EVA: Geometric Inverse Design for Fast Protein Motif-Scaffolding with Coupled Flow
Authors: Yufei Huang
Yunshu Liu
Lirong Wu
Haitao Lin
Cheng Tan
Odin Zhang
Zhangyang Gao
Siyuan Li
Zicheng Liu
Yunfan Liu
Tailin Wu
Stan Z. Li
Venue: ICLR 2025 Poster (2025)
Keywords: Generative Model
Protein Structure Generation
Training-free Conditional Generation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: A training-free method for motif-scaffolding and conditional generation with flow-based model

Abstract: Motif-scaffolding is a fundamental component of protein design, which aims to construct the scaffold structure that stabilizes motifs conferring desired functions. Recent advances in generative models are promising for designing scaffolds, with two main approaches: training-based and sampling-based methods. Training-based methods are resource-heavy and slow, while training-free sampling-based methods are flexible but require numerous sampling steps and costly, unstable guidance. To speed up and improve sampling-based methods, we analyzed failure cases and found that errors stem from the trade-off between generation and guidance. Thus we proposed to exploit the spatial context and adjust the generative direction to be consistent with guidance to overcome this trade-off. Motivated by this, we formulate motif-scaffolding as a Geometric Inverse Design task inspired by the image inverse problem, and present Evolution-ViA-reconstruction (EVA), a novel sampling-based coupled flow framework on geometric manifolds, which starts with a pretrained flow-based generative model. EVA uses motif-coupled priors to leverage spatial contexts, guiding the generative process along a straighter probability path, with generative directions aligned with guidance in the early sampling steps. EVA is 70× faster than SOTA model RFDiffusion with competitive and even better performance on benchmark tests. Further experiments on real-world cases including vaccine design, multi-motif scaffolding and motif optimal placement searching demonstrate EVA's superior efficiency and effectiveness.

Published as a conference paper at ICLR 2025
EVA: GEOMETRIC INVERSE DESIGN FOR FAST MOTIF- SCAFFOLDING WITH COUPLED FLOW
Yufei Huang ?2 Yunshu Liu", Lirong Wu Haitao Lin Cheng Tan Odin Zhang', Zhangyang Gao Siyuan Li Zicheng Liu 12, Yunfan Liu Tailin Wu?, Stan Z. Li Zhejiang University, Hangzhou
2 Al Lab, Research Center for Industries of the Future. Westlake University huangyufei @west lake. edu. cn; cairt @eeet edu. cn
ABSTRACT
Motif-scaffolding is fundamental component of protein design, which aims to construct the scaffold structure that stabilizes motifs conferring desired functions. Recent advances in generative models are promising for designing scaffolds, with two main approaches: training-basse and sampling-basee methods. Training-based methods resource-heavy and slow, while training- frren sampling-base based methods are flexible but require numerous sampling steps and costly, unstable guidance. To speed up and improve sampling- methods, analyzed failure cases and found that errors stem from the trade-off between generation and reconstruction. Thus we proposed exploit the spatial context and adjust the generative direction to be consistent with guidance to overcome this trade-off. Motivated by this, Geometric Inverse Design task inspired by the image inverse problem, and present Evolution- reA.ssconttiuiio (EVA), a novel sampling-bas based coupled flow framework on geometric manifolds, which starts with a pretrained flow-based generative model. EVA uses motif-aligned priors to leverage spatial contexts, guiding the generative process along straighter probability path, with generative directions aligned with guidance in the early sampling steps. EVA is 70x faster than SOTA model RFDiffusion with competitive and even better performance world cases including vaccine design and multi-motif scaffolding demonstrate EVA's 's superior efficiency and competitive performance.
we formulate motif-scaffolding
benchmark tests. Further experiments real-
INTRODUCTION
An important task in protein design is the generation of structural protein fragments, named scaffold, to support and stabilize a target motif. Here, motifs are structural protein fragments that carry the desired biological functions. Scaffolds, together with motifs, form complete, stable, and designable functional proteins. Motif-scaffolding achieves great success in applications to vaccine and enzyme design (Procko et al., 2014, Correia et al., 2014 Jiang et al. 2008 Siegel et al. 2010). is similar out-painting images (as illustrated in Fig. 11. which aims to generate high-fidelity and consistent images with the desired pixel patches (Chung et al., 2022) Song et al.) 2022). What makes soaff-scaffolding more complex is its 3D spatial and geometric nature, where we need to consider more factors (Castro et al. 2024 Wang et al. 2021). including spatial relationships, layout, physics, and so on to obtain realistic samples ls for real world applications.
to the task of in-painting
Generating diverse and designable scaffolds with accurate motifs of desire is highly challenging. With the development of techniques in protein structure generation, generative models such as Diffusion Probabilistic Models (DPM) have been successfully applied to motif-scaffolding (Trippe et al. 2023
Wu et al. 2023, Watson et al. 2023). Generative methods can be divided into two families: training- based and sampling- based. Training-baa -based methods (Watson et al., 2023] Yim et al.) 2024 Didi ctt al. 2023) train a motiffconditioned generative model that takes the motif directly as input and generates the remainder of the protein the scaffold. These methods are computationally intensive with slow inference speed and inflexible to take advantage of different pretrained generative models,
"Corresponding Author

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as they always need additional training. Sampling-based methods (Trippe al. 2023 Wu et al. 2023) employ guidance (c.g., constrained gradients) in the sampling process of pre-trained generative models, bypassing the dependency on extra or specific training. These methods typically complete two processes at the same time. One is the reconstruction process, which ensures the accurate display of desired motifs with the aid of guidance. The other is the evolutionary (i.e., generative) process which helps to generate overall designable proteins from a broad distribution defined by pretrained generative models. Although training-free and flexible, sampling-based methods require numerous diffusion sampling steps (500-1000 steps) to gradually shift the generative path to the desired distribution, along with computationally expensive and unstable guidance steps, such as backpropagation or sequential Mont-Carlo steps. The slow inference speed impedes the application of both training- and cempling-baseee methods.
To further speed up and improve the sampling- maapli method, we investigate two typical types of failure for sampling-base based methods: samples with low designability or inaccurate motif, as illustrated in Fig. We attribute these emrors to the inherent trade-off between generation (overall protein generation) and guidance (desired motif reconstruction) in sampling-base -based methods, which is also identified for similar image inverse problem solvers (Chung al. 2022 2024b). Thus, overcoming this trade-off and adjusting the generative direction to be consistent with guidance in earlier sampling stages the keys speeding up and improving success rates of sampling-based methods. To achieve this, we propose leveraging the distinct spatial context offered by the explicit point- cloud-like representations of proteins. Previous methods focused on applying sampling theories the motif-scaffolding with little exploration into the spatial contexts of protein point clouds with actual moordinates-sude as the orientation and other geometric properties of desired motifs and generated structures to better align the generative process with guidance and overcome the trade-off.
Motivated by the analysis above, we frame motif cotif-scaffolding as a Geometric Inverse De- sign problem, providing a geometric perspec- tive alongside the traditional posterior sampling view. We introduce Evolution ViA reconstruc- tion (EVA), fast, sampling- based method that uses a novel coupled flow framework on geomet- ric manifolds, starting from a pretrained flow- based generative model. EVA uses motif- moiin- priors to exploit spatial contexts and steers the generative process onto straighter probability path, where the generative directions are aligned with guidance at early sampling steps. EVA is 70x faster than current state-of-the-an RFDif- fusion with competitive and even better perfor- mance. It can finish sampling in only 100 steps with nearly cost-free and highly stable guidance steps, which include simple spatial alignment and interpolation.
Measurement
Recon. Evo.
Evo. Recon.
Image
Inverse Problem
Low Fideliy
Not Consistent
Geametric
Not Designable
Not Accurate
Inverse Design
Devistion
Desired Motif
sc_ 3.500) moli rmadd 0.79 motif_ mmmdd 2.1004)
Two failures in the image inverse problem and geometric inverse design (motif- scaffolding). Recon. is short for motif reconstruc- tion and Evo. is short for overall evolution (i.c, generation). Given partial observations, that is, measurement (motif), the image inverse problem imotti--ccafolldigg aims produce high-fidelity (designable) and consistent (accurate) complete images (protein structures). The images and pro- teins are generated with Chung et al. (2024b) and TDS (Wu et al., 2023). We interpret the 'undes- ignability issue as the tangible low-fidelity issue as they both seek for an overall harmonization.
Figure 1:
In detail, we utilize flow-based methods pre- trained with the flow matching objective (Lip- man et al. 2022 Tong et al. 2024] Yim et al. 2023) for faster inference and more flexibility to control the sampling path. Instead of ran- dom noises, EVA starts with motif-aligned prior which leverages the spatial alignment for adjust- ing the global orientation and center of mass of initial point clouds to be aligned with given mo- tif structure. This could reduce extra variance induced by a random alignment between motifs and prior point clouds, and avoid a twisted and unstable sampling process. At steps of sampling, EVA will estimate posterior means (i.e., predicted noise-free overall structures) that are consistent with desired motif via simple yet effective spatial interpolation of C2 coordinates and residue orientations. The estimated posterior will give generative direction consistent with motif reconstruction.
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To evaluate our method, conducted experiments on the RFDiffusion dataset (Wasson al. 2023) and benchmarked EVA with various methods including state-of-the-an gradient correction methods (Pokle et al.] 2024 Chung et al. 2022) extended to the geometric manifold. We also designed a new vaccine design benchmark that reflects real-world scenarios with our curated datasets. Furthermore, test EVA in cases requiring to find optimal motif indexes multi-motif relative positions. These conditions are provided in benchmark tests, rarely explored before, but actually not easily accessible real-world cases. EVA shows comparative results in all benchmark tests with much reduced inference time. These results suggest our potential for realistic applications.
