ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protein Backbone Generation
Authors: Angxiao Yue
Zichong Wang
Hongteng Xu
Venue: ICML 2025 poster (2025)
Keywords: protein backbone generation
quaternion representation
flow matching
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Protein backbone generation plays a central role in de novo protein design and is significant for many biological and medical applications.
Although diffusion and flow-based generative models provide potential solutions to this challenging task, they often generate proteins with undesired designability and suffer computational inefficiency.
In this study, we propose a novel rectified quaternion flow (ReQFlow) matching method for fast and high-quality protein backbone generation. 
In particular, our method generates a local translation and a 3D rotation from random noise for each residue in a protein chain, which represents each 3D rotation as a unit quaternion and constructs its flow by spherical linear interpolation (SLERP) in an exponential format.
We train the model by quaternion flow (QFlow) matching with guaranteed numerical stability and rectify the QFlow model to accelerate its inference and improve the designability of generated protein backbones, leading to the proposed ReQFlow model. 
Experiments show that ReQFlow achieves on-par performance in protein backbone generation while requiring much fewer sampling steps and significantly less inference time (e.g., being 37$\times$ faster than RFDiffusion and 63$\times$ faster than Genie2 when generating a backbone of length 300), demonstrating its effectiveness and efficiency.

ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protein Backbone Generation
Angxiao Yue Zichong Wang²
Hongteng Xu 134
Abstract
1. Introduction
De novo protein design (Ingraham et al., 2023; Lin & Alquraishi, 2023) aims design rational proteins from scratch with specific properties functions, which has many biological and medical applications, such devel- oping novel enzymes for biocatalysis (Kelly et al., 2020) and discovering new drugs for diseases (Teague, 2003; Silva et al., 2019). This task is challenging due to the extremely huge design space of proteins. For simplifying the task, the mainstream de novo protein design strategy takes protein backbone generation (i.e., generating 3D protein structures without side chains) as the key step that largely determines the rationality and basic properties of designed proteins.
Protein backbone generation plays central role
in de novo protein design and is significant for many biological and medical applications. Al- though diffusion and flow-based generative mod- els provide potential solutions to this challeng- ing task, they often generate proteins with un- desired designability and suffer computational inefficiency. In this study, we propose novel rectified quaternion flow (ReQFlow) matching method for fast and high-quality protein back- bone generation. In particular, our method gen- erates a local translation and a rotation from random noise for each residue in a protein chain, unit quater-
Focusing on protein backbone generation, many deep gen- erative models, especially those diffusion and baseb models (Ho al., 2020; Watson et al., 2023; Lin et al., 2024; Yim al., 2023b;a; Bose et al., 2024; Huguet al., 2024; Lipman et al., 2023), have been proposed potential solu- tions. However, these models often generate protein back- bones with poor designability (the key metric indicating the quality of generated protein backbones), especially for pro- teins with long residue chains. addition, diffusion or flow models often require many sampling steps to generate pro- tein backbones, resulting in high computational complexity and long inference time. As result, the above drawbacks generation quality and computational efficiency limit these models in practical large-scale applications.
which represents each 3D rotation nion and constructs its flow by spherical linear interpolation (SLERP) in an exponential format. We train the model by quaternion flow (QFlow) matching with guaranteed numerical stability and rectify the QFlow model to accelerate its in- ference and improve the designability of gener- ated protein backbones, leading the proposed ReQFlow model. Experiments show that Re- QFlow achieves on-par performance in protein backbone generation while requiring much fewer sampling steps and significantly less inference time (e.g., being 37x x faster than RFDiffusion and 63x faster than Genie2 when generating back- bone of length 300), demonstrating its effective- and efficiency. Code is available at ht tps: / //github... www.ccom///coeeeeee aoree [ReQFlow
To overcome the above challenges, we propose a novel rec- tified quaternion flow (ReQFlow) matching method, achiev- ing fast and high-quality protein backbone generation. As illustrated Figure la, our method learns model to gener- ate local 3D translation and 3D rotation respectively from random noise for each residue in protein chain. Different from existing models, the proposed model represents each rotation as a unit quaternion and constructs its quaternion flow in SO(3) by spherical linear interpolation (SLERP) in an exponential format (Sola, 2017), which can be learned by our quaternion flow (QFlow) matching strategy. Fur- thermore, given trained QFlow model, leverage the rectified flow technique in (Liu, 2022), re-training the model based on the paired noise and protein backbones generated by the model itself. The rectified QFlow (i.c., ReQFlow)
'Gaoling School of Artificial Intelligence, Renmin Uni- versity of China, Beijing, China "School of Statistics, Ren- min University of China, Beijing, China Beijing Key Lab- oratory of Research on Large Models and Intelligent Gover- nance "Engineering Research Center of Next-Genern Intel- ligent Search and Recommendation, MOE, Correspondence to: Hongteng Xu <hongtengxu ruc. edu.en....>.
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Random
QFlow
Protein Backbone
Noise
R ReQFlow L-50c
ReQF low L-50
ReQFiow
FoldFlow
Genie
Bucildian
Dentarmion
AFDilusion
Fraction
Framel low
FoMFlow-OT
0.8
Recttfying Paired Data
Of
Exponential
H50
0.7
FoldFio SFM
ReQFlow
The Frame of Residue
0.6
Base
FrameD
T= q) SE(3)
0
0.0
0.5
1.0
1.5
2.0
2.5
Euolidean Flow
Gustemion Flow SO33
Transformation
Log10 Inference Time (s)
(a) The scheme of rectified quaternion flow matching
(b) Comparisons for various methods
Figure 1. (a) An illustration of our rectified quaternion flow matching method, in which each residue represented as a frame associated with a local transformation. (b) For each method, the size of its circle indicates the model size, and the location of the circle's centroid indicates the logarithm of the average inference time when generating a protein backbone with length. N = 300 and the Fraction score of designable protein backhones. For QFlow and ReQFlow, we set the sampling step L E {20,50 50, 5003, respectively.
model leads to straightened sampling paths in R3 and SO(3), respectively, when generating translations and rotations. As result, we can apply fewer sampling steps to accelerate the generation process significantly.
tations, respectively, Following the same framework, flow- based methods like FrameFlow (Yim al., 2023a) and FoldFlow (Bose et al., 2024) replace the stochastic diffusion processes with deterministic flows.
We demonstrate the rationality and effectiveness of Re- QFlow compared to existing diffusion and flow-based meth- ods. In particular, thanks to the exponential format SLERP, ReQFlow is learned and implemented with guaranteed nu- merical stability and computational efficiency, especially when the rotation angle is close 0 ग. Experimen- tal results demonstrate that ReQFlow achieves competitive performance, generating high-quality protein backbones with significantly reduced inference time. Furthermore, Re- QFlow consistently maintains effectiveness and efficiency in generating long-chain protein backbones (c.g., the pro- tein backbones with over 500 residues), where all baseline models suffer severe performance degradation. As shown in Figure lb, ReQFlow outperforms existing methods and gen- erates high-quality protein backbones, whose designability Fraction score is 0.972 when sampling 500 steps and 0.912 when merely sampling 50 steps.
For the above methods, many efforts have been made to modify their model architectures and improve data rep- resentations, e.g.. the Clifford frame attention module in GAFL, (Wagner et al., 2024) and the asymmetric protein representation module in Genie (Lin & Alquraishi, 2023) and Genie2 (Lin al., 2024). In addition, some methods leverage large-scale pre- pre-trained models to improve gener- ation quality. For example, RFDiffusion (Watson et al., 2023) utilizes the pre-trained RoseTTAFold (Back et al., 2021) as the backbone model. FoldFlow2 (Huguet et al.,
2024) improves FoldFlow by using a protein large lan- guage model for residue sequence encoding. Taking scal- ing further and adopting different architectural approach, Proteína (Geffner et al., 2025) developed large-scale, flow- based generative model using non-equivariant transformer operating directly on C-alpha coordinates.
Currently, the above methods often suffer the conflict on computational efficiency and generation quality. The state- of-the-art methods like RFDiffusion (Watson et al., 2023) and Genie2 (Lin al., 2024) need long inference time to generate protein backbones with reasonable quality. Frame- Flow (Yim et al., 2023a) and GAFL (Wagner et al., 2024) significantly improve inference speed while lag behind RFD- iffusion and Genie2 in protein backbone quality. Moreover, these methods suffer severe performance degradation when generating long-chain protein backbones. These limitations motivate to develop the proposed ReQFlow, improving the current flow-based methods and generating protein back- bones efficiently with satisfactory designability.
2. Related Work and Preliminaries
2.1. Protein Backbone Generation
Many diffusion and flow-based methods have been proposed to generate protein backbones. These methods often param- eterize protein backbones like AlphaFold2 (Jumper et al., 2021) does, representing each protein's 's residues as set of SE(3) frames (Yim et al., 2023b;a; Wang et al., 2025). Ac- cordingly, FrameDiff (Yim et al., 2023b) generates protein backbones by two independent diffusion processes, gener- ating the corresponding frames' local translations and ro-
2

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
2.2. Quaternion Algebra and Its Applications
et al., 2020; Wang al., 2023) and the quaternion gener- ative models for molecule generation (Köhler al., 2023; Guo al., 2025). However, the computational quaternion techniques are seldom considered in protein-related related tasks. Our work fill this blank, demonstrating the usefulness of quaternion algebra in protein backbone generation.
The proposed ReQFlow is designed based on quaternion algebra (Dam al., 1998; Zhu et al., 2018). Mathematically, quaternion is an extension of complex numbers into four- dimensional space, consists of one real component and three orthogonal imaginary components. A quaternion is formally expressed as 9 = s + æi + yj + 2k E H. where H denotes the quaternion domain, and 8, x. y, E IR. The imaginary components {i. J. j.k} k} satisfy 1² j² = k2 ijk 11 Each q E H can be equivalently represented as vector 9 [s, | E I where u [zz, y, 2] Given
3. Proposed Method
3.1. Protein Backbone Parameterization
We parameterize the protein backbone following (Jumper et al., 2021; Yim et al., 2023b;a; Bose et al., 2024). As illustrated in Figure la, each residue is represented as a frame, where the frame encodes a rigid transformation start- ing from the idealized coordinates of four heavy atoms: [N' C. C* O*) E 3x3x In this representation, C² = [0,0, 0J is placed at the origin, and the transformation in- corporates experimental bond angles and lengths (Engh & Huber, 2012). We can derive each residue's 's frame by
qi = [s, u and Q2 = [s2, u2l their multiplication is achieved by Hamilton product, i.c.,
8182 - u u2
q 6 Q2-
=
(I)
S2 + un U2 $1 U2
where x denotes the cross product. Quaternion is a powerful tool to describe 3D rotations. For a 3D rotation in the axis-angle formulation, i.e., w = &uu E IRS where the unit vector u and the scalar & denote the rotation axis and angle, respectively, we can convert to a unit quaternion by an exponential map (Sola, 2017):
N' C, C² O T 1 C C", C".
(5)
where T E SE(3) is the local orientation orientation-preserving rigid transformation mapping the idealized frame to the frame of the i-th residue. In this study, we represent T = (æ*, q²), where æ² E RR represents the 3D translation and unit quaternion q¹ E S3, which double-covers SO(3), represents 3D rotation. According to Eq. (4), the action of T coordinate E Rd can be implemented as
exp ) = cos
[[oos
Φ sin
2 u
T E
ESS,
(2)
2
where 3 = (q E R |lqll2 1 is the 4D hypersphere. The conversion from a unit quaternion to an angle-axis representation is achieved by a logarithmic map:
w 2 (o)(
(3)
Too= + Im(q @ |0, (
(6)
Suppose that rotate a point i1 E R to U2 by w, we can equivalently implement the operation by
Note that, for protein backbone generation, we can use the planar geometry of backbone to impute the coordinate of the oxygen atom O' (Yim et al., 2023a; Watson et al., 2023). do not need to parameterize the rotation angle of the bond "C C". As result, for protein backbone with N residues, we have collection of N frames, resulting in the parametrization set e = [T Therefore, we can formulate the protein backbone generation problem as modeling and generating (T* ) automatically.
= lm(q q & [0, v 1
(4)
where -- [cos S. - sin Gu T is the inverse of and "Im(-)" denotes the imaginary components of a quaternion (i.e., the last three elements of the corresponding 4D vec- tor). The quaternion-based rotation representation in Eq. (4) offers several advantages, including compactness, computa- tional efficiency, and avoidance of gimbal lock (Hemingway & O'Reilly, 2018), which has been widely used in skele- tal animation (Shoemake, 1985), robotics (Pervin & Webb, 1982), and virtual reality (Kuipers, 1999).
3.2. Quaternion Flow Matching
We decouple the translation and rotation of each frame, establishing two independent flows in R and SO(3), re- spectively. Without the loss generality, we define these two flows in the time interval [0, 1]. When t O, we sam- ple the starting points of the flows random noise, i.e., To = (mo, ạo) ~ To x Co. where To = =N(00, I3) is the Gaussian distribution for translations, and Qo = IG is the isotropic Gaussian distribution on SO(3) for rota- tions (Leach et al., 2022), corresponding to uniformly sam- pling rotation axis E $2 and rotation angle á E [0,
Besides computer graphics, some cinternioo-bbased ma- chine learning models have been proposed for other tasks, e.g., image processing (Xu et al., 2015; Zhu al., 2018) and structured data (e.g.. graphs and point clouds) analy- sis (Zhang et al., 2020; Zhao et al., 2020). Recently, some quaternion-bar models have been developed for scientific problems, e.g., the quaternion message passing ( (Yue et al., 2024) for molecular conformation representation (Gasteiger
3

