Prior-Guided Flow Matching for Target-Aware Molecule Design with Learnable Atom Number
Authors: Jingyuan Zhou
Hao Qian
Shikui Tu
Lei Xu
Venue: NeurIPS 2025 poster ()
Keywords: Structure-based drug design
Molecule Generation
Generative models
Flow Matching
Computational Biology
Primary area: machine_learning_for_sciences
TL;DR: We propose a 3D all-atom flow matching model with prior interaction guidance and a learnable atom number predictor  for target-aware molecule generation.

Abstract: Structure-based drug design (SBDD), aiming to generate 3D molecules with high binding affinity toward target proteins, is a vital approach in novel drug discovery. Although recent generative models have shown great potential, they suffer from unstable probability dynamics and mismatch between generated molecule size and the protein pockets geometry, resulting in inconsistent quality and off-target effects. We propose PAFlow, a novel target-aware molecular generation model featuring prior interaction guidance and a learnable atom number predictor. PAFlow adopts the efficient flow matching framework to model the generation process and constructs a new form of conditional flow matching for discrete atom types. A protein–ligand interaction predictor is incorporated to guide the vector field toward higher-affinity regions during generation, while an atom number predictor based on protein pocket information is designed to better align generated molecule size with target geometry. Extensive experiments on the CrossDocked2020 benchmark show that PAFlow achieves a new state-of-the-art in binding affinity (up to -8.31 Avg. Vina Score), simultaneously maintains favorable molecular properties.

Prior-Guided Flow Matching for Target-Aware Molecule Design with Learnable Atom Number
Jingyuan Zhou', Hao Qian', Shikui Tu 1 Lei Xu
School of Computer Science, Shanghai Jiao Tong University ?Gunggoogg Laboratory of Artificial Intelligence and Digital Economy (SZ) {zjoyuan0930, qhonearth, tushikui, leixu]@sjtu edu. cn
Abstract
Structure-based drug design (SBDD), aiming to generate 3D molecules with high binding affinity toward target proteins, is vital approach in novel drug discovery. Although recent generative models have shown great potential, they suffer from unstable probability dynamics and mismatch between generated molecule size and the protein pockets geometry, resulting in inconsistent quality and off-tt effects. We propose PAFlow, novel target-aware molecular generation model featuring prior interaction guidance and a learnable atom number predictor. PAFlow adopts the efficient flow matching framework to model the generation process and constructs a new form of conditional flow matching for discrete atom types. A protein-ligand interaction predictor incorporated to guide the vector field toward higher-affinity regions during generation, while an atom number predictor based on protein pocket information is designed to better align generated molecule size with target geometry. Extensive experiments the CrossDocked2020 benchmark show that PAFlow achieves a new state-of-the-art binding affinity (up to -8.31 Avg. Vina Score), simultaneously maintains favorable molecular properties. The code of the paper is provided at http://ggtuubcco/CMCCHHHOSO
Introduction
As subfield of deep learning in drug discovery, structure-based drug design (SBDD) is considered as rational and challenging approach for developing novel drugs [[2] aims to generate drug-like molecules with stable 3D structures and high binding affinities conditioned on the target proteins, which can be formulated as a conditional generation problem. In the past few years, an increasing number of generative models have been applied to SBDD task. One category involves autoregressive models that generate 3D molecules atom by atom 3 or fragment by fragment [6. Another category includes diffusion- based methods 11. which gradually denoise standard Gaussian noise to predict full-atom distributions by leveraging both local and global information. Other models, such Rectified Flow [22 and Bayesian Flow Networks (13,, have also been employed in SBDD [1] [5]. Although these methods have demonstrated the potential to generate targer-aware molecules, they still face several limitations: (1) autoregressive sampling suffers from unnatural generation orders, leading to unrealistic fragments and error accumulation; (2) the denoising trajectory in diffusion models highly stochastic, resulting in unstable molecular quality; (3) to the best of our knowledge. all current con-auvoreeressivv methods determine atom numbers in generated molecules by sampling from predefined distribution, which relies on prior knowledge from reference ligands and often causes mismatches between ligand size and binding site geometry.
*Correspondence authors Shikui Tu and Lei Xu.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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The recently proposed Flow Matching (FM) framework [16] has shown promising initial results and demonstrates potential to serve as an effective solution for SBDD problems. Specifically, FM is a simulation-frs free approach for training Continuous Normalizing Flows (CNFs) that exhibits notable generative capabilities. is compatible with the Gaussian probability paths employed in diffusion models for transitioning between noise and data samples, while also supporting non-Gaussian paths as conditional probability trajectories. During the sampling phase, the adoption of ordinary differential equation (ODE) solvers enables fast and stable generation.
Inspired by the recent advancement of FM, we propose PAFlow, a 3D all-atom Flow matching model with Prior interaction guidance and a learnable Atom number predictor. which is designed to address the above drawbacks of existing methods. To overcome limitations (1) and (2). PAFlow adopts the FM as the framework for molecule generation. Specifically, we employ the established Variance Preserving (VP) probability path [16] for generating continuous atomic coordinates while deriving new form of Conditional Flow Matching (CFM) for discrete atom type. To further enhance the binding affinity between molecules and target proteins, a protein-ligand interaction predictor is incorporated during generation, which guides the vector field using prior binding knowledge towards poses with tighter interactions. Regarding limitation (3), we develop an atom number predictor that only utilizes protein pocket information rather than reference ligand priors to estimate the appropriate number of atoms. Extensive empirical evaluation on the CrossDocked2020 dataset [17] demonstrates that PAFlow generates molecules with not only significantly improved binding affinity compared to all baseline methods but also maintaining desirable molecular properties.
Our main contributions can be summarized as follows:
We propose SBDD generative model based the FM framework, where atomic coordi- nates and atom types are modeled using the existing VP path and a newly developed CFM, respectively.
A protein-ligand interaction predictor is integrated into the generation process to introduce prior binding knowledge, guiding the vector field toward directions that correspond to higher binding affinity.
To address the mismatch between molecule size and binding site geometry when sampling from the predefined distributions, we introduce an atom number predictor that estimates the appropriate number of atoms only using the binding site information.
Extensive experiments the CrossDocked2020 benchmark demonstrate that PAFlow can generate molecules with -8.31 Avg. Vina Score, significantly outperforming other strong baselines by a large margin (-1.24) and establishing a new oftt--of-the-rts while maintaining favorable molecular properties.
2 Related Works
Structure-Based Drug Design Structure- drug design is fundamental task in drug discovery, which aims to generate molecules that specifically bind with high affinity to a given protein pocket [18). Early efforts such as [19 20 generate 1D SMILES strings based on protein context, while (2] fit the distribution of target sequence embeddings in latent space to enable target-aware molecular graph generation. Motivated by the advances in 3D and geometric modeling, numerous works have attempted to address the problem directly in the 3D space. 222 voxelizes molecules in atomic density grids to generate 3D molecules in conditional VAE framework.
3 propose iteratively generate
atoms and bonds through autoregressive sampling within target binding site. [6] 7 generate ligand molecules motif by motif utilizing chemical priors of molecular fragments. However, the unnatural generation order during autoregressive sampling leads to unrealistic fragments and severe error accumulation. In recent works, diffusion models have been extensively applied ligand molecule generation, which denoises atom types and coordinates sampled from prior distributions using SE(3)- equivariant networks II 23 24). [[5] also explore the application of Bayesian Flow Networks (BFN) for molecular generation. However, these noo-auutorerressvve methods typically acquire the number of atoms the generated molecules by sampling from a predefined distribution, which depends on reference ligand information and often results in mismatches between the molecular size and the protein pocket. In this work, we aim to address this issue by introducing learnable atom number predictor.

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Flow Matching Flow Matching has recently attracted considerable attention and has demonstrated its potential in various domains such as image generation 25 26 and biomolecule design [27 28 As simulation-free approach for training continuous normalizing flows (CNFs), it is compatible with diverse probability paths and achieves better sampling efficiency through ODE solvers. For instance, [16] apply Flow Matching to the Variance Preserving diffusion path and formulate the VP Conditionnl Flow Matching (CFM), which addresses the unstable probability dynamics in conventional diffusion models. Other works [12, 26 [29] have explored transporting the prior distribution to the data distribution along straight line paths as much as possible. Although such paths offer shorter distances and lower computational cost, they lack the capacity to effectively model complex tasks like SBDD. For instance, [14] adopt this strategy for SBDD but achieve suboptimal performance. Therefore, we adopt the more robust VP path to model continuous atomic coordinates and construct novel CFM tailored for discrete atomic types.
3 Preliminaries
Problem Definition The SBDD task can be defined as conditional generative problem, where the goal is to generate ligand molecules that specifically bind to given protein target. A target protein P is represented as set of atoms P {(xp) ap of the i-th protein atom, ap N is a one-hot vector representing its features (e.g.. element type, amino acid type), Np is the number of atoms contained in the protein and Ny is the feature
E R33 is the 3D coordinates (1)) No where Xp (t)
dimension of protein atom. Correspondingly, a binding molecule M = {(x()) (1) ) comprises
(5) Nas atoms with 3D coordinates x E R and atom types aM RRK where K is the molecule atom type dimension. Then molecular representation can be simplified as m = [x ay]. where
XM E Na aM N and [ -] is the concatenation operator. Similarly, the binding pocket is denoted as p xp, ap]. where Xp E IR Np33 and ap E R RN
Preliminaries on Flow Matching In this section, summarize how the general flow matching method is implemented based on [16). Let Id represent the data space with data points æ (x'......t) 2²)) Rd. 9 is the data distribution, 21 denotes data point from & and To represents sample from the prior distribution Pu- The dime-dependent probability density path is defined as
Prefo.1] Fd + R> and the corresponding vector field is defined as ure[0.1] Fd + Rd The vector
field can construct a unique time-dependent flow 4/rejo.1] Rd + Rd defined by the ODE:
d
d the (x) = u, (v) (z)),
1 (x) = æ.
FM seeks to regress the target vector field with a neural network uo t):
LFM (f) px(x)) |l (x 1) ue(xr) 11
However, the lack of knowledge about what an appropriate Pt and , makes Eq. 2 infeasible to apply in practice. [16] proposed an alternative that employs a definable conditional probability path Pe (alaa)))) ) with conditional vector field un (x((a). The Conditional Flow Matching (CFM) objective
formulated as follows:
LCPM Et (11)). (xla1) uo(x, t) u4 (x(22)) 11
The CFM objective is tractable and shares the same gradient LFM 16 30) thus optimizing the CFM objective is equivalent to optimizing the FM objective. As for inference phase, Eq. I can be solved using ODE solvers: 21 ODESolve(2u. Vp, 1).
4
Method
This section elaborates on the implementation details of PAFlow. First, different probability paths are used to model continuous atomic coordinates and discrete atom types within the FM framework while maintaining SE(()-equivariance during ODE-based sampling. Based on this backbone, a protein-ligand interaction predictor is employed guide the vector field during generation to further enhance molecular quality. Additionally, to address the geometric incompatibility between generated molecules and target proteins, atom number predictor that relies solely target protein infor-
3

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Generation Process
T =
T
T At
Single Sampling Step
Me-de m, vgdr
d' Rq)
6
vv
Update
T=1
(Vōx
Qc
noise
Interaction Predictor Po = 1/m,)
Vlog 2
Protein Features
unguided path
predictor
guided path
SE(3)-Equivarie Equivarint GNN
Summatior over inpurs
volume
surface
Figure I: Overview of the PAFlow generation process. The atom predictor first estimates the number of atoms in the generated molecule based on the target protein information, which then used to initialize the molecule. SE(3)-EGNY, is applied to predict vector fields for atomic coordinates and atom types, while a protein-ligante interaction predictor provides prior binding guidance for the coordinate vector field. The final ligand molecule with high binding affinity is obtained through iterative updates. For simplicity, d1 is denoted as a.
mation-ratho than prior knowledge from reference ligands-is proposed. The whole training and sampling procedures are summarized in the Appendix
4.1 Flow Matching in Molecule Generation
For PAFlow, a data point from the prior distribution po is m. and the target data distribution p1 is the ligand molecule mı that can specifically binds the target protein. The molecular probability path can be formulated as a product of atom coordinate distribution and atom type distribution following
g
(c m1 p) Pe (x(xx11,) p) Pr(a (1, p).
For brevity we use x in place of X.M [31] proposes that in the multivariable case the probability path for each variable can constructed separately, which allows us model the two variables and a belonging to different data types-ussee separate probability paths. The continuous atomic coordinates x are modeled using the Variance Preserving path, with the probability path and target conditional vector field defined as follows [16]:
(x(xxx,, p) (Vai-1) a1 1-
(x X1, p) = (X (1- (I)
(Val-ix x1), (5)
1-
where NV is Gaussian distribution and the fixed variance schedules are defined as B, (t 0, At,... 1). with a, the derivative with respect to time t. For the discrete atomic types a, the probability density path is defined according [8] categorical distribution C:
3e,
IT'= 3. 1 ā, following (8]. (Vā1-.)/ denotes
(a/a p) ((a,c(aaaa) a1),
where c(a, a1) = ā1-1a1 + (1-61-0//K
(6)
In practice and follow different schedules, but retain the same notation for conciseness. Since this probability path has not been constructed as CFM in previous work, we derive its conditional vector field as below:
2 (c(a, a1 a1, p) ai={(a1 - -a))
(7)
The complete derivation is given in Appendix A.2 In fact, the probability paths for atom coordinates and types are identical to those applied in { however, the generation strategies adopted by the two methods are fundamentally different. In 8 generation is performed by iteratively denoising

