Piloting Structure-Based Drug Design via Modality-Specific Optimal Schedule
Authors: Keyue Qiu
Yuxuan Song
Zhehuan Fan
Peidong Liu
Zhe Zhang
Mingyue Zheng
Hao Zhou
Wei-Ying Ma
Venue: ICML 2025 poster (2025)
Keywords: Structure-Based Drug Design
Bayesian Flow Network
Noise Schedule
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: Twisted probability path induced by noise schedule with the optimal VLB

Abstract: Structure-Based Drug Design (SBDD) is crucial for identifying bioactive molecules. Recent deep generative models are faced with challenges in geometric structure modeling. A major bottleneck lies in the twisted probability path of multi-modalities—continuous 3D positions and discrete 2D topologies—which jointly determine molecular geometries. By establishing the fact that noise schedules decide the Variational Lower Bound (VLB) for the twisted probability path, we propose VLB-Optimal Scheduling (VOS) strategy in this under-explored area, which optimizes VLB as a path integral for SBDD. Our model effectively enhances molecular geometries and interaction modeling, achieving state-of-the-art PoseBusters passing rate of 95.9\% on CrossDock, more than 10\% improvement upon strong baselines, while maintaining high affinities and robust intramolecular validity evaluated on held-out test set. Code is available at https://github.com/AlgoMole/MolCRAFT.

Piloting Structure-Based Drug Design via Modality-Speciine Optimal Schedule
Keyue Qiu 12
Yuxuan Song 12
Zhehuan Fan
Peidong Liu
Zhe Zhang?
Mingyue Zhengè 3
Hao Zhou'
Wei-Ying Ma
Abstract
PB-Valid-Mol (ID)
97.3
Structure-Baser Drug Design (SBDD) is crucial for identifying bioactive molecules. Recent deep generative models are faced with challenges in geometric structure modeling. A major bottle- neck lies in the twisted probability path of multi- iiltaaities-omtinnuee
90-
PB-Valid-Mol (OOD)
89.3
79.1
70
68.1
60
57.6
57.7
51.0
50-
3D positions and discrete 2D wohhh jointly determine molecu- lar geometries. By establishing the fact that noise schedules decide the Variational Lower Bound (VLB) for the twisted probability path, we pro- pose VLB- L.ttOpl Scheduling (VOS) strategy in this under-explored area, which optimizes VLB as path integral for SBDD. Our model effec- tively enhances molecular geometries and interac- tion modeling, achieving a state-of-thhearr Pose- Busters passing rate of 95.9% on CrossDock, more than 10% improvement upon strong base- lines, while maintaining high affinities and robust intramolecular validity evaluated test set. Code is available at https: : //github. com/ A1 Aow.llllle/oooooool.ooollloeelel /aw.oooooeeMMMoooo MooeeeeMolalaaa
40
30 TargelDiff
DecompDiff
MoICRAFT
Ours
Figure I. PoseBusters passing rates (%) for non -utoorrrrrresivv models, where ours maintains the best ID and OOD performance.
ID: in-distributional CrossDock test; OOD: out-of-distriuutionall PoseBusters test. PB- Valid-Mol: intramolecular validity. Detailed results can be found in Appendix
Recent advances in geometric deep generative models have centered non-autoregressive methods such diffusion (Guan et al., 2023) and Bayesian Flow Network (BFN) (Qu et al., 2024), showing promise by capturing the structures at the global level. However, when evaluated with a rigorous out-of- difriisttiittion (OOD) test on PoseBusters benchmark (Buttenschoen et al., 2024), Fig. shows a notable perfor- mance drop in intramolecular validity, suggesting that cur- rent global generation strategies may inadequately capture the fine- fine-grained molecular geometries.
held-out
1. Introduction
Structure-Based Drug Design (SBDD) plays pivotal role in the discovery of bioactive molecules, leveraging the knowl- edge of protein-ligand interactions to identify potential ther- apeutic compounds. At the core of SBDD is the accurate modeling of 3D protein-ligand geometries, as only when bioactive compounds can bind effectively to their target re- ceptors can they elicit their therapeutic effects (Isert et al., 2023). Despite its importance, achieving high-fidelity inter- action modeling remains a significant challenge, primarily due to the complexity of the underlying binding dynamics.
One key factor in geometric structure modeling is the inter- twined probability path of different modalities: continuous atom positions and discrete molecular topologies. While these modalities jointly determine molecular geometries and protein-ligand interaction types, there lacks system- atic understanding in designing the twisted probability path for SBDD. Prior works (Peng et al., 2023; Vignac et al., 2023; Guan et al., 2023) adopt sophisticated noise schedules adapted for different modalities, and propose to generate 3D positions first. In our preliminary studies (Sec. 2), we identify the potential problem of prioritizing the 3D modal- ity, where the model cannot effectively utilize 2D topology information in the generation, suggesting that the current noise schedule is suboptimal. Despite its importance, the design of optimal noise schedules for twisted probability path remains a largely underexplored area.
'Equal contribution 'Institute for AI Industry Research (AIR), Tsinghua University 'Department of Computer Science and Tech- nology, Tsinghua University 'Shanghai Institule of Materia Med- ica, Chinese Academy of Sciences "Sichuan University. Corre- spondence to: Hao Zhou <zhouhao @ air www.ssssinndu...oooom edu.en>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
To address the gap left by previous methods, we aim at a systematic solution for the optimal schedule for the twisted

---

Piloting Structure-Baser Drug Design via Modality-Speciffe Optimal Schedule
Density CC
Reference AR QSD 0.611
0.20
Reference
TergetDiff (JSD 0.37)
0.20
Ours (JSO 0.20)
Reference
Pocket2Mal (SD 4.4
0.15
DecompDint (SS 0.33)
0.15
MoICRAFT USD 0.32)
0.10
0,10
0.05
0.05
0.0
0.00
0.00
1.2
1.3
1.4
1.5
1.6
1.7
1.2
1.3
1.5 Bond length (A)
1.4
1.6
1.7
1.2
1.3
1.4
1.5
1.6
1.7
Bond length (A)
Bond length (A)
1e-3
1e-3
1e-3
Reference
Reforence
Reference Ours IJSI 0.23)
Density of
1.5
(JSD 0.37)
1.5
TargetDirt (SD 0.34)
1.5
Packet2Moll USD 0.38)
DecompDirt (JSD 0.34)
1.0
1.0
MoiCRAFT USD
1.0
0.5
0.5
0.5-
100
110
120
130
140
100
110
120
130
140
100
110
120
130
140
Bond angle (*)
Bond angle (*)
Bond angle (*)
Figure 2. Visualization of the bond length and bond angle distribution for the most frequent bond types in CrossDock test. More types are shown in Appendix D.2, where our MolPilot consistently captures the molecular geometries most accurately.
probability path of different modalities, and emphasize the need for principled metrics relevant to generative model- ing in order to evaluate schedule quality. EquiFM (Song et al., 2024b) introduces improved hybrid probability paths aligned with information- thcoretic heuristics but lacks a theoretical foundation for optimality. To rigorously define optimality, we analyze the Variational Lower Bound (VLB) under varying noise schedules. Crucially, we prove that in the multi-modality generation, the VLB becomes a path- dependent integral; its value depends on the entire noise schedule, not just endpoints (Kingma et al., 2021).
We establish the theoretical link between noise sched- ules and VLB in multi-modality probabilistic model- ing. Unlike prior heuristic approaches, our principled method reveals path-dependent VLB dynamics and provides insights for modality-speciin scheduling.
inforoation-thhorrtiic
Integrated with advanced frameworks, our proposed MolPilot achieves SOTA in de novo design with a re- markable PoseBusters passing rate of 95.9% on Cross- Dock, and competitive in local docking with 44.0% RMSD
2Å on PoseBusters.
This motivates our ILB-Optimal Scheduling (VOS), elimi- nating heuristic noise schedule design and directly linking it the theoretical guarantees of the VL.B landscape. Specifi- cally, we combat the combinatorial complexity in schedule design spuce when navigating and optimizing for VLB. To make it tractable to exhaust the possibilities, we demonstrate that the design space of multi-modality noise schedules can be reduced to a two-dimensional plane. Building on this insight, we develop a generalized objective, showing that decoupling timesteps during training allows a single model to implicitly cover this space, generalizing beyond fixed schedules. This framework enables efficient interpolation and extrapolation of schedules at inference time, bypassing costly retraining for new design constraints. By advanc- ing the underexplored area of modality specciicc optimal scheduling, we address key shortcomings in SBDD models such as strained conformations and suboptimal interactions, substantially improving the geometric structure modeling.
2. Issues with Current Probability Path
In this section, we introduce the formulation of Structure- Based Drug Design (SBDD) and demonstrate that the de- fault noise schedule in current generative models leads to a 3D-dominant probability path. Through preliminary experi- ments, we demonstrate that the model trained along this path cannot adequately capture the interdependence between 2D and 3D modalities, which motivates the need for an optimal probability path facilitated by corresponding noise schedule.
2.1. Problem Formulation
Structure-based Drug Design (SBDD) involves model- ing the conditional probability P(x (r)), where XMM (rM, hM, Aw) represents the N-atom molecular geome- try, and xp (rp, hp) represents the protein target. Here, r E Nx3 denotes continuous atom positions, h E NxKa encodes discrete atom types, and A NNN-1)) en- codes discrete bond types, with K₁ and KA denoting the number of atom types and bond types, respectively. In molecular generation, r from the continuous modality is usually described as the 3D geometry, and (h, A) from the discrete modality as the 2D topology of the molecular graph.
Our contributions can be summarized as follows:
We introduce VLB-Optimal Scheduling (VOS), a novel method for systematic noise schedule design in SBDD, achieving fine- fine-graine grained control over multi-n malii-modallty in- terdependence and improved interaction modeling.
2

