Controllable 3D Molecular Generation for Structure-Based Drug Design Through Bayesian Flow Networks and Gradient Integration
Authors: Seungyeon Choi
Hwanhee Kim
Chihyun Park
Dahyeon Lee
Seungyong Lee
Yoonju Kim
Hyoungjoon Park
Sein Kwon
Youngwan Jo
Sanghyun Park
Venue: NeurIPS 2025 poster ()
Keywords: Structure-based Drug Design
3D Molecule Generation
Generative Model
Primary area: machine_learning_for_sciences
TL;DR: This study proposes a Bayesian Flow Network integrated with gradient-based guidance to generate molecules satisfying diverse properties.

Abstract: Recent advances in Structure-based Drug Design (SBDD) have leveraged generative models for 3D molecular generation, predominantly evaluating model performance by binding affinity to target proteins. However, practical drug discovery necessitates high binding affinity along with synthetic feasibility and selectivity, critical properties that were largely neglected in previous evaluations. To address this gap, we identify fundamental limitations of conventional diffusion-based generative models in effectively guiding molecule generation toward these diverse pharmacological properties. We propose $\texttt{CByG}$, a novel framework extending Bayesian Flow Network into a gradient-based conditional generative model that robustly integrates property-specific guidance. Additionally, we introduce a comprehensive evaluation scheme incorporating practical benchmarks for binding affinity, synthetic feasibility, and selectivity, overcoming the limitations of conventional evaluation methods. Extensive experiments demonstrate that our proposed $\texttt{CByG}$, framework significantly outperforms baseline models across multiple essential evaluation criteria, highlighting its effectiveness and practicality for real-world drug discovery applications.

Controllable 3D Molecular Generation for Structure-Based Drug Design Through Bayesian Flow Networks and Gradient Integration
Seungyeon Choi', Hwanhee Kim', Chibyun Park Dahyeon Lee", Seungyong Lee Yoonju Kim', Hyoungjoon Park', Sein Kwon', Youngwan Jo Sanghyun Park If Yonei University 2 UBLBio Kangwon National University {tmddus (
sanghyun) @yonsei. ac ac, kr
Abstract
Recent advances in Structure-based Drug Design (SBDD) have leveraged gen- erative models for 3D molecular generation, predominantly evaluating model per- formance by binding affinity to target proteins. However, practical drug discovery necessitates high binding affinity along with synthetic feasibility and selectivity, crit- ical properties that were largely neglected in previous evaluations. To address this gap, we identify fundamental limitations of conventional diffusion- generative models in effectively guiding molecule generation toward these diverse pharmaco- logical properties. We propose CBYG, a novel framework extending Bayesian Flow Network into a gradient- based conditional generative model that robustly integrates property-specific guidance. Additionally, we introduce comprehensive evaluation scheme incorporating practical benchmarks for binding affinity, synthetic feasibility, and selectivity, overcoming the limitations conventional evaluation methods. Ex- tensive experiments demonstrate that our proposed CBYG framework significantly outperforms bascline models across multiple essential evaluation criteria, highlight- ing its effectiveness and practicality for real-world drug discovery applications.
Introduction
In Structure-based Drug Design (SBDD), generative models capable of designing 3D molecules
that selectively bind target proteins have emerged as essential tools in drug discovery [3, 50). Initial approaches primarily utilized voxel-grid representations [33], evolving through autoregressive archi- tectures [34, 43], to recent high-performing non-autoregressive. diffusion-basse methods [20, 21, 19]. Despite significant progress, practical drug discovery requires more than just binding affinity; viable drug candidates must also satisfy critical pharmacological constraints such as synthetic feasibility and selectivity [29, 26]. To address this, recent diffusion- based studies have incorporated gradient-basee guidance strategies for enhanced property control during generation [11, 22).
However, it remains unclear whether diffusion- based guidance represents optimal and practically reliable strategy for controllable molecule generation. Moreover, widely used evaluation metrics in existing SBDD research have rarely been rigorously examined regarding their adequacy in assessing realistic drug candidate properties. this study, we systematically investigate the theoretical and practical limitations of diffusion-batee based guidance strategies and conventional evaluation metrics. To overcome these problems, we introduce CByG (Controllable Bayesian Flow Network with Integrated Guidance), extended Bayesian Flow Network [18] framework utilizing gradient-based conditional generation, accompanied by comprehensive evaluation benchmarks targeting synthetic feasibility, selectivity, and binding affinity. Our contributions provide essential insights and methodological
"Correspondence to: sanghyun @ ac..ei.ac.kr.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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advancements toward more robust, practical, and effective Al-driven molecular generation for real- world drug discovery applications.
2 Rethinking of 3D Molecule Modeling in SBDD
Problem of guidance in hybrid modalities. In SBDD, generating molecules with desired properties that bind to target proteins is pri- mary objective. To this end, employing guid- ance strategies with external predictors within diffusion models is considerable approach for property-driven sampling [22]. However, the hybrid nature of 3D molecular data, which com- prises continuous Cartesian coordinates (typi- cally modeled with Gaussian distributions) and categorical atom types (modeled with categorical distributions), presents significant limitations to the conventional application of guidance. The primary challenges are as follows:
Propoeed rammwrrk Gradant
Offiatoo Gradint
Figure I: Gradient trajectory for target properties throughout the Generative Process for Proposed model and diffusion model. Further Interpretation these re- sults provided in Section 6.4.
1. Difficulty in Capturing Interactions: As continuous coordinates and categorical atom types are sampled from fundamentally different types of distributions, the guidance mechanism may fail to accurately reflect the interactions between them [35, 46]. This can lead to a loss of crucial chemical context during the generation process.
2. Ineffective and Unstable Categorical Variable Guidance: Applying gradient-based guidance to categorical variables in diffusion models is inherently problematic. The argmax operation in the reverse process [19, 20, 21]. coupled with the non-differentiability of categorical distributions, renders guidance signals either ineffective or a source of instability, while workaround strategies often introduce unnatural representations and increase model complexity, as shown in Figure 1. 3. Loss of Chemical and Structural Validity: Direct injection of gradients calculated during the denoising process into the sample space can readily compromise the chemical and structural validity of 3D molecules, which are highly sensitive to numerical perturbations. This complicates property control and can result in the generation of unstable molecular structures.
Recent studies have begun exploring Bayesian Flow Network (BFN) [18] as a solution to these issues in 3D molecular generation [46, 37]; however, these typically treat BFN as independent generative frameworks distinct from diffusion models, leaving gradient- based conditional generation (guidance) underexplored both theoretically and practically. Therefore, systematic theoretical clarification and practical expansion of gradient guidance mechanisms within BFN, particularly in comparison with well-established diffusion- based guidance strategics, remain essential research directions. A detailed discussion this point is provided in the Appendix J. Necessity of Posterior Sampling in Guidance. In gradient-based generative frameworks such as diffusion models, conditional generation typically leverages poste- rior conditioned on labels (attributes) 1, known as the con- ditional score function xx, xz). which is learned via dedicated neural network [1 45, 22]. While effective in image domains, where intermediate noisy states remain semantically informative, this approach is unsuitable for 3D molecular generation because noisy intermediate struc- tures lack chemical validity and making reliable attribute prediction difficult, as shown in Figure 2. Recent studies have proposed posterior sampling methods utilizing pre- dicted final states (Xn) to address this challenge [7, 23]; however, these methods focus primarily on general conditional molecular generation tasks, not specifically on structure-based drug design (SBDD). Furthermore, existing methods unnaturally discretize categorical atom types into continuous variables and neglect inherent prediction uncertainties of the final state. Thus, further research and methodological refinements are required to effectively employ bo-ba guidance for SBDD tasks. A detailed discussion on this subject is provided in the Appendix J.
Image Domrin
in 3800 bemalt
pbln)
Figure 2: A comparative illustration of at- tribute conditioning using p(1 x.) versus p(1 xu) in the image and SBDD domains.
Need for Selectivity dataset and various evaluation metries. Previous research in SBDD has predominantly relied on binding affinity measurements, typically using AutoDock Vina [12], to evaluate generated 3D molecules, introducing potential evaluation biases. To enhance reliability, incorporating diverse docking algorithms for affinity evaluation is essential.

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Another key consideration is Synthetic Accessibility (SA) scores [14], combining structural complex- ity and fragment contributions into a single numerical value between 0 and have frequently been employed to assess synthetic feasibility. However, high SA scores often do not guarantee practical synthetic routes, highlighting a critical gap accurately evaluating real-world synthesizability.
Moreover, selectivity (ensuring that molecules bind specifically to target proteins without off-target interactions) is equally important as affinity 26]. Poor selectivity can cause unwanted side effects, and while recent studies have proposed diffusion-based guidance methods [17, 6]. these rely heavily on pretrained classifiers predicting true binding molecules. The widely used CrossDocked2020 [15] dataset, initially intended for general docking research, is unsuitable for selectivity evaluations without substantial additional docking computations. Furthermore, unclear criteria for identifying true binders and significant data imbalance hinder obtaining reliable guidance signals. Consequently, establishing biologically relevant benchmark datasets and developing effective controllable generation strategies for selectivity is urgently required. A detailed discussion on this subject is provided in the Appendix J.
The Contributions of this research addressing the aforementioned points are as follows:
We introduce novel approach that integrates BFN with diffusion models from guidance perspective to circumvent the limitations of diffusion guidance SBDD. This approach explicitly formulates gradient-based guidance mechanism within Bayesian update process and rigorously establishes its theoretical foundations.
We provide a comprehensive analysis of how injecting guidance into generative models affects posterior sampling cansinng-based based predictions and uncertainty in SBDD. This study offers new insights into the interplay between guided generation and uncertainty quantification.
We revisit the limitations of existing evaluation metrics in SBDD and propose new set of essential metrics for evaluating practical molecular generative models, including binding affinity, Synthetic Accessibility (SA), and selectivity.
Through extensive and comprehensive experiments, we demonstrate that the proposed framework CBYG outperforms existing baseline models by a substantial margin. Notably, it achieves state-of- the-art performance under both conventional and newly introduced evaluation criteria.
3 Background
comprehensive overview of related work pertinent to this study provided in the Appendix C.
3.1 Bayesian Flow Network in 3D Molecule Modeling
Bayesian Flow Network (BFN) [18] synthesize data through iterative Bayesian refinements in unified parameter space, distinct from diffusion models which operate directly in data space. Unlike conventional methods that separately estimate distributions for continuous atomic coordinates and categorical atom types. BFN jointly model both modalities within single, tractable distribution (typically Gaussian). Formally, the data distribution P& (m) is approximated by progressively refining parameters 0 via Bayesian updates:
P& (m) = = f ps palm O (x))(()) II Po
Each iterative refinement involves three transition kernels:
Sender distribution Ps(y; m: az): injects controlled noise into the molecular representation. enabling gradual data transformations.
Output distribution (i 44): uses neural network V to reconstruct molecular features based on previous parameter states, ensuring coherence despite noise.
Receiver distribution Pa(y: 0;- f4, a;): predicts noisy observations by marginalizing over the output distribution.
Bayesian updates are computed by incorporating noisy observations y; into parameters 0,:
(u 0 10-1;;0 = (m).._........)))) [ppxy. mma.) [5( - h(0,,1,,,, [3))]]
(2)
BFN ultimately optimize the KI. divergence between sender and receiver distributions, progressively refining parameters toward accurate data approximations. Therefore, this framework uniformly handle discrete and continuous data through modality-agnostic framework operating entirely
3

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distribution parameters, thus avoiding discontinuities found in discrete diffusion models, After training, sample generation involves iteratively refining parameters 0,: 1) Draw intermediate sample m ~ pa( 0,-1,4) l.). 2) Obtain noisy observation yi Ps(. m (1)). 3) Update parameters via Bayesian inference: 0: = 1 0z).
This iterative Bayesian refinement yields progressively accurate parameter representations, enabling coherent and robust data generation. For detailed explanation of this part, recommend referring to the original paper [18] that proposed this model, as well as the Appendix provided in this paper.
Unified Framework of Bayesian Flow Network and Score Based Diffusion
4.1 Overview
Building on the previously outlined theoretical foundations of score-based models [22], particularly the concept of guidance, we now discuss how this framework can be integrated into the inference process of BFN across different data modalities.
4.2 BFN for Continuous Variables from Score Gradient
When modeling continuous variables (e.g., atomic coordinates) using a Gaussian distribution N [X] 80 pI)). I), the parameters 0 that BFN seeks to optimize can be defined as the mean and covariance matrices of the distribution:
pp-- The bayesian update function of update distribution (Equation (2)) whose fundamental objective find the optimal 0 that best describes the data distribution, is defined as below:
0 PPs-1 yau
h ((0 Pi-1 a.) [0%,p:) where - 0i, Or
Pi
Proposition 4.1. According Graves [18), when the sender distribution for sampling 0% is defined as ps(y| aI) - (X,a='I) the updated O resulting from the update function can be expressed in the following normal distribution form:
0 (0--1P (0N000PPP--1 ax
0% ~~~N(
Oi pp:
44)
Ps
By applying the reparameterization trick to the normal distribution above, it can be reformulated as the following linear Gaussian model: 0% oX-00_- transformation from above linear gaussian model the following equivalent formulation
vo: z for (o,, The
P:
x~N N (2 (p.0r - 05 -1
(5)
is obtained by rearranging the terms explicitly express in terms of the updated parameters.
The resulting distribution allows us approximate the mean of x as follows: 1 Furthermore, this can be alternatively interpreted using Tweedie's 's formula from different perspec- tive.
o (p,0% 0% Pi-1).
Definition 4.1eTweeds'' ' Formula [13, 28)). Let z be a Gaussian random variable following the distribution N Px, Ex). Then, Tweedie's 's formula states that:
E[px|xi = x + Σx Vx log p(x)
(6)
where p (X) denotes the marginal distribution of x: f(x) = S do(x - -))g()dd, here &o (1)
=
(2na²2) -1/2 exp (=22////02)
Based on the principles outlined in Definition 5.1, the corresponding relationship for the normal
distribution in Equation (5) can be formally established as follows: (p.0% 0% PP--)) =
x x+ Vx p(x). The derived formula enables us to reinterpret the Bayesian update function in Equation (4) from the perspective of the score function.
h (0,_,, =
0% 1Pi-i + you
ax 0
Pi-1
+
ya
Pi
Pi
Pi
Pi-1
0x
Vxlogp(x)
P:
Pi
Ps

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Using the reparameterized Bayesian update function derived above, we can rewrite the Bayesian update distribution Pu from Equation (2) follow:
Pu (0 10 1)))) Do (m
E (y, im:oo))
(5: 1akk00-110 01-11
where (0, 0 Ya. 0r x Pi-1 0-111 + Pi
Vxlogp(x)
(8)
Pi
where we define a compact notation for convenience: 8 80% (0r 10
< (0x 1, oz) Interestingly, the above derivation demonstrates that we can update the pa-
rameter 0 by means of the gradient taken with respect to x.
4.3 BFN for Discrete Variables from Score Gradient
Similar to the modeling of continuous variables, and recalling that the fundamental objective of BFN is to identify parameters that effectively encapsulate the data, this section explores whether the Bayesian update process for discrete types can be reinterpreted in gradient form. According to Graves,
the Bayesian update function for discrete variables is defined as follows: h (0,-1, Y, a) = 0,
e' ERS 21)0 Furthermore, the sender distribution, which governs the sampling of the perturbed state y of the discrete variable, is given by:
Ps (y x; (y a (Kex akK
where 1 a vector of ones, I is the identity matrix, and e; E R is vector defined as the projection from the class index i to a length K one-hot vector (details provided by [18]. It is important to note that ex a one-hot vector only when it serves as the initial input.
Proposition 4.2. Similar to Proposition 4.1, we can reformulate the relationship between ex and y using the repurameterization trick, as expressed follow: ex ~ N' (g y R . II The essence of the Bayesian update function for discrete variables lies in applying sofimax operation to the previous timestep parameter @ and the perturbed variable y. Notably, this allows to express y in gradient form, enabling a formal characterization of the relationship between the update function and the score gradient.
Similarly. applying Tweedie's 's formula to the normal distribution from which is sampled, as derived above, yields the following relationship for the computed mean: o y + = ex ak Vex log p(ex). Utilizing this result, the update function for discrete variables can be extended into the following gradient form:
h (0 (0,-1,8, = v
= Softmax(e" .0;-1)
(10)
Σ 0h (0,-1)
where y= ex
-x VoK
e = a (K ex 1) V., logp(ex)
Similarly, as with the case of continuous variables, the Bayesian update distribution for discrete vari- ables can be expressed using Eq. 10 follows (for notational convenience, 5 (0, h (0.-1.y., a.))
is denoted as &x (0, 0,-1.y.: as) in the discrete variable case):
Po (Ø) 0 az)==
[sv (0 1.. as)] pa m oz (mla)
(11)
where Sv (0 Yi, 0r - Softmax (ea(Koor-1)+ 1)+ loR p(ex) 0;-1)
Through this, we have demonstrated that the Bayesian update process for both continuous and categorical variables can be effectively reformulated into gradient-based representation. In addition, we provide detailed theoretical discussion and analysis of the relationship between our extended BFN and diffusion models in the Appendix B. In the following section, we will incorporate this concept specifically into the context of molecular generation for SBDD, introducing a strategy to generate molecules with desired target properties.
5