RELATED WORKS
Motif-Scaffolding Wang et al. (2021) first proposed to use deep learning for
Recently, deep generative models (Trippe et al. 2023 Wu et al. 2023) Yim et al. (024) have been applied for motif-scaffolldng which can be divided into two families: training- maining-based and sampling- based. SMCDiff (Trippe et al. al.]|22222 (023) first proposed the sampling- based training- frrenng--reeeeeo method with diffusion and Sequential Monte Carlo(SMC), TDS (Wu et al. (023) then improved upon SMCDiff using reconstruction guidance for each particle in SMC. These SMC-based method require much more network calls than directly sampling and thus (2024) propose to use gradient correction for motif reconstruction. It is more efficient but requires numerous sampling steps and costly, unstable guidance. Training-basse methods (Yim al. 2024 Didi et al., 2023) like RFDiffusion (Watson et al., 2023) achieve state-of-the-an (SOTA) results on the motif-scaffolding benchmark. However, they rely on expensive fine-tuning of complex model architectures for conditional generation and have slow inference speeds. Our method lies in the sampling-bas category with competitive performance and much improved efficiency, and thus has potential for various realistic applications.
tine-consuming [W et al. (2023) Yim et al.
Image Inverse Problem Image inverse problem aims to recover the original image given some measurement (e.g. part of the image, noisy image). Recent methods solve this problem in plug- and-play fashion via providing gradient correction by differentiating through the diffusion model in the form of reconstruction guidance (Ho et al., 2022), which is further extended in DPS (Chung
ett al. 2022) to nonlinear inverse problems. IIGDM (Song et al. (022)) introduces pseudo-inverse guidance that improves the guidance approximation by inverting the measurement model. Pokle et al. (2024) extends gradient correction to the flow-based model. Gradient- methods heavily rely on a deep gradient approximation of the intractable posterior score, which costly to compute and crude for non-small noise levels at many steps of the diffusion process (Mardani et al. 2023
Chung et al. 2024a). General conditional generation methods can also be applied. SDEdit (Meng et al. (2022) is based on the eplacement-basse method which replaces the measurement-region of the generating samples with noisy given measurement. Our methods is as simple as replacement method with much better performance, which interpolates the predicted clean data with the clean given measurement. More related works can be found in the Appendix A.3
3 BACKGROUNDS
3.1 FLOW MATCHING
In this section, we provide overview of the general flow matching method to introduce the necessary notations and concepts based on (Pooladian al. 2023 Tong et al., 2024 Lipman et al.] [2022].
Definition. Flow Matching is a family of simulation-free training objectives for continuous nor- malizing flow (CNF), which proposes to learn the -eeeddeendennt vector field v(2, t) Rd + IRd to transform sample 2g Rdd from an easy-to-sample prior distribution po to data point 21 from the data distribution Pt. The vector field v determines a unique time-varying flow ere(o.1) Rd Rd through the Ordinary Differential Equation (ODE):
depo(zo) ((be(2)) with boundary condition (o(=)) 2g-
dt
Flow Matching, One wishes to find a vector field e that pushes the flow eh to reach the desired data distribution P1. i..., 2/1 P1. Generally, such vector field is intractable, but can be learned by regressing the tractable conditional vector fields u(2,, t/20, 21 = 4 z where 2, = v (2ol2n)) interpolates between endpoints 2g Po and 21 P1. Theoretically, this interpolation defines the
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time-dependent distributions p(za|20 21). which are referred to conditional probability paths. We focus on the probability path for the conditional Optimal Transport (OT) path (Lipman et al. 2022):
v <>x 21) 21 (1 z) 12ı, p(=x|20, 21) p(==|=1) = (((axi
where 0g = t, and 11- - The conditional OT path has been demonstrated to have good empirical properties, including faster inference and better sampling in practice. Based on the conditional OT path, the conditional vector fields and Flow Matching (FM) regression objective are:
u(2t, 2| 21) n( 2121 21 24
== pu (s)) []0 t) u( =1
(3)
where (2,, t) is a neural network to regress the conditional vector field. The optimal & that minimizes CFM includes the posterior of conditional OT path and takes the form the below:
21 2t -
8(2,, t)
21 = [=
Guidance-based conditional inference The equation above also holds with extra conditions (Pokle et al.) 2024). The optimal vector fields for conditional inference could be obtained by replacing the unconditional posterior with conditional one Ep|21|22 c] in Eq.
Given pretrained unconditional generative models that are trained to approximate EE [21 [x]. guidance-
based conditional inference methods typically convert the conditional posterior W., 2 24, C approxi-
mation into guidance term approximation problem via Tweedie's identity (Yim et al. 2024 Chung
al. 2022 Song et al., 2022] Pokle et al., 2024). Applying this identity under Conditional OT path
in Eq. and simplifying gives (Pokle et al.
2024
Epl21 21 + of 2. In 1((()2)).
Ep b 24, c)
at
(5)
In(as/or) v. In p(c/
0(2,, t, c) = 0(2,, t) to dt
Thus, current guidance-based based conditional inference methods focus on approximating the guidance term v., In p(c|2;) in training-free manner. This forms the basis for various training-free conditional inference methods, including the state-of-the-art sampling-base methods for motif-scaffolding.
3.2 FLOW MATCHING ON THE PROTEIN GEOMETRIC MANIFOLD
The atom positions of each residue in protein backbone structure could be represented as a residue frame tuple g (z, r) (Jumper et al. 2021b), where z IR3 and r E SO(3) is the translation vector (i.e., the coordinate of the residue Ca atom) and rotation of a residue frame. The protein backbone consists of N residues, thus it can be represented by N residue frames as (g(), g )))). We use bold face for collections of elements, superscripts for residue indices, and subscripts for time.
Notation.
The protein geometric manifold MM is product space G of the 3D translation subspace T and the 3D rotation group 50(3) of N residues in the protein backbones. Here T is the Zero Center of Mass (CoM) subspace of R Nx3 which means that the average of N translation vectors should be zero to avoid the extra translation induced by the global rotation. As a start point, we first build the flow matching on this geometric manifold based on Riemannian flow matching (Chen & states that the time-dependent vector field in a manifold M can be defined as: v(z,t)= M x R T&M where t E [0, 1] is the time step and T&M is the tangent space of the manifold at 2 E M. This vector field determines a CNF (ite : M + M. Since G is a product manifold, the tangent space is a direct sum: TGG = T e T,3) z R @ RNNx3 where g (x, r). and the flow ODEs proceed independently in the each manifold. Thus the construction of the flow on G is equivalent to building flows on each geometry independently. We can define the
Lipman 2023).
conditional vector field u(gt, t (gg)) ) = (Mg, (ax,, l[r1), u,, (/ri) of Riemannian flows on with
Conditional Optimal Transport path as follows (Chen & Liimann 2023 Yim et al., 2023):
logr, (ri)
Ua (2, t|x,) = 1-t
Uy (T,, t(ri)
(6)
1-t
The first equation straightforward. Since T Nx3 it trivial to build the flow matching on translation in the same form in Eq. The second equation involves projecting elements in SO(3) to its Lie algebra so s3) using logarithmic mapping. Its core intuition is very similar to the case of T, which computes an evolutionary direction from noisy data point 2; pointing to denoised data æ1.
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with Deelied Molike
Decked
Exolstion VIA raconetrucion
Deaignable Protein
MM
Shorter Sampling
Rardfrr Prior
With Desined
Twikted Part
Coupled
Actuaal Updatee Over HOunl
Nati'
Shorter Path
Shorler hoster
Speed-up
Mu
xo Quskanne
Mestf Allgned Prtor 65
Prior sats
reation A2
Marsdon
Mesil Allgoed Price
Dioignedtt with destred
Falkum Evo. Bro.-Bacon
Evo diecdiecoon
Algreed Prior of
Mavifidd
(e) Speciad Most Prior with Prior Data Distante Winimization Objective
Evenuaiion [Laft, Two casmee--...AA (oura) Recontehuctoo, Generstive Direction allgned with gui guidanne for campling speed
(0) Matt--rteppzheeed Poxteeior concillored sumpligg
The illustration of EVA framework. Our key designs are twofold: (a) The random prior is first aligned with the desired motif through rotation (formally, the Eq. (8) to minimize the distance between them, resulting in a shorter sampling path and accelerated sampling. The aligned prior is referred to as the (Spatial) Motif Aligned Prior. (b) The conditional posteriors g(g,, m) is approximated as the interpolation between unconditional posteriors and motifs m. This novel geometric solution could give generative directions consistent with motif reconstruction (formally, the coupled vector fields in Eq. 12 in early sampling stages.(c) Compared with gradient guidance-based methods, EVA could alleviate the conflict between generation and reconstruction (the failure cases 1 and 2, which require numerous sampling steps to gradually correct in gradient guidance-baa based methods) and thus need less sampling steps for better notif-scaffolding.
Figure 2:
METHOD
This section describes the Evolution ViA reconstruction (EVA) framework. Our method draws inspiration from recent advances in flow matching (Song et al. 2023 Tong et al. 2024), which straighten flows during training for faster inference and better sampling practice. We extend this intuition from training to conditional sampling for the speed-up of motif scaffosoffll which seeks to straighten the sampling path with given motifs. We achieve this through coupled flow, using spatial motif-aligned prior that shortens the sampling path and motif-interpolated posterior, which aligns the generative direction with motif-reconstruccion in earlier sampling stages. We summary our algorithm in Alg. and a high-level schematic is provided in Fig. 2
4.1 THE GEOMETRIC INVERSE DESIGN PROBLEM
We formulate motif-scaffoldiin as a Geometric Inverse Design problem. Formally, we observe motifs gMd {g"),... g²*² remaining residues. Motifs meet the following conditions:
length where {i.,..,; C [I,...,,) and the scaffold g all the
x M AxI,
exp (Alog (r1)).
where g
ri) T x SO(3) is drawn from an unknown data distribution Pt, Nm is the total length of motifs and A E R is a known estimated motif mask matrix. The second equation involves converting between elements in SO(3) and its Lie algebra (3( using Logarithmic and Exponential Mapping (Chen & Lipman 2023 Bose al. 2023 Yim et al. 2023).
Given a pretrained flow model with ((:,,), which is trained to approximate the unconditional posterior Ep .9 [9t] as in Eq. 4 and the motifs gM, our goal is to generate designable protein backbones with accurate motif reconstruction via directly approximating the conditional posterior Ep 1g1g119. g M] in geometric space.
