Rationalized All-Atom Protein Design with Unified Multi-Modal Bayesian Flow
Authors: Hanlin Wu
Yuxuan Song
Zhe Zhang
Zhilong Zhang
Hao Zhou
Wei-Ying Ma
Jingjing Liu
Venue: NeurIPS 2025 poster ()
Keywords: Bayesian Flow Networks
Protein Generation
Primary area: machine_learning_for_sciences

Abstract: Designing functional proteins is a critical yet challenging problem due to the intricate interplay between backbone structures, sequences, and side-chains. Current approaches often decompose protein design into separate tasks, which can lead to accumulated errors, while recent efforts increasingly focus on all-atom protein design. However, we observe that existing all-atom generation approaches suffering from an information shortcut issue, where models inadvertently infer sequences from side-chain information, compromising their ability to accurately learn sequence distributions. To address this, we introduce a novel rationalized information flow strategy to eliminate the information shortcut. Furthermore, motivated by the advantages of Bayesian flows over differential equation–based methods, we propose the first Bayesian flow formulation for protein backbone orientations by recasting orientation modeling as an equivalent hyperspherical generation problem with antipodal symmetry. To validate, our method delivers consistently exceptional performance in both peptide and antibody design tasks.

Rationalized All-Atom Protein Design with Unified Multi-Modal Bayesian Flow
Hanlin Wu Yuxuan Song!.2 Zhe Zhangl. Zhilong Zhang'
Hao Zhou 1sa Wei- Ma' Jingjing Liu'
Institute of AI Industry Research (AIR), Tsinghua University
Dept. of Comp. Sci. & Tech., Tsinghua University
wuh124@mails tsinghua. edu.
{songyuxuan, 2aongyuuuan,zhhuhua zhouhao, i jjliiljllallle Sairr. tsinghua. edu. cn
Abstract
Designing functional proteins is critical yet challenging problem due to the intri- cate interplay between backbone structures, sequences, and side-chains. Current ap- proaches often decompose protein design into separate tasks, which can lead ac- cumulated errors, while recent efforts increasingly focus on all-atom protein design. However, we observe that existing all-atom generation approaches suffering from an information shortcut issue, where models inadvertently infer sequences from side-chain information, compromising their ability to accurately learn sequence distributions. To address this, we introduce novel rationalized information flow strategy to eliminate the information shortcut. Furthermore, motivated by the advan- tages of Bayesian flows over differential equation based methods, propose the first Bayesian flow formulation for protein backbone orientations by recasting orien- tation modeling
equivalent hyperspherical generation problem with antipodal symmetry. To validate, our method delivers consistently exceptional performance in both peptide and antibody design tasks. Our code, checkpoint, and designed PDBs can be found in https: / thub. com/GenSI- Bte///GSSI-TTUATRRRRO THUATR PProaaayrr...B.s....
Introduction
Proteins are fundamental to life, carrying out essential functions in biological processes [Clark and Pazdernik, 2012, Whitford, 2013]. Designing functional protein stands as grand scientific challenge with great potential to modulate life processes Notin et al. [2024]. Prevalent approaches to protein design often decompose this task into set of more manageable sub-problems, i.e. generating partial protein components (e.g. backbone), and subsequently designing the remaining components (e.g. sequence and side-chain) conditioned on those generated sub-units [Watson al., 2023, Yim et al., 2023a, Bose et al., 2023, Lin and AlQuraishi, 2023].
However, in such loosely connected pipeline, each compartment can only capture partial protein component, which may lead to an incongruous whole. Furthermore, the three-dimensional structure of protein simultancously determined by both the backbone and the side chain [Schulz and Schirmer, 2013] while each model in pipelines can only capture and design partial protein component. Moreover, early fixed inaceurate partial components accumulate and propagate errors downstream [Kong et al., 2023]. These issues all call for a unified end-to-end all-atom protein design model.
Understandably, achieving all-atom modeling is inherently challenging, due dependency between the sequence and the coordinates (namely, the number and arrangement of side-chain atoms vary with the sequence. e.g. Lysine residue has four side-chain rigid groups while the Valine residue only has
Correspondencce Hao Zhou aatouhao@airtsinggheee
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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one, see Fig. 1). We observe that existing all-atom modeling practices [Li al., 2024, Chu et al., 2024, Kong et al., 2023, Zhu et al., 2024] suffer from an information shorteut issue: when the network is trained to jointly denoise both the sequence and the side chain, the network can infer sequence from even noised side-chain for hacking the training loss, thereby failing to properly learn the sequence distribution. In this paper, we propose to address this by designing rationalized information flow, which avoids such problem while still preserving network's 's ability to capture all-atom geometry.
Furthermore, current works employ stochastic differential equations (SDE) (e.g. diffusion models [Song et al., 2020, Ho et al., 2020]) or ordinary differential equations (ODE) (e.g. flow matching [Lipman et al., 2022, Chen and Lipman, 2023]) to generate each protein component. However, these SDE and ODE methods are known be prone to discretization errors, which can hinder both the quality and efficiency of the generation process [Karras et al., 2022, Tong et al., 2024, Sabour et al., 2024]. Bayesian Flow Networks (BFNs) [Graves et al., 2023] are new class of generative models that build a generation process through Bayesian inferences over noised data, which effectively circumvents the discretization error without the need to discretize continuous differential equations and has been verified to be effective in molecule generation tasks [Song et al., 2023, Wu et al., 2025].
Motivated by this, we introduce ProBayes, a new all-atom protein design model with Bayesian flows. A key theoretical challenge needs to be tackled: The orientations of the protein backbone frame are SO(3) matrices [R: R E RR33 RR I, det( R) I], while there no existing Bayesian flow workable on SO(3). Directly conducting Bayesian inference on SO(3) is impractical, because current SO(3) distributions (e.g. IGSO(3)) cannot guarantee conjugacy under Bayesian inference, which is pivotal to build Bayesian flow. To overcome this issue, we propose to transform the problem into an equivalent hypersphere one, by generating SO(3) elements on hypersphere with antipodal symmetry. Additionally, ProBayes demonstrates clear advantages over the current prevalent SO(3) flow matching method [Yim et al., 2023a]. which relies on a hand-crafted inference annealing technique to adjust the generation flow. Our contributions can be summarized as follows:
We introduce ProBayes, new protein generation approach with Bayesian flow, by transform- ing SO(3) generation into an equivalent problem of hypersphere generation with antipodal symmetry.
We identify and resolve the issues of information shorteut in all-atom protein generation with novel rationalized information flow.
Extensive experiments demonstrate that ProBayes consistently achieves significant improve- ments over existing methods on peptide design and antibody design tasks, egg, achieving DockQ score of 0.74 on PepBench.
2 Preliminary
Hypersphere and hypertorus Given an Eu-
S,=LYS
clidean space R", the hypertorus T is repre- sented as the Cartesian product of flat toruses Ti= {0 0 E [0, }. The hypersphere 1d-1 located in RR is defined as: gd-1 = (x x E Rd ||>|| fz|122 = 1}. Superscript d denotes the de- gree of freedom.
N
R
CG
CD
X1
X2
NZ
CA
CB
X4
CE
0
Protein representation A protein P biomolecule composed of I. amino acid residues, each defined by its type, backbone frame, and side-chain angles Fisher [2001]. All amino- acid types form the protein sequence S = [$1...., St), where 31 E {I,..., 20), VI E L}. The backbone frame of each residue is parameterized by the position vector of the C. atom, p E R3, an orientation matrix
S VAL
N
CA
R
*
CG2
CB
CG2
Figure 1: Illustration of protein representation. Note the pattern that different residue types have different numbers of side chain torsion angles.
'There exists a study [Xue et al., 2024] attempting to connect continuous-tine BENs with SDEs. However, we clarify that the discrete-ting time BFN functions as gradient- free generative models involving no differential operations, which is fundamentally different from SDEs. Further discussion is provided in Appendix G.

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R E SO(3) that determines the relative positions of the N and C atoms, and a torsion angle 1b which govemns the position of the O atom [Engh and Huber, 2006, Jumper et al., 2021, Yim et al., 2023b]. The side-chain can be represented by up to four rigid groups with x denoting the flexible torsion angle between rigid groups. The number of side chain torsions and arrangements depend on the specific residue type as shown in Fig. 1.
Bayesian flow networks As new class of generative models, Bayesian flow networks (BFNs) [Graves et al., 2023] build a generation process through Bayesian inferences using noised data, which fundamentally different from diffusion models [Song and Ermon, 2019, Ho et al., 2020, Song et al., 2020] and flow matching [Lipman et al., 2022].
Formally, BFN parameterizes a belief for clean data over the space with distribution family, e.g. Gaussian distribution for x E R. The belief starts from an uninformative prior lg, which is updated according the received noised data y: ps(y (x), signal-to sigmal-to-noise ratio parameter (x.. The update transition from 0 to 0, is termed Bayesian update function O, = h h0 Yı, ai), which is derived from the Bayesian theorem according to the form of the belief distribution and noise distribution ps. With each 0.-1 as input, the neural network y is trained to transform the belief 0:-1 1 as receiver distribution pR. minimize the evidence lower-bound based training loss:
{1, 2, ........... n. with different levels of
V n Ei Bo....p Kr. (p x a:)ln)() (0,-))). a:)l
E
where pF is the Bayesian flow distribution representing the distribution of 0 at time step i. Com- pared to diffusion models and flow matching, BFN enjoys the following advantages: (1) free from discretization error without the need to discretize continuous differential equations; (2) precisely adjusted step sizes according to the entropy parameter ag, allowing fast and accurate generation demonstrated in [Song et al., 2023, Wu et al., 2025]. Motivated by these advantages, we aim to design a new (3(3) Bayesian flow as a more effective and efficient approach to protein design.
3 Bayesian flow for SO(3) generation
In this section, we introduce the first known Bayesian flow for generating SO(3) data, which is derived from constructing a Bayesian Flow on the hypersphere 3 with antipodal symmetry.
3.1 Theoretical challenge in directly building SO(3) Bayesian flow
Difficulty in guaranteeing Bayesian conjugacy The Bayesian inference approach used in BFN requires the Bayesian conjugacy of the distribution, i.e. the posterior distribution after observing the noised data should be in the same distribution family as the prior belief distribution. However, existing SO(3) distributions (e.g. the IGSO(3) distribution used in Yim et al. [2023a,b], Bose et al. [2023]) are intractable to derive analytical Bayesian update to ensure its conjugacy due to its infinite sum. Furthermore, other well-known existing statistical distributions over rotation matrix [Watson, 1966, Downs, 1972, Khatri and Mardia, 1977] also involve intractable probability density function to guarantee their Bayesian conjugacy, e.g. matrix Fisher distribution.
Therefore, we resort to other widely used rotation representations such as Euler angles, axis-angle and unit quaternions, instead of the rotation matrix. As discussed below, we demonstrate that both Euler angles and the axis-angle representation fail to adequately capture the group structure of SO(3), whereas unit quaternions are able to do so effectively, which is used in this paper. Additionally, the unit quaternion's 's intrinsic manifold, i.e. the hypersphere, is significantly simpler than the structure of SO(3), and there exists a normal distribution over the hypersphere which satisfies Bayesian conjugacy.
Non-covering of Euler angle and axis-angle representation We observe that some rotation rep- resentations cannot capture the group structure of SO(3). For example, Euler angles suffer from the singularity problem of gimbal lock (Hoag, 1963]. From the perspective of topology, such prob- lems reveal the fact that the hyper-torus To cannot cover the SO(3). Practically, we validate the ineffectiveness of Euler angles and axis-angle representation via toy data generation experiment (see Appendix C). The definition of the cover map is as follows:
2 Although the position vector and the orientation matrix form SE(3), measure on SE(3) can be disintegrated into measure on S3( and I [Yim et al., 2023a, Bose et al., 2023]. Therefore, we can focus on SO(3) generation only.
3