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the Gaussian noise. Instead, our work formulates the generation process as the integration of the ODE dm,
di = va(mi (m, t) from t = 0 to t = 1 using an Euler solver [32] starting from an initialized
ligand molecule mo, where the vector field approximated with a neural network parameterized by 0. Specifically, for x and a we have:
(8) where Xy is initialized with a standard Gaussian distribution inside the protein pocket and a are initialized with a uniform distribution. The generative process is expected to be invariant to translations and rotations of the -rogind complex, an essential inductive bias when generating 3D molecules (3) Given the evidence that an invariant distribution composed with an equivariant invertible function will result in an invariant distribution 33 we have the following proposition (proof Appendix ..
X++AI = X + vg (m, p. t),
c(ar+Ar, a1) = c(a,, a1) + (m, p. t),,,,
Proposition 1.
Denoting the ttunffrronooooaaation as Tg and (wg 1, ug ( p. ()
vo (m,e; ; p. [), if we shift the Center of Mass (CoM) of protein aloms zero and parameterize no (mee P. 1) with an -qqiivrana network, then the generation process is invariant w.r. Tp on the protein-ligand complex.
There are different ways to parameterize vp. In this work the neural network is designed to predict [x1, ai], which are then processed through Eq. 5 and Eq. 7 to obtain vz and og respectively. To guarantee the invariance constraint, the generation process is parameterized using an SE(3)- Equivariant GNN Φo:
[$1 à. = 20 t.),
(9)
where the 1-th layer works as:
h171 = h
Σ
In [n(aity h/: hýj,
h). ezy;
0n)
(10)
JEV
++ x
Σ
(x (dij. h1 .n,t7 eiy; 0n) lmask
(11)
jEViij
I denotes the layer index and V the set atoms from both the protein and the ligand molecule.
d, ||x; xyyl and e;y represents the relative distance and option edge features between atom i
and atom j. Imask is a mask applied to the ligand atoms to keep the protein atom coordinates fixed.
fa and fz are graph attention networks. å can be obtained by input h [hf,... .... - into multi-layer perceptron and a softmax function.
4.2 Prior-Guided Generation
10 24 that utilize prior knowledge to guide the denoising process of diffusion models, we leverage predicted protein-ligand interactions, i.c. binding affinity, as guidance for the vector field, thereby further improving the quality of generated molecules. The ground truth of binding affinity is denoted y. The binding affinity of ligands in the training set normalized to [0, 1], thus y = 1 indicates a strong interaction between the molecule and the target protein. To predict the binding affinity ý between generated molecule and target protein, the final atom hidden embedding h containing useful global information is used following 8 and Eq. can be rewritten as:
Inspired by previous works 23
NM-1
X1 â1 Ø] = db (xx,, ay), t,p,,
9 =
Σ
(MLP(h')).
(12)
NM
==0
where a is a sigmoid function. According to Lemma / from generation process follows:
37
we derive the predictor guidance
& v log Amm
X++Ai + (ng (m;, p,t) 2ā1-1
(13)
c(a;-Ar, a1) c(a,, a1) + og (m, p, At.
(14)
Detailed derivation can be found in Appendix A..3 sealing factor y is added to control the gradient strength and the likelihood function can be defined as:
po(y 1m x exp(- 1)),
(15)
5

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where & R + is the mean square error (MSE) loss function to evaluate the deviation between the predicted score and y =
Explicitly guiding discrete atom types through gradients in molecular generation is challenging. However, in this work, sampling time step I, the sampled atom type at+ is obtained from X and via the SE(3)-Equivariant GNN. As a result, the optimized x, naturally influences &t thereby providing the implicit guidance for atom types. This guidance strategy consistent with the approach employed in 24
4.3 Learnable Atom Number Predictor
To generate molecules that better match the size of the protein pocket and avoid relying prior knowledge from reference ligands, an atom number predictor ip with neural network is trained using only target protein information. [38 suggests that there is certain correlation between the volume of the binding pocket and the ligand size. To provide the network with comprehensive information about the binding pocket and improve prediction accuracy, the surface area of the binding site A, the number of atoms within the pocket Np, and the space size 5Swihchh is used by existing models to also utilized inputs. Besides, dataset consisting of binding site
determine atom numbers- information--igantt atom number pairs is constructed for training. To enhance training stability and accelerate convergence, normalized ligand atom numbers used training labels, then the network outputs must be denormalized to obtain the final predicted number of atoms:
(16) 39 propose that introducing noise in generative models can enhance both the diversity and quality of discrete outputs. We observe the improved molecular generation performance when small Gaussian noise term r ~ N(0, 6²) is added to the predictor's 's output, which resembles the reparameterization trick used in variational autoencoders. Neural networks inherently involve uncertainty their predic- tions, and the injection of Gaussian noise may reflect this uncertainty. As form of regularization, this strategy helps mitigate overconfidence in potentially inaccurate point estimates and encourages exploration of broader solution space. We further prove the effectiveness of noise injection in mathematical form in Appendix AA.4
Nas vM (N
N
NM = ([NP, V, A, +T
Experiments
5.1 Experimental Setup
Dataset We train and evaluate PAFlow on the CrossDocked2020 dataset [17]. Following the same strategy as in R, binding poses with RMSD greater than Á and protein pairs with sequence identity over 30% are excluded. Similar to 23 binding affinities in the CrossDocked2020 dataset are normalized to the range [0, 1] for training the protein-ligand interaction predictor, where higher values indicate better binding affinity. We then randomly sample 100,000 complexes for training and select 100 complexes with distinet target proteins for testing. We further process the data and construct a dataset in the form of D {[NM, Np. V, A, S)}, where V and A are computed by the PyKVFinder [40] package. The dataset consists of 98k+ training samples and 100 testing samples for the atom number predictor.
Baselines PAFlow is compared with the following baseline methods: LiGAN 22 is CNN-Based conditional VAE generating voxelized atomic density. AR 3 and Pocket2Mol 4 are autoregressive generative models where atoms are sampled one by one. TargetDiff & DecompDiff (9) IPDiff 100. MoGMol
24 and ALiDiff [11 are diffusion-based models that generate 3D molecules in non-autoregressive manner. MolCRAFT [15] generates molecules in the continuous parameter space using Bayesian Flow Network (BFN) framework. FlowSBDD [14] is based the rectified flow model, another form FM that transports prior to data along linear paths, with a novel bond distance loss.
Evaluation metrics The generated molecules are evaluated from two aspects: binding affinity with the target protein and molecular properties. We employ the widely adopted AutoDock Vina 41 to estimate the mean and median values of affinity-based metrics (Vina Score, Vina Min, Vina Dock and High Affinity). Vina Score directly measures the binding affinity based on the generated poses;

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Vina Min optimizes the pose through local minimization before estimation; Vina Dock performs re-docking to reflect the optimal binding affinity; High Affinity computes the percentage of generated molecules that bind better than the reference ligands per test protein. Following 3 [2]. QED [42 (drug-likeness; SA 45 molecular properties.
(synthesize accessibility), and diversity are adopted to evaluate critical
Table I: Summary of binding affinity and molecular properties of reference molecules and molecules generated by PAFlow and other baselines. (t)/(L) denotes a larger smaller number is better. Top results are highlighted with bold text and underlined text, respectively.
Vina Score (1)
Viza Min (1)
Vina Dock (1)
High Affinity (*)
QED (f)
SA (1)
Diversity (f) Med.
Method
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Med.
-6.36
-6.71
-6.49
-7.45
-7.26
0.47
0.73
CVAB
LiGAN
-6.33
-6.20
21.1%
11.1%
0.39
0.39
0.59
0.57
0.66
0.67
Auto-
AR
-5.75
-5.64
-6.18
-5,88
-6.75
-6.62
37.9%
31.0%
0.51
0.50
0.63
0.63
0.70
0.70
regressive
Pocker2Mol
-5.14
-.700
-6.42
-5.82
-7.15
-6,79
48.4%
51.0%
0.56
0.57
0.74
0.75
0.69
0.71
TargeiDiff
-5.47
-6.30
-6.64
-6.83
-7.80
58.1%
59.1%
0.48
0.48
0.58
0.58
DecompDiff
-5.77
-7.04
-7.09
-8.43
71.0%
0.43
Diffusion
IPDif
-6.42
-7.01
-7.45
-7.48
-8.57
-8.51
69.5%
75.5%
0.52
0.53
0.61
0.59
0.74
0.73
TAGMal
-7.02
-7.77
-7.95
-8.07
-8.59
-8.09
69.8%
76.4%
0:55
0.56
0.56
0.56
0.69
0.70
ALiDEE
-7.07
-7.95
-8.09
-8.17
-8.90
-8.81
73.4%
81.4%
0.50
0.50
0.57
0.56
0.73
0.71
BFN
MoCRAFF
-6.59
-7.04
-7.27
-7.26
-7.92
-8.01
59.1%
62.6%
0.50
0.51
0.69
0.68
0.73
0.73
FlowSBDD
-3.62
-6.72
+6.60
-8.50
63.4%
70.9%
FM
Ours
-8.31
-8.92
-8.79
-8.96
-9.46
-9.49
80.8%
93.7%
0.49
0.50
0.57
0.57
0.71
Median Vina Score
-10
14:
proteinO
protein10
protein20
protein30
protein00
proteinb0
protein60
protein70
protein80
protein90
(lowest in
MoICRAFT (lowest 9%)
(lowest 77%)
Figure 2: Median Vina energy for different generated molecules (ALiDiff, MoICRAFT, PAFlow) across 100 testing binding targets. The proteins are sorted by the median Vina energy of molecules generated from PAFlow.
5.2 Main Results
Target Binding Affinity and Molecular Properties We comprehensively evaluate the performance of PAFlow by comparing against four types of SBDD methods: autoregressive, diffusion, BFN, and FM methods. 100 molecules are sampled for each test protein. As shown in Tab, I PAFlow significantly outperforms all baseline methods on all binding-related metrics, where the Vina score is computed without conformational optimization and serves as the most critical metric. Specifically, PAFlow surpasses the strong diffusion method ALiDiff by large margin of 17.5%, 8.7%, and 6.3% on Avg. Vina Score, Vina Min, and Vina Dock, respectively. Furthermore, it also achieves superior performance over the novel BFN method MoICRAFT by 26.1 %, 20.9%, and 19.4% on the same metrics. In terms of high-affinity binders, an average of 80.8% of the PAFlow molecules exhibit higher binding affimity than the reference ligands, significantly exceeding all other basclines. Notably, the performance of wwwSSBDDDanthhe FM-based model-is unremarkable in binding affinity due to the use linear probability paths, inadequately capturing the complexity inherent to SBDD. In contrast, PAFlow designs appropriate probability paths for atomic coordinates and atom types, effectively generating molecules with high binding affinity. Fig. presents the median Vina energy of all generated molecules for each binding pocket, comparing PAFlow with two SOTA methods ALiDiff and MolCRAFT, PAFlow achieves the highest binding affinity on 77% of the targets, significantly exceeding the other two methods.
Flowow generates molecules with higher binding affinity in Tab. D while maintaining comparable QED, SA, and diversity to ALiDiff. These properties are not explicitly emphasized during generation,
7

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as in real-world drug discovery they are typically used for rough filtering and are considered acceptable as long as they fall within reasonable range. Based on the PAFlow framework, extending the molecule-protein interaction predictor to molecular properties holds promise for further improving these metrics. Fig. presents examples of generated molecules along with their properties. The results show that the molecules generated by PAFlow exhibit favorable properties while maintain reasonable structures, suggesting strong potential as candidate ligands. We further conduct structural analysis experiments, with the results presented in Appendix E.7
4YH]
302H
Reference
0.7
10064
Figure 3: Visualizations of reference molecules and molecules generated by ALiDiff and PAFlow for protein pockets (4YHJ, 3DZH, 273H and 2JJG). Vina Score, QED and SA reported below.
Sampling Efficiency We select the most efficient model from each category of baselines compare sampling speed with PAFlow. Additionally, we report the sampling efficiency of PAFlow using reduced number of steps T 20. Fig. shows the inference time for generating 100 molecules on average. While TargetDiff and Pocket2Mol require 3968s and 4009s respectively, PAFlow only takes 717s, achieving 5.5x speedup. Although PAFlow is slightly slower than MolCRAFT, it produces molecules with significantly higher binding affinity, which compensates for the additional time cost. Moreover, when using fewer steps T 20, PAFlow becomes even faster than MoICRAFT (402s vs 287s) while still generating molecules with better binding performance (see Appendix E..). This allows the sampling step size to be adjusted according to the trade-off between generation speed and molecule quality in different scenarios.
4008
Podee
Brewthir
200
1340
560
Figure 4: Average time required by different methods to generate 100 molecules for a target protein, with shorter times indicating higher pling efficiency.
5.3
Ablation Studies
Effectiveness of FM Framework As described in Sec. 4.1 PAFlow and TargetDiff share the same probability paths, but differ in their sampling strategies. To highlight the advantages of FM- based generation, we construct a variant denoted as PAFlow w/o PA, by removing prior guidance and replacing the atom count predictor with predefined sampling. This essentially corresponds to replacing the denoising strategy in TargetDiff with ODE-based sampling while keeping all other components unchanged. Evaluation results for the generated molecules are reported Tab, 2 PAFlow w/o PA generally outperforms TargetDiff the affinity-related metrics. Moreover, while maintaining the same SA score, it achieves even better QED and diversity than TafgetDiff. These results demonstrate the superior generative performance of FM compared to diffusion-
denoising. Fig. 5 further visualizes the trajectories of atomic coordinates during generation, clearly showing that compared to diffusion the smoother probability path of FM could result in the better molecular quality.
Impact of Prior Guidance To validate the effectiveness of guiding the vector field with the protein- ligand interaction predictor, we generate 1,000 molecules in all test proteins with PAFlow w/o P (which excludes prior guidance during sampling) and PAFlow. As demonstrated in Tab. 3 when guidance is applied, the generated molecules exhibit substantial improvements across all affinity- related metrics, with increases of 60.4%, 22.5%, and 13.0% Avg. Vina Score, Vina Min, and Vina Dock, respectively. Improvements are also observed in SA and Diversity. Although QED shows a slight decrease, remains within an acceptable range. Given the superior performance of guidance strategy, extending the predictor to molecular properties holds great promise for further