Our perspective is different from previous training- based and sampling -bsed methods. Training- based methods (Watson et al., (023) directly train a specific conditional model vo (g, g that only samples the scaffold fragment while our formulation needs to sample the entire protein. As mentioned Sec 3.I and Eq. 5 the sampling-bass conditional sampling while our formulation aims to directly approximate the conditional posterior Ep [g 9tt g M as a concrete protein point clouds using geometric methods. By leveraging the spatial context of explicit representations, our formulation offers geometric solution to the generation vs. reconstruction trade-off, while the implicit approximation via guidance term is more exposed to the risk of the inherent trade-off.
methods use a proxy guidance gradient for
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4.2 THE GEOMETRIC SOLUTION
(Noisy) protein structures are not abstract latents or pixels but point clouds with specific coordinates and geometrie properties (e.g.. global orientations). The proposed EVA framework leverages these spatial contexts as a breakthrough point to overcome the trade-off and align the generative direction with motif reconstruction. Our geometric solution includes the spatial motif-aligned prior and moter-interpolated posterior.
The isotropic Gaussian prior used in the pretrained flow model may cause issues (Song al. 2023) when directly applying the vector fields in Eq. 6 Since Zy (the point cloud from the prior) and x (the motif structure) are independent, random alignment between them introduces extra variance, leading to twisted and unstable sampling process. To mitigate this issue, we introduce mapping that aligns the global orientation and CoM between x. and 2o to the 'closest' or optimal configuration, thereby reducing variance. The geometrically modified prior straightens the sampling path and paves the way for geometric interpolation as the orientations of data and prior are aligned. Additionally, noisy motif structure information provides an initial estimate for the motif parts of the prior. Further details can be referred in Sec.
In Sec.
introduce geometric method for conditional sampling in the EVA framework. Instead of complex models or guidance terms, we directly use geometric interpolation between the unconditional posterior [g ( and the target motif AM to approximate the conditional posterior Eg gt. (... This gives with motif reconstruction from the start by reconstructing the motif in the denoised protein ğ1 (i.e., unconditional posterior), which we term Evolution ViA reconstruction. It is straightforward yet effective. Further analysis shows our approximation is equivariant to adding an extra OT conditional vector field corresponding to the motif reconstruction process. Thus, EVA steers the sampling process into a coupling of two conditional OT paths as in Eq. 6 which we call the Coupled Flow.
clear evolutionary (i.e., generative) direction that aligns
4.3 SPATIAL MOTIF-ALIGNER PRIOR
Starting from prior go (æu, ra) (3S) N @ @N Iz)N following (Yim al. 2023). , where (3)) is uniform distribution over SO(3) and (0, I3) is the isotropic Gaussian with Zero CoM, we make it coupled with the motif from two aspects of orientation and specific coordinates.
Motif Gearetry-aware Prior At the start of the sampling, we have access to the specific motif backbone coordinates xM and the prior point clouds coordinates Zo we seek to estimate mapping that aligns the global orientation and CoM between Mf and ! Formally for the global orientation, need to solve the equivariant optimal transport mapping as:
R argmin| ( Rag, Rog Ro (s
(8)
R.
where π is a selection of residues as the motif in Eq. 7 out of N total residues (i.e. defining the motif placement or indexes in the overall protein) and R E x3x3 represents a rotation matrix in the 3D space. With both sides in the zero mass space (by subtracting CoM first), the mappings in Eq. are optimal for E(3) -quuiiaavvr transformations on either side of the point clouds (Song et al. 2023). Thus, we refer to these mappings as equivariant optimal transport (EOT).
The equivariant optimal transport finds the minimum straight-line distance between the paired atom coordinates upon all the possible rotations and alignment. Applying these mapping, we can obtain the optimized prior Go- (mo, Fo) = (I' æo, R ro). with the selected motif indexes x," " For solving this EOT, we first enumerate the motif region using sliding windows on the entire protein sequences and then conduct Kabsch algorithm (Kabsch 1976) to solve the optimal rotation matrix based on the atom alignments. We will try different motif regions in different samples of one target to finally obtain designable proteins. For multi-motif with unknown relative positions, we extend EOT in the prior to the generative process. More details can be referred to the Appendix C.T
Geometric Interpolation Given the known motif structure g = (a r ), we can initialize the motif part (the other is still original prior) of the protein with noised motif structure at some time steps along the Gaussian probability path which interpolates the prior and the data, we note:
a = (1 - t) a +
F = expay (t- logags (r M M)).
(9)
This interpolation transforms the initialization of motif part into the conditional OT probability path of motif reconstruction time f. This initialization gā prepares for the subsequent coupling of the

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overall structure conditional OT path and the motif reconstruction OT path, and also provides the prior with specific coordinate information of the desired motifs.
Algorithm A training-fres Coupled Flow-based Framework for fast Motif- Scaffo-iaffo (EVA)
Input: Pretrained flow model (o 1) which can estimate posterior ÿi (gt); Desired Motif Structure m as in Eq. ? Enumerated or given (as in the benchmark test) motif region indexes n².
Output: Designable backbones 91 with desired motif's.
1: Sample: original prior go. initial time lg-
2: Obtain motif-aligned prior 9ô via solving Eq. with Kabsch algorithm.
3: for each time step E [to, 1] of ODE integration do
4: Get predicted unconditional posterior Φ1 (g:) = (f1. .1) from pretrained va (g, t);
5: Predict intrifiinteppolated conditional posterior as follow:
6: Ep rr, M] = exPĄi (3: logap (r")), )), Ep (z) æx, a= = (1 Be) @ + Br M
7: Calculating estimated conditional vector fields 0 (at, MM and (r(, r as in Eq. 12
8: Continuing the ODE update step and get intermediate results 9r:
9: end for
10: return Designable backbones gi with desired motifs.
4.4 EVOLUTION VIA RECONSTRUCTION
Recall the vector field we defined in Eq. For conditional sampling on the OT path as in Section 3.1 we can parameterize the conditional vector field as:
Use (xe, ar æ1 (x Ix
Uy (rg M) logm, (f1
(10)
1-t where 인1 = - A1) is the denoised prediction of gt. When Si (gt. M E.p 9 gt. g..9 field v(g..g ) is the optimal, the same equations hold without M (Pokle et al. 2024).
M the vector
We propose to explicitly approximate the conditional posterior 19lg1lg..g' MI via geometric inter- polation between unconditional posterior 9i (g.) given by pretrained unconditional flow model Ex 9 [g.] and gM, instead of implicit approximation via estimating the gradient of guidance and then combining gradients as in Eq. []] Further analysis shows that this spatial interpolation approximation will lead to adding an extra vector field corresponding to a motif reconstruction OT path from the start. Thus, EVA steers the sampling process onto coupling of two OT paths and estimates the evolution direction via directly reconstructing motif in g1.
Motiffinterpolated Posterior For simplicity, we denote the unconditional posterior 91 (g:) = (.1, f1). During conditional inference ODE, the optimal vector fields (M) are follows:
(golg))) dt
Conditional Inference ODE:
M)
vo Vo To.
(11)
Ep
[22 [] æ (r(re, xM log,, (Eplrilre, MD)
dn (x (x
(12)
1-t
1-t
We approximate the posterior E,p, and obtain the estimated conditional vector fields as follows:
E, Ir,, N1 = expa2 (3: logagr (T)
(13)
Ep |2,, 11 = (1 3t) - + B, M.
(14)
where Br = 1 yt E [0, 1] is the coupling strength and 7 is an extra scaling factor that can be adjusted case by case. The scaffold region is still approximated with the unconditional posterior.
Coupled Flow Taking a closer look into its formula, we find the vector field is the coupling of the unconditional vector field and the motif reconstruction vector field, which are both corresponding to the conditional OT paths. Taking 0z example, inserting the expressions in Eq. i4to Eq. gives:
(1 - t) (xt (r = (1 - -3)) aM + B x xx
= (1 -B)) aM + Br x M - [11- BB) x. + Be x; M
(ax (a=
(1 ()) ( + B M (x:").
M
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Success Rate (%)
Designable (9)
mRMSD %cc
scRMSD &
Method
Solved
Time(s)
<
100
>=
100
All
<100
>= 100
All
<100
>= 100
All
<100
>= 100
All
FF-G
18
4,0
13
86
86
87
29
7.0
21
52
36
47
153
18
15
RFDiff
87
90
141
20
EVA
89 86 88
53
20
0.87
Table 1: Performance the RFDiffusion benchmark of 24 targets. The best metrics are marked by bold and the second best metrics are marked by underline. < <100 refers to targets with total length less than 100 residues and >= 100 refers targets with total length greater equal 100 residues.
Unconditional Generation Metrics
Succ. (%)
Des. (%)
mRMSD
scRMSD
0.8
Framelliow
160
Method
OT-Hlow
(%<1)
(%<22
0.6
SDEdit
2.0
32
7.0
18
DPS
10
67
11
43
0.4
FF-G
12
85
RFDiff
16
91
19
45
0.2
TDS
14
83
22
22
20
EVA
17
94
18
41
0.0
Des.
Nov.
Dhv.
Metrics
Table 2: Performance on the Vaccine Design bench- Figure 3: Unconditional Generation Results. The mark. The best metrics bold. EVA achieve results suggest that both the flow models can better balance and thus higher success rate.
serve as strong unconditional foundations.
where y.M (xxh) ) is the vector fields determining conditional OT path towards x for motif recon- struction and (n (2;") is the original unconditional vector fields evolving on another conditional OT path for overall designable proteins. This coupling works the motif regions. In early sampling steps, the motif reconstruction vector field quickly evolves the sampling motif towards the desired structure align the overall generative direction with motif reconstruction, since the gradually reconstructed motif structure will influence the prediction of .t. Later, the overall generative vector fields take over, enabling the designability of the entire protein. From the perspective of manipulating 1 beyond motif-scaffolding, other conditional generation tasks that can formulate constraints a can utilize our framework. simply requires modifying & using gradient-baser (simply on ô not deep gradients on 2;)) or direct modification methods and blending the modified with the original. We leave this for future work.
5 EXPERIMENTS
We justify the advantages of EVA with comprehensive experiments. The experimental setup is introduced in Section As a start-up, we also report the unconditional generation performance for pretrained flow models. We aim to answer five research questions. Q1: How effective is EVA for motif-scaffolding against other training-fire -free inverse problem solvers? Q2: How does EVA perform compared the sate-offthheart motif-scaffolding models in efficiency and results Q3: Can EVA generalize well to more challenging and realistic tasks (e.g. vaccine design)? Q4: How do key designs impact the performance of EVA? Q5: what's the quality of generated samples of EVA?