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hypersphere S distribution w/o antipodal symmetry p(g) * p ((-9).q
hypersphere @ distribution
(3) distribution
w/ antipodal s
(R)) R. E SO(3)
p(4)= p(-q) e p(R), (Gg) = ((-g))
4
Figure 2: Intuitive illustration of the equivalent transformation from SO(3) generation to hypersphere generation with antipodal symmetry. Extending the modeled space to hypersphere yields a more symmetric and tractable manifold.
Definition 1 (Covering Map, Hatcher [2005]). Given topological space X and Y, a covering map is a surjective continuous map f X + Y such that: every element E Y has an open neighhorhood U S Y where (Uy) is disjoint union of open sets, each of which is mapped by f homeomorphically onto Uy.
The quaternion representation of rotation can be used to build an SO(3) Bayesian flow because the space of this representation (i.e. the hypersphere) can cover SO(3). Intuitively, we aim to build a generative model in a larger topological space while preserving the original group structure of SO(3).
3.2 Transforming SO(3) generation into hypersphere 3 generation
Fig. 2 provides an intuitive illustration of transforming SO(3) generation into hypersphere S genera- tion. Firstly, we introduce the group structure of unit quaternion rotation representation, i.e. SU(2) group and its corresponding manifold hypersphere $ Then, we explain the proposed key constraint of antipodal symmetry to equivalently transform SO(3) generation into hypersphere $ generation, and introduce a simple approach to satisfy such constraint.
SU(2) group SU(2) is the special unitary group composed by unitary matrices U E {U U C2x2, UU I, dUt = 1}. SU(2) is diffeomorphic to the hypersphere 3d [Hall, 2013]. Hence, we can represent SU(2) elements using four real numbers, namely the quaternion a + bi -c + di
[a, b, c, d) with
unit 2-norm constraint: SU(2) = (
c di a a- bi : (,, E R. + 62 + + + =
Double covering between SO(3) and SU(2) In fact, SU( is the double covering group of SO(3) [Hall, 2013] and there exists a surjective homomorphism @ SU(() + SO(3) such that for every (su(2) E SU()), we have (((su(2)) = (--9su()) = 9so(3) E SO(3). Therefore,
transform the representations of these two groups between one another. Specifically, given a unit quaternion q bi + cj + dk, the corresponding rotation matrix R ((q) c5( can be obtained by:
1 - 26² - 2d
2bc - 2da
2bd + 2ac
R
2bc + 2da
25² - 2ff
2ed - 2ab
[Shoemake, 1985]
(2)
2bd 2ac
2ed + 2ab
26 20²
Inversely, rotation matrix R can be transformed into the corresponding double unit quaternion - (a + bē + cj + ddk) from Eq. (2):
q1 = a + bi + ej + dk and q2
1 b = s R322 R2a), s s(R13 - Rai), d s(R21 - Ri2) (3)
8
a
48"
As illustrated above, each rotation matrix corresponds to two unit quaternions q and -q, q, which are equivalent when representing the three-dimensions rotation operation. This phenomenon is termed as SU(2)'s double-covering of SO(3) [Altmann, 2005], which results in the discrepancy between generating rotation matrices and directly generating unit quaternions. We propose natural solution to this problem, by adding an antipodal symmetry constraint to the modeled unit quaternion distribution on the hypersphere S3
Definition 2 (Antipodal Symmetry). For distribution p(x) supported on Sd, we say this distribution antipodal invariant p(x) = p(-x), Va E Sd. Correspondingly, we say a function I operating 3 is antipodal equivariant if f( q) -f(g).
4

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Now, with antipodal symmetry, we can transform the generative modeling problem of SO(3) data into an equivalent one on S with antipodal symmetry. We formulate the proposed transformation with the following preposition:
Proposition 1. Every distribution (so(3)R)) supported on SO(3) can be bijectively mapped to a distribution pao (q) supported on 3 which satisfies p&> (q) = Pga (-9). q)-
In fact, we find that antipodal equivariance can be easily ensured by implementing the Invariant Point Attention (IPA) module [Jumper et al., 2021] ] rotation matrix) without extra operations or complexity.
based on quaternion representation (instead the
3.3 Bayesian flow on hypersphere gd
Equipped with Proposition 3, we can now derive a Bayesian flow for generating SO(3) on the hypersphere 33. In this section, introduce the first derivation of the Bayesian flow on Sd, which can be applied to generate unit quaternions with d = 3 for SO(3) generation.
Von Mises Fisher distribution and Bayesian update function h We choose von Mises Fisher (vMF) distribution [Mardia and Jupp, 2009] to parameterize the belief over S because of its tractability and Bayesian conjugacy. With the location parameter μf and concentration parameter n; the probability density function of the von
distribution is:
R
fp(a;; pa, K) = e M F az) (a, x) = (p()s) exp(11]" x)) = (2mr (R) exp exp(Fpp x)),
where =
1, | d/2 - 1, 2 0, (x) denotes the modified Bessel function of the first kind at order Given the prior belief parameterized by vMF distribution with parameter 14- Ki- I the noised sample ys from sender distribution with unknown clean æ, and the accuracy a; describing the entropy of ys the Bayesian update function h for von Mises Fisher distribution is deducted as:
(5) The proof of the above Eq. (5) can be found in Appendix B. This update function operates in an intuitive manner, analogous to the moment equation in physics, where the accuracy/entropy
h( (((1:11, 14-1) Yr, as)) = {Mu, k. where Kipii = 14-14:-11 + a,y;
def
parameters к and represent the mass, and μ and y denote the velocity. We define kp to compactly denote the parameter of the vMF distribution {K, 1k).
Bayesian flow distribution Based the Bayesian update function, the Bayesian flow distribution hypersphere dd over vMF distribution can be obtained by conducting a series of Bayesian inferences.
With the notation ar (a1, a......, Appendix B):
def
ar), the Bayesian flow distribution is expressed (details in
pr (0:(aa o1:)) vM (MFF(lmm,)).... ((g.
8(0;
Σ Eoj9;)
Now with Eq. (6) and Proposition 3, we are able to generate the orientation of the backbone frame with Bayesian flow, which is free from discretization error and enjoys entropy-adjusted dynamic step size. The antipodal symmetry of the modeled distribution is guaranteed as follows:
Proposition 2. With V as an antipodal equivariant function, i.c. VI(09) = -00, and uniform prior p(03) on $3, the marginal distribution defined by multiple Markov transitions po (0g) = (g)
(O")))) 1x-1 is antipodal invariant, which is equivalent to a distribution pe R))-
Note that, as the rotation and orientation are ubiquitous in geometric deep learning, such SO(3) Bayesian flow can be used other applications, e.g., molecular docking [Corso et al., 2022].
4 Rationalized all-atom protein design
In this section, we present our approach to jointly generating the complete protein structure, including the sequence, backbone and sidechain. We emphasize that such a co-design process is inherently tricky, and naive joint training risks information shortcut between sequence and side-chain.
5

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ProBayes parameterizes the belief over the protein space via each modality's belief parameter @ {0s, 0", Q 0000000000 with a predefined number of generation steps n. For training, we first sample the time index ~ U(1,...:: ) and 0 ~ pr for each modality. The neural network (0 is trained to minimize the training loss according to Eq. (1). After sufficient training, the data distribution Pdara is well captured by the neural network V. As for sampling, each modality starts from an uninformative prior 0% which is to be updated according to the network output v (0",, i)
and the Bayesian flow distribution till
4.1 Bayesian flow for each protein modality
We explain the definition of the Bayesian flow for each protein modality. For detailed training and sampling algorithm, please see Appendix D.
Bayesian flow for orientation matrix R generation We utilize the proposed Bayesian flow Eq. (6) to generate the orientation of the protein backbone frame R. We begin by uniformly converting the orientation matrix R into its corresponding quaternion representation, yielding one of the two equivalent quaternions {g. - q} with equal probability. The SO(3) Bayesian flow distribution is:
E O0 R R g~U((-1((0) WM FM
E
8(0)
Σ Cajy)
where of is the predefined accuracy to make the entropy of Eq. (7) linearly decrease with respect to time step Sampling 0 from p/. the training objective is derived based on Eq. (1) and the KI. divergence between vMF distributions (details in Appendix B):
Lu+1(a?) (1 dot(q, (0p .))) I(aa)
E C Uo-Pan.i~UU1.
(8)
~pp
where dot represents the dot product between two quaternions, and 4 represents the quaternion part of the network prediction. We prove in Appendix that such loss is proportional to the geodesic distance between the predicted and ground truth orientation matrix up to a constant, which further solidifies the transformation between SO(3) and SU(2).
Bayesian flow for C. position p generation The C. positions of the backbone frame are in Euclidean space p R Therefore, we use the Gaussian distribution Bayesian flow [Graves et al., 2023] to generate them. The Bayesian flow distribution of C. position p is:
bf (1² p) = ((tp)(t))p (() - I))) where ((t) = of
(9)
where 01 is the hyperparameter controlling the entropy of the Bayesian flow at = The variance of the belief Gaussian distribution In (t) In(1 (t))/t does not need to be modeled because it is deterministic w.r.f. time steps.
Bayesian flow for sequence S generation Given that sequences are discrete variables, we utilize the discrete Bayesian flow [Graves al., 2023] to generate the sequence. With the notation of projection from the class index j to the length K one-hot vector (ej)ı def' 8yū, where ej E R es
der
(e.,..... Csy,) E TR. K the Bayesian flow distribution of sequence is defined as:
(00--
5 (08 S;t)
0s
(10)
(O (Bt
Σk=1 e (0n)
where (t) is the predefined accuracy schedule.
Bayesian flow for angles vi X generation The torsional angles vo and x are periodic variables the hypertorus. Hence, we use the periodic Bayesian flow [Wu et al., 2025] to generate them:
E (m xi axx "Miyn bx )... ((
6(m atan2
cos yj Σ a sin 9,)) (11)
We illustrate our method using X; the Bayesian flow for &i has the same form. While both lie on the same manifold, unlike Li et al. [2024]. we treat them differently. Specifically, we distinguish x from the backbone torsion vb because X carries partial sequence information. We expand on this in the next section.

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Train Seq Loss
Shorra Sharset
+.0 ShorcLt Shortcrt
-Mildd Storcet Shortot
2.5
0.6
0.4
0.3
Step
64
6.8
50k 2006 150K 200k
2504
0
50k
100k
1506
200k
250k
50k 100K 1536 200k
250k
(a)
(b)
(c)
(d)
Figure 3: (a): Illustration of information shorteut measured by KL divergence. (b)-(d): Training curves demonstrating the negative impact of the shortcut. As shown in Fig. 3b, the network with a shorteut achieves a smaller loss initially, but exhibits a larger loss towards the end. As for the metric. the shortcut results in worse validation AAR in Fig. 3c. Additionally, we observe that the shortcut also hurts the performance of other modalities, such the SO(3) in Fig. 3d.
4.2 The information shorteut problem
As discussed in Sec. 2 and Fig. I, the residue type S reflects the types of the side-chain with varying numbers of side-chain torsion angles from 0 to 4 [Jumper et al., 2021, Li et al., 2024]. However, diffusion-like models (e.g. FM and BFNs) are inflexible in capturing varying numbers of random variables. Prior practices [Li al., 2024, Zhu et al., 2024] tried to address this by padding the number of variables with a fixed padding value (e.g. 0 in [Li et al., 2024]), then mixing the side chain information with the sequence and backbone.
Shortcut induced by paddings However, we prove that such padding and mixing operations cause information leakage from the even noised side chain information to the sequence prediction. To illustrate the idea, we design a simple threshold based shortcut function to "predict" the ground truth sequence for residue Alanine and Glycine, which have side chain torsion angles iee. X [0, 0, 0, O]:
[12 1 O,...... 20 20
if circmean (0x) < 0.5
shortcut(02)
=
1
otherwise
(12)
20
where circmean is the circular mean of periodic variables [Mardia and Jupp, 2009], with the first two elements representing the predicted probability of Alanine and Glycine, respectively. We compare the KL divergence between the predicted categorical distribution shortcut one-hot distribution S, against the KL divergence between the noised sequence 8r and S, as shown in Fig. 3a. The results demonstrate that the shorteut prediction achieves significantly lower KL divergence than the noised sequence most of the time. This indicates the network can exploit shortcut information from the number of side-chain angles/paddings than genuinely denoising the sequence by learning the underlying sequence distribution.
and the ground-truth
minimize sequence training loss, rather
Shortcut induced by inherent side-chain distributions Moreover, emphasize that the informa- tion shortcut can still exist even in the absence of padding. This is because the side-chain torsion distributions themselves (i.e., the non-padded values) are inherently dependent on the residue types, as shown in Fig. 4. Consequently, the network can exploit these statistical regularities in the side-chain distributions reduce the sequence prediction loss, thereby hindering the proper learning of the true sequence distribution.
Rationalized information flow Therefore, we propose a rationalized network information flow to avoid the possibility of shortcuts, as illustrated in Fig. 5. Concretely, we avoid involving side-chain information with the sequence S v (0S 0P 0t 0V, 88)), while maintaining network's 's capacity to capture the all-atom geometry by the backbone and side-chain mixing module for denoising other modalities. For side-chain prediction, the loss is computed only for non-padded values to prevent the network from learning harmful patterns. The detailed implementation is provided in Appendix F.
Excluding side-chain information from sequence prediction does not negatively impact the perfor- mance ceiling sequence generation. This stems from the fundamental unfair relationship between sequence and side-chain in the data distribution: The sequence serves as the primary determinant of residue types, while side-chain configurations are dependent on the sequence for both side-chain atom types and conformational specifications.
7

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om1.1.61
"=152.9"
p=126.3'
130.4"
Dersity
Dencht
Densiit
Arginine
100
100
100
100
100
0
100
100
0
100
Angler
Angla!
Angle"
Angler"
j=-4,2"
2.66.66
Density
Proline
-100
100
-100
100
Angler
Angles
Figure 4: Visualization of the side-chain angle distributions by residue type in PepBench. The side-chain angle distributions of the two residues differ significantly, and the network can similarly exploit the statistical characteristics of side-chain distributions to "predict" the sequence just like Eq. (12).
Sequence
g
Seq. Head
S ..... Info. Shortcut
Seq. &Backbone
Rational Info.
Position OP
P
Mixing Module
Orientation
09
Backbone &Sidechain
q
Owygen Torsion 0%
Mixing
ip
Module
Sidechain Torsion
0x
Sidechain Embedder
X
Figure Illustration of the proposed rationalized information flow.
Related work
Protein generation There is body of meaningful protein generation practices focusing on single protein component, backbone only Yim et al. [2023b], Bose et al. [2023]. Geffner et al. [2025], Watson et al. [2023], sequence only Madani et [2020], Dauparas al. [2022], and side-chain only Zhang et al. [2023]. Recently, motivated by the advantages of joint modeling, an increasing number of sequence-strucee structure co-design approaches have emerged, e.g., backbone sequencce co- design [Campbell et al., 2024, Lisanza et al., 2024] and all-atom protein generation [Martinkus et al., 2024, Li al., 2024, Chu et al., 2024, Kong et al., 2024, Zhu et al., 2024]. However, the increased information of all-atom geometry also brings challenges: We observe that all-atom generation is prone to the information shortcut problem due to the dependency between sequence and side-chain, which is highlighted and resolved in this paper.
BFN for molecule generation Inspired by BFN's unified generative modeling for continuous and discrete data, GeoBFN [Song et al., 2023] first incorporates molecular geometric constraints into the BFN framework and achieves exceptional generation performance and efficiency. Recently, Wu et al. [2025] proposed to build BFN on the hypertorus for periodic material generation, demonstrating a superior sampling efficiency. However, the rigidity the protein backbone requires generative modeling SO(3), which is inherently difficult due to the intractability of ensuring Bayesian conjugacy with SO(3) distributions. this paper, we address the problem by transforming the modeled space into simpler thereby allowing Bayesian inferences.