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Generative process
CBy6 Generative process
Sample space
Figure 3: Schematic illustration comparing the guidance propagation mechanisms of diffusion- models and CByG in target protein-awarg molecular generation. The figure shows how guidance propagation occurs in the molecular distribution p(x) across the 3D molecular space for each model type. Further interpretation of this schematic illustration provided in the Appendix E.
Guided Bayesian Flow Networks for 3D Molecule Generation in Structure-Based Drug Design
5.1
Notation
The primary objective this study is to advance SBDD by developing generative models capable of producing molecules specifically tailored to bind to given protein target. A target protein is represented as P = {(xP, vp)}. and our goal to generate ligand molecules denoted by m {(M,, VM)).
Each atom in molecules and proteins is described by its 3D position E TR& and its chemical type E IR* where K represents the total number of possible atom types. Molecules and proteins can thus be effectively represented as matrices: m XM. VMj and p = [xp, vp]. with dimensions XM E IR Nux3 VM E IR Ny K Xp E R- N3. and vp E R N For brevity, we denote molecules as M [x, v). Here, NM and Np denote the number of atoms in the molecule and the protein, respectively.
In the following section, we introduce proposed model, CBYG. A general schematic comparison between CBYG and diffusion models is provided in Figure 3.
5.2 Bayesian Update Distribution in SBDD
To enable property-controlle generation without retraining, we leverage the gradient-based Bayesian update formulation (Section 4) explicitly conditioned on target properties 1: 0, <- h(0,-1, Q4, 1). Instead of embedding properties directly into the output distribution (which would require retraining), we integrate pretrained BFN with BNN (Bayesian Neural Network) based external property predictor. This conditional sampling approach closely aligns with methods used in conditional diffusion models and guided generative frameworks [45, 4]. Formally, extending Equation (1), the conditional molecular distribution is defined as follows:
/ Pol (m On. p. (),))) II 0..
n
Ps (m p.
/ Pa Pox(x(0%PP.) 1)p(6))
II Pu 6x p. 1.
If D 1)
II Pa
(12)
The Bayesian update distribution Pv, incorporating explicit conditioning on protein and property 1, given by:
O0 10 p.l: a:)) = Epo .p:t.) [ ( [<x xt
Pu (0 1N
Bm ppt.)
..t..)
(v) [Gx 1. v, 1]]]
(13)
The sender distribution for atom positions defined as: Ps(y x; = Nxx where a controls the noise intensity around clean coordinates x. The corresponding output distribution

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is modeled via a neural network V: Pm 0;-1.P;t) = (0..1:...5 which predicts atom coordinates and types each time step t,, conditioned on protein p. For atom types, the sender distribution embeds discrete types, v into continuous space: Ps(y | v; = Wy a( Kee 1). I), with ey as one-hot vector, the number categories, and a scaling the noise.
Employing score-bas based guidance [45, 11], we embed property conditions into the gradient updates of identified in Section explicitly guiding molecule generation towards desired properties. Thus, our guided Bayesian updates for continuous and discrete variables are respectively defined as:
6x (0, 0 1. x, 04 p. 0:
Pi-1
0
+
1 Vxlog 1)
Pi
Pi
Pr
03
y
Pi-1
Vx logp(11 x)
(14)
Pi
Unconditional Generation
Controllable Guidance
Sv (0X 1o p.1) 0 Softmax
ear(K- ex logp(ex(2). 0
Softmax
ex 0
Vex log p(1le>)
(15)
Uncendational Generation
Controllable Guidance
The n-step sampling process of CBYG begins with an initial parameter Oo. accuracy parameters an and time steps 44. This process sequentially generates oj...., On to produce the final sample. First, using 0;-1 and 1 a sample is drawn from the output distribution Po (- 0;-1, P; t.). Next, based on x and a,, noisy sample y is drawn from the sender distribution ps(: X;-1, p: az). The parameter then updated via Pu(- 0 y: p, 1: ax). Finally, the sample generated from Pa(- θn, f. using the refined parameter f.. Detailed descriptions of the Bayesian neural kework-based property predictor for quantifying predictive uncertainty (guidance reliability), the sampling procedure of CBYG that incorporates this uncertainty, and the SE(3)-equivarianc) of the proposed guidance method are provided in Appendix D)
Experiment
To thoroughly evaluate the effectiveness and practicality of our proposed framework, we define 4 key research questions essential to the domain of structure-based drug design: 1) Are conventional metrics sufficient for reliably evaluating the predicted binding affinity of generated molecules? 2) the widely-used Synthetic Accessibility (SA) score an absolute indicator for practical synthetic feasibility? 3) Does the guidance mechanism incorporated into our Bayesian Flow Network (BFN) framework significantly enhance controllable generation performance? 4) Can our generative frame- work effectively handle molecular selectivity, key property required for generating novel drug candidates? In the subsequent sections, we systematically address each of these research questions through rigorous empirical analyses and extensive evaluations.
6.1 Experimental Setup
We the CrossDocked dataset [15] for training and testing, which originally contains 22.5 million protein-ligand pairs, and after the RMSI--based filtering and 30% sequence identity split by Luo et al. [31], results in 100,000 training pairs and 100 test proteins. For each test protein, sample 100 molecules for evaluation. Baseline models used for comparison are described in the Appendix F,
6.2 Comprehensive Evaluation Including Binding Affinity of Generated Molecules
Previous evaluations of generative models for target-specific molecular generation have typically relied on binding affinity metrics derived primarily from AutoDock Vina [12]. introducing inherent biases due to dependence on single algorithm. To address this limitation, we incorporate additional docking tools, namely SMINA [30], which captures broader physicochemical interactions, and GNINA [32], employing deep learning-based scoring function, to enhance the reliability and comprehensiveness of the evaluation. Furthermore, shifting away from the narrow focus on binding affinity alone, we introduce the PoseBusters [5] benchmark, which assesses molecular validity and
7

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Table Summary of binding affinity and molecular properties of reference molecules and molecules generated by CBYG and baselines. (T) (1) denotes whether a larger / smaller number is preferred. Top 2 results are bolded and underlined, respectively.
Methods
SMINA (
GNINA (
Vina (L) Dock.
High Aftimit(())
SA(T)
PB- Validr1) Med.
Diversity( Med.
Dock
Avg.
Med.
Avg.
Med.
Avg.
Avg.
Refevence
6.37
7.92
-7.06
-7.61
66.36
-7.45
0.73
0.74
95.0%
95.0%
AR[31]
-5.44
7.18
-..66
6..31
-5.75
66.75
37.9%
31.0%
0.63
0.63
88.3%
90.0%
0.70
0.70
Pocket2Mol [54]
-1.8
-7.55
-5.50
-6.30
-5.14
-7.15
18.45
51.0%
0.74
0.75
72.5%
86 5%
0.69
0.7]
Gen.
IugetDiff
33.6
-5.50
-3.47
-7.80
58.1%
59.15
0.58
67.0%5
0.72
DecompDiff [21]
4.48
-3.89
-5.77
64.4%%
0.68
MaICRAFT [37]
5.03
3.56
-6.02
-7.26
-6.61
-7.59
63.4%
70.0%
0.68
0.67
74.0%
82.5%
0.70
0.73
RGA [16]
22.61
-7.25
19.99
-5.28
20.58
88.11
64.4%
89.3%
0.71
0.73
91.3%
94 1%
0.41
0.41
Gon.
DevomppOpt [S1]
-3.54
-8.10
-5.26
-5.87
8.96
73.5%
93.3%
0.65
0.65
68.7%
77.P%
0.60
ThcoGFNN (42)
39.80
32.90
32.63
03.656
89.34
0.56
Ope.
ALIDiE
-5.08
.6.85
-2.02
73.4%
41.4%
0.73
TaggatOpt
-5.1
8.39
6.12
7.01
6.96
8.87
75.61%
82. 22.3%
0.71
0.69
78.8%
$1.35%
0.50
0.59
CByG
-7.74
99.61
-7.63
-8.33
-8.00
-9.16
93.6%
100.%
0.84
0.87
94.9%
96.0%
0.61
0.62
stability across 17 metrics, including chemical consistency and ligand stability. Detailed descriptions of the evaluation metrics and experimental setup are provided in the Appendix G.
Results. As shown in Table. Our proposed model substantially outperforms baseline methods across 12 key metrics related to binding affinity and molecular properties, confirming effective gradient-based guidance during generation. While baseline models exhibit significant differences between pre- (Score. in Iable. and post- (Dock. in Table.1) docking scores, our model maintains consistently high affinity scores even before docking, indicating enhanced capability to implicitly identify favorable binding poses and explicitly capture protein-ligand interactions. The robustness of our model's is superior binding affinity is further supported by its consistent outperformance across all three docking tools, unlike baseline methods whose relative rankings fluctuate across these tools. Although our method does not achieve state-of-th offthhean art performance in molecular diversity, this limitation naturally arises from conditional generation, which inherently targets narrower distributions to meet specific binding criteria. Futhermore, We assessed energetic stability of top-ranked molecules from CBYG and baseline models using the PoseCheck [24] benchmark. Figure. shows consistently low energies, indicating stable binding poses. Additional results are provided in the appendix. Detailed interpretations and results, including comprehensive score comparisons before and after docking and ablation studies on guidance scale variations within the CBYG framework, are provided in the Appendix
6.3 Evaluation of Realistic Synthetic Feasibility for Generated Molecules
To achieve a more precise assessment of the realistic synthesizabil- ity of generated molecules beyond conventional Synthetic Acces- sibility (SA) scores (as discussed in Section 2), we introduced the AiZynthFinder benchmark, which utilizes Monte Carlo Tree Search (MCTS) to systematically identify viable retrosynthetic pathways. This benchmark quantitatively evaluates practical synthetic feasi- bility through six distinct metrics (Detailed descriptions and values provided in the Appendix G).
Condlatiee hrdakihr
HoCret
Dear 200
AL DV1
Results. Our experimental results (Table. 3) demonstrate that de- spite exhibiting high binding affinity, our proposed model achieves near state-of-the-ae performance in AiZynthFinder evaluations, in- dicating its ability to generate molecules that are both efficacious and realistically synthesizable. Interestingly, we observed no clear correlation between SA scores and actual retrosynthetic feasibility (as measured by the Solved metric), underscoring the need for broader and more comprehensive
Strale
Figure Cumulative distribu- tion function of strain energy
Reference
ALIDiff
DecompOpt
MoLCRAFT
CByG
4F1M
Figure 4: Visualizations of reference molecules and generated ligands for protein pockets (PDB ID: 4F1M) generated by CBYG, ALIDiff, DecompOpt, and MoICRAFT.

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Table 2: Evaluation results of SA for molecules generated by baseline models and the proposed model using the AiZynthFinder benchmark.
0.8
0.6
Solved
Top Score
Solved (t)
Routes (t)
Metric
Routes
(1)
(1)
Cuisanne
0.4
Reference
0.410
294.1
11.43
0.826
0.2
AR
0.194
227.2
3.616
0.744
Pocket2Mol
0.373
150.6
7.743
0.797
TargeDDiff
0.082
233.5
1.800
0.695
0.0
Diftuston
DecompDiff
0.038
202.7
1.165
0.703
MoICRAFT
0.196
242.2
4.554
0.745
ALIDiff
0.079
321.8
400
2000
RGA
0.487
157.9
3.292
0.822
DecompOpl
0.080
201.9
1.205
0.713
Figure 6: Guidance score dynamics of three model types throughout the generation pro-
TacoGFN 0.072
255.4
0.530
0.743
CBYG 0.487
334.4
18.90
0.853
cess
evaluations of synthetic accessibility in future SBDD research. Notably, models with high SA scores exhibited practical retrosynthesis success rates of less than 50%, emphasizing the necessity of placing greater emphasis on realistic synthetic feasibility within the SBDD field. Further detailed discussion provided in the Appendix H.
6.4 Assessment of Guidance Effectiveness Across Model Types
To validate the effectiveness of mrudient-based guidance injection within CBYG framework, compared it empirically against a diffusion- model. Specifically, we analyzed the dynamics of guidance scores (p(1 m)), representing molecular property predictions each timestep during controllable generation (Diffusion model implementation details provided in Appendix E).
Results. As shown in Figure. 6, our CBYG guidance approach consistently achieves higher and more stable guidance scores compared diffusion- based methods. While these method utilizing predicted final states (Xo) can yield higher absolute scores, they exhibit increased variance due to unstable potit-estimations from intermediate noisy states. Unlike diffusion models operating directly in sample space, CBYG conducts continuous gradient calculations within parameter space (including categorical variables) thereby enabling more stable and effective controllable generation. Further detailed analyses and discussions on this comparative experiment are provided in Appendix H.
6.5 Evaluation of Selectivity Control Capability
In this experiment, we evaluated whether our proposed controllable generation strategy effectively enhances molecular selectivity, a critical factor in practical drug discovery. We constructed a biologically relevant selectivity test set from prior pharmacological studies [27, 10] involving binding data of 38 kinase inhibitors across panel of 317 kinases (Refer to Appendix for detailed methods and dataset construction). Our results demonstrate that the proposed BFN-based guidance method significantly outperforms diffusion- based approaches in selectivity met- rics (Succ. Rate and Score), with greater improvements upon guidance injection. These findings strongly sup- port the practical applicability of BFN-based guidance in real-world drug development scenarios. Further detailed metrics and analyses are provided in the Appendix H.
Table 3: Evaluation of Selectivity Opti- mization Capability between Diffusion-based Guidance Methods and the CBYG Model. "CBYG w/o G" denotes the CByG model without guidance injection.
Suce. Rate (t)
Metric
A Score (L)
TargetDiff TargetOpt
58.2
-0.78
68.6
CBYG w/o CBYG
62.6
-1.39
78.3
7 Conclusions
In this paper, we introduced CBYG, generative framework designed to produce 3D molecules satisfying multiple critical properties, thereby enhancing practical applicability in structure- based drug design (SBDD). We discussed inherent limitations of conventional diffusion- based guidance methods and theoretically integrated a gradient-based guidance mechanism within the BFN framework to address these issues. Empirical evaluations, guided by the key objectives highlighted in Section 2, demonstrated that our proposed model significantly outperforms comparative baselines across various evaluation metrics. Nevertheless, the finding that approximately half of the molecules generated

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(even by advanced models including CBYG) remain synthetically infeasible offers critical insight and underscores a significant area for further investigation within this research domain.
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A Detail description of Bayesian Flow Network
A.1 Key Distribution
A Bayesian Flow Network (BFN) maintains four distributions at each step: the input distribution Pr(m 0), the output distribution Po 0, t). the sender distribution ps m; ai), and the receiver distribution pa 0, i, at). Semantically, the input distribution represents the model's current belief (prior or posterior) over the data; the output distribution is the network's predicted distribution (allowing context) given the input parameters; the sender distribution is noisy perturbation of the true data; and the receiver distribution is the model's predicted distribution over such noisy messages (averaging over the output distribution). Mathematically, all four are factorized over the data dimensions for tractability.
Input distribution pr(m 0). This is factorized distribution over the data (m())... ...... m (2)
with parameters 0 (0(1) ((D)). For example, each ((d) might parametrize univariate Gaussian categorical for m( Initially pr is simple prior (e.g. (0,, or uniform categorical). During sampling, pr is updated via Bayes's 's rule when new observations arrive, so its parameters become progressively more informative about m.
Output distribution Pa (m t). Given imput parameters 0 and the current (discrete or continuous) process time f, a neural network computes an output vector that parameterizes Pa- However, unlike Pr it integrates context across dimensions via the network: the parameters po depend jointly on all of 0 and I, allowing modeling of correlations among data dimensions. In effect, Po represents the empirical sample distribution accumulated up to step t, providing comprehensive prediction that integrates the gathered evidence together with contextual information, and will be used predict the clean data distribution.
Sender distribution Pa(y mm;a) a). This is the distribution over noisy observation y given the true data m, with accuracy (noise level) az3. is also factorized across dimensions: Ps m;aa) = I Ps (y!d) m a.:) The parameter og controls informativeness: at the sender's sample is pure noise (uninformative about m), and as i > Do the sample concentrates on practice, a increases through the transmission steps, so that each sender sample carries more refined information about the true data. Intuitively, Ps defines the "message" drawn from data: Alice adds controlled Gaussian or categorical noise to x to form the sender distribution and then draws a sample y (: Iz; a.) (For the Alice-and and-Bob example, recommend consulting the original BFN paper [18]).
m. In
Receiver distribution Pa 0, f, a;). This is the model's predictive distribution over possible noisy observations y given the output distribution. Formally, obtained by marginalizing out E Ps (y rn; a;). In other words, for the unknown true data m: Pa(y; 0 0,-1it,, a;) f 1:4a)
every candidate m, we consider the sender distribution ps(y m; a:) that would have been used if m the truth, and weight these by the network's 's probability Po(m 0,t). The receiver distribution
thus captures both the "known unknown" due to the sender noise (entropy of ps) and the "unknown unknown" from the output distribution's uncertainty.
Bayesian update distribution po(O; 0.-1;a;). This distribution specifies how the parameter vector of the input distribution evolves after assimilating a noisy observation. Concretely, let the closed-form update rule be 0; = 1(00;-1. yi, a;;) where y; ~ ps(y; m; az). Conditioning on the current parameters 0 and the accuracy level aq, the update distribution is defined as the push-forward of the sender distribution:
Pu (0; 0-11:::) po(m[00-11t.)
E
E
[8 (8; h (0.11. ai)]
(16)
[P (y: (m:a.)
Because h(-)) is applied component wise under the factorisation of ps, Pu factorises across parameter dimensions, preserving tractability. The role of Pa is to enact a one-step Bayesian posterior update on 0: at early iterations, when o; is small, Pa is broad and admits substantial parameter movement, whereas at later iterations, larger o, makes the update sharply concentrated, refining 0 toward the values most consistent with the clean data. Thus Pu provides the formal bridge between each noisy message and the progressively more informative input distribution maintained by the network.
14