5.1 EXPERIMENTAL SETUPS
Datasets. We conduct sampling for motif soaffl-caafolddig on the RFDiffusion Benchmark (Watson al. 2023 of 24 targets following previous works (Trippe al. 2023 Wu et al. 2023 Zheng et al.
2024). For vaccine design, we established an in silico benchmark test comprising 10 vaccine design targets addressed in recent publications, including epitopes from the respiratory syncytial virus (RSV)
fusion protein (RSVF) that can produce neutralizing effect (Castro et al. 2024 More details can be referred in the Appendix B
Metrics and Sampling. We use the self-consistency evaluation protocol following (Trippe et al., 2023 Yim al.) 2024 Zheng et al. 2024), including metrics as self-consistency backbone root-mean- squere deviation (scRMSD), self self-consistency motif rmsd (mRMSD), Designability (Des.), Diversity (Div.) and Novelty (Nov... Given generated protein sample, the self- self-consistency evaluation includes three stages: 1) inverse folding generated backbones with ProteinMPNN to get designed sequences.
We predict 8 designed sequences per backbone following (Yim al. 2024 2) refolding the
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Performance Inverse Problem Solvers
Suec.
Des.
mRMSD
scRMSD
180.0
Method
(%)
(%)
(%<1)
(%<
160.0
FF-G
12
85
16
RFDiff
27
85
27
45
Divensty
140.0
TDS
10
87
13
25
120.0
EVA
28
83
28
100.0
SDEdi
DPS EVA
Table 3: Performance on targets with or more discontinuous motifs in the RFDif- fusion Benchmark. The best metrics are marked by bold. EVA can achieve better bal- Figure 4: Performance of inverse problem solvers on ance on challenging targets and thus higher the RFDiffusion benchmark.
80.0
10
12
14
18
20
Num Solved Targets
overall success rate.
designed sequences via ESMFold. 3) calculating various metrics including RMSD, TM-score (Zhang Skolnick] 2005), to evaluate the consistency (i.e., structural similarity) between the generated sample and the refolded sample. We use self-consistency IMM-scoresccTMM for evaluating the overall similarity between generated samples and refolded structures. scRMSD, which more sensitive to minor structural variances, is also applied for a more stringent measurement. In addition, to judge whether the motif scaffolding was successful or not, we calculate the motif RMSD between the predicted design structure and the original input motif (mRMSD). A protein scaffold is designable if meets: scTM >0.5, pLDDT > 700 (pAE <5), which are confidence metrics employed in ESMFold AlphaFold2 to ascertain the reliability of the self-consistency metrics. A designable scaffold with mRMSD <1 is considered a successful motif-scaffold.
We generate 100 scaffolds per target with 100 time-steps using Euler integrator, perform the self- consistency evaluation above on every scaffold, and report the percentage of samples that designable and successful (designable rate, Des. and success rate, Succ.), the percentage of samples with scRMSD <2 and mRMSD <1. Furthermore, we also report diversity in designable samples, which the cluster number with TM-score Threshold set 0.5. Novelty (Nov.) also reported for the unconditional generation, which is the average TM-score of each sample to its closet protein in the PDB computed using FoldSeek (van Kempen et al., 2022). Average metries across targets are reported. We evaluate the efficiency with the average time (over 100 samples in total for each method) for generating a 100 amino acid backbone in the same device.
Baselines We compare EVA with state-of-the ate motif-scaffolding methods including RFDiffusion (short as RFDiff) (Watson et al. 2023). TDS (Wu et al. (002), and Foi-ooooo-Giicceeee (FF-G for short) (Yim al. 2024) (and its pretrained flow model, FrameFlow (Yim et 2023)). and inverse problem solvers including SDEdit (Meng et al. 2022) (the Replacement method), DPS (Chung et al. 2022) (the Gradient Correction method), which we both extended to the geometric manifold. We use the original sampling set ups in their paper for fair comparison. Unless otherwise specified, all conditional generation experiments are based on the pretrained FrameFlow (Yim et al., 2023) model. In addition, we have reproduced the OT-Flow model (Bose et al., 2023) to test the flexibility of our approach. More details can be found in Appendix B It is noted that we are not comparing the FrameFlow and OT-Flow. As long as they sampling. We add OT-Fllow as another choice just for the Ablation study. 5.2 RESULTS ON UNCONDITIONAL GENERATION
successfully Ily trained, we can use them for conditional
To demonstrate the flexibility of EVA, we additionally reproduced a protein flow model OT-Flow (Bose ct al.) 2023) and evaluated its performance. The unconditional generation result in Fig. shows that we are using reliable pretrained generative models for training-free methods, which builds basic intuitions for the performance of the following conditional sampling. We evaluate these models by sampling 100 samples from lengths 70, 100, 200, 300 following (Watson et al. 2023
5.3 RESULTS OF INVERSE SOLVERS (Q1)
Fig. 4 shows the results of different inverse problem solvers for motif-scaffolding Our EVA signifi- cantly outperforms all inverse problem solvers including replacement based and gradient correction methods, highlighting the better balance between reconstruction and evolution of EVA. The results imply when the unconditional generative bases are not as strong as in the image domains, training-free methods need carefully reach trade-off.

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mRMSD (%<1)
Des.
Method
Solved
(%)
EVA
20
42
88
EVA (OT-Hlow)
20
90
MRMSD=0.68,
MAMSD=0055
DPS
14
ScRMSD=1.41
Case1: RSV-F SiteIV (Vaccine Design)
GcRMSD=1.10
w/o our Prior
18
39
87
Interp. to DPS
14
15
69
setting 1.1 19
40
91
setting 0.9
20
41
86
Table 4: Ablation study for designed com- ponents on motif-scaffolding in RFDiffu- sion Benchmark. The best metrics are marked by bold. Des. is the short for Des- ignable Rate.
MRMSD-0.31 31
mRMSD=0.77
scRMSD=0.84
scRMSD=1.95
Case2: Double EF-hand motif (Multi-Motif)
Figure Case Study on RSVF4 and 1PRW.
5.4 RESULTS ON REDIFFUSION BENCHMARK
methods in RFDiffusion benchmark in Table 1 EVA is 70x We compare EVA with -frth RFDiffusion, which supports our analysis about the trade-off' faster than current state-of-the-ar and demonstrates the effectiveness of the spatial context. As training-free approach, we achieve comparable performance with current SOTA training-based methods, RFDiffusion and superior performance targets with total length <100. It should be noted that RFDiffusion is based much more complex protein structure prediction model, which better at generating larger proteins and requires time-consuming structure prediction pretraining and generative fine-tuning. We also achieve leading performance across training-free methods, including SOTA particle-based SMC methods, TDS, which enjoys good performance in the cost of intensive guidance computation.
5.5 PERFORMANCE COMPARISON ON REALISTIC TASKS (Q3)
Vaccine design and multi-motif scaffolding both important and challenging real-world tasks (Cas tro et al. 2024). We test EVA on those challenging tasks as in Table 2 and Table 3 Our methods has achieved competitive performance approximating the posterior mean using spatial context is simple yet effective. We also tested cases requiring to find optimal motif indexes multi-motif relative positions. More results and details be referred in the Appendix C
the benchmark test, suggesting
scheme of directly
5.6 ABLATION STUDY (Q4)
As Table 4 suggests, components of EVA. all contribute our superior performance. The results highlight the flexibility of EVA, which could take advantage of different unconditional generative foundations. As guidance the essential for motif-scaffolding, we replace the coupled approximation module within EVA with DPS guidance to study the importance of motif notif-aligned approximation.
5.7 CASE STUDY (QS)
EVA can generate designable scaffolds with accurate motif reconstruction. Fig. 5 shows EVA can scaffold the epitope of RSVF-site4 and the double motif EF hand of Protein 1PRW successfully, suggesting the potential our method for real-world tasks. More cases can be found in the Appendix. EVA can generate diverse scaffolds Fig. 5 shows different scaffolds with various lengths and secondary structure for two targets. Diversified seaffolds can improve the experimental success rate.
CONCLUSION
We formulate the motif-scafiii scaffolfifffnggooc as Geometric Inverse Design and identify its inherent trade-off between generation and motif reconstruction. Tailored to this perspective, we present EVA, novel coupled flow framework on geometric manifolds, which uses motif-aligned priors and steers the generative process onto straighter probability path, where the generative directions are aligned with guidance at early sampling steps. EVA is 70x faster than current sfate-of-the-ar RFDiffusion with competitive and and effectiveness. Limitations still exist, including insufficient exploration of general conditional sampling and motif layouts.
better performance. The results on benchmarks demonstrate efficiency
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7 ACKNOWLEDGMENTS
Many thanks to Prof. Dapeng Li for helpful discussions and constructive feedback. This work was supported by National Science and Technology Major Project (No. 20222D0115101,, National Natural Science Foundation of China Project (No. 624B2115, No. U2IA20427), Project (No. WU2022A009) from the Center of Synthetic Biology and Integrated Bioengineering of Westlake University, Project (No. WU2023C019) from the Westlake University Industries of the Future Research Funding.
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A BACKGROUNDS
A.1 MORE ANALYSIS
key challenge in motif-scaffolding lies the trade-off between preserving accurate motifs and generating overall designable proteins (illustrated in Fig. 6 Traditional machine learning methods use time-consuming stochastic search techniques, taking hours to generate one scaffold (Wang et al. 2021. This hinders the creation of diverse scaffolds essential for experimental success.
ES
Scoffold
Eppo
Ephope
Epitope (Unstable):
Design Cane: Vaccinea RBV-F SILEE
Figure 6: Cases requiring to find optimal motif indexes or multi-motif relative positions
Traditional machine learning- based methods rely on stochastic search techniques that require hours of computation to generate single plausible scaffold, which is not desirable as sets of diverse scaffolds are needed increase the likelihood of success experimental validation.
Training or fine-tuning conditional generative models is natural and effective idea. While these methods need lots of computing resources and a large number of protein motif-scaffold pairs for training, resulting in high computational costs. Moreover, training-based methods are inflexible to take advantage different pretrained generative models and incorporate various practical constrains like spatial arrangements layout for real-world applications, as they always need extra training fine-tuning.