---

Piloting Structure- -aaedd Drug Design via SSpelit Optimal Schedule
that shows superiority over diffusion counterparts. In this section, introduce the noise schedule and define the cor- responding noising process, paving the way for Variational Lower Bound (VLB) analysis w.r.t. noise schedules.
MSE for Continuous (3D)
Cross Entropy for Discrete (2D)
5.0-
sync
3.25
sync
w/ 20
3.00-
w/ 3D
2.75
0.0
0.5
0.5
Similar to diffusion, BFN involves a noising process @ for data x to obtain a temporal sequence of observed latents Vim := {y1, : Jn and optimizes the VLB:
Figure 3. Validation loss curves default schedule w.r.t. time in generation. Sync: Modalities at the same timestep. w/2: Discrete modality t 1. w/ 3D: Continuous modality L.
logp(x) 2 VLB = B (x) p( Yim)
- DKL (Pl,(((11)))) (((y)m//))))))) (x)()))(p(().)))
(1)
where the noising process q is defined by the variational distribution Q(y1:n x) IT= g(y, x) that depends on the noise schedule over timesteps:
2.2. 3D-driven Probability Path
Fig. 2 suggests that models have yet to accurately capture the molecular geometry, potentially resulting in 3D molecular structures that are incompatible with their 2D molecular graphs (Appendix D). To investigate this, we visualize the modalitt-speciffi validation losses for vanilla BFN (Qu al., 2024) trained with default noise schedule.
[N(xx | qly; x) [Kaa -1),
continuous data discrete one-hot
(2)
where K is the number of classes, ex is Kronecker function, i.e., the projection from a class index x = j to a one-hot vector E R with the j-th value equal to 1. For time step i, the discretized a, obtained from modality-specifie noise schedules, defined as monotonically increasing differen- tiable functions over / E [0, 1J:
As shown in Fig. 3. the continuous modality loss decreases before the discrete modality, verifying that the twisted prob- ability path is driven by 3D modality. However, the problem lies in that the model leverages cleaner 3D input to denoise the 2D topology effectively (Right), but fails the other way around (Left), as performs worse when exposed to less noisy 2D input, suggesting that it unable to benefit from cleaner 2D information. This highlights that the current noise schedule favors a twisted probability path driven by 3D modality, inducing a generative process functioning with lower-noise 3D conformation and higher-noite noise 2D topology.
Bc(1) = 0 2 1,
(3)
where 01 E R+ controls the noise for continuous atomic positions, and 31 E R is the hyperparameter for discrete topology. The noise level a(I) = (1) = d/((t/dl.
BFN differs from diffussions in the generative process, where instead of noisy latents y, the network is informed by a lower-varianee posterior 0 given the observed latents, from the Bayesian flow distribution of Gaussian Dirac d:
Intuitively, a well-balanced probability path should allow the models to leverage cleaner input from either modality to inform the generative process effectively, thereby accurately capturing the molecular geometries required for effective drug design. The problem of current twisted probability path is strengthened by the observation of inaccurate geo- metric structure modeling, where SBDD baselines exhibit abnormal bond lengths and angles (Fig. 14-16), suggest- ing inconsistency between modalities. We hypothesize that the distorted molecular geometry originates from the gap between modalities in the twisted probability path, which inspires our search for optimal scheduling.
[N 7(1 - -)) continuous
Pr x;
(4)
8(0 softmax(y)), discrete
where 7 = I+9.(0) follows the discrete case in Eq. 2.
8.(1)
The network (0, t) is trained denoise x given instead of y, and thus constructs receiver distribution PR(y 0;t) q (0, t)). We show in Appendix B.
that the single-modality VLB objective simplifies to:
Lv = -Fggisi... (x) logp(x Yiss)
We desire the optimal probability path defined by the op- timal noise schedule to benefit from mutually informed modalities, with the potential to improve the generation of chemically valid and spatially accurate molecules. ulti- mately enhancing protein-ligand interaction modeling.
n
+ > (Kyy x) ply: Yis-1))
(5)
and can be optimized by the continuous-time loss:
L (x)
E (~
DRI(gly xx Pn(y (;1))
3. Background
(1- 8'(1]xx X (0,
E
(6)
Our work is built within the framework of Bayesian Flow Network (BFN) (Graves et al., 2023), the SOTA model for 3D molecular generation (Song al., 2024a; Qu et al., 2024)
'The actual noise prediction model is specified as x0 (0,,t,xx xp) with the protein structure input. We omit xp for simplicity.
3

---

Piloting Structure-Based Drug Design via SSpelity-Specifie Optimal Schedule
A. KL Loss Surface
Optimal Path
B.
Default Noise Schedule B
VS.
8 with Optimal VLB
Projected Path
Sampled r. Trajectry
Sampler v, Tagectory
636
0.4
A6
0.R
2.0
0.5
0.5
(ontinunun)
4. (continuous)
(continuous)
2a (discrete)
(dliscrete)
4s (discete)
0.0
0.2
0.4
0.6
0.3
1.0
0.0
0.2
0.11
0.6
0.8
1.0
Figure 4. Our proposed VL VIB-Opttima Scheduling (VOS) that works by estimating the loss landscape and deriving an optimal noise schedule. A. Visualization of the loss surface over the function space. B. Upper: Visualization of the prohahility path of continuous data Xc. Lower: The equivalent time-rescaling functions t. = f(t). = g()).
where we redefine x == VKex for the discrete data for sim- plicity. Based on the monotonic and thus invertible function
(2) In Sec. 4.2, we formalize the entire function space of pos- sible noise schedules, and propose to navigate the space by time rescaling. (3) In Sec. 4.3, we propose a path-invariant generalized loss objective that enables efficient estimation of the VLB landscape over the entire function space of noise schedules. Then, we describe the method to search for the optimal schedule on the landscape that maximizes the VLB,
u = =t),, we perform a change of variables similarly to Kingma et al. (2021) as = 3 3=)( and rewrite the net- work as x (0, u). Therefore, the loss in Eq. is equivalent to the continuous-time loss expressed by noise schedule 3(t) [0, - R
1 ((1)
(x (x)
E
|x Es (0,
4.1. Path-dependent VLB for Joint Noise Schedule
B(O) pr (0):))
In this section, we establish key result for the twisted prob- ability path, where the joint noise schedule B(t) [0. 1] +
In the single-modatie case, this states that the VLB is in- variant to the shape of noise schedule functions except for the endpoints (Kingma et al., 2021). However, we will see in the following section that this invariance no longer holds for multi-modality cases, where the VLB becomes:
[IR y)² induces path-dependent VLB.
A key foundation in single modality is that the continuous- time loss, i.e., VLB, remains invariant to the shape of noise schedule function 3()), except for the endpoints 3(0), 3(1) (Kingma et al., 2021). allowing the design of different sched- ules for efficient training. As natural extension, gener- alize this invariance to multi-modalities given Be Ba:
1 (x) = 2 (21x0((0))))) J. 18(()) (0, (()(1
+ +1((0) (xa - (0, ((00)
(8)
1 (1)
(a(1)
which is simply summed along each modality due to the factorized nature of q. yet this results in path-dependent VLB which is closely related to the design of joint noise
(x)
2 (.)() B2(0) PF (0
2.
E
x x (0. 13.
This equation shows that the generalized loss is invariant to the decoupled schedules as a surface integral over the plane [B)(0), (1)] x [8a(0), (a)].
schedule (()) [0, 1] 22.
4. Methodology
However, this generalized loss i no longer corresponds to the objective of generative modeling, where the VLB should be path integral along line such plane:
We propose VLB-Optimal Scheduling (VOS), principled methodology for optimal probability path in SBDD, by ana- lyzing and identifying the optimal noise schedule for both discrete 2D topologies and continuous 3D atomic positions.
1
- (x)
E B.,Ba (0
|xx
X
(10)
Overview (1) In Sec. 4.1, we theoretically analyze the path-dependent VLB for a given twisted probability path induced by specific noise schedule that function space.
This curve is equivalent to specific coupled noise schedule B E Z. Therefore, we conclude that the VLB varies for different coupling even with the same endpoints (())

---

Piloting Structure-Bascl Drug Design via SSpelity-Specifie Optimal Schedule
Algorithm 1 Deriving Optimal Schedule
4.3. Navigating for the VLB-optimal Schedule
Input: Multi-modallty data x, default noise schedule / =
The first obstacle in identifying the optimal (3r obtain a model (8. ,3) that can be evaluated over all possible B Z. We facilitate the VLB analysis of different joint noise schedules by training (0 t) to minimize the generalized loss F described in Eq. 9 through the change of variables
(Be. Ba), grid resolution M, step N, step scale K.
Output: Optimal schedule (3", generative model x t)-
Train (,,0, 1) to minimize the generalized loss 2 (Eq. 14)
2 (x) = 2 Jo So
E pr(0)x:))
|x
Discretized grid g (te, MxM -
[ M..... ........1. 1/2
Le = = (t), ld = f (t), thereby the objective becomes:
foreach (fc, (2) in do
L (x) = 2 [ E x- (O, (0
L Compute cost C(te, td) + 211x (s)1
,-+ + DD(CC,N,) Optimal schedule (t) + (Bolt2), Bu(t)))
Optimal path {t- = f(t), ty = g(t)),,,'
(14)
for which the proposition holds (proof in Appendix B.4):
(,(()),B((O)) and ((1) = (a(()), and it is no longer agnostic to the intermediate trajectory of ((t), which is essentially different from the generative modeling within single modality.
Proposition 4.3. Suppose we have a model x t) trained by Eq. 14, and let Be(t), Ba(() be any monotonically in- creasing functions in X. Then the line integral in Eq. 10 corresponds to the negative VLB for ((t) (Ba(t), Ba(t)).
The fundamental challenge, then, lies in identifying an opti- mal joint schedule 3 in the design space Z with the best possible VLB integrated along that path:
Then, we present the method to navigate the function space of arbitrary B. Note that from Remark 4.2, we can discretize the function space through discretized combinations of bey la- Therefore, can formulate finding the optimal 3 search problem for the solution with the minimal cumulative cost along the discretized N-step trajectory ftc, tut from [0, 0] to [1, 1].
/ (p)(0))))))
3 = arg min
xx x (0,3)11
(11)
BEZ
4.2. Design Space of Joint Noise Schedules
Depicted in Fig. 4, we estimate the cost matrix across a grid of possible te and ta values by evaluating the KL divergence over batch of samples given the L-trained model ( B3,, and fit smooth loss surface using B-spline interpolation. Definition 4.4 (Cost). The cost a given noise level tor ta as implicit functions over t is defined as:
To formalize the design space of noise schedules, we de- fine the space of monotonically increasing functions X'
[o, 1] + R+, thus the coupled function space 2 is:
(3(t) = (t), Bd(t)) VBe, 3u E X & > 0, > 0}.
(12)
where (1) and Ba(() are monotonic schedules for the con- tinuous and discrete modalities, respectively.
Clte: ta) = f1xxx - (0,
(15)
To explore arbitrary schedule configurations, we introduce time-rescaling functions t. = ftt, t4 = g(t), allowing for arbitrary B expressed in predefined Pe, 3a as in Eq. 3 and varying forms of implicit functions/ f and g by:
We employ dynamic programming to solve the search prob- lem for a minimal cumulative cost J:
(ter ta) = min (J(to - Ec, to (ecifa)
(16)
(t)) = (3.(f(t), Ba(((t)
(13)
where a = (²()) is the accuracy level, and the valid ranges of (Ee, ea) are approximated by the gradient of the smooth surface. It is guaranteed that the method yields schedule with maximal VLB given the corresponding accuracy level:
for which we have the following theorem that guarantees this general form of B expressed in terms of Lez la is sufficient to cover the entire function space Z.
Remark 4.5. (11, 1) corresponds to an unbiased Monte- Carlo estimate of the optimal VLB among all generative models trained by arbitrary , E 2, and backtracking the path {te = f(t), ts = g()) from [1, 1] to [0, 0] yields the optimal schedule 3 (1) = (3.().), Ba(La)).
Theorem 4.1. Suppose we have monotonic function Bm)(): [0, + Rt, and let Bm)(t) be any such mono- tonic function. Then there exists a time-rescaling function ty = f() such that Bm.(t = (tm)) In fact, f() 3- (Sm.at) has the same monotonicity as Pm (t), Bmn(t) Remark 4.2. It follows from the monotonicity of the sched- ule functions that we can obtain arbitrary combination of noise levels simply by setting tey La E [0, 1] separately.
Table empirically validates the derived schedule with bet- ter VLB than default. The overall procedure is summarized in Algorithm 1, with function DP specified in Algorithm 2.
5