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A.2 Modality-Agnostic Representation and Progressive Refinement
Because BFN operate on distribution parameters rather than raw data, the same framework applies uniformly discrete, discretized, and continuous modalities. For example, discrete data are repre- sented by categorical distribution parameters (probabilities), which lie on probability simplex and thus provide continuous inputs to the network. fact, the parameters of categorical distribution real-valued probabilities, so the inputs to the network are continuous even when the data is discrete. Likewise, continuous data use Gaussian or other continuous distributions. all cases model processes continuous-valuee parameters, avoiding discontinuities common in discrete diffusion models. This modality-agnostt formulation means that the same Bayes-and-netwonn machinery can generate text, images, other data with only minimal adaptation.
A.3 Generative Sampling Process
After training. sample generation proceeds by iteratively refining the distribution parameters via Bayesian updates. Starting from an initial parameter 0g (c.g.. prior at initial noise level to), the model performs N iterations of the following procedure for N:
1. Sample from output distribution: An intermediate sample m is drawn from the output distribution m ~ Po(- 0;-1,44 given the current parameter 0;-1 and scheduled noise level f,.
2. Sample from sender distribution: noisy observation y; is then drawn from the sender distribution, conditional on m4, via y ~ ps(- m/, os), where o; is the accuracy (inverse noise variance) prescribed for step
3. Bayesian parameter update:
The distribution parameter is updated by incorporating the observation y; through the Bayesian update function: 0, h, 0z)- Here h(0,-1, a;) computes the posterior parameter after observing ys with precision a,, given the prior 0,-1-
Repeating the above steps yields a final parameter ON after N iterations. This O characterizes highly concentrated distribution (approximately a Dirac delta distribution in the limit of large N). from which the final data sample can be obtained by drawing m po 0. txx ). Importantly, the receiver distribution Pa. is not explicitly used during generation its effect is implicitly achieved by the two-step sampling (Po followed by ps) each iteration. This ensures that sampling relies solely on the forward update pattern m² - Y: - 0, described above, in line with the canonical BFN [18] formulation. simplified schematic illustration of this is provided in Figure 7.
B Score-Driven Sampling in DDPM vs. BFN: A Rigorous Comparison
B.1
Continuous Variable Modeling
Diffusion based sampling
1 1- Va:
Xx v1 aiyar
ED +02
Vag
(1 ax-1)
vāa-1 (1 ar)
Xx
xo + 0,
1 1-
as Vx, logol (xt) 0,
Xx
+
Vat
Vat
1
1-a4 at xx, 1
guidance
X1
logg (x,
9) 0
(17)
Vat
Va:
1
1 1- L- logp (x1)
1 1- 1- logp (y) x,) + 0,2
x
(18)
vaz
var
Voz
1 aq
1 a Vx, logp Xo) z
X₁
+
Vx, (x))
+
(19)
Van
Vav
15

---

BFN based sampling
0:
Pi-1
0
+
-y
y
(20)
Pi
pi
Pi-l
02
X0
1
V log p(xo)
(21)
Pi
Pi
Pi
Pi-1
α
guidance
01-17
+
X0
+
xx log p(xo 8)
(22)
Pi-1
0:-13
+
1
Xxo log
+ 1 V logply xo)
(23)
Pi
Pi
Pi
Pi
Pi-1
0:-1
y
1
Xx logp(y xo)
(24)
Pi
P:
Pi
B.2 Categorical Variable Modeling
Diffusion based sampling
X C (xx-1 ppost (xe, xto)
(25)
[alx] +
1-ax K
1x0 +
1-6e-1
K
where posa (Xe, 2o)
(26)
E ( + 1-01
1- K
Here. 3 denotes element-wise multiplication, ensuring the probabilities normalized over all categories. And, C denotes categorical distribution to model discrete atom types v. BFN based Sampling
0: = Softmax (6"
-0:-1)
(27)
Softmax Softmax(c -1)+ Vo+ logp(ex) 0;-1)
(28)
guidance + Softmax (x loR p(e)())))) 0i-1)
(29)
Softmax
e 0 e Vex log((ee)))
(30)
B.3 Theoretical Distinctions between Diffusion Models and Bayesian Flow Networks
Bayesian Flow Networks (BFN) differ fundamentally from Diffusion Models (DMs) in their mecha- nism of uncertainty injection and the domain of parameter updates. Unlike diffusion- baseed generative processes, which explicitly manipulate the data sample æ and inject Gaussian noise at each step to maintain stochasticity, BFN iteratively update distribution parameters 0 through Bayesian inference, integrating noisy observations yi drawn from a sender distribution Ps(y æ;; ox.)
In contrast to diffusion methods, BFN have no explicit forward noise injection process. Instead, the generation is achieved by deterministic parameter update conditional upon the observed
Bayesiun Liddate for Continous Variuble
Bayesian Update for Categorical Variable
Sanght ppee
Przametonth 2012
Figure 7: Schematic illustration of Bayesian updates for each variable type
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sample Vs, implicitly encoding uncertainty through the random sampling of 3/- Specifically, for continuous variables, the sender distribution Ps(y; a;);) is typically Gaussian, with precision (inverse variance) controlled by 0:- A small value of o; implies high noise, rendering the observed value yi nearly independent of the true data T; whereas large a; indicates a highly informative (low-noise) observation. Formally, the BFN update can be expressed as:
0, = 0,=(1-
Ci Pi
0
Vi, where == Pi- 0g--
(31)
Pi
This update rule can be directly interpreted through the lens of Kalman filtering for scalar Gaussian models. In this analogy, each update of 8; is drawn toward the observed sample proportionally to the precision parameter a;, thus progressively reducing uncertainty as Pa increases. Given the sampled yc. the Bayesian update function remains strictly deterministic, and no additional random noise is explicitly introduced beyond the implicit stochasticity already present in yi-
This principle of implicit uncertainty introduction also extends naturally to categorical variables. Suppose the parameter vector 0; represents a categorical probability distribution over K classes, i.e.,
0, = (p) ()) Ppi ...... .....p; pt ( In this case, the sender distribution can be viewed as emitting symbols y: according to a confusion matrix parameterized by a, Specifically, we have:
(2)
when a, = 0 (uninformative observation),
Ps(y:
k;)
(32)
1e=ks as 0r + 0o (perfectly informative).
Upon observing the symbol y: = c, the Bayesian update for the parameter 0g follows:
0++(i) x 0.(i) Pr[Y = c x=
= =
(33)
Although y: is originally sampled as continuous vector from Gaussian distribution centered around a one-hot encoding of the true class, the Bayesian update interpretation treats y; as if it were decoded arg max; Vk- For rigorous derivation and proof of this decoding
into discrete class index interpretation, we refer the reader to Graves et al. [18]. The resulting posterior update normalizes probabilities over all classes j. illustrating explicitly how the sampled observation y; reweights prior probabilities (j) based on the likelihood of observing Yi under each class hypothesis X = j. Thus, the sampled variable yi serves directly as informative mediator driving the parameter dynamics.
Unlike the arbitrary Gaussian noise used in diffusion processes, the sampled variable yı in BFN is inherently linked to the unknown true data through the sender distribution (3 z). As the inference procedure progresses (i.e., as increases and noise decreases), Yi provides increasingly precise information, causing the parameter 0; to converge toward a distribution sharply concentrated around the true data. This iterative Bayesian updating mechanism effectively transfers information from the sample space into the parameter space.
In summary, two critical distinctions between BFN and diffusion- based generative methods emerge from our theoretical analysis:
1. Implicit versus Explicit Noise Injection: BFN deterministically update parameters given noisy observations, implicitly capturing uncertainty. In contrast, diffusion models explicitly add random noise to samples each step to maintain stochasticity.
2. Parameter-space versus Sample-space Updates: Diffusion models perform both training and inference entirely within the sample space. Conversely, BFN operate fundamentally within the parameter space, integrating information from the sample space through occa- sional noisy observations, thus creating a natural integration of information between the two spaces.
C Related Work
C.1 Molecule Generation in Structure-based Drug Design
With the rapid accumulation of protein structural data, generative methods for molecule design have become increasingly important in structure-based drug discovery. Initial approaches, such as [44], employed sequence-based generative models to produce SMILES representations informed by protein binding sites. However, the advent of powerful geometric and 3D modeling methods has shifted the paradigm toward directly constructing molecules within three-dimensional spaces. For
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example, [38] represented molecules using, voxelized atomic density grids and utilized Variational Autoencoders (VAEs) for molecular synthesis. Meanwhile, [31], [34], and [36] introduced sequential. autoregressive approaches for placing atoms or functional groups step-by-step into target binding pockets. Building upon these autoregressive techniques, fragment-base based strategies, as seen in FL.AG [49] and DrugGPS [48], integrated chemically meaningful fragments, thereby improving the structural realism of generated ligands.
In parallel, diffusion-based generative methods have emerged, achieving remarkable success across various generative tasks such as image and text synthesis. Adaptations of these models in molecular contexts, exemplified by recent studies [20, 39, 25, 21], iteratively refine atom identities and coordi- nates, leveraging
architectures such as SE(3)-equivariint neural networks to ensure chemical validity and structural accuracy.
Despite substantial advancements, current generative frameworks frequently encounter difficulties in simultaneously optimizing multiple pharmacologically relevant properties, including binding affinity, synthetic accessibility, and low toxicity. In practical drug development scenarios, the simultaneous control and optimization of these attributes are typically mandatory requirements rather than optional criteria [9]. Thus, developing generative strategies capable of effectively incorporating multiple property constraints remains a critical challenge in the field.
C.2 Optimization based Molecule Generation in Structure- based Drug Design
Recent research in molecular generative modeling has moved beyond approximating training data distributions toward explicit optimization strategies aimed at producing molecules with desirable properties, such as high target protein binding affinity and synthetic accessibility (SA).
For example, RGA [16] employs genetic algorithms specifically tailored to structure-based drug design (SBDD), explicitly incorporating target protein structures into molecular optimization. De- compOpt [51] combines pre- trained, structure-aware equivariant diffusion model to initially identify suitable molecular substructures complementary to target binding pockets, followed by a docking- based greedy iterative optimization to enhance binding affinity. TacoGFN [43] leverages generative flow networks (G-FlowNett) to identify pharmacophoric interactions with target proteins, subse- quently utilizing reinforcement learning optimize binding affinity and synthetic accessibility. Additionally, AL.IDif [19] introduces a preference-basee optimization method that aligns pre-trained generative models to specified molecular properties through fine-tuning.
Despite their demonstrated effectiveness, these approaches share an intrinsic limitation: since op- timization methods are tightly integrated into their training processes, adapting the models to new property requirements inevitably necessitates retraining. In contrast, our proposed approach dis- tinguishes itself by leveraging pre-trained generative model, enabling effective multi-property optimization directly through modifications at sampling time, thus avoiding costly retraining and enhancing flexibility in targeting diverse pharmacologically relevant properties.
C.3
Bayesian Flow Network
Recently, Bayesian Flow Network (BFN) have gained attention as effective models for protein sequence modeling [2] and molecular structure generation [46, 37], demonstrating promising capabil- ities particularly the generation of realistic three-dimensional molecular structures. Despite these advancements, the development of controllable generation methods within BFN, aimed at optimizing diverse molecular properties required for viable drug candidates, remains largely unexplored. Thus, establishing theoretical link between generative models based on maximum likelihood estimation, such as diffusion models and BFN, and subsequently formulating the BFN framework, represents an essential step forward. Such advancements could significantly inform future directions in the field of structure-based drug design.
-uidiance strategies within
D Implement Detail
D.1 Predictor with Bayesian Neural Network for Uncertainty
To achieve property-driven generation without retraining the generative model, we introduce an external Bayesian neural network (BNN) predictor modeling the conditional distribution p(1 m. p)
as Gaussian distribution A (1; (m,,, p), (m,,,)mm))))) p) with mean μ4 (m, p) and variance a (m, p)²
Unlike point estimates, this BNN provides ancer predictions for properties (e.g., binding affinity), enabling informed guidance. During molecule generation, the gradient V, log m.p). p).
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which incorporates predictive uncertainty, guides the generative process. This uncertainty integration allows the model to appropriately moderate its guidance, preventing overly confident predictions toward regions unsupported by the predictor.
We approximate the overall predicted label distribution Gaussian with predictive mean and variance computed from these samples. In particular, the mean is (x. p) =M -M-- Mo, p), and the predictive variance oj p) M: (x),()) + (x) (x) combines the average of the BNN's output variances with the variance of its output means, Using the law of total variance, we decompose the predictive uncertainty into aleatoric and epistemic components:
d M Σ D. +M- Σ (m.
[ m. p)]
+ [ m p)] (m p)]
(34)
E (m.
Var [p(mm.))
Aleatoric Uncertainty
Epistemic Uncertainty
where & represents the BNN's parameters (random due to the weight posterior). The first term alcatooic the expected predictive variance (reflecting inherent noise or irreducible uncertainty in the property given (x, p)), while the second term o eppiscmiic is the variance of the predicted means (reflecting uncertainty in the model parameters due to limited training data). Our CBYG can thus modulate the influence of the guidance signal in proportion to the predictor's confidence, improving robust controllability.
We train the property predictor on an external dataset of protein-ligand complexes, allowing to learn a mapping from 3D structures to property values independent of the generative model. In particular, we use the CrossDocked2020 dataset [15] to train the predictor. Each training sample provides a protein structure a ligand m, and a ground-truth label I (c.g.. an experimental or docking-derive affinity score). We optimize the BNN by maximizing the likelihood of the true labels under its predicted Gaussian distribution. The negative log- oggliikelhood (NLL) loss for single sample is given by (For notational simplicity, we omit p):
LNLL (Mn: log oj (xx). +
(peD (Xn) - yn)²²
(35)
2
2 (x,)
where we omit the constant 1 log(2rr) ) for brevity. In addition to the standard NLL, we also employ a 8- weighted NLL variant (denoted 3-NLL) improve training stability in the presence of heteroscedastic
uncertainty [41], as follow:
DDL NLL (yn, x,) stop (o PNLL (y Xn),
(36)
for some hyperparameter B O. Setting B O recovers the original NLL loss, while positive B increases the relative penalty for large predicted variances.
D.2 Implement detail
The generative backbone architecture employed in our proposed CByG framework directly inherits the architecture and pretrained weights from MolCRAFT [37]. For the property predictor backbone, we adapted the graph transformer architecture from TargetDiff [20] by removing the equivariant head, resulting in novel inherent symmetry of pootein-ligane complexes, where binding affinity and synthetic accessibility scores remain invariant to rotations and translations.
graph transformer. This design choice naturally aligns with the
Specifically, the property predictor comprises 16 attention heads, with each attention block consisting of three SE(3)-invariant layers featuring a hidden dimension of 64. Key and value embeddings are generated through a two-layer MLP. Layer normalization uniformly applied throughout the network, and the Swish activation function is adopted. The learning rate is exponentially decayed by factor of 0.6, with a lower bound set at 10- This decay is triggered if no improvement is observed in the validation loss for ten consecutive evaluations, which occur every 1000 training
steps. Finally, the property predictor's 's scoring function is defined as Score = -20 x SA). where DS means docking score and SA means synthetic accessibility score.
D.3 Sampling
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Algorithm Sampling Procedure CBYG
1: Input:
2: Protein pocket p.
3: Pre-trained output network:
4: (m 0,-1, p: &- W Woutput(0 p.t:),
5: Property predictor network providing property predictions and uncertainty:
6: [1 m. p) < N (1: p.y (m. p): (m, ))
7: Pre-defined precision schedule for coordinate and type (axx; (v,,)) according [18),
8: Guidance scale for coordinate and type t- Ax, Av
9: 0% 0g
0, [ N
for to N do
11:
tt i-1
12:
m : [x, v] Po (m 0;.1,);t
13:
Yx,i Pyly axxx))
14:
Yv.i py (i
ps, + p1 m. p)
Mean and uncertainty from property predictor
0R a
p. Yx.; + Pi-1 0xx + o
ix
V (1
[& v).
Pi
17:
OY Softmax (eXvi 0 ee
where h =ō dv Ve, log
[ vI. p), e, GumbelSoftmax((?)
18: end for ~ Po (m 1N.P:t) 20: return 8 V)
D.4 SQ(3)-Equivarincce
Since our proposed guidance injection strategy integrated into an SE())eqqivariant generative model, the designed guidance itself must inherently preserve the SE(3))-quivariance property. As described by [40], aligning the protein-ligand complex center the pocket the origin removes translation equivariance, requiring only equivaarrae The following provides a formal proof verifying this property.
Proposition D.1. Suppose the property preditor 1// prrop VM]: Xp, vpl) + P1 (1 m. is invariant such that 4/ pprop vM), [Xp, vpl) V Vprop([2 [T, (XM), VM]: [T, (xp), Denoting Tg as the group of transfrtanoo T2 (x(x) Rx, where R R33 is the rolation matrix, and b RS is the translation vector. Then, gradient guidance function is orthogonal equivariant such that xu propp(1 (T, (xM), T, (xp)) = T (Vx. propp(xx Xp)). Variables corresponding to type v are xu
omitted from the notation, as they remain unaffected by 0(3) transformations.
Proof. Given the invariance of the property predictor V proop (XM, Xp) under O(3) transformations, follows that V "roop RRM, Rxp) = Vp (XM, Differentiating both sides of the equation with
respect to XM yields:
"roppl XMM, xp) 1 prop (RxM, Rxp)
(37)
XM prop (MM, Xp) = Vxx proo (RxM, Rxp)
(((RXM)
Vxu L prop RRMM Rxp)
txm
= R Vxu Wp prop
RXM. Rxp)
(38)
R Xu Lp prop (XM, Xp) = RR T V v prop RXM. Rxp)
V xM ! prop [RxM. Rxp)
(39)
Therefore, V xss V (prop() (-) exhibits equivariance under the transformation T, T,(,), completing the proof.
E TargetOpt Implement Detail
To evaluate the effectiveness our proposed guidance injection within the Bayesian Flow Network framework, implemented comparable guidance method within diffusion- based generative
20