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Table 2: Evaluation of generation results on TargetDiff and PAFlow w/o PA. PAFlow w/o PA is derived from TargetDiff by retaining all components unchanged except for replacing the sampling strategy with ODE-based sam- pling.
25
Metries
Tirfe
PAFlow w/o PA
Avg.
Med.
Avg.
Med.
(a)
(b)
Vina Score
-5.47
-6.30
-5.13
-6.35
Vina Min
-6.64
-6.83
-6.76
-7.12
Figure 5: Atomic coordinate trajectories from
Vina Dock
-7.80
-7.91
-8.08
-8.18
I == 0 to / 1 under different sampling strate- gies with the same probability paths. (a) shows the generation trajectory of TafgetDiff, while (b) presents that of PAFlow w/o PA.
QED
0.48
0.48
0.53
0.53
SA
0.58
0.58
0.58
0.58
Diversity
0.72
0.71
0.73
0.72
enhancing these metrics. Additionally, the extent of improvement is influenced by the guidance strength y, with related results presented in the Appendix E.3
Table 3: Effect of using the molecular- protein interaction predictor for guidance. PAFlow w/o refers to the variant that does not incorporate prior knowledge to guide the vector field during the generation process.
Score
Vina
Vina Dock (L)
High Affinity (1)
QED (1)
SA (1)
Diversity (f)
Methods
Avg.
Med.
Avg.
Mod.
Avg.
Med.
Avg.
Med.
Avg.
Mod.
Avg.
Med.
Avg.
Med.
PAFlow w/o
-5.05
.6.78
-8.3
70.3%
0.56
0.69
PAFlow
-8.10
-8.83
-8.50
-8.80
-9.62
-9.39
80.7%
100.0%
0.49
0.50
0.57
0.57
0.70
0.70
Effect of Atom Number Predictor To assess the impact of atom number predictor, we design three variant models for comparison. All modifications are based on PAFlow w/o P to eliminate the influence prior-guided sampling: (1) predefined: atom numbers are sampled from the predefined distributions: (2) 8 O: atom numbers are directly predicted by the predictor without added noise; and (3) 0.01: Gaussian noise (O, 0.013) is added to the predicted result. We generate 1,000 molecules for each setting on the test proteins, and the evaluation results are summarized in Tab. 7 Using the predictor yields substantial improvements in all affinity-related metrics compared to sampling from the predefined distributions. For molecular properties, QED and SAA are improved while maintaining diversity. This demonstrates that molecules generated with predicted atom numbers exhibit more appropriate molecular sizes and thus perform better overall. Furthermore, adding small Gaussian to the predicted result leads to additional gains in performance. This randomness offers robustness that deterministic outputs lack and helps compensate for minor prediction errors. Moreover, introducing noise allows model to explore a broader design space and potentially discover molecules with superior performance.
Table 4: Comparison of different strategies for determining the number of atoms in molecules generated by PAFlow w/o
Vina Score (4)
Vina Min (L)
Vina Dock (1) Avg.
High Affinity (t)
QED
SA (1)
Diversity (1)
Methods
Avg.
Med.
Avg.
Med.
Med.
Avg.
Med.
Avg.
Mod.
Avg.
Med.
Avg.
Med.
predefined
-4.99
-6.50
-6.33
-7.28
-7.62
61.2%
66.7%
0.52
0.57
0.57
0.73
0.71
8 6=0
-5.72
-6.86
-7.30
-7.52
-8.30
-8.48
70.4%
88.9%
0.55
0.56
0,58
0.58
0.72
0.71
0.01
-6.17
-7.03
-7.50
-7.74
-8.70
-8.61
72.8%
88.9%
0.55
0.56
0.57
0.57
0.71
0.71
6
Conclusion
In this work, we propose PAFlow, a flow matching- baased SBDD method that separately models continuous atomic coordinates and discrete atom types using the VP path and a newly constructed CFM. By incorporating protein-ligand interaction predictor, PAFlow effectively guides the vector field toward higher binding affinity. Additionally, an atom number predictor is introduced to determine the number of atoms in generated molecules, addressing the geometric incompatibility between generated molecules and target proteins and eliminating the dependence prior knowledge from

---

reference ligands. Extensive experiments on the CrossDocked2020 benchmark demonstrate that PAFlow achieves state-of-the-arg performance terms of binding affinity, with up -8.31 Avg. Vina score, while maintaining desirable molecular properties. In future work, we plan to extend the interaction predictor to molecular properties, enabling multi-objective optimization over both binding affinity and critical molecular properties for enhanced drug discovery potential.
7
Acknowledgements
This work was supported by the National Natural Science Foundation of China (grants No. 62172273), the Science and Technology Commission of Shanghai Municipality (grants No. 24510714300), and the Shanghai Municipal Science and Technology Major Project, China (Grant No. 2021SHZDZX0102).
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19

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16. Declaration of LLM usage
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20

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A Proofs
A.1 Proof of Equivariant Generation
An essential requirement for generated molecules translational and rotational equivariance to the protein complex. Since atom types always invariant to SE(3) transfrrtooo,
ligand-protein
only need to consider the transformation behavior of atomic coordinates. As the probability path used during training is identical that of TargetDiff (8) and the path of TargetDiff is invariant by design-we only have to focus on the generation process. Let T denote the group of SE-(3) transformation, e.g. Tg (x) Rx + b, where R E 3x3 is the rotation matrix and E IR is the translation vector.
First, we move the complex to achieve zero Center-of-Mass (CoM) on protein positions by the following linear transformation:
Q = I3 & [NNa
NF Nar 17 Np 1N Np
RM, xp] = (xMM,
where
(17)
INp
Then flow can be initialized from standard Gaussian, and during the generation process Xo can be sampled from a standard Gaussian distribution. Meanwhile, for any T, applied to the protein and molecule, have T, ( invariance by definition. For brevity, we denote XM, Xp by XM, Xp in the following.
Thus Xas and &p inherently satisfy translational
Given that do is an SEQ))--quivariant GNN, it follows from Eq. 9 that x1 is SE-(3) equivariant to and Xp. At t = 0, based on Eq. 5 and S we obtain
XAt = Xo + ut
(vān-t)) (V01-t) (Vai_ - &1).
Xo
(18)
Q1
We can prove that X is SE(3)-equivariant to Xo and x2 as follows
Ty (x(xxi(X0, xp)) -xy(xn) + (Vāi-t))
(V01- (x(()))
51
Rxo b (Vā1-1)
(Vā1- vāi_ Ro b)
(Vo1-t) R(vā1-ix, - xt
Rx 1 Ō1-1
XXX xp)) +b.
(19)
where 5 - (Vaa(vaa)))))) (a) b. Since translation invariance is achieved by moving
CoM of the protein atoms to zero, b can be omitted thereby Ty (x (Xp, xP)) = X xp)) = XAi (T, (x0, xp)). Following the same reasoning, it can be derived that X++ is SEuiieequiaaiian to X2 and Xp. By mathematical induction, we conclude that X₃ is SE(u)-equivariant to X and xp. Consequently, the entire generation process is -quiivaquvvariant with respect Xy and xy.
A.2 Vector Field of Type Flow
Theorem in [66] propose that for a flow the form (x())) = r((x1)x sx(21,, the corresponding vector field is given by:
r'(xx1) (x))
(z sx1)) (1()))
(20)
where re(a1) and sx(11) can be arbitrary differentiable function satisfying the desired boundary conditions. Let s, āj-(aj, Ti āi-c. then Eq. 20 is rewritten as
(x(())) = ai-t
(x an a1-(01) + 21-1011
(21)
ā1 1-
21

---

For atomic types we have c(a, a1) = â1-(a1 + +1 0 ā1-1)au, which can be substituted into Eq. 21 obtain:
01 (o1-ta1 aai + (1 -1--)a0- an-1a1) + á
(<(c(a, a1)la)) = 1 d1-1
01 (1 - 1-1111a + a
- Ø1-!
01-1(a1 an).
(22)
A.3 Prior Guidance of Vector Field
According to Lemma in [37. the guided vector field for a Gaussian Path pt(a(z1)
Ma I) can be expressed as:
(x(2l)) Uz + V. logp(rely),
(23)
b. = (140, - mx0,)) Tt Pl
where 0t
(24)
1t:
The probability path for atomic coordinates follows the Gaussian distribution with μ4, / and or = /1 ā1-t- Substituting into Eq. 24
yields:
(Va1-xX1) -1X1)
(25)
1X1
201
I 011
01 01
V-
2
2 a1
I a
a1-1
1 1-
---1 +1
2
di-t.
(26)
2ā
& 20.-.
Thus we have a; = b
Substituting this into Eq. 23 gives the guided vector field for atomic
coordinates as follows:
0i-c (xt + v lxxl(x))))).
m p.
(27)
According to the Bayes' rule, log-probability function can decomposed as:
V log y) V Vlogg((x,) + log (x)) logr()))
V log p(x;) v log (())
(28)
where V log p(y(x)) can be calculated using predictor with Eq. as:
15 Then we can reformulate Eq. 27
a'-t (xt + logp(xx)) +
0' logp(y|x:)
by
2
2
á'_ log p(y x:)-
= og (m;, 1, +
(29)
2
Moreover, scaling the predictor gradient by a constant factor > 1 is necessary to enhance the alignment of the generated samples with the desired condition. Similar to [44) we know that ? v log p(y)x,) = v log 1 >((//x)", where Z is an arbitrary constant. Consequently, the conditioning procedure remains theoretically grounded in a re-normalized predictor distribution proportional to p(y|x,)) Higher probabilities are emphasized by the exponent when 7 > 1, leading distribution more peaked than the original (x)). Thus increasing the gradient seale focuses more on the modes of the predictor, which promotes the generation of molecules with higher binding affinity.
22

---

A.4 Proof of Noise Injection Effectiveness
As introduced in Sec. 4.3 the predicted normalized number of atoms n. in the generated molecule obtained via the atom number predictor. For simplicity, we denote as n the following. Using n directly corresponds to deterministic sampling, i.c., nt n, whereas adding small Gaussian noise corresponds to stochastic sampling with ń (n,, 0²).. Let f(n) denote the binding affinity as function of the atom number n. The objective is to maximize the expected binding affinity, defined as R = E3 [r]]].
Under deterministic sampling, have
R= [n]]([(a)] f(n).
(30)
When Gaussian noise is injected,
R = (o,
(31)
The second-order Taylor expansion of f(ñ) ñ is given by:
f(n) = f(n) + (') - n) + f (m) -n)2.
(32)
Since ñ ~ (n, (2), the expectation can be rewritten as
Ex ~ f(n) (m))
(33)
As result, the expected affinity increases over the deterministic value f(n) if f"(n) > 0, i.e., if the function is locally convex around n. In practice, we observe that adding small amount of noise (e.g., 0.01) consistently improves the binding affinity of the generated molecules (see Tab. (7).
This implies that lies near local minimum of the affinity function, and noise injection effectively allows exploration nearby data points that lead to better binding performance.
B Training Objective
The training objectives for atomic coordinates x, atomic types a, protein-ligand interactions y, and atom numbers nM are defined follows:
c(a a1)x
C, x1
x1 1|
C. = Ed(as, a1)s log
(34)
c(ar, a1)x
C, || -
C, = 1| - n 1
(35)
where the loss for a is KL-divergence of categorical distributions, while others are optimized using Mean Squared Error. KGDiff demonstrates that jointly training x, and y is more effective than training separately. Following this strategy, we adopt a joint training scheme with the combined loss defined as:
L= L. XCa why:
(36)
where À and w are scaling factors. The atom number predictor is trained independently.
C Algorithm
The training and sampling procedure of PAFlow summarized below.
23

---

Algorithm Training Procedure of PAFlow
Input: Protein-ligand complex {p. m, y neural network do, atom type loss weight and
predictor loss weight
while Φg not converage do
2: Sample U( 1)
3: Shift the complex to make the CoM of protein atoms zero
4: Obtain X; and a,:
5:
X = vai-ix1 + v(1 -1-1)s where & I0,,
6:
log = 1og + (1 k
7:
a; = one har(_haaa,r,aa,,..
+ log cil), where ~ Gumbel(O, 1)
8:
Predict [x1, à u) with dha [81, â . = do an), 4)
9:
Compute the loss function using Eq. 36
10: L = X1 x1 11 AKL(c(a, a llcc)ll.a...... ān)) + w 11 0|
11: Update by minimizing C
12: end while
Algorithm 2 Sampling Procedure of PAFlow
Input: Protein pocket p. the learned model de, pretrained atom number predictor yee, scale factor
standard deviation of Gaussian ô, total number of sampling steps T
Output: Generated ligand molecule m
1: Obtain the number of atoms in using Eq. 16
2: Move the CoM of protein atoms
3: Initialize ligand atom coordinates Xo and atom types ag
4: steps < t + 0, At 1/T
5: while steps
do
6: Predict [*, å,, Ø] with Φq: [3,, .0 dol ( agl.
7: Compute vector fields:
8:
(vā--+)' i0 1-a
(V11-ix 81) + ai
log 1(m;)
26
9:
og αi- an)
10:
Update x,+ and at-
11:
Xt+A: X₁ + ug At
12:
c(at+ár, â = c(a, â1) + At
13:
Sample a, from C(a++A:) c(a
14:
end while
D Implementation Details
the protein atom features include a one-hot indicator of the element type (H, C, O, S, Se), one-hot vector indicating the amino acid type (20 dimension), and one-dim flag indicating whether the atom belongs to the protein backbone. Ligand atom types are also represented as one-hot vectors that encode the element type (C, N, O. F, P, S, CI) along with aromatic information. Two separate single-layer MLPs are utilized to embed the protein and ligand features into a 128-dimensional latent space.
Data Following [8)
The protein-ligand complex adaptively represented as a k-nearest neighbors (knn) graph at the [-th layer, constructed using the known protein atom coordinates and the current ligand atom coordinates generated by the 1-th layer. k is set experiments. The edge features are obtained as the outer product between distance embeddings and bond type, while distances are expanded using radial basis functions centered at 20 points distributed between 0 À and 10 Á. Bond types are encoded as 4-dimensional one-hot vector indicating whether the connection is protein-protein, ligand-ligand, motein-to-llignnd or ligand-to-prote