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
with the density:
Besides the above MSE losses, we further consider the auxiliary loss proposed in (Yim et al., 2023b), which dis- courages physical violations, e.g., chain breaks or steric clashes. Therefore, we train the model by
f()) 1 COS Φ
30 Σ 1=0
(21 + 1)e (00 sin(( =)))
sin(
ming Lgs + L3o(3) + ax ((- < <(} L (12)
Based on Eq. (2), we convert the sampled axis and angle
to To- When & the ending points of these two flows,
where 0 is the weight of the auxiliary loss, 1 is an indicator, signifying that the auxiliary loss is applied only when / is sampled below a predefined threshold Ç-
denoted as T = (z), qı), should be the transformation of a frame. We denote the data distribution of T1 as Ti x Q1.
Linear Interpolation of Translation. For 2o ~ To and 31 ~ Ti, we can interpolate the trajectory between them linearly: for t E [0, 1].
Inference Based on QFlow. Given a trained model, we can generate frames of residues from noise with the predicted velocities. In particular, given initial (æo. qu) To X Qo- the translation is generated by Euler solver with L. steps:
F = (1 - 20 + tz;,
with constant translation velocity: v = 21 Fz-
Tr+A: 3t + U0 AL,
(13)
where the step size At 1.- The quaternion of rotation is generated with an exponential step size scheduler: We modify Eq. (8), interpolating qe with acceleration
SLERP of Rotation in Exponential Format. For unit quaternions Qo ~ La and Qi ~ 21, we interpolate the
trajectory between them via SLERP in an exponential for- mat (Sola, 2017):
q: qu @ exp((1 e - log(a @ 9q)),
(14)
q: = qo @ exp(t & qı)), with constant angular velocity: w ou.
(8)
where y controls the rotation accelerating, and we empiri- cally set 7 10. Then, the Euler solver becomes:
Here, 4u q1 [cos (6/2). sin (d/2) u J and w 2 log(%) (o & )1). exp(-) and log(-) are exponential and log- arithmic maps defined in Eq. (2) and Eq. (3), respectively.
q = q: @ exp exp(( (xpp2_At-
At Aat.7e- ge- wo..)
(15)
where yo- e is the adjusted angular velocity. Previous works (Bose et al., 2024; Yim et al., 2023a) have demon- strated that the exponential step size scheduler helps reduce sampling steps and enhance model performance.
Training QFlow Model. In this study, we adopt the SE(3)--quivariant neural network in FrameFlow (Yim et al., 2023a), denoted Mo, to model the flows. Given the transformation at time f, i.e., T,, the model predicts the transformation at / = 1:
3.3. Rectified Quaternion Flow
Te,1 (0.1 Mo...
(9)
Given the trained QFlow model Ma. we can rectify the flows in E and SO(3), respectively, by applying the flow rectification method (Liu, 2022). In particular, we gen- erate noisy T (# %6 To x Qo and transfer to T { TI x Q1 by MM. Taking Mg as the initialization, we use the noise-sample pairs, i.e., (Ty, Ty) ) to train the model further by the same loss in Eq. (12) and derive the rectified QFlow (ReQFlow) model.
We train this model by the proposed quaternion flow (QFlow) matching method, In particular, given the frame Ti (z). we first sample a timestamp t Uniform([0, 1)) and random initial points To = (æo, qo) Tox Qo. Then, we derive (æz, v) and ()) via Eq. (7) and Eq. (8), respectively. Passing (mt, qt, f) through the model Mg. we obtain IΦ.1 and ge,1, and derive the translation and angular velocities at time by
The work in (Liu, 2022) has demonstrated that the rectified flow of translation in R preserves the marginal law of the original translation flow and reduces the transport cost from the noise to the samples. We find that these theoretical properties are also held by the rectified quaternion flow under mild assumptions. Let (qu, q1) ~ Qo x Q1 be the pair used to train QFlow, and (g6- 9 be the pair induced from (go, qi) by flow rectification. Then, we have
L0,1
2 2 q. @ go.1)
V0.
Wa.t
(10)
1
Based on the constancy of the velocities, we train the model Me by minimizing the following two objectives:
Theorem 3.1. (Marginal preserving property). The pair (g6. qi) is a coupling of Qu and Q1. The marginal law of qt equals that of qu at everytime, that is Law = Law((e).
Lga = E.,To.T, T |||e= - 1.
(11)
Lsos3) = E.,e.. e,ll,,lllla -|lW
1.)
4

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Mean Round-trip aindttipppp.... Error
0.7
0.66
6.3
SCOPe
# Small Angles (<
Frequency
0.4
0.33
Retattion Matri
0.3
Gusternion
0.2
2-
1.7
0.1
0.06
0.00
0.03
4-10
1-10 m-gp
0.0
23
100
200
300
400
500
Large Rotation Angle (x-e)
# Large Angles (2 2)
Length
(a)
(b)
(c)
Figure 2. (a) Mean round-trip emrors from ग - 1001 to π
10-7. (b) The frequency of suffering large rotation angles per protein when
training the two datasets. (c) The average numher of small rotation angles per protein when generating backhones for each length.
Table 1. Comparisons for various rotation interpolation methods.
Method
Matrix Geodesic
SLERP (Add. Format)
SLERP (Exp. Format)
Interpolation
Formula
Ro expar (t [Ro R.))
(in((1))
9
sin(t
sin
q
qo @ exp(t log( &g: ))
Tp. @cccel lg1 cool( (1- 11)))))))) (go)
Velocity
Ω logar (R. R1)
2 log(g0 (1)))
sin
Euler Solver
Update
R.+ = R. expM (AL (AC. 92)
qi+ar = q: + Al Tp
4:+ = @ exp( Al w
No Renomalization
Numerical Stability
Φ 2 x 10-2
x
6 < 10-6
FrameFlow (Yim al., 2023a), FoldFlow (Bose et al., 2024)
AssembleFlow
QFlow (Ours), ReQFlow (Ours)
Application Scenarios
(Guo et al., 2025)
Theorem 3.2. (Reducing transport costs). The pair (4o= qi) yields lower or equal convex transport costs than the in- put (oo, Q1). For any convex c: R3 - R, define the cost as Cg q1) = c (log(an" 8 q1)). Then, we have E|C(db di) S B[C(gu.q.) q:)l.
3.4. Rationality Analysis
Most existing methods, like FrameFlow (Yim et al., 2023a) and FoldFlow (Bose et al., 2024), represent rotations as 3 x matrices. Given two rotation matrices Ro and R,, they con- struct a flow in SO(3) with matrix geodesic interpolation:
R Ro exp (t (R, R1)).
(16)
Theorem 3.2 shows that the coupling (16- Φ either achieves a strictly lower or the same convex transport cost compared to the original one, highlighting the advantage of the quaternion flow rectification in reducing the overall rota- tion displacement cost without compromising the marginal distribution constraints (Theorem 3.1.. In addition, we have
where exp (-) and log (-) denote the matrix exponen- tial and logarithmic maps, respectively. The corresponding angular velocity Ω log [R, R1). Different from exist- ing methods (Yim et al., 2023a;b; Bose al., 2024), method applies quaternion-based rotation representation and achieves rotation interpolation by SLERP in an exponential format, which achieves superior numerical stability and thus benefits protein backbone generation.
Corollary 3.3. (Cost Reduction with Nonconstant Speed). Suppose the geodesic interpolation qi between qo and qı has constant axis u, but its speed is nonconstant in time, i.e, a(L)u. The quaternion flow rectification still reduces or preserves the transport cost.
To verify this claim, we conduct round-trip error exper- iment: given an rotation w in the axis-angle format, we convert it to rotation matrix R and quaternion q, re- spectively, and convert it back to the axis-angle format, denoted as WR and wa. respectively. Figure 2a shows the round-trip errors in L2 norm for large rotation angles (e.g., @ E [T -0- 2 TT). Our quaternion based method is numer- ically stable while the matrix-based representation suffers
This corollary means that when applying the exponential step size scheduler (i.e., Eq. (15)), the rectification still reduces or preserves the transport cost.
5

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
severe numerical errors. When training a protein back- bone generation model, the numerical stability for large rotation angles is important. Given the frames in the Pro- tein Data Bank (PDB) (Burley et al., 2023) dataset and the SCOPe (Chandonia et al., 2022) dataset, we sample a ran- dom noise for each frame and calculate the rotation angle between them. The histogram in Figure 2b shows that when training an arbitrary flow-based model, the probability of suffering at least one large angle per protein is 0.59 for PDB and 0.34 for SCOPe, respectively. It means that the matrix- based representation may introduce undesired numerical errors that aggregate and propagate during training.
Table 2. Comparisons for various models on PDB. For each des- ignahility metric, we hold the hest result and show the top-3 results with a blue background. In the same way, we indicate the best and top-3 diversity and novelty results among the rows with Fraction > 0.8. The inference time corresponds to generating protein
backbone with length N = 300.
Efficiency
Designability
Diversity Novelty
Method
Step Time(s) Fractiont scRMSDJ
TMJ
TMJ
RFDiffusion
50
66.23
0.904
1.102
0.382
0.527
Genie2
1000
112.93
0.908
1.132
0.370
0.475
500
55.86
0.000
18.169 9.63
FrameDiff
500
48.12
0.564
2.93613.008
0.441
0.591
In addition, a very recent work, AssembleFlow (Guo et al., 2025), also applies quaternion-based rotation representation and SLERP when modeling 3D molecules. In particular, it applies SLERP in additive format:
FoldFlowsue
500
43.52
0.624
3.080
0.469
0.645
FoldFlowsrM
500
43.63
0.636
3.031 23,59
0.411
0.604
FoldFlowor
500
43.35
0.852
1.760 2,99
0.434
0.617
FoldFlow2
50
6.35
0.952
1.083 1.330
0.373
0.527
20
2.63
0.644
3.060
0.339
0.492
FrameFlow
500
17.05
0.872
1,380 1.392
0.346
0.562
q: sin((1 - 2
(t
9o +
91
(17)
200
6.77
0.864
1.542 889
0.348
0,564
sin( (6)
sin
100
3,46
0.708
2. 167 2.167 22.333
0.332
0,560
50
1.73
0.704
2.63943.om
0.334
0,536
20
0.71
0,436
4.652
0.319
0.501
and updates rotations linearly by the following Euler solver:
10
0.37
0.180
7.343 12.115
0.317
0.482
QFlow
500
17.37
0.936
1.163
0.356
0.635
200
7.10
0.864
1.400
0.344
0.620
g+ = q: + At-n: ne
(18)
100
3.48
0.916
1.342.1.364
0.348
0.614
50
1.77
0.812
42.15
0.344
0.571
Here, nh is the instantaneous velocity in the tangent space of qt, which is derived by the first-order derivative of Eq. (17). However, this modeling strategy also suffers numerical is- sues. Firstly, although the additive format SLERP can gen- crate the same interpolation path as ours in theory, when rotation angle á is small (e.g.. & E 10- Eq. (17) of- ten outputs "NaN" because the denominator sin( () tends to zero. The exponential step size scheduler leads to rapid convergence when generating protein backbones, which fre- quently generates rotation angles below the threshold 10-6 (as shown in Figure 2c) and thus makes the additive format SLERP questionable in our task. Secondly, the Euler step in Eq. (18) makes |/q+ * 1, so that renormalization is required after each update. Table 1 provides comprehen- sive comparison for the three rotation interpolation methods, highlighting the advantages of our method.
20
0.73
0.604
3.090
0,325
0.537
10
0.38
0.332
5.032.14.008
0.313
0.528
ReQFlow
500
17.42
0.972
1.071.0.482
0.377
0.645
200
6.94
0,932
1.160
0,384
0.648
100
3.58
0.928
1,245
0.369
0.629
50
1.78
0.912
1,254
0.369
0.608
20
0.72
0.872
1.418 10,998
0.355
0.581
10
0.38
0.676
2.443
0.337
0.540
4.1. Experimental Setup
Datasets. We apply two commonly used datasets in our experiments. The first is the 23,366 protein backbones collected from Protein Data Bank (PDB) (Burley al., 2023), whose lengths range from 60 to 512. The second is the SCOPe dataset (Chandonia et al., 2022) pre-processed by FrameFlow (Yim et al., 2023a), which contains 3,673 protein backbones with lengths ranging from 60 to 128.
4. Experiment
To demonstrate the effectiveness and efficiency of our meth- ods (QFlow and ReQFlow), we conduct comprehensive experiments compare them with state-of-thh-arr protein backbone generation methods. In addition, we conduct abla- tion studies to verify the usefulness of the flow rectification strategy and the impact of sampling steps on model per- formance. All the experiments are implemented on four NVIDIA A100 80G GPUs Implementation details and ex- perimental results are shown in this section and Appendix C.
Baselines. The baselines of our methods include diffusion- based methods (FrameDiff (Yim al., 2023b). RFDiffu- sion (Watson et al., 2023), and Genie2 (Lin et al., 2024)) and flowd methods (FrameFlow (Yim et al., 2023a), FoldFlow (Bose et al., 2024), and FoldFlow2 (Huguet et al., 2024)). In addition, we rectify FrameFlow by our method (i.e., re-training FrameFlow based on the paired data gen- crated by itself) and consider the rectified FrameFlow (Re- FrameFlow) as a baseline as well.
6