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model. Specifically, we adopted the TargetDiff [20] architecture, enhancing it with gradient- gradient-based guidance strategies. Two distinct gradient guidance approaches were considered: (1) gradient computation based property prediction at arbitrary intermediate time steps x,, and (2) gradient computation via posterior sampling, leveraging property prediction at the fully denoised state Xo. For fair comparison, the model utilized for property prediction at Xy was identical to that employed in our CBYG model. Below, present the unconditional denoising procedures used by TargetDiff for each variable type; additional details can be found in the original paper (20).
q (x+-1 x,, xo) = =N (x1-1: jae (x,. xo) Bi), 9 (V:-1 vv. vo) = C (ve-1 ê (Ve, Vo))
(40)
where ju (x),
1-a: Xo + vor(1- Xr, 1-ār-1 Bc, and è (V1, vo) 1-a:
a-a.
2² & and c² (v) = lavV + (1) / KK] [ā=-ivo + (1 a-1)
Under the scenario of property prediction arbitrary intermediate time steps x,, the update procedure for atomic coordinate and type can be modified as follows.
Vau-18
V Vot (1 0i-
+ - al Vx. logr m.
in Xo)
X0
(41)
1 āt
1- Oz
Von
à (Va, vo)=
c²
Prppllivs.))
(42)
1 Ck
Additionally, when employing posterior sampling for gradient computation, the gradient form can be reformulated analogous to DPS as follows.
V x. log (1 xx)= ~ v log Exx [p (y &o)]
(43)
Vv, logp (1 vz) ~ v log E ppovlll)) [p (y vo)]
(44)
F
Baseline Model for Evaluation
For a fair comparison, we categorize baseline models into two groups: "Only Generation Type" and "Optimization based Generation Type". Movevoer, selected state-of-the-art generative models exhibiting competitive performance as representative models for the "Only Generation Type" category.
Only Generation Type
AR [31] leverages Markov chain Monte Carlo techniques to sequentially infer molecular structures from spatial atomic density representations.
Pocket2Mol [34] incrementally synthesizes molecules through sequential prediction of atoms and associated bonds using an E(3-eqquivariant architecture, selectively expanding frontier atoms to significantly improve sampling efficiency.
TifgetDiff [20] enhances dual-modality diffusion methodologies, distinctly processing continuous and discrete modalities through parallel diffusion pipelines, demonstrating improved outcomes over purely continuous formulations such DiftSBDD.
DecompDiff [21] adopts molecular decomposition techniques, separating the molecular structure into functional arms and connective scaffolds, thereby integrating chenically-informedd priors within diffusion-based generative mechanisms.
MolCRAFT [37] exploits Bayesian Flow Networks coupled with sophisticated sampling schemes, showing significant enbancements relative to contemporary diffusion- based methodologies. We directly utilized the publicly available molecules provided through their respective GitHub reposi- tories for our experiments.
Optimization based Generation Type
RGA [16] extends the evolutionary optimization approach of AutoGrow4 by integrating rein- forcement learning-basse based policy conditioned on target binding pockets, effectively constraining exploratory randomness during molecular search procedures. Since the molecules generated by these models for CrossDocked2020 benchmark are not publicly available on GitHub, we trained these models ourselves to produce molecules for evaluation.
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Dpt [51] employs conditional generative modeling of chemically meaningful fragments aligned with receptor sites, iteratively optimizing molecular generation via guided resampling within structured diffusion latent space, informed by fragment- based oracle rankings. Since the molecules generated by these models for CrossDocked20200 benchmark are not publicly available on GitHub, we trained these models ourselves to produce molecules for evaluation.
TacoGFN [42] leverages G-FlowNNetbaase based autoregressive approach, incrementally assembling molecular structures fragment-by by-frameettt while identifying key pharmacophorie interactions with target proteins. It integrates a reward- -aased optimization mechanism, simultaneously promoting advantageous properties such as binding affinity and synthetic accessibility (SA score) throughout the generative process. We directly utilized the publicly available molecules provided through their respective GitHub repositories for our experiments.
ALIDiff [19] an -qquivaaan diffusion generative model that incorporates recent reinforee- ment learning from human feedback techniques, leveraging Energy Preference Optimization to effectively generate molecules exhibiting superior properties such as enhanced binding affinity and other desirable attributes. We directly utilized the publicly available molecules provided through their respective GitHub repositories for our experiments.
generative model developed specifically in this study to enable gradient- brasde guidance propagation, serving as a comparative baseline for evaluating the ef- fectiveness of guidance mechanisms between BFN and diffusion-based frameworks. Although several studies have adopted similar optimization strategies, we exclude these approaches from consideration, as they either exclusively optimize for binding affinity or neglect guidance on categorical atom types.
TargetOpt is a diffusion-
G Additional Experiment Metrics
G.1 Experimental Setup of Section 6.2
We employ multiple metrics to comprehensively evaluate the binding affinity and intrinsic properties of the generated molecules. Specifically, we utilize SMINA, GNINA, and AutoDock Vina to independently calculate two distinct forms of binding affinity: (1) the intrinsic docking scores of the generated molecules themselves (denoted as Score.), and (2) the affinity scores obtained via re-docking procedures (denoted as Dock.). Additionally, we introduce the High Affinity metric, defined as the percentage of generated molecules exhibiting superior binding affinity compared to a given reference ligand for each target protein. To quantify the intrinsic molecular properties, we measure the Synthetic Accessibility (SA) and Diversity metrics. Finally, to capture the overall stability and validity of generated molecules (both intrinsically and in complex with the target protein) we utilize the PB-valid score from the PoseBusters benchmark. The metric PB-valid denotes the proportion of generated molecules considered valid under the PoseBusters benchmark, assuming that violation of any of its 17 evaluation criteria renders the molecule invalid.
G.2 Experimental Setup
of Section 6.3
We quantitatively evaluate the synthetic feasibility of generated molecules using six key metrics provided by AiZynthFinder (Solved, Routes, Solved Routes, and Top Score). The reported values in Table 2 are averages computed across all generated molecules per model. Specifically, Solved indicates whether AiZynthFinder successfully identified at least one valid retrosynthetic route for a given molecule. A higher number of nodes indicates more extensive exploration of possible reaction pathways by the algorithm, which may reflect the inherent complexity of the target molecule, diversity in applicable reaction templates, increased search depth. While large node count can imply thorough and comprehensive search, it might also signal inefficiencies numerous unproductive pathways were evaluated. Thus, this metric provides valuable insights into the balance between computational effort and search comprehensiveness.
Routes counts the total number of distinct retrosynthetic pathways (complete reaction sequences from purchasable precursors to the target molecule) identified by AiZynthFinder. This metric quantifies the diversity of retrosynthetic solutions identified by the algorithm. A higher number of identified routes suggests multiple viable synthetic strategies for the target molecule, providing chemists with alternative synthetic options. Nevertheless, not all identified routes may be equally practical or feasible; thus, this metric should be interpreted alongside complementary measures such as the number of solved routes or the top-scoring pathways.
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Solved Routes is the subset of these routes comprising exclusively purchasable precursors listed in commercially available databases (e.g., ZINC), thus representing practically realizable synthetic pathways. This metric enables rapid assessment of synthetic feasibility given predefined inventory least one retrosynthetic pathway is successfully identified, the
of building blocks. Specifically, target molecule is considered synthesizable, making this a straightforward, high-level indicator of synthetic achievability. However, as it does not capture pathway quality or diversity, it should be interpreted in conjunction with complementary metrics.
Lastly, Top Score reflects the highest-ranked synthetic route evaluated by AiZyythFinder's scoring function, which aggregates criteria such as precursor availability, reaction step count, and reaction feasibility (e.g., average template frequency). This metric quantitatively represents the quality of the highest-ranked synthetic route, assisting chemists in prioritizing retrosynthetic pathways for consideration. higher score reflects routes with greater feasibility, efficiency, and desirability. This measure is particularly useful for comparing alternative pathways or selecting the most promising candidates for subsequent experimental validation.
H
Additional Experiment result
H.1
Experimental Analysis of Section 6.2
Our proposed model consistently outperforms baseline methods across 12 evaluation metrics covering binding affinity and intrinsic molecular properties, as shown Table 1. In particular, our model significantly outperforms baseline methods in the 'Score' metric, which measures the binding affinity of generated molecules prior to any docking procedure. This result indicates that our model inherently generates molecules possessing high binding affinity, even without additional docking optimization. Similar superior performance is observed in the Synthetic Accessibility (SAA) and PoseBusters validity (PB- Valid) metrics, indicating that gradients derived from binding affinity and SA scores are effectively propagated during the guidance-based sampling process.
Furthermore, we observe that different baseline models excel depending on the affinity evaluation tool used; specifically, ALIDiff outperforms DecompDiff when evaluated by SMINA, whereas DecompDiff achieves better results when GNINA is used. Considering the variability performance across different evaluation metrics, the consistently strong performance of our proposed model across all three docking tools (Vina, SMINA, and GNINA) highlights its robustness and reliability in generating molecules with high binding affinity, as well as its generalizable efficacy across diverse evaluation standards, Additionally, our model uniquely exhibits minimal differences in binding affinity between pre-docking and post-docking evaluations. This minimal discrepancy indicates our model's ability to intrinsically predict stable and energetically favorable binding poses, explicitly capturing meaningful pootein-ligane interactions.
Table 4 demonstrates the effectiveness of jointly applying gradient-based guidance to both atomic coordinates and atom types. As clearly indicated by the results, simultancous guidance across both modalities consistently outperforms methods employing guidance on single modality alone. This outcome aligns with fundamental chemical principles, as molecular properties and the corresponding energy landscape inherently depend upon intricate interactions between atomic types and their spatial configurations.
In addition to the results presented in Figure 5, conducted supplementary experiments using the PoseCheck benchmark to further assess the generated molecules' structural stability and validity. Crucially, molecules evaluated in this additional experiment were directly sampled from generative models, without employing docking. This methodological choice ensures that the evaluation reflects the inherent capability of the generative models, rather than improvements arising from docking based optimizations or adjustments by docking tools. Their reliance on external docking for generating final 3D conformations makes it unsuitable accurately evaluate these models from the perspective of generating intrinsically stable 3D molecular structures. Consequently, models such as RGA and TacoGFN, which initially generate molecules as SMILES strings 2D graphs and subsequently rely on docking software to derive the final 3D conformations, were excluded from this comparative analysis.
Experimental results (Table 5) indicate that the CByG model outperforms baseline models in terms of the "Clash" and "Strain Energy" metrics, whereas DecompOpt achieves the best performance in the "Intermolecular Interaction" " metric. Considering that DecompOpt explicitly optimizes molecules by fixing protein-interacion fragments, this result aligns naturally with its optimization strategy.
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Table 4: Summary of binding affinity and molecular properties of reference molecules and molecules generated by CBYG and baselines. (T)(C) denotes whether a larger/smaller number is preferred. Top results are bolded and underlined, respectively.
Methods
SMINA (1)
GNINA (U)
Vina (L)
SA (1)
PB-Valid (7)
Score.
Dock.
Score.
Dock.
Score.
Dock.
Avg.
Med.
Avg.
Med.
Reference
-6.77
-7.92
-7.06
-7.61
-6.36
-7.45
0.73
0.74
95.0%
95.0%
CBYG
-7.74
-9.61
-7.63
-8.33
-8.60
-9.16
0.84
0.87
94.9%
96.0% %
CBYG w/o pos guidance
-7.05
-8.60
-6.88
-7.42
-7.74
-8.12
0.78
0.79
90.4%
91.1%
CBYG w/o type guidance
-6.92
-8.45
-6.70
-7.20
-7.60
-7.95
0.76
0.77
88.3%
89.0%
CBYG w/o uncertainty
-7.25
-8.81
-7.10
-7.64
-7.90
-8.31
0.75
0.76
87.1%
87.7%
CBYG (Ax 40,
40)
-7.74
-9.61
-7.63
-8.33
-8.60
-9.16
0.84
0.87
94.9%
96.0%
CBYG (Ax 30, Av 30)
-7.13
-8.64
-6.95
-7.47
-7.79
-8.17
0.79
0.81
91.0%%
91. .1%
CBYG (Ax 50, Av 50)
-7.33
-8.94
-7.22
-7.79
-8.07
-8.49
0.74
0.76
85.6%
86.2%
CBYG (Ax 30, Av 40)
-7.28
-8.89
-7.15
-7.68
-7.98
-8.42
0.77
0.78
89.2%
89.7%
CBYG (A2 40 30)
-7.10
-8.59
-6.92
-7.42
-7.76
-8.10
0.80
0.82
91.8%
92.3%
Table 5: Posebusters results for all methods.
CByG
TargetDiff
DecompDiff
MolCraft
DecompOpt
ALIDiff
Avg. Clash (U)
3.71 5.85 10? 15.65
10.54
13.66
5.72
17.05 1.18x 10t1 18.77
8.71
Avg. Strain Energy (L)
x.4 17.15
1.44x10* 18.26
7.57 x 1
6.22x 1016
Avg. Interaction (1)
15.92
17.74
Table 6: PB- Valid results for all metrics.
Metric
Valid Score
Mal Prad Loaded Sanitization Arovatic Ring Flatnasa
1,000
All Atone Connected
0.985
Band Length & Bond Angle
0.995
Internal Steric Clash
0.993
Double Flatnass
0.998
Internal Enargy
0.971
Liggein-Ligand Mexinun Distance Xinimun Distance to Organic Cofactors & Minimum Distaance Waters
1,000
Micimun Distance
Miniman bistanca to Tnarganic, Cafactora
0.994
Volume Overlap with Protein & Volume Overlap with Organic Cofactora
1,000
Volune Overlap Tnorganic Cofactora Valune Overlap Watera
H.2 Experimental Analysis of Section 6.3
In this experiment, both our proposed model and the RGA model exhibited overall high performance. Here, it is important to consider molecular complexity in relation to binding affinity. Typically, molecules with greater complexity tend to possess higher binding affinity due to increased oppor- tunities for intermolecular interactions with target proteins; conversely, simpler molecules usually exhibit lower affinity. Given this, the performance of the RGA model on the AiZynthFindes bench- mark aligns logically with its relatively lower binding affinity scores reported in Table 1. Applying the same perspective to our proposed model, it is particularly noteworthy that model not only achieves top-tier performance in binding affinity but also exhibits near state-of-the-ar results on the AiZynthFinder benchmark. This indicates the ability of our approach to generate molecules that are simultaneously effective in terms of biological efficacy and practical retrosynthetic feasibility.
Interestingly, we observe that several models show no clear correlation between their performance on the AiZynthFinder benchmark and the SA score reported in Table 1. This highlights the necessity for evaluating synthetic feasibility in SBDD research using multiple diverse criteria beyond just the SA score. A notable concern is that despite high SA scores (approaching 0.8 for RGA and surpassing 0.8 for our proposed model) the fraction of molecules classified as synthetically feasible under the AiZynthFinder 'Solved' metric remains below 50%. This outcome suggests that synthetic feasibility deserves greater attention in future SBDD research, underscoring the need for broader consideration and deeper analysis of retrosynthetic practicality.
H.3 Experimental Analysis of Section 6.4
As demonstrated in Figure 6, guidance scores obtained using the BFN-based approach consistently surpass those derived from diffusion-based methods across the entire generative trajectory. Fur- thermore, guidance scores from diffusion models utilizing predictions of the final clean state (Xn) exhibit marked instability, underscoring the robustness of the BFN-guided generation procedure.
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Notably, diffusion basedd methods (green and gray plots) yield higher absolute guidance scores when employing predictions of the final molecular states; however, these scores simultaneously exhibit increased variance as the generative process advances. This behavior highlights a fundamental trade-off within diffusion-basee guidance strategies in 3D molecular generation: gradients derived from predicted clean states facilitate higher guidance scores but necessitate point estimation toward the final state in the sample space, inherently introducing instability into intermediate guidance steps. In contrast, BFN operates in continuous parameter space rather than directly in sample space, enabling stable and continuous gradient propagation even for categorical variables. Consequently, BFN- based guidance using predicted final states provides inherently more stable gradient trajectories, making it particularly advantageous for robust and controllable 3D molecular generation.
H.4 Experimental Analysis of Section 6.5
To evaluate the selectivity control capabilities of the proposed CBYG model, we conducted exper- iments using the selectivity benchmark set specifically constructed for this study. We primarily assessed and compared selectivity performance before and after applying guidance in two generative model categories: diffusion-baas -basd models and Bayesian Flow Network-based models. Notably, optimization-basee generation models were excluded from this comparison due to their intrinsic requirement for retraining to optimize for different molecular properties, highlighting the versatility of our proposed model in addressing diverse property optimization objectives.
Molecule generation was directed toward enhancing binding affinity to designated on-target proteins, while guidance was explicitly designed to minimize binding affinity to specified off-target proteins. In Table 3, the "Succ..Rate" represents the proportion of generated molecules demonstrating superior affinity for the on-target protein relative to the off-target, whereas the "A Score" quantifies the differential affinity between on-target and off-target interactions.
Experimental results revealed that even without explicit dnlectivity-guidaaae both model categories produced molecules with superior affinity toward the on-target protein in more than half of the gener- ated cases. This outcome can be attributed to the inherent advantage of structure- models, which explicitly encode and leverage the structural context of the target proteins during molecule generation. Nevertheless, Bayesian Flow Network-based models consistently demonstrated superior performance compared to diffusion-based models, and this advantage was markedly ampli- fied when selectivity guidance was employed. These findings collectively underscore the efficacy and versatility of the proposed CBYG framework in achieving controlled generation not only for conventional metrics such as synthetic accessibility (SA score) but also for critical properties such as selectivity.
generative
H.S Comparative Visualization of Generated Ligands
I
Selectivity Dataset
We constructed a selectivity-focunt dataset based on kinase inhibitor selectivity data. Initially, we identified the selectivity profiles of 285 proteins across 38 kinase inhibitors, reported in a study on the quantitative analysis of kinase inhibitor selectivity [27, 10]. Subsequently, for each inhibitor, we categorized the proteins into on-target and off-target groups and extracted their corresponding Entrez Gene Symbols. These gene symbols were then used to systematically gather protein-related data from the UniProt [8] database via REST APIs. The UniProt web crawling process was structured in three stages. First, the Entrez Gene Symbols were URL-encoded and filtered by the human taxonomy ID (9606) to retrieve corresponding UniProt IDs in ISON format. Second, protein sequences were obtained by querying the UniProt FASTA API. with FASTA headers subsequently removed. Third, ATP binding site information coemmatioonnnlludinn binding site positions and sequences was extracted from the UniProt feature sections. The collected data, comprising UniProt IDs, sequences, and binding site details, were integrated using the initial set of 285 Entrez Gene Symbols as a reference. Proteins lacking ATP binding site data (six in total) were excluded, yielding a final dataset of 279 proteins prepared for further analysis. Protein structures were predicted using AlphaFold3 [1], and model structure files were retrieved in CIF format. These files were converted to PDB format using the PDo module of Biopython. The ATP binding site information was then employed to define and extract protein pocket structures, specifically targeting the binding site residues and the surrounding region within a 5Á radius. Structural similarity among the protein pockets was evaluated using TM-score and RMSD metrics. The TM-score quantifies topological similarity between protein structures, with values ranging from (0, 1]; scores above 0.5 typically indicate identical protein folds,
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Reference
ALIDiff
DecompOpt
MoICRAFT
CBy6
4QLK
3DAF
3696
Figure 8: Visualization of generated ligands for protein pockets, with a reference molecule (left) and corresponding outputs from ALIDiff, DecompOpt, MoICRAFT, and CBYG.
while scores below 0.17 suggest unrelated structures. We classified and organized the extracted protein structures into directorics based on a TM-score threshold of 0.4 and an RMSD 1Å, reflecting structurally similar protein pockets suitable for downstream selectivity analyses. This process ultimately facilitated the construction of on-target (primary) and off-target pairs.
J
Rethinking
J.1 Addressing Fundamental Challenges of Difflsioo--bssd Guidance in 3D Molecular Generation
core objective in Structure- Drug Design (SBDD) is generating molecules that bind specifi- cally target proteins while simultaneously satisfying desired properties. Diffusion-based generative models are particularly suited to this task, as they can incorporate external predictors for property- guided sampling. Specifically, these models leverage guidance derived from predictors direct the generative process toward property-specific regions of molecular space. However, as briefly men- tioned in previous sections, 3D molecular structures inherently comprise hybrid data types, consisting of continuous variables (e.g., Cartesian coordinates) typically modeled by Gaussian distributions, and categorical variables (e.g, atom types such as oxygen nitrogen) typically represented by categorical distributions. This hybrid nature presents fundamental challenges for conventional gradient- based guidance approaches.
First, since coordinates and atom types are sampled from fundamentally distinct distributions, guidance gradients tend to propagate independently across these data types. Consequently, the guidance mechanism often fails to accurately capture the critical chemical interdependencces between atomic coordinates and categorical atom identities, thereby undermining the chemical coherence of the generated molecules.
Second, categorical variables in diffusion models rely on discrete sampling processes involving an argmax operation each reverse sampling step. Due the discrete nature of argmax operations, direct application of gradient- based guidance becomes infeasible, minor guidance gradients typically do not influence the argmax outcome unless excessively amplified. Yet, increasing the guidance scale excessively can cause the distribution during reverse sampling to become dominated by guidance gradients, resulting in unstable and unrealistic molecular structures. Alternatively, attempting to circumvent this issue by artificially converting categorical distributions into continuous or discretized variables introduces unnatural assumptions and significantly increases the complexity of model design.
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Lastly, injecting guidance gradients directly into the denoising process, which operates in the molecu- lar sample space, risks destabilizing intermediate molecular configurations. This instability arises due to the numerical sensitivity of 3D coordinates, potentially causing molecules to lose chemical validity and structural coherence during intermediate generation stages. Thus, this approach substantially hinders effective controllability over molecular properties, and leads to unstable molecular outcomes. In the context of 3D molecules (with continuous coordinates and discrete atom types), diffusion-basel generative guidance methods face several fundamental limitations.
First, guidance gradients often fail to capture interdependencies between modalities (e.g. coordinate updates and stom-type assignments may be misaligned if treated separately) as evidenced by the need for separate latent spaces or noise schedules for different variable types in prior diffusion approaches. Second, guidance in categorical diffusion unstable and often ineffective: choosing atom types via an argmax during the denoising process introduces discontinuous, non-differentiable operation that disrupts gradient-based optimization. Third, the denoising of spatial coordinates is structurally fragile adding noise to atomic positions can break chemical bonds or distort interatomic distances beyond physical limits, leaving intermediate states chemically invalid and uninformative. Given these challenges, BFN offer promising alternative for property-guided molecular generation. BFN operate in a fully differentiable parameter space and provide a unified probabilistic treatment of continuous and categorical modalities, thereby inherently modeling cross-modal dependencies and avoiding the need for modality- speciffc hacks. In contrast to diffusion, BFN do not require per-step argmax sampling for atom types; instead, they maintain probability simplex representation for categorical variables, preserving gradient information throughout the generative process. This unified and differentiable approach enables stable pradient-based guidance on molecular properties, making BFN a robust paradigm for 3D molecule generation under complex hybrid objectives.
J.2 Rethinking Posterior Guidance: From Intermediate States to Predicted Final Structures
In generative frameworks such as diffusion models, conditional generation typically involves control- ling the generative process by leveraging gradient guidance from posterior conditioned on labels (attributes) 1. According to the theoretical foundations of the reverse process, the introduced posterior term can be represented as x(1 xx). commonly known as the conditional score function, which is usually learned using dedicated neural network. Here, it is crucial to reassess whether the interme- diate state Iy, employed as input for the conditional score function, a sensible variable for attribute prediction [11, 45, 22]. In domains like image generation, where score-based diffusion models were initially introduced, intrinsic structural characteristics of the data enable meaningful predictions of labels even from intermediate noisy states. Thus, utilizing the conditional score function in the form p(1 xx) has proven reasonable in these contexts.
However, unlike image data, intermediate states of 3D molecular structures with added noise lose their chemical validity, rendering derived molecular properties essentially meaningless. Consequently, employing a posterior conditioned directly on the intermediate noisy state x,, i.e., p(1 xt). is fundamentally unreasonable as guidance for molecule generation tasks. To overcome this limitation, recent studies have adopted a posterior sampling strategy, originally proposed in inverse problems within the image generation domain [7, 23]. Specifically, these methods predict the final, free molecular structure Zo and leverage this prediction to guide gradient- based generation, i.e., p(1 &o).
Further efforts have also extended such posterior sampling approaches to the conditional generation of 3D molecules. However, existing methods primarily focus on general conditional molecular generation rather than the specialized task of structure-based drug design (SBDD), which involves molecular binding to specific proteins. Therefore, substantial modifications and further methodologi- cal advancements are necessary for applying these approaches to the SBDD task. Notably, existing frameworks discretize categorical variables representing atom types into continuous representations, which is inherently unnatural given the data's discrete characteristics.
In summary, predicting the posterior for the final molecular structure xo and subsequently using it for calculating guidance gradients is more principled in the context of SBDD tasks. We propose that this principle is broadly applicable across generative modeling frameworks, including not only diffusion models but also BFN.
J.3 Limitations of Current Evaluation Methods
In previous research on structure-based drug design (SBDD), evaluating the binding affinity between generated molecules and their target proteins has been common practice to assess model perfor- mance. Most studies traditionally relied heavily AutoDock Vina to measure binding affinity.
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Although AutoDock Vina provides three distinct scoring metrics, these metrics inherently depend on the same underlying computational algorithm, potentially introducing bias due to reliance on single scoring method. To enhance the generalizability and reliability of affinity assessments, incorporating multiple docking algorithms in the evaluation process necessary. Accordingly, Section 6.2, we present a detailed experimental setup employing several docking tools, including AutoDock Vina, to enable a more comprehensive and robust evaluation of generative model performance.
Furthermore, previous SBDD research has commonly utilized the Synthetic Accessibility (SA) score to evaluate the synthetic feasibility of generated molecules. The SA score quantitatively integrates chemical fragment contributions and structural complexity penalties into a single metric ranging between 0 and 1, with higher scores indicating greater synthetic accessibility. However, molecules possessing very high SA scores (c.g., greater than 0.9) frequently lack viable retrosynthetic pathways, making their actual synthesis infeasible. Regardless of molecule's theoretical efficacy, its practical value is severely limited without an achievable synthetic route. Therefore, rigorously assessing realistic synthetic accessibility is critical, although research addressing this aspect has been relatively limited. Recognizing the importance of this issue, we introduce the AiZymthFinder benchmark, an evaluation method for retrosynthetic analysis based on practically available chemical building blocks.
From the viewpoint of practical drug development, selectivity is equally important as binding affinity for identifying promising drug candidates. Selectivity refers to the ability of a candidate molecule to specifically bind to its intended target protein without significant interactions with off-target proteins. Molecules lacking sufficient selectivity may interact with unintended proteins, potentially causing side effects or adverse reactions, thereby reducing or negating the desired pharmacological effects. Recently, selectivity has received increased attention in the field of 3D molecular generation, and several diffusion-based guidance strategies have been proposed to address this requirement.
However, existing dllectivtt-foccssee strategies typically require prior training of classifiers that distinguish between positive (binding) and negative (non- binding) protein-ligane pairs. Furthermore, the CrossDocked2020 dataset, commonly used in docking studies, was not originally constructed for selectivity evaluations. Thus, leveraging this dataset for selectivity assessments necessitates extensive additional docking computations. Moreover, the absence of clear criteria for identifying true binding molecules and the significantly greater number of false binding molecules relative to true binding molecules pose substantial challenges for obtaining generalizable guidance signals. Consequently, deriving selectivity metrics based solely on the CrossDocked2020 dataset inherently risks bias due to these intrinsic dataset limitations. Most critically, the CrossDocked dataset may not adequately reflect biologically meaningful selectivity, limiting its practical utility for reliable selectivity assessment. Therefore, establishing rigorous, standardized benchmark datasets capable of objectively evaluating selectivity is essential. Additionally, there is an urgent need to develop novel, efficient controllable generation strategies capable of effectively ensuring molecular selectivity.
K Justiffcation for Applying Tweedie's Formula
Justification for Applying Tweedie's Formula in Continuous Variables In our framework, the sender distribution for continuous variables is explicitly modeled as a Gaussian:
ps x; ai) -Ny x1)a"1
Given a prior over (possibly Gaussian), the Bayesian update for the parameter 0 given noisy observation y from the sender is:
p(a y) x Ps(y) x; a) p(x) If we assume the prior p(x) N(x po, Eo). then the posterior p(x (g is also Gaussian, due to conjugacy:
p y) N (xr Mpout: Eppus)
where
Spo - + al, Ppost Eposs (27 no + aay
Thus, the Bayesian update function for the mean parameter becomes a linear function of the previous mean and the new observation y:
0 Pi-l 0, + S
Pi
P:
with P Pi-1 + ag, which matches the classic Kalman filter update for Gaussian models.
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Conclusion:
Since the posterior is exactly Gaussian, Tweedie's formula
E v| = y + a V log p(v)
strictly valid, and our use of Tweedie's formula to obtain a gradient-based update is mathematically justified.
Justification for Applying Tweedie's Formula in Categorical Variables For categorical variables, the latent class indicator ezr is originally a one-hot vector, i.e., 6z E (0, 1] with Ek=1 (k) = 1. However, in framework, reparameterize e, to continuous relaxation (e.g., using the Gumbel- softmax trick), and the sender distribution is modeled as multivariate Gaussian over this relaxed variable:
Ps(y (x; = N (y a(Kez - aK])
where y a continuous vector and e₂ is now allowed to be a point in the probability simplex. Given this sender structure, the "pseudo-posterior" p(ez y) (formally, the conditional distribution in
the continuous relaxation) is also multivariate Gaussian, due the properties of conjugacy between Gaussians.
Thus, the Bayesian update for the categorical parameter 0 (in the reparameterized space) can be written function of y and the previous parameter:
Pi-1
cxi++
C +
y/:
Pi
Pi
with P: P:-1 + ag. which is the direct analogue of the update for the continuous case.
Applicability of Tweedie's Formula:
Since both y and ez are continuous and (conditionally)
Gaussian under the sender, Tweedie's s formula is applicable:
Eie, vl = y (aK) 'V, log pwy
and, by chain rule,
Ve, logp(e>)) V,
where Jem-oy is the Jacobian matrix of the sender's 's mapping.
By reparameterizing the categorical variable into continuous, sender- Gaussianized latent variable, all required conditions for Tweedie's formula (Gaussianity and differentiability) are satisfied in the update step. This mathematically justifies the use of Tweedie's formula for gradient-based guidance in categorical settings, just as in the continuous case.
Conclusion:
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36