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Table Evaluation on peptide co-design. On each target, 40 candidates are generated for evaluation.
Energy
Native Likeness
Design
Dataset
Method
AG(L)
Success(1)
DockQ(1)
RMSDe,
Valid (1)
V&Div(t)
V&Novel(())
PepBench
Test Set
35.25
95.70
1.0
0.0
0.989
RFDiffusion
24.61
69.86
0.265
4.12
0.943
0.439
0.670
PepFlow
-23.09
59.20
0,848
0,453
0.623
PepGLAD
-21 .44
55.97
0.656
3.35
0.330
0.245
0.226
ProBayes
-28.77
72.85
0.742
2.27
0.998
0.449
0,727
PepBDBB
Test Set
-35.96
95.
1.0
0.0
PepFlow
-22.45
65.60
0.517
0.615
0.379
0.521
PepGLAD -24.53
48.47
0.472
5.44
0.165
0.163
0.128
ProBayes
-30.55
85.78
0.605
3.39
0.959
0.423
0.725
Table 2: Evaluation binding conformation generation. Baseline results are from Kong al. [2024].
PepBench
PepBDB RMSD DockQ(1)
Model
RMSDe,())
RMSD
DockQ(t)
RMSDe,
AlphaFoold 2
8.49
9.20
0.355
DiffAh
4.23
7.60
0.586
13.96
13.12
0.236
PepGLAD
4.09
5.30
0.592
8.87
8.62
0.403
ProBayes
2.62
3.81
0.672
3.64
4.91
0.563
Experiments
We verify the effectiveness of ProBayes on two protein design tasks, including peptide design and antibody design. For peptide design task, our evaluation configurations follow Kong et al. [2024], including peptide co-design and peptide binding conformation generation. The description of metrics and implementation details can be found in Appendix F. We also include toy dataset experiment in Appendix C of SO(3) Bayesian flow compared to SDE ODE methods [Bose et al., 2023].
6.1 Peptide design
Datasets The datasets used for evaluation include PepBench and PepBDB [Wen et al., 2019]. PepBench is constructed from the Protein Data Bank [Berman al., 2000], containing 6105 complexes and the LNR dataset Tsaban et al. [2022] is utilized as the test set with 93 complexes. PepBDB protein-peptide complex dataset containing 8434, 370, and 190 items for training, validation, and test, respectively.
Baselines We evaluate approach against the following baselines: PepFlow [Li et al., 2024], peptide design model utilizing Riemannian flow matching; PepGLAD [Kong et al., 2024], which employs latent diffusion techniques for all-atom peptide modeling; and RFDiffusion, multi-stage protein design framework [Watson et al., 2023] integrated with ProteinMPNN [Dauparas et al., 2022] for sequence generation.
Results We report the results on PepBench and PepBDB in Table and Table for seq-structure co- design and binding conformation generation task, respectively. Across both PepBench and PepBDB datasets, ProBayes outperforms existing methods in energy, native structural fidelity, and design quality. For peptide binding conformation generation task, method significantly outperforms all baselines. It achieves the lowest RMSDD,,, RMSDusmm and DockQ, indicating ProBayes's 's superior all-atom structure generation accuracy.
6.2 Antibody design
In this experiment, we focus on designing the third Daleemmennarity-eettmnnnn Region of the antibody heavy chain (CDR-H3), as exhibits the greatest variability and plays dominant role in antigen binding [MacCallum al., 1996].
Table 3: Results of antibody design task RAbD.
Method
AAR 1
RMSD
Epsa
AG
Test Set
100.0
0.00
-16.76
-15.33
dyMEAN
40.05
2.36
1239.29
612.75
DifFAb
35.04
2.53
495.69
489.42
AbX
41.27
2.40
281.29
237.22
Dataset We utilize the SAbDab database et al., 2014] of antibody-antigen com- plexes as our training dataset and evaluate model performance using the RAbD bench-
AbDPO
31.29
2.79
270.12
116.06
ABDPO++
36.25
2.48
338. 13
223.73
ProBayes
42.12
2.27
54.68
44.12

---

mark WAdolf-Bryfope et al., 2018]. The data- droepesees procedures, evaluation configurations, and baseline results are from [Ye al., 2024]
Baselines We compare our method against three existing approaches: dyMEAN [Kong et al., 2023] employs a full-atom geometric encoder with iterative non-autoregressive generation; DiffAb [Luo et al., 2022] jointly diffuses categorical residue types, Ca coordinates, and residue orientations; AbDPO and its variant AbDPÔ++ [Zhou et al., 2024] are methods tailored for optimizing antibody energy based on direct preference optimization.
With consistently better AAR and RMSD, ProBayes also demonstrates significant improvements in energy-based metrics, achieving state-of-the results with lower order of magnitude, surpassing even the energy poorgy-opiiiizaiionffo focussed models AbDPO and AbDPO++
Results The results are listed in Tab. 3.
6.3 Ablation study
Table Results of ablation study on the antibody design task.
We validate our design considerations through the following ablation experiments using the antibody design task. Results can be found in Tab. 4: (1) antipodal symmetry: we break the model's 's antipodal symmetry by removing equiv- ariance in the Markov transition i.e. we replace our quaternion- base implementation with the original IPA implementation [Jumper al., 2021], which leads to a significant drop in energy-related metrics, demonstrating the necessity of enforcing the proposed constraints. (2)information shortcut: we reintroduce the information shortcut by incorporating side-chain information during sequence prediction. This experiment results in 100% drop in AAR and hurts performance across other metrics, indicating the detrimental impact of such shortcuts on model performance. (3) all-atom capability: we replace model-generated side-chains with those obtained from Rosetta's side-chain packing [Alford et al., 2017]. Our model achieves superior performance on energy-based metrics, demonstrating its capability to generate high-fidelity side-chain structures.
Method
AAR1
RMSD
Epail |
AG |
ProBayes
42.12%
2.27
54.68
44.12
w/o anti. symm.
40.17%
2.40
171.33
104.15
w/ shortcut 32.51%
2.42
203.29
86.76
w/ Rosetta pack 42.12%
2.27
76.11
49.28
7 Conclusion
We introduce ProBayes, a novel all-atom protein design model using Bayesian flows, which effectively resolves critical issues of information shortcut that plagues existing methods. Achieving significant improvements over prior methods on peptide and antibody design tasks, ProBayes demonstrates more robust pathway towards generating functional proteins.
Acknowledgments
This work is supported by the National Science and Technology Major Project (2022ZD01 17502), the National Natural Science Foundation China (Grant No. 62376133), the Wuxi Research Institute of Applied Technologies, Tsinghua University under Grant 20242001 120.
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Contents
Introduction
2 Preliminary
3 Bayesian flow for SO(3) generation
3.1 Theoretical challenge in directly building SO(3) Bayesian flow
3.2 Transforming SO(3) generation into hypersphere @ generation
3.3
Bayesian flow on hypersphere Sd
Rationalized all-atom protein design
4.1
Bayesian flow for each protein modality
4.2
The information shorteut problem
5
Related work
Experiments
6.1 Peptide design...
6.2 Antibody design
6.3 Ablation study
10
7 Conclusion
10
Proofs of propositions
15
Detailed derivation of SO(3) Bayesian flow
16
B.1 Hypersphere and von Mises Fisher distribution
16
B.2 Bayesian update function h...............................
16
B.3 Bayesian flow distribution pr (0.lz;
17
B.4 Discrete-time loss with n steps (x(x)
17
C SO(3) toy data experiment
18
D Detailed training and sampling procedure
19
Limitations and broader impacts
20
F
Experiment details
21
F.1
Rationalized information flow implementation
F2
Peptide design
22
F3 Antibody design
22
F.4 Other implementation details
23
G Comparison between BFN and ODEs/ODEs
23
14

---

H More results
23
H.1 Error bars
23
H.2 Visualization
24
I Further discussion of information shortcut
25
A Proofs of propositions
Proposition 3. Every distribution Pso(3)()) supported on SO(3) can be bijectively mapped to distribution paa (q) supported on @ which satisfies psa (q) Pga (q))
Proof. Given each distribution PSo(3) [)) support on the SO(3) group, we can build the correspond- ing @ measure P5o(3) (R) using the surjective homomorphism ø SU(2) + SO(3):
bq
s Psa (q) == (5
(13)
Now we prove that pea (g) satisfies all the probability distribution axioms and the antipodal symmetry:
1. Non-negativity: Vqq E 33 Psa (q) Po
2. Normalization: fga Psa (g) = 1/ 1s3 pl (R)ds Sgz pl (R)dSO(3) = 1
3. Antipodal symmetry: Pas (q) (.POO3)) ( (R)) p33 (-q)
For (g) = (-a(- we can also build pso R = 2p (g) = 2pga ( 4). VR E SO(3). We check that PSo(3) (R) 2 0 and Ssor3) PSo(s) (R))SO(3) = 1. Therefore, PSo(3) is probability
density function SO(3).
Proposition 4. With V as an antipodal equivariant function, i.c 4 (0 and
uniform prior p p(0)) on $3, the marginal distribution defined by multiple Markov transitions antipodal invariant, which is equivalent to pv (0%) p(0) I (00
distribution pa R)..
Proof.
Po (-09) = p(-6%) /
Po
8 m:
-0%) 1n-1
n-
=p(-08)
II
po
09 01 00,
000 fmm-1
n-
ppp 08)
II
Pr
09 1+1
(-
01:))
1m-1
t=
T-
-(-08)
II
PF(
89 1+1
(000
a1.;)
do0
1mm-1
t=
=p(-08
II t=
PF (000
=P(0%)
II t=0
Py (0%).
Therefore, the marginal distribution p(0) is antipodal invariant.
15

---

x0)=1 1 33
b=)= μ= レ
(c) 100
kd)=
# = [0,0,1
Figure 6: Visualization of the Mises Fisher distribution with different parameters with d 2. Those figures are from Frisch and Hanebeck [2023].
B Detailed derivation of SO(3) Bayesian flow
B.1 Hypersphere and von Mises Fisher distribution
Given Euclidean space Rd the hypersphere g located in I is defined as:
sd-1 fa x E Rd
< = 1}
(14)
The Mises Fisher distribution [Mardia and Jupp, 2009] is uni-modal distribution S with the location parameter and concentration parameter к. The parameters μu and и are analogous to the mean μ and variance 1/o² in the normal distribution: μ represents the central location around which the distribution is concentrated, and serves as measure of concentration. We provide a visualization of vMF distribution with different μ and in Fig. 6. The probability density function of vMF distribution is:
R²
fa nt, K) uMM uMFFw (x(() exp(xu²" x) (2m) [2) exp(s1² x))
(15)
where |||||| = ||>||| --v= = d/2 1, 2 (x) denotes the modified Bessel function of the first
kind at order v. The differential entropy of vMF is:
(() - lu+i(x) (u (x)
(v 1) 2n + (x)) - v In
(16)
We vMF's entropy Eq. (16) determine the accuracy schedule in the following sections.
B.2 Bayesian update function
Sender distribution For each time step i, the sender corrupts the clean data Pasta using the sender distribution Ps and the signal-to-nonl ratio parameter a, gets the noised information y ~ Ps- and send y the receiver. Here, we use the vMF distribution instantiate ps for hypersphere data E Sd-1.
ps ps(p; a) = uMF(yl a))
(17)
Input distribution As for the receiver, its belief over the ground truth data x is formulated by distribution family with parameter 0, e.g., Gaussian distribution with l and for Euclidean space data E R!, which can be expressed as:
pz vMF FMF((z),
(18)
Bayesian update function For each time step the receiver updates its prior belief 0:-1 according to the observed noised data y: with known accuracy a; according the Bayesian theorem. This process is formulated as the Bayesian update function 0, h(0.11. yr, a;). The Bayesian update
16