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
FoldFlow
FrameFlow
RFDiffusion
ReQFlow
PDB
Strand percent
0.8
0.10
0.4
0.05
0.0
0.0
0.2
0.4
0.8
0.8
1.0
0.0
0.2 0.4
0.8
0.8
1.0
0.0
0.2
0.4
0.8
1.0
0.0
0.2
0.4
0.8
1.0
0.0
0.4
0.8
1.0
0.00
Helix percent
Helix percent
percent
percent
Helix percent
Figure 3. The distribution of protein backbones with respect the percentages of their secondary structure.
25
1.0
Metrod
DFlow
20
ReOFlow
Fiection
6.8
FramoDif
RF DFDiilusion
FadFlsw-OT
TrmmeFow
Langth
Langth
Length
0.4
OFiw
teGhHow
0.2
0.0
300
350
400
450 Length
500
350
800
100
200
300
400
500
600
Length
Longth 400
Length 600
Length 600
(a) Fraction
(b) scRMSD
(c) Samples generated by ReQFlow
Figure 4. The comparison for various methods on the designability of generated long-chain protein backhones.
Implementation Details. For the PDB dataset, we uti- lize the checkpoints of baselines and reproduce the sults shown in their papers. Given the QFlow trained PDB, we generate 7,653 protein backbones with lengths in [60, 512] from noise and then train ReQFlow based on these boike-backbone pairs. For the SCOPe dataset, we train all the models from scratch. Given the QFlow trained on SCOPe, generate 3,167 protein backbones with lengths in [60, 128] from noise and then train ReQFlow based these
2) Diversity: Given designable protein backbones. whose scRMSD S 2', quantify structural diversity by av- eraging the mean pairwise TM-scores computed for each backbone length.
3) Novelty: For each designable protein backbone, we com- pute maximum TM-score to data PDB using Fold- seek (van Kempen et al., 2022). The average of the scores reflect the novelty of the generated protein backbones.
pairs. When training ReQFlow, apply structural data filtering, selecting training samples based on scRMSD (<2 and TM-score (20.9 for long- chain proteins) and removing proteins with excessive loops (>50%) or abnormally large radius of gyration (top 4%).. ReFrameFlow trained the same way.
4) Efficiency: We assess the computational efficiency of each method by the number of sampling steps and the aver- age inference time for generating 50 proteins at two lengths: 300 residues for PDB and 128 residues for SCOPe.
4.2. Comparison Experiments on PDB
Evaluation Metrics. Following previous works (Yim et al., 2023a; Bose al., 2024; Geffner et al., 2025), evaluate each method in the following four aspects:
Generation Quality. Given the models trained on PDB, we set the length backbone E {100, 150, 200, 250, 300), and generate protein backbones for each length. Table shows that ReQFlow achieves state-of-the-an performance in designability, achieving the highest Fraction (0.972) among all models, significantly outperforming strong petitors such as Genie2 (0.908) and RFDiffusion (0.904). Additionally, it achieves the lowest scRMSD (1.07110:482)
1) Designability: As the most critical metric, designability reflects the possibility that generated protein backbone can be realized by folding the corresponding amino acid sequence. is assessed by the scRMSD between the gen- erated protein backbone and the backbone predicted by ESMFold (Lin et al., 2023). Given set of generated back- bones, we calculate the proportion of the backbones whose scRMSD < 2 (denoted as Fraction).
with a notably smaller variance compared to the other meth- ods, highlighting the model's 's consistency and reliability in generating high-que high-quality protein backbones. Meanwhile, Re-

---

ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
QFlow maintains competitive performance in diversity and novelty.
Table 3. The Fraction scores of ReQFlow under different settings when generating hackbones with 300 residues by 500 and 50 steps. For each method, we evaluate the results on five checkpoints.
Computational Efficiency. Moreover, ReQFlow achieves ultra-fast protein backbone generation. Typically, ReQFlow achieves a high Fraction score (0.912) with merely 50 steps and 1.78s, outperforming RFDiffusion and Genie2 with 37x and 63x acceleration, respectively. The state-of-the- art methods like Genie2 and FoldFlow2 suffer severe per- formance degradation in designability when the number of steps is halved, while ReQFlow performs stably even re- ducing the number of steps from 500 to 20. In addition, although the main speed bottleneck is model prediction, ReQFlow's 's rotation update can be ~200 faster than Frame- Flow's update because of utilizing the quaternion-ban computation (Appendix C.3).
Exponential Scheduler
Flow Rectification
Data
Sampling Steps
Filtering
500
50
X
0.143440079
0.047
x
x
0.910
0. 0.79510005
V
0.519 0.612
V
0.969
0.9322a0m2
1.0
0.9
0.8
Fraction
0.7
Given generated protein backbones, we record the percentages of helix and strand, respectively, for each backbone, and visualize the distribu- tion of the backbones with respect to the percentages in Figure 3. The protein backbones generated by ReQFlow have a reasonable distribution, which similar those of RFDiffusion and FrameFFlow and comparable to that of the PDB dataset. However, the distribution of FoldFlow is sig- nificantly different from the data distribution and indicates mode collapse risk- - the protein backbones generated by FoldFlow are always dominated by helix structures.
Fitness of Data Distribution.
FrameFlow
0.6
ReFrameFlow
QFlow
0.5
ReQFlow
0.4
500
400
300
200
100
50
20
10
Num Steps
Figure 5. A comparison for various methods on their designability with the reduction of sampling steps. Original data is in Table 8.
Effectiveness on Long Chain Generation. Notably, Re- QFlow demonstrates exceptional performance in generating long-chain protein backbones (e-g., N > 300). As shown in Figures 4a and 4b, ReQFlow outperforms all baselines generating long protein backbones and shows remarkable robustness. Especially, when the length N > 500, which is out of the length range of PDB data, all the baselines fail to maintain high designability while ReQFlow still achieves promising performance in Fraction score and scRMSD and generates reasonable protein backbones. This generaliza- tion ability beyond the training data distribution underscores ReQFlow's 's potential for well-wrldd applications requiring robust long-chain protein design.
4.3. Analytic Experiments on SCOPe
Universality of Flow Rectification. Note that, the flow rec- tification method used in our work is universal for various models. As shown in Table and Figure 5, applying flow rectification, we can improve the efficiency and effective- ness of FrameFlow as well. This result highlights the broad utility of flow rectification an operation that can enhance the performance of flow models on SO(3) spaces.
Superiority of Exponential-Forma SLERP. The results in Table 4 and Figure 5 indicate that QFlow and ReQFlow outperform their corresponding counterparts (FrameFlow and ReFrameFlow) terms of designability across all sam- pling steps. As we analyzed in Section 3.4, the superiority of our models can be attributed to the better numerical sta- bility of quaternion calculations compared to the traditional matrix geodesic method.
Ablation Study. We conduct an ablation study to evaluate the impact of different components in the ReQFlow model. The results in Table 3 reveal that similar existing meth- ods (Yim et al., 2023a; Bose et al., 2024; Huguet et al., 2024), the exponential step size scheduler is important for ReQFlow, helping generate designable protein backbones with relatively few steps. Additionally, the data filter is necessary for making flow rectification work. In particu- lar, rectifying QFlow based on low-quality data leads to a substantial degradation in model performance. In contrast, after filtering out noisy and irrelevant data, rectifying QFlow based on the high-quality data boosts the model performance significantly.
5. Conclusion and Future Work
In this study, we propose rectified quaternion flow match- ing method for efficient and high-quality protein backbone generation. Leveraging quaternion-based representation and flow rectification, our method achieves encouraging performance and significantly reduces inference time. In the near future, we plan to improve our method for gen- erating high-quality long-chain protein backbones. This
8

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Table Comparisons for various models on SCOPe. Five checkpoints per model evaluated show statistical significance. We
indicate the best and top-3 results in the same way as Table 2 does.
Designability
Diversity Novelty
Designability
Diversity Novelty
Step Epoch
Step Epoch
Fractiont scRMSDL
TM
TM4
Fraction scRMSDI
TM4
TMJ
FrameFlow 500
187
0,880
4.418
0.399
0.711
QFlow
500
181
0.899
1.240
0.392
0.695
189
0.849
1.44
0.397
0.713
185
0.913
1.210
0.393
0.719
193
0.851
1.39
0.390
0.720
195
0.907
1.263
0.389
0.712
195
0.845
1.427:1001
0.397
0.713
197
0.893
1.285 10.29
0.401
0.716
199
0.830
1. 1
0.379
0.712
199
0.852
1. 4,444
0.385
0.693
187
0.797
1. 1.:38
0.382
0.686
50
181
0.849
1.447
1
0.379
0.648
189
0.820
1. 1.54611316
0.379
0.685
185
0.875
1,389
1:1669
0.386
0.687
193
0.836
1. 1.504
0.375
0.680
195
0.872
1.389
0.371
0.674
195
0.817
L.528
0.390
0.682
197
0.823
1.517
0.384
0.682
199
0.787
1.60
0.366
0.676
199
0.849
200700
0.378
0.682
20
187
0.739
L888 1552
0.373
0.656
20
181
0.778
1.746
0.369
0.620
189
0.713
1.918 ,955
0.362
0.656
185
0.778
1.68
0.377
0.655
193
0.723
1882
0.371
0.645
195
0.764
1. 1.76441..22
0.367
0.646
195
0,687
2.035
0.377
0.650
197
0.746
4.834
0.374
0.652
199
0.677
2.
1..770
0.369
0.636
199
0.742
+.802
0.373
0.648
ReFrameFlow 500
2
0.912
1 1.205 1:0.755
0.406
0.713
ReQFlow 500
0.939
1. 1:.677
0.411
0.704
3
0.933
1,21 10631
0.410
0.713
0.952
1.088
+0.52a
0.404
0.703
4
0.923
1.201 +0,000
0.411
0.709
0,939
1.149 +0.995
0.405
0.693
5
0.923
1. 1.270
0,399
0.701
0.949
1. 40.555
0.402
0.690
6
0.930
1.178 1-0622
0.407
0.709
6
0.955
1. 1.143108119
0.407
0.688
50
2
0.903
1.267 70.704
0.407
0.691
50
0.928
1.:57
0,411
0.687
3 0.897
1.262
0.408
0.692
0.919
1.17940762
0,410
0.688
4
0.909
1,271 4-0.74
0.408
0.690
0.916
1. 19450
0.413
0.676
5
0.906
1.270
+-0.13
0.406
0.690
0,933
1.184 184 4-0091
0.407
0.670
6
0.914
0.406
0.684
0,914
1.229 200768 768
0.414
0.681
20
2
0.877
1.43
0.400
0.669
20
0,929
1.267
40244
0.407
0.678
3
0.878
1.378
0.410
0.673
0.913
1,25640760
0.404
0.674
4
0,899
1.354
0.405
0.679
4
0.913
1.232.1072
0,406
0.671
5 0.877
1.4 1.440
0.400
0.672
0.893
1.322
0.404
0.670
6
0.888
1. 1.303
0.409
0.675
0.900
1,331 10.85
0.405
0.663
will involve constructing a larger training dataset, build- ing on approaches such as Genie2 (Lin et al., 2024) and Proteína (Geffner et al., 2025). Additionally, we plan to refine our model architecture through two key strategies: in- creasing model capacity via parameter scaling and exploring non-equivariant design, drawing inspiration from the archi- tecture of Proteína (Geffner et al., 2025). Furthermore, we intend to leverage the knowledge embedded in large-scale pre-training models (Li et al., 2025; Huguet et al., 2024), such as FoldFlow2 (Huguet et al., 2024), which incorporated sequence information using ESM2 (Lin et al., 2023). As long-term goals, will extend method to conditional protein backbone generation and explore its applications in side-chain generation and full-atom protein generation.
Renmin University of China, and the Public Computing Cloud, Renmin University of China. We also acknowledge the support provided by the fund for building world-class universities (disciplines) of Renmin University of China and by the funds from Beijing Key Laboratory of Research on Large Models and Intelligent Governance, Engineering Research Center of Next fext-Gener Intelligent Search and Recommendation, Ministry of Education, and from Intelli- gent Social Governance Interdisciplinary Platform, Major Innovation & Planning Interdisciplinary Platform for the "Double-First Class" Initiative, Renmin University of China.
Impact Statement
This paper presents work whose aim is to advance the field of Machine Learning and AI for Science, especially the task of protein design. There are many potential societal conse- quences of our work, e.g., accelerating drug development and contributing to healthcare. None of them we feel must be specifically highlighted here.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (92270110) 10), the Fundamental Research Funds for the Central Universities, the Research Funds of