---

Piloting Structure-Bascd Drug Design via SSpelit Optimal Schedule
We visualize the derived optimal schedule in Fig. 4B. Intu- itively, this time rescaling corresponds to a two-stage proba- bility path: (1) Shape-driven sketching: First, the generation predominantly focuses on generating the continuous atom positions and largely ignores the discrete topology. Then, when t E [0.3, 0.8). the model starts to fit a possible 2D molecular graph into the rough shape of fixed 3D conforma- tion. (2) Topology-drives docking: at the last stage > 0.8, the generation enters into the docking stage, altering the conformation according to the discrete molecular topolo- gies. By facilitating such principled probability path, the optimal noise schedule effectively harnesses different modal- ities, achieving the best VLB and sample quality.
For interaction modeling, we emphasize RMSD, the root- mean-squared distance between generated and ground truth pose in docking. We also report the self-consistency RMSD (scRMSD) in de novo design calculated between gener- ated and Vina Dock pose as an indicator of binding mode consistency upon redocking. Affinity metrics are calcu- lated by AutoDock Vina (Eberhardt et al., 2021), including binding affinities where Vina Score directly scores the gen- erated molecular pose in the pocket, Vina Min quickly optimizes the pose in-place and scores the minimized pose, and Vina Dock exhaustively searches for optimal pose to obtain the lowest energy.
(3) For molecular properties, we report drug- (QED) and synthetic accessibility (SA), which are desired to fall within reasonable ranges. We addi- tionally report the percentage of fully connected molecules (Connected) to show the performance of generation suc- cesses, together with the average number of atoms of those successfully generated molecules (Size).
5. Experiments
5.1. Experimental Setup
We conduct main experiments within the broader scope of SBDD: (1) de novo design in the in-distributional (ID) and out-of distribbtbibl (OOD) settings, and (2) molecular docking. The same model checkpoint trained with general- ized loss is evaluated throughout both main experiments.
5.2. De novo Design
We report the main results for ID and OOD design in Table I by sampling 100 molecules for each test protein.
Dataset. Following SBDD conventions (Luo et al., 2021), adopt the same split of CrossDock (Francoeur et al., 2020) to train and validate our model, which consists of 100,000 training poses and 100 validation poses. (1) For de novo design, we evaluate on an OOD subset of PoseBusters (Buttenschoen et al., 2024) in addition to the ID Croks test set. We cluster all chains via MMseqs2 (Steinegger & Söding, 2017) and filter out test proteins with any chain > 30% sequence identity w.r.t. CrossDock training sequences, obtaining 180 test proteins. This avoids possible information leakage from the original data splits (detailed discussion in Appendix C.I), and serves as a more reliable held-out test, where the structures of protein-ligand complexes are experimentally determined. (2) For molecular docking, after removing 10 proteins that cannot be processed for non- standard residues from the PoseBusters V2, we test on the remaining 298 protein-ligame complexes ground-truth- based evaluation.
MolPilot achieves the most accurate molecular geome- tries. For intramolecular validity, we achieve the highest passing rate of internal energy, demonstrating our conforma- tion stability and showing robustness in both ID and OOD settings. We additionally report the bond length, angle, and torsion angle distributions in Fig. 14, 15, 16, further underscoring its superiority.
MolPilot generates the best binding poses. For protein- ligand intermolecular validity, we excel at the highest PB- Valid, ensuring reasonable binding poses. For binding affini- ties, our method outperforms all other models with an av- erage Vina Score of -6.88 (ID) and -7.45 kcal/mol (OOD). very close to Vina Min and Dock.
CrossDock benchmark may not be challenging enough. In Table I, we observe that our rate of PB-Valid and internal energy passed even outperforms the CrossDock test set, sug- gesting potential problems with this synthetic dataset (Qu et al., 2024). We dive deeper into the benchmarking statis- tics, and identify possible data leakage in Appendix C.I.
Baselines. We consider the following SBDD baselines: (1) Autoregressive models including AR (Luo et al., 2021), Pocker2Mol (Peng et al., 2022), (2) Diffusion-based Target- Diff (Guan et al., 2022). DecompDiff (Guan et al., 2023), and (3) BFN-based MoICRAFT (Qu et al., 2024). A detailed list of baselines can be referred to in Appendix C.3.
Models underperform on the PoseBusters held-out test. In Table 1, we select competitive baselines on CrossDock, and run the sampling given their released codebases and checkpoints. Sampling efficiency measured in the ratio of successfully generated molecules (Connected) shows that autoregressive models severely degrade on these challenging targets. Among non -a-autrregresivve baselines, DecompDiff exhibits the most prominent degeneration in Vina affinities,
Metrics. We employ the following metrics: (1) For con- formation quality, we report PoseBusters passing rate (PB- Valid), and Strain Energy Passed denotes the ratio of molecules passing PoseBusters internal energy check. (2)
6

---

Piloting Structure-Basce Drug Design via Modality-Speciffe Optimal Schedule
Table 1. Performance CrossDock in an in-distribution (ID) setting and PoseBusters in an out-of-distrihution (OOD) setting, where MolPilot shows robust results. V results cited from Qu et al. (2024). 1: results calculated by us using the official samples. c results calculated by us using the official code. Top-2 results highlighted in bold and underlined, respectively.
PB- Valid
Vina Score (L)
Vina Min (4)
Vina Dock (4)
scRMSD
Energy
Connectedi Avg. (1)
QED
SA
Div
Size
Methods
Avg. (*)
Avg.
Med.
Avg.
Med.
Avg.
Med.
< (t)
Passed (t)
Avg.
Avg.
Avg.
Avg.
CrossDock (ID)@
95.0%
-6.3
-6.46
-6.71
-6.49
-7.45
-7.26
34.0%
98.0%
0.48
0.73
22.8
AR
59.0%
-5.75
-5.64
-6.18
-5.88
-6.75
-6.62
36.5%
84.9%
93.5%
0.51
0.63
0.70
17.7
Pocket2Mol?
72.3%
-5.14
.4.70
-6.42
-6.79
32.0%
97.3%
96.3%
0.57
0.76
17.7
FLAGO
16.0%
45.85
36.52
9.71
-2.43
-4.84
-5.56
0.3%
83.4%
97.1%
0.61
0.63
0.70
16.7
TargetDiftV
50.5%
-5.47
-6.30
-6.64
-6.83
-7.80
-7.91
37.1%
69.8%
90.4%
0.48
0.58
0.72
24.2
DinSBDD
37.6%
-1.44
.4.91
-4.52
-5.84
-7.14
-7.3
18.7%
74.0%
93.2%
0.47
0.58
0.73
24.4
DecompDiff
71.7%
-5.19
-5.27
-6.03
-6.00
-7.03
-7.16
24.25
84.1%
82.9%
0.51
0.66
0.73
21.2
MolCRAFTV
84.6%
-6.55
-6.95
-7.21
-7.14
-7.67
-7.82
46.8%
91.1%
96.7%
0.50
0.67
0.72
22.7
MolPilot (Ours)
95.9%
-6.88
-7.03
-7.23
-7.27
-7.92
-7.92
41.1%
98.5 %
97.9%
0.56
0.74
0.69
22.6
PoseBusters (OOD)
98.9%
-7.06
-7.05
-7.50
-7.41
-7.98
-7.82
59.4%
100%
0.40
0.72
25.7
AR
54.7%
-5.45
-5.17
-5.67
-5.38
-6.18
-5.94
35.3%
77.7%
39.1%
0.50
0.67
0.76
13.6
Pocke12Mol)
63.6%
-5.39
-5.03
-6.64
-6.24
-7.40
-7.03
37.8%
97.4%
67.7%
0.57
0.74
0.73
17.4
TargeiDiffO
32.3%
-6.57
-6.78
-7.16
-7.31
-8.18
-8.20
32.3%
65.2%
81.3%
0.41
0.55
0.67
27.0
DecompDiftt)
40.2%
-3.14
-3.02
4.03
-4.11
-5.06
-5,40
17.0%
80.1%
82.9%
0.47
0.66
0.81
19.3
MolCRAFT
57.8%
-7.29
-7.11
-7.44
-2.22
-7.95
-7.73
46.4%
71.6%
97.0%
0.41
0.65
0.65
23.8
MolPilot (Ours)
79.1%
-7,59
-7.54
-7.74
-7.67
-8,20
-7.99
56.1%
98.1%
97.3%
0.49
0.72
0.67
23.5
Blind docking
Local docking
SBDD docking (N-298)
60
RMSD= 2
58.1
59.7
58.1
Table 2. Tanimoto similarity of interaction profiles between PoseBusters reference and gen- crated PB-Vali molecules.
50
& PB-Valid
RMSD 24
54.5
44.0
%6 RMSD s -2A
40
38.0
13.1
30
Methods
Sim. (f)
21.8
20
15.9
AR
0.221
Pocket2Mol
0.436
10
1.9
3.2
12.7
5.2
1:9
0.8
TurgetDiff
DecompDitf
0.374
0.458
EquiBind
Tankeind
DiffDock
DeepDock
Uni-Mol
Gold
Vina
MolCRAFT
Ours
MoICRAFT
0.498
MolPilot (Ours)
0.551
Figure 5. PoscBusters V2 structural accuracy (RMSD) and validity (PB- Valid) for docking methods (in gray, cited from Abramson al. (2024)) and SBDD methods (in blue, calculated by us).
while TirgetDiff displays a notable drop in PB-Valid and ranks the least, and MoICRAFT shows significantly more strained structures reflected by the tail distribution of Strain Energy in Fig. 8. This suggests that PoseBusters can serve as a challenging OOD test for de novo design.
might be reaching its limit in evaluating the pose generation accuracy. This brings us step further towards ground truth-based evaluation, i.e., molecular docking.
5.3. Molecular Docking
We note that the model trained with C in Eq. 9 is also available for multimarginal generative modeling (Campbell et al., 2024), i.e., it can be used for molecular docking by an induced P(r (h, A), xp).
Our method is robust on the difficult PoseBusters test. MolPilot reliably generates physically valid conformations within the protein pocket with impressive 56.1% % of cases with scRMSD < 2A, comparable to the 59.4% obtained by redocking the co-crystallized ligands, which indicates its ability to capture the interaction patterns and maintain binding pose consistency. We additionally calculate the Tanimoto similarity of interaction profiles between gener- ated and ground-trute truth molecules via ProLIF (Bouysset & Fiorucci, 2021). Table 2 shows that ours best matches the genuine interaction pattern of the co-crystallized structures.
For a deeper understanding our model's ability in cap- turing genuine spatial interaction, we additionally evaluate its docking accuracy on PoseBusters V2 dataset with holo pocket residue struetures specified, and report the RMSD results. While we did not specifically optimize our design w.r.t. molecular docking, Fig. 5 suggests that our method is a competitive SBDD model even compared with the local docking methods. This capability in recovering ground truth binding poses serves as an indicator of capturing true inter- actions, which is essential for designing bioactive molecules
Furthermore, considering that the accuracy of Vina docking tool evaluated on ground-truth complexes in fact places an upper bound, our result of 56. 1 % suggests that Vina
7