---

function h for von Mises Fisher distribution is derived as follows:
p( 3) p(y) a)p(a; (1-1, p
x plyla a)ppa; M1-1.
uM F.MF a)u F (1-11
x exp(ox" y + - 11=113
vMM p4, K.))
where K;d4; 81-14i-1 a.y, Therefore, the Bayesian update function is:
M(fp:-1,K--1b 04) {us k.). where K;4; = Ki-14--1 + a,y;
(19)
We define 0 def ku to compactly denote the parameter of the vMF distribution {k, 14).
B.3 Bayesian flow distribution pF(0:(;; a1:))
Given time index i and clean data æ; the Bayesian flow distribution for 0 is defined as the marginal distribution after conducting multiple Bayesian updates over noised data y
pF (0 oM
E
8(0;
Σ ag
(20)
For determining the accuracy schedule Ona according to entropy, we first formalize the entropy of the Bayesian flow:
E (1)) Pr(0, x;
H(pr(-10)) where i = ml.
(21)
The entropy of the Bayesian flow is dominated by the accuracy schedule 01:n- Although Eq. (21) is not tractable, we evaluate its value each step given the accuracy schedule 01.z- For synchronizing the entropy across each modality, we find the accuracy schedule that @1m such that the entropy of this Bayesian flow linearly decreases with respect time:
H(t) = (1 (0) + (1
(22)
Although Eq. (22) is not analytically tractable, solve this equation using binary search find each a; at each step i.
B.4 Discrete-time loss with n steps (x)
Receiver distribution Given 0 from the Bayesian flow distribution as input, the receiver uses the network to improve its belief considering interdependency across modalities and dimensions, which is referred to as the output distribution. We choose parameterize the output distribution as delta distribution to directly predict the ground truth æ following [Graves al., 2023]:
po V (O,
(0,,t,)) t.)) = 5(x t (0:
(23)
To approximate the sender's 's distribution, the receiver's distribution is obtained by convolving the output distribution with the sender's ' distribution:
0 t, a.)
ax) uMF (0, (0,,tt)), a.)
(24)
Discrete-time loss with steps The discrete time loss defined the KL divergence between sender and receiver distribution:
L*(x)= 71
DEL (Ps (+1
æ;; a;) PP |
(25)
(F
04 Io+1(as) (1 dot(2r, (00+))))
n
E (0.
(26)
In fact, the term (1 dot(æ,, (( ())) in Eq. (25) is proportional to the geodesic distance between the corresponding orientation matrix R σ(x) and R (#) (0,
17

---

Proposition 5. The training loss defined in Eq. (8) is proportional to the geodesic distance between the predicted orientation matrix R = o T (O? t)) and the ground truth orientation matrix R to a constant.
Proof. We first introduce the (w, v) representation for unit quaternions. Specifically, for unit quaternion [a, b, c, dJ"", we define the scalar part so that the quaternion can be written as q = w + v, where i,j, are the standard basis elements corresponding to the three Cartesian axes. This representation particularly convenient because the rotation angle 1 of the rotation matrix = Φ I (q) satisfies the relation
and the vector part bi ej + dk,
cos 0 0/2) [Shoemake, 1985]
(27)
Using the (w, v) representation, the geodesic distance between two unit quaternions 00 + V1 and q = W2 22 3 is given by:
dga (q. q) arccos ((g. q)) arceos (wi W2 + (21,
(28)
where ( denotes the standard Euclidean inner product.
The geodesic distance between the corresponding rotation matrices R ④(q) and R = (()) in the rotation group S0(3) the rotation angle of the relative rotation R, = R R. This angle can be computed via the quaternion representation of R,:
= o [R,)
= q q- x q = [ + (v1, U2), W1 12 + W222 + 02 x U2l
(29)
The corresponding rotation angle 9 of R, is then:
dso(3) (R R) I arccos (wi + (v1, v2)) = 2 dea (q. q).
(30)
The proof is done.
C SO(3) toy data experiment
In this section, we use the synthetic dataset Brofos et al. [2021] to verify the effectiveness of the proposed SO(3) Bayesian flow over SDE/ODE based methods following [Bose et al., 2023].
Toy dataset network The antipodal equivariance of the toy dataset network is achieved by the following formula:
T toy (09, t) Normalize(ja @ MLP( ks + 11)
(31)
where @ denotes the element-wise product between two vectors, and Normalize = q/ 4/l///12-
Metrics The metrics W1 and W2 refer the Wasserstein p-distance between the generated distri- bution and the ground truth test distribution with = 1 and p = 2, respectively. These two metrics measure the distributional distance between the generated distribution and the ground truth data distribution.
Results We present the visualized generated results in Fig. 7 and the corresponding evaluation metrics in Tab. 5. As shown in Fig. 7, ProBayes produces higher-quality samples with fewer outliers compared to the SDE and ODE baselines. This observation consistent with the improved W1 and W2 scores reported in Tab. 5, further demonstrating the effectiveness of the proposed SO(3) Bayesian flow.
We further validate the ineffectiveness of the Euler angle and axis-angle rotation representations in Fig. 7f and Fig. 7g. as discussed in the main text. Specifically, we employ the hypertorus BFN [Wu et al., 2025] to generate the Euler angle representation, and a combination of the hypertorus and hypersphere BFN to produce the axis-angle representation. As shown, both representations fail to capture the underlying SO(3) group structure, resulting poor SO(3) generative modeling performances.
18

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data dist.
FoldFlow-base
(c) FoldFlow-OT
(d) FoldFlow-SPM
(e) ProBayes
(I) Euler angle
(g) Axis-angle
Figure 7: The data is visualized using the Euler-angle representation of the rotation matrices. (a)-(d) are from Bose et al. [2023]
Table 5: Toy dataset performance comparison. Baseline results are from [Bose al., 2023].
Method
W1 (x10-2, ()
W2 (x10011 1
FoldFlow-base
5.39 + 0.88
1.52 0.27
FoldFlow-O1
4.96 + 0.27
1.25 = 0.12
FoldFlow-SFI
4.92 + 1.56
1,26 = 0.49
Simulated SDE
5.13 + 1.36
1.33 = 0.44
ProBayes
3.23 = 0.68
0.99 = 0.33
D Detailed training and sampling procedure
We describe the training loss functions used for the remaining modalities:
C. position p training loss Sampling μº= from pf. the training loss for position p is calculated as the discrete time loss for continuous Euclidean space variable:
VP 10P, .111
n1 Cp
2/n 2/ (1 71
E (-((L.)) p
(32)
2
2/n
Sequence S training loss Sampling 0 from ps, the training loss is the n-step discrete-time loss for discrete variable:
Ls = n
In (y as (Kes 1). (a;KK)K))
WU (05)): (S:t: (KW
K
En
Σ
( 0 [::44-)W ( (1(r) as (Kor - 1), (x
(33)
where I E RRXLLL and ER
Angle X, v training loss Sampling 0x from pxz the training loss is the n-step discrete-time loss for periodic variable:
(as)) (1 cos(x ((0P(),t-))) 0: Io(a;)
Cx = 1~ (1,2)))) (2 (0x
(34)
Note that for X. we only calculate the loss for those non-padded values. Also, sampling 0 from p the training loss for v0 is:
04i (as)) (1 cos(i- 1(0,_,44-11 Io(a;)
C = n
(35)
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Algorithm Training
1: Require: n, K 2, E R* E ax ai. ii,,
oi. E R* Aq; As, As, Ax; tu Aso RR
Input: sequence S. position p. orientation R. backbone torsion sidechain torsion X. X mask Mask (i=1)
3: Sample 011,m),
4: # randomly select one from the two quaternions
5: q ~ (())
6: # sampling from the Bayesian flow distribution of each modality
7: 09 p (0 (
8: 0 ~ ~Pg (0
S;t)
9: nº ~ ( p: 1P t- {uP}
10: m (m X: as) c px
11: m. +m, Maskx(y e²ll Mask - 1, 0x + {m?
12: m (mp Wi and of ~ (c Wi; aj.) 0" + [c,
13: feed all protein modalities into the network
14: 0" -(0-, 0r, RR .0}
15: ws uP, ya yr, 4x + vi ,t)
16: # calculate losses
1estia (1 - dot(g, (2((""))) 17: La < a, Ivla))
18: Ls NBBiyll, 1), In (y a (Kes - a),k,
19:
-E In [E Po
(k (k 0 :t- (v(0) | (Kee - 1), 1),,
20: Cp n (1 (1-of 2/n) p- 100
21: fx + n a. In(ax (1 cos(x 4x vx.
[1(
22: fx + fx .Msk xaxxx((S (S)
23: Ly f- n a Inla?)
[1(a;)
(1 - cos(th - (0,
24: Optimize Cq AsLs Aphp Axxx du du
Algorithm 2 Sampling
Require: n, K E 2, 01 E R B1 E Rt, aat, a alm E 1R+
# initialize the prior parameters
n² Uga k² 0, 8t {n², x²); 0. Lixis 4² 0, (1²}
mx U(, 2m) x 0, 0x + (mx c*); m + (0,277), 0x {m c*} for irritl 1,... ntt i-1
# use network to do inter-dependency modeling across dimensions of all modalities g + {oS OP, 0 0 (x)
S yp, R yo t v .t)
ifi n then
break end if
update each modality with Bayesian flow
8a ~ (0 R. a
0 pe ( ((:))
@P - u2 Pe p)t)
m ~ pf (m xi at)), cx ~ px (c (x
mx t mx Mask (w)) c1 (VS))( 1, 0x (mX ox)
m. py (m/ an of (" 00 + {m, c
for
Return V App R VV, x
E Limitations and broader impacts
Limitations Despite the promising results ProBayes, we acknowledge that the binding affinities and complex stability assessed by Rosetta serve as an initial screening step. Real-world properties of the generated candidates still require validation through costly wet-lab experiments. However, this challenge is shared across the research community.
20

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Broader impacts Our work contributes to the advancement of unified, end-to-end protein design, potentially enabling more accurate generation of functional proteins. This has implications for therapeutic discovery, enzyme engineering, and synthetic biology. However, as with any generative biological technology, careful oversight is essential to prevent unintended misuse, such as the design of harmful or dual-use proteins.
F Experiment details
Algorithm 3 Invariant point attention with quaternion implementation (IPAq)
Require: {s:]: ): (T:²), Nhead 12, c 16, Nquery points
4, poiint values
1: q. k o = (inearNoBias(ss))
{q) k V E R., h E {I,.... Nead } hp k hp E R . {1, .I..... N query points q,
2: q. hp k LinearNoBias(s:)
3: o LinearNoBias(s))
hp E R .pE E {1, {l..... point values
4: b LinearNoBias(z2) (2ig)
5: we
9 query
6: WL
T oq. hp - - o k/pp )²)
7: 2 = softmax,
A
WL
1 vogt Tk bbj - we
2 Ep
8: 0 E; a Ris
9. of Ej a m)
10: 0
Ap = (T E, ai T T o
11: 8; Linear concataa. (0)) of " (1,;
12: return
}
We provide our code in https: het//ggtttbbb wtt.au/..c.a//eaSS-TUAAAAA/A//PP
F1 Rationalized information flow implementation
In fact, the rationalized information flow can be implemented in various ways using different parame- ter sizes by mixing modules, with the only constraint being the input and output specifications of each module. Here we illustrate our instantiations for each module. Firstly, we explain the key component, Le., IPA with quaternion implementation.
Invariant point attention with quaternion implementation We illustrate the invariant point atten- tion [Jumper et al., 2021] ]] implemented by the unit quaternion in Algorithm 3 using the orange color to denote the modification compared to the original IPA. Specifically, we only replace the original imple- mentation of the SE(3) transformation based rotation matrix and translation T, {R, t with the quaternion-based one e {q. t). For point with homogeneous representation p [zz, y, z, 0)"
and e the transformation is:
T opp q x -1
(36)
where x is the Hamilton product for two quaternions [Hazewinkel et al., 2006] and the inverse (7,))1=((())--
transformation (1,8)1 = ((g)=1,--0))00 -,, Such operations are antipodal invariant because q x p x
-q)-1= (-4)))()))))))) x x Therefore, the IPAq module defined in Algorithm is antipodal invariant transformation.
Sequence&BBcckbone Mixing Module implementation The implementation of this module builds upon the graph attention-based architecture FramePred [Yim et al., 2023b], with the following modifications: 1) The initial node embedding is mixed with the noised sequence 08, the noised backbone torsion angles 0*, and the timestep via an MLP as follows:
ho MLP(ho, e (aabeedd(000 (0S), embed (0*) (0*)),emeddd" embedd (t))
(37)
21