This limitation has its root in the Evolution Verse Reconstruction trade-off process mentioned before. The evolutionary process tries to sample from unconditional distribution to generate overall stable and designable proteins, while the guidance term focuses the accurate reconstruction of desired motifs. Thus, the direction of Evolution and Reconstruction could be conflict. This often leads to two types of unsatisfied samples with low designablity inaccurate motif integration (as illustrated in the 1ig.1). The conflict demonstrated the fact that current conditional generative frameworks fail to fully utilize motif' information to guide the evolutionary path of the generation process. Additionally, the geometric information. In addition, motif geometry information such as orientation can provide clues to the evolution of the overall protein, and existing methods fail explicitly account for these geometric constraints.
we formulated the motif-scaffolding problem as the Geometric Inverse Design task and identified the Evolution Verse Reconstruction issue in this task. We introduce EVA with Evolution ViA reconstruction, a flexible and novel coupled flow matching framework on geometric manifolds, (which uses measurement-coumee gradient) with the measuremennt-couplee Evolution ViA reconstruction plugin-and-play module, on E(3) Equivariant Optimal Transport Path. EVA better balance the evolution and reconstruction process within generation and achieve comparable performance with greater flexibility.
priors and steers the generation process explicitly (vs implicitly as
A.2 FURTHER COMPARISON WITH GRADIENT GUIDANCE
For ease of comparison, the following analysis will be conducted under the unified formulation of flow-based models.
In algorithmic sense, grad guidance includes two step: a. estimate the unconditional vector field and unconditional posterior d b. estimate the guidance gradient V., log p(a"M (ar.) and the composition
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of the two:
In(as/oz) 22, poz (x,)) V.,, ",,11=='' -MM
6(2,, 1/x i( () o+ dt
Vx, log p(x (0gg (ate); Mare,
(15)
Where the guidance gradient is usually estimated by the assumption that p(x (are) so p(x ( a Guassian distribution. This is the origin of this term Vx, 1| 2/
is
Our method differs from grad guidance in the second step, given the estimated #1, we directly estimate the conditional posterior Ep (an lare, MM via interpolation between 4/ and x.M Then the conditional update direction given by:
M Eplar [x) [re, x (1- -3)) 1/1 + 3r x:
6(x,, 6(1, t (2) |2,,, xx
(16)
1
Our method doesn't rely on any assumption (e.g. gaussian distribution assumption in FramiFlow-Cuudance Chroma) about the form of p(a (x)) estimate its gradient. Instead of two independently estimated gradients, our method gives the overall update direction directly.
Furthermore, our vector field for motif scafffssffolliiigg can be viewed composition of vector fields for overall generation and motif reconstruction:
Ü (2 (xM) (1 - -3) (-x ) + 3- MM (x
M It
(17)
MM (x:a)) 1
A.3 MORE RELATED WORK
Motif-Scaffolding SMC-Diff and TDS utilize Sequential Monte Carlo (SMC) algorithm to re- purpose pretrained DPM for motif-conditioned sampling. Though training-free, these SMC-based methods need much more network calls to incorporate heuristic approximations, which may result in unsatisfied and slow generations. Gradient correction-based methods frame the notif-conditioned sampling as a problem of posterior sampling. Multi-motif scaffolding (Ke et al.) 2024) was also explored, which treat the motifs as floating anchors. We refer recent benchmark work (Zheng et al. 2024) for further reading.
Inverse problem Recently there also many training-based methods solve the inverse problem to seek for effectiveness. Notably, Diffusion Bridge-based method (Chung et al. (023a) applies the similar idea of data-coupling as our methods. Improved gradient- mooreett-ooreectoon method has also appeared (Chung et al. 2023b) which uses geometric decomposition.
Protein Structure Prediction The solficonditinnnn approach (Stark et al.,
2023 Huang et al.
2023) is from the recycling mechanism in AlphaFold2 (Jumper al. 202 la Huang et al. 2024a) for improved structure prediction (Hu et al. 2023: Huang et al. 2024b, Hu et al. 2022). takes the previous predicted posterior as model input and encodes the posterior to embedding for conditioning next iteration. In contrast, our method just takes the posterior for interpolation which will not be taken model input for embedding.
B EXPERIMENTS SETUP
Inference set-up All baselines and our approach are implemented using the PyTorch 1.6.0 library with Intel(R) Xeon(R)Gold6240R @ @2.40GHz CPU and NVIDIA 100 GPU. For fair comparison, we follow the original inference set-up of all baselines (Yim et al. 2024 Wu et al., 2023 Watson (et al. 2023). Previous sampling-basee methods usually require 500 inference steps for reproducing their original performance, more than that in EVA. All sampling- based methods use a batch of 25 for every case, generating 100 scaffolds in total in a single 1.00 GPU. Because of the memory burden of running RFDiffusion,
instead use batch of in single A100 GPU. The inference code framework is kept the same as TDS (Wu et al. 2023). The only difference for each sampling-bart based method is the time-step number and the conditional sampling implementation.
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Data Curation We download the RFDiffusion benchmark sets via PDB. The data list can be found in (Wasson et al.) 2023). The vaccine dataset (Watson et al.] 2023| Castro et al. 2024 Correia al. 2014) include targets follows: RSV Fusion Protein Site 0 (4JHW), Site 2 (3IXT), Site 4 (4ZYP), Site 5 (5TPN), RSV G-Protein 2D10 Site (5WN9); epitopes of DENV2 (pdb id: 6FLA) that
corresponding to neutralizing antibodies: 3E31, Z004, 7021, 3H5, 2C8, 2D73). We illustrated these neutralizing antibodies below in Figure
Neutralizing antibodies targeting DENV2 EDIII
2004
2021
3H15
(ADE
DEN)
3E31
2073 (ADE DENV
Figure 7: Neutralizing antibodies targeting DENV2 ED3
Baselines Implementation We implement all the inverse solver baselines according to their official GitHub repositories and change the data structure from images to protein. The only modification the normal distribution defined in different manifold (Id SO(3), E(3)). We follow the normal distribution definitions of TDS (Wu et al., (023) and FrameFlow (Yim et al. 2023 2024). The OT-Flow reproduced based the FoldFlow (Bose et al. 2023). using their official Code Repository for the SO(3) optimal transport path version. As we are not comparing the OT-Flow and FrameFlow, we can just use the default training settings in their original papers. As long as the pretrained flow model is capable of generating high-quality unconditional backbones (which we test in the Fig. 5). we can use them as generative foundations for training-free motif-scaffolding.
Solving Equivariant Optimal Transport Our proposed algorithm solves Eq.7 approximately based on enumerated motif selections with sliding windows (i.e. enumerating the start indexes of motif segments) and kabsch algorithm for optimal rotation. The enumeration used here is for estimating multiple possible motif selections to increase the diversity of generated backbones, which contributes to the overall success rate. It's efficient since the sequential enumeration space is not big (total length motif length or constrained by prior) and single enumeration can determine motif selection for N samples (by simply selecting the top N), allowing the computation time be amortized across multiple samples. Kabsch algorithm is highly efficient and the runtime is negligible. Additionally, for long scaffolds with multi-motif and no prior constrains, we will apply variant of iterative closest point algorithm (Song et al. 2023), where it iteratively obtains motif selection and Rotation.
C MORE RESULTS
We show the numerical results of unconditional generation and inverse problem solvers in Table 5 and T More case studies are introduced below.
C.1 MULTI MOTIF SCAFFOLDING
As mentioned in the main text, we also test two cases requiring to find optimal motif indexes or multi-motif relative positions. These conditions provided in benchmark tests, rarely explored before, but actually not easily accessible in real-world cases. As these are not our main focus, we just want show the flexibility and extensibility of our method in proof-of-concept manner. We
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Method Des.(1)
Div.(t)
Nov.(J)
RFDiff
0.87
156
0.64
FrameFlow
0.80
171
0.61
OT-Flow
0.82
167
0.63
Table 5: Unconditional generation results.
Method
Solved(1)
Div(t)
SDEdit
10
79
FF-G
18
153
DPS
14
126
EVA
20
173
Table 6: Performance of inverse problem solvers on the RFDiffusion benchmark. The best metrics marked by bold.
conduct the case study on protein 1PRW, we simulate the two scenarios by not providing the model with the motif index from existing cases or randomly rotating or translating the given motif structures. The results are show in Fig g For the case with unknown motif indexes, they involve multiple factors: 1) the spatial location of the motif, 2) the length of the motif itself, and 3) the distances between motifs. The final evaluation criterion is whether a designable backbone with the desired motif can be generated, and there is currently no numerical standard for this. Here, we present heuristic enumeration method. We assume that the relative positions of multiple motifs are known
Algorithm 2 A training-free Coupled Flow-based Framework for fast Motif-Scaffolding (EVA)
Input: Pretrained flow model (g, t) which can estimate posterior gi (gt); Desired Motif Structure m as in Eq 7 Enumerated given (as in the benchmark test) motif region indexes n.. Output: Designable backbones with desired motifs.
1: Sample: original prior go- initial time to E 10.1, 0.2).
2: Obtain spatial motif-aligned prior gi with Kabsch algorithm.
3: for each time step E [fo, 1] of ODE integration do
4: Get predicted unconditional posterior 91 (g.) = (f1, .t) from pretrained ve (g. t);
5: Adjusting motif relative positions according to Eq. 18 and get updated 9m-
6: Predict motif-interpolated conditional posterior as follows:
7: (v rM] = exppar (3, loggęr (xrM)) )), [x [2,, a² (1 Bc) - B, a
8: Calculating estimated conditional vector fields 0g (2,, Fm) and i (r,, m)) in Eq. 12
9: Run the ODE update step as in Eq. 12 and get intermediate results g:;
10: end for
11: return Designable backbones 91 with desired motifs.
(the case where the relative positions motifs can vary independently will be discussed later). For a single motif, only need to enumerate its starting index using sliding window method, ensuring it remains within the protein index range. For multiple motifs, we first calculate the number of intervening amino acids based on the spatial distance between motifs plus their interaction radius (approximately 2-5 A), divided by the average amino acid length. Since this value only simulates the scenario where amino acids are arranged in straight line, represents a minimum. We then enumerate the relative number of amino acids between each motif, i.c., the index difference between each motif. Once we have enumerated the index differences between motifs, the problem converts back to the single motif case, and we can continue using the sliding window for enumeration. For the case with unknown relative motif positions, we assume known motif indexes. Or it's little complex and needs slow iterative algorithm. The algorithm first conducts the Hungarian algorithm to align the atoms between the initial geometry from Pu and the ground truth geometry from 9I; and then conducts the Kabsch algorithm to solve the optimal rotation matrix based on the atom alignment. The proposed algorithm asymptotically converges to the optimal solution. But it's slow for accurate exploration of the motif indexes and motif relative positions product space.