---

Piloting Structure- Beaee Drug Design via Modality-Seecffie Optimal Schedule
Vina Score
Vina Min
Vina Dock
PB-Valid (%)
Connected (%)
100
85
95
-6.5
-6.5
66.5
80
90-
-7.0
70
85
-7.5
-7.5
-7.5
65
80
75-
-8.0
-8.0
50
70
Default VLB
Default Generalized
Optimal p
Generalized
Optimal 8 in VLB
Figure 6. Ablation studies regarding loss objectives and noise schedules. Training with VLB (Eq. 10) as line integral requires specific schedule in training and sampling, while with generalized loss (Eg. 9) surface integral only requires ( for test time.
Table 3. Results of applying VOS to diffusion model (TargetDiff)) on CrossDock. Top-1 results highlighted in bold.
PB-Valid;
Vina Score (L)
Vina Min (L)
Vina Dock ())
scRMSD
Energy Passed (1)
Connectedf Avg. (1)
QED
SA
Methods
Avg. (f)
Avg.
Med.
Avg.
Med.
Avg.
Med.
<2 (t)
Avg.
Avg.
TargetDiff' TurgeDDiff L
50.5%
-5.47
-6.30
-6.64
-6.83
-7.80
-7.91
37.1%
69.8%
90.4%
0.48
0.58
53.7%
-6.27
-6.31
-6.82
-6.78
-7.87
-7.90
36.8%
70.3%
89.2%
0.50
0,62
TargeDDiff 8" (VOS)
58.1%
-6.46
-6,53
-7.04
-7.09
-8.04
-8.12
40.2%
73.2%
93.4%
0.49
0,59
with potential biological efficacy. We believe that incorpo- rating molecular docking as a subtask of SBDD also sheds light on the model's ability of interaction modeling.
denoising model effectively allocates more capacity in opti- mizing the VLB, leading to sigmificantly faster convergence in about half the training time with generalized objective é.
5.4. A Ablation Studies
5.5. Generality of VOS
We conduct ablation studies by sampling 10 molecules for each 180 proteins in PoseBusters OOD test, showing the significance of our VOS method in Fig. 6.
To demonstrate VOS's broader applicability, we have inte- grated with the diffussoon- framework Tiffet We train for 140k steps following the default configuration with our generalized objective L, and then derive the optimal schedule that proves to resemble the shape Fig. 4B.
Effect of generalized loss C The generalized training ob- jective boosts the generative model's 's ability in capturing spatial interactions, indicated by the improved Vina affinities especially the Vina Score that directly scores the generated poses. We attribute this gain to the fact that this loss objec- tive forces the model to learn better-balanced probability path for multi-modality molecular data as shown Fig. 11, especially the previously missing ability denoise continu- ous 3D positions with cleaner discrete 2D information the region of nigser-noise 3D and lower-noise 2D data.
We sample 10 molecules per target CrossDock and re- port the results in Table 3, showing that VOS successfully enhances conformation quality for diffusion models as well, with generated poses achieving Vina Scores closely match- ing Vina Min values, indicating near-optimal realistic poses.
6. Related Works
Structure-based Drug Design (SBDD) SBDD genera- tive models focus on the joint generation of discrete molec- ular topology and continuous conformation conditioned on pockets. Recent approaches have centered on capturing the critical protein-ligand interactions for the generation of high-affinity molecules. Autoregressive mod- els (Luo et al., 2021; Peng et al., 2022: Liu et al., 2022) gen- erate molecules nomm-by-toon while preserving geometric equivariance but are computationally expensive. Fragment- based methods (Powers et al., 2022; ZHANG et al., 2023; Lin et al., 2023) improve efficiency by generating motifs
Effect of optimal schedule 3 When training under the generalized objective C and sampling under different noise schedules B, it can be seen that optimal schedule f consis- tently achieves better performance than default noise sched- ule. Moreover, the optimal noise schedule can be further adopted in training under the VLB objective, yielding a slightly improved performance. This is due to the fact that VLB remains invariant as a line integral along the path de- termined by the optimal noise schedule function, yet the
8

---

Piloting Structure-Based Drug Design via SSpelity-S Optimal Schedule
instead of atoms, though they often require post-processing to mitigate error accumulation. Non-autoregrressive mod- els such diffusion-based approaches (Schneuing et al., 2022; Guan et al., 2022; 2023) and Bayesian Flow Networks (BFNs) (Qu et al., 2024) focus on full-atom generation, en- abling scalability and improved controllability. Another line of works incorporates interaction context or property guidance signals such as binding affinity, to enhance the profile of generated ligands (Huang et al., 2024a;b; Qiu et al., 2025). While significant progress has been made, challenges in conformation quality and interaction model- ing remain central to advancing structure-based drug design (Harris et al., 2023).
search Institute of Applied Technologies, Tsinghua Uni- versity (20242001 120). The authors would like to thank Jingjin Gong, Hanlin Wu and Zhilong Zhang for their help- ful comments on this work.
References
Abramson, J., Adler, J., Dunger, J., Evans, R., Green, T., Pritzel, A., Ronneherger, O., Willmore, I.,, Ballard, A. J., Bambrick, J., et al. Accurate structure prediction of biomolecular interactions with alphafold 3. Nature, pp. 1-3, 2024.
Bouysset, C. and Fiorucci, S. Prolif: a library to encode molecular interactions as fingerprints. Journal of chemin- formatics, 13(1):72, 2021.
Molecular Docking In the context of SBDD, molecular docking concerns predicting the 3D conformation given 2D molecular topology and the protein pocket structures. Traditional search-based local-docking approaches include AutoDock Vina (Eberhardt et al., 2021), Gold (Jones al., 1997). Deep learning- based methods are divided into blind docking methods with holo protein structures specified, such as DiffDock (Corso et al., 2023), and local docking methods such as DeepDock (Méndez-Lucio et al., 2021) and Uni- Mol (Zhou et al., 2023). These methods additionally rely on the RDKit-iittialized molecular conformation as input.
Buttenschoen, M., Morris, G. M., and Deane, C. M. Pose- busters: Ai-based docking methods fail to generate physi- cally valid poses or generalise to novel sequences. Chem- ical Science, 15(9):31300 3300-3199 2024.
Campbell, A., Yim, J., Barzilay, R., Rainforth, T., and Jaakkola, T. Generative flows discrete state-spaces: Enabling multimodal flows with applications to protein co-design. In Salakhutdinov, R., Kolter, Z., Heller, K., Weller, A., Oliver, N., Scarlett, J., and Berkenkamp, F. (eds.), Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pp. 5453-5512. PMLR, 21-
7. Conclusion
This work addresses critical challenge in Structure- Based Drug Design (SBDD) by introducing dynamics in the generative process of multi-modalities. Equipped with an optimal noise schedule in terms of Varia- tional Lower Bound (VLB), MolPilot achieves a SOTA Pose- Busters passing rate of 95.9% CrossDock with improved molecular geometry and interaction modeling, offering promising step forward in the field of drug discovery.
27 Jul 2024. URL ht propeedicceeeeesooooo mmlr. press/ / v235/ campbe/ 124a. html.
Corso, G., Stärk, H., Jing, B., Barzilay, R., and Jaakkola, T. Diffdock: Diffussin steps, twists, and turns for molecular docking. International Conference Learning Repre- sentations (ICLR), 2023.
Dieleman, S., Sartran, L., Roshannai, A., Savinov, N., Ganin, Y., Richemond, P. H., Doucet, A., Strudel, R., Dyer, C., Durkan, C., et al. Continuous diffusion for categorical data. arXiv preprint arXiv:2211. 2022.
Impact Statement
This work makes a contribution to the field of computational drug discovery by improving the modeling of protein-ligand interactions, which are crucial for identifying therapeutic compounds. Our approach can be generalized to other tasks involving multimodal generative modeling such as material design. Ultimately, we hope this research could accelerate the discovery of bioactive compounds by enabling more accurate and reliable in silico design of grotein-binding ligands, leading to more effective therapeutic interventions.
Eberhardt, J., Santos-Marting D., Tillack, A. F., and Forli, S. Autodock vina 1 1.2. O: New docking methods, expanded force field, and python bindings. Journal of chemical information and modeling, 1198),88,, 2021.
Francoeur, P. G., Masuda, T., Sunseri, J., Jia, A., Iovanisci, R. B., Snyder, I., and Koes, D. R. Thrre--imennionnl convolutional neural networks and a cross-docked data set for structure- based drug design. Journal of chemical information and modeling, 60(9):4200-47) 4215, 2020.
Acknowledgments
This work is supported by the Natural Science Founda- tion of China (Grant No. 62376133) and sponsored by Beijing Nova Program (20240484682) and the Wuxi Re-
Graves, A., Srivastava, R. K., Atkinson, T. and Gomez, F. Bayesian flow networks. arXiv preprint arXiv.2308.07037, 2023.

