”Enzymatic transition states and transition state analogues.”\nCurrent opinion in structural biology 15.6 (2005): 604-613.", "theoretical_claims": "Although the authors mostly adopt theoretical foundations from previously published\nwork Komorowska et al. (2024), they propose to learn the conditional loss term directly from the\npredictions of denoised protein structures made by the generative model via a\ngraph neural network. The authors demonstrate that this approach performs\nbetter than estimating the guidance term analytically as in Komorowska et al. (2024).\nThe goal is to sample from the conditional posterior p(y | x0) , where y encodes\nthe eigenvectors of a Hessian matrix from the NMA computation. Instead of\nlearning the eigenvectors directly, the authors propose to learn a correction term\nto the unconditional noise which best matches the eigenvectors of\nthe precomputed lowest modes.", "experimental_designs_or_analyses": "• Table 1: For reproducibility, I suggest adding full inference settings of\nRFdiffusion beyond the noise scale (e.g., number of timesteps). It would\nbe informative to include novelty and diversity metrics (see example definitions\nin [4]) for generated protein structures. Lower designability of\nNMA-tune compared to unconditional generation might stem from novel\nbackbone geometries.\n\n• Figures 2 and 3: Report Cα-Cα distances and secondary structure distributions\nwithout filtering for designability. I would like to see distributions\nfor only designable samples.\n\n• Inference Time: Absolute inference time for each sampling method\nwould be informative.\n\n• Section 5.2: MolProbity and SwissModel scores may be unfamiliar to\nsome readers. An appendix explaining these metrics would be useful.\n\n• Figure 1: Good visualization of the sc-cossim, but the colors of the arrows\nshould be adjusted for better readability, especially in Figure 1(b), where\nit is difficult to see the direction of the arrows.\n\n[4] Yim, Jason, et al. ”Fast protein backbone generation with se (3) flow matching.”\narXiv preprint arXiv:2310.05297 (2023).", "relation_to_broader_scientific_literature": "To the best of my knowledge, no existing generative models condition protein\nstructure generation on dynamics observables. The only relevant baseline is\nNMA-guidance, which inspired the refinement proposed in this paper.", "essential_references_not_discussed": "Another recent work which focuses on high designability perhaps worth mentioning is\nWagner et al., Generating Highly Designable Proteins with Geometric Algebra Flow Matching, Neurips 2024.", "other_strengths_and_weaknesses": "Strengths:\n• The paper is very well-written with smooth transitions between sections.\nThe authors state the limitations of the proposed method.\n\nWeaknesses:\n• The novelty of the theoretical contribution is limited.", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "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": "xf4AGhS9KN", "reviewer": "Reviewer_xx2G", "summary": "This paper introduces a training-based method to address the problem of dynamic-conditioned generation of proteins. Specifically, they replace the prior-guided term with a simpler, more computationally efficient one to improve sampling speed, and they introduce a small network to learn such conditioned mappings.\n\n## update after rebuttal\n\nI keep my original rating.", "questions": "", "limitations": "", "rating": 1, "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": "Thank you for providing your valuable feedback. Firstly, let us motivate again the usage of the trainable conditioner in NMA-tune. The analytical form of the NMA-loss that we use for loss-guidance might steer the generation into structures that have the ideal NMA-loss, but do not resemble proteins at all. Whether the balance between conditional and the unconditional terms can even be achieved under this loss formulation is the question we are tackling in this work. Most importantly, we show that NMA-tune achieves this balance, and optimises a number of metrics simultaneously, while NMA-guidance without a trainable conditioner struggles to do so.\n\nNext, let us address the other crucial points you make.\n\n1. “However, as shown in Table 1, the designability is notably worse compared to the guided method.”\n\nThe designability as measured by sc-RMSD is an important metric to understand the effects of conditioning. However, the most important metric that determines the method’s success is not just the number of designable samples, but the number of designable and successfully conditioned samples as measured by sc-RMSD and sc-cossim. **It is not enough to obtain a high quality sample - a sample must be both designable and meet the conditions,and with NMA-tune we show that we can achieve designability whilst meeting these conditions.**\n\n2. “Furthermore, there are no metrics provided that consider sampling speed.”\n\nPlease see our response to Reviewer FHJ7, where we provide the comparison of sampling speed, and show NMA-tune is much faster than NMA-guidance.\n\n“Moreover, the evaluation is conducted on a relatively small dataset, which may not accurately reflect real-world performance” and “This paper evaluates methods on a very small subset of dynamic proteins, whereas prior work uses a much larger dataset (10,037 protein structures).”