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mRMSD=0.87. scRMSD=11. 14
Case Study 1: 1PRW scaffolding with
Case Study 1PRW scaffolding positions
Figure 8: Cases requiring to find optimal motif indexes or multi-motif relative positions On the other hand, if the motifs are independent, we can assume a relative far sequential distances between the motifs. This intuition comes from the fact that these motifss don't t need to interact with each other and their relative positions are not necessarily to be close. So we just keep the independent motifs independent and assign them motif indexes that are far in sequential order. Note the independent motifs could still be spatially close although they are sequentially far. But doesn't violate our assumptions of independent motifs. Therefore, we can simply enumerate some motif indexes that are relatively far apart in the sequence, and then focus on exploring their relative positions.
We can explore the motif relative positions with a modified version EVA. This time, we need to adjust the EOT objective in Eq.
18
π R argmin
|| [z) 1... xT¹ - (Rx). R
Ro Nm )
(18)
n.Rl
In every step (or in just some selected steps for efficiency), we take the intermediate sampling result T, relative reference and the Kabsch algorithm based on Eq. [8] to compute the optimal rotation for each motif relative to its corresponding prior part. Then, we apply the optimal rotations and perform Geometric Interpolation calculate the conditional posterior. This method implicitly leverages the pretrained generative model to explore the relative positions of the motifs. As the protein structure gradually folds into a stable state during generation, the generative model makes spatial adjustments for each motif part. Our approach aligns with these adjustments step by step through spatial alignment, while also ensuring that the motif structure itself conforms to the given structure via Geometric Interpolation. We omit the trivial CoM alignment for simplicity. We summary the algorithm in Alg. 2
C.2 MORE ADVANCED RESULTS
More ablations. We provide the performance of FrameFlow against sampling steps in Table 7 more ablation study of hyperparameters of EVA in Table g and more case study results for the RFDiffusion benchmark Table 9 The ablations suggest the hyperparameters could control the strength of motif reconstruction and the designability and users could adjust them case by case. We find simply setting 7 1, 4o 0.1 is enough for most cases.
Method
Solved
mRMSD(%<<1)
Des(%)
Time(s)
FrameFlow-guidancee (100 steps)
16
14
79
3.1
FrameFlow-guidanee (300 steps)
16
83
9.7
Framullow-guidance (500 steps)
18
21
87
15
EVA (ours,100 steps) EVA (ours, 500 steps)
20
42
88
0.87
21
47
88
4.6
Table 7: The performance of FrameFlow-guidance against sampling steps.
Image In-painting. To demonstrate the broader applicability of EVA's geometric method, we conducted image in-painting study using EVA algorithms adapted for image tasks. The results
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Hyper-parameter
Solved
mRMSD(%<<) Des(%)
coupling strength gamma 0.8
17
45
79
coupling strength gamma- 1.2
38
starting time t0=0.15
20
43
86
starting time t0=0.05
19
39
88
EVA (original, gamma=1.0, (0=0.1)
20
42
88
Table 8: More ablation study of EVA.
IYCR
6EXZ short
6E6R med
STRV short
3IXT
6EXZ long
IBCF
2KL8
LPRW
5TRV
med
success.
0.67
0.33
0.08
0.08
designable_rate
0.92
1.00
1,00
1,00
0.67
0.92
0.92
1.00
0.83
1.00
motif rasd_rate
0.75
0.42
0.50
1.00
0.08
1.00
1,00
0.50
0.33
scumsd_ratee
0.75
0.83
0.58
0.75
0.08
0.42
0.58
1.00
0.58
0.75
Table 9: More case study results for the RFDiffusion benchmark.
in Fig. [highlighh the effectiveness of leveraging spatial context for conditional generation. The mod- ification of EVA for image in-painting task includes: a. The pretrained generative model is changed the DDPM model pretrained in FFHQ dataset (Chung et al., 2022) b) The rotation alignment is no longer needed since it's standard 2D image task without any misalignment. Interpolation between masked image and random noise kept for a better initialization. c) The inference setting and implementation for image in-painting follows DPS (Chung et al. 20022. For the comparison against DPS (Table we report the LPIPS (lower better) on FFHQ 256 x256 in different sampling steps (i.e., Neural Function Evaluations, NFE).
Original Image
Masked Image
Inpainted with EVA
Figure 9: Image in-painting cases of EVA
It is noted that EVA performs better in low NFE setting (NFE<100), while DPS is better in high NFE setting (DPS's original setting). This aligns with the efficient and direct design of our geometric solution. When the number of sampling steps increases, methods like DPS, which are based on analytical solutions for image in- paintiiing, can better balance the guidance and unconditional scores, thereby achieving superior performance. Since images, unlike proteins, not 3D geometric entities, making it challenging to fully leverage the advantages our geometric method. This comparable performance is acceptable and demonstrate our efficiency and effectiveness.
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DPS
EVA
NFE=50
0.36
0.31
NFE=100
0.28
0.26
NFE=500
0.21
0.23
Table 10: The performance of EVA in image in-painting. report the LPIPS (lower is better) FFHQ 256 x256 in different sampling steps (i.e., Neural Function Evaluations, NFE).
Random Error Analysis. To demonstrate the impact of random errors on the model, report the success rates and random errors for fixed targets. Under the original inference setting, we generated 100 scaffolds for each target, calculated the number of samples that successfully achieved the motif- scaffolding task, and reported the success rate. We now run this process five times for five targets (with different random seeds as required). Each target yields five success rates, and we report the average success rate and standard deviation (std) for each target in Table I1
6EXZ
6E6R
2KL8
IBCF
IYCR
Success Rate(%)
1.
0355
10010.40
58
67.1.2
Table 11: The success rates and random errors for fixed targets. The headers are the PDB ID of five targets.
The results show that EVA performs consistently the same case, benefiting from the generation process of the flow-based ODE. As for the ablation study, it is worth noting that single metric being close does necessarily mean there is no difference between the methods. For example, higher designability might result from ignoring the motif constraints. Our method successfully addresses the most
targets and achieves the best motif reconstruction performance.
Trajectory Analysis We provide Trajectory Analysis of FrameFlow-guidane and EVA in Table 12 Empirical results indicate that EVA generates straighter and shorter trajectories.
Method
1=0. (Prior)
t=0.4
t=0.7
t=1.0
Frameflow-Guidance (100 inference steps) EVA (100 inference steps)
11.1 (7.33)
7.2 (5.14)
4.6 (3.01)
2.8 (1.74)
9.3 (6.21)
5.6 (3.36)
2.1 (1.49)
1.3 (0.82)
Table 12: Empirical Analysis of Sampling Trajectories for EVA and FrameFlow FlanGlllo-G We calculated the average RMSD between the noisy point clouds and the fully denoised ground truth point clouds for 100 sampling trajectories the target IYCR. The values outside the parentheses represent the backbone RMSDD while those inside the parentheses represent the motif RMSD.
D IMPACT STATEMENT
While there exists the potential risk that motif-scaffolding methods could be misused to develop harmful drugs, it's important note that drug development subject to stringent oversight globally. This rigorous regulatory environment ensures that such misuses can be effectively managed and controlled.