Reviews:
[{"review_id": "QSEMShSSfX", "reviewer": "Reviewer_dDy1", "summary": "This paper addresses a practical and important challenge in structure-based drug design: generating 3D molecules that are not only high-affinity binders, but also selective and synthesizable. The authors propose CByG, a novel framework that integrates Bayesian Flow Networks with gradient-based guidance. Using Tweedie’s formula, they reinterpret Bayesian updates as gradient steps, allowing controllable generation in parameter space. The paper also presents a well-rounded evaluation framework. Results show CByG outperforms existing methods across key metrics.", "questions": "1. The \"Solved\" rate in AiZynthFinder is below 50% despite high SA scores. Could the authors briefly discuss why this gap exists?\n2. No runtime comparison is provided. Can the authors include training/inference time versus diffusion models?\n3. Line 302 incorrectly refers to “Table 3” for AiZynthFinder results. Please correct.\n4. Line 300 mentions six metrics, but only four are shown in Table 2. Please clarify.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n• Clear problem statement and strong motivation\n• Strong theoretical link between Bayesian updates and gradient guidance\n• Addresses practical needs beyond binding affinity by incorporating synthesizability and selectivity into both modeling and evaluation\n• Clear writing and well-organized structure\nWeaknesses\n• The discussion of synthesizability results is somewhat limited.\n• The paper does not provide analysis of training or inference efficiency.", "was_revised": true, "final_justification": "The authors’ rebuttal offers a clear and nuanced explanation of the SA-score / AiZynthFinder relationship, promises careful corrections to citations and table labels, and provides useful runtime comparisons. With these points now clarified, I am happy to continue supporting acceptance.", "rebuttal": "We have carefully reviewed the valuable comments you provided, and have summarized the main points for revision as follows.\n\nPlease note that, due to space constraints, we have grouped similar issues into broader categories for clarity. If any of these points have been misunderstood, we would appreciate your feedback.\n\n1. Discrepancy between SA score and AiZynthFinder\n\n2. Citation errors regarding AiZynthFinder and label errors in Table 2\n\n3. Experimental comparison of computational efficiency between our proposed model and diffusion-based approaches\n\n---\n\n***1. Discrepancy between SA score and AiZynthFinder “Solved” rate***\n\nThe gap between the SA score and the AiZynthFinder “Solved” rate, as pointed out by the reviewer, stems from fundamental differences in the evaluation mechanisms and interpretations of these two metrics. Below, we provide a detailed explanation of these differences and how they should be interpreted:\n\n*SA (Synthetic Accessibility) Score*\n\n* Evaluation Method:\nThe SA score statistically estimates the synthetic accessibility of a molecule by decomposing it into fragments and quantifying how frequently these fragments appear in large chemical databases (e.g., PubChem). Penalties are applied for structural features that increase complexity, such as stereocenters, spiro rings, and macrocycles. The score reflects “average synthetic accessibility” and is normalized between 0 (very difficult) and 1 (very easy).\n\n* Interpretation:\nThe SA score is a statistical predictor of how easy a given molecular structure might be to synthesize chemically. It does not account for the actual existence of synthetic routes, available starting materials, or practical experimental constraints.\n\n*AiZynthFinder “Solved” Rate*\n\n* Evaluation Method:\nAiZynthFinder is a procedural retrosynthetic tool that uses Monte Carlo tree search and neural network-based reaction predictors to iteratively break down a target molecule into commercially available starting materials (stock molecules). A molecule is marked as “Solved” if a complete synthetic route is found; otherwise, it is “Unsolved.”\n\n* Interpretation:\nThis metric reflects practical synthesizability given current reaction databases, available stocks, and model performance—in essence, whether a concrete, feasible synthesis route exists under realistic laboratory or industrial conditions.\n\n***Key Causes for Discrepancy***\n\n1. Stock Compound Limitations:\nThe SA score is agnostic to the availability of stock chemicals, while AiZynthFinder requires the route to begin with a predefined set of purchasable starting materials. Thus, a simple molecule may still be “Unsolved” if a necessary intermediate is not available in the stock set.\n\n2. Fragment Popularity vs. Reaction Feasibility:\nThe SA score rewards frequent fragments but does not guarantee that the combination of those fragments is represented in known reactions. AiZynthFinder, on the other hand, demands feasible reaction pathways, so rare or unlearned connection strategies will result in “Unsolved” outcomes.\n\n3. Multistep or Protection/Deprotection Complexity:\nEven seemingly simple molecules may require complex multistep syntheses or specialized strategies. The SA score does not reflect this procedural complexity, but AiZynthFinder will directly be affected by such challenges when searching for routes.\n\nSA score and AiZynthFinder “Solved” rate complement each other but fundamentally reflect different aspects of synthetic feasibility.\nA high SA score suggests structural simplicity and ease of synthesis, but a low “Solved” rate may still result if there are strict constraints imposed by stock compound availability or reaction databases.\nThus, both metrics should be interpreted together, and reliance on either metric alone is insufficient for a comprehensive assessment of synthesizability.\n\n---\n\n***2. Citation errors and labeling issues in Table 2 for AiZynthFinder***\n\nWe will immediately correct the cited reference errors to ensure there are no mistakes in the revised version.\nAdditionally, we acknowledge the error in reporting six metrics in the text, and will ensure that such inaccuracies do not persist in the updated manuscript.\nThank you for your careful attention to these details.\n\n---\n\n***3. Experimental comparison of computational runtime***\n\nWe have measured the training and inference times of TargetDiff [1], the most widely cited diffusion-based model.\nAdditionally, as described in Appendix E, TargetOpt is a guidance model based on TargetDiff that we developed in this study specifically for comparison with CByG , which implements diffusion-based guidance methods.\nBelow, we provide a comparative table of computational speed between the diffusion-based models and our approach.\n||Training Time (h)|Inference Time (h)|\n|:-:|:-:|:-:|\n|TargetDiff|11.2|0.95|\n|TargetOpt|13.4|1.67|\n|CByG|8.7|1.01|\n\n#### [1] Guan, Jiaqi, et al. \"3D Equivariant Diffusion for Target-Aware Molecule Generation and Affinity Prediction.\" The Eleventh International Conference on Learning Representations.\n\n---\n\nYour insights and constructive feedback have played a crucial role in enhancing the quality of our manuscript. We are deeply grateful for the careful attention and time you have invested in reviewing our work.\n\nIf you require further information or would like to discuss any aspect in more detail, please let us know at any time."}, {"review_id": "FDgwnX4miw", "reviewer": "Reviewer_kfDY", "summary": "The authors identify fundamental concerns with gradient-based guidance applied in sample space (e.g., when using diffusion models for structure-based drug discovery (SBDD)) and subsequently propose a Bayesian flow network for parameter space gradient-based guidance in SBDD. Empirical results for existing and new benchmarks are compelling and further suggest that future work (for all generative methods) is needed to improve the practical synthesizability of each method's generated molecules.", "questions": "Could the authors discuss what (practical or empirical) disadvantages they have found when developing Bayesian flow networks for molecular data? Are they harder to train than diffusion models, for example?", "limitations": "The authors currently only include one core SBDD benchmark dataset for general sample scoring (i.e., CrossDocked). Including another dataset for benchmarking would improve the rigor of the authors' experiments.", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Points of strength:**\n1. The authors identify and propose a solution to the challenges of sample space gradient-based guidance in SBDD (e.g., with diffusion models).\n2. The authors' proposed CByG network achieves strong empirical results for existing and new SBDD benchmarks.\n3. The authors' writing and methodological rationale are clear and convincing.\n\n**Points for improvement:**\n1. Some of the authors' claims may not be entirely accurate. For example, in Section 6.2, the authors state that they introduce PoseBusters docking scores [1] into SBDD evaluation criteria. However, previous works in SBDD with diffusion models have already done so, as shown in works such as [2] (and such baselines are missing from the results in Table 1).\n2. The authors should consider including a second benchmark dataset for SBDD sample scoring, such as Binding MOAD, PDBBind, or PLINDER. This would help to address a concern that CByG is simply overfitting to CrossDocked-like protein-ligand complexes.\n\n**References:**\n\n[1] Buttenschoen, M., Morris, G. M., & Deane, C. M. (2024). PoseBusters: AI-based docking methods fail to generate physically valid poses or generalise to novel sequences. Chemical Science, 15(9), 3130-3139.\n\n[2] Morehead, A., & Cheng, J. (2024). Geometry-complete diffusion for 3D molecule generation and optimization. Communications Chemistry, 7(1), 150.", "was_revised": true, "final_justification": "The authors have addressed my concerns regarding overstatements, additional benchmarking, and the scalability of Bayesian flow networks in general.", "rebuttal": "We have carefully reviewed your valuable comments and, as such, have summarized below the areas for improvement that you highlighted.\nWe hope you will understand that, due to space limitations, we have grouped similar questions into broader categories rather than addressing each in exhaustive detail.\nIf any of our interpretations are inaccurate, please let us know.\n\n***1. Overstatement regarding the introduction of PoseBusters and omission of GCDM as a baseline***\n\nWe thank the reviewer for this precise feedback. The use of “introduce” in Section 6.2 was indeed inaccurate. We acknowledge that the PoseBusters docking score has already been utilized in prior SBDD works, such as [1]. We will revise the language to “adapt” or “incorporate” as appropriate.\n\nWe also agree that omitting the diffusion-based baseline (e.g., GCDM-SBDD) from Table 1 is a significant oversight. As the generated molecule samples for GCDM-SBDD are not publicly available, we reproduced the model using the released weights and generated samples in our experimental environment. We evaluated these samples using PoseBusters under the same benchmarking protocol, and have added the results to Table 1 and the main text.\n#### HA refers to \"High Affinity\" , and PV refers to \"PB-Valid\"\n|Methods|SMINA Score|SMINA Dock|GNINA Score|GNINA Dock|VINA Score|VINA Dock|HA  |SA  |PV |\n|-|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|\n|Reference| -6.37| -7.92| -7.06| -7.61| -6.36| -7.45|-|0.74|95.0%|\n|CByG|-7.74|-9.61|-7.63|-8.33|-8.60|-9.16|93.6%|0.84|94.9%|\n|GCDM-SBDD|4.34|-6.55|4.34|-4.91|3.19|-5.79|17.1%|0.58|21.2%|\n\n\nFor the reproduced GCDM-SBDD-cond(Ca) model, we found that the AutoDock vina scores could be reasonably replicated. In the original GCDM paper’s Table 4, the “Vina” column corresponds to the AutoDock(Dock) score in our results table.\nMoreover, Table 4 of the original GCDM paper reports PB-Valid scores as 38.1 / 15.7, where 38.1 excludes the ‘protein-ligand steric clashes’ check, and 15.7 includes all metrics. The original text in *GCDM* paper is as follows:\n> Additionally, the PB-Valid metric is defined as the percentage of generated molecules that pass all docking-relevant structural and chemical sanity checks proposed by ref. 23, with the validity ratio to the left (right) of each/denoting the percentage of valid molecules without (with) consideration of protein-ligand steric clashes.\n\nAs our PB-Valid scores were computed with all PoseBusters checks included, comparing our results to the 15.7 benchmark is most appropriate, and our reproduction appears consistent in this context.\n\nBy addressing these points, we have made our benchmarking and result comparisons more comprehensive and transparent. Thank you again for your meaningful feedback; we will continue to ensure that our evaluations remain consistent and reproducible.\n\n***2. Experiments on additional benchmark sets***\n\nWe greatly appreciate this important suggestion. To further assess the generalization of CByG and potential overfitting to the CrossDocked dataset, we are currently establishing an evaluation pipeline on the Binding MOAD benchmark.\n\nFirst, we would like to share the interim results that have been completed so far(The experimental setup for Binding MOAD was preprocessed as described in Schneuing et al. [2], resulting in a benchmark of 130 test proteins, with 100 ligands generated per target pocket. All other settings were kept consistent with those used in Table 1 of the submission.). We are doing our best to generate reliable results in a timely manner. As soon as all evaluations on the Binding MOAD benchmark are finished, we will transparently summarize and share the results and analysis before the discussion period ends.\n\n||Vina Score ↓|Vina Dock ↓|\n|:-:|:-:|:-:|\n|Ref|-6.942|-8.491|\n|CByG|-8.415|-9.594|\n\nWe ask for your patience as we finalize these experiments, and we are confident that these results will adequately address your concerns regarding generalizability.\n\n***3. Empirical limitations of Bayesian Flow Networks***\n\nThank you for this important question. Bayesian Flow Networks (BFNs) have demonstrated several practical advantages over diffusion-based generative models. For instance, the model converges stably with relatively few (e.g., 100) timesteps, and sampling can be performed efficiently in significantly fewer steps, resulting in lower computational costs and generation times.\n\nTo date, we have not observed any significant increase in training difficulty or major practical limitations compared to diffusion models. In fact, the reduced timesteps and numerically stable closed-form update rules have contributed to overall training efficiency.\n\nHowever, we have identified a limitation regarding the propagation of multiple guidance signals. While controlling a single property (e.g., binding affinity) is effective, generating molecules that simultaneously satisfy multiple guidance properties (e.g., binding affinity and synthetic accessibility) remains challenging. This difficulty is particularly relevant in drug discovery scenarios, where multi-property optimization is essential. We recognize this as a crucial open challenge.\n\nIn summary, the primary empirical limitation of BFNs lies in the optimization difficulty and fine-grained trade-off control when propagating multiple guidance signals. We plan to address these limitations in future work.\n\n#### [1] Morehead, A., & Cheng, J. (2024). Geometry-complete diffusion for 3D molecule generation and optimization. Communications Chemistry, 7(1), 150.\n#### [2] Schneuing, Arne, et al. \"Structure-based drug design with equivariant diffusion models.\" Nature Computational Science 4.12 (2024): 899-909.\n\n---\n\nYour comments have been instrumental in significantly improving the quality of our submission. We sincerely appreciate the time and effort you have devoted to reviewing our work.\n\nShould you have any additional inquiries, we would be happy to address them at your convenience."}, {"review_id": "5nQN3N9IdA", "reviewer": "Reviewer_aH1P", "summary": "In this work, the authors present a gradient-based representation for Bayesian Flow Networks suitable for diffusion-guided structure-based molecular generation. As their main contributions, the authors highlight specific challenges surrounding diffusion-guidance for SBDD applications (invalid intermediate samples, synthesizability, selectivity), propose a novel update for bayesian flows and apply this model to molecular generation tasks. The authors evaluate the proposed model against a number of SBDD baselines and show favourable performance across binding affinity and synthesizability evaluations.", "questions": "1. AIzynthfinder is used to assess synthesizability in Section 6.3. Is a different model used for guidance?\n2. How distinct from the training set are the molecules generated by the model?\n3. While the authors present an improved diffusion-based method for synthesizability guidance, it is not clear how this synthesizability evaluation can be rooted in specific chemistry capabilities. The use of generic synthesizability scores would lead to compounds that are believed to be synthesizable \"in general\", or simply similar to known compounds, but specific chemistry routes still have to be designed by hands (or through retrosynthesis) for every generated compound, without the possibility of binding the generation process to a particular vendor's catalog. Given this limitation, it appears that the method is primarily aimed a low-throughput, custom chemistry stages rather than large-scale hit identification?", "limitations": "Not quite. I found little discussion on the limitations of this line of work in general (structure-based continuous molecular generators) and for the presented method in particular (except about one line in the conclusion).", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\nOverall, the manuscript is of good quality and generally well written. In Section 2, the authors clearly identify a number of challenges motivating the presented work. The proposed perspective for the bayesian update is presented in sufficient detail and the evaluation procedure addresses important challenges of molecular generation, especially for struture-based continuous methods. Figures 2, 3, 4 effectively support the main text.\n\n## Weaknesses\n\n### Lack of justification and context for diffusion-based models for drug design\n\nOne of my main critiques is the absence of context to justify SBDD relying on diffusion models. In particular, the authors highlight in Section 2 a number of challenges relating to guidance, synthesizability and multi-parameter optimization which affect diffusion-based models for drug design applications. Some of these challenges are absent or considerably mitigated with discrete optimisation approaches such as Reactor [1], Synthemol [2], Synflownet [3] which can incorporate synthesizability constraints directly into the design of the MDP and use a reinforcement learning paradigm which more naturally lend itself to multi-objective optimization. This contrast is not being discussed either in the introduction, problem presentation or related work. I believe it is important to situate the reader with respect to the pros and cons of both family of approaches (structure-based continuous generators v.s. RL-based constrained generators).\n\n[1]: Horwood, J., & Noutahi, E. (2020). Molecular design in synthetically accessible chemical space via deep reinforcement learning. ACS omega, 5(51), 32984-32994.\n\n[2]: Swanson, K., Liu, G., Catacutan, D. B., Arnold, A., Zou, J., & Stokes, J. M. (2024). Generative AI for designing and validating easily synthesizable and structurally novel antibiotics. Nature Machine Intelligence, 6(3), 338-353.\n\n[3]: Cretu, M., Harris, C., Igashov, I., Schneuing, A., Segler, M., Correia, B., ... & Liò, P. (2024). Synflownet: Design of diverse and novel molecules with synthesis constraints. arXiv preprint arXiv:2405.01155.\n\n### Other elements hindering clarity\n\n- I believe the Related Work should be presented in the main paper. It is reasonable to extend the related work section in appendix but in its current version the manuscript lacks context and references to previous work in the main text.\n- In Figure 1: the axes are not identified, not clear what we are looking at. The caption is not particularly helpful either in fully understanding what is being shown. This Figure needs to be clarified.\n- It is unclear from the main text which models are used for synthesizability and selectivity guidance for the experiments carried out in Sections 6.3 and 6.4.