---

Piloting Structure-Bass Drug Design via SSpecit Optimal Schedule
Guan, J., Qian, W. W., Peng, X., Su, Y., Peng, J., and Ma, 3d equivariant diffusion for target-aware molecule genera- tion and affinity prediction. In The Eleventh International Conference on Learning Representations, 2022.
Information Processing Systems, volume 36, pp. 34603- 34626. Curran Associates, Inc., 2023. URL https: / // //arxiv..
Liu, M., Luo, Y., Uchino, K., Maruhashi, K., and Ji, S. Generating 3d molecules for target protein binding. In International Conference on Machine Learning, 2022.
Guan, J., Zhou, X., Yang. Y., Bao, Y., Peng, J., Ma, J., Liu, Q.. Wang, L., and Gu, Q. DecompDiff: Diffu- sion models with decomposed priors for structure-based drug design. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.), Pro- ceedings of the 40th International Conference on Ma- chine Learning, volume 202 Proceedings of Machine Learning Research, pp. 11827-1 11 1846. PMLR, 23-29 Jul 2023. URI.httss: ptt.//eooeeeeieooooocooocooee..c ngs ml mol...rrsss. press/ #202/guan23a .tmm..
Luo, S., Guan, J., Ma, J., and Peng, J. A 3D Gen- erative Model for Structure- Based Drug Design. Ad- vances in Neural Information Processing Systems, 34: 6229-6239 2021. URL 2203.1044
Méndez-Lucio, O., Ahmad, M., del Rio-Chanona, E. A.,
and Wegner, J. K. A geometric deep learning approach to predict binding conformations of bioactive molecules. Nature Machine Intelligence, ((1)): 1039, 2021.
Harris, C., Didi, K., Jamasb, A. R., Joshi, C. K., Mathis, S. V. Lio, P., and Blundell, T. Benchmarking generated poses: How rational is structure- drug design with generative models? arXiv preprint 2rXiv:2308.07411 2023.
Peng, X., Luo, S., Guan, J., Xie, Q., Peng, J., and Ma, J. Pocket2Mol: Efficient molecular sampling based on 3D protein pockets. In Chaudhuri, K., Jegelka, S., Song, I.., Szepesvari, C., Niu, G., and Sabato, (eds.), Pro- ceedings of the 39th International Conference on Ma- chine Learning, volume 162 of Proceedings of Machine Learning Research, pp. 17644-17655 PMLR, 17-23 Jul 2022. htthttps: http::::///pcceeeeeeeeee ml www..rrss.p/r/p..mmmmmmmmmmmmm v1 62 http/ppeg22b.... html.
Huang. Z., Yang, L., Zhou, X., Qin, C., Yu, Y., Zheng, X., Zhou, Z., Zhang, W., Wang, Y., and Yang, W. Interaction-based retrieval-augment -angentdd diffusion models for protein-specific 3d molecule generation. In Forty- first International Conference on Machine Learning, 2024a. URL https: http//peerevieeeeeeeeeeeeeeeeeee www.rreeiieeee..e....ee wet./ee///ooo.e forum? id=ee
Peng, X., Guan, J., Liu, Q., and Ma, J. Moldiff: Addressing the atom-bond inconsistency problem in 3d molecule diffusion generation. In International Conference on Machine Learning, pp. 27611-27622 PMLR, 2023.
Huang, Z., Yang, L., Zhou, X., Zhang. Z., Zhang, W., Zheng, X., Chen, J., Wang, Y., Bin, C., and Yang, W. Protein- ligand interaction prior for binding-aware 3d molecule diffusion models. In The Twelfth International Confer- ence on Learning Representations, 2024b.
Powers, A. S., Yu. H. H. Suriana, A., and Dror, O. Fragment-based ligand generation guided by geometrie deep learning protein-ligand structures. In ICLR2022 Machine Learning for Drug Discovery, 2022. URL https:/ /ttpnr/////eeeewwww net forum? 10=1929ccr-HHU
Isert, C., Atz, K., and Schneider, G. Structure-based drug design with geometric deep learning. Cur- rent Opinion in Structural Biology, 79:102548, April 2023. ISSN 0959440X.
doi: 10. s0.1016/j.sbi.2033. wwetrieeweppi//0054443332
102548. URL Ltt els.e..e.r.guubeeeeeeee.....
Qiu, K., Song. Y., Yu, J., Ma, H., Cao, Z., Zhang, Z., Wu, Y., Zheng, M., Zhou, H., and Ma, W.-Y. Empower truuttre-based molecule optimization with gradient guid- ance. arXiv e-prints. pp. MxXiv22411, 2025.
Jones, G., Willett, P., Glen, R. C., Leach, A. R., and Taylor, R. Development and validation of a genetic algorithm for flexible docking. Journal of molecular biology, 267(3): 727-748, 1997.
Qu, Y., Qiu, K., Song. Y., Gong, J., Han, J., Zheng, M., Zhou, H., and Ma, W.- MoICRAFT: Structure-basee drug design in continuous parameter space. In Forty- first International Conference on Machine Leurning, 2024. URL https:/ /pppneeveewwwweee//e/eeeeeeeeewe net forum? 1 d==aAQU5ENUU.
Kingma, D., Salimans, T., Poole, B., and Ho, J. Varia- tional diffusion models. Advances in neural information processing systems, 34:21696-21777, 2021.
Lin, H., Huang. Y., Zhang, O., Liu, Y., Wu, I.., Li, S., Chen, Z., and Li, S. Z. Functional-groue grau------d diffu- sion for pocket-specific molecule generation and elabo- ration. In Oh, A., Naumann, T., Globerson, A., Saenko, K., Hardt, M., and Levine, S. (eds.), Advances in Neural
Schneuing, A., Du, Y., Harris, C., Jamasb, A., Igashov, I., Du, W., Blundell, T., Lió, P., Gomes, C., Welling, M., Bronstein, M., and Correia, B. Structure-based
10

---

Piloting Structure-Based Drug Design via Modality-Seciffte Optimal Schedule
Drug Design with Equivariant Diffusion Models, Oc- tober 2022. URL http::ttt///t....tt http///axvvvv.. wttpiv..rrg/b//2///// 13695. arXiv:2210.13695 [es, q-bio].
Song. Y., Gong, J., Qu Y., Zhou, H., Zheng, M., Liu, J., and Ma, W.-Y. Unified generative modeling of 3d molecules via bayesian flow networks. arXiv preprint MrXiv:2403.15444 2024a.
Song, Y., Gong, J., Xu, M., Cao, 7.,, Lan, Y., Ermon, S., Zhou, H., and Ma, W.-Y. Equivariant flow matching with hybrid probability transport for 3d molecule generation. Advances in Neural Information Processing Systems, 36, 2024b.
Steinegger, M. and Söding, J. Mmseqs2 enables sensi- tive protein sequence searching for the analysis of mas- sive data sets. Nature biotechnology, 35(11):1026-1028 2017.
Vignac, C., Osman, N., Toni, L., and Frossard, P. Midi: Mixed graph and 3d denoising diffusion for molecule generation, 2023. URL 2302. 09048.
abs/
ZHANG, Z., Zheng, S., Min, Y., and Liu, Q. Molecule gen- eration for target protein binding with structural motifs. In International Conference on Learning Representations, 2023, URL https: / http//pppeeeeeeeeeeeeeeeeeeeeeea htt.riiwwweeeeeeeeeeeeeeeeeeeeee forum?
Zhou, G.. Gao, Z., Ding, Q., Zheng, H., Xu, H., Wei, Z., Zhang, I.., and Ke, G. Uni-mol: A universal 3d molecu- lar representation learning framework. In The Eleventh International Conference on Learning Representations, 2023. URL https:/ open httpp::/opeereeieeeeeeeee ew n forum? 6d=6K2RM6VVKKu.

---

Piloting Structure-Base Drug Design via Sppelit Optimal Schedule
A. Details about the Proposed Method
A.1. Dynamic Programming for Optimal Schedules
Algorithm 2 Dynamic Programming for Optimal Path
Input: Cost matrix C' E R. Xx2, step budget I., step scale K, default noise schedule (t)
Output: Optimal path, minimal cumulative cost J
Initialization:
Fit a smooth loss surface Ć using B-spline interpolation.
Jc: M,: M,: N + 1] < o0, J[0 0, +
prev[: M,: M,: N + 1,: 2]
+-1
active states {(0,00 0, 0)}
size + 1+ where ||Vč1| =
aC + (g))
Dynamic Programming:
for / +- to L - 1 do
states +
foreach (x,, y, E active states do foreach Cy) E valid stlidsstepsssizelz.. vl) do
Te. V1 5 x + Co. + Ey
a + (cle))-fole) (ell))
cost + -|z, y. + a Cla,, yel cost then
if J Jar, V1,1 +
J[z, Ne. + + cost
prev Fe, V1,1+ + 1] t (x,y) y) states s- states U (1te yt, + 1))
active staates states
y. path + M - 1, MM - 1,0
+ arg min [ vl
while 12 0 do path.append(<r,)) y)) (z, - prpveeee
+- 1 1-1 return path, min J[M 1, MM 1,
Convergence guarantee As the grid resolution N - oo, the discrete solution converges to the continuous VLB-optimml path (1). Since the VLB for a joint schedule (()) is line integral over the model's loss field in the 2D) noise space (Appendix B.4), and training with the generalized loss _ ensures the model's predictions are accurate everywhere in this space, enabling the evaluation of VLBs for arbitrary paths (()) (Appendix B.5), the dynamic programming solution recovers the optimal path asymptotically, maximizing the VLB over the function space Z.
Time complexity The computational complexity is O(NM2K), which does not add too much computational overhead and can be solved within a few minutes on single CPU. Once solved, the optimal schedule can be adopted for test time, enabling the VLB-optimal generative process, where the multi-modalities effectively inform each other at the optimal accuracy level. However, in order to estimate the cost matrix over M x M discretized grid, the same number of evaluations of the generative model X₂ is required, which scales quadratically to the discretization M and involves considerable GPU computation. Empirically, we found that the loss surface is smooth enough to allow for effective interpolation, therefore M = 20 would yield an accurate enough estimate, taking less than 20 minutes.
Table 4. Validation loss the sum of scaled KI. divergence sample time for different scheduling.
Schedule
Default
Optimal
Learned
Validation Loss
5.51
4.23
5.39
12