---

Model Information The SE(3) network Og consists of 9 equivariant layers, where each layer is implemented as transformer with hidden dim 128 and n head 16. The key/value embeddings and attention scores are generated through 2-layer MI.ss with ReI.U activation and Layer Normalization. The protein-ligand interaction predictor employs a 2-layer MLP with ShiftedSoftplus activation. For the atom number predictor (ee. we adopt a 4-layer MLP with hidden dimensions of 128. 256, and 128. Each hidden layer is followed by Batch Normalization and ShiftedSoftplus activation, along with Dropout for regularization. The final output is produced by a linear projection to a scalar. We apply a sigmoid schedule with 31 = 1e-7 and 30 = 2e-3 for atom coordinates, and cosine 3 schedule proposed by [45] with = 0.01 for atom types.
Training Details When training the SE(3)-Equivariant network, the Adam optimizer is employed speed up convergence with init_ caaf_eearningggar rate 5e 4, betas (0.95, 0.999), batch_size 4 and bii_graadent, gradient_norm _oom = 8. The learning rate is scheduled to decay exponentially with a decay factor of 0.95, and we the minimum learning rate be le 6. We decay the learning rate if the validation loss is not improved for 15 consecutive evaluations. Following [23], we set A, the weight corresponding to the atom feature term in loss function, 100, and w. the weight corresponding to the expert network, is 1. When generating, the scaling factor is 350 and the number of sampling steps is set to 50.
During the training of the atom number predictor, we use the Adam optimizer with init_ fiit_learning_rate
5e 4, batch_size 256, and betas (0.95, 0.999). The learning rate follows an exponential decay schedule with a decay factor of 0.8, and a minimum learning rate of le-5 enforced. The learning rate is decayed when the validation loss does not improve for consecutive evaluations. All models are trained on one NVIDIA A100 GPU (40 GB).
Table 5: Overview of the properties of the reference ligands and the molecules generated by different methods on the Binding MOAD dataset. (t)/(L) denotes larger / smaller number is better. Top results are highlighted with bold text and underlined text, respectively.
Vina Score (1)
Vina Min (L)
Vina Dock (L)
High Affinity (t)
QED (t)
SA (t)
Diversity (1) Med.
Method
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Med.
Avg.
Ref
-6.32
-5.82
-7.22
-6.65
-7.95
-7.67
0.65
0.63
0.35
0.34
Pocker2Mol
.4.85
4.55
-5.95
-5.54
-6.66
-6.31
46.2%
33.5%
0.62
0.62
0.83
0.84
0.83
0.83
TargetDiff
-5.60
-5.49
-6.48
-6.15
-7.47
-7.20
62.3%
63.0%
0.55
0.62
0.76
IPDiff
-6.64
-6.80
-7.50
-7.31
-8.57
-8.25
77.6%
86.4%
0.53
0,53
0.58
0.58
0.74
0.74
MolCRAFT
-7.33
-6.90
-7.54
-7.10
-7.90
-7.50
65.3%
77.4%
0.56
0.70
0.69
0.74
0.76
PAFlow
-9.12
-8.87
-9.18
-8.78
-9.69
-9.19
83.4%
96.6%
0.48
0.46
0.58
0.58
0.70
0.70
E
More Experimental Results
E.1 Results on Binding MOAD
To investigate the performance of PAFlow across different datasets, we evaluate it the Binding MOAD dataset [46) without additional retraining, which is a commonly used benchmark comprising experimentally determined protein-ligant complexes. We apply filtering based on the proteins' enzyme commission numbers [47] and exclude entries that could not be processed. resulting in 100 protein-ligand pairs for testing following previous work (48]. PAFlow and the baseline methods generate 100 molecules for each protein pocket, with the evaluation results summarized in Table 5 PAFlow achieves the best performance across all aanding-related metrics, outperforming the second- best method by 24.4%, 21.8%, and 13.1% 1% on Avg. Vina Score, Vina Min, and Vina Dock, respectively. Although there is a slight decline molecular properties, they remain within a reasonable range. It is worth noting that PAFlow is not trained on the Binding MOAD dataset, indicating its strong generalization capability. Furthermore, it is reasonable to expect that performance could be further improved by training on this dataset.
25

---

E.2 Effect of Sampling Steps
We conduct an ablation study on the sampling steps of PAFlow by generating 10 molecules each for 100 test proteins using sampling strategies with different step (20, 50, 80, 200). The resulting Vina Score, Vina Dock, QED, and SA curves are shown in Fig. 6 can be observed that PAFlow achieves very competitive results on affinity-related metrics even with only 20 steps, outerforming all baseline methods listed in Tab. D contrast, diffusion-barn indicating that PAFlow significantly improves sampling efficiency. Moreover, increasing the sampling step further enhances performance across all metrics, since more sampling steps allows model to more closely approximate the probability flow, leading to smoother and more accurate trajectory toward the target distribution. Considering the trade-off between sampling efficiency and overall molecular quality, the number of sampling steps is set to 50 for PAFlow.
models typically require 1000 sampling steps,
g Lik
6.27
6.10
Figure 6: Ablation study sampling steps. The performance of Vina Score, Vina Dock, QED, and SA under different numbers of sampling steps (20, 50, 80, and 200) is reported. Lower Vina Score and Vina Dock, well higher QED and SA, indicate better results.
E.3 Effect of Scaling Factor in Guidance
The scaling factor y, introduced Sec. 4.2 is utilized to control the strength of guidance applied to the vector field. We generate 10 molecules each for 100 test target proteins using PAFlow with different y values (0, 10, 50, 150, 250, 350, 400). The results are presented in Fig. 7 A larger ? corresponds to stronger guidance, which leads higher binding affinities for the generated molecules. However, clear trade-off between binding affinity and QED can be observed, where as binding affinity improves, QED tends to decrease. This is consistent with the discussion in Sec. A.3 that larger gradient scales concentrate more on the modes of the predictor, potentially at the expense of other molecular properties. Considering both binding affinity and molecular properties, = 350 for PAFlow. Additionally, y can be tuned during molecule generation to achieve different trade-offs between binding affinity and molecular properties, allowing adaptation to various drug design scenarios.
select
012
26
100
20
10
300
100
Figure 7: Ablation study of PAFlow under different guidance scaling factor 7. The performance on Vina Score, Vina Dock, QED, and SA is reported. In the SA plot, only one curve is shown because the average and median SA are identical.
E.4 Effect of the Gaussian standard deviation
To investigate the impact of different ó on molecular quality, we generated 10 molecules for each of 100 test proteins, where 5 is the standard deviation of small Gaussian noise added the output of the atom count predictor. Fig. g reports the averages of Vina Score, QED, SA, and the proportion of complete molecules under (= 0.02, 0.04, 0.06, 0.08, 0.1). When small, both binding affinity and the proportion of complete molecules improve while maintaining molecular properties. This
26

---

No 66
0.30
Figure 8: Ablation study using Gaussian noise with different standard deviations added to the predicted atom number. Vina Score, QED, SA, and the percentage of complete molecules are reported.
is because the added randomness introduces robustness that deterministic outputs lack, helping to compensate for minor prediction errors. However, when the standard deviation larger, the noise injected overly disturbs the predicted values, leading to decline in molecular quality, For PAFlow, we chose á 0.01.
E,5 Factors Inftuencing the Number Atoms Ligands
To validate the effectiveness of the protein features used for predicting the atom numbers generated molecules (including the number of pocket atoms Np, binding site volume V. binding site surface area A, and space size S), conduct study their relationships ing the training dataset. and are computed using PyKVFinder, while S is the median value of the top 10 largest pairwise distances among protein atoms. No- tably, existing oooeautoreeressvve methods commonly use S alone to determine the atom number. The rela- tionships between these features and Nar shown in Fig. 9 All features exhibit significant correlations (p < 0.05) with NA, supporting the reasonableness of using them for atom number prediction. Additionally, we compare the performance of using only V (denoted as Volume) versus using all four features (denoted All), as reported in Tab. 6 The test loss refers to the loss of the trained predictor evaluated on the test set. Results show that incorporating all features leads to a lower test loss and better molecule quality, indicating that providing richer protein information allows the predictor to more accurately infer the appropriate Nas
Table 6: Comparison of evaluation results using only binding site volume versus using all four features.
Metrics
Volume
All
Test Loss (e-3)
9.03
3.80
Vina Score
-4.88
-5.72
QED
0.59
0.55
SA
0.56
0.58
Diversity
0.72
Complete 94.0%
94.7% 4.7%
Figure 9: Relationships between ligand atom number and pocket atom number, binding site volume, surface area, and space size in the training set. Red lines show regression fits. Correlation coefficients and p-values (all < le 10) are reported.
E.6
Validation for Atom Number Predictor
established a strong positive correlation between ligand size (i.e., number of atoms) and binding affinity. However, the performance gain in our method does not stem from simply generating molecules with more atoms. Instead, results from generating molecules whose atom counts closer to the reference ligand.
Prior research
49
27

---

To more intuitively demonstrate the effectiveness of our predictor, we compare it with the predefined sampling by generating 1.000 atom numbers for each of three randomly selected test proteins. For each protein, we define 20% and 30% intervals around the reference atom number and compute the proportion of generated samples falling within these ranges. As shown in Table 7 most atom numbers predicted by our model fall within the 20% range of the reference ligand's atom count, without producing significantly larger molecules. In contrast, the traditional method-ww -hhlod samples atom numbers from predefined distributions--yille range, Such deviation increases the likelihood of generating molecules that are either too large or too small to fit well within the binding pocket, thereby negatively affecting the protein- -lgaadd interaction.
substantial portion of results outside the 30%
Table Comparison between predefined sampling and our predictor on atom number. For each of the three randomly selected test proteins, 1,000 atom numbers are generated.
PDB ID
<70%
170%, 80%)
180%, 100%)
[100%, 120%)
[ |120%, 130%)
> -300%
Predefined
4.7%
10.9%
26.2%
34.2%
12.5%
11.5%
4YHJ
Predict
64.0%
36.0%
Predefined
7.4%
7.8%
17.1%
17.4%
13.7%
36.6%
2H24
Predict
0
0
2.9%
81.4%
15.5%
0.2%
Predefined
25.2%
11.1%
29.8%
23.1%
5.8%
5.0%
3NFB
Predict
0
0.4%
97.2%
2.4%
E.7
Structure Analysis
To evaluate whether PAFlow can generate valid molecular conformations, a series of structure-related experiments are conducted. We report the average Jensen-Shae Maasenn-hannon divergence (ISD) between the bond length, bond angle, and torsion angle distributions of the generated and reference molecules. where lower JSD indicates distribution closer to that of real structures. In addition, provide the ring size distribution of generated molecules. Together, these evaluations comprehensively assess substructure stability. To evaluate global structural stability, the strain energy (SE) and steric clash (Clash) statistics are also reported, which are calculated by PoseCheck [50). As shown in Table 8 10 PAFlow exhibits comparable performance to existing methods across both local and global structural metrics. Although classifier guidance known to sometimes produce ill-formed conformations, PAFlow is still able to generate molecules with stable and chemically plausible geometries both the local and global levels. future work, we plan to incorporate physical rules and energy- based constraints further improve molecular structural quality.
Table Results of molecular conformation evaluation. (4) denotes that lower values indicate better performance.
Methods
Length (J)
Angle (J)
Torsion (L)
AR
0.554
0.467
0.519
FLAG
0.511
0.284
IPDiff
0,402
0.415
0.386
ALiDiff
0.445
0,422
0.422
PAFlow
0.507
0.461
0.424
F
Discussion
F1
Difference Between PAFlow and Other FM-Based Methods
Flow Matching a simulation-free approach for stably training continuous normalizing flows that demonstrates strong generative capabilities. Various types of probability paths can be used within the FM framework as summarized in Table of [29], and the choice can be flexibly adapted based on the specific task requirements.
28