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Reviews:
[{"review_id": "J9kQMZvRy9", "reviewer": "Reviewer_sMzV", "summary": "The paper addresses the motif scaffolding problem - one of the most challenging problems in protein design, where one wants to find a 3D protein structure accommodating a certain functional motif. The authors define motif scaffolding as a Geometric Inverse Design problem on the SE(3) manifold. They approach this using a flow matching, sampling-based strategy that does not require training and can be implemented within any pre-trained unconditional model, which they call Evolution-ViA-reconstruction (EVA). Their sampling strategy adjusts the overall generative direction with the guidance of motif residues with the aim of defining a straighter probability path towards meaningful posterior distribution. They achieve so by implementing coupled flow with the spatial motif-aligned prior and motif-interpolated posterior. The resulting approach matches the performance of state-of-the-art pre-trained models and sampling-based algorithms while significantly reducing inference time.", "questions": "**Conclusion**\n\nOverall, the approach introduced and its implementation in the paper is interesting and seems to be efficient. However, the paper is hard to follow. The concepts underlying the choice of the prior and the interpolation strategy are formulated in a very convoluted way, which can be done in simpler terms and explicitly explained in the text accompanying equations, e.g. see [3]. Many crucial details required for reproducibility of the experiments are absent from the paper. One key prior work that introduced similar ideas is not mentioned at all. **I'm willing to raise the score if the authors address all issues raised above, especially:**\n\n1. Improve the readability of the text, improve figures and conduct additional experiments (mentioned above) and introduce explanations. Please, also structure Appendix and properly refer to the concrete Appendix sections within the main text.\n2. Critical comparison with Chroma's analytical solution to the motif guidance.\n3. Detailed statement on inference settings, training details of the OT-Flow model and explicit explanation why EVA is so much faster.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. Authors identify failure cases in motif scaffolding and draw an informative analogy with the image inverse problem.\n2. The authors clearly define the questions they aim to address in their experiments.\n3. Indeed, EVA achieves the best performance or is on par with both SOTA sampling and training-based methods on well-established benchmarks.", "weaknesses": "1. The core ideas, the motif-aligned prior and interpolated posterior, are barely explained, if at all. The concept and reasoning behind choosing a motif-aligned prior appear to be justified by leveraging some spatial context; however, what this spatial context means is never explained. Figure 2, which attempts to build intuition for the modified prior, is highly convoluted and unreadable (please improve this). The authors never showed (and to the best of my knowledge, there is no empirical evidence) that their choice of prior would lead to a straighter probability path *per se*. I believe the optimal transport performance of EVA could be assessed in terms of the normalized path energy and compared to the baselines, similar to [1].\n\n2. The strategy of pre-aligning the prior with the motif does not appear to be novel. Moreover, the authors never referred to the Chroma paper [2], which first introduced the concept of motif-aligned priors and optimal motif index selection. EVA employs a very similar approach to Chroma (Appendix K.2) by aligning the initial prior $\\mathbf{g_0}$ with the motif $\\mathbf{m}$. The only significant difference is that, in Chroma, classifier guidance is used to approximate the conditional probability $p(\\mathbf{g_1} \\mid \\mathbf{g_t}, \\mathbf{m})$, with an analytical classifier that quantifies the distance between the predicted and target motifs to estimate $p(\\mathbf{m} \\mid \\mathbf{g_t})$. EVA is expected to be faster than Chroma, but the novelty of the spatial motif-aligned prior is questionable.\n\n3. I think the authors' statement that they directly approximate $\\mathbb{E}_{q} \\left[ \\mathbf{g_1} \\mid \\mathbf{g_t}, \\mathbf{m} \\right]$ is not self-explanatory. At least from the paper, it is not clear how exactly their sampling procedure fundamentally differs from gradient guidance as in [3]. EVA will most certainly have a shorter length of the motif residues trajectory, but how much better is it than FrameFlow-guidance? One could compute the trajectory length of the motif residues in terms of rotation and translation for EVA and compare with the baselines.\n\n4. The authors trained their own protein structure generative model, OT-Flow, and reported both its unconditional and motif-conditioned generation performance to demonstrate that EVA's sampling strategy is compatible with and efficient for any pre-trained model. However, it is absolutely unclear how the model was trained; neither there are any details of the training procedure in the appendix, which the authors referred to. Referring to the original FrameFlow [4] and FoldFlow [5] papers for training details is insufficient; I checked and they have different training set cut-off dates and hyperparameters. Inference settings are not mentioned either...\n\n5. From Table 1, it follows that EVA is about 70x faster than RFdiffusion. For vanilla FrameFlow [4], the inference speed with the same number of timesteps for unconditional generation of a protein with 100 amino acids is about 6 seconds, making EVA more than 5x faster than the underlying flow model. Again, the authors never specify the inference settings in this experiment. Is there any batching? Can unconditional FrameFlow sampling achieve the same performance given EVA's inference settings? Is FrameFlow-guidance significantly slower because the gradient computation is needed on top of the unconditional prediction? Further studies, including a timestep study (with success rate as a function of timesteps), would be informative.\n\n6. It is unclear why in Section 5.7 the authors only qualitatively assess the method's performance on two motifs, despite performing computations for the RFdiffusion and vaccine design benchmarks. A table summarizing the metrics would be informative.\n\n**References**\n\n[1] Tong, Alexander, et al. \"Improving and generalizing flow-based generative models with minibatch optimal transport.\" arXiv preprint arXiv:2302.00482 (2023).\n\n[2] Ingraham, John B., et al. \"Illuminating protein space with a programmable generative model.\" Nature 623.7989 (2023): 1070-1078.\n\n[3] Yim, Jason, et al. \"Improved motif-scaffolding with se (3) flow matching.\" arXiv preprint arXiv:2401.04082 (2024).\n\n[4] Yim, Jason, et al. \"Fast protein backbone generation with SE(3) flow matching.\" arXiv preprint arXiv:2310.05297 (2023).\n\n[5] Bose, Avishek Joey, et al. \"SE(3)-stochastic flow matching for protein backbone generation.\" arXiv preprint arXiv:2310.02391 (2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wndoTyQKAg", "reviewer": "Reviewer_3CsL", "summary": "This paper introduces a method that improves the flow-based generative model for motif-scaffolding (a kind of protein design). The proposed algorithm, Evolution ViA Reconstruction (EVA), leverages the protein’s geometric properties through two main strategies: (1) motif-aligned priors for improved initialization, and (2) motif-reconstruction guidance, a time-varying adjustment from the condition (motif) that enables adaptive guidance during the sampling process. EVA demonstrated faster sampling speeds than baseline methods on two motif-scaffolding benchmarks, including a newly introduced vaccine design benchmark.", "questions": "1. Lines 121-123 say that the training-based methods \"rely on expensive pre-training\". But, as far as I understand, EVA also uses a pre-trained model (FrameFlow). In what sense can we say the pre-training of a conditional flow model is more \"expensive\"?\n2. What makes EVA much faster than others? Is it primarily due to using fewer timesteps? I’d appreciate an algorithm-level explanation that compares EVA to both sampling-based and training-based methods.\n3. The performance, except the sampling speed, seems not so remarkable. Does the sampling time really matter in motif-scaffolding (i.e., was the sampling time a bottleneck)? Can EVA generate better protein by running the algorithm longer (e.g., with more fine-grained timesteps)?\n4. What is the $\\alpha_t$ in line 239?\n5. Does the proposed algorithm solve Eq.7 exactly or approximately? And, how is the $\\pi^*$ selected?\n6. Is the runtime for obtaining $R^*$ with the Kabsch algorithm negligible?\n7. How was $t_0$ in Algorithm 1 set in each experiment?  \n8. How sensitive is the algorithm against the hyperparameter $\\gamma$ and $t_0$?  \n9. Do you have a specific plan to release the source code and the vaccine design benchmark?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. They frame the motif-scaffolding problem as an inverse-design problem, primarily studied in computer vision, and effectively leverage geometric properties, which appear to have significant practical value.\n\n2. The idea of using geometric properties to improve prior and guidance is both straightforward and novel. It's particularly interesting that the resulting guidance is heavily influenced by the condition (motif) during the early sampling steps, with this effect gradually decreasing over time.\n3. This paper introduces a new dataset, which is a valuable contribution to the research community.\n\n4. The proposed method demonstrated improved efficiency without compromising the quality of the designed protein.", "weaknesses": "First, I want to note that I’m not an expert in this domain (protein design or geometric deep learning), so I may not have fully understood all aspects of the paper.\n\nMy concerns about the paper are as follows:\n1. **Scope and Applicability**: I wonder if the proposed method could be applied to general inverse problems beyond motif scaffolding, such as inverse imaging. Since ICLR is a machine learning conference, a problem-specific approach may have limited impact. Note that I am questioning the broader applicability of the proposed method, rather than suggesting the authors need to show state-of-the-art performance in various domains.\n2. **Lack of Experimental Details**: Important experimental details are lacking. For example, the paper doesn’t specify the number of random seeds tested or the values of key hyperparameters (e.g., $t_0$ and $\\gamma$). Additionally, standard deviations are not reported, making it difficult to assess the significance of the results against baselines, such as the similarity between “w/o our Prior” and EVA results in Table 4.\n3. **Writing and Presentations:** (including minor ones)  \n3.1. Lines 114, 119, 130, 151, and etc: Inappropriate citation command is used. I think when you want to use the cited paper as a subject or object in a sentence, you may use \"\\citet\" command in latex.  \n3.2. Line 554: Year is missing. The BibTeX must have some problem.  \n3.3. In section 3, there is only one subsection 3.1. You may just replace \"3. Backgrounds -> 3. Flow Matching for Non-geometric Domains\" without using a subsection. Another option is to make Section 3.2 with the first paragraph of Section 4.2 (lines 250-269) since it can be considered as a background and is not a contribution of this work.  \n3.4. Line 183: Subscripts should be removed, i.e., $p^{batch}_0$ => $p^{batch}$ and $q^{batch}_1$ => $q^{batch}$  \n3.5. Line 269, 315, 518, and maybe more: \"Appendix\". An indication of a specific part of the appendix is needed.  \n3.6. Line 279: $y$ seems to be a typo. Maybe it should be $x_m$.  \n3.7. Line 265: \"The formula of transitional vector field seems **sightly** different ...\" -> \"The formula of transitional vector field seems **slightly** different ...\".  \n3.8. In Eq. 7, is **x**$_0=(x^1, x^2, \\ldots, x^N)$ without subscript 0? Also, the subscript m is used inconsistently: Eq. 4 uses a plain $m$ but Eq. 7 uses a bold **m**.  \n3.9. Some inconsistent usage of $x$ and $x_0$ in Eq. 1 and Eq. 2 (and maybe more elsewhere).  \n3.10. Line 367: $\\hat{\\pi}^*$ -> $\\bar{\\pi}^*$  \n3.11. Line 58: the -> The  \n3.12. Line 503: \"Table.1.\" -> \"Table 1.\"; BTW, why the Table 1 is at page 8 when they are referred at page 10 for the first time?  \n3.13. I don't think experiment 5.2 provides any message or insights about the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "M1xYXsM17H", "reviewer": "Reviewer_3tvr", "summary": "In this paper, motif-scaffolding is formulated as a geometric inverse design task, with a lot of inspiration from the image inversion problems. Their approach, namely evolution via reconstruction, proposes a sampling-based coupled flow framework. The method is built upon a pretrained flow-based generative model, and does not train the model any further. Instead, EVA employs motif-coupled priors to utilize spatial contexts, causing the generative process to follow a more \"direct\" probability path, where the generative directions are more aligned to \"guidance\" during the initial sampling steps.", "questions": "1. Since the authors propose an example from the vision area, I would like to see how the proposed method at reconstructing the same example image for a frame of reference. I understand the authors mainly provide this example to support reader's understanding, but I think for this comparison to be relevant, the image in-painting example using the proposed method of the paper should be also demonstrated. I'm afraid there is too much discussion about \"image inverse problems\" without any related data about this context **in this paper**.