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ReQFlow:
Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
A. Proofs of Key Theoretical Results
A.1. The Angular Velocity under Exponential Scheduler
Proposition A.1. For spherical linear interpolation (SLERP) with angular velocity w, when applying an exponential scheduler during inference:
Q -0 exp ((1 - 6 log(an" @g1)),
(19)
the resulting angular velocity evolves as W; Y ye w.
Proof. The standard SLERP formulation in exponential form is:
q Qo @ exp (llog(90 @ q))),
(20)
where the relative rotation qrel = qo 2 qı has logarithm map log = jou. The angular velocity is:
2 log(gre)) du.
(21)
Introducing an exponential scheduler <(t) = 1 e 7 with derivative x'(t) = 7e ye- the modified SLERP becomes:
q: no X exp (xr(1) log(qra))
(22)
Differentiating with respect to time using the chain rule:
qi qo exp (x(t) log((r)))
? rt log quel) @ qo & exp (M(() log(qe))
(23)
7 nt log((
(rel) @ qr-
Applying the quaternion kinematics equation q = 10, w J @ a (Sola, 2017), we solve for the effective angular velocity:
[0, [T = 24r @ qi
2ye log @ q: @ qi
(24)
>y log(gr))
Substituting the angular velocity from Eq. (21) yields:
w = ye "'.
(25)
A.2. Proofs of The Theorems in Section 3.3
Our proofs yield the same pipeline used in (Liu, 2022). The proofs are inspired by that work and derived based on the same techniques. What did is extending and specifying the theoretical results in (Liu, 2022) for $3, The original rotation process is {qe he Ij. where each is unit quaternion representating rotation in SO(3), w, E IR3 is the angular velocity at time f. The quaternion dynamics are given by
ク 10, w
& q: E T (8*),
(26)
where Ty. (S²) is the tangent space at We write go 2o. qi Q for the initial and target distributions. For a given input coupling (qo. q2), the exact minimum of (3o(3) in Eq. (11) is achieved if
W (g t) = E - q: = ql E R
(27)
which is the expected angular velocity at point q. time t. We now define the rectified process {dohetoo1 by
dl = 10 wr I < dí. d ~ Co.
(28)
2
12

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
1.2.1. PROOF OF THEOREMS 3.1
Proof. Consider any smooth function h 3 IR. By chain rule:
B [q:] E ((sgh)() ac)
(29)
where Vgah the gradient on the manifold. From the definition in Eq. (26), since we is random, we rewrite inside the expectation by conditioning on Zt:
E V q:) = vg - [(w)l))))))) @ }q)
(30)
because wr](q: = q) has conditional mean (g l),
d Elh(g)) ([Vgah(q:)) 11 (g; sq.
(31)
di
This evolution is exactly the weak (distributional))om) of the continuity equation:
O: Mr + V ( (q. ().a))))
(t) = 0,
(32)
where μ4, Law(gi (:).. According to Eq. (28). That exactly the same weak-form evolution equation satisfied by the di process, where simply replaced by Wr initial data Pn nlo- On compact manifold like SO(3), the continuity equation has unique solution given initial distribution. Hence at all times t. That is,
let v4 Law(g)). solves the same continuity equation with the same
Law ( = Law(q:) for all e [0. 1].
(33)
A.2.2. PROOF OF THEOREMS 3.2
Proof. The net rotation from qu to qı can be given by integrating the angular velocity we E RR3.
1 / w: dl,
log 10
1o =Qq)) -
(34)
and similarly,
log 0 - 9í) = Wr t) dt,
(35)
Strictly speaking, one must keep track of the axis direction to ensure consistency, but the geodesic assumption here handles that. The rectified angular velocity W. = E[ss [q.] implies that the total rotation in the rectified process a conditional expectation of the original rotation:
1 .( w. dt = F S wer dt (a6)
log (g) 3 q/)
(36)
2
2
Applying Jensen's 's inequality to the convex cost over this conditional expectation:
e (log (go - @ 4()) = ((2 [ wi dt (q:3)
(f wrat)
|163
E 2
(37)
Taking the total expectation on both sides:
1 2 (S, ur (erdt) [C((o. q1)]
E [C d')
(38)
This final inequality establishes that the rectified coupling (g6. gi) achieves equal or lower expected transport cost than the original coupling (qu. q.).
13

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
A.2.3. PROOF OF COROLLARY 3.3
Proof. Suppose the original process has the nonconstant angular velocity w: a(t)u (fixed axis), with T 2 So a(t)dt.
1 2 I/ wi dt = 2 2m
S. a(t) dt Tu
log (gg 83 (1)
(39)
Recall that the rectified angular velocity is:
(. [|w]]g]
(40)
Since wi = a(t)u, we simply get:
3,(g, 1) B[a(t))
(41)
The total rotation from d to the of in the rectified process satisfies:
1 {
2 ( Ela(t) q/)ctt
log(o6 -t 2 (i)
w.(d,,
(42)
2
Let = 2 J0 E[a(t) Al Thus,
log(g) 1- 2 411) r'u
(43)
Because T= So a(I)dl, +' So B[a(() q:]dl, and Eq. (36) in Theorem 3.2, we
r'u= = (L. Bia(t) |//dtt)
S a(t)ud dt { E[ru|{ghH
= 2 E
(44)
For the coupling (46- ), the cost is:
C(go= d)) c(r'u).
(45)
Since r'u = "ruu=B/rulleee convexity of e implies Jensen's inequality in conditional form:
c(r'u) = c(E(Tul( s Elc(ru))(4())
(46)
Next, take unconditional expectation on both sides. By the law of total expectation (tower property),
Bic(r'u)] - B[c(Tu)].
(47)
Since c(ru) c(log (qo @ q:)) C'go, 91) and c(r'u) C'a': df) Therefore,
E|C(d d) < B[C(go. 9:)).
(48)
B. Implementation Details
B.1. Ensuring The Shortest Geodesic Path on SO(3)
When we interpolate two quaternions by using SLERP in an exponential format (Eq, (8)), due to the double-cover property of quaternions (where every 3D rotation is represented by two antipodal unit quaternions), it is possible that the inner product (go ) < O, which means that qo and lie in opposite hemispheres. In such a situation, we apply -qi in Eq. (8), ensuring the shortest geodesic path on SO(3).
B.2. Auxiliary Loss
We adopt the auxiliary loss from (Yim et al., 2023b) to discourage physical violations such as chain breaks or steric clashes. Let A = [N, C., OJ be the collection of backbone atoms. The first term penalizes deviations in backbone atom coordinates:
N I|an a.
Lab
(49)
4N
#=
a
14

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
where an is the ground-truth atom position, On predicted position, N represents the number of residues. The second loss is a local neighborhood loss pairwise atomic distances,
Ldis
Σ
<0.6)]ldbb
dan 11?
(50)
Z
n,mm a.BEA
2
Σ
Σ
1b < <0.6) N,
(51)
1,m=1 O.EEAA
where dab = |a,n - -bmll and dmm ||ā, - bm.ll represent true and predicted inter-atomic distances between atoms a, b E A for residue n and m. is an indicator, sigmifying that only penalize atoms within 0. 0.6nm(6A). The full auxiliary loss can be written
Laux
Lbb + Ldis-
(52)
B.3. The Schemes of Training and Inference Algorithms
The schemes of our training and inference algorithms are shown below.
Algorithm Training Procedure of QFlow
Require: Training dataset TP {(T] = (x) (
P
model Mg, number of epochs N
1: Initialize model parameters 0 N do
2: for epoch
3: for each mini-batch T C TTP do
4: Sample 8 (0 1]. TØ. To x Qo
5: Interpolate translations: = Linear( af 15)
Eq. (7)
6:
Interpolate rotations: qe = SLERP-EXE Exp( qo qi
Eq. (8)
7:
Predict targets: 2².1- f0.1 = Mo (5))
8:
Compute loss C(0; R q. x 1
Eq. (12)
9:
Compute gradient vo
10:
Update parameters: + Vo
11: end for
12: end for
13: Return: Trained model parameters l
Algorithm Inference
Require: Trained model Mg, noise To To Qo. number of steps L, rotation acceleration constant ?
1: Initialize 0, At 1
2: for step = 1 to I. do
3: Predict targets: 20,1, 90,1 ()
4: Compute velocity: 80 Wo,t
Eq. (10) Eq. (13)
5: Update translations: d1+A1 f- It U8,1 At
6: Update rotations: 94+1 f- qi 2 exp
2 At . ye ()e
Eq. (15)
7: Update states: t + At, T, t T,
8: end for
9: Return: Generated backbone frame T
B.4. Data Statistics and Hyperparameter Settings
We follow (Yim et al., 2023b) to construct PDB dataset. The dataset was downloaded on December 17, 2024. We then applied length filter (60-512 residues) and a resolution filter (< 5 Á) to select high-quality structures. To further refine the
15

---

ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Algorithm 3 Training Procedure of ReQFlow
Require: Trained QFlow model Ma. number of epochs N
Sample noise T To x Qo
2: Create flow rectification pairs: (To: T))
Alg.
3: for epoch to N do
4: for each mini-batch (Th, T C (Ty: T2 do
5: Sample 1 0(0 1]
6: Interpolate translations: at = Linear a 1. 18)
Eq. (7)
7:
Interpolate rotations: q = SLERP-ENs Saap-Exx(ggg (0 91 ;B)
Eq. (8)
8:
13 Predict targets: a: q..1 Mo (T?: (5)
9:
Compute loss (0; x q. 2 98.1)
Eq. (12)
10:
Compute gradient VoL
11:
Update parameters: 0 t- 0 - V
12:
end for
end for
14: Return: Trained model parameters
PDB
SCOPe
250
2000
200
Count
Count
1000
100
500
0
0
200
400
500
70
80
110
120
Length
Length
Figure The length distribution of PDB and SCOPe dataset we use for training.
dataset, we processed each monomer using DSSP (Kabsch & Sander, 1983), removing those with more than 50% loops to ensure high secondary structure content. After filtering, 23,366 proteins remained for training. We directly use the SCOPe dataset preprocessed by (Yim et al., 2023a) for training, which consists of 3,673 proteins after filtering. The distribution of dataset length is shown on Figure
When conducting reflow, first generated large amount of data to create the training dataset and then applied filtering to refine it. The filtering criteria were as follows: for proteins with lengths < 400, we selected samples with scRMSD < 2; for proteins with lengths 2 400, we included samples with either scRMSD < or TM-score 2 0.9. We also remove those with more than 50% loop and those with max 4% radius gyration. For the PDB dataset, we generated 20 proteins for each length in {60, 61...., 512}. resulting in reflow dataset containing 7,653 sample-noise pairs. For the SCOPe dataset, we generated proteins for each length in (60, 61. 128}, producing reflow dataset with 3,167 sample-noise pairs.
B.5. Metrics
Following existing work(Geffner et al., 2025; Yim et al., 2023b;a; Bose et al., 2024; Huguet et al., 2024), we apply the metries below evaluate various methods.
16