---

Table 9: Ratios of ring sizes for reference ligands and molecules generated by different methods.
Ring Size
Ref
IPDiff
TAGMoL
PAFlow
1.7%
0.0%
0.0%
0.0%
0.0%
3.2%
4.0%
3.4%
30.2%
25.8%
29.5%
19.6%
67.4%
42.4%
39.0%
38.4%
0.7%
18.6%
19.7%
26.0%
0.0%
7.0%
5.9%
9.2%
0.0%
2.9%
1.9%
3.4%
Table 10: Results on molecular conformation stability. For SE, the 25th percentile, median, and 75th percentile are reported.
SE (4)
Clash (L)
Methods
25%
50%
75%
Avg
Med
TargetDiff
363
1233
13773
10.77
7.00
ALiDiff
1607
56055
5779799
8.68
5.00
KGDiff
566
84186
124750452
7.82
5.00
PAFlow
3339
90239
5906610
7.74
5.00
FlowSBDD employs Rectified Flow to learn the transport mapping of atom coordinates and types from the prior to the data distribution, aiming for straight-line trajectories between the two. FlexSBDD(51 adopts similar modeling approach. While such straight-line paths are computa- tionally efficient, they lack the capacity to model complex generation tasks effectively. contrast, PAFlow utilizes different probability paths tailored to the distinct characteristics of atom coordi- nates and types. Specifically, continuous coordinates are modeled using Gaussian distributions with pariance-preserving (VP) trajectories derived from diffusion models, while discrete atom types are modeled with categorical distributions, for which we specifically construct FM formulation and conditional vector field (detailed derivations in Appendix A.2). In addition, prove that the resulting generative process consistent with transforroooooaaa invariance, offering theoretical justification for modeling strategies (see Appendix A.I).
Under the same experimental settings, a comparison between the results of these methods presented. As shown in the Table 11 below, PAFlow significantly outperforms FlexSBDD and FlowSBDD on all affinity-related metrics while maintaining reasonable molecular properties. that FlexSBDD generates molecules with modeling protein flexibility. whereas PAFlow performs generation based on rigid proteins. Despite this difference. PAFlow still achieves superior perfor- mance, suggesting the greater effectivvnees of our modeling strategy. Overall, although both PAFlow and FlexSBDD are built within the FM framework, they differ fundamentally in the design proba- bility paths and modeling strategies. We hope these extensions offer useful insights and contribute meaningfully to the advancement FM-based molecular generation.
worth noting
Table 11: Comparison of PAFlow with other FM-based methods. The table presents the mean values of each metric. The best results are highlighted in bold.
Methods
Vina Score (4)
Vina Min (4)
Vina Dock (1)
High Affinity (t)
QED (t)
SA (t)
Div (f)
TiffetDiff
-5.47
-6.64
-7.80
58.1%
0.48
0.58
0.72
FlowSBDD
-3.62
-6.72
-8.50
63.4%
0.47
0.51
0.75
Flex
-6.64
-8.27
-9.12
78.5%
0.58
0.69
0.76
Flllow
-8.31
-8.79
-9.46
80.8 %
0.49
0.57
0.71
29

---

F.2 Reasons for the Better Binding Affinity
There are two main reasons why the molecules generated by PAFlow achieve lower Vina scores compared to the ground truth in the test set. Firstly, we calculate the average and median Vina scores of the protein-ligamm pairs in the training set be 8.19 and 8.32 respectively, which are comparable to those of the molecules generated by PAFlow. This indicates that PAFlow effectively learned the binding patterns contained the training data. Secondly, the interaction predictor is employed to guide the generation process. Since the predictor is trained using the Vina scores of the training set as labels, captures prior domain knowledge about contiin-ligane binding embedded in these scores. By incorporating this learned prior into the generation process, the model benefits not only from the information the training data also from the guidance, which together enable PAFlow to generate molecules with better binding affinity than those in the training set. Actually, the reference ground-truth ligands are excellent binders for their corresponding protein target according to certain experiments predictions, but they may not be the optimal ones with the best Vina scores for the target.
Limitations, Future Work and Broader Impact
While PAFlow demonstrates outstanding performance, there remain several potential limitations that we aim to address future work. First, volume and surface area binding pockets used train the atom number predictor are computed using PyKVFinder, which provides only approximate estimation and may introduce bias. In the future, we plan to incorporate experimentally measured pocket information improve the performance of the predictor. Additionally, have attempted to apply flow matching to rigid proteins and observed promising results. We intend to extend this approach to flexible proteins, which better reflect realistic scenarios in structural biology. Lastly, since PAFlow only focuses on optimizing binding affinity, we plan to extend the guidance strategy to include molecular properties for multi-objective optimization to generate ligands with higher pharmaceutical potential.
Our work holds promise for improving the efficiency of drug design and advancing the pharmaceutical industry. can help streamline drug development process, reducing both time and resource costs. In addition, regulatory oversight of structure-bare drug design (SBDD) technique is necessary to ensure they are used for socially beneficial purposes and to prevent potential misuse that could lead to the generation of harmful molccules.
F.4 More Examples
The visualization of more ligand molecules generated by PAFlow, comparing to reference, ALiDiff and MoICRAFT, are provided in Fig. [1 We also present several unreasonable molecules generated by PAFlow as edge cases Fig. To illustrate its limitations, including improper double bonds, large rings, and fused rings, which occasionally occur. Incorporating physical rules and energy constraints expected to alleviate this issue and improve the structural quality of the generated molecules.
unrealistic double bonds
large ring
fused ring
Figure 10: Some edge cases of molecules generated by PAFlow.
30

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2Z3H
ALIDIff
MolCRAFT
Vina: -8.82 QED: 0.11 SA: 0.56
Vina: -6.32 QED: 0.31 SA: 0.55
Vina: -5.56 QED: 0.48 SA: 0.50
Vina: 100.30 QED: 0.61 SA: 0.57
2JJG
Referrence
ALiDiff
MoICRAFT
PAFlow
Vina: -5.84 0.67 SA: 0.80
Vina: -6.4
Vina: -6.54 SA:
Vina: -7.04 QED: 0.64
1A2G
Referrence
ALiDiff
MolCRAFT
PAFlow
Vina: -5.48 QED: 0.58 SA: 0.83
Vina: -8.51 QED: 0.56 SA: 0.61
Vina: -6113 QED: 0.58 SA: 0.57
Vina: -9.70 QED: 0.79 SA: 0.68
5MMA
Referrence
ALIDiff
MolCRAFT
PAFlow
Vina: -3.08 QED: 0.67 SA: 0.57
Vina: -0.35 QED: 0.44 SA: 0.47
Vina: -4100 QED: 0.25 SA: 0,51
Vina: -4.14 QED: 0.50 SA: 0.66
5808
MoICRAFT
Vina: -667 QED: 0.67 SA: 0.87 Vina: -5.14 QED: 0,27 SA: 0.31 Vina: -4.83 QED: 0.58 SA: 0.70
Vina: -7.35 QED: 0.75 SA: 0.91
Figure 11: More visualizations of generated molecules and reference ligands for protein pockets. Carbon atoms of the reference ligands and molecules generated by AliDiff, MolCRAFT, and PAFlow are colored yellow, blue, orange, and green respectively. The corresponding Vina Score, QED, and SA for each molecule are also reported.
31