---

where embed", embed", embed" are embedders for sequence, angles, and time, respectively. 2) The IPA module modified as stated above and also in Algorithm 3. 3) The prediction of &a is removed and repositioned at the end of the Backbone & Side-chain Mixing module.
Backbone&Sidechaal Mixing Module implementation To capture all-atom geometry during backbone prediction, we introduce an additional multilayer perceptron (MLP) that embeds sidechain features. This sidechain embedding is then passed to the BackboneUpdate module [Jumper et al., 2021], enabling refined updates to the backbone frame. The sidechain embedding module com- prises three linear layers with bias terms, followed by LayerNorm layer. Conversely, to in- corporate backbone information into sidechain design, we employ another MLP-based module consisting of linear layers with bias terms- which jointly encodes the node embeddings from the Sequence&Backoone Mixing Module alongside the sidechain features.
F.2 Peptide design
Metries Consistent with previous research [Kong et al., 2024], we generate 40 candidate structures receptor for the sequence-structury co-design task and 10 candidates for each receptor-ligand pair the binding conformation generation task. The evaluation metrics are detailed as follows:
A G [Kong et al., 2024] The binding energy (in kcal/mol) calculated by Rosetta [Alford et al., 2017] evaluate the binding affinity of the generated peptide using the function InterfaceAnnalyzer.
Success rate [Kong et al., 2024] The proportion of successful designs i.e., those with AG < 0, out of all generated candidates.
DockQ [Basu and Wallner, 2016] comprehensive metric that evaluates the all-atom similarity at the interface between candidate and the reference complex.
RMSDc, [Kong al., 2024] The root mean square deviation (RMSD) of the C, coordinates between candidate and the reference structure, measured in Angströms (A).
Validity [Lin et al., 2024] refers to the proportion of peptides that are chemically valid, determined by whether the generated atomic bond lengths are within 0.5A above and below the ideal value.
Diversity [Lin al., 2024] is measured by the average pairwise distance of the generated peptides using TM score and sequence similarity.
Novelty [Lin et al., 2024] A generated peptide is considered novel if both its TM-score and sequence overlap with the reference are both below 0.5.
Note that the diversity and novelty are evaluated combining the validity to avoid false positives.
RMSD stom [Kong al., 2024] on all atoms measure the quality of the all-atom geometry for the binding conformation generation task.
Baselines For RFDiffusion, set the number of cycles to one and disable the empirical force field refinement to ensure a fair comparison with other methods. For PepFlow, we use the official implementations and retrain the models on the same datasets using the default hyperparameters provided in its repository. For PepGLAD, most evaluation metrics are taken directly from the original paper, while the design metrics are computed using the official checkpoint available in its repository.
F.3 Antibody design
Following Ye et al. [2024], we generate 64 candidates for each receptor in the sequence-structure co-design task and report the average performance across these candidates.
Metries The metrics for antibody design include:
AAR [Ye et al., 2024] The amino acid recovery rate, calculated as the number of residues in the generated CDR-H sequences that match the reference antibody;
RMSD [Ye et al., 2024] The root-mean-soe -quure deviation, measured between the generated and natural antibodies using the Ca coordinates of the CDR-H3 region.
Footal Ye al., 2024] The total energy computed using Rosetta's 's full-atom score function with the default REF15 weight set, evaluated on the CDR-H3 regions.
22

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AG [Ye et al., 2024] The binding energy, analogous to that used in peptide design, but here evaluated between the CDR-H3 region and the antigen.
Baselines We directly borrow the baseline results reported in [Ye et al., 2024].
F.4 Other implementation details
Computation resources All experiments in this paper conducted node with 8 NVIDIA A100 80GB. Each training task requires GPUs.
Relaxation For peptide design, we adopt the relaxation procedure described in Kong et al. [2024], which optimizes the full all-atom structure. In contrast, for antibody design, we employ the protocol from Ye et al. [2024], which performs relaxation exclusively side-chain atoms while keeping the backbone atoms fixed.
Hyperparameters For all experiments, the network is configured with node embedding size of 128 and an edge embedding size of 64. Node embeddings the IPAq module have 128 dimensions, and edge embeddings have 64 dimensions. The IPAq attention mechanism comprises heads, with 8 query-key points and 12 value points for geometric attention. Additionally, sequence transformer is integrated into the encoder, featuring 4 attention heads and 2 layers to enhance contextual representation. The entire encoder architecture consists of 6 stacked IPAq blocks, enabling deep and expressive modeling of the input molecular graph structure. Finally, the network consists of 7202 million parameters. For optimization, we set the learning rate to 10 and use a batch size of 40 per distributed node with Adam optimizer.
G Comparison between BFN and SDES/ODES
A recent study by Xue et al. [2024] explores potential connections between continuous-time Bayesian Flow Networks and SDEs. However, we clarify that the BFNs are fundamentally distinct from SDE/ODE with evidence as follows:
1. The Bayesian flow distribution, as defined in Graves al. [2023], does not involve differen- tial terms, which are central to the formulation of SDEs and ODEs.
2. The Bayesian flow formulation employed in this work is discrete-time and does rely on any discretization of continuous-time dynamics, in contrast to SDEs- and ODEs-based approaches.
3. In both the hypertorus Bayesian flows [Wu et al., 2025] and the hypersphere Bayesian flows introduced in this paper, the accumulated accuracy parameters к; are inherently stochastic. These stochastic components cannot be fully represented or captured within conventional SDE framework.
4. The connection proposed in Xue et al. [2024] is not mathematically rigorous: it introduces a truncation of the time domain from [0, 1] to [0, 1011) 7/. where 1) a parameter, thereby deviating from the standard continuous-tima SDE formulation.
H
More results
H.1
Error bars
Table 6: Peptide design task results with error bars representing standard deviations from three independent experiments using distinct random seeds.
Dataset
Method
Energy Success(1)
DockQ(t)
Native Likeness
RMSDe
Valid
V&Div(1)
Design
V&:Novek
PepBench
ProBayes -28 03
72.92-0.79
0.740=0.037
2.28-0036
0.000-
0.449-0.0021
0.728+0.0014
We report the error bars for the protein design task, as shown in Tab. 6, based on three independent experiments using different random seeds.
23

---

H.2 Visualization
We provide the visualization of designed peptides and antibodies in Fig. and Fig. 9.
(a) Reference:
AG -34.21,
(b) Generated: AG -39.42,
(c) PepGLAD: AG 300.1, PDB=4cu4
PDB=4cu4
PDB=4cu4
m
mm
(d) Reference:
AG = -33.32,
(c) Generated: AG = -33.88, PDB=3pkn
(f) PepGLAD: AG 11339, PDB=3pkn
PDB=3pkn
Figure 8: Visualization designed peptides.
(a) Reference: AG 18..76, PDB=5j13
(b) Generated: AG -23.64, PDB=5j13
(c) dyMEAN: AG +23911 1.25. PDB=5j13
(d) Reference:
AG -0.45.
(e) Generated: AG -7.42,
(I) dyMEAN: AG
PDB=4dvr
PDB=4dvr
+11654.77, PDB-4dvr
Figure 9: Visualization of designed antibodies.
24

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I Further discussion of information shortcut
Protpardelle [Chu et al., 2024] tries remedy the information leakage by noising all 37 unique atom position inputs instead of only the atoms corresponding to the sequence. However, we have proved in the main text that even noisy side-chain information can create an information shorteut for the sequence prediction. Therefore, the information shortcut still exists in Protpardelle.
25