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Table 5. Training Hyperparameters
Hyperparameters
Value
aux loss_r pass (time threshold)
PDB=0.5, SCOPe=0.25
aux loss loss_weghht
1.0
batch size
128
squared
TDB=:00000,SCPP==500
max epochs
1000
learning rate
0.0001
interpolant.min.. min.t
0.01
Designability. We use this metric evaluate whether protein backbone can be formed by folding amino acid chain. As shown in Figure 7, for each backbone, generate 8 sequences with ProteiMMPNN(Daupara. et al., 2022) at temperature 0.1, and predict their corresponding structures using ESMFold(Lin et al., 2023). Then we compute the minimum RMSD (known as scRMSD) between the predicted structures and the backbone sampled by the model. The designability score (denoted as "fraction" in this work) is the percentage of samples satisfying scRMSD <A.
Diversity. This metric quantifies the diversity of the generated backbones. This involves calculating the average pairwise structural similarity among designable samples, broken down by protein length. Specifically, for each length L under consideration, let Sy be the set of designable structures. We compute TM(b,, by) for all distinct pairs (b,, by) within SL. The mean of these TM-scores represents the diversity for length L. The final diversity score is the average of these means across all tested lengths L. Since TM-scores closer to indicate higher similarity, superior diversity is reflected by lower values of this aggregated score.
Novelty. We evaluate the structural novelty by finding the maximum TM-score between a generated structure and any structure the Protein Data Bank (PDB), using Fova Kempen al., 2022). lower resulting maximum TM-score signifies more novel structure. The ceefaer et al., 2025) utilized for this Foldseek search configured as follows:
foldseek easy-search <pdb_path> <database> <aln_file> <tmp_folder>
--alignment-type 1
--exhaustive-searcct
max-seqs 10000000000
tmacore-threshold 0.0
format-out query, query,target target, alntmscoore,dddt evalue
According to the issue of FoldSeek mentioned in htttpp///thhhoocccmmeeeeeeeeeees 323, we use the E-value column to report the TM-score.
Efficiency. To ensure fairness, we measure inference time on idle GPU and CPU systems. For PDB-based models, we sampled 50 proteins of length 300 and reported the mean sampling time. Similarly, for SCOPe-based models, we sampled 50 proteins of length 128 and reported the mean sampling time. File saving and self-consistency calculations were excluded from the timing.
B.6. Baselines
We compare our work with state-of-the-ar methods in the community, including Genie2, REdiffusion, FoldFlow/FoldFlow2, FrameFlow, and FrfmeDiff. We use the default checkpoints and parameters provided in these methods' repositories for comparisons.
C. More Experimental Details
C.1. Hyperparameter Settings
We adopt the the same hyperparameter settings as FrameFlow for a fair comparison, and the key parameters are shown in Table
17

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Table Computation time breakdown (in seconds) for different methods, datasets, and sampling steps. The time corresponds generating a backbone with length N = 128 for SCOPe and N 300 for PDB.
Datasets
Methods
Steps
Model Prediction Rotation Update
Translation Update
Total Time
FrameFlow
500
16.308
0.608 +0000
0.033 000000
17.053
50
1. 20013
0.0591000
0.003
1.727
20008
0.02441000
0.001 10,00
0.713
PDB
QFlow
500
16.732.10088
0.49221000
0.036_00000
17.370
50
1.670
0.048
0.003
1.776
20
0.653
0.019
0.001:0.000
0.726
FrameFlow
500
11.947 10.125
0.601 1:0000
0.033
100.24
50
1. 1.166
0.009
0.003
1.215
20
0.471 40002
0.025
0.001 20:000
0.53
SCOPe
QFlow
500
11. 20037
0.483
0.034_20.00
12.602
50
1. 16066
0.048
0.003.20.00
1.262
20
0.466
0.019
0.001 10,00
0.528
Compute Self-Consistency (TM-Score, scRMSD)
QFlow ReQFlow
Inverse Folding
(ProteinMPNNN
ENTIL
Prediction (ESMFold)
Structure
Noise
Backbone
Sequence
Backbone
Figure 7. Illustration the designability computation pipeline.
C.2. Checkpoint selection strategy
For each method, after observing loss convergence, select checkpoints based the metrics of the generated protein validation set. We choose the checkpoint where the valca_valid_peerentt > 0.99 and the proportions of secondary structures are closest to the dataset's average values.
C.3. Detailed Speed Comparison
As shown in Table 6, we record the runtime (second) on generating protein of length 300 in the PDB experiment and length 128 in the SCOPe experiment for a detailed speed comparison. The neural network feedforward computation is the main computational bottleneck. However, the quaternion operations are 15~20%% faster than rotation matrix- based operations (see the Rotation Update column).
C.4. Detailed Comparisons Based on SCOPe
Table 7 reports the mean and standard deviation of the results corresponding to Table 4. The checkpoints of ReFrameFlow and ReQFlow used here are selected from epochs 2 to of rectification training for fair comparison. The checkpoints of FrameFlow and QFlow used for rectification are from epoch 189 and epoch 195, respectively.
Table 8 presents comprehensive results from the SCOPe experiment using fine-grained step size corresponding Figure 5, Note that the chekponts of ReFrameFlow and ReQFlow here different from Table 4. We select checkpoints following the criteria in Appendix C.2. Even with generation process as concise as 10 steps, ReQFlow achieves designable fraction of 0.848. This highlights the efficiency and effectiveness of ReQFlow in generating feasible protein structures. Additionally, both QFlow and ReQFlow models produce proteins with reasonable secondary structure distributions, indicating their capability to generate structurally plausible proteins. These findings underscore the potential these models to significantly advance the field of protein design by balancing computational efficiency with structural accuracy.
18

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Table 7. Comparisons for various models on SCOPe. For each metric of generation quality, we indicate the best and top-3 results in the same way Tahle 2 does. The inference time corresponds generating backhone with length N 128. For each method, we evaluate the results on five checkpoints and compute their mean and standard deviation.
Efficiency
Designability Diversity
Novelty
Method
Step Time(s)
Fraction
TM
TM4
FrameFlow
500
12.69
0.85 10006
0.392
0.714
50
1.28
0.81 1 10017
0.378
0.6821.0044
20
0.54
0.708 20:003
0.370
0.649
ReFrameFllww
500
12.77
0.924
0.407 0.70910.04
50
1.26
0.906
0.407 1:000
0.68910000
0.52
0.884
0.405
0. 0.714:10.00
QFlow
500
12.60
0.893 100002
0.392 0. 1:.01
1.26
10.011
0.380
1.000
20
0.53
0.762
0.372 0.604
ReQFlow
500
12.52
0.947 1.00007
0.406
0.696-40.00
50
1,30
0.922++0:00
0.411 #0002
+.000
20
0.53
0.910
0.405
0.671 1.0005
C.5. Comparisons on Model Size and Training Data Size
The comparison of model size and training dataset size is listed in Table Model sizes in the table refer to the number of total parameter. FoldFlow2 utilizes a pre-trained model, thus having 672M parameters in total. The number of trainable parameters is 21M
C.6. Visualization Results
We use Mol Viewer (Schnal al., 2021) to visualize protein structures generated by different models, as shown in Figure 8 and Figure 9. In Figure 8, all proteins originate from the same noise initialization generated by QFlow, whereas in Figure the initialization is generated by FoldFlow. Each method follows its own denoising trajectory, leading to distinct structural outputs. FoldFlow2 adopts a default sampling step of 50, while all other methods use 500 steps. Due to architectural differences, the final structures vary across models, but within the same model, different sampling steps generally yield similar structures.
Among all models, ReQFlow exhibits the most stable and robust performance, maintaining low RMSD and variance across different sampling steps while demonstrating resilience to varying noise inputs. In contrast, other methods show significant limitations. FoldFlow-OT is highly sensitive initial noise, displaying drastically different performance in Figure 8 and Figure 9- evidenced by substantial variance across sampling steps when using QFlow noise.
Moreover, FoldFlow-OT tends to overproduce e-helices-coilee spiral-like stractures-resulting in high designability scores but deviating from realistic protein distributions. This pattern suggests a high risk of mode collapse, where the model predominantly learns a specific subset of protein structures, thereby lacking diversity and novelty in its predictions. Conversely, ReQFlow and QFlow generate higher proportion of f-strands, which appear as extended, ribbon-like structures, indicating closer alignment with natural protein distributions.
Furthermore, sampling steps decrease, most baseline models experience sharp deterioration in quality: RMSD values increase, rendering the structures non-desig.. In extreme cases, some samples exhibit severe fragmentation or disconnected backbones (e.g., the dashed regions in FoldFlow2 at 20 steps, Figure 8), highlighting instabilities in their sampling dynamics.
19

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Table 8, Unconditional protein backbone generation performance for 10 samples each length in {60, 61, ... 128). We report the metrics
from Section 4.1 we indicate the best and top-3 results the same way as Table does.
Designability
Diversity
Novelty
Sec. Struct.
Step
Fraction (t)
scRMSD (L)
())
TM (4)
Helix
Strand
Scope Dataset
0.330
0.260
FrameFlow
500
0.849
1.448
0.397
0.713
0.439
0.236
400
0.864
40390
0.390
0.713
0.452
0,229
300
0.861
1.422
0.389
0.715
0.449
0.230
200
0.842
1,496+11411
0.387
0.704
0.437
0.237
100
0.823
1.517
0.388
0.697
0.426
0.238
50
0.820
1.546
0.379
0.685
0.441
0.228
20
0.713
1.918 495
0.362
0.656
0.416
0,219
10
0.504
2.924
0.381
0.626
0.363
0.213
ReFramcFlow
500
0.897
1,368
0.403
0.700
0,501
0.187
400
0.893
1,328
0.405
0.698
0.489
0.202
300
0.888
1.313
0.405
0.697
0.485
0.199
200
0.907
1.326+47761
0.408
0.689
0.482
0.206
100
0.886
1.322 +000
0,410
0.690
0.499
0,201
50
0.903
1.291 40763
0.404
0.685
0.504
0.202
20
0.871
1 1.416 40.80
0.406
0.675
0.528
0.190
10
0.806
1.696410093
0.390
0.650
0.496
0.192
QFlow
500
0.907
1.263
0.389
0.712
0.498
0.214
400
0.907
1.199
0.394
0.711
0.476
0.223
300
0.910
1.243 1.007
0.393
0.710
0.503
0.209
200
0.877
1,309 4-2.00
0.389
0.714
0.481
0.224
100
0.903
1.283
0.390
0.702
0.476
0.225
50
0.872
1.389
0.371
0.674
0.491
0.206
20
0.764
1.764
0.367
0.646
0.492
0.192
0.565
2.589 42.216
0.374
0.614
0.467
0.167
ReQFlow
500
0.972
1.043 10.416
0.418
0.703
0.507
0.228
400
0.962
1.050
0.417
0.697
0.523
0.212
300
0.962
1076445188
0.421
0.702
0.498
0.233
200
0.948
1.0844050
0.407
0.696
0.513
0.218
100
0.933
1.123
0.425
0.695
0.514
0.310
50
0,932
1.162
0.422
0.693
0.491
0.237
20
0,929
1.21444063
0,409
0.670
0.514
0,307
10
0.848
1.546
0.416
0.662
0.518
0.195
Table 9. Model Sizes and Training Dataset Sizes
Model
Training Dataset Size
Model Size (M)
RFDiffusion
208K
Genie2
590K
15.7
FrameDiff
23K
16.7
FoldFlow( (aFToo/(BBBe,O
23K
17.5
FoldFlow2
160K
672
FrameFlow
23K
QFlow
23K
ReQFlow
23K+7K
When training ReQFlow, we first apply 23K samples PDB to train QFlow, and then we use additional 7K samples generated by QFlow in the flow rectification phase.
20

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
20 Steps
50 Steps
100 Steps
300 Steps
500 Steps
ReQFlow
RMSD=1.08
RMSD=1.02
RMSD=1.09
RMSD=1.02
RMSD=0.91
GRAD Caza
Creet
BOOOS
QFlow
RMSD=4.17
RMSD=1.57
RMSD=2.16
RMSD=1.14
RMSD=1.39
C00O EOLA
wwwm
FrameFlow
FORDDOS
RMSD=7.06
RMSD=6.01
RMSD-2.65
RMSD=1.25
RMSD=1.12
FrameDiff
0000
RMSD=12.71
RMSD=7.16
RMSD=9.11
RMSD=9.95
RMSD=15.53
MAMMNNA
FoldFlow-OT
RMSD=9.19
RMSD=1.61
RMSD=10.42
RMSD=5.81
RMSD=4.39
O
FoldFlow2
N
RMSD-2.18
RMSD=0.65
RMSD=0.60
RMSD=2.94
RMSD=3.41
Figure 8. Visualization of different methods on length 300. Sampling start with same noise generated by QFlow.
21