Reviews:
[{"review_id": "g8b4foxrJ1", "reviewer": "Reviewer_AuA9", "summary": "This paper proposes PAFlow, a novel structure-based drug design (SBDD) model built on the flow matching (FM) framework. The method separately models continuous 3D atomic coordinates and discrete atom types using tailored probability paths and incorporates two key components: a protein-ligand interaction predictor that guides generation toward high-affinity regions, and a learnable atom number predictor that aligns molecule size to the target binding pocket. The method achieves strong results on CrossDocked2020, outperforming many baselines.", "questions": "All added within \"Strengths And Weaknesses\"", "limitations": "All added within \"Strengths And Weaknesses\"", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The idea of using interaction prediction to steer the vector field during flow-based generation is clever and effective. It directly targets the main objective of SBDD, tight binding, instead of relying only on general chemical priors.\n\nThe inclusion of a learnable atom count predictor is a surprisingly impactful addition. Most generative models just sample atom counts from empirical priors, which often leads to pocket-size mismatches. Here, by directly predicting atom counts from protein geometry (volume, surface area, atom count), the model creates molecules that better fit into the pocket. The ablations support this clearly.\n\nThe use of conditional flow matching (CFM) for discrete atom types is new and nicely derived. While [8] and others model discrete atoms in diffusion frameworks, this work generalizes the vector field formulation to discrete types in flow matching, which is a non-trivial extension.\n\nOn empirical performance, PAFlow leads across most binding affinity metrics (Vina Score, Vina Min, Vina Dock), outperforming strong baselines like ALiDiff and MolCRAFT. It also achieves good QED and SA scores while preserving diversity. The figure showing that PAFlow achieves best performance on 77% of targets is especially compelling.\n\nThe sampling efficiency is another win. Despite using ODE-based generation, PAFlow is significantly faster than TargetDiff and Pocket2Mol, and even competes with MolCRAFT when run with fewer steps.\n\nThe paper is technically detailed and derivations are thorough. The proofs in the appendix about SE(3) equivariance and the effect of noise injection in the atom number predictor are solid and well-grounded.\n\nWhile the results are strong, the novelty of the base generation framework is incremental. Flow Matching has already been applied in this space (e.g. FlowSBDD), and the main novelty lies in the protein-guided vector field and atom count handling. That’s still a valuable contribution, but not a foundational shift.\n\nThe protein–ligand interaction predictor is trained on normalized binding scores. It would help to discuss how sensitive the model is to this normalization choice, and whether a more realistic affinity predictor (e.g. based on physical interactions) could further improve things.\n\nThe molecular property control (QED, SA, etc.) is not explicitly incorporated but still performs reasonably. Since the authors already use a binding affinity predictor to guide generation, adding similar predictors for QED or SA in future work seems like a straightforward and useful extension.\n\nThere’s no mention of how well this method handles large or irregular binding pockets. A few edge-case visualizations where PAFlow fails or generates oddly-shaped molecules would help clarify model limitations.\n\nRuntime stats are useful, but it might help to report average time per molecule instead of (or alongside) time for 100 samples; just easier to interpret.\n\n[Final Thoughts]: I have raised my score based on the sufficient author responses.", "was_revised": true, "final_justification": "[Final Thoughts]: I have raised my score based on the sufficient author responses.", "rebuttal": "**Q1**:\tWhile the results are strong, the novelty of the base generation framework is incremental. Flow Matching has already been applied in this space (e.g. FlowSBDD), and the main novelty lies in the protein-guided vector field and atom count handling. That’s still a valuable contribution, but not a foundational shift.\n\n**A1**: Thank you for acknowledging our contributions in the design of the prior-guided vector field and atom number prediction. While both PAFlow and FlowSBDD are built upon the Flow Matching (FM) framework, their modeling strategies differ substantially. Flow Matching is a simulation-free approach for stably training continuous normalizing flows (CNFs) that demonstrates strong generative capabilities. Various types of probability paths can be used within the FM framework as summarized in Table 1 of [1], and the choice can be flexibly adapted based on the specific task requirements.\n\nFlowSBDD employs Rectified Flow to learn the transport mapping of atom coordinates and types from the prior to the data distribution, aiming for straight-line trajectories between the two. FlexSBDD adopts a similar modeling approach. While such straight-line paths are computationally efficient, they lack the capacity to model complex generation tasks effectively. In contrast, PAFlow utilizes different probability paths tailored to the distinct characteristics of atom coordinates and types. Specifically, continuous coordinates are modeled using Gaussian distributions with variance-preserving (VP) trajectories derived from diffusion models, while discrete atom types are modeled with categorical distributions, for which we specifically construct a FM formulation and conditional vector field (see lines 144–149 on page 4, detailed derivations in Appendix A.2). In addition, we prove that the resulting generative process is consistent with SE(3)-transformation invariance, offering a theoretical justification for our modeling strategies (see Appendix A.1). We compare these methods under the same experimental settings. As shown in the table, PAFlow significantly outperforms both FlowSBDD and FlexSBDD across all affinity-related metrics while maintaining favorable molecular properties, demonstrating the effectiveness of our modeling approach. Overall, although all these methods fall within the FM framework, they differ fundamentally in the design of probability paths and modeling strategies. We hope these extensions offer useful insights and contribute meaningfully to the advancement of FM-based molecular generation.\n\n| Methods| | Vina Score↓ | Vina Min ↓ | Vina Dock ↓ | High Affinity ↑ | QED ↑ | SA ↑ | Diversity ↑ |\n|-|-|-|-|-|-|-|-|-|\n| TargetDiff      | Avg.        | -5.47                     | -6.64                   | -7.80                    | 58.1%                     | 0.48             | 0.58            | 0.72                   |\n|                 | Med.        | -6.30                     | -6.86                   | -7.91                    | 59.1%                     | 0.48             | 0.58            | 0.71|\n| FlowSBDD| Avg. | -3.62| -6.72| -8.50 | 63.4% | 0.47| 0.51| 0.75|\n|                 | Med.        | -5.03                     | -6.60                   | -8.36                    | 70.9%                     | 0.48             | 0.51            | 0.75                   |\n| FlexSBDD        | Avg.        | -6.64                     | -8.27                   | -9.12                    | 78.5%                     | 0.58             | 0.69            | 0.76                   |\n|                 | Med.        | -7.25                     | -8.46                   | -9.25                    | 84.2%                     | 0.59             | 0.73            | 0.75                   |\n| PAFlow          | Avg.        | **-8.31**                 | **-8.79**               | **-9.46**                | **80.8%**                 | 0.49             | 0.57            | 0.71                   |\n|                 | Med.        | **-8.92**                 | **-8.96**               | **-9.49**                | **93.7%**                 | 0.50             | 0.57            | 0.70                   |\n\n\n**Q2**: The protein–ligand interaction predictor is trained on normalized binding scores. It would help to discuss how sensitive the model is to this normalization choice, and whether a more realistic affinity predictor (e.g. based on physical interactions) could further improve things.\n\n**A2**: Thank you for your valuable suggestion. During the training of the interaction predictor, we apply the widely-used min-max normalization to linearly map the affinity labels into the [0, 1] range. To examine the sensitivity to the choice of normalization, we additionally normalize the labels to the [-1, 1] range by dividing them by the maximum absolute value, and train a new model named PAFlow_abs. The results comparing molecules generated by both methods are shown in the table, where PAFlow_abs achieves comparable performance to PAFlow. For efficiency, PAFlow_abs uses the same scaling factor for guidance as PAFlow. The performance of PAFlow_abs can be further improved by tuning a more suitable scaling factor for its specific normalization. These results suggest that as long as the normalization does not distort the relative distribution of the original labels, the predictor’s effectiveness remains stable. In addition, we agree that using a more realistic predictor based on physical interactions may provide more accurate guidance and lead to improved generation quality. We plan to explore this direction in future work by integrating physical modeling and energy-based prediction.\n\n| Methods| Vina Score ↓| Vina Min ↓| Vina Dock ↓| High Affinity ↑| QED ↑| SA ↑| Diversity ↑|\n|-|-|-|-|-|-|-|-|\n|PAFlow| -8.10| -8.50| -9.62 | 80.7% | 0.49| 0.57| 0.70|\n| PAFlow_abs| -7.74| -8.05| -8.32| 74.6% | 0.43| 0.59| 0.71|\n\n**Q3**: The molecular property control (QED, SA, etc.) is not explicitly incorporated but still performs reasonably. Since the authors already use a binding affinity predictor to guide generation, adding similar predictors for QED or SA in future work seems like a straightforward and useful extension.\n\n**A3**:\nWe strongly support your suggestion to extend the guidance for QED and SA.  We conduct preliminary experiments where, in addition to affinity, predictors for QED and SA are trained, and a simple gradient summation is employed to simultaneously guide all three properties. This approach is named PAFlow-multi. The experimental results are shown below, where “backbone” denotes the baseline without any guidance. The Hit Rate represents the percentage of generated molecules that satisfy QED ≥ 0.4, SA ≥ 0.5, and Vina Dock ≤ -8.18 [2]. The results indicate that incorporating the property predictor guidance leads to improvements across all metrics, along with a higher rate of the hit molecules, which demonstrates the effectiveness of the guidance strategy. Furthermore, jointly optimizing binding affinity, QED, and SA is essentially a multi-objective optimization problem, where conflicts among these properties inherently exist. Simply summing the gradients is insufficient to fully address this issue. Recent work (e.g., MoC [3]) has made initial attempts to balance such trade-offs, but this remains an important direction for future research.\n\n| Methods| Vina Score ↓| Vina Min ↓| Vina Dock ↓| High Affinity ↑| QED ↑| SA ↑| Diversity ↑| Hit Rate ↑|\n|-|-|-|-|-|-|-|-|-|\n|backbone|-5.24|-6.90|-8.30|64.7%|0.52|0.56|0.71|24.9%|\n| PAFlow-multi | **-6.73** | **-7.74**|**-8.75**|**75.1%**|**0.53**|**0.58**| **0.72**|**29.3%**|\n\n\n**Q4**: There’s no mention of how well this method handles large or irregular binding pockets. A few edge-case visualizations where PAFlow fails or generates oddly-shaped molecules would help clarify model limitations.\n\n**A4**: Thank you for your valuable suggestion. We fully agree with your point. Since the OpenReview does not allow uploading images, we will supplement the visualizations of failure cases or irregularly structured molecules generated by PAFlow to provide a clearer illustration of its limitations in the revision.  \n\n**Q5**: Runtime stats are useful, but it might help to report average time per molecule instead of (or alongside) time for 100 samples; just easier to interpret.\n\n**A5**: Thank you for your suggestion. We report the average generation time per molecule as a supplement to Fig. 4. We hope this table provides a more intuitive comparison across different methods and further highlights the high efficiency of PAFlow.\n\n|Methods|Pocket2Mol|TargetDiff|MolCRAFT|PAFlow|PAFlow T=20|\n|-|-|-|-|-|-|\n| **Avg Time (s)** |39.7|40.1|4.0|7.2|**2.9**|\n\n[1] Tong A, Fatras K, Malkin N, et al. Improving and generalizing flow-based generative models with minibatch optimal transport[J]. arXiv preprint arXiv:2302.00482, 2023.\n\n[2] Dorna V, Subhalingam D, Kolluru K, et al. Tagmol: Target-aware gradient-guided molecule generation[J]. arXiv preprint arXiv:2406.01650, 2024.\n\n[3] Zhou J, Zhao D, Qian H, et al. Multi-Objective Structure-Based Drug Design Using Causal Discovery[J]. IEEE Transactions on Computational Biology and Bioinformatics, 2025."}, {"review_id": "IuP1ZhtWzu", "reviewer": "Reviewer_ns3U", "summary": "This paper introduces PAFlow, a novel target-aware generative model for structure-based drug design. PAFlow integrates prior protein-ligand interaction knowledge and a learnable atom number predictor into a conditional flow matching framework to generate 3D molecules with improved binding affinity. The model addresses ligand-pocket size mismatches and guides generation toward higher-affinity regions.", "questions": "- How did you obtain the binding affinity labels to train the guidance network? How do you ensure the affinities are comparable across different targets?  \n- How does the model perform on molecular stability metrics, such as bond length accuracy?  \n- How is pocket size calculated in Appendix E.5?", "limitations": "yes", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strength**  \nThis paper addresses the problem of undetermined atom numbers by introducing an additional predictor.  \nInstead of relying on explicit priors as in previous works, it trains a binding affinity predictor and uses it to guide flow matching sampling toward higher-affinity distributions.\n**Weakness**  \nThe novelty of the flow matching framework is limited, as flow matching has already been applied to SBDD in prior works (e.g., *FlexSBDD: Structure-Based Drug Design with Flexible Protein Modeling*, NeurIPS 2024).  \nMoreover, although the authors claim their method generates stable molecules of high quality, the visualized results in Figure 3 include unrealistic double bonds. The work should include more results on bond length and bond angle accuracy as evaluation metrics.  \nWhile the method generates molecules with low Vina scores, these scores are even lower than the ground truth from the test dataset, which requires discussion in the paper.", "was_revised": true, "final_justification": "Authors have addressed my concerns and questions regarding the validity of the generated molecules and provided additional experiments. I really appreciated their efforts.\nHowever, regarding the novelty of using flow matching frameworks and training an additional Vina score predictor for guided generation, the originality of this work is still limited. As a result, I will maintain my rating.", "rebuttal": "**Q1**: The novelty of the flow matching framework is limited, as flow matching has already been applied to SBDD in prior works (e.g., FlexSBDD).\n\n**A1**: Thank you for raising this important point. Flow Matching (FM) is a simulation-free approach for stably training Continuous Normalizing Flows that exhibits strong generative performance. Within the FM framework, various probability paths can be used for modeling. Tab. 1 in [1] summarizes available options, including diffusion and optimal transport paths, with the appropriate choice depending on the specific application.\n\nAlthough both FlexSBDD and PAFlow are built upon the FM framework, their modeling strategies are fundamentally different. FlexSBDD employs the optimal transport displacement interpolant as the probability path for transferring atom coordinates and types, aiming for straight flows from prior to data. FlowSBDD[2] adopts the same modeling approach. While such paths offer shorter distances and reduced computational cost, they are limited in their capacity to effectively model complex tasks. In contrast, PAFlow utilizes different probability paths tailored to the different characteristics of atom coordinates and types. For continuous coordinates, we use the Variance Preserving (VP) path derived from diffusion models. For discrete atom types, we model them using categorical distributions and specifically design a novel FM formulation along with a conditional vector field (see lines 144–149 on page 4), where detailed derivations are provided in Appendix A.2. Furthermore, we demonstrate that this generation process is consistent with SE(3)-transformation invariance (see Appendix A.1), providing a theoretical justification for the validity of our modeling approach.\n\nUnder the same experimental settings, a comparison between the results of these methods is presented. As shown in the table below, PAFlow significantly outperforms FlexSBDD on all affinity–related metrics while maintaining reasonable molecular properties. It is worth noting that FlexSBDD generates molecules with modeling protein flexibility, whereas PAFlow performs generation based on rigid proteins. Despite this difference, PAFlow still achieves superior performance, suggesting the greater effectiveness of our modeling strategy. Overall, although both PAFlow and FlexSBDD are built within the FM framework, their implementation details differ substantially. We believe that PAFlow introduces meaningful and technically valuable contributions in the design of the probability paths.