\n\n**Please note that NMA-guidance does not evaluate the conditioning method on 10,037 structures.** Authors of NMA-guidance train their own unconditional protein generative model on 10,037 structures, while we use the already established and thoroughly evaluated RFdiffusion as our unconditional model. For the evaluation of the dynamics-conditioning on its own, the authors use 600 randomly selected targets, which roughly matches the number of samples we take for our carefully selected, real world targets. For that evaluation part, NMA-guidance also uses a different, less restrictive NMA-loss formulation (disregarding orientation of the conditioning residues).\n\nRegarding your concern that our method might not translate into real-world performance, please note that our MD evaluation is designed to tackle exactly this question. MD simulations are regarded as a very close proxy to the real-world behaviour of molecules, and to the best to our knowledge there are no better in silico approximations to the real-world motions than MD simulations. In our experiments Section, we show that the target motion is indeed present in the MD trajectory, therefore showing the success will be translated to real life.\n\n3. “In the MD-evaluation experiment, there is a lack of baselines.”\n\nOur MD simulation experiments were designed to prove the link between presence of the target normal mode in the designed sample and the MD trajectory (close approximation to real life). Most importantly, they show that our *in silico* metrics (in which NMA-tune performs best) translate to real life, and therefore any designable, high quality and successfully dynamics-conditioned sample will exhibit the targeted motion in the MD trajectory. This is the crucial link for assessing  the method’s performance in biological tasks, but since 1) the link *in silico* metrics and MD trajectory is proven; 2) NMA-tune performs best as measured by *in silico* metrics; and 3) MD simulations are incredibly computationally expensive (one simulation can take about a day) and we have limited bandwidth, we deemed it sufficient to include only NMA-tune in the MD evaluation part.", "claims_and_evidence": "The authors claim that their tuning-based method offers superior performance and accelerates sampling. However, as shown in Table 1, the designability is notably worse compared to the guided method. Furthermore, there are no metrics provided that consider sampling speed. Given that the tuning-based method incurs additional training costs, I am not convinced that it is a better strategy than the guided approach.", "methods_and_evaluation_criteria": "I question the necessity of training a network for this purpose. Moreover, the evaluation is conducted on a relatively small dataset, which may not accurately reflect real-world performance.", "theoretical_claims": "I have reviewed the theoretical claims and find them to be correct.", "experimental_designs_or_analyses": "I have examined the experiments and compared them with prior works: \n\n1.\tI am concerned about the performance gap between the tuning-based method and the prior guidance method. Although the paper asserts that the learned guidance term is better, the actual performance is worse.\n\n2.\tThis paper evaluates methods on a very small subset of dynamic proteins, whereas prior work uses a much larger dataset (10,037 protein structures). I recommend evaluating the proposed method on this larger dataset.\n\n3.\tIt would be valuable to include more methods for comparison. For instance, beyond RFDiffusion, other protein generative models such as Frameflow, Framediff, and Genie (as used in the NMA-Guidance paper) should be considered. Additionally, I suggest including other guidance methods, such as classifier-based, classifier-free, and loss-guidance methods.\n\n4.\tIn the MD-evaluation experiment, there is a lack of baselines.\n\n5.\tI recommend incorporating more dynamic-related metrics in addition to protein design metrics, since the paper claims to achieve more accurate dynamic-conditioned generation.", "relation_to_broader_scientific_literature": "The idea of using loss-guidance is well-established (Song et al., ICML 2023), but this method employs a trainable network to approximate this term. The study of dynamic-conditioned protein generation has been explored in the NMA-guidance paper (Urszula et al., ICLR 2024).", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "1.\tThe writing and presentation need significant improvement. Sections 3 and 4 are particularly difficult to read, and there is no main figure in the method section.\n2.\tThe intuition behind using a trainable network is unclear, and the results do not sufficiently support the claims.\n3.\tThe experiments lack numerous baselines and should be conducted on a larger dataset.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "NMIMWTzvp2", "reviewer": "Reviewer_k2U2", "summary": "The paper introduces NMA-tune, a plug-and-play modification to the RFDiffusion framework aimed at enhancing protein design by integrating Normal Mode Analysis (NMA)-inspired diffusion conditioning correction. The proposed method introduces a computationally efficient conditioning term that utilizes the fully denoised RFDiffusion sample prediction to correct the diffusion trajectory. The authors demonstrate an improvement in motif scaffolding tasks and analyze the approach’s performance from a Molecular Dynamics (MD) perspective.", "questions": "", "limitations": "", "rating": 4, "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": "Thank you for providing your constructive critique of our work. We are happy to hear that you appreciate the strong points of our paper, particularly you note “the experimental section is strong. The authors present three case-study proteins with well-documented hinge motions in the literature.”  