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
20 Steps
50 Steps
100 Steps
300 Steps
500 Steps
ReQFlow
RMSD=1.32
RMSD=1.56
RMSD=0.87
RMSD=0.79
RMSD=0.87
QFlow
RMSD=3.94
RMSD=1.66
RMSD=2.49
RMSD=4.16 4.16
RMSD=1.55
FrameFlow
AAAS
RMSD=12.73
RMSD=14.73
RMSD=8.54
RMSD=3.92
RMSD=0.92
FrameDiff
RMSD=6.03
RMSD=4.66
RMSD=4.01
RMSD=5.80
RMSD=3.08
FoldFlow-OT
RMSD=3.21
RMSD=1.61
RMSD=1.47
RMSD=2.33
RMSD=4.36
FoldFlow2
RMSD=1.48
RMSD=0.77
RMSD=0.64
RMSD=1.15
RMSD=3.59
Figure 9, Visualization of different methods on length 300. Sampling start with same noise generated by FoldFlow.
22

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ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protcin Backbone Generation
Table 10. Comparisions for various methods on their performance (Fraction Score) in long backbone generation. The lengths of the generated backbones range from 300 to 600. We generate 50 samples for each length. We bold the best result and show the top-3 results with blue background.
Length N
300
350
400
450
500
550
600
RFDiffusion
0.76
0.70
0.46
0,36
0.20
0.20
0.10
Genie2
0.86
0.90
0.74
0.58
0.28
0.12
0.10
FoldFlow2
0.96
0.88
0.70
0.56
0.60
0.26
0.16
FrameDiff
0.24
0.18
0.00
0.00
0.00
0.00
0.00
FoldFlow-OP
0.62
0.48
0.04
0.00
FrameFlow
0.72
0.74
0.48
0.28
0.24
0.10
0.00
QFlow
0.88
0.78
0.54
0.50
0.30
0.02
0.00
ReQFlow
0.98
0.96
0.78
0.76
0.70
0.56
0.10
23