\n\n| Methods | Vina Score↓ | Vina Min ↓ | Vina Dock ↓ | High Affinity ↑ | QED ↑ | SA ↑ | Div ↑ |\n|-|-|-|-|-|-|-|-|\n| TargetDiff | -5.47| -6.64| -7.80 | 58.1% | 0.48 | 0.58  | 0.72  |\n| FlowSBDD| -3.62| -6.72| -8.50| 63.4%  | 0.47 | 0.51 | 0.75\n| FlexSBDD| -6.64| -8.27 | -9.12 | 78.5% | 0.58| 0.69 | 0.76\n| PAFlow | **-8.31**  | **-8.79** | **-9.46** | **80.8%**| 0.49 | 0.57| 0.71|\n\n**Q2+Q6**: **Q2** Moreover, although the authors claim their method generates stable molecules of high quality, the visualized results in Figure 3 include unrealistic double bonds. The work should include more results on bond length and bond angle accuracy as evaluation metrics. **Q6** How does the model perform on molecular stability metrics, such as bond length accuracy?\n\n**A2+A6**:\nThank you for your valuable comment. We report the average Jensen-Shannon divergence (JSD) between the bond length, bond angle, and torsion angle distributions of the generated and reference molecules to assess conformational stability. The JSD is computed between the estimated distributions from generated and reference molecules, where a lower JSD indicates closer alignment with realistic molecular geometries. As shown in the table, PAFlow achieves stable performance across all metrics, indicating the geometric reliability of the molecular structures.\n\nWe acknowledge that a few visualized molecules may exhibit unrealistic double bonds, which is likely due to the fact that we do not explicitly model chemical bonds but instead rely on OpenBabel for bond assignment. However, the overall statistical results suggest that the structural quality remains stable. We appreciate you pointing out this potential issue, and in future work, we plan to enhance the molecular structure modeling by incorporating bond representations and introducing energy-based constraints to improve chemical plausibility.\n\n|Methods|Length|Angle|Torsion|\n|-|-|-|-|\n|AR|0.554|0.467|0.519|\n|FLAG|0.511|0.284|-|\n|IPDiff|0.402|0.415|0.386|\n|ALiDiff|0.445|0.422|0.422|\n|PAFlow|0.507|0.461|0.424|\n\n**Q3**: While the method generates molecules with low Vina scores, these scores are even lower than the ground truth from the test dataset, which requires discussion in the paper.\n\n**A3**: Thank you for your valuable suggestion. There are two main reasons why the molecules generated by PAFlow achieve lower Vina scores compared to the ground truth in the test set. Firstly, we calculate the average and median Vina scores of the protein–ligand pairs in the training set to be -8.19 and -8.32 respectively, which are comparable to those of the molecules generated by PAFlow. This indicates that PAFlow effectively learned the binding patterns contained in the training data. Secondly, the interaction predictor is employed to guide the generation process. Since the predictor is trained using the Vina scores of the training set as labels, it captures prior domain knowledge about protein–ligand binding embedded in these scores. By incorporating this learned prior into the generation process, the model benefits not only from the information in the training data but also from the guidance, which together enable PAFlow to generate molecules with better binding affinity than those in the training set. Moreover, many existing methods [3-5] have achieved Vina scores better than the ground truth in the test set, which is a reasonable outcome resulting from effectively leveraging information of the training data. Actually, the reference ground-truth ligands are excellent binders for their corresponding protein target according to certain experiments or predictions, but they may not be the optimal ones with the best Vina scores for the target. We will include a corresponding discussion in the revised manuscript.\n\n**Q4**: How did you obtain the binding affinity labels to train the guidance network? \n\n**A4**: Thank you for your question. The binding affinity labels used to train the interaction predictor are Vina Scores computed by AutoDock Vina, following the setup in [4]. To facilitate stable learning, these scores are further normalized to the [0, 1] range using Min-Max normalization, where higher values indicate stronger binding affinity.\n\n**Q5**: How do you ensure the affinities are comparable across different targets?\n\n**A5**: Thank you for raising this point. We address this question in two parts.\n\n(1) Regarding the training process, the binding affinities used to train the property predictor are all obtained from Vina scores computed by AutoDock Vina. Since these scores are derived based on a consistent set of physicochemical principles, they share a unified numerical scale and physical interpretation across all protein–ligand pairs.\nFurthermore, we analyze the distribution of Vina scores in the training set below and observe substantial overlap across different protein–ligand pairs, with most scores falling within the range of [-12, -4]. This indicates that the binding affinities across different targets are defined on a consistent scale. In addition, prior works [4-6] have also adopted similar strategies of training on binding affinities across different targets.\n\n|Range|< -12|[-12,-4]|> -4|\n|-|-|-|-|\n|Percentage|4.40%|91.78%|3.82%|\n\n(2) For generation, the High Affinity in Tab. 1 measures the percentage of generated molecules that bind more strongly to a given target than the reference ligand, with molecules grouped according to their respective target proteins. As shown in Tab. 1, PAFlow significantly outperforms other methods in High Affinity, indicating its strong capability to consistently generate tightly binding molecules across different targets. Fig. 2 presents the median Vina binding energy of molecules generated by different methods for each of the 100 test proteins. PAFlow achieves the best results on 77% of the targets, which further suggests that PAFlow has the potential to maintain strong performance for a wide range of target proteins. Additionally, except for High Affinity, the other affinity-related metrics reported in Tab. 1 are calculated over molecules generated for all test proteins collectively, without restricting to individual targets. This evaluation approach is commonly adopted by most structure-based drug design methods [2-6], and therefore, we follow the same convention.\n\n**Q7**: How is pocket size calculated in Appendix E.5?\n\n**A7**: Thank you for your question. Pocket size is characterized using four features: the number of pocket atoms $N_P$, binding site volume $V$, binding site surface area $A$, and space size $S$. Among them, $N_P$ is obtained by directly counting the atoms in the pocket, while $V$ and $A$ are computed using pyKVFinder. The space size $S$ is the median value of the top 10 largest pairwise distances between protein atoms. These features can provide a comprehensive characterization of pocket size.\n\n[1] Improving and generalizing flow-based generative models with minibatch optimal transport. \n\n[2] Rectified Flow For Structure Based Drug Design. \n\n[3] Flexsbdd: Structure-based drug design with flexible protein modeling. \n\n[4] KGDiff: towards explainable target-aware molecule generation with knowledge guidance. \n\n[5] Tagmol: Target-aware gradient-guided molecule generation. \n\n[6] Multi-Objective Structure-Based Drug Design Using Causal Discovery."}, {"review_id": "CRnI5G35js", "reviewer": "Reviewer_cwgo", "summary": "The authors aim to solve the problem of structure-based drug design. Upon this, the authors noticed limitations of existing generative methods, such as unstable probability dynamics and molecular size mismatches. To address such limitations, the authors introduce PAFlow, a novel target-aware molecular generation model designed for structure-based drug design. PAFlow addresses such limitations by leveraging the flow matching framework with a new conditional flow matching for discrete atom types, incorporating a protein-ligand interaction predictor for guiding generation towards high-affinity regions. Additionally, it integrates a learnable atom number predictor, relying solely on protein pocket information, which significantly improves the alignment of generated molecule size with target geometry, achieving state-of-the-art binding affinity results while maintaining favorable molecular properties.", "questions": "- Looking at Figure 5, the generation trajectory of the target diff appears much messier compared to the proposed method, yet the sampling efficiency shows slightly better results than the proposed method. How can we understand this phenomenon?\n- What insights might there be if coordinate and atom type probability density paths were modeled jointly rather than separately?\n- Regarding prior guidance, how dependent is the method on prior quality?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n- Extensive evaluation and good performance: The paper contains a comprehensive comparison with a number of baselines on a large-scale benchmark with comprehensive metrics, and necessary ablation studies of the proposed method, which helps to understand the empirical robustness of the proposed method.\n- Overall decent recipe to solve SBDD: The proposed method appears to be a well-developed methodology that efficiently utilizes prior work formulations (probability density path formulation, prior guidance) to solve current SBDD problems, while adding unique methods (proposition, learnable atom predictor) to overcome the limitations of prior work.\n- Efficient improvement over previous methods: The introduction of an atom predictor effectively resolves the issue that non-autoregressive previous works for SBDD problems required reference information for generated ligands. In addition, the sampling efficiency over baseline is comparable to the most efficient baselines.\n\n## Weaknesses\n\n- Incremental methodological contribution: The method appears to be largely similar in framework to [1] in terms of using flow matching and prior guidance, and to [2] in terms of probability density path formulation, suggesting that the novelty may be limited to incremental improvements.\n- Lack of formulation justification: In Equation (4), the probability density is defined as the product of probability distributions of atom type and atom coordinate. Given that these two features have strong correlations with each other, it would seem more natural to formulate it as $p_t(x, a| x_1, a_1, p)$. Is there a reason why this approach is difficult? The text simply explains that it follows [3] without specific justification, which seems insufficient beyond merely following prior work.\n- Typo:\n    - In line 143, it appears that the definition of $\\hat{\\alpha}$ is missing.\n\n[1] Z. Zhang, M. Zitnik, and Q. Liu, “Generalized protein pocket generation with prior-informed flow matching,” arXiv preprint arXiv:2409.19520, 2024.\n\n[2] J. Guan, W. W. Qian, X. Peng, Y. Su, J. Peng, and J. Ma, “3d equivariant diffusion for target349 aware molecule generation and affinity prediction,” arXiv preprint arXiv:2303.03543, 2023.\n\n[3] Y. Song, J. Gong, M. Xu, Z. Cao, Y. Lan, S. Ermon, H. Zhou, and W.-Y. Ma, “Equivariant flow\n407 matching with hybrid probability transport for 3d molecule generation,” Advances in Neural\n Information Processing Systems, vol. 36, pp. 549–568, 2023.", "was_revised": true, "final_justification": "The paper proposes a solid framework to solve the SBDD problem with extensive evaluation to prove superiority over prior works, and the authors' rebuttal well addressed my concerns.", "rebuttal": "**Q1**: Incremental methodological contribution\n\n**A1**: Thank you for your comment. While both PAFlow and PocketFlow [1] are built upon the flow matching (FM) framework, their implementation details are entirely different. FM is a simulation-free approach for stably training continuous normalizing flows (CNFs), and various types of probability paths can be employed within this framework, as summarized in Tab. 1 of [3]. PocketFlow is designed for generating protein pockets conditioned on given ligand molecules, which can be considered a dual problem of structure-based drug design (SBDD). It models the optimal transport mapping from the prior to the data distribution, aiming for straight-line trajectories between them. Similar modeling strategies have also been adopted in SBDD tasks by methods such as FlexSBDD and FlowSBDD. Although these straight-line paths offer shorter transport distances and computational efficiency, they lack the capacity to effectively model complex tasks such as SBDD. In contrast, PAFlow uses different probability trajectories tailored to the distinct characteristics of atom coordinates and types. For continuous atomic coordinates, we employ the variance-preserving (VP) path derived from diffusion models. For discrete atom types, we model them using categorical distributions and specifically construct a corresponding FM formulation and conditional vector field (see lines 144–149 on page 4, detailed derivations in Appendix A.2). Moreover, we demonstrate that the resulting generative process is SE(3)-transformation invariant (Appendix A.3), providing a theoretical justification for the soundness of our modeling approach. We compare these methods under the same experimental settings. As shown in the table, PAFlow significantly outperforms both FlowSBDD and FlexSBDD across all affinity-related metrics while maintaining favorable molecular properties, demonstrating the effectiveness of our modeling strategy.\n\n|Methods|Vina Score↓|Vina Min ↓|Vina Dock ↓|High Affinity ↑|QED ↑|SA ↑|Div ↑|\n|-|-|-|-|-|-|-|-|\n|TargetDiff|-5.47|-6.64|-7.80|58.1%|0.48| 0.58|0.72|\n|FlowSBDD|-3.62|-6.72|-8.50|63.4%|0.47|0.51|0.75\n|FlexSBDD|-6.64|-8.27|-9.12|78.5%|0.58|0.69|0.76\n|PAFlow|**-8.31**|**-8.79**|**-9.46**|**80.8%**|0.49|0.57|0.71\n\nPocketFlow also incorporates prior guidance during generation, and both PocketFlow and PAFlow use predictors to guide the process. However, the underlying probability trajectories are fundamentally different, leading to distinct mathematical formulations and derivations. In the derivation, PocketFlow starts from a general ODE formulation, while PAFlow is based on the guided vector field for Gaussian trajectory. Regarding implementation, the predictor in PocketFlow is trained on binary labels (0 for below-average affinity and 1 for above-average affinity) aiming to estimate the probability that affinity equals 1. In contrast, PAFlow uses vina scores normalized to the [0, 1] range as labels and directly predicts affinity values. This allows PAFlow to learn a smoother affinity distribution, enabling more accurate and fine-grained guidance during the generation process.\n\nIn terms of the probability density path, PAFlow modeling under the FM framework adopts the Gaussian trajectory used in [2], where TargetDiff is introduced as a diffusion-based method. We can mathematically prove that when using the same Gaussian path, FM and diffusion share the same mean trajectory, but FM exhibits significantly lower variance (Please let us know if you need the full proof). As a result, the denoising trajectories in diffusion (Fig. 5a) tend to be highly stochastic, leading to unstable probability dynamics and high computational cost. In contrast, the FM-based path (Fig. 5b) is much smoother and more stable due to its low variance. This smoother path enables PAFlow to reduce the number of sampling steps and achieve a 5.5× speedup in efficiency (Page 8, Line 275; Fig. 4), while also delivering better generation quality (Tab. 2).\n\nOverall, although both PAFlow and [1] adopt the FM framework with prior guidance, our implementation differs substantially from [1]. While we share the same probability path as [2], the significantly lower variance of FM enables PAFlow to achieve a smoother and more stable generation process. We believe the improvements introduced in PAFlow are meaningful and technically valuable extensions for the SBDD task.\n\n**Q2**: Lack of formulation justification \n\n**A2**: Thank you for this valuable suggestion. We fully agree with the perspective that atom coordinates and atom types are inherently interdependent in molecular structures, and that jointly modeling them is more reasonable. However, directly modeling the joint distribution $p_t(x, a \\mid x_1, a_1, p)$ faces several practical challenges. Specifically, atom coordinates are 3D continuous variables, while the atom types of ligands and proteins are 27-dimensional and 13-dimensional discrete variables, respectively. Learning a joint distribution in such a high-dimensional space with heterogeneous dimensions and different modalities introduces computational complexity and lacks effective modeling mechanisms. Existing flow matching methods also lack theoretical foundations and practical implementations for constructing mixed-modality vector fields. Therefore, to facilitate both the modeling and the computation of the vector field, we factorize the joint probability into the product of a Gaussian distribution and a categorical distribution, allowing us to separately handle the 3D continuous coordinates and high-dimensional discrete atom types. \n\nIt is important to emphasize that we do not assume independence between atom coordinates and types. Instead, they are conditioned on the same contextual information and their interdependence is considered in the network architecture. We employ an SE(3)-Equivariant GNN for modeling, where atom types are associated with nodes and coordinates with edges. Through the message passing mechanism in the graph structure, the two components influence each other. During the generation process, the spatial positions of nodes and the types of edges are updated jointly, which implicitly captures the dependencies between atom coordinates and atom types. This modeling strategy has been adopted by most existing non-autoregressive structure-based drug design methods, and our experiments also demonstrate its effectiveness in achieving strong performance. Joint modeling undoubtedly presents a valuable research direction that could significantly advance the field of drug design, and we are willing to further explore it in future work.