---

Piloting Structure-Based Drug Design via Modality-Speciffe Optimal Schedule
Validation Loss
Train with static ( VLB Train with dynamic optimized by VLB
Train with default B in generalized loss Train with default 8 in VLB
10
6
5
Training Step
20k
40k
60k
80k
100k
120k
Figure 7. Validation losses for different configurations over the course of training.
A.2. Why Learning for Optimal Schedule Might Fail
Given the obtained optimal schedule (t), we can train the generative model with invariant optimal VLB (Eq. 10) more effectively, allocating the model capacity towards learning to denoise only along the selected path. We found this results in earlier convergence, taking around half the previous training time for generalized objective (Eq. 9) with similar performance.
We would like to take one step further to see if we can directly optimize the joint noise schedule ((t) in generative training with the negative VLB objective in Eq. 10, i.e., optimizing the generative model and the noise schedule simultaneously through the same loss objective.
We parametrize the learnable time-rescaling function by
((1) = + (1 sigmoid((.m)))))))
(17)
where hys (1) is monotonic neural network which we choose to be three-layered MLP with Softplus activation before the final output. This rescaling function is strictly monotonically increasing with endpoint constraints, i.e., f(0) = 0, f(1) = 1.
However, we found that the learned noise schedule does not change much from its initialized shape, and the corresponding VLB proves to be suboptimal in Table 4. We hypothesize that there are several challenges learning for the optimal schedule during training: (1) Large combinatorial space with limited exposure to possible noise schedules in the process of active learning. Unlike single-modality models, where the range of noise levels is easily explored within the modality (Dieleman et al., 2022), multi-modality generative models can suffer from exposure to only subset of the possible noise combinations. This incomplete exploration makes it challenging to optimize directly for the best schedule, as large regions of the design space remain unvisited. It impacts the VLB estimation since the model X is still learning to adapt to the scaled noisy inputs during training, and is not invariant w.r.t. noisy input at different scales. (2) Optimization difficulties. Directing taking the gradient w.r.t. ley la introduces numerical instability, particularly when 3. involves exponentiation, which can grow or shrink very rapidly for certain values of tz. leading to vanishing or exploding gradients. Moreover, the VLB optimization is sensitive to the trajectory defined by the noise schedule, and the optimization landscape of may have many local minima, especially due to the coupling of noise schedules, which can make it difficult to find the global optimum.
Addressing these might require more sophisticated methods such as bilevel optimization, which we leave for future work.
A.3. Sensitivity to Optimal Path Choice
Effect of Interpolation We conduct additional experiments interpolating with a coefficient e between identity time function (c = 0) and our derived optimal time-rescaling functions t,2 (c 1), by setting time functions t = c 4 + (1 c) Our findings show a clear trend of improving performance as we move toward the optimal schedule.
Table 5 shows generally monotonic improvement in binding pose quality on both datasets as we move from identity to optimal scheduling functions, with the most significant gains occurring when e E [0.75, 1]. Additionally, the case with
13

---

Piloting Structure-Based Drug Design via SSpelity-Specifie Optimal Schedule
Table Results of interpolating between identity time function (c and optimal time-rescaling functions (c protein target 10 molecules are sampled.
where for each
Vina Score (L)
Vina Min ())
QED
SA
Connected Avg. (t)
PB-Valid
Avg.
Med.
Avg.
Med.
Avg.
Avg.
Avg. (1)
CrossDack (ID)
-6.36
-6.46
-6.71
-6.49
0.48
0.73
0.95
0 (Identity)
-6.63
-6.44
-7.06
-7.06
0.55
0.76
0.95
0.95
0.25
-6.66
-6.88
-7.02
-7.01
0.55
0.76
0..9
0.96
0.5
-6.70
-6.82
-7.04
-6.94
0.55
0.76
0.95
0.97
0.75
-6.87
-6.97
-7.21
-7.11
0,55
0.76
0.96
0.96
(Optimal)
-6.92
-7.02
-7.23
-7.18
0.55
0.75
0.96
0.95
PoseBusters (OOD)
-7.06
-7.05
-7.50
-7.41
0.40
0.72
0.99
0 (Idemtity)
-7.35
-7.38
-7.64
-7.48
0.48
0.73
0.94
0.79
0.25
-7.42
-7.37
-7.66
-7.50
0.48
0.73
0.94
0.79
0.5
-7.20
-7.36
-7.56
-7.51
0.48
0.73
0.95
0.78
0.75
-7.44
-7.43
-7.75
-7.54
0.48
0.73
0.95
0.79
(Optimal)
-7.52
-7.52
-7.79
-7.65
0.49
0.73
0.95
0.80
= O also demonstrates the performance boost brought by our generalized training objective compared with the default objective (not shown in the table, please refer to Fig. 6).
Effeet of Accuracy Level a as Scaling Factor The desired scaling factor ought to be the derivative of joint schedule function B, which is not known before solving the search problem. To determine the appropriate accuracy level in practice we have empirically found that we can actually approximate the accuracy by taking the derivative directly from the default schedule function ((t) (t)( (t)), instead of taking the slope ((t)- and this applies both to the experiments with BFN and diffusion models like TafgetDiff. Note that by setting the accuracy level as such, the optimization objective no longer corresponds to the exact likelihood, but a rescaled sum KL divergence terms that put more weight on the continuous variable. We hypothesize that this suggests gap between likelihood estimation and sample quality, where the latter is more influenced by the 3D structure part.
A.4. Model Architecture
We employ the equivariant Graph Transformer architecture similar to Guan (2023), with few modifications attention score calculations designed save memory. Denote node positions as gg one-hot atom types h E Nx and one-hot bond types A E R NN where K₁, and K are the numbers of atom and bond types, respectively. We brief the model architecture below.
Heterogeneous Message Passing for Node Update The heterogeneous update for node representation consists of two parts: (1) K-nearest neighbors graph Gx built for interactions within the protein-ligand complex, and (2) fully connected ligand graph 9r. within ligand atoms for ligand bond-based interactions.
Abko
Σ (EN
Øb (XK X. concat(dyy,h,,) (edd))), Koy
AbL
Σ JENL(()
Φhe. (X X h
where the input features for complex graph X concat(h,, hj, odge X concat(h;; hods), d.j = ||r; ry the pairwise distance, and elge is the one-hot encoding for edge types between protein- atoms (protein-protein protein- ligand, brgand-proteni or ligand-ligand). For ligand graph, the input features is defined as X L = concat(h, b where havnn denotes flattened bond embeddings.
leu = concat(h, (h, h,, hbon),
The node-level representation updated by aggregating the heterogeneous messages:
h h; + FFN (AhKa + Ahr,))
14

---

Piloting Prtucture-Bascee Drug Design via -Sdelity-Specifie Optimal Schedule
Intra-Ligand Message Passing for Bond Update
The bond representation is updated by directional message passing:
bond
Σ
Φ
Ox Joy X (dgi)).
MENZG)
where the input features defined X con hg, mk3). X concat(hk)) (hhond con aijk; hgs hy),
and X F co bond h.). a is the Gaussian smearing function applied to the pairwise distance, and a is the angular encoding for the bond triplets.
Heterogeneous Message Passing for Position Updates
Similarly, the positions are updated from heterogeneous messages:
Ark,
Σ (:) o drx (X kut X q concat(dij, hi )) ArL,
Σ
- (o-rr)orz (X kuz X h hond)
jENx(1)
(EN2(()
given the updated node features h and bond features h b The final position is updated by
r, f- r, + (Ark, + ArL.)) m
where I mot is the boolean mask for ligand atoms, the protein atoms need to be fixed ensure equivariance.
Attention Mechanism The function @ is implemented as a multi-headed attention mechanism. The query, key, value are obtained by passing imput features through linear layer without bias to save memory:
K
ke Wk.
Xiv
Q = Q=x, Wg
The attention scores are computed using:
(k Waj)
Oij
Wij tanh(e;; W Wedge);
vdn
where dy is the head dimension, Wij is the edge-specific weights modulating the contribution of distant neighbors, and O denotes Hadamard product.
A.5. Implementation Details
Training and Inference We use Adam optimizer with learning rate 5c-4, batch size of 16, and fit the model with 3.1 million parameters on one NVIDIA 80GB A100 GPU. We set 31 = 1.5 for discrete atom types and bond types, 01 = 0.05 for atom coordinates. The training converges in 200K steps (around 24 hours). For inference, we use the exponential moving average of the weights from training that updated at every optimization step with a decay factor of 0.999. We run inference with 100 sampling steps with the same variance reduction sampling strategy Qu al. (2024).
Hyperparameters for Network We set the network to be kNN graphs with k 32, N layers with d = 128 hidden dimension, 16-headed attention, and dropout rate 0.
Featurization We describe our atom-level and edge-level featurization for the protein-ligand complexes. At the atom-level, each protein atom is represented with one-hot element indicator (H, C, N. O, S. Se), a 20-dimensional one-hot amino acid type indicator, and 1-dimensional backbone flag. Ligand atoms are featurized with one-hot element indicator (C, N, O, F, P. S. Cl) coupled with aromaticity flag. For edge-level featurzation in the heterogencous protein-ligand graph, edge types are encoded as d-dimensional one-hot vector indicating whether the edge is between ligand atoms, protein atoms, higand-protein atoms, or protein-ligand atoms. For the ligand graph, bonds are represented with 4-dimensional one-hot bond type vector (non-bond, single, double, triple), where aromatic bonds are kekulized using RDKit.
B. Proof
B.1. Derivation of Eq. 6 as single-modality VLB (Eq.5) 5)
We begin with the negative Variational Lower Bound (VLB) in Eq. 5:
LVLB(X) -Eg(y bo/ logp(x Y1.)) +
Σ DKu D (glys x) pla; (ys-=)))
15

---

Piloting Structure-Bass Beaeeurr--Baeddde Drug Design via Modality-Seciffee Optimal Schedule
In further analysis, we foeus on the treatment of the second term on the right-hand side. For convenience in the subsequent derivation, we define:
7
(x)
Buly (x)
Σ
DKulaly: x) p(y [x)=1,
Bl 1x) [ x) ply: y1s-1;
(18)
Next, specifyy the conditional distribution ply; y1=-1, we utilize ply: M1S-1, the following decomposition:
p(yn)) ( and incorporate
ply1s x) II Epr pip.te---ii,s--pppp 10--1,
(19)
follows that:
ply: Y)--1, por (i (pnlr 0
(20)
Substituting this result into Eq. 18, we obtain:
(x) = n. Be..
DRI(a(y: x) |l PR(Y; 1,,,, (044-1))
(21)
Given our parametrization with neural network, we can further replace pn(y; 0, with g (0,
L (x) (.m. DKa(g(y; x) || < (0,
(22)
Following Graves et al. (2023), different noising processes are specified for for different modalities. For continuous data, q'y x) = -xxx a I). For discrete data, g(y x) = NN(a(eex 1), a)I), where we define x == VKex. Denoting
(x) lim Ld (x)))) = 3(t) - B(t ( c), we have
lim 1 c-0 6
(x)
(L 1) DKI,(a x) 1 (0 ())
lim (n) ctt,c) ||x- (0.tt))
--0
2e
(... (t)lx X (0,
(23)
which establishes the fact that Eq. 6 corresponds to the single-modality VLB.
B.2. Equivalence between Eq. 6 and Eq.
Rewrite the expectation in the above equation into an integral form and make the substitution u = =B(t). This substitution can be transformed as follows: 3(t) + du 3 (t)dt + dt
du
and we directly rewrite (0, as (0,0, u).
1
(x)
2 1 (p
B((1)lxx (2( dt
1 ((1)
du
8(8-1(1)) (0, u)l? 3'(B-(w)) d0
2 8(0)
(0)
1
((1)
11 (0, u) || 2 d0
2
B(0)
((
1
((11
(0
1|x
(0)
(24)
2
B(0)
which gives the form of Eq. 7.
16