Reviews:
[{"review_id": "ejRS3io6r5", "reviewer": "Reviewer_L7Bp", "summary": "This paper introduces a new flow matching method for unconditional protein backbone generation, based on quaternion representations and rectified flows.\nMore specifically in this work, the rotational part of the backbone residues is represented as unit quaternion, instead of an SO(3) matrix which is the canonical choice in most previous work. It is demonstrated that this choice improves the numerical stability of calculations, leading to improved backbone quality as well as a speed up of the inference process. \nSecondly the “reflow” method from rectified flows is applied to pretrained models, leading to straighter flow trajectories and further improving the quality of sampled backbones.\n\n## Update after rebuttal\n\nI thank the authors for their detailed response and the additional \nexperiments. The new results, in my view, enhance the clarity of the \npaper and further highlight its core contribution: a robust and \nnumerically stable framework for protein backbone design. I will thus \nincrease my score and recommend to accept the paper.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your positive and constructive comments. \n\n**Correction of long-chain results:** Before resolving your concerns, we would like to report that we found a bug in our script in the rebuttal phase and correct the results of ReQFlow on long-chain generation in the anonymous link https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/01_revised_longchain.pdf. **Note that, the new results are lower than the wrong ones in the submission, but they are still significantly better than those of baselines, and thus, do not affect our main claims and contributions.**  \n\nBelow we try to resolve your concerns one by one.\n\n**Q1. Hyperparameters of QFlow/ReQFlow**\n\n**A1.** We adopt the the same hyperparameter settings as FrameFlow for a fair comparison, and the key parameters are shown below:\n|Hyperparam.|Value|\n|-|-|\n|aux\\_loss\\_t\\_pass (time threshold)|PDB=0.5, SCOPe=0.25|\n|aux\\_loss\\_weight|1.0|\n|batch size|128|\n|max\\_num\\_res\\_squared|PDB=1000000, SCOPe=500000|\n|max epochs|1000|\n|learning rate|0.0001|\n|interpolant\\_min\\_t|0.01|\n\nThe batch size of FrameFlow is originally set to 128 for PDB and 100 for SCOPe. We retrained FrameFlow on SCOPe in our work and set the batch size to 128, the same as QFlow/ReQFlow. This doesn't influence the comparison's effectiveness and fairness.\n\nIn addition, the value \"0.6\" in Eqs. (50, 51) does not represent a time threshold but rather an inter-atomic distance threshold, with the unit nanometers (nm). \n\n**Q2. Clarification on SO(3) prior**\n\n**A2.** Following FrameFlow, we use an IGSO(3) prior for training and a uniform prior on SO(3) for inference. We apologize for the mistake on omitting the density. The text on the right side of lines 149-150 actually means \"uniformly sampling ... with the following density:\" \n$$\nf(\\omega) = \\frac{1 - \\cos\\omega}{\\pi} \\sum_{l=0}^{\\infty} (2l + 1) e^{-l(l+1)\\epsilon^2} \\frac{\\sin\\left(\\left(l + \\frac{1}{2}\\right)\\omega\\right)}{\\sin(\\omega/2)}.\n$$\nThis description is the same as Leach, Adam, et al.'s definition. We will clarify this point in the revised version.\n\n**Q3: Issue on Fold-seek whether affect novelty results.**\n \n**A3:** Thank you for pointing out this known issue (Foldseek Issue #323). We were aware of this issue where the `evalue` column reports the TM-score in TM-align mode (`--alignment-type 1`). To avoid this issue, as shown in our supplementary command, we requested the TM-score using the `--format-output ... alntmscore, ...` mode. Therefore, we utilized the correct `alntmscore` column for our analysis, ensuring our reported novelty results based on TM-scores are accurate and unaffected by this issue.\n\n**Q4. Why numerical stability on large ablges (close to $\\pi$) increases designability.**\n \n**A4.** As mentioned in Section 3.4, lines 258–260, during the inference, the probability of suffering at least one large angle ($\\geq \\pi - 10^{-2}$) per protein is 0.59 for PDB and 0.34 for SCOPe. Meanwhile, Figure 2(a) illustrates that when the angle is close to $\\pi$ ($\\geq \\pi - 10^{-2}$), the matrix’s mean round-trip error increases dramatically. Because this round-trip step (i.e., conversions between rotation angles and rotation matrices/quaternions) is a critical and high-frequent operation of interpolation during inference, the round-trip errors propagate and lead to significant orientation deviations in residues during inference.   \n\n**Q5. Why QFlow/ReQFlow speedup over FrameFlow**\n\n**A5.** We had an in-depth analysis of the code, and the speed improvements can be attributed to the following 3 components. \n\n**1) Fewer float-point operations:** When describing a 3D rotation, the matrix multiplication (27 mul., 18 add.) is intrinsically more computationally expensive than a quaternion multiplication (16 mul., 12 add.).\n\n**2) Fewer matrix-vector multiplications:** For each residue, matrix-based interpolation performs 3 matrix multiplication (i.e., computing relative rotation matrix, implementing the Rodrigues formula, and applying one rotation matrix to the initial matrix) while quaternion-based performs 2 quaternion multiplications (i.e., two Hamilton products).\n\n**3) Cheaper nonlinear operations:** The matrix-based log/exp maps require to handle numerical issues (although still failed on angles $\\geq \\pi - 10^{-2}$ ), e.g., using truncated Taylor approximation. In contrast, quaternions rely on simple mathematical operations, such as acos, sin, cos, and normalization (sqrt/division).\n\n**Q6. Ablation results on more checkpoints**\n\n**A6.** For the ablation results, we use 5 checkpoints to evaluate their statistical significance, demonstrating their consistency and stability.\n|Exponential Scheduler|Flow Rectification|Data Filtering|500|50|10|\n|-|-|-|-|-|-|\n|x|x|x|0.143±0.079|0.047±0.030|0.002±0.002|\n|✓|x|x|0.910±0.029|0.795±0.051|0.309±0.058|\n|✓|✓|x|0.612±0.084|0.519±0.154|0.385±0.136|\n|✓|✓|✓|0.969±0.027|0.932±0.022|0.698±0.041|\n\nWe hope the above answers help resolve your concerns and enhance your confidence to support our work.", "claims_and_evidence": "The claims about increased numerical stability and higher quality backbones are supported by sensible experiments and results.\nThe application of the reflow procedure on a curated set of training samples significantly increases designability while trading off diversity and novelty, as is expected.\nThe comparison of ReQFlow and ReFrameFlow indicates an increase in designability of approximately 8% and a 25% speedup of ReQFlow compared to Frameflow. Unfortunately no detailed summary of training hyperparameters is provided, which would allow for a better understanding of the differences between the two models. The time threshold of 0.6 for the auxiliary loss in eq. (50) and (51) indicates that the hyperparameters are not the same as in FrameFlow.", "methods_and_evaluation_criteria": "The comparison of numerical stability of matrix and quaternion implementations in terms of a round-trip error is appropriate, given that similar operations are performed when calculating the geodesics on SO(3). \nThe benchmark datasets SCOPe and the curated version of PDB are widely adopted in the field.\nThe performed experiments cover most interesting benchmarks for unconditional protein backbone generation models, including evaluations of designability, diversity, novelty and secondary structure content in various settings.", "theoretical_claims": "The theoretical claims of this paper are mainly concerned with repeating the proofs of Liu et al. for the case of quaternion algebra and seem correct.\n\nLiu, Xingchao, Chengyue Gong, and Qiang Liu. \"Flow straight and fast: Learning to generate and transfer data with rectified flow.\" arXiv preprint arXiv:2209.03003 (2022).", "experimental_designs_or_analyses": "The experimental design choices seem sensible and are in accordance with common choices in the field of unconditional protein backbone generation. In particular, the definition of the metrics designability, diversity and novelty as well as sampling steps and choices of backbone lengths to generate are the same as for multiple other baselines.", "relation_to_broader_scientific_literature": "The proposed method directly builds on established flow matching models for protein backbone generation in particular FrameFlow. The usage of reflow is a direct application of the ideas in Liu et al. The idea of using quaternion representations and SLERP in exponential format for handling the rotations of residue frames is novel and could be readily applied also to many other models in the field which work with frame representations for backbone residues.\n\nLiu, Xingchao, Chengyue Gong, and Qiang Liu. \"Flow straight and fast: Learning to generate and transfer data with rectified flow.\" arXiv preprint arXiv:2209.03003 (2022).", "essential_references_not_discussed": "All important references are included.", "other_strengths_and_weaknesses": "Weakness:\nThe paper claims that the IGSO(3) prior corresponds to uniformly sampling rotation axis and rotation angle, which is not the case (see e.g. discussion in Leach et al. and Yim et al.). Crucially many other baselines use the IGSO(3) prior during training and a uniform prior on SO(3) during inference. Clarification on what is implemented for QFlow would be desirable.\n\nLeach, Adam, et al. \"Denoising diffusion probabilistic models on so (3) for rotational alignment.\" (2022).\nYim, Jason, et al. \"Fast protein backbone generation with se (3) flow matching.\" arXiv preprint arXiv:2310.05297 (2023).", "other_comments_or_suggestions": "For the computation of novelty, Fold-seek was used. Fold-seek has an issue where the TM-Score is provided in the wrong column of the output. The command provided in the supplementary suggests that this error might affect the novelty results reported in the tables? (see https://github.com/steineggerlab/foldseek/issues/323)", "questions_for_authors": "1.\tCould you please provide a more detailed list of training hyperparameters and indicate if and how the QFlow models are different from FrameFlow? In the case of equal hyperparameters could you explain how the increased numerical stability for rotation angles close to π can lead to such a significant increase in designability?\n2.\tThe reported 25% speedup over FrameFlow seems quite large. Could you provide more details on the specific computational bottlenecks that QFlow optimizes and the corresponding speedup for each of these individual components?\n3.\tFor the ablation results in Table 4 I would like to see results on more than one checkpoint to evaluate the statistical significance of the reported metrics. This would also allow for a better comparison between the models.\nI am willing to raise my score if the above points are addressed in the rebuttal.", "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": "ro31IXipRi", "reviewer": "Reviewer_TTp8", "summary": "The paper proposes a method to train a generative model of protein backbones. They follow previous work in parameterizing protein backbones using a translation and a rotation. They have two main contributions:\n\n- The use of normalized quaternions to parameterize rotations (most previous work use 3x3 rotation matrices)\n\n- The application of the ReFlow process to enable fast sampling through additional training on synthetic data. The paper also extends the theoretical results from ReFlow (diminishes transport cost).\n\nThe empirical evaluation shows promising results for the proposed method.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your positive feedback and constructive comments.\n\nBefore resolving technical concerns, \n\n**1) Apology for reusing sentences in Proteina.** We sincerely apologize for reusing sentences from Proteina in Appendices B.5 and B.6 without proper citation. We read this paper when it was under review anonymously. We did not mention it in the related work because we focus on the tech route using frame-based representation + flow-based generative learning, while Proteina applies a very different route and did not release code at that time. In the appendix, originally, we had a sentence in the beginning of B.5, saying we follow existing evaluation methods, with the citations of Frameflow, FoldFlow, Genie2, and Proteina. However, it is accidentally commented when we adjusted the layout of Figure 6 and B.5. We thank the reviewers for pointing out our oversight. \n\nWe take this matter seriously. **In the revised paper, we will 1) rewrite B.5 and B.6 completely in a different logic flow; 2) mention Proteina in the related work and introduce its tech route briefly; and 3) take the comparison with Proteina as our future work in the conclusion.** \n\n**2) Correction of long-chain results:** We found a bug in our script in the rebuttal phase and correct the results of ReQFlow on long-chain generation are in https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/01_revised_longchain.pdf. **Note that, the new results are lower than the wrong ones in the submission, but they are still significantly better than those of baselines. This correction does not affect our main claims and contributions.**\n\nBelow we try to resolve your technical concerns one by one.\n\n**Q1. More metrics on long chain experiment**\n\n**A1:** As shown in https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/03_long_chain_metrics.pdf, both QFlow and ReQFlow outperform FrameFlow on designability, though their diversity and novelty metrics drop. \n\nWe think this is reasonable because compared with FrameFlow, both QFlow and ReQFlow generate more designable proteins. There are more proteins used to compute novelty and diversity, which may lead to higher TM-scores because of the inherent trade-off among the metrics.\n\nIn addition, currently, ReQFlow is not as good as larger models because 1) our training utilized only 23k samples, a small subset of the PDB, far fewer than Genie2 (590k), FoldFlow2 (160k), and RFDiffusion (>208k); 2) our model has much fewer parameters than RFDiffusion and FoldFlow2. \n\nTraining a larger QFlow/ReQFlow model on a larger dataset can be one of our future work, but it is out of the scope of this work.\n\n**Q2. The reason for the stability of using quaternion.**\n\n**A2:**  When angle is close to $\\pi$，the quaternion logarithm is simply `log(q) = (π/2)u`. This calculation is direct and numerically stable; there's no division by a small number.\n\nWhen angle is close to $0$, quaternion logarithm `log(q)` also faces potential instability. It is handled by clamping the divisor much simpler than the Taylor expansion.\n\n**Q3. A larger synthetic dataset for rectifying flow**\n\n**A3:** We use 17.6k synthetic data to train ReQFlow. However, using a larger dataset does not affect results significantly. As shown in **A1**, this is due to 1) a trade-off among the metrics and 2) the limited model size and data we used.\n\n||Steps|Designability|Diversity|Novelty|\n|-|-|-|-|-|\n|Original: 7.7k|500|0.972|0.377|0.828|\n||50|0.912|0.369|0.810|\n||10| 0.676|0.337|0.760|\n|Larger: 17.6k|500|0.968|0.381|0.832|\n||50|0.932|0.379|0.825|\n||10|0.724|0.360|0.793|\n\n**Q4. The rationality and usefulness of ReFlow on designable proteins.**\n\n**A4 (Short answer):** Selecting designable proteins matters, but the self-distillation of QFlow on such proteins may suffer mode collapse. ReQFlow avoids this issue to some extent and maintains competitive results. Please refer to our response **A2 to Reviewer K7Q3** for detailed answer. https://openreview.net/forum?id=f375uEmYDf&noteId=40R3BVTqCr\n\n**Q5. The computation of the average of pairwise tm-scores**\n\n**A5:** The score is first done per length, and then averaged across lengths. We will revise our paper to make it clear.\n\n**Q6. Setting of baselines.**\n\n**A6:** All the baselines are evaluated using the default settings in their repos. For Genie2, we use the default setting (noise scale=0.6). We will revise our paper to make it clear.\n\n**Q7. Explain the power of exponential step scheduler.**\n\n**A7 (Short answer):** **current method models velocity field backward**, making the prediction at the starting point difficult and inaccurate. Exponential scheduler allows models to approach the ending point with few steps, avoiding severe error accumulations near the starting point. Please refer to our response **A4 to Reviewer K7Q3** for more details. https://openreview.net/forum?id=f375uEmYDf&noteId=40R3BVTqCr\n\nIn summary, we hope our responses can resolve your concerns and enhance your confidence to further support our work.", "claims_and_evidence": "Partly. As I state in the sections below, I think a more careful discussion of the results (including all metrics), would be useful, and an extended analysis of the numerical stability of using unit quaternions (vs 3x3 rotation matrices) would strengthen the paper.", "methods_and_evaluation_criteria": "The evaluation criteria follows standard practices in unconditional protein backbone design, training on a few subsets of the PDB, and evaluating widely used metrics such as designability, diversity, novelty, and secondary structure content.", "theoretical_claims": "The theoretical claims in the paper are extensions from the ReFlow [1] results (keeps correct marginals, reduces transport cost). I did not check the proof in detail.\n\n[1] “Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow”, by Liu et al.", "experimental_designs_or_analyses": "As mentioned above, the metrics reported in the tables are very reasonable and widely used in the literature. The authors, however, appear to heavily rely on the designability metric for most of its analyses and claims in the text, which could be problematic. Most methods (including, for instance, Genie2) are able to achieve very high designabilities by changing the temperature parameter during sampling. However, there’s usually a tradeoff between designability and diversity, and reducing sampling temperature often leads to reduced diversity. While the tables include several metrics (designability, diversity, novelty), it would be good for the text to discuss the results for all metrics more. For instance, line 375 (left column) claims that ReQFlow outperforms Genie2 and other baselines, but the comparison is only done in terms of designability. Taking the same lines in the table referred to in the text, it can be seen that ReQFlow, for that configuration, achieves a worse diversity and novelty than Genie2. Having a more comprehensive analysis, commenting on these tradeoffs, and all metrics, would be good.\n\nAnother example, table 7 in the appendix only includes designability. Adding other metrics to get a more holistic view of the methods’ performance would be informative.", "relation_to_broader_scientific_literature": "As I mentioned above, the paper has two main contributions.\n\n(1) Using unit quaternions to represent rotations in protein backbone design.\n(2) Using the ReFlow procedure to rectify the flow and get better generation with fewer sampling steps.\n\nRegarding (1). This idea attempts to improve backbone design methods that represent residues as a translation+rotation. However, instead of representing rotations using 3x3 rotation matrices, the paper proposes to use unit quaternions, claiming they provide benefits in terms of stability. I consider this to be the main reason why (1) is a relevant contribution (but please correct me if I’m wrong). This being said, I feel section 3.4, which comments on stability and does a small empirical study for both representations, could be a lot more detailed. The paper “Exploring SO(3) logarithmic map: degeneracies and derivatives” by Nurlanov discussed in quite some detail the instabilities of the log map when dealing with 3x3 rotation matrices. In certain cases, a Taylor approximation can be used to avoid instabilities (eqs 9a and 9b in that paper). In other cases, due to numerical reasons, when the rotation angle is close to pi, things can only be determined up to a global sign (see paragraphs right before section 2.2). Which, if any, of these instabilities are being addressed by using unit quaternions? I think providing more details about this would be informative in understanding exactly which instabilities are being addressed, and how / why. Also, would be interested to have a clear discussion of which instabilities are left when working with unit quaternions, to fully understand how the two methods compare.