\n\n### Insufficiently supported claims\n\n- In the abstract, line 12, the claim that the limitations of previous evaluation methods have been \"overcomed\" is too strong. I would suggest tuning it down -- there certainly remains numerous limitations to the proposed evaluation paradigm.\n- On lines 93-94: \"However, high SA scores often do not guarantee practical synthetic routes, highlighting a critical gap in accurately evaluating real-world synthesizability.\" -> this observation has been frequently made and would benefit from pointers to the relevant litterature. Scrutiny into SA score is again overclaimed on lines 253-257: \"To thoroughly evaluate the effectiveness and practicality of our proposed framework, we define 4 key research questions essential to the domain of structure-based drug design: [...] 2) Is the widely-used Synthetic Accessibility (SA) score an absolute indicator for practical synthetic 257 feasibility?\". This question has been thoroughly studied and almost always answered negatively in previous work (for exemple but not limited to: [4, 5]). While it is relevant for the authors to highlight the limitations of the SA score, this phrasing makes it sound as if this investigation was an orginal contribution of the presented work. However, nothing forces the author to even reference that metric which is largely seen as outdated. It is indeed crucial to consider synthesizability challenges of SBDD applications. I would simply recommend to authors to cite the relevant and most up-to-date litterature for synthesizability scoring (AIzynthfinder, as the authors use in Section 6.3 is a reasonable choice). Finally, again on lines 306-307, the authors portray as \"novel\" an observation which has been repeatedly made in previous work: \"Interestingly, we observed no clear correlation between SA scores and actual retrosynthetic feasibility 307 (as measured by the Solved metric)\".\n\n[4]: Coley, C. W., Rogers, L., Green, W. H., & Jensen, K. F. (2018). SCScore: synthetic complexity learned from a reaction corpus. Journal of chemical information and modeling, 58(2), 252-261.\n\n[5]: Liu, C. H., Korablyov, M., Jastrzebski, S., Włodarczyk-Pruszynski, P., Bengio, Y., & Segler, M. (2022). RetroGNN: fast estimation of synthesizability for virtual screening and de novo design by learning from slow retrosynthesis software. Journal of Chemical Information and Modeling, 62(10), 2293-2300.\n\n### Typos and syntax\n\n- The hyperlinks seem to be missing (on citations, section numbers, etc.). I cannot click on them to jump to other sections.\n- In lines 69-86, I'd suggest using a different letter than $l$ for the labels/properties (ambiguous).\n- Figure 3 is not referenced anywhere in the text.\n\n## Conclusion\n\nI believe the presented work is of good quality, and of interest to the NeurIPS community, but that it could also be improved by better contextualizing this line of work (diffusion models for structure-based molecular generation) and tuning down some of its claims.", "was_revised": true, "final_justification": "The authors' response mostly addressed my concerns and I have updated my score to 5: accept.", "rebuttal": "We sincerely appreciate your careful reading of our manuscript and your invaluable feedback. In response, we have organized and summarized the main points for improvement, as you suggested.\n\nWe ask for your understanding that, due to space limitations, we have re-categorized similar comments under broader themes rather than addressing each item individually. If we have misrepresented any aspect, please let us know and we will gladly revise it.\n\n---\n\n***1. Discussion of the Relationship Between Our Proposed Approach and RL-based Constrained Models***\n\nThank you for this important observation. Structure-based continuous generative models and RL-based discrete optimization methods (e.g., Reactor, Synthemol, Synflownet) each play unique roles in drug discovery, and it is essential to clarify their respective strengths and application perspectives. The essential distinctions can be summarized as follows:\n\n* *Explicit vs. Implicit Utilization of Target Protein Structure.*\nStructure-based continuous generative models leverage explicit 3D structural information to directly design ligands optimized for the target binding pocket. By contrast, RL-based discrete optimization models generally do **not** incorporate **target structures** as direct model inputs, instead using learned predictors (e.g., binding affinity) or property scores as rewards. This leads to limitations in optimizing for structural constraints or binding affinity, and inference-time reliance on repetitive docking is computationally expensive and suboptimal.\n\n* *Generation Process.*\nRL-based approaches generate molecules **autoregressively**, typically by sequentially attaching fragments or atoms. This introduces **cumulative errors** and makes it difficult to control **global molecular structure**. In contrast, structure-based continuous models can update all parameters simultaneously (non-autoregressively), enabling more effective global constraint enforcement.\n\n* *Generalization and Retraining Requirements.*\nRL-based methods generally require **retraining for each new target**, necessitating redefinition of reward/state/action spaces. In contrast, structure-based generative models (such as our proposed model) can be directly applied to arbitrary target structures after initial training, offering superior transferability and generalization.\n\nWe also recognize that structure-based models rely on protein–ligand complex data and may **struggle with out-of-distribution generalization**, while **RL approaches can more flexibly** encode custom constraints or reward schemes. Recently, hybrid models that combine the advantages of both are being developed (e.g., TacoGFN, included in our experimental comparisons), and we acknowledge that integrating and benchmarking such approaches is an important future direction.\n\nWe will ensure that these points (comparative strengths, limitations, and the need for integration) are explicitly added to the “Related Work” sections of our revised manuscript.\n\n---\n\n***2. Placement of Related Work, Abstract Overstatement, Line 306 Interpretation, Hyperlink Omissions, Labeling, and Figure References***\n\nWe thank the reviewer for these detailed technical comments.\n\nWe **fully agree** that the related work section should be incorporated into the main text, not only in the appendix. Due to initial page limitations, we placed it in the appendix, but we will move and expand the related work discussion in the main paper.\n\nRegarding the use of \"overcomed\" in the abstract and other overstatements (e.g., in line 306), we concur that such expressions are too strong and will replace them with more measured terms such as **“mitigated”** or **“addressed”** to better reflect the inherent limitations that remain.\n\nWe will also cite the relevant prior work on the limitations of the SA score at the appropriate points, e.g., \\[1,2]. We will also replace the phrase “interestingly” with a more precise statement referencing prior studies, such as “in line with previous findings,” and cite relevant literature instead.\n\n\nConcerning internal hyperlinks and references: our initial submission was prepared as a single LaTeX file where all links worked. Upon splitting the main text and appendix for submission, some hyperlinks (especially to the appendix) may have broken. If resubmission as a single file becomes possible, we will ensure all links are active. If the reviewer identifies any specific missing or erroneous links, please notify us and we will address them promptly.\n\nRegarding Figure 3, our manuscript explicitly refers to it at line 222 (on page 6). If there are additional referencing or labeling issues we have missed, we would greatly appreciate further clarification so we can correct them.\n#### [1]: Coley, C. et al,(2018). SCScore: synthetic complexity ~ . Journal of chemical information and modeling, 58(2), 252-261.\n#### [2]: Liu, C. H. et al, (2022). RetroGNN ~. Journal of Chemical Information and Modeling, 62(10), 2293-2300.\n---\n\n***3. Clarification of Figure 1***\n\nWe apologize for any confusion caused by the insufficient explanation of Figure 1.\n\nThis figure presents t-SNE projections of the **model-computed gradients** during sampling, grouped by model type. The axes represent t-SNE dimensions; color encodes timestep progression, with darker colors indicating later steps. The gradients depicted for the Bayesian Flow Network correspond to those described in Equations 14 and 15 of the main text: $\\[\\nabla\\_{\\mathbf{x}}\\mathrm{log}p(\\mathtt{l}\\mid \\mathbf{x}), \\nabla\\_{e\\_\\mathbf{x}}\\mathrm{log}p(\\mathtt{l}\\mid e\\_\\mathbf{x})]\\$ (with $\\[\\cdot, \\cdot]\\$ denoting concatenation).\n\n---\n\n***4. Models Used for Guidance***\n\nWe apologize for omitting this from the main text (it was moved to the appendix due to length limits). The property predictors are based on the TargetDiff model architecture, trained to predict both binding affinity and synthetic accessibility [SA score] (see Appendix D.2). To account for predictive uncertainty, we adopted a Bayesian Neural Network (see Appendix D.1). The final guidance signal is the product of the two predicted properties, incorporated as a condition for generation.\n\nSelectivity guidance is implemented, as described in Appendix H.4, by assigning positive weights to on-target binding affinity and negative weights to off-target affinity in the guidance loss. We will add this information to the main experimental methods section to prevent any further confusion.\n\n---\n\n***5. Deeper Discussion of the SA Score***\n\nWe appreciate the reviewer’s attention to this issue.\n\nWe acknowledge that our presentation may have incorrectly implied a novel contribution in questioning the SA score, and that our citations of earlier work on its limitations were insufficient. As the reviewer noted, the inadequacy of the SA score as an absolute metric of synthetic feasibility is well documented (\\[1,2], etc.), and we fully agree that it is now considered outdated in the SBDD field.\n\nOur intent, however, was to draw attention to the fact that **many recent studies on 3D molecule generation for SBDD continue to rely almost exclusively on the SA score** for practical synthesizability evaluation. Our motivation was to highlight this “lazy evaluation” trend and to advocate for broader adoption of more practical synthesis metrics (such as AiZynthFinder), especially since, to our knowledge, there has been little direct discussion of the reliability of the SA score in SBDD 3D generation.\n\nWe will revise our manuscript to avoid the implication of novelty in this discussion, provide relevant and up-to-date references, and clarify that our goal is to objectively **inform the community (3D Mol generation in SBDD)** about the limitations of current evaluation practices. We will also stress the importance of incorporating metrics like AiZynthFinder for more robust synthetic accessibility assessment in SBDD applications.\n\n---\n\n***6. Clarification of the Role of the Proposed Model***\n\nThank you for this insightful observation.\n\nOur adoption of SA score guidance is rooted in the clear trade-off between synthetic accessibility and binding affinity: molecules with higher binding affinity are often more complex and thus receive lower SA scores; simpler molecules tend to have less optimal binding. Our approach aims to balance these factors to generate candidates with both high affinity and practical (if not absolute) synthetic tractability.\n\nWe agree with the reviewer that the SA score cannot explicitly optimize for custom chemistry or specific retrosynthetic routes, and that fully automated end-to-end design of synthesizable (considering AiZynthFinder), high-affinity hits remains a grand challenge. Our practical aim is to **reduce the incidence of implausible or synthetically impossible structures** among candidates and to provide a more realistic starting point for hit selection.\n\nIn this response, the term “hit discovery” is interpreted not as exhaustive high-throughput screening, but as the generation of a set of optimized candidate molecules under multiple property constraints, conditional on the target protein. If this does not fully address your question, please feel free to let us know at any time. Furthermore, we fully recognize and will explicitly state that integrating more advanced retrosynthetic constraints and property optimization remains a crucial area for future work.\n\n---\n\nIf there are any misrepresentations or if you would like us to elaborate further on any topic, please let us know. We are committed to addressing all feedback thoroughly in our revised submission."}, {"review_id": "sN9uGwQvMa", "reviewer": "Reviewer_59Bc", "summary": "This paper presents CBYG, which extends Bayesian Flow Networks (BFNs) into a gradient-based conditional generative model that robustly integrates property-specific guidance. The authors also introduce new metrics to include synthetic feasibility and selectivity, and show clear performance gains.", "questions": "1. I am curious about the benefit of BFN over diffusion. The authors claim in Appendix B.3 that BFN involves deterministic parameter update that remains strictly deterministic given y (containing the implicit stochasticity), and that the update rule can be directly interpreted through the lens of Kalman filtering for scalar Gaussian models. Can the authors elaborate on this point? What advantages does this bring to the training and sampling of BFN over diffusion / SDE? I know this may be hard, but can the authors provide a more quantitative analysis based on their theoretical construction, or just shed some light on it?\n2. Given the Tweedie's formula, it seems that we need to perform guidance over y for discrete data, as done in MolJO. Can the authors correct their derivation based on that observation or in some other way? How would that compare with guidance over $\\theta$ in the continuous case?\n3. How is AiZynthFinder's score correlated with real-world synthetic feasibility?\n4. How is PB-Valid calculated? The authors mention that they adopt 17 metrics, but PoseBusters test suite has more than 17 checks. Can the authors provide detailed validity ratio for each metric? Otherwise I remain doubtful of the reproducibility of PB-Valid results, as the PB-Valid of CBYG largely exceeds its backbone MolCRAFT (94.9% vs. 74.0%) without explicitly incorporating structural validity into its optimization objective, which seems unrealistic to me. Moreover, is RGA or TacoGFN generating molecules in the 3D space? If not, what's the point of calculating their structural validity?", "limitations": "This paper misses at least the mentioning of the validation and credibility of AiZynthFinder benchmark, which will make it more self-contained, and better indicative of its contribution towards addressing the limitation of previous SA metrics. Besides, the authors mention that approximately half of the molecules generated by CBYG remain synthetically infeasible without further explanation.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n- This paper moves beyond mere binding affinity and incorporate synthetic feasibility (via AiZynthFinder) and selectivity into SBDD evaluation.\n- This paper reformulates Bayesian updates for both continuous and discrete variables into a gradient-based form via Tweedie's formula, which I think brings theoretical insight for Bayesian Flow Network (BFN), and the authors have empirically demonstrated its superiority over gradient guided diffusion frameworks.\n\nWeakness:\n- Clarity of derivations can be further improved. $\\mu_x$ and $x$ in Equation 6 is not defined, and it seems that the notation $x$ is abused for both the noised version and the ground truth. For Equation 7, I recommend the authors to split the first line and the second line because they are essentially coming from different derivations, and it's best not to confuse them. $\\delta$ in line 177 is not clearly presented, either. It's difficult to understand why it is a compact notation because it obviously introduced more notations than the right-hand-side. Overall I think section 4 should be explained more clearly to make this paper more accessible to broader audiences.\n- Lack of referencing a directly relevant work MolJO, which is also a gradient guided BFN method [1].\n- Derivation of Section 4.3 (BFN for Discrete Variables from Score Gradient) is confusing and seems incorrect. The score $\\nabla_{e^x}\\log p(e^x)$ in L196 and Eq. (10) is undefined, as L188 states that $e^x$ is a one-hot vector. Tweedie's formula does not hold for non-Gaussian variables.\n\nI will consider raising my score if my concerns are adequately addressed.\n\n[1] https://arxiv.org/abs/2411.13280", "was_revised": true, "final_justification": "I thank the authors for their thorough and constructive rebuttal, which has resolved most of my initial concerns regarding the paper's theoretical and methodological foundations. I hope the authors will follow through on the remaining points, which will strengthen the empirical claims of the paper and make it a solid contribution to the field.\n\nResolved issues:\n- The authors have successfully addressed the major theoretical questions, including the Tweedie's formula for Gaussian $e_x$ and Gumbel-Softmax one-hot $e_x$ for property predictor, which carried the most weight in my evaluation. \n- The authors provided a clear comparison to MolJO, highlighting the differences in guidance approaches.\n- Further explanations of BFN properties are referred to appropriate literature.\n\nRemaining points for revision:\n- The authors have committed to providing PB-Valid ablation study, as this is necessary for understanding the effectiveness of the dual-guidance approach.\n- The authors have provided additional reference to AiZynthFinder and will further explore and validate its use as a reliable indicator of synthesizability.", "rebuttal": "Thank you for your valuable comments. We have grouped similar points into key themes due to space constraints. If there are any misunderstandings or omissions, please let us know.\n\n---\n\n***1. Definition of Equation 6 and Notational Consistency for x***\n\nEquation 6 in our manuscript is a concise adaptation of Tweedie’s formula for posterior mean estimation, as originally described by Robbins (1956). Here is an excerpt from the original paper on Tweedie’s formula that I referenced.\n>Herbert Robbins (1956) credits personal correspondence with Maurice Kenneth Tweedie for an extraordinary Bayesian estimation formula. We suppose that $\\mu$ has been sampled from a prior “density” $g(\\mu)$ and then, $\\mathbf{x} \\sim \\mathcal{N}\\left(\\mu, \\sigma^2\\right)$ observed, $\\sigma^2 $ known, $\\mu \\sim g(\\cdot) \\textrm { and } \\mathbf{x} \\sim \\mathcal{N}\\left(\\mu, \\sigma^2\\right) .$\nLet’s $f(\\mathbf{x})$ denote the marginal distribution of $\\mathbf{x}$,\n\n>$f(\\mathbf{x})=\\int_{-\\infty}^{\\infty} \\varphi_\\sigma(\\mathbf{x}-\\mu) g(\\mu) d \\mu$\n,where $\\varphi_\\sigma(\\mu)= \\left( 2 \\pi \\sigma^2\\right)^{-1 / 2}  \\exp (-\\mathbf{x}^2 / \\sigma^2)$\n\n>Tweedie’s formula calculates the posterior expectation of $\\mu$ given $\\mathbf{x}$ as follow:\n$E[{\\mu} \\mid \\mathbf{x}]=\\mathbf{x}+\\sigma^2 l^{\\prime}(\\mathbf{x}) \\quad \\text { where } l^{\\prime}(\\mathbf{x})=\\frac{d}{d \\mathbf{x}} \\log f(\\mathbf{x})$\n\nWe recognize, however, that our notation for $z$ and $\\mathbf{x}$ was inconsistent, and will revise Definition 4.1 and the marginal distribution to consistently use $\\mathbf{x}$. This should resolve the notational ambiguity—thank you for highlighting this.\n\n---\n\n***2. Notation at Line 177***\n\nWe apologize for the confusion regarding the notation $\\zeta(\\theta_i^x \\mid \\theta_{i-1}^x, y_i, \\alpha_i)$. This was intended as a more explicit function notation for the Dirac delta distribution, $\\delta(\\theta_i^x - h(\\theta_{i-1}^x, y_i, \\alpha_i))$, to make the update rule accessible to readers unfamiliar with delta functions. Moving forward, we will consistently use the standard $\\delta(\\cdot)$ notation in both text and equations to avoid ambiguity.\n\n---\n\n***3. Score Definition for Categorical Variables at Line 196***\n\nWe thank the reviewer for raising this important theoretical point and agree that Tweedie’s formula cannot be directly applied to strictly discrete, non-Gaussian variables like one-hot vectors.