---

Piloting Structure-Based Drug Design via Sppelit Optimal Schedule
B.3. Equivalence between Eq. and 14 Recall that the generalized loss over B E Z is:
- (x) = 1
E |xx
2 Se,Be (0(3):
and the function space reparameterized into product of uniform distributions over te, ta yielding
(x) = 2 [ (0
x
xx(0,)) (txt dt,
this is equivalent to implicitly sampling f(t), covverinn The joint distribution over (a)
covering all possible time-rescaling functions as stated Remark 4.2.
Ba) is thus
= -().))))=)()000001 (0,
(25)
where 3.(1) = (t) and Ba(t) = Ba(g(t)).
Therefore, we see that the time-rescaling functions f(t), gt) act latent variables defining the coupling of (e, 3r. Integrating over (fe, marginalizes over all possible B3 ensuring the model is trained to denoise all possible combinations of noise levels. By demonstrating this equivalence, we reframe the invariant objective described in Eq. 9 as Eq. 14.
B.4. Derivation of Eq. 10 multi-modality VLB
To rigorously prove that the VLB corresponds to line integral along 1D) submanifold (path) in the 2D) noise schedule space, recall that the joint noise schedule be parameterized path in the joint function space Z:
(t) (at))), e [0, 1],
where Ba(t),Ba(t) are monotonically increasing functions with fixed endpoints:
Be(0) = Be,,. (e(1) = Se,1, Ba(0)= = Ba,,, Ba(1) = 3a,1-
Following Appendix B. and the factorized nature of q. the VLB for a joint schedule (3t) and multi modality x is:
(vLB(x)= = - 2 (pp(0(x:)) (x:0)) L'' [8(()) (0.
- (x,,),))))
where X4 (So,e, x is the denoising model output for continuous (c) and discrete (d) modalities.
The VLB can be interpreted line integral over the trajectory B(1) 2. Define the vector field F(3e, Ba) as:
Xx= x (0,g)]1))1 |Ixa (0,
F F(Be, =
The VLB then becomes:
1 IE / F(3e, Ba) dB (a(()
(vLB(x)
where d3 (t)dt, (t) is the differential vector along the path B(t), and denotes inner product. By redefining (e (3 == F(3,, we obtain Eq. 10.
B.5. Derivation of Proposition 4.3
Proof. Since the model (0, t) is trained by the generalized loss _ which is equivalent to integrating the scalar field |x 122 over the entire function space Z, it is exposed to all pairs (Be, Ba) Z within the region (O), (.)) x [B), (1)] with 3c, Ba defined in Eq. 3. Therefore, the model achieves minimal prediction error everywhere in the joint noise space, allowing accurate computation for any specific path (t).
17

---

Piloting Structure- Based Drug Design via -odelity-Specffte Optimal Schedule
Train:
4BW3 A (gray)
Train: 5APO A (gray)
Test:
5MGL_A (purple)
Test:
1PHK_ (purple)
Figure Structural alignment randomly selected cases where training sequence and test sequence exhibit 30% identity, showing nearly identical protein structures well as ligand hinding pockets.
Following Appendix B.4, the line integral (x) corresponds to the VLB for a given path (t). Since the integral is equivalent to restricting the generalized loss ~ (x) to the ID manifold defined by 3()), and the model's predictions are accurate along any submanifold within the space, evaluating [ (x) for a specific (3t) uses the model's pre-optimized predictions each point the path, thereby yielding valid VLB estimate. This establishes that the generalized training enables estimating VLBs for arbitrary joint schedules B E Z.
C. Details Benchmarking
C.1. Problems with CrossDock
Table Test protein chains that exhibit sequenee identity higher than 30% any chain of training sequences. For those with more than 10 overlapping training chains, we randomly select 10 items to display, and report the total number of unique overlapping chains. Each item characterized by its PDB ID chain ID.
Test PDB
#Overlap
Training PDB(s) with Sequence Identity >30%
IPHK A
104
2WQO A, 3TKU B. 3WYY A. 3WYX A. 4C4E A. 4BHZ A. 2XKD A. 4ZEG A. 4CV8 A. SEI8 A
JUMD_B
INI4_B
5NGZLA
5
SP6Y ST66WA LA, SP6DLA, SFGELA, SP6ULA
SLIV_A
5
4DNILA, IZ8QLA, IZ80JA, IEGY IJIPA
IFMCB
8
5L7T_A, SICS,C, SICM,A, 51.7_A, SHS6.A, SEN4.A, S186.A
4PX7_A
4734_A, SWIV_A, 4736_A
3HY9 H
1R55 A
4AZF A
SJIV B. SX8LB 289J A, IZS7 S.G.C
3PDH
4
IYC2.D, IYC2 187G.B, 187G.C
SMGL A
127
SUVW A. SDOC A, 407B A. SKUS A. 5DX4 A. 4A9F A. SCS8 A. SD3L A, 407C A. SEOR A
4XLLB
43
4OTF_A. 4Y93.A, AY95 SJRSLA, 3GENLA. 5P9HLA, 4NWMAA 4ZLY JPTY 4RFY
4AUA_A
75
2R3Q_A, 3EZR_A, 3TN8 A, SPGK,A, 2R3MLA, SIDN 2YIY SHBLA, 4CFULA, 3DOGLA
S10B_A
100
5OPB A. SOPR.A, 2YM4 2C3K_A, 3169 4FTTLA, A,,O 3TKHLA, 2BRG.A, 2YDK_A
4FIMAA
SLPV A. SLPW.A, SLPY 40H4 4QSLA, SLPB_A
2CY0LA
10
3PHLA, 4FQ8_B, 3PHHLA, 3PGLCC, 3DON_A, 4ROS.A, 3PHGLA, 3D00LA, 4FR5_B, 4FPX.A
The CrossDock dataset, first processed by Luo et al. (2021), is commonly used benchmark in the SBDD field. Luo et al. (2021) claimed to have employed the MMseqs2 method to filter the test set with 30% sequence similarity threshold. However, upon rigorous examination, we found that the test set still contains proteins with sequence similarity higher than 30%.
Table 6 summarizes the calculated statistics for these similar proteins, where we only show the randomly sampled 10 PDB
18

---

Piloting Structure- Baaecu Drug Design via Modality-Seecffie Optimal Schedule
IDs when there are more than 10 sequences that exceed 30% sequence identity. Fig. 8 illustrates structural alignments between some random proteins in the test and training sets that exhibit high sequence similarity. The near-identical overlap suggests that the dataset split may not be challenging enough. To address these potential problems with CrossDock evaluation, we propose using PoseBusters (Buttenschoen et al., 2024) as a held-out test set.
C.2. Curation of the held-out PoseBusters test set
The PoseBusters Benchmark (Buttenschoen et al., 2024) set exclusively contains complexes released since 2021. In contrast, the CrossDock dataset comprises data collected from the PDB before 2020. This makes PoseBusters challenging time-split dataset, ideal for evaluating the generalizability of models to real-world, unseen scenarios. We further apply MMseqs clustering (Steinegger & Söding, 2017), and filter any test protein with any chain that has > 30% sequence identity threshold to CrossDock training sequences. This process leaves us with 180 data points, making most of the baselines directly available for the curated held-out test. Additionally, the protein- ligand complex structures CrossDock are mainly generated using docking software, which inevitably introduces noisy poses (Francoeur et al., 2020). In contrast, PoseBuster contains real-world crystal structures resolved from wet-lab experiments, making it suitable for evaluating the SBDD model's ability to capture genuine molecular interactions accurately.
C.3. Experimental Setup
Baselines We provide brief overview of all SBDD baselines follows:
Autoregressive methods: AR (Luo et al., 2021) utilizes MCMC sampling to reconstruct molecules atom-by-atom based voxel-wise density predictions. Pocket2Mol (Peng et al., 2022) generates molecules atom-by-atom with bonds using an E(3)-eqqivariant network, predicting frontier atoms improve sampling efficiency. FLAG (ZHANG et al., 2023) is a fragment-based model that assembles molecular fragments by predicting their positions and torsion angles.
Diffusion: DiffSBDD (Schncuing et al., 2022) constructs an E())-vquivariant continuous diffusion model for full-atom generation, applying noise to both atom types and coordinates, while TargetDiff (Guan et al., 2022) adopts hybrid diffusion process separately handle continuous coordinates and discrete atom types. DecompDiff (Guan al., 2023) incorporates chemical priors by decomposing molecules into scaffolds and contact arms.
BFN: MoICRAFT (Qu et al., 2024) uses Bayesian Flow Network (BFN) with advanced variance reduction sampling technique, demonstrating notable improvements over diffusion- based models in molecular design.
Metrics We evaluate the generated molecules using the following commonly adopted metrics:
Affinity Metrics are calculated using AutoDock Vina (Eberhardt et al., 2021), these include Vina Score as the raw binding energy of molecular pose in the pocket, Vina Min as the binding energy after local energy minimization of the molecular pose, and Vina Dock as the lowest binding energy obtained after an extended search for the optimal pose.
Molecular Properties including QED for drug-likeness, and SA (synthetic accessibility score) are calculated using RDKit. They are desired to fall within reasonable ranges.
Connected Ratio is the percentage of fully connected molecules.
Diversity assesses the variety of generated molecules for each binding site by averaging Tanimoto similarity over Morgan fingerprints across all test proteins, following Luo et al. (2021). It is worth noting that this is not necessarily the higher the better, as the desirable bioactive compounds against a protein target often cluster in the molecular space.
Key Interactions such as the formation of critical non-covalent interactions with protein binding sites, calculated using ProLIF (Bouysset & Fiorucci, 2021).
Strain Energy reflects the internal energy of generated poses, indicating pose quality by Harris et al. (2023).
RMSD reports the percentage of molecules where the RMSD between generated poses and ground-truie or Vina redocked poses is within 2 Å, indicating consistent binding modes. For better differentiation, we refer to the latter as self-consistency RMSD (scRMSD).
19