As you suggest, in the future we would like to expand NMA-tune to operate with other motif scaffolding approaches as well. While it is not feasible in this short rebuttal period, and the strength of our method lies in dynamics rather than just structure conditioning, using other motif scaffolding models would be a great contribution to the community.\n\nWe will revise for clarity the Tables’ descriptions and the findings of the MD evaluation. Please let us know if you have any direct comments on the writing style of the results Section.", "claims_and_evidence": "The claims regarding improved motif scaffolding are supported by experiments involving three case-study proteins with well-documented hinge motions. However, the interpretation of the MD analysis results is not entirely clear, and additional references to tables and figures would strengthen the conclusions.", "methods_and_evaluation_criteria": "The proposed method aligns well with the problem statement. Still, a direct comparison with alternative motif scaffolding approaches, such as SMCDiff [a], would further substantiate the claim of performance improvement.\n\na. Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem, Trippe et al., 2022", "theoretical_claims": "The mathematical description and algorithm appear correct.", "experimental_designs_or_analyses": "The experimental section is strong. The authors present three case-study proteins with well-documented hinge motions in the literature. However, the results from the MD analysis could be better explained to highlight key findings. Furthermore, the paper would benefit from a direct comparison with other motif scaffolding approaches to establish a more comprehensive baseline.", "relation_to_broader_scientific_literature": "The paper advances motif scaffolding and plausible protein structure generation by introducing a plug-and-play conditioning correction for RFDiffusion, guiding diffusion toward the lowest non-trivial normal mode. This novel and original approach opens a new direction for integrating physical constraints into generative models.", "essential_references_not_discussed": "All essential references are discussed.", "other_strengths_and_weaknesses": "The paper is well-written but not always easy to follow. While the method description is clear, additional intuition and explanations for some experimental choices would improve clarity. Including more visualizations of generated proteins in the main text would enhance the presentation.", "other_comments_or_suggestions": "I suggest to:\n1. Include column descriptions in Tables 2 and 3, as they are not immediately obvious and require searching within the text.\n2. Revise the MD analysis section to highlight key findings and provide references to tables for clarity.\n3. Compare NMA-tune with other motif scaffolding approaches, such as SMCDiff, to strengthen the empirical evaluation.", "questions_for_authors": "All suggestions were stated in other sections.", "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": "EEBEHug5mw", "reviewer": "Reviewer_uR1t", "summary": "NMA-tune is a new method that incorporates dynamic information into protein design by conditioning backbone generation on the lowest normal mode of oscillation. It extends RFdiffusion as a plug-and-play component, improving the proportion of samples with high structural quality and desired dynamics. The approach addresses challenges in generating molecules with both designable quality and targeted motions. Molecular Dynamics simulations confirm the presence of the targeted modes.", "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": "Thank you for carefully going through our manuscript and giving us your constructive feedback. As you mentioned, we tackle “challenges in generating molecules with both designable quality and targeted motions”, and we are happy to see you note the strength of the MD simulations evaluations. Let us address your questions one by one.\n\n1. “[...] the results in Table 1 indicate that the performance of “tune” is significantly inferior to that of “guid” on the designable metric. The authors need to demonstrate that this discrepancy is not due to an improperly tuned scaling factor affecting conditional and …” (though the sentence seems unfinished, we guess it was supposed to say “conditional and unconditional terms”)\n\nIt seems to us that a couple of your concerts across your review sections are strongly related to this final point you make. Please note that the final criterion for assessing the method’s performance is a simultaneous optimisation of both designability and NMA-loss. The guidance scale of NMA-guidance was fine-tuned for optimal performance in both of those metrics, and it struggled to achieve the balance between conditional and unconditional terms. Fine-tuning even the most advanced protein generative models is often a necessity (e.g. Proteina fine-tunes CFG and auto-guidance weights [1]). While NMA-tune also requires fine-tuning, it achieved much better balance in those metrics than NMA-guidance, which further motivates the usage of the trainable component.