\n\n2. In Table 1, it seems that the proposed method is having a poor performance under long residue lengths. This is quite concerning as high-dimensional porteins are somewhat the main target application for this class of generative models, and specifically, speed-ups are more essential for this type of problem.\n\n3. There are too many metric \"hyper-parameter\" choices that may affect the downstream conclusions. Examples are the \"mRMSD%<1\", \"scRMSD%<2\", and \"TM-score threshold of 0.5\" for diversity. I understand some of the choices, e.g., \"scTM>0.5\", \"pLDDT>70\", and \"pAE<5\", were referred back to ESM/AlphaFold. \n\n  * How are the arbitrary thresholds chosen? \n  \n  * How sensitive are the conclusions with respect to these choices?\n\n  * Can you provide other non-parametric metrics for evaluation?\n\n4. The 70x speed-up claim is unjustified in my opinion. The RFDiffusion method is producing much higher quality results according to the results in this paper, and is competing in a different league. The proposed method is advertised as a \"computationally efficient\" option, and the speed up must be reported w.r.t. the much faster FF-G method, for instance. \n\n5. Since the inference speed is emphasized as the main contribution, the paper should also consider more computationally efficient baseline methods and approximations and a more thorough set of time-complexity analyses must be included. Just reporting three time statistics without any confidence intervals, categorizations, hardware-related considerations, etc. is awfully inadequate.\n\n6. I think the language in the trailing sentence of the abstract, i.e., \"multi-motif scaffolding and motif optimal placement searching demonstrate EVA's superior efficiency and effectiveness\" should be modified and toned down; the presented experimental results do not suggest \"superior effectiveness\" by any measure, for instance.\n\n## Minor Comments\n\n1. Line 90: \"In detail, we exploit flow-based method pre-trained with flow matching objective\" is grammatically incorrect.\n\n2. Line 95: In the Figure caption, the \"MCG\" acronym was never defined in the paper.\n\n3. Line 314: \"desingable\" should be \"designable\".", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. The authors did try to motivate the work by identifying a problem in the data, relating it to sibling areas in machine vision problems, visualizing it on a few examples, and then proposing a solution.\n\n2. The suite of experiments support the computational efficiency of the proposed method.\n\n3. The core idea of straightening the inference paths and making the optimization process \"easier\" is reasonable at its base.", "weaknesses": "1. The paper is based more around \"proposing a solution\", than following the proper scientific steps. There are too many moving parts and it is difficult to identify how much each of the \"motif-aligned prior\", the \"motif-aligned posterior\", or the other numerous approximations are contributing or inducing risks to the end result. It is difficult to assess the quality of the proposed method in other contexts out of this paper.\n\n2. Unfortunately, the paper is not in a proper shape of writing in my humble opinion; it was quite a difficult read. The mathematical definitions are quite clunky, and the in-line style of latex math expressions made it too dense and difficult to comprehend, refernce, or look up. I cannot excuse this problem for page limitations, as many other ICLR papers handle the same problem much better. Referencing equations by number is almost impossible due to this style of writing. Irrelevant and secondary information also keeps distracting the reader from the main trajectory.\n\nThe notations are also odd at numerous instances:\n\n  * The \"batch\" superscript in $p^{batch}$ and $q^{batch}$ is too long and distracting. \n\n  * The notations, subscripts, and superscripts are inconsistently italic or upright, e.g., $\\sigma_{\\min}$ in Line 165 vs $\\sigma_{min}$ in Line 170.\n\n    * Both the italic and upright versions can be found in the same Equation (1). \n\n    * $Exp$ and $Log$ are both italic in Equation 8. Equation 4 uses $\\exp$ and $\\log$ instead. This is inappropriate.\n  \n  * $x$ is sometimes bold-faced, e.g., Line 191, and sometimes not, e.g., Line 152.\n    \n    * if bold-facing is linked to the $N$-scaled dimensionality, then how come \"0\" is bold-faced in both Line 165 and 202?  \n\n  * $m$ is oddly both a subscript and a vector in Line 205.\n\nHaving coding-style comments in a latex algorithm description is inappropriate; see Lines 329 and 333. If the algorithm has different sections, you may want to consider breaking the algorithm to smaller modules. Coding-style comments in a formal algorithmic description are not a proper solution for this problem.\n\n## Recommendation\n\nFor now, I kindly decided to give the work a weak rejection. That being said, I'm keeping an open mind, and I would be happy to raise my score after (1) hearing the other reviewers' viewpoints, (2) whether the authors deliver a helpful rebuttal, and (3) delivering on any improvements to the state of the paper before the final decisions (i.e., I cannot accept promises of post-acceptance improvements and would like to see any efforts delivered before then).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "x5qLhvN6Uk", "reviewer": "Reviewer_xJVQ", "summary": "The paper introduces a flow-based generative model guidance method named EVA for motif scaffolding in protein design. EVA claims to achieve faster inference times while maintaining performance comparable to SOTA methods. Unlike previous approaches that rely on approximating the distribution of the motif given the prior using Tweedie's formula, EVA approximates the distribution of correct residue frames conditioned on both the prior and the motif. Overall, the method presented seems to be novel; however, I found the paper to be a bit difficult to follow, particularly the correspondence between Figure 2, Algo 1, and the text body. Please see the questions below.", "questions": "1. How are the residues of the prior selected for comparison to the motif residues in Eq. 7? Does this refer to \"Motif index selection\" in Figure 2?\n2. Is the Eq number in line 1 of Algo 1 correct?\n3. Does the order of the steps in Algo 1 match Figure 2?\n4. More elaboration on the far-right plot in Figure 2 would be helpful.\n5. It would be helpful if the text mentioned in Figure 2 matched 1 to 1 with the text in the body of the paper. E.g. \"compositional gradient\" is not mentioned anywhere else. It would also be helpful to put (ours) next to the \"coupled vector field\".\n6. Can you elaborate on the cost of computing the unconditional posterior at every time step? Especially in comparison to other works?\n7. How do you decide the coupling strength in Eq 12 - 13?\n8. How do your vector fields in Eq 9 and 11 relate to the self-conditioning approach of https://openreview.net/forum?id=XTrMY9sHKF and references therein?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Faster inference, on-par results with SOTA.\n2. Does not require retraining.", "weaknesses": "1. Requires running the pre-trained model inference at every time step. Would like elaboration on this.\n2. Difficult to following the correspondence between the paper's main body, Figure 2, and Algo 1.\n3. Notation a bit confusing at times.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4JQx5bflxZ", "reviewer": "Reviewer_m3UM", "summary": "This study models motif-scaffolding as a Geometric Inverse Design task and proposes a sampling-based flow framework, Evolution-ViA-reconstruction (EVA), that incorporates spatial information of the motif. The reconstructed motif in the early stage guides the whole generation process by aligning generative directions with motif reconstruction. Experimental results show that EVA generates desirable proteins with significantly reduced time.", "questions": "- In Algorithm 1, \\pi* is assumed to be given. Is this assumption realistic in practice? If not, I wonder whether obtaining \\pi* and R* in Eq. 7 is tractable for large proteins.\n- If one prefers to generate a more desirable protein with more accurate motif reconstruction, even if it requires more computation time, is this feasible with the proposed method?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The idea of modeling motif-scaffolding as a geometric inverse design task is intriguing, and addressing the trade-off between generation and reconstruction is important in this context.\n- Directly utilizing the spatial information of the target motif and guiding the generative process through motif reconstruction seems practical.\n- The inclusion of real-world tasks enhances the applicability of the proposed method.", "weaknesses": "My primary concern is that the experimental results do not sufficiently demonstrate the advantages of the proposed method over existing approaches, particularly RFDiffusion, for the following reasons:\n\n- The results are reported based on a single run, and the performance differences are not significant enough to conclusively determine which method performs better.\n- In Table 1, RFDiffusion, a training-based method, appears to perform better with a computation time of 61 seconds. These computation times seem acceptable, especially when considering subsequent synthesis processes.\n- The mRMSD scores of EVA are not higher than those of other methods in Table 1 (≥100) and Table 2, even though accurate reconstruction is crucial in this work.\n\nAdditionally, the manuscript could benefit from improved consistency in terminology, clearer notations, and enhanced coherence. Please refer to minor comments for further suggestions.\n\n- There is inconsistent use of terms regarding the trade-off, such as \"trade-off between generation and guidance,\" \"generation vs. reconstruction trade-off,\" and \"Evolution Verse Adoption trade-off\" (mentioned in the Appendix).\n- The term \"Evolution via adaptation\" appears in the appendix.\n- Section 5.4 discusses the fast computation of EVA rather than directly addressing Question 2: \"Can EVA generate designable proteins of various lengths with accurate motif reconstruction?\"\n\n\n### Minor comments\n\n- Please use \\citet for textual citations. For example, (Wang et al., 2021) → Wang et al. (2021) in line 114\n- It would also be helpful to provide specific information about the appendix, such as \"Appendix A,\" rather than just saying \"Appendix.\"\n- Please place the table captions above the tables, not below them.\n\n### Typos\n- Line 58: the → The\n- Line 167: : → .\n- Line 367: I guess it is pi bar, not pi hat", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OOXT8hNLf6", "reviewer": "Reviewer_DcMW", "summary": "The paper studies the problem of motif scaffolding, which requires the construction of scaffold structure stabilzing motifs corresponding to desired functions. Towards this goal, the work leverages spatial context in order to guide the generation process. Motif scaffolding is interpreted as a Geomtric Inverse Design problem which leads to the formulation of Evolution-Via Reconstruction (EVA). EVA is a sampling-based coupled flow framework which utilizes motif-coupling in its prior as spatial contexts. This leads to the construction of a straighter probability path wherein generative directions are aligned with the sampling steps. EVA improves sampling time by a signficant margin while presenting competitive performance on RFDiffusion benchmark and in-silico vaccine design task.", "questions": "Please refer to the weaknesses section above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The paper is well written and easy to understand.\n* The paper tries to address an important and challenging problem from a geometric perspective.", "weaknesses": "* **Benefits of Motif Geometry:** My main concern is the benefits of motif geometry. Addition of motif-aligned prior and motif-interpolated posterior only benefits the model at sampling time. If we are using motif-coupled information with a rich prior, this should improve sample quality (especially designability) as well. However, empirical evaluation on benchmarks in Tables 1, 2 and 3 shows that additiona of motif-coupled information deteriorates designability of sampled complexes. What benefits does motif information provide for sampling novel complexes aside from faster sampling? How can motif information aid in the designability of sampled complexes?\n* **Improvement over OT-Flow:** The work presents empirical evaluation for motif-coupling on the RFDiffusion benchmark and an in-silico vaccine design benchmark. In most experiments on the RFDiffusion benchmark, backbone models (such as OT-Flow) are better for designability and success rates. What improvements does EVA offer over the backbone models? If the main contribution of the work is the addiiton of geometry information and motif-interpolation, then do these design choices benefit over prior backbones?\n* **Unconditional Generation:** Figure 3 and Table 2 present results for the backbone flow models. I am struggling to understand the presence of these results in the main text. If these experiments are not directly related to EVA, could they be better placed in the appendix? From my understanding, these results can be understood as ablations over backbone models in determining the choice of flow for motif coupling.\n* **Scallbility:** While EVA demonstrates consistent improvements on the RFDiffusion benchmark, it still remains unclear whether the method is scalable for real-world applicability. Table 1 demonstrates that EVA performs best on smaller lengths of residues (< 100). Intuitively, motif geometry is informative only for smaller complexes. Can the authors comment on the use of motif coupling for larger residue lengths? How can EVA be adopted/improved for larger complex sizes?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)