\n\nRegarding (2). I see this contribution as orthogonal to the method introduced, as most existing methods for protein backbone design could potentially benefit from ReFlow. However, to my knowledge this is the first paper to explore the use of ReFlow in this domain, showing promising results. However, doing ReFlow appears to affect the method’s performance (see “Other strengths and weaknesses” section below) quite a bit, not always positively.", "essential_references_not_discussed": "Most relevant references are included. However, please see the section “Other comments and suggestions” for some issues connected to an existing paper.", "other_strengths_and_weaknesses": "One weakness that came up when looking at the results in some detail, about the application of ReFlow. Using ReFlow requires creating a synthetic dataset, where each sample is obtained by fully simulating a pre-trained flow model.\n\n(w1) The synthetic datasets created are filtered to keep only designable samples. Without this filter performance of the ReFlow-ed model takes a bit of a hit. Can the authors explain why this may be the case? Under perfect ReFlow marginals should be preserved? Is the important component here the ReFlow or the designability filter?\n\n(w2) Generating the ReFlow synthetic dataset is quite expensive. In fact, the datasets created in the paper are somewhat small, consisting of roughly 5k samples. After ReFlow, designability goes up (this is likelt thanks to the self-distillation effect achieved by filtering the synthetic data for only designable samples), but diversity and novelty get worse. The authors don’t really comment on this. Could this be addressed by generating a larger synthetic dataset?", "other_comments_or_suggestions": "There is some discrepancy between the description of diversity in the main paper and in the appendix. The main paper states that the pairwise tm-scores are averaged, while the appendix states that this is done per length, and then averaged across lengths. Which one is used for the results shown in the paper?\n\nAdditionally, Sections B.5 and B.6 in the Appendix appear to be copied from the paper “Proteina: Scaling Flow-based Protein Structure Generative Models”. While there are small changes (a few words here and there), there are entire paragraphs that are almost an exact copy. Since this is just describing metrics and baselines, and not a core part of the method, I don’t consider this to be very problematic. But copy / pasting entire paragraphs, especially without citing / mentioning the source (!) (the original paper, which also deals with backbone design using flow matching, is not even mentioned), seems in general unacceptable, even more so for an ICML submission.\n\nOn this line, the Genie2 description in B.6 states that the noise scale was set to 1 for full temperature sampling. However, this full temperature sampling is not included in any of the results shown in the paper.", "questions_for_authors": "Reading other papers, such as FrameFlow and FoldFlow, I was always curious about the gamma parameter to control the rotation generation speed. Do you have anh intuitions why this works so well? Most methods, including the one presented in the paper, perform quite poorly without it.", "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": {"ethical_review_flag": "Flag this paper for an ethics review.", "ethics_expertise_needed": ["Other expertise"]}}, {"review_id": "bpmT1yWlGC", "reviewer": "Reviewer_K7Q3", "summary": "This paper focuses on the task of protein backbone generation. It proposes quaternion flow (QFlow) and rectified quaternion flow (ReQFlow) for generative modeling on a translation/rotation manifold. In particular, in contrast to previous work, QFlow models the rotations with quaternions and the authors introduce quaternion flow matching. In contrast to previous works, the quaternion formulation avoids numerical instabilities and is an elegant way to describe rotations. The authors then use the rectified flow method to rectify and accelerate their models, making it possible to sample with relatively few steps. The paper then applies the method to protein backbone generation, where each protein residue is represented by its backbone coordinate and a rotation on SO(3). The paper generally achieves appealing results, slightly outperforming or being on-par with previous methods on standard benchmarks and evaluation metrics.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your positive feedback and constructive comments.\n\n**Correction of long-chain results:** Before resolving your concerns, we would like to report that we found a bug in our script in the rebuttal phase and correct the results of ReQFlow on long-chain generation are in https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/01_revised_longchain.pdf. **Note that, the new results are lower than the wrong ones in the submission, but they are still significantly better than those of baselines on designability and efficiency and comparable on novelty and diversity. This correction does not affect our main claims and contributions.**\n\nBelow we try to resolve your concerns one by one:\n\n**Q1. Trade-off among different metrics and the definition of SOTA performance**\n\n**A1.** In our work, the \"SOTA performance\" means that **ReQFlow achieves higher designability and computational efficiency than the baselines and obtains comparable diversity and novelty**. We emphasize on designability because novelty and diversity are computed based on designable proteins. We can obtain very low TM-scores even if the protein backbones are random noise, which, however, is meaningless.\n \nTo be clear, it does not mean that we do not care about novelty and diversity. We have offered experimental results for all metrics in Tables 2 and 4. In addition, although the diversity score degrades slightly after rectifying QFlow, the data distribution of ReQFlow in Figure 3 is similar to that of PDB. It means that ReQFlow has a low risk of mode collapse and achieves reasonable diversity.\n\nWe will claim the trade-off among the metrics and eliminate any potential misleading content in the revised paper.\n\n**Q2. The significance of Reflow and designable samples** \n\n**A2.** In our opinion, both Reflow and designable samples are important. \n\nApplying Reflow without selecting designable samples means fine-tuning the model under the supervision of noisy samples, which naturally leads to performance degradation (see Table 3). **However, it does not mean that Reflow is useless.** To verify our claim, we fine-tune a trained QFlow on generated designable samples, leading to \"self-distill QFlow\". The comparison between the self-distill QFlow and ReqFlow is in https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/02_reflow_vs_selfdistill.pdf. Although self-distill QFlow obtains competitive results (comparable designability, slightly worse novelty and diversity), it changes the data distribution and suffers a high risk of mode collapse. \n\nIn summary, our contributions include 1) first introducing quaternion algebra into flow-matching and protein design; and 2) making the first attempt to apply Reflow to protein design. For the second contribution, it is a potential technical route. Even if it is not consistently better than self-distillation, it does not mean it is not worth to explore. \n\nSelf-distillation v.s. ReFlow, which one is better? It is interesting and can be our future work, but it is out of the scope of this work.\n\n**Q3. Conditional generation.**\n\n**A3.** Existing methods (FrameFlow, FoldFlow, etc.) focus on unconditioned backbone generation and design experiments accordingly. We just follow the same setting for a fair comparison.\n\nNevertheless, we follow your suggestion, training QFlow and FrameFlow on SCOPe like [1] did. We select three motifs (4JHW, 5IUS and 1PRW) and report the success rate (%) of generation (i.e., scRMSD≤2A, motifRMSD≤1A). The results show that QFlow is at least comparable to FrameFlow.\n\n||4JHW|5IUS|1PRW|\n|-|-|-|-|\n|FrameFlow|4|76|99|\n|QFlow|8|76|99|\n\n[1] Improved motif-scaffolding with SE(3) flow matching\n\n**Q4. Why does exponential step scheduler matter?**\n\n**A4.** Given a trained model, at the early stage of inference, i.e., $t\\in [0, 0.25]$, the loss is significantly higher than that near the endpoint (i.e., $t\\in [0.75, 1]$), as shown in https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/rot_loss.pdf. It means that **the vector field is not well-learned in the early phase.** The reason for this phenomenon might be **the models like FoldFlow and FrameFlow learn the velocity field backward.** This strategy makes the predictions far from the endpoint challenging.\n\nExponential scheduler allows the model to rapidly approach the endpoint with few steps, reducing error accumulation caused by the imprecise samples at the early stage. \n\n**Q5. Joint interpolation time? and acceleration in training.**\n\n**A5.** Yes, we train the model with one joint interpolation time. As noted in the Eq.(7) of the FrameFlow paper, they attempted to accelerate the training phase as well, but this made training too easy with little learning happening. This aligns with our observations in **A4**: accelerating training further exacerbates the difficulty of learning the velocity field at the starting point.\n\nIn summary, we hope the above responses can resolve your concerns and help enhance your confidence to raise your score.", "claims_and_evidence": "The authors claim that the proposed method, i.e. the quaternion flow matching and rectification, improves protein designability, accelerates generation, and overall achieves state-of-the-art performance. While the proposed method generally shows good performance, I think the \"state-of-the-art\" comment might be exaggerated.\n\n- Generally, it is well-known that there is a trade-off between designability and diversity and novelty, and different methods achieve different trade-offs along this Pareto frontier. The state-of-the-art claim would be appropriate if the method clearly achieved both the best designability *and* diversity/novelty in the same setting, but this is not the case. After all, in theory, we can achieve 100% designability trivially by always outputting the same designable protein, in the extreme case.\n  - We see that QFlow gets best diversity in Table 2, but not designability.\n  - ReQFlow gets very good designability (in part potentially due to designable protein filtering, see comment below), but shows reduced diversity. A similar trend is also visible in Table 3.\n  - Hence, it would be more appropriate to claim that the method performs \"on-par\" with existing works. \n- Moreover, ReQFlow, in fact, not only does rectification but also blends in a distillation procedure where the rectification is only carried out on designable high-quality proteins. The fact that ReQFlow without this does not work well is concerning. Also, none of the baselines is trained on 100% designable protein backbones, but these baselines would potentially benefit from training on purely designable proteins, too. This makes the comparisons and claims somewhat questionable.", "methods_and_evaluation_criteria": "In general, all proposed methods and evaluation criteria do make sense. My more specific concerns are outlined above.", "theoretical_claims": "The theoretical claims and proofs in the appendix all seem correct to me. These are mostly generalizations of the derivations done in the rectified flow paper to SO(3).", "experimental_designs_or_analyses": "I do not have any concerns regarding the general experimental design or analyses and no reason to believe that any experiment should be incorrect. My concerns are described above.", "relation_to_broader_scientific_literature": "The paper's key contributions are appropriately positioned with respect to the broader literature. In particular, previous works that conduct flow matching on rotation manifolds are cited and discussed, and works that leverage similar quaternion formulations in other machine learning areas are also cited and discussed. The most relevant related works in protein generative modeling are also discussed and cited.", "essential_references_not_discussed": "I cannot identify any essential references that are not discussed.", "other_strengths_and_weaknesses": "**Strengths:**\nThe paper is generally well-written and well-presented and easy to read. The proposed quaternion flow matching is novel and original, and successfully validated. I think the proposed quaternion approach makes sense to model rotations in this setting. As the authors pointed out, similar methods have been used elsewhere in other domains.\n\n**Weaknesses:**\n- As mentioned above, I think the paper is a bit \"overclaiming\" when saying it gets state-of-the-art results.\n- As discussed above, the ReQFlow method suffers from reduced diversity. There are trade-offs between designability and diversity.\n- The paper focuses on simple protein backbone generation and, as discussed, performs approximately on-par with similar work overall, or slightly better when looking at individual metrics like designability only. In practical settings, protein generation is almost always carried out in a conditional setting. For instance, there is a target protein, for which we want to generate a binding protein, or there is a motif, and we want to generate the scaffold. Unfortunately, the paper does not study any such tasks.\n\nOverall, I think the proposed method can be broadly useful to the community for modeling proteins with frame-based representations, including rotations on SO(3). While I do have criticisms, I do not see any fundamental flaws and hence I am generally leaning towards suggesting acceptance. I would be willing to raise my score for a rebuttal addressing my questions.", "other_comments_or_suggestions": "I do not have any further comments or suggestions.", "questions_for_authors": "1. The exponential step size scheduler for generating the rotations plays a critical role. This is well-known and similar to previous works. Nonetheless, why exactly do you think that this is the case? \n\n2. Related, did you train the model with one joint interpolation time $t$ for translations and rotations, or did you sample separate independent $t$ for rotations and translations during training?\n\n3. If you used one joint interpolation time $t$, did you consider not only running inference with the accelerated rotation schedule but to also train the model with the accelerated schedule (accelerated relative to the translations)?", "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": "nSGOyrwAO7", "reviewer": "Reviewer_oNdZ", "summary": "The paper introduces ​ReQFlow, a novel method for protein backbone generation. To ​address numerical instability in matrix-based representations, the main innovation is to use quaternions to model the rotations and spherical linear interpolation in the flow matching training. The paper also extends the rectified flow techniques to quaternion space. The resulting model achieves more than 30x speedup over RFDiffusion/Genie2 while maintaining high designability.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your comments.\n\n**Correction of long-chain results:** Before resolving your concerns, we would like to report that we found a bug in our script in the rebuttal phase and correct the results of ReQFlow on long-chain generation in the anonymous link https://anonymous.4open.science/r/6342_ReQFlow_Rebuttal-62BB/01_revised_longchain.pdf. **Note that, the new results are lower than the wrong ones in the submission, but they are still significantly better than those of baselines, and thus, do not affect our main claims and contributions.**\n\nBelow we try to resolve your concerns one by one:\n\n**Q1. The novelty of our work, especially the theoretical part.**\n\n**A1.** As we claimed in the supplementary file, our theoretical result is a natural extension of Rectified Flow [1] and its proof applies the same pipeline. However, we indeed makes the first attempt to extend the theoretical result in [1] to the SO(3) scenario. In addition, technically, our approach is novel for the following reasons:\n\nFirst, we innovatively replace rotation matrices with quaternions. Thanks to the quaternion representations, our QFlow significantly enhances designability compared to its counterpart FrameFlow, without sacrificing diversity and novelty (see Tables 2 and 4). Moreover, QFlow exhibits superior efficiency, achieving approximately 10% and 25% speedups on the PDB and SCOPe datasets, respectively (see lines 358-370, Tables 2 and 4).  \n\nSecond, although the rectified flow technique was proposed in [1], we pioneeringly apply it to the protein generation and provide theoretical guarantees on SO(3). As demonstrated in the table below, ReQFlow markedly improves efficiency while maintaining the other three metrics on par with other advanced models.\n\n||training set|model size|steps|time (s)|designability|diversity|novelty|\n|-|-|-|-|-|-|-|-|\n|Genie2|590k|15.7|1000|112.93|0.908|0.370|0.759|\n|RFDiffusion|>208k|59.8|50|66.23|0.904|0.382|0.822|\n|FoldFlow2|160k|672|50|6.35|0.952|0.373|0.813|\n|ReQFlow|30k|16.7|50|1.81|0.912|0.369|0.810|\n\n[1] Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow\n\n**Q2. More experimental details**\n\n**A2.** We adopt the the same hyperparameter settings as FrameFlow for a fair comparison, and the key parameters are shown below:\n|Hyperparameters|Values|\n|-|-|\n|aux\\_loss\\_t\\_pass (time threshold)|PDB=0.5, SCOPe=0.25|\n|aux\\_loss\\_weight|1.0|\n|batch size|128|\n|max\\_num\\_res\\_squared|PDB=1000000, SCOPe=500000|\n|max epochs|1000|\n|learning rate|0.0001|\n|interpolant\\_min\\_t|0.01|\n\nNote that the batch size of FrameFlow is originally set to 128 for PDB and 100 for SCOPe. We retrained FrameFlow on SCOPe in our work and set the batch size to 128, the same as QFlow/ReQFlow. This doesn't influence the comparison's effectiveness and fairness. \n\nAs for when to stop training, we found that the loss converges after 600 epochs on the PDB dataset and 200 epochs on the SCOPe dataset.\n\n**Q3. Improve the diversity of training dataset by sampling**\n\n**A3.** For a fair comparison, we set our training dataset and training pipeline to be the same with those of the baseline models (e.g., FrameFlow, FoldFlow, and FrameDiff), so that the performance of various methods is achieved under the same setting and comparable. To our knowledge, existing methods, including the baselines mentioned above, train their models on the raw PDB dataset. Because the experimental setting is same, so the superiority of our method can be attributed to our technical contributions clearly. \n\nIn addition, as shown in Figure 3, the distribution of generated proteins obtained by our method is comparable to that of PDB, which does not suffer severe mode collapse issue as FoldFlow does. The diversity and novelty scores shown in Tables 2 and 4 also indicate that the proteins generated by our method have reasonable diversity and novelty.\n\nSampling PDB data may mitigate the inductive bias in the dataset and further improve our model performance, but it is out of the scope of our work at the current stage.\n\nIn summary, we hope the above responses can resolve your concerns and help you re-evaluate our work. We would appreciate it if you consider raising your score based on our reply. Feel free to contact us if you have any other questions.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes, for theorems 3.1 and 3.2, they are very similar to the results in the original rectified flow paper (https://arxiv.org/abs/2209.03003 theorems 3.3 and 3.5).", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "ReQFlow builds upon and advances several prior works in protein generation, flow-based models, and quaternion representations.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "1. The theoretical part of the paper is substantial, but the innovation is a bit limited.\n2. Some parts of the experiments are not clear. e.g., when to stop training?\n3. The training dataset can be improved. PDB data biases the model toward naturally abundant folds. Or sampling strategies to ensure structure diversity can be applied.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This submission proposes ReQFlow, a new flow matching framework for protein backbone generation. ReQFlow uses unit quaternions to represent 3D rotations, which improves numerical stability and enables faster computation. It also incorporates rectified flows to enhance sampling quality and efficiency.

The reviewers have recognized the significance of the paper’s contributions and acknowledged the strong empirical results. Reviewer TTp8 noted that a couple of paragraphs describing the evaluation metrics were copied from prior work. I agree with the reviewer that this issue is not substantial enough to warrant rejection, but I strongly encourage the authors to revise this in the final version or give propert citation to the relevant work. 

As pointed out by reviewers TTp8 and K7Q3, the authors should soften their claims from "state-of-the-art" to "on-par" and include a discussion or visualization of the trade-offs between designability, novelty, and diversity.

Overall, the strengths of this submission outweigh its weaknesses, and I am happy to recommend acceptance.
Decision: Accept (poster)