\n\n[1]. Proteina: Scaling Flow-based Protein Structure Generative Models,  https://arxiv.org/abs/2503.00710\n\n2. “Additionally, the evaluation metrics currently only consider designability and sc-cossim; it would be important to also report on novelty and diversity.”\n\nPlease take a look at our response to Reviewer FHJ7, where we provide those metrics.\n\n3. “In Section 5, the authors have altered the threshold for sc-RMSD for two targets, which compromises the fairness of the experiment”\n\nSince NMA-tune and NMA-guidance are based on the same base model RFdiffusion, which struggles exactly in the same way to perform structure conditioning of the difficult targets, we believe the comparison of two dynamics-conditioning methods remains fair even with the adjusted sc-RMSD criterion. Ideally, we would evaluate using a standardised benchmark of dynamical-motifs, like RFdiffusion benchmark for static motifs. Currently, no such benchmark exists, and the number of targets we can use is limited.", "claims_and_evidence": "In the introduction part, the author argued that NMA-guidance has relatively poor performance because of the balance between conditional and unconditional terms. However, it is hard to tell that the model gains performance because of trainable conditioning networks since additional loss terms are introduced in Section 4.2. Experimental evidence is needed to support this. Besides, even if they claim they could adjust the unconditional and conditional terms, they are still upscaling the conditional term in the experiments.", "methods_and_evaluation_criteria": "The novelty of the methodology lies in its ability to decouple the conditional score from the large diffusion model during training, eliminating the need to backpropagate through RFdiffusion. Additionally, the evaluation metrics currently only consider designability and sc-cossim; it would be important to also report on novelty and diversity. Moreover, the experiments are limited to only four targets, which may constrain the robustness of the findings presented in the paper.", "theoretical_claims": "In Eq. 6 and Eq. 7, the author mentions that the Jacobian term can be disregarded. I am curious about how the model can accommodate this, considering that ( x_t ) is not input into the correction network. A more theoretical discussion on this topic would be appreciated.", "experimental_designs_or_analyses": "The results presented in Section 4 are quite limited. Additionally, the section on evaluation metrics is poorly articulated; it only includes two metrics, which could be effectively conveyed by simply stating the relevant thresholds. In Section 5, the authors have altered the threshold for sc-RMSD for two targets, which compromises the fairness of the experiment. It would be more appropriate to select other, more suitable targets for experimentation.\nFurthermore, the results in Table 1 indicate that the performance of “tune” is significantly inferior to that of “guid” on the designable metric. The authors need to demonstrate that this discrepancy is not due to an improperly tuned scaling factor affecting conditional and", "relation_to_broader_scientific_literature": "This paper builds significantly upon the earlier work titled “Dynamics-informed Protein Design with Structure Conditioning,” presented at ICLR 2024. Many of the theoretical concepts are derived from that foundational research.", "essential_references_not_discussed": "The authors have provided a fair related work discussion section, however, some of the statements are not correct. For example, NMA-guidance also has a trainable GVP based model for denoising. A short literature review of condition guided diffusion model papers is missing here too.", "other_strengths_and_weaknesses": "Weaknesses:\nThe paper lacks organization, with an excessive amount of pages devoted to background and related work. It would be more effective to retain only the essential elements of this content in the main text.", "other_comments_or_suggestions": "Table 2 and Table 3 should have a head row to show which model is being used.", "questions_for_authors": "All questions are listed in the above sections.", "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: The paper presents NMA-tune, a method for conditioning protein structure generation on structural dynamics. Reviewers had a mix of positive and critical views. Consensus points included the methodological soundness, strong experimental aspects, and good connection to the literature. Concerns include that reviewers are not convinced by the lower designability, questioned the need for a training network, and pointed out the small dataset size.  Overall, despite weaknesses like limited novelty and performance concerns, the paper's strengths, such as its experimental design and novel approach, are significant. The recommendation is to accept the paper with the condition that the authors address remaining issues in a revised version, including clarifying the MD analysis, evaluating on a larger dataset, and justifying the designability trade-off.
Decision: Accept (poster)