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Reviews:
[{"review_id": "FBsRfZDQHe", "reviewer": "Reviewer_R3Sb", "summary": "The paper introduces ProteoBayes, a Bayesian Flow Network-based all-atom protein design model. It takes use of the homomorphism from SU(2) to SO(3) and maps the orientation matrix into a 4-dimensional sphere, which makes Bayesian posterior computation tractable. Further, it resolves the issues of information shortcut in all-atom protein generation by a novel information flow. It also achieves significant improvements in peptide design and the antibody design task.", "questions": "1. Does information shortcut exist in the next time step? The sequence parameter current time will meet the Sidechain Torsion parameter in the next time step.\n\n3. Most said BFN is equivalent to diffusion in certain conditions. In the last paragraph in the Appendix, it says BFN does not involve differential terms and is discrete-time in this work. Can the differential term cancel out by score matching in the diffusion model? Can a discrete-time BFN be a discrete version of a continuous-time diffusion model?", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n1. The paper is logically coherent and well written.\n\n2. It gives a rigorous proof of homomorphic mappings and carefully constructs an Antipodal distribution on S^3.\n\n3. The work discovers and proves the existence of an information shortcut and designs an information flow that separates information from sequence and that from sidechain torsion.\n\nWeaknesses\n1. This work does not provide the training and sampling time cost and memory cost.\n\n2. It lacks comparison with other methods in the visualization part. \n\n3. The paper lacks detailed information regarding the calculation of evaluation metrics, such as the specific parameters used.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful feedback and thoughtful suggestions. Your comments will be invaluable in improving the quality of our work. Our responses to your concerns are as follows:\n\n> **W1: About the training, sampling and memory cost**\n\nWe appreciate the reviewer's suggestion in our computational resource specifications which will further improve the quality of our paper. Our framework utilizes four A100 GPUs for each training task and one A100 GPU for inference. Below, we provide comprehensive details regarding the time and memory requirements for both training and sampling processes (these specifications will be incorporated into the revised manuscript):\n\n|                           |          | Batch Size | Time     | GPU Memory    | RAM Memory   |\n| ------------------------- | -------- | ---------- | -------- | ------------- | ------------ |\n| Peptide Design (PepBench) | Training | 40 x 4     | ~ 55h    | ~ 25663 x4 MB | ~3216 x 4 MB |\n|                           | Sampling | 40         | 1h 33min | ~ 3294 MB     | ~3122 MB     |\n| Antibody Design           | Training | 12 x 4     | ~ 51h    | ~45237 x 4 MB | ~3297 x 4 MB |\n|                           | Sampling | 128        | 4h 53min | ~ 49362 MB    | ~3846 MB     |\n\n> **W2: About the comparison with other methods in the visualization part**\n\nWe are truly grateful for your valuable suggestion, which will enhance the quality of our paper providing readers with a clearer understanding of our approach's advantages. However, in accordance with official policy: \n\n*\"Because of known concerns on identity leakage, we prohibit using any links in the rebuttal, including but not limited to anonymous or non-anonymous URL links, or updating your submitted GitHub repository. We understand that 1) and 2) effectively prevent using image/video or other rich media as a communication protocol between reviewers and authors. We are sorry about that.\"*\n\nWe are unfortunately unable to provide the requested visual comparison at the rebuttal stage.\n\nNevertheless, we would like to promise that we will include the visual comparison with existing methods in the future version of the paper.\n\n> **W3: Detailed information on calculating evaluation metrics**\n\nThank you for pointing out the need for a more detailed explanation of our evaluation process. The comprehensive information regarding our evaluation metrics calculation process below: During the training stage, for each task, including peptide co-design, peptide binding conformation generation, and antibody co-design, we re-train a randomly initialized model on the specific training set (PepBench, SAbDab). \n\n- Peptide Co-Design\n\n  The procedure for calculating metrics follows the approach outlined in [1] and [2]. Specifically, our evaluation metrics code is sourced from the official PepGLAD and PPflow repositories.\n\n  1. We sample  $N_{samples}=40$  peptides based on the binding pocket of each receptor in the test set, as described in Algorithm 2, Appendix D, with 200 steps.\n  2. After peptide sampling, we perform Rosetta relaxation (with parameter `-relax:default_repeats 2`) and use the `InterfaceAnalyzer` function in PyRosetta (the version we use is 2024.35) to compute the binding energy using the PepGLAD’s script in its official repository.\n  3. We then use the `cal_metrics` function in PepGLAD’s script to calculate DockQ and $\\text{RMSD}\\_{\\text{C}\\_\\alpha}$.\n  4. For design-related metrics, we rely on scripts from PPFlow to calculate validity, sequence similarity, and structure similarity. Finally, we average these design metrics across the  $N_{samples}$  and average the metrics across receptors.\n\n- Peptide Binding Conformation Generation\n\n  The procedure for calculating metrics follows the methodology described in [1].\n\n  1. We sample  $N_{samples}=10$  peptides based on the binding pocket of each receptor and peptide sequence in the test set.\n  2. After sampling, we conduct Rosetta relaxation and use the `InterfaceAnalyzer` function in PyRosetta to compute the binding energy following the same protocol as PepGLAD’s script.\n  3. Additionally, we use the `cal_metrics` function in PepGLAD’s script to calculate DockQ, $\\text{RMSD}\\_{\\text{C}\\_\\alpha}$, and $\\text{RMSD}\\_{\\text{atom}}$.\n\n- Antibody Design\n\n  This process follows the procedure described in [3].\n\n  1. We sample $N_{samples}=64$ CDR-H3 sequences for each antigen in the test set.\n  2. Next, we apply the relaxation protocol from [3], which relaxes only the side-chain atoms while keeping the backbone atoms fixed.\n  3. Subsequently, we use the `InterfaceAnalyzer` and the `create_score_function('ref2015')` function in PyRosetta to compute both the binding energy and total energy.\n  4. To calculate RMSD and AAR, we use the `cal_metrics` function in dyMEAN’s [8] script.\n\nIf you require additional information or clarification on any of the methodologies described above, please do not hesitate to let me know!\n\n> **Q1: The possibility of an information shortcut in the next time step?**\n\nThank you for your insightful question! We find this question particularly meaningful and worthy of discussion. \n\nWe understand your concern: during sampling, given the dependency relationship $\\theta^\\chi\\_{i-1} \\overset{\\Psi}{\\leftarrow} \\hat{\\textbf{p}}\\_i \\overset{h}{\\leftarrow}\\theta^{\\textbf{p}}\\_{i} \\overset{\\Psi}{\\leftarrow} \\hat{\\mathcal{S}}\\_{i+1}$, it appears that the sequence prediction $\\hat{\\mathcal{S}}\\_{i+1}$ might depend on side-chain information via the backbone position $\\hat{\\textbf{p}}\\_i$ from two time steps prior $\\theta^\\chi\\_{i-1}$. \n\nHowever, this does not actually create an information shortcut learning problem for the following reasons: \n\n- During training, the loss function is based solely on a single network prediction $\\theta^\\chi_{i-1} \\overset{\\Psi}{\\leftarrow} \\hat{\\textbf{p}}_i \\overset{\\textbf{p}}{\\leftarrow} \\mathcal{L}^{\\textbf{p}}$ rather than on multiple network predictions. Therefore, the network is unable to learn the correlation across two steps.\n- During sampling, although the network  $\\Psi$ may potentially utilize the sequence information from the previous step, the network parameters are not updated (meaning the network can not utilize the side-chain sequence relationship across two steps), so from this perspective, no information leakage occurs.\n\n> **Q2: Discussion of the equivalence between BFN and diffusion model**\n\nThat's a great point to discuss where we are very happy to share our views on different generative models!\n\n- **Can the differential term cancel out by score matching in the diffusion model?**\n\n  We humbly believe that the differential term can not be cancelled out because the definition of the ground truth score $\\nabla_{\\mathbf{x}} \\log p(\\mathbf{x})$ in score matching still involves taking the derivative of a probabilistic density function, while the Bayesian update function $h$ does not include derivative terms. \n\n- **Can a discrete-time BFN be a discrete version of a continuous-time diffusion model?**\n\n  From the perspective of marginal distributions, we believe a discrete-time BFN can be approximated by carefully constructing a discretization of SDE that shares the same marginal distribution $p_t(\\textbf{x})$ as seen in the approach in [4], particularly for Gaussian distribution-related BFNs.\n\n  However, we must emphasize that **sharing the same marginal distribution does not establish the exact equivalence** between BFNs and diffusion models as generative modeling approach because:\n\n  1. BFN offers a discretization-free and analytical approach to precisely update states according to the noised sample and its accuracy based on the Bayesian theorem, whereas diffusion models require carefully designed discretization schedules [5, 6].\n  2. SDEs cannot fully capture all Bayesian flows. The SO(3) Bayesian flow used in this paper incorporates accumulated accuracy parameters $\\kappa_i$ that are inherently stochastic and cannot be fully represented within a conventional SDE framework.\n  3. Sharing the same marginal distribution does not guarantee equivalent generation quality, as seen with SDE and ODE: Although the SDE and its corresponding probability flow ODE share the same marginal distribution [7], their generation quality differs in practice [5].\n\n[1]  Kong, Xiangzhe, et al. \"Full-atom peptide design with geometric latent diffusion.\" *Advances in Neural Information Processing Systems* 37 (2024): 74808-74839.\n\n[2] Lin, Haitao, et al. \"Ppflow: Target-aware peptide design with torsional flow matching.\" *arXiv preprint arXiv:2405.06642* (2024).\n\n[3] Fei, Y. E., et al. \"ProteinBench: A Holistic Evaluation of Protein Foundation Models.\" *The Thirteenth International Conference on Learning Representations*.\n\n[4] Xue, Kaiwen, et al. \"Unifying bayesian flow networks and diffusion models through stochastic differential equations.\" *arXiv preprint arXiv:2404.15766* (2024).\n\n[5] Karras, Tero, et al. \"Elucidating the design space of diffusion-based generative models.\" *Advances in neural information processing systems* 35 (2022): 26565-26577.\n\n[6] Tong, Vinh, et al. \"Learning to discretize denoising diffusion odes.\" *arXiv preprint arXiv:2405.15506* (2024).\n\n[7] Song, Yang, et al. \"Score-based generative modeling through stochastic differential equations.\" *arXiv preprint arXiv:2011.13456* (2020).\n\n[8] Kong, Xiangzhe, et al. \"End-to-end full-atom antibody design.\" Proceedings of the 40th International Conference on Machine Learning. 2023."}, {"review_id": "T6KBoPkSAN", "reviewer": "Reviewer_2sZG", "summary": "The paper proposes a Bayesian flow method for protein sequence-structure co-generation, including sidechains. Authors propose a new Bayesian flow formulation for SO3 generation, needed for protein backbones. They show, that sequence-structure models that incorporate sidechains, can provide a shortcut for sequence prediction, that negatively impacts model training. They adapt their architecture to avoid this, by sequence part of the model being blind to the current sidechain positions. The method is benchmarked on peptide and antibody (CDR H3) design tasks. Where they show superior performance to baselines.", "questions": "Are there some other (simpler) solutions for the shortcut issue? For example one could use a training mask and pad with arbitrary (or random) values, that would still be further noised/denoised, just without training loss applied to them during training. At least equation 12 would then stop working. Such non-0 padding strategies have been used, like with 'ghost' sidechain atoms in AbDiffuser (https://arxiv.org/pdf/2308.05027, it also makes sense to include it in the related work). Although never with the aim of improving sequence prediction performance. It would be curious to see what fraction of the issue comes from padding, and which from the simple fact that noised non-padded angles always will carry some information about the sequence, even if it's hard to pull out.\n\nSpeaking about shortcuts, why in Fig 3 b) the training performance with the shortcut is so much worse? One would assume that a properly tuned model should still fit the training set just as well with extra (shortcut) information that makes the task easier.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "To the best of my knowledge this is the first work using Bayesian flows for protein generation. The proposed formulation adapting BFNs to S03 looks sound and is new as far as I know. I think trying out new types of generative models for proteins is quite valuable for the field.\nThe paper is well written and easy to follow.\n\nThe insight made about shortcut learning is quite valuable and new. It also is nicely backed by in-silico KL divergence experiments. \n\nThe experiments are quite standard in the field and reasonable extensive. Including ablations of the proposed method. The antibody experiment (Table 3) experiments could include some newer and harder baselines, like AbX (https://openreview.net/pdf?id=1YsQI04KaN), that now is not cited.\nBut in the end I can't see much wrong or lacking in the paper.", "was_revised": true, "final_justification": "The authors addressed the main points I raised, thus i recommend acceptance now.", "rebuttal": "We sincerely appreciate the reviewer’s recognition of our methodology and the identification of the information shortcut. We address your concerns as follows:\n\n> **W1: About including AbX as a baseline**\n> \n\nThank you to the reviewer for providing this valuable suggestion, we will add the citation and comparison to AbX in the future version. Our experiment on including AbX is as follows:\n\nWe use the checkpoint from AbX's code repository to perform inference, relaxation (only relax side-chain) and calculate metrics under our experimental setup. The results are as follows:\n\n| Method | AAR ↑ | RMSD ↓ | E$_{\\text{total}}$ ↓ | $\\Delta \\text{G}$ ↓ |\n| --- | --- | --- | --- | --- |\n| Test Set | 100.0 | 0.00 | -16.76 | -15.33 |\n| dyMEAN | 40.05 | 2.36 | 1239.29 | 612.75 |\n| DiffAb | 35.04 | 2.53 | 495.69 | 489.42 |\n| AbDPO | 31.29 | 2.79 | 270.12 | 116.06 |\n| AbDPO++ | 36.25 | 2.48 | 338.14 | 223.73 |\n| AbX | 41.27 | 2.40 | 281.29 | 237.22 |\n| ProteoBayes | **42.12** | **2.27** | **54.68** | **44.12** |\n\nBased on the table, AbX demonstrates strong performance among the baseline methods, while ProteoBayes still consistently outperforms all competitors across each evaluated metric.\n\n> **Q1: Will using training masks or other/random padding values work? Are there other simpler solutions for the shortcut issue?**\n> \n- **Will using training masks or other/random padding values work?**\n\nThank you for your insightful idea. You've raised an excellent point that we will address to improve our paper's clarity in the future version. And thanks for pointing out a relevant work AbDiffuser, which we will include in the related work in the future version. \n\n- For training mask, it is hard to apply an appropriate mask\n    \n    This is because in the sequence-structure co-design scenario, we face a fundamental challenge, i.e., sequence determination only occurs at $t=1$. Prior to this point, we work with a probability distribution $\\theta^{\\mathcal{S}}_t$ across 20 possible residue types for $t\\in[0,1)$. Consequently, we cannot definitively establish the length of $\\chi_t$ or implement appropriate masking during training. Therefore, current approaches [3, 4] remedy this by modeling the maximum possible number of random variables throughout both training and sampling processes. \n    \n- For random padding, it might alleviate the problem while patterns still exist\n    \n    We acknowledge that the random padding might make it more difficult to directly utilize sequence information in the side chains (making Eq 12 stop working). A fixed random distribution's pattern is still possible to be utilized. \n    \n\nMore importantly, we would like to further explain that:\n\n- **The information shortcut still exists** even without the padding issues.