\n**Q3**: Looking at Figure 5, the generation trajectory of the targetdiff appears much messier compared to the proposed method, yet the sampling efficiency shows slightly better results than the proposed method. How can we understand this phenomenon?\n\n**A3**: As shown in Fig. 5(a), the unstable probability dynamics of diffusion models lead to noisy and disordered generation trajectories. In contrast, Fig. 5(b) demonstrates that the flow matching approach produces much smoother trajectories by updating molecular positions through the vector field. Such smooth trajectories enable PAFlow to reduce the number of sampling steps while preserving generation quality. As illustrated in Fig. 4, PAFlow requires only 717s on average to generate 100 molecules, whereas TargetDiff takes 4009s, yielding a 5.59× speedup. Regarding the quality of generated molecules, Tab. 1 shows that PAFlow substantially outperforms TargetDiff on all binding-related metrics, while maintaining comparable molecular properties. These results clearly demonstrate that PAFlow achieves both efficient sampling and high-quality generation.\n\n**Q4**: What insights might there be if coordinate and atom type probability density paths were modeled jointly rather than separately?\n\n**A4**: Thank you for the thought-provoking question. Jointly modeling the coordinate and atom type probability densities could indeed provide a more expressive framework that captures deeper interactions between molecular geometry and atom semantics. It has great potential to lead to enhanced chemical validity and structural coherence in the generated molecules. As discussed in **A2**, joint modeling faces many computational challenges due to the different modalities and heterogeneous dimensions of the data. Nevertheless, this remains a valuable direction for future research, and we appreciate your insightful comment highlighting its importance.\n\n**Q5**: Regarding prior guidance, how dependent is the method on prior quality?\n\n**A5**: Thank you for your question. According to [5], the CrossDocked2020 dataset mainly consists of complexes with moderate binding affinities, so the prior quality we use is not particularly high. As shown in Tab. 3, the generated molecules exhibit substantial improvements across all affinity-related metrics when prior guidance is employed, compared to the unguided setting. This demonstrates that even our predictor trained on moderate-quality data can effectively guide the generation process, indicating a certain level of robustness to the prior. In addition, high-quality prior information can help the predictor capture more accurate protein–ligand binding patterns, thereby providing more precise guidance during generation. Designing more effective priors through better data or modeling is an important direction for further improvement.\n\n[1] Generalized protein pocket generation with prior-informed flow matching.\n\n[2] 3d equivariant diffusion for target-aware molecule generation and affinity prediction.\n\n[3] Improving and generalizing flow-based generative models with minibatch optimal transport.\n\n[4] Equivariant flow matching with hybrid probability transport for 3d molecule generation.\n\n[5] Tagmol: Target-aware gradient-guided molecule generation."}, {"review_id": "ZvV8RJD9Ip", "reviewer": "Reviewer_CJX1", "summary": "This paper proposes PAFlow, a novel flow matching (FM)-based generative model for structure-based drug design (SBDD) that addresses key limitations of existing autoregressive and diffusion-based approaches, such as unnatural generation orders, unstable denoising trajectories, and reliance on predefined ligand priors for atom number prediction. PAFlow introduces a conditional flow matching (CFM) method for discrete atom types alongside variance-preserving (VP) paths for continuous coordinate generation, integrates a protein-ligand interaction predictor to enhance binding affinity, and employs a geometry-aware atom number predictor to eliminate dependency on reference ligands. Experiments on CrossDocked2020 demonstrate state-of-the-art performance, achieving an average Vina score of -8.31 (a 1.24 improvement over baselines) while maintaining favorable molecular properties, establishing FM as a robust alternative for 3D molecule generation in drug discovery.", "questions": "See in weakness.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n1. The proposed method demonstrates outstanding performance in terms of binding affinity scores on the CrossDocked2020 benchmark, surpassing existing baselines by a substantial margin.\n2. The flow-based generative framework employed in the model exhibits superior sampling efficiency compared to diffusion-based counterparts, as evidenced by the reported generation time.\n## Weaknesses:\n1. Although the model achieves high predicted Vina scores under classifier guidance, it is well-known that such guidance often leads to ill-formed molecular conformations, such as abnormal bond lengths, distorted bond angles, and overly twisted structures. The paper lacks essential evaluation metrics that are standard in recent works [1,2,3], including bond length/angle JSD, validity, ring size distribution, strain energy, and atom clash statistics.\n2. The authors claim that the learnable atom number predictor addresses the mismatch between ligand size and binding pocket geometry caused by prior-based methods. However, the experimental section does not include any results or ablation studies to validate this claim.\n3. Prior research [4] has established a strong positive correlation between ligand size (i.e., number of atoms) and binding affinity. It remains unclear whether the performance gain attributed to the proposed atom number predictor stems merely from generating larger molecules. The paper does not analyze whether the predicted atom counts differ significantly from traditional methods or how this affects binding affinity.\n4. The protein–ligand interaction predictor used for classifier guidance appears to be functionally equivalent to the affinity predictor introduced in TAGMol. Given this similarity, it is questionable to position this component as a major contribution of the paper.\n\nReferences:\n\n[1] Guan et al., DecompDiff: Diffusion Models with Decomposed Priors for Structure-Based Drug Design. arXiv:2403.07902\n\n[2] Zheng et al., PoseCheck: Robust Pose Filtering for Diffusion-Based Molecular Docking. arXiv:2308.07413\n\n[3] Qu et al., MolCraft: Structure-based Drug Design in Continuous Parameter Space. arXiv:2404.12141\n\n[4] Wang et al., CBGBench: Fill in the Blank of Protein-Molecule Complex Binding Graph. arXiv:2406.10840", "was_revised": true, "final_justification": "Most of my concerns are addressed.", "rebuttal": "**Q1**: Lack of structural evaluation metrics.\n\n**A1**: Thank you for your valuable comments. We report the average Jensen-Shannon divergence (JSD) between the bond length, bond angle, and torsion angle distributions of the generated and reference molecules, where a lower JSD indicates a distribution closer to that of real structures. In addition, we provide the ring size distribution and validity of generated molecules. Together, these evaluations comprehensively assess the substructure stability. To evaluate global structural stability, we also report strain energy and steric clash statistics, including the 25th, 50th (median), and 75th percentiles. As shown in the tables, PAFlow exhibits comparable performance to existing methods across both local and global structural metrics. Although classifier guidance is known to sometimes produce ill-formed conformations, PAFlow is still able to generate molecules with stable and chemically plausible geometries at both the local and global levels. We sincerely appreciate you pointing out this important issue and will include the above evaluation metrics in the revised manuscript. In future work, we plan to incorporate physical rules and energy-based constraints to further improve molecular structural quality.\n\n|Methods|Length ↓|Angle ↓|Torsion ↓|\n|-|-|-|-|\n|AR|0.554|0.467|0.519|\n|FLAG|0.511|0.284|-|\n|IPDiff|0.402|0.415|0.386|\n|ALiDiff|0.445|0.422|0.422|\n|PAFlow|0.507|0.461|0.424|\n\n|**Ring Size**|Ref|IPDiff|TAGMol|PAFlow|\n|-|-|-|-|-|\n|3|1.7%|0.0%|0.0%|0.0%|\n|4|0.0%|3.2|4.0|3.4|\n|5|30.2%|25.8|29.5|19.6|\n|6|67.4%|42.4|39.0|38.4|\n|7|0.7%|18.6|19.7|26.0|\n|8|0.0%|7.0|5.9|9.2|\n|9|0.0%|2.9|1.9|3.4|\n|**Validity**|-|98.3%|92.6%|94.7%|\n\n|Methods||SE ↓| |Clash ↓| |\n|-|-|-|-|-|-|\n| |25%|50%|75%|Avg|Med|\n|TargetDiff|363|1233|13773|10.77|7.00|\n|ALiDiff|1607|56055|5779799|8.68|5.00|\n|KGDiff|566|84186|124750452|7.82|5.00|\n|PAFlow|3339|90239|5906610|7.74|5.00|\n\n**Q2**: Lack of validation for atom number predictor.\n\n**A2**: Thank you for your question. In Tab. 4 on page 9, we compare the results of two approaches: sampling atom numbers from the predefined distribution (i.e., the prior-based method referred to as \"predefined\" in the table) and using the atom number predictor (the last two rows in the table). As shown in the results, molecules generated using the predictor achieve substantial improvements over the prior-based method in both affinity-related metrics and molecular properties, while maintaining diversity. To more intuitively demonstrate the effectiveness of our predictor, we compare it with the predefined sampling by generating 1,000 atom numbers for each of three randomly selected test proteins. For each protein, we define ±20% and ±30% intervals around the reference atom number and compute the proportion of generated samples falling within these ranges. As shown in table, a considerable portion of the results from the predefined sampling fall outside the ±30% range, which may lead to overly large or small molecules that poorly match the pocket geometry. In contrast, the samples acquired by our predictor are concentrated within the ±20% range, suggesting a stronger potential for generating molecules that are structurally compatible with the target pocket.\n\n| | |<70%|[70%, 80%]|[80%, 100%]|[100%, 120%]|[120%, 130%]|>130%|\n|-|-|-|-|-|-|-|-|\n|4YHJ|Predefined|4.7%|10.9%|26.2%|34.2%|12.5%|11.5%|\n| |Predict|0|0|64.0%|36.0%|0|0|\n|2E24|Predefined|7.4%|7.8%|17.1%|17.4%|13.7%|36.6%|\n| |Predict|0|0|2.9%|81.4%|15.5%|0.2%|\n|3NFB|Predefined|25.2%|11.1%|29.8%|23.1%|5.8%|5.0%|\n| |Predict|0|0.4%|97.2%|2.4%|0|0|\n\n**Q3**: It remains unclear whether the performance gain attributed to the proposed atom number predictor stems merely from generating larger molecules. The paper does not analyze whether the predicted atom counts differ significantly from traditional methods or how this affects binding affinity.\n\n**A3**: Thank you for your valuable suggestion. Although larger molecules tend to exhibit higher binding affinity, the performance gain in our method does not stem from simply generating molecules with more atoms. Instead, it results from generating molecules whose atom counts are closer to the reference ligand. As shown in the table in **A2**, most atom numbers predicted by our model fall within the ±20% range of the reference ligand’s atom count, without producing significantly larger molecules. In contrast, the traditional method—which samples atom numbers from predefined distributions—yields a substantial portion of results outside the ±30% range. Such deviation increases the likelihood of generating molecules that are either too large or too small to fit well within the binding pocket, thereby negatively affecting the protein–ligand interaction.\n\nThe traditional method employs a set of predefined atom number distributions, each associated with a specific interval of pocket space size. When determining the number of atoms to generate, the space size of the given pocket is first calculated, and then a corresponding distribution is selected based on which interval the space size falls into. This coarse-grained approach makes it difficult to accurately capture the appropriate atom number range for molecules targeting different pockets. To address this limitation, our proposed predictor estimates the atom number based on multiple geometric features of the pocket, including the pocket atom number, volume, surface area, and space size. As shown in Fig. 9 (page 19), these factors are strongly correlated with the atom number of the reference ligands. By leveraging this pocket-specific information, our predictor provides a smoother and more informed estimation strategy, which has a higher potential to yield molecules that effectively interact with the target proteins. We will include the relevant analysis in the revised version of the paper.\n\n**Q4**: The protein–ligand interaction predictor used for classifier guidance appears to be functionally equivalent to the affinity predictor introduced in TAGMol. Given this similarity, it is questionable to position this component as a major contribution of the paper.\n\n**A4**: Thank you for your comment. We acknowledge that both the interaction predictor used in PAFlow and the affinity predictor in TAGMol can be categorized as classifier guidance. However, their underlying principles and effects differ significantly. TAGMol is built upon a diffusion-based framework, where the probability path of atom coordinates is defined as $q(x_t|x_0)=\\mathcal{N}(\\sqrt{\\bar{\\alpha_t}}x_0,(1-\\bar{\\alpha_t})\\mathbf{I})$, with $t=0$ corresponding to data and $t=T$ to the prior. The generative process without guidance in TAGMol is defined as $q(x_{t-1}|x_t,x_0)=\\mathcal{N}(\\tilde{\\mu_t},\\tilde{\\beta_t} \\mathbf{I})$, where $\\tilde{\\mu_t}=\\frac{\\sqrt{\\bar{\\alpha_{t-1}}}\\beta_t}{1-\\bar{\\alpha_t}} x_0+\\frac{\\sqrt{\\alpha_t}(1-\\bar{\\alpha_{t-1}})}{1-\\bar{\\alpha_t}}x_t$, $\\tilde{\\beta_t}=\\frac{1-\\bar{\\alpha_{t-1}}}{1-\\bar{\\alpha_{t}}}\\beta_t$. In generation, the coordinate difference between two consecutive timesteps is $\\mathrm{d}x=x_{t-1}-x_t=\\tilde{\\mu_t}-\\tilde{\\mu_{t+1}} + \\sqrt{{\\tilde{\\beta_t}}+ {\\tilde{\\beta_{t+1}}}} \\mathbf{I}$. By substituting the expression of $\\tilde{\\mu}$, we can simplify the result to\n\n$\\mathrm{d}x=(\\sqrt{\\bar{\\alpha_{t-1}}}-\\sqrt{\\bar{\\alpha_t}})x_0+\\sqrt{\\sigma_{t-1}^2+\\sigma_t^2+{\\tilde{\\beta_t}}+{\\tilde{\\beta_{t+1}}}}\\mathbf{I}$,\n\nwhere $\\sigma_{t}=\\frac{\\sqrt{\\alpha_{t+1}}(1-\\bar{\\alpha_t})}{\\sqrt{1-\\bar{\\alpha_{t+1}}}}$. In PAFlow, the generative process is defined with $t=1$ corresponding to data and $t=0$ to the prior. We discretize and reverse the time steps to align with the time series of TAGMol to facilitate the comparison. According to Eq.5 on page 4, we obtain: $\\mathrm{d}x =\\frac{\\sqrt{\\bar{\\alpha_{t}}}-\\sqrt{\\bar{\\alpha_{t-1}}}}{1-\\bar{\\alpha_t}}(\\sqrt{\\bar{\\alpha_t}}x_t-x_0)$. By substituting $x_t=\\sqrt{\\bar{\\alpha_t}}x_0 + \\sqrt{1-\\bar{\\alpha_t}}$, the expression can be simplified as:\n\n$\\mathrm{d}x =(\\sqrt{\\bar{\\alpha_{t}}}-\\sqrt{\\bar{\\alpha_{t-1}}})x_0+\\frac{\\bar{\\alpha_{t-1}}(\\alpha_t-\\sqrt{\\alpha_t})}{\\sqrt{1-\\bar{\\alpha_t}}}\\mathbf{I}$.\n\nClearly, without guidance, both methods share the same mean in their $\\mathrm{d}x$ expressions. By plotting the variance curves, we observe that PAFlow exhibits a much smaller variance term than TAGMol, resulting in a smoother and more stable generation trajectory.\n\nWhen guidance is applied, the generation in TAGMol is defined as $q(x_{t-1}|x_t,x_0)=\\mathcal{N}(\\tilde{\\mu_t}+\\tilde{\\beta_t}\\nabla \\log p_{\\theta}(y|x_t), \\tilde{\\beta_t} \\mathbf{I})$, and $\\mathrm{d}x$ can be expressed as:\n\n$\\mathrm{d}x=(\\sqrt{\\bar{\\alpha_{t-1}}}-\\sqrt{\\bar{\\alpha_t}})x_0 + \\frac{1-\\bar{\\alpha_{t-1}}}{1-\\bar{\\alpha_{t}}}\\beta_t\\nabla\\log p_{\\theta}(y|x_t)+\\sqrt{\\sigma_{t-1}^2+\\sigma_t^2+{\\tilde{\\beta_t}}+ {\\tilde{\\beta_{t+1}}}}\\mathbf{I}$.\n\nFor PAFlow with guidance, based on Eq.13 on page 5, $\\mathrm{d}x$ can be written as:\n\n$\\mathrm{d}x =(\\sqrt{\\bar{\\alpha_t}}-\\sqrt{\\bar{\\alpha_{t-1}}})x_0+\\frac{\\sqrt{\\bar{\\alpha_t}}-\\sqrt{\\bar{\\alpha_{t-1}}}}{2\\bar{\\alpha_t}}\\nabla\\log p_{\\phi}(y|x_t)+\\frac{\\bar{\\alpha_{t-1}}(\\alpha_t-\\sqrt{\\alpha_t})}{\\sqrt{1-\\bar{\\alpha_t}}}\\mathbf{I}$.\n\nIt is evident that after incorporating guidance, the mean paths of PAFlow and TAGMol differ, indicating that the two methods employ mathematically distinct guidance strategies. Furthermore, as shown in Tab. 1 on page 7, molecules generated by PAFlow significantly outperform those of TAGMol across all affinity-related metrics, demonstrating the greater effectiveness of PAFlow’s guidance approach. The smoother probability path of PAFlow without guidance also contributes to improved stability during guidance. \n\nOverall, although both PAFlow and TAGMol utilize classifier guidance, they differ fundamentally in their mathematical formulation and performance."}]

Author rebuttal:

Meta-review: The paper introduces a target-aware generative model using flow matching for structure-based drug design (SBDD). The proposed approach has shown to be effective through relatively solid empirical evaluations. There are shared concerns regarding relatively incremental methodology contribution as flow-matching has already been used in solving SBDD, such as FlowSBDD, FlexSBDD and PocketFlow. The author provided additional results and resolved most of the concerns. While reviewer **ns3U** remain less enthusiastic regarding the limited novelty, the rebuttal convinced two reviewers to raise their scores. The AC tends to be on the positive side and believes the paper contributes valuable insights and effective approach to the ML and SBDD community, thus recommending acceptance. The AC has discussed the case with the SAC, and that the decision has been confirmed by the SAC. As suggested by the reviewers, the final version should include more detailed clarifications and discussion to better highlight the model innovation.
Decision: Accept (poster)