\n\nTo clarify:\n\nAs stated in line 186, $e_x$ is initially a one-hot vector representing a categorical variable. From Proposition 4.2 (line 190) onward, we apply a reparameterization trick: $e_x$ is treated as a sample from a multivariate Gaussian distribution, making it continuous and differentiable.\n\n$e_x \\sim \\mathcal{N}\\left(\\frac{1}{\\alpha K} \\cdot y + \\frac{1}{K}, \\frac{1}{\\alpha K} \\mathbf{I}\\right)$\n\nWe further use the **Gumbel-softmax** to bridge the gap between discrete and continuous spaces, allowing $e_x$ to retain interpretability while enabling gradient-based optimization.\nThus, after initialization, $e_x$ is a continuous, differentiable latent variable, which justifies the application of Tweedie’s formula. The update is theoretically valid for the relaxed, continuous form of $e_x$, not for the original strictly discrete one-hot vector. We will clarify this distinction more explicitly in the revised manuscript.\n\n---\n\n***4. Advantages of Our Bayesian Update Process Over Diffusion***\n\nWe appreciate your question about the theoretical advantages of Bayesian Flow Networks (BFNs) compared to diffusion/SDE-based models:\nDiffusion models inject random noise at every step, resulting in highly stochastic and potentially unstable sampling paths with variance accumulation.\nBFNs, in contrast, introduce stochasticity only through $y_i$ (noisy observation), with all parameter updates being deterministic (closed-form Bayesian update). This leads to lower sampling variance, stable trajectories, and greater numerical robustness.\nThe BFN update for continuous variables mathematically matches the mean update in Kalman filtering, providing a solid foundation for recursive, optimal variance reduction.\nIn both sampling and training, BFNs are more stable and efficient, avoiding the high computational overhead and instability of repeated noise injection.\nWe will clarify these points and their practical implications more explicitly in the main text and appendix.\n\n---\n\n***5. Discussion of Related Work and Guidance Calculation***\n\nThank you for highlighting the need for a rigorous comparison with related work such as MolJO. We believe it is essential to clarify the core theoretical differences.\n\nMolJO introduces guidance by multiplying an energy-based term, $p_E(\\theta^x, y^v) \\propto e^{-E(\\theta^x, y^v, t)}$, with the BFN transition kernel in an unnormalized fashion. The $E$ function is defined arbitrarily, and multiplying it with the original kernel does not guarantee that the resulting kernel preserves the probabilistic properties required by BFN, such as normalization, convergence, or the maintenance of Markovian/Bayesian structure. As a result, there is no mathematical guarantee that the newly defined kernel is still a valid transition kernel or that the sampling trajectories remain consistent with the data-generating distribution.\nAdditionally, the Taylor expansion used to approximate the energy function is a heuristic, valid only for small steps, and may induce significant errors or artifacts when the energy landscape is highly nonlinear. There is no quantitative analysis provided for these approximations.\n\nIn contrast, our approach is based on Tweedie’s formula and the reparameterization trick, providing mathematically grounded, gradient-based guidance that strictly preserves the Bayesian probabilistic semantics. This ensures normalization, stability, and interpretability, and applies equally to both continuous and categorical variables. While MolJO may have empirical utility, it remains an ad hoc modification rather than a principled Bayesian update, whereas every step of our method is theoretically justified and robust.\n\nFurthermore, as discussed previously, our proposed approach is formulated in a differentiable form for both continuous and categorical variables, so applying guidance to either \\$x\\$ or \\$e\\_x\\$ poses no theoretical or practical issues.\n\n---\n\n***6. Relationship Between AiZynthFinder and Actual Synthetic Feasibility***\n\nIn our team’s experience, we previously relied on statistical metrics such as SA score to assess synthetic accessibility, but even molecules with high SA scores (>0.9) often proved impossible to synthesize when we commissioned CROs, mainly due to missing viable synthetic routes or specific chemical constraints. This limitation led us to adopt retrosynthesis-based tools like AiZynthFinder.\n\nWhen we synthesized molecules classified as “solved” by AiZynthFinder, a significant portion were indeed experimentally accessible, with a much higher success rate compared to selection by SA score alone—though not all were synthesizable. Thus, while AiZynthFinder “solved” predictions are a strong indicator of synthetic feasibility, they are not foolproof due to limitations in the starting material database, reaction diversity, and model imperfections. We believe that using both SA score and retrosynthesis-based metrics in combination is currently the most practical approach, which is reflected in our study’s evaluation strategy.\n\n---\n\n***7. PB-Valid Reliability and Reproducibility***\n\nThank you for your detailed inquiry regarding the calculation and reproducibility of PB-Valid results.\n\nThe official PoseBusters test suite includes 18 criteria, but we used 17 for PB-Valid, excluding “File loads.” Since our workflow operates directly on generated molecular structures in memory, “File loads” is always trivially satisfied and thus omitted for meaningful evaluation.\n\nAdditionally, we provide the average validity statistics(18) across all molecules instead of listing scores per pocket.\n\n|Metric|Valid Score|\n|-|-|\n|mol_pred_loaded\\|sanitization\\|aromatic_ring_flatness|1.000|\n|all_atoms_connected|0.985|\n|bond_lengths|0.995|\n|bond_angles|0.995|\n|internal_steric_clash|0.993|\n|double_bond_flatness|0.998|\n|internal_energy|0.970|\n|protein-ligand_maximum_distance\\|minimum_distance_to_organic_cofactors\\|minimum_distance_to_waters|1.000|\n|minimum_distance_to_protein|0.997|\n|minimum_distance_to_inorganic_cofactors|0.994|\n|volume_overlap_with_protein\\|volume_overlap_with_organic_cofactors\\|volume_overlap_with_inorganic_cofactors|1.000|\n|volume_overlap_with_waters|1.000|\n\nRegarding the reviewer’s concern that CByG achieves unusually high PB-Valid compared to MolCRAFT without explicit structural validity optimization: our core contribution is the simultaneous optimization of binding affinity (BA) and synthetic accessibility (SA) during generation. While PB-Valid integrates both intra- and intermolecular structural validity, our dual-guidance approach naturally favors the generation of chemically reasonable and synthetically accessible molecules, as well as more plausible binding poses. Although we do not claim a direct 1:1 mapping or strict causality between guidance and PB-Valid, this combined optimization process enhances both aspects.\n\nFinally, models like RGA and TacoGFN generate 2D molecular graphs autoregressively and then use external docking tools to obtain 3D structures. Thus, the PB-Valid score for these models reflects the validity of post-processed 3D structures, not the intrinsic 3D generation capability. For fairness and reproducibility, we followed standard benchmarks but acknowledge that this is a limitation in direct comparability.\n\nThank you again for your insightful review. We believe PB-Valid results are fully reproducible and have provided all details for transparency and fair comparison. We are committed to revising the manuscript to address all your points and improve the clarity, rigor, and reproducibility of our work."}, {"review_id": "uJ4UizVDAA", "reviewer": "Reviewer_wjHx", "summary": "This paper presents a generative framework for structure-based drug design that combines Bayesian Flow Networks with gradient guidance. It handles both the prediction of atomic types and 3D coordinates and enables property-driven molecule generation. It outperforms existing methods in binding affinity, synthetic feasibility, and selectivity, validated through diverse docking tools.", "questions": "1)The theoretical derivations in Section 4, especially the use of Tweedie’s formula and the transition from Bayesian updates to gradient-based forms, lack formal justification. Key assumptions—such as distributional properties and whether gradients are taken in data or parameter space—are not clearly stated. A more rigorous derivation with explicit assumptions would greatly strengthen the theoretical foundation and improve the credibility of the method.\n\n2)Could you provide a detailed ablation study focusing specifically on the key hyperparameters (e.g., gradient guidance strength, Bayesian neural network design choices)? Clarifying how sensitive your results are to these parameters would significantly enhance the robustness and credibility of your findings. Providing clear criteria or thresholds demonstrating stability or sensitivity would positively influence my evaluation.\n\n3)Could you provide an analysis of computational cost in different experimental settings to elaborate the efficiency of the method?\n\n4)Could you elaborate the treatment of categorical variables in Section 4.3 as discrete variables can be handled through gradient-based updates contradicts the fundamental nature of categorical distributions. The softmax operation in Equation (10) does not resolve the non-differentiability issue that the authors initially identified as a problem with diffusion models.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1)The integration of gradient-based guidance into Bayesian Flow Networks for 3D molecule generation is novel and conceptually sound as well as addresses fundamental limitations in classical diffusion-based models. \n\n2)The work includes multiple baselines and evaluation metrics (e.g., GNINA, SMINA, Vina, PoseBusters, AiZynthFinder), offering a comprehensive evaluation of the generative capabilities.\n\nWeakness:\n1)Each individual component is known separately. The novelty primarily lies in their theoretical integration and application context rather than entirely new algorithm. Specifically, the novelty lies more in their theoretical unification and application context than in introducing fundamentally new mechanisms.\n\n2)The paper mentions at the end of Section 5.2 and in Appendix D the use of a Bayesian Neural Network to quantify predictive uncertainty and incorporate it into the sampling process. This is a sophisticated design that potentially makes the guidance more robust. However, the description in the main text is too brief for readers to understand. I believe that the specifics of this mechanism's implementation (e.g., how does uncertainty modulate the guidance scale?) and its contribution to the final performance (e.g., an ablation study comparing it to deterministic guidance without a BNN) are critical details of the CByG framework. These deserve more space in the main text rather than being relegated entirely to the appendix.\n\n3)Theoretical derivations in Section 4, especially 4.2 and 4.3, lack rigor due to missing key assumptions. For instance, the use of Tweedie’s formula (Eq. 6) to reinterpret Bayesian updates as gradient-based guidance lacks justification—there’s no discussion of required conditions like distributional assumptions or continuity. In Eq. 5–8, the transition from Bayesian update to score-based form is asserted without formal derivation. Additionally, it’s unclear whether gradients are taken in data or parameter space, which weakens the theoretical clarity of the proposed method.", "was_revised": true, "final_justification": "My concern has been addressed, and I raised the score.", "rebuttal": "We have carefully reviewed your valuable comments and have summarized the key areas for improvement you highlighted below.\nIn fact, while we would have liked to address each of your points in detail, we hope you understand that due to space constraints we have reorganized similar questions into broader categories.\n\nIf you find that we have misunderstood or omitted any of your concerns, please let us know.\n\n---\n\n***1. Placement of the detailed implementation and contribution of BNN in the design of sophisticated guidance***\n\nWe greatly appreciate your careful feedback. We agree that the quantification of predictive uncertainty, the mechanism for guidance scaling via BNNs, and the results of ablation studies are core contributions of the CByG framework. Unfortunately, these points were not sufficiently covered in the main text due to length limitations, so we included the relevant details in the appendix.\n\nTo address this, we will restructure the main text to include:\n(i) a concrete description of how predictive uncertainty is estimated using BNNs and how this adjusts the guidance scale, and\n(ii) ablation results comparing BNN-based guidance to a non-uncertainty approach.\nThese results are shared below for your reference.\n|Methods|SMINA↓ Score.|SMINA↓ Dock.|GNINA↓ Score.|GNINA↓ Dock.|Vina↓ Score.|Vina↓ Dock.|SA↑ Avg.|SA↑ Med.|PB-Valid↑ Avg.|PB-Valid↑ Med.|\n|-|-|-|-|-|-|-|-|-|-|-|\n|Reference| -6.37| -7.92| -7.06| -7.61| -6.36| -7.45|0.73| 0.74| 95.0%|95.0%|\n|**CByG**|**-7.74**|**-9.61**|**-7.63**|**-8.33**|**-8.60**|**-9.16**|**0.84**|**0.87**|**94.9%**|**96.0%**|\n|CByG w/o pos guidance|-7.05|-8.60|-6.88|-7.42|-7.74|-8.12|0.78|0.79|90.4%|91.1%|\n|CByG w/o type guidance|-6.92|-8.45|-6.70|-7.20|-7.60|-7.95|0.76|0.77|88.3%|89.0%|\n|CByG w/o uncertainty|-7.25|-8.81|-7.10|-7.64|-7.90|-8.31|0.75|0.76|87.1%|87.7%|\n|**CByG (λx=40,λv=40)**|**-7.74**|**-9.61**|**-7.63**|**-8.33**|**-8.60**|**-9.16**|**0.84**|**0.87**|**94.9%**|**96.0%**|\n|CByG (λx=30,λv=30)|-7.13|-8.64|-6.95|-7.47|-7.79|-8.17|0.79|0.81|91.0%|91.4%|\n|CByG (λx=50,λv=50)|-7.33|-8.94|-7.22|-7.79|-8.07|-8.49|0.74|0.76|85.6%|86.2%|\n|CByG (λx=30,λv=40)|-7.28|-8.89|-7.15|-7.68|-7.98|-8.42|0.77|0.78|89.2%|89.7%|\n|CByG (λx=40,λv=30)|-7.10|-8.59|-6.92|-7.42|-7.76|-8.10|0.80|0.82|91.8%|92.3%|\n\n---\n\n***2. Theoretical Justification for Gradient-Based Interpretation of Bayesian Update***\n\nWe sincerely thank the reviewer for this important point.\n\nWe will expand the main text to clarify the distributional assumptions (e.g., continuity) underlying the gradient-based interpretation of Bayesian updates, as follows:\n\nFor continuous variables, The sender distribution for continuous variables is explicitly modeled as a Gaussian:\n$\\[p_{\\mathtt{S}}(y \\mid x; \\alpha) = \\mathcal{N}(y \\mid x, \\alpha^{-1} I)\\]$\nGiven a prior over $x$ (possibly Gaussian), the Bayesian update for the parameter $\\theta$ given a noisy observation $y$ from the sender is:\n\n$p(x \\mid y) \\propto p_{\\mathtt{S}}(y \\mid x; \\alpha) \\cdot p(x)$\n\nIf we assume the prior $p(x) = \\mathcal{N}(x \\mid \\mu_0, \\Sigma_0)$, then the posterior $p(x \\mid y)$ is also Gaussian, due to conjugacy: \n\n$p(x \\mid y) = \\mathcal{N}\\left(x \\mid \\mu_{\\text{post}}, \\Sigma_{\\text{post}}\\right)$\n\nwhere $\\Sigma_{\\text{post}}^{-1} = \\Sigma_0^{-1} + \\alpha I,\\quad\n\\mu_{\\text{post}} = \\Sigma_{\\text{post}}\\left(\\Sigma_0^{-1}\\mu_0 + \\alpha y\\right)$\n\nThus, the Bayesian update function for the mean parameter $\\theta$ becomes a linear function of the previous mean and the new observation $y$:\n\n$\\theta_{i+1} = \\frac{\\rho_{i-1}}{\\rho_i}\\theta_i + \\frac{\\alpha_i}{\\rho_i}y_i$\n\nwith $\\rho_i = \\rho_{i-1} + \\alpha_i$, which matches the classic Kalman filter update for Gaussian models.\n\nSince the posterior is exactly Gaussian, Tweedie’s formula\n\n$\\mathbb{E}[x \\mid y] = y + \\alpha^{-1}\\nabla_y \\log p(y)$\n\nis strictly valid, and our use of Tweedie’s formula to obtain a gradient-based update is mathematically justified.\n\nFor categorical variables, the latent class indicator $e_x$ is originally a one-hot vector, i.e., $e_x \\in \\{0,1\\}^K$ with $\\sum_{k=1}^K e_x^{(k)}=1$. However, in our framework, we reparameterize $e_x$ to a continuous relaxation (e.g., using the Gumbel-softmax trick), and the sender distribution is modeled as a multivariate Gaussian over this relaxed variable:\n\n$p_{\\mathtt{S}}(y \\mid e_x; \\alpha) = \\mathcal{N}\\left(y \\mid \\alpha (K e_x - \\mathbf{1}), \\alpha K I\\right)$\n\nwhere $y$ is a continuous vector and $e_x$ is now allowed to be a point in the probability simplex.\n\nGiven this sender structure, the \"pseudo-posterior\" $p(e_x \\mid y)$ (formally, the conditional distribution in the continuous relaxation) is also a multivariate Gaussian, due to the properties of conjugacy between Gaussians.\nThus, the Bayesian update for the categorical parameter $\\theta$ (in the reparameterized space) can be written as a function of $y$ and the previous parameter:\n\n$e_{x,i+1} = \\frac{\\rho_{i-1}}{\\rho_i} e_{x,i} + \\frac{\\alpha_i}{\\rho_i} y_i$\n\nwith $\\rho_i = \\rho_{i-1} + \\alpha_i$, which is the direct analogue of the update for the continuous case.\n\nSince both $y$ and $e_x$ are continuous and (conditionally) Gaussian under the sender, Tweedie’s formula is applicable:\n\n$\\mathbb{E}[e_x \\mid y] = y + (\\alpha K)^{-1} \\nabla_y \\log p(y)$\n\nand, by chain rule,\n$\\nabla_{e_x} \\log p(e_x) = J_{e_x \\to y}^\\top \\nabla_y \\log p(y)$, \nwhere $J_{e_x \\to y}$ is the Jacobian matrix of the sender’s mapping.\n\nThis demonstrates the mathematical **justification** of applying the Tweedie formula to the Bayesian update process.\n\n\nAlso, regarding whether the Bayesian update is performed in data space or parameter space, we would like to answer as follows.\n\nIn our framework, we combine the distributional properties of the sender distribution with the Tweedie formula to derive the Bayesian update rule in a gradient-based form. Using Bayes’ rule, we logically derive how the guidance term for arbitrary conditions can be obtained.\n\nFor continuous variables, differentiation is performed with respect to $\\mathbf{x}$.\n\nFor categorical variables, differentiation is performed with respect to $\\mathbf{e}_x$.\n\nThese derivations and distinctions are detailed in the main text (pp. 4–5) and in Appendix B. Notably, Section B.3 explicitly clarifies that the Bayesian update is performed in parameter space, while the computation of the guidance term (i.e., gradient calculation) is carried out in data space.\n\nWe hope these clarifications address your concerns, and we will revise the manuscript accordingly.\n\n---\n\n***3. Experimental comparison of computational runtime***\n\nWe have measured the training and inference times of TargetDiff [1], the most widely cited diffusion-based model.\nAdditionally, as described in Appendix E, TargetOpt is a guidance model based on TargetDiff that we developed in this study specifically for comparison with CBYG, which implements diffusion-based guidance methods.\nBelow, we provide a comparative table of computational speed between the diffusion-based models and our approach.\n||Training Time (h)|Inference Time (h)|\n|:-:|:-:|:-:|\n|TargetDiff|11.2|0.95|\n|TargetOpt|13.4|1.67|\n|CByG|8.7|1.01|\n\n#### [1] Guan, Jiaqi, et al. \"3D Equivariant Diffusion for Target-Aware Molecule Generation and Affinity Prediction.\" The Eleventh International Conference on Learning Representations.\n\n---\n\n***4. Regarding the treatment of discrete variables***\n\nWe appreciate the reviewer’s insightful comment and concur that Tweedie’s formula is not directly applicable to truly discrete, non-Gaussian variables such as one-hot encoded vectors.\nHowever, we do not perform differentiation on non-differentiable one-hot vectors.\n\nTo clarify:\n\nAs stated in line 186, $e_x$ is initially a one-hot vector representing a categorical variable.\nFrom Proposition 4.2 (line 190) onward, we apply a reparameterization trick: $e_x$ is treated as a sample from a multivariate Gaussian distribution, making it continuous and differentiable.\n\n$e_x \\sim \\mathcal{N}\\left(\\frac{1}{\\alpha K} \\cdot y + \\frac{1}{K}, \\frac{1}{\\alpha K} \\mathbf{I}\\right)$\n\nWe further use the **Gumbel-softmax** operation to bridge the gap between discrete and continuous spaces, allowing $e_x$ to retain interpretability while enabling gradient-based optimization.\nThus, after initialization, $e_x$ is a continuous, differentiable latent variable, which justifies the application of Tweedie’s formula. The update is theoretically valid for the relaxed, continuous form of $e_x$, not for the original strictly discrete one-hot vector. We will clarify this distinction more explicitly in the revised manuscript.\n\n---\n\nThank you sincerely for highlighting this essential aspect of the theoretical validity and logical consistency of our manuscript.\n\nWe are more than willing to provide any supplementary information or engage in a more detailed discussion whenever needed."}]