\n\nThis is because the side chain torsion distributions themselves (those non-paddings values) are **inherently dependent on the specific residue types**. \n\nFor example, the $\\chi_2$ distribution of Tyrosine exhibits $\\pi$-rotation symmetry, i.e., $p^{\\text{Tyr}}(\\chi_2)=p^{\\text{Tyr}}(\\chi_2+\\pi)$ , while this symmetry does not exist for Isoleucine $p^{\\text{Ile}}(\\chi_2) \\neq p^{\\text{Ile}}(\\chi_2+\\pi)$  [1, 2].\n\nSuch dependency between residue type and side-chain conformation even forms the foundation of the rotamer library concept [3, 4], where scientists determine possible side-chain conformations based on the sequence and search for the lowest energy combinations to predict protein conformation.\n\nConsequently, the network can still exploit this sequence-sidechain correlation as an information shortcut to \"predict\" residue types, regardless of whether training masks or alternative padding values are used. In the future version, we will include visualizations of torsion angle distributions from the PepBench training dataset to illustrate this (as providing images is officially forbidden during the rebuttal stage). These visualizations clearly show that side chain torsion distributions display distinct patterns based on residue type, representing **another source of information shortcut** beyond padding.\n\nFinally, we sincerely appreciate your thoughtful feedback.\n\n- **Are there other (simpler) solutions for the shortcut issue?**\n\nWe deeply appreciate your recognition of our identification of this issue. \n\nFor now, we believe there are two other possible solutions: \n\n1. One promising approach would be introducing a neural network module $\\Psi^{\\text{detach}}(\\theta^{\\chi})=h^{\\text{noseq}}$ to detach and output sequence-agnostic features $h^{\\text{noseq}}$ from side-chain embedding $\\theta^{\\chi}$. This module could be adversarially trained to minimize the classification accuracy of a discriminator $\\text{Classifier}(h^{\\text{noseq}})$ that attempts to predict the residue type $\\mathcal{S}$ according to the detached sequence-agnostic features $h^{\\text{noseq}}$.\n2. Another approach would be to carefully design a noise schedule or flow path for side-chain information such that the sequence information implied in side-chain input $\\theta^{\\chi}_t$ will never exceed that contained in sequence input $\\theta_t^{\\mathcal{S}}$. Intuitively, this approach aim to moving the cross point in Figure 3(a) to occur at $t=0$. \n\nHowever, these two methods also have potential problems:\n\n1. Adversarial training is often unstable and prone to degradation. How to prevent the adversarial loss from affecting the overall optimization objective is an important issue.\n2. Such a schedule design would be ad hoc, limiting design flexibility and possibly affect the quality of sequence or side-chain generation.\n\nCompared to these alternatives, our current information flow approach is more fundamental, easy to implement, and elegant.\n\n> **Q2: Why is the training performance with the shortcut so much worse?**\n> \n\nThis is an excellent question! First, we consider that your intuition is quite reasonable—providing more information to the network should not decrease training performance. However, the optimization dynamics might not be that ideal in practice.\n\nOur explanation for this phenomenon is that: In a single-modality scenario, providing additional information during training would typically lower the loss. \n\nHowever, in the co-design setting, the sequence prediction head leveraging the shortcut rapidly converges toward a **local optimum**. Additionally, the representations developed for prediction with the shortcut deviate substantially from those of other modalities that are genuinely attempting to predict the ground truth, resulting in gradient conflicts between the different optimization objectives. \n\nThese conflicting gradients make it difficult for the network to **compress all inputs for a general meaningful representation**, and ultimately leading to suboptimal performance.\n\nThis explanation is supported by the loss curve in Fig. 3(b) and the gradient norm curves (which we will include in the future version, as updating images is officially forbidden during the rebuttal stage). During the later stages of training, networks with shortcuts consistently show significant fluctuations in the loss function, accompanied by higher gradient norms. This indicates that the network is actively trying to escape the local optimum but remains trapped within it.\n\n[1] Zhang, Yangtian, et al. \"Diffpack: A torsional diffusion model for autoregressive protein side-chain packing.\" *Advances in Neural Information Processing Systems* 36 (2023): 48150-48172.\n\n[2] Jumper, John, et al. \"Highly accurate protein structure prediction with AlphaFold.\" *nature* 596.7873 (2021): 583-589.\n\n[3] Dunbrack Jr, Roland L. \"Rotamer libraries in the 21st century.\" *Current opinion in structural biology* 12.4 (2002): 431-440.\n\n[4] Bhuyan, Md Shariful Islam, and Xin Gao. \"A protein-dependent side-chain rotamer library.\" *BMC bioinformatics* 12.Suppl 14 (2011): S10."}, {"review_id": "vHeSKYyp9R", "reviewer": "Reviewer_XpUT", "summary": "This paper presents ProteoBayes, an end-to-end framework for all-atom protein design based on a unified multi-modal Bayesian flow. The method jointly generates amino acid sequences, backbone structures, and side-chain conformations while addressing challenges related to modeling inconsistency and information leakage. A key contribution is the formulation of a Bayesian flow over SO(3), achieved by mapping rotations to unit quaternions on the hypersphere $S^3$ and enforcing antipodal symmetry to ensure valid generation. Additionally, the authors propose a rationalized information flow design that prevents side-chain torsion angles from leaking residue-type information during sequence prediction, improving generalization. Experimental results show that ProteoBayes outperforms prior methods in peptide and antibody design tasks.", "questions": "- Why did the authors choose to represent side-chain torsion angles with a fixed-length $\\chi$ vector padded to four angles, rather than using a residue-type-dependent dynamic length or masking? This design appears to introduce an artificial information leakage issue rather than reflecting an inherent modeling challenge.\n- The paper does not clearly specify the training setup: was a single universal model trained across all design tasks, or were separate task-specific models used? Please clarify the datasets, model checkpoints, and training regimes.\n- In the design setting, what inputs are provided to the model? If receptor structure and sequence are included, the current evaluation does not fully validate the framework in realistic pipeline scenarios—can the authors elaborate on this?\n- Given that the model predicts backbone positions p and orientations R (or quaternions q), why is it necessary to explicitly model the backbone dihedral angle $\\phi$? This variable may be redundant unless justified.\n- Have the authors considered refolding the generated sequences using AlphaFold 3 and computing structural quality metrics (e.g., RMSD, TM-score, binding energy)? Such an experiment could strengthen the validation of both peptide and antibody design results.", "limitations": "- The framework primarily integrates existing techniques—such as quaternion-based rotation modeling and antipodal symmetry handling—without introducing substantial theoretical innovation. \n- Experimental details are insufficient, and the evaluation would benefit from folding the designed sequences using AlphaFold3 to verify structural and functional fidelity.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1. Strong alignment between BFN and protein design modalities\n    The paper systematically applies the BFN framework to all-atom protein design, effectively modeling diverse data types: vMF distributions on $S^3$ for backbone rotations, circular distributions for side-chain torsions, and discrete BFNs for residue types. This highlights BFN’s flexibility and simplicity across continuous, periodic, and discrete modalities in structural biology.\n\n2. Theoretically grounded modeling with principled handling of symmetry and information flow\n    The method extends BFN to non-Euclidean spaces via antipodal symmetry constraints, enabling valid generation over SO(3) through the unit quaternion space $S^3$. \n\nWeaknesses\n\n1. The information leakage issue stems from the authors’ modeling choice rather than the task itself. By representing all residues with fixed-length $\\chi$ vectors padded with zeros, the model implicitly encodes residue-specific torsion patterns into the input. This design choice introduces leakage by construction. The issue could be fundamentally avoided by using dynamic χ lengths or residue-type-based masking.\n\n2. The SO(3) to $S^3$ transformation lacks novelty. Modeling rotations with quaternions and addressing antipodal symmetry are well-established techniques. The paper does not introduce new mathematical insights, but rather applies standard tools within the Bayesian flow framework—an engineering integration rather than a theoretical contribution.", "was_revised": true, "final_justification": "The rebuttal provided clear clarifications on the source of information leakage, demonstrating it is partly intrinsic to the data, and added AlphaFold3 refolding experiments that strengthen validation. Details on datasets, training setups, and input configurations resolved earlier ambiguities. The SO(3) $\\rightarrow$ quaternion projection work is a genuine theoretical contribution in this context, though its novelty remains moderate given prior uses in other domains. Overall, most technical concerns were addressed, but scope limitations and moderate novelty lead me to retain my original score.", "rebuttal": "Thank you for your thorough review and insightful suggestions. Your feedback will help us significantly improve our work. We address your concerns as follows:\n\n> **W1 & Q1: About the source of information leakage: author’s modeling choice or the task itself?**\n> \n\nWe greatly appreciate your insightful observation, which raises an excellent point that we will clarify and address to improve our paper's presentation! \n\n- **Why do we use a fixed-length $\\chi$ vector rather than a dynamic length or masking?**\n\nThis is because: In the sequence-structure co-design scenario, we face a fundamental constraint: sequence determination only occurs at $t=1$. Prior to this point, we work with a probability distribution $\\theta^{\\mathcal{S}}_t$ across 20 possible residue types for $t\\in[0,1)$. Consequently, we cannot definitively establish the length of $\\chi_t$ or implement appropriate masking during training. Therefore, current approaches [3, 4] remedy this by modeling the maximum possible number of random variables throughout both training and sampling processes. \n\nNevertheless, we would like to emphasize that even in the absence of this issue, the information shortcut remains an intrinsic property of the underlying protein data distribution, as detailed below:\n\n- T**he information shortcut can still exist even without paddings / with masks**\n\nWe would like to further explain that **the information shortcut can still exist even without paddings with masks**. \n\nThis is because the side chain torsion distributions themselves (those non-paddings values) are **inherently dependent on the specific residue types**. Such dependency even constitutes the idea of the rotamer library [5, 6], where scientists determine a set of possible side-chain conformations according to the sequence and search for the lowest energy combinations to predict the protein conformation. \n\nAs a distinct example, the $\\chi_2$ distribution of Tyrosine is $\\pi$-rotation-symmetric, i.e., $p^{\\text{Tyr}}(\\chi_2)=p^{\\text{Tyr}}(\\chi_2+\\pi)$ , while this symmetric property does not hold for Isoleucine $p^{\\text{Ile}}(\\chi_2) \\neq p^{\\text{Ile}}(\\chi_2+\\pi)$  [1, 2]. \n\nTherefore, the network can still utilize such sequence-sidechain correlation as an information shortcut to “predict” the residue types even with dynamic $\\chi$ length or masking. \n\nFinally, we apologize for any insufficient explanations that may have caused misunderstandings, and we sincerely appreciate your thoughtful feedback.\n\n> **W2&L1: About the novelty of projecting SO(3) to $S^3$**\n> \n\nWe thank the reviewer for pointing out the literature to better highlight our contribution. \n\nWe would like to clarify that although rotation quaternions are widely used in computational graphics and other areas, how to correctly use them in generative modeling remains unclear. **Our theoretical contribution primarily lies in identifying its value in constructing conjugate Bayesian update, find the flaw in naive projection**, i.e., $p_{\\Psi}(-\\textbf{q})\\ne p_{\\Psi}(-\\textbf{q})$, and addressing this issue by proposing a simple yet effective approach to solidify and ground such transformation in generative modeling scenario. Table 4 demonstrates the necessity and effectiveness of the proposed techniques.\n\n> **Q2&L2: Clarification on training and evaluation setups: datasets, model checkpoints, and training regimes**\n> \n\nWe apologize for the caused confusion and thank you for pointing out the need for a more comprehensive explanation of our methodology. The detailed information regarding our training and evaluation setups are provided below: \n\n- Training\n    - For datasets\n      \n        For peptide design, following [9], we use two datasets: PepBench and PepBDB. For antibody design, we follow prior practices [11] utilizing the SAbDab database of antibody–antigen complexes as our training dataset and evaluate model performance using the RAbD.\n        \n    - Model checkpoints\n      \n        We train a randomly initialized neural network for each protein design task from scratch and use every task’s own checkpoint to evaluate.\n        \n    - Training regimes\n      \n        Our training regimes follow standard settings in the field. We use the AdamW optimizer with an initial learning rate of 5e-4 that gradually decays to 5e-6. Each training task runs for 100k steps.\n    \n- Evaluation\n    - Peptide Co-Design\n      \n        Our evaluation metrics code is sourced from the official PepGLAD and PPflow repositories\n        \n        1. We generate 40 peptides for each receptor in the test set .\n        2. After sampling, we perform energy minimization through Rosetta relaxation and calculate binding energy using PepGLAD’s scripts and settings.\n        3. We compute metrics like DockQ and RMSD $\\text{C}_{\\alpha}$  for each generated sample. For design-related metrics, we rely on scripts from PPFlow. Finally, we average these design metrics across the samples  and average all metrics across receptors.\n    - Peptide Binding Conformation Generation\n        1. We sample 10 conformations for each receptor-peptide pair in the test set.\n        2. After Rosetta relaxation, we calculate binding energy for each sample.\n        3. We evaluate structural accuracy using DockQ, RMSD $\\text{C}_{\\alpha}$ , and all-atom RMSD metrics.\n    - Antibody Design\n        1. We generate 64 CDR-H3 sequences for each antigen.\n        2. We apply a side-chain-only relaxation protocol while keeping the backbone fixed.\n        3. We evaluate both binding energy and total energy using the `ref2015` score function and assess design quality through RMSD and amino acid recovery (AAR) metrics using dyMEAN’s [11] scripts.\n\nIf you require any additional information or clarification regarding the methodologies described above, please don't hesitate to let us know!\n\n> **Q3: Elaboration on the input of the design setting**\n> \n\nWe appreciate the reviewer's feedback regarding realistic pipeline frameworks.  \n\nWe understand that there might exist gap between current in-silico protein design approaches and real-world implementation requirements. However, for fair methodological comparison with existing in-silico methods, we maintain consistency with established methodological practices for comparative evaluation purposes.  \n\n- Peptide Design\n  \n    Following [8, 9, 12], we use the receptor pocket's sequence and structure as input to our model. \n    \n- Antibody Design\n  \n    Following [4, 11], our model takes as input the sequence and structure of the antigen-antibody complex, excluding the CDR-H3 region that is being designed.\n    \n\n> **Q4: About the usage of the variable $\\phi$**\n> \n\nWe apologize for any confusion caused by our notation.\n\nThe backbone dihedral angle we model is $\\psi$, which determines the position of the O atom in the backbone frame as used in [2, 7, 8]. Figure 1 in [7] illustrates that: Although $\\textbf{R}/\\textbf{q}$ and $\\textbf{p}$ can determine the positions of $\\text{C}_{\\alpha}, \\text{C}$ and $\\text{N}$ atoms in the backbone, the oxygen atom in the backbone still has a torsional degree of freedom. Therefore, the variable $\\psi$ is needed to further determine its position.\n\nIn this draft, the notation $\\phi$ does not refer to the backbone dihedral angle. Instead, it denotes the surjective homomorphism $\\phi:\\text{SU}(2)\\rightarrow \\text{SO}(3)$ (defined in Section 3.2).\n\nTo avoid confusion, we will use a different notation for this mapping in the future version.\n\n> **Q5&L2: Refolding generated sequences using alphafold3**\n> \n\nThank you for your valuable suggestion on improving the quality of our evaluation. \n\nUsing the PepBench co-design task’s result, we fold the peptide sequences designed by ProteoBayes and PepFlow along with the reference receptor sequences using AlphaFold3 (no MSA for faster inference), and evaluated the structural similarity between the folded structures and AlphaFold3-folded structures. For the structures generated by AlphaFold3, we use the same relax protocol and evaluate their PyRosetta binding energy:\n\n|  | RMSD $\\text{C}_{\\alpha}$ ${\\downarrow}$ | RMSD$_{\\text{atom}}$${\\downarrow}$ | TMScore ${\\uparrow}$ | AlphaFold3 dG | Designed dG |\n| --- | --- | --- | --- | --- | --- |\n| PepFlow | 3.247 | 4.619 | 0.390 | -17.88 | -23.09 |\n| ProteoBayes | 2.370 | 4.070 | 0.445 | -17.93 | -28.77 |\n\nOur AlphaFold3 refolding experiments show ProteoBayes outperforms PepFlow on all structural metrics: lower RMSD values, higher TMScore, confirming ProteoBayes produces structures with better AlphaFold3 alignment than PepFlow.\n\n[1] Zhang, Yangtian, et al. \"Diffpack: A torsional diffusion model for autoregressive protein side-chain packing.\" *NeurIPS* (2023).\n\n[2] Jumper, John, et al. \"Highly accurate protein structure prediction with AlphaFold.\" *nature* .\n\n[3] Li, Jiahan, et al. \"Full-atom peptide design based on multi-modal flow matching.\" *ICML* (2024).\n\n[4] Zhu, Tian, et al. \"Antibody design using a score-based diffusion model guided by evolutionary, physical and geometric constraints.\" *ICML* (2024).\n\n[5] Dunbrack Jr, Roland L. \"Rotamer libraries in the 21st century.\" *Current opinion in structural biology* 12.4 (2002).\n\n[6] Bhuyan, Md Shariful Islam, and Xin Gao. \"A protein-dependent side-chain rotamer library.\" *BMC bioinformatics*.\n\n[7] Yim, Jason, et al. \"SE (3) diffusion model with application to protein backbone generation.\" *ICML* (2023).\n\n[8] Li, Jiahan, et al. \"Full-atom peptide design based on multi-modal flow matching.\" *ICML* (2024).\n\n[9] Kong, Xiangzhe, et al. \"Full-atom peptide design with geometric latent diffusion.\" *NeurIPS*  (2024).\n\n[10] Ye, Fei., et al. \"A holistic evaluation of protein foundation models.\" *ICLR* (2025).\n\n[11] Kong, Xiangzhe, et al. \"End-to-end full-atom antibody design.\" *ICML* (2023).\n\n[12] Lin, Haitao, et al. \"Ppflow: Target-aware peptide design with torsional flow matching.\" *ICML* (2024)."}, {"review_id": "nXWChvZdDf", "reviewer": "Reviewer_r2HE", "summary": "This paper proposes a novel all-atomic Bayesian flow framework for SO(3) generation. It has a rational information flow design to avoid the information shortcut from the side-chain to the sequence in joint generation. The work provides theoretical proofs and achieves good results on multiple benchmarks.", "questions": "1. The theory demands $f(-q)=-f(q)$, how the network architecture or training enforces this?\n\n2. Bayesian flows avoid fine-grained ODE/SDE integration and thus should be faster or use fewer steps. Is there any efficiency comparison to diffusion baselines?\n\n3. How sensitive are results to the choices of $\\alpha_i$ for the vMF noise, which controls how “noisy” each update is? \n\n4. Fully excluding side-chain inputs from sequence prediction removes the shortcut. Would this also discards useful structures like sequence correlations? Is there a balance of properly using side-chain hints?", "limitations": "The work has discussed its limitations in the appendix.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Introduces Bayesian flow on the rotation group by mapping SO(3) generation to antipodally symmetric hypersphere (S³) sampling, avoiding the numerical discretization errors typical of diffusion or flow-matching approaches.\n\n2. The authors identify and eliminate the “side-chain→sequence” shortcut problem by masking side-chain inputs in the sequence branch. Their ablation demonstrates that this design prevents degenerate learning.\n\n3.  Both the hypersphere and SO(3) bijection (Proposition 1) and the vMF Bayesian update rule (Eq. 5) are stated and proved, giving strong mathematical grounding. \n\nWeaknesses:\n\nThe motivation for applying Bayesian Flow Networks to protein design is important but not adequately explained. The details of the discretization errors and the problems they introduce to protein design should be discussed.", "was_revised": true, "final_justification": "I would like to keep my score of 4 for acceptance.", "rebuttal": "Thank you for your thoughtful comments and suggestions. We greatly appreciate your feedback, which will help us improve our manuscript. Our responses to your concerns are outlined below:\n\n> **W1: Discussion of discretization error and the motivation for applying BFN to protein design**\n> \n- The details of discretization error\n  \n    Due to the unknown analytical solution, using ODE/SDE requires numerically solving the continuous equations, which produces only an approximation of the true continuous trajectory. \n    \n    However, at each step, the solver introduces discretization error that accumulates over $n$ steps. The local error typically scales super-linearly with respect to step size. The commonly used Euler's method is a first-order ODE solver with $O(h^2)$ local error relative to step size $h$ [2]. \n    \n    As for BFN, the Bayesian update is analytical given conjugate priors, which avoids such numerical approximation.\n    \n- The problem discretization error introduce to protein design\n  \n    Unlike images which are relatively robust to noise, protein structure and function can deviate significantly when atoms' coordinates undergo small disruptions [3]. For example, in enzyme active sites, minor structural perturbations involving just a few key atoms at the angstrom scale can disrupt the geometric configuration of the active site, resulting in dramatic reduction or complete loss of enzymatic activity and catalysis [4]. \n    \n    Therefore, it would be ideal to avoid such sources of systematic error.\n    \n- Motivation for applying BFN to protein design\n    - Theoretically, BFN would bypass the discretization error, resulting in higher protein design quality.\n    - Practically, BFN has already demonstrated success in the general molecular generation domain, including applications with small molecules [5] and crystals [6].\n\n> **Q1: How does the network architecture $\\Psi$ enforce $f(-q)=-f(q)$ ?**\n> \n\nSorry for causing confusion. We find that such antipodal equivariance  **$f(-q)=-f(q)$** can be easily ensured by implementing the Invariant Point Attention (IPA) module [1] based on quaternion (instead of the original rotation matrix) without extra operations or complexity. We denote this module as IPAq module. We provide a detailed explanation of this IPAq module with quaternion as the rotation operator in Algorithm 3 in Appendix F. \n\nHere we explain the content related to antipodal equivariance.\n\nFor each point $\\textbf{x}=(x,y,z)$ with its homogeneous representation $\\textbf{p}=[x,y,z,0]^T$, each rotation operation $\\textbf{R}\\circ\\textbf{x}=\\textbf{Rx}$ based on the rotation matrix $\\textbf{R}$ is replaced by a quaternion-based rotation $\\textbf{q} \\circ \\textbf{p}$ which is antipodal invariant:\n\n$$\n\\textbf{q} \\circ \\textbf{p}=\\textbf{q} \\times \\textbf{p} \\times (\\textbf{q})^{-1} = \\textbf{-q} \\times \\textbf{p} \\times (\\textbf{-q})^{-1}=(-\\textbf{q}) \\circ\\textbf{p}\n$$\n\nTherefore, the node embedding $\\textbf{s}_i$ based on this IPAq module is antipodal invariant, which is mapped as an update quaternion:\n\n$$\n\\textbf{q}\\_{\\text{update}}=\\text{MLP}(\\textbf{s}\\_i)=\\text{MLP}(\\text{IPAq}(\\textbf{q}\\_{i},...))\n$$\n\nWe design a residual update based on this update quaternion:\n\n$$\n\\textbf{q}\\_{i+1} = f(\\textbf{q}\\_{i},...)=\\textbf{q}\\_{i} + \\textbf{q}\\_{i}\\times \\text{MLP}(\\text{IPAq}(\\textbf{q}\\_{i},...))\n$$\n\nHence, the update transformation is antipodal equivariant:\n\n$$\n\\begin{align*} f(-\\textbf{q}_{i},...)\\&=-\\textbf{q}\\_{i} - \\textbf{q}\\_{i}\\times \\text{MLP}(\\text{IPAq}(-\\textbf{q}\\_{i},...)))\\\\ \n=-\\textbf{q}\\_{i} - \\textbf{q}\\_{i}\\times \\text{MLP}(\\text{IPAq}(\\textbf{q}\\_{i},...)))\\\\\n=-(\\textbf{q}\\_{i} + \\textbf{q}\\_{i}\\times \\text{MLP}(\\text{IPAq}(\\textbf{q}\\_{i},...)))\\\\\n= -\\textbf{q}\\_{i+1}=-f(\\textbf{q}\\_{i}) \\end{align*}\n$$\n\n> **Q2: Efficiency comparison to diffusion baselines?**\n> \n\nThank you for your suggestion, which will allow us to more thoroughly demonstrate the efficiency advantages of our proposed model.\n\nWe compared our SO(3) generation algorithm with the SO(3) diffusion algorithm proposed by FrameDiff [7] using the synthetic dataset experiment described in Table 5 of Appendix C. To measure generative modeling capability, we calculated the Wasserstein distance between the generated data distribution and the test data distribution. The results are shown in the table below:\n\n| steps | 10 | 50 | 100 | 150 | 200 |\n| --- | --- | --- | --- | --- | --- |\n| ProteoBayes | 0.290 | 0.155 | 0.115 | 0.104 | 0.099 |\n| FrameDiff | 0.335 | 0.254 | 0.144 | 0.152 | 0.129 |\n\nIt can be seen that our model achieves better results with fewer steps compared to diffusion baselines, and it is worth noting that previous experiments in Euclidean space [5] and on periodic variables [6] have also similarly demonstrated that BFN can achieve better results with fewer steps than diffusion baselines.\n\n> **Q3: How sensitive are results to the choices of $\\alpha_i$ for vMF noise?**\n> \n\nWe appreciate the reviewer's interest in examining the robustness of our method with respect to hyperparameters. \n\nOur experimental results demonstrate that the outcomes are not highly sensitive to the choices of $\\alpha_i$ for vMF noise. The experiment is conducted as follows:\n\nWe adjust the scale of $\\alpha_i, i\\in[1,...,200]$ by modifying $\\beta_1$, which represents the expected accumulated accuracy at $t=1$.\n\n| $\\beta_1$  |  $\\alpha_1$ | $\\alpha_2$ | … | $\\alpha_{199}$ | $\\alpha_{200}$ |\n| --- | --- | --- | --- | --- | --- |\n| 100 | 0.478 | 0.515 | … | 3.116 | 3.149 |\n| 1000 (used in reported results) | 0.610 | 0.656 | … | 30.404 | 31.295 |\n| 10000 | 0.720 | 0.776 | … | 394.137 | 410.890 |\n\nTo evaluate the performance of different choices, we conducted experiments on peptide conformation generation using the PepBDB dataset:\n\n| Model | RMSD$\\text{C}_{\\alpha}$ ↓ | RMSD$_{\\text{atom}}$↓ | DockQ ↑ |\n| --- | --- | --- | --- |\n| DiffAb | 13.96 | 13.12 | 0.236 |\n| PepGLAD | 8.87 | 8.62 | 0.403 |\n| ProteoBayes ($\\beta_1$=100) | 3.54 | 4.69 | 0.587 |\n| ProteoBayes ($\\beta_1$=1000, reported) | 3.64 | 4.91 | 0.563 |\n| ProteoBayes ($\\beta_1$=10000) | 3.58 | 4.70 | 0.593 |\n\nGenerally, the results remain relatively stable despite significant variations in the scale of $\\alpha_i$. \n\nIt is also worth noting that these alternative parameters yielded slightly better results than reported because we did not perform an exhaustive hyper-parameter search process.\n\n> **Q4: Would fully excluding side-chain inputs from sequence prediction discard useful structures?  Is there a balance of properly using side-chain hints?**\n- **Would this also discards useful structures like sequence correlations?**\n\n  Thank you for raising this important methodological question. \n\n  Based on our analysis, excluding side-chain information from sequence prediction does not negatively impact the performance ceiling of sequence generation. \n\n  This stems from the fundamental unfair relationship between sequence and side-chain in the data distribution: The sequence $\\mathcal{S}$ serves as the **primary determinant** of residue types, while side-chain configurations are **dependent on the sequence** for both side-chain atom types and conformational specifications. \n\n  Consequently, side-chain information is not essential for sequence prediction. On the contrary, the network must develop the capability to modify side-chain states in accordance with sequence parameters.\n\n  Nevertheless, we recognize that when properly utilizing side-chain input, the network could leverage side-chain states to guide sequence prediction, potentially leading to better structural alignment between sequences and side-chain.\n\n- **Is there a balance of properly using side-chain hints?**\n  \n    That’s a quite insightful observation and question! \n    \n    Although our current information flow architecture has demonstrated effectiveness in generating all-atom geometry, we believe there exist a way to properly incorporate side-chain information into sequence prediction for better performance as a promising future work. \n    \n    From a technical perspective, this is because the side-chain angles $\\chi$ or Euclidean atom coordinates is composed of partial discrete residue type information and partial side-chain geometric information.\n    \n    A promising approach would be to develop a proper dis-entangled representation for sequence and side-chain, effectively jointly modeling these two information bypassing the information shortcut challenge.\n    \n\n[1] Jumper, John, et al. \"Highly accurate protein structure prediction with AlphaFold.\" *nature* 596.7873 (2021): 583-589.\n\n[2] Karras, Tero, et al. \"Elucidating the design space of diffusion-based generative models.\" *Advances in neural information processing systems* 35 (2022): 26565-26577.\n\n[3] Karplus, Martin, and Gregory A. Petsko. \"Molecular dynamics simulations in biology.\" *Nature* 347.6294 (1990): 631-639.\n\n[4] Frauenfelder, Hans, Gregory A. Petsko, and Demetrius Tsernoglou. \"Temperature-dependent X-ray diffraction as a probe of protein structural dynamics.\" *Nature* 280.5723 (1979): 558-563.\n\n[5] Song, Yuxuan, et al. \"Unified generative modeling of 3d molecules with bayesian flow networks.\" *The Twelfth International Conference on Learning Representations*. 2024.\n\n[6] Wu, Hanlin, et al. \"A periodic bayesian flow for material generation.\" *arXiv preprint arXiv:2502.02016* (2025).\n\n[7] Yim, Jason, et al. \"SE (3) diffusion model with application to protein backbone generation.\" *arXiv preprint arXiv:2302.02277* (2023)."}]

Author rebuttal:

Meta-review: The paper proposes a unified all-atom protein design framework introducing Bayesian flow network for backbone orientations and preventing information shortcuts (side-chain torsion angles from leaking residue-type information during sequence prediction) via a rationalized information flow. The reviewers generally appreciate the paper as the first work to apply Bayesian flow network for protein generation.

Reviewer R3Sb is the only negative reviewer. The reviewer's concerns were about lack of details, lack of visualizations, and some technical confusion. Since reviewer R3Sb did not follow-up during the rebuttal phase, I checked the discussions and can verify that the concerns are resolved, given that the authors provided the promised visualizations in the final manuscript. 

Overall, I do not see additional critical concerns about the paper. I recommend acceptance.
Decision: Accept (poster)