---

Piloting Structure- Based Drug Design via SppecittySSeeiiiie Optimal Schedule
Percentage of molecules passing the check
0.9
0.8
0.7
AR
0.6
Packet2Mol
FLAG
0.5
TargetDiff
DiffSBDD
PB-Valid Mol
PB-Valid Dock
DecompDiff
0.4
MoICRAFT
MolPllat (Ours)
0.3
Mol pred loaded
Mol cond loaded
Sanitization
Inchi convertible
All atoms connected
Bond lengths
Internal steric clash
Bond angles
Aromatic ring flatness
Double bond flatness
Protein-ligand maximum distance
Minimum distance to organic cotactors Minimum distance inorganic cofactors
Internal energy
Minimum distance to protein
Minimum distance to waters
Volume overlap with organic cofactors
Volume overiap with protein
Volume overlap with inorganic cofactors
Volume overlap with waters
Figure 9. Percentage of generated molecules that have passed the PoseBusters validity checks ID CrossDock test set. PB- Volid Mol: intramolecular validity. PB- Valid Dock: intermolecular validity. Reported PB- Ba-Valid: PB- Valid Mol & PB- Baldd Dock.
D. More Evaluation Results
D.1. Dynamics of Better-balanced Modalities
We visualize the modality-specific validation loss curves for the model trained under generalized loss (Eq. 9) in Fig. 11, which shows that the denoising model effectively learns to mutually benefit from cleaner information either modality.
D.2. In-Distribution De novo Design
We report the results on CrossDock (Francoeur et al., 2020) an in-distribution (ID) setting. We sample 100 molecules for each of the 100 test proteins, and the results in Table show that our model consistently performs exceptionally across most metrics leading in PB- Valid, overall Vina affinities, and achieving comparable drug-like properties w.r.t. reference molecules.
For conformation quality, our model achieves the best performance with the highest PoseBusters passing rate, closely matching the reference value of 95.0%. Details of each validity check are shown in Fig. 9, and it can be seen that the leading factors affecting the overall performance are bond lengths and angles (AR, TargetDiff), internal energy (MoICRAFT, DecompDiff) for intramolecular validity, and minimum distance to protein (Pocket2Mol) for intermolecular validity.
For molecular geometries, we visualize the distributions of bond length, bond angle, and torsion angle, between generated molecules and reference molecules in the test set in Fig. 14, 15, 16 for the top-5 frequent types, and summarize the overall Pensen-Shannov Divergence (JSD) in Table 7, averaged over all types with frequency > 100. While the previous strong-performing method MoICRAFT captures the bond length distributions relatively well, it cannot fit all the bond angle or torsion angle distributions, displaying for example non-standard C-C-C bond angle, smoothed C-C-N bond angle, and diverges more severely in C-C-C-C and C-C-N-C torsion angles compared with reference. Our method excels at modeling all these distributions, underscoring its ability to capture the molecular geometries.
For binding affinities, our model demonstrates the leading performance with an average Vina score of -6.88 kcal/mol,
20

---

Piloting Structure- Based Drug Design via Modality-Seciffee Optimal Schedule
Percentage of molecules passing the check
0.9
0.8
0.7
0.6
AR
0.5
Pocket2Mol
TargetDiff
PB-Valid Mol
PB-Valid Dock
DecompDiff
0.4
MoICRAFT
MolPllat (Ours)
0.3
Mol pred loaded
Mol cond loaded
Sanitization
Inchi convertible
All atoms connected
Bond lengths
Internal steric clash
Bond angles
Aromatic ring flatness
Double bond flatness
Protein-ligand maximum distance
Minimum distance to organic cotactors
Internal energy
Minimum distance to protein
Minimum distance inorganic cofactors
Minimum distance to waters
Volume overlap with organic cofactors
Volume overiap with protein
Volume overlap with inorganic cofactors
Volume overlap with waters
Figure 10. Percentage of generated molecules that have passed the PoseBusters validity checks OOD PoseBusters test set. PB Valid Mol: intramolecular validity. PB- Valid Dock: intermolecular validity. Reported PB Valid: PB- Valid Mol & PB-Valid Dock.
maintaining the closest gap between Vina Score, Vina Min, and Vina Dock. This shows our model's superiority in capturing spatial interactions in the generated pose.
For molecular properties like QED and SA, our model displays competitive performance, and ranks at the top among all non-autoregressive models.
D.3. Out-of-Distribution De novo Design
We select competitive baselines in the ID settings, and evaluate them on the OOD PoseBusters test set. We sample 100 molccules for each of the 180 test proteins, and report the overall results in Table
We provide more detailed statistics of PoseBusters passing rate in Fig. 10. It can be seen that the performance drop in PB-Valid is mainly attributed two major factors: (1) Intramolecular validity drops due distorted molecular geometry such non-standard bond lengths, bond angles, ring flatness and the most prominently, internal energy as the overall indicator of molecular geometries, where all the non-autoregressive baselines and AR are around below 80%. (2) Intermolecular validity drops due to clahses w.r.t. protein surface. TargetDiff appears to be mostly affected, and nearly all models cannot effectively avoid clashing by maintaining a feasible distance to protein atoms, waters or other cofactors. For protein atom clashing, it worth pointing out that the high performance over CrossDock might come from potential information leakage caused by data splits (Appendix C. Thus passing rates on held-out PoseBusters test can serve as more informative indicator the performances between different methods, showing that voxel-based AR and OFFN-ased models are performing better than diffusion- based models. However, it should be noted that though with better clashing performance, AR generates significantly fewer molecules that are also with smaller sizes, undermining its practicality. Besides, current SBDD models have not yet taken waters and cofactors into consideration, suggesting that there might be a calling for an advanced and comprehensive formulation of SBDD.
be viewed
For a finer grained quantification the internal energy, we employ PoseCheck test suite (Harris et al., 2023) to measure the Strain Energy of generated molecules, and report the percentiles of distributions in Table associated with the internal
21

---

Piloting Structure-Based Drug Design via Modality-Specifte Optimal Schedule
MSE Loss for Continuous (3D)
Cross Entropy Loss Discrete (2D)
3.4
sync
sync
w/ 2D
3.2
w/ 3D
3.0
2-
2.8
2.6
0.4
0.8
Figure 11. Validation loss for the model trained under generalized objective timestep in the generative process. Sync: Modalities at the same timestep. w/ 2D: Discrete modality always at t w/ 3D: Continuous modality always at 1.
Table Results of Jenson- Shannon Divergence (JSD) of bond length, bond angle, torsion angle distributions between generated molecules and Docks reference molecules, averaged over all types with frequency > 100.
Length (4) Avg. JSD
Angle (L) Avg. JSD
Torsion (4) Avg. JSD
Methods
AR Pocke12Mol
0.544
0.507
0.545
0.472
0.482
0.467
TargetDiff
0.365
0.435
0.411
DecompDiff
0.332
0.410
0.338
MoICRAFT
0.318
0,384
0.322
Ours
0.252
0.351
0.287
Table Strain Energy results calculated by PoseCheck v1.1, and PoseBusters internal energy passing rate for SBDD baselines.
Strain Energy
Methods
25%
50%
75%
Passed
Reference
2.6
7.1
18.9
100%
AR
5.6
40.1
238.8
77.7%
Pocket2Mol
1.0
4.4
28.5
97.4%
TargetDiff'
41.4
158.2
625.1
65.2%
DecompDiff'
6.8
41.5
249.2
80.1%
MolCRAFT
5.3
46.5
27397.8
71.6%
Ours
2.7
10.6
38.1
98.1 %
HBDonor
HBAcceptor
XBDonor
PiCation
Interaction Count
Interaction Count
1.0
0.020
Interaction
0.5
0.005
0.0
Mot
Decongoii
MaICRAFT
Ours
Rar
Packet2Med
TargeDDff
DecompDiff MaiCAAFT
Ours
0.000
Ruden12Me
Targetai
Decongooi
MaCRRAFT
Ourx
0.000
Ref
4A
Mol
Taroneer
DecongDir
MorCRAFT
Ours
Figure Detailed interaction counts generated molecules on PoseBusters test. Only PB- Valid molecules are considered.
22

---

Piloting Structure-Basse Drug Design via Modality-Specffte Optimal Schedule
energy passing rate in PoseBusters checks. shows that MolCRAFT generates considerably more strained structures as suggested by the tail distribution, and MolPilot and Pocket2Mol maintains the robust performance.
We provide the detailed interaction counts compared with ground-truth complex structures in PoseBusters. Fig. 12 shows that MolPilot matches the genuine interaction profiles relatively well, especially in Cation-T interactions. We additionally report in Table 9 the full statistics of interaction profile similarity calculated from different poses. It can be seen that our MolPilot and MoICRAFT closely match interaction fingerprint, but MoICRAFT suffers from strained poses, resulting in unrealistic interactions for generated poses with high internal energy. In contrast, our model is able to capture the true interactions reliably.
Table 9. Tanimoto similarity of the interaction profiles between ground-truth structures in PoseBusters and molecules generated by SBDD models. Gen: interactions directly calculated from generated poses. Redock: interactions calculated from Vina redocked poses. Gen PB- Valid: interactions directly calculated from generated poses that have passed the PoseBusters validity checks.
Methods
Sim. (Gen)
Sim. (Redock)
Sim. (Gen & PB-V PB-Valid)
AR
0.394
0.344
0.221
Pocket2Mol
0.447
0,405
0,436
Tiff
0.477
0.415
0.458
DecompDiff
0.438
0.332
0.374
MoICRAFT
0.560
0.444
0.498
Ours
0.552
0.477
0.551
MelPliat (Ours)
AR
Pocker2Mol
TargetDiff
DecompDiff
MclCRAFT
A
RMBO: 0.36A
RMSD 6.57A
RMSD 5.4AA
RMSD: 4.19A
RMSD: 5.84A
RMSD: 1.90:3.00 3.49A
B
PS-Vaii
Strai:
X
Non-slaner aromasco
X
Clash with protein
X
Non standard band length Ideal C-D: 1.4A
X
Non-standart bond angle Ideal C-C-C:-
X
Figure 13. Illustration molccular geometrics for SBDD models, A. Binding mode consistency upon redocking, measured by scRMSD between generated poses (Green) and redocked poses (Silver). B. Generated molecules with unrealistic geometries, thus failing pass PoseBusters validity checks.
D.4. Molecular Docking
We repurpose the SBDD models from the joint codesign model P(x, h A xP) to the conditional marginal P(x
h, A, xp). This actually corresponds to another specific line on the loss surface described by Eq. 10, with 3a(() = (1). We employ the joint schedule B(t) (t), (1)) with 3 being the default noise schedule for continuous modality in Eq. 3.
23

---

Piloting Structure-Basse Drug Design via Sopeality-Speefffe Optimal Schedule
Reference AR (.62)
Reference
Raference
Refarence
0.30
0.20
TargetDii (50 0,371
0.20
MoICRAFT USD 0.32)
0.20
Ours USD 0.20)
0.25
Pocket2Hol 6.49)
DecomgD (50 0:331
Denstty
0.20
Density
0.15
Density
0.15
Donsity
0.15
0.15
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
0.00
0.00
0.00
1.2
1.4
1.6
1.2
1.4 Bond length (A)
1.6
1.2
1.4
1.6
1.2
1.4
1.6
Bond length (A)
Bond length
Bond longth (A)
(a) C