Author rebuttal:

Meta-review: This paper focuses on the structure-based drug design problem and has two main contributions: 
1. extend and advocate Bayesian flow networks for structure-based drug design
2. call for practical evaluation for structure-based drug design 

The rethinking section J is interesting, in particular J.1 and J.2. However, I could see it stronger as main motivation in the main paper (just a suggestion), this relates to the question reviewers asked about the advantage of BFN over diffusion models. It is argued in the paper J.1 can be solved with BFN, but it seems J.2 is not? J.3 missed citing many relevant efforts from the community, see below.

The reviewers had several questions about (1) claims and related work discussions, (2) additional evaluation on the proposed methods, and (3) theoretical justification of the proposed framework. After rebuttal, all the concerns were addressed and all reviewers vote for acceptance of this paper. 

In addition to what several reviewers have mentioned, I would like to point out the study to consider synthesizability [2], selectivity [3] and pharmacophoric constraint [1] in drug design have been extensively studied in the literature, just list a few examples. 

I encourage the authors to integrate the reviewers' comments into the revision, especially the discussion to related work and importance missing details, e.g. the experiment setup for selectivity problem.

[1] Adams, K., Abeywardane, K., Fromer, J.C. and Coley, C.W., 2025. ShEPhERD: Diffusing shape, electrostatics, and pharmacophores for bioisosteric drug design. In ICLR 2025.

[2] Liu, S., Zhang, D., Tu, Z., Dai, H. and Liu, P., 2024. Evaluating Molecule Synthesizability via Retrosynthetic Planning and Reaction Prediction. arXiv preprint arXiv:2411.08306.

[3] Schneuing, A., Harris, C., Du, Y., Didi, K., Jamasb, A., Igashov, I., Du, W., Gomes, C., Blundell, T.L., Lio, P. and Welling, M., 2024. Structure-based drug design with equivariant diffusion models. Nature Computational Science, 4(12), pp.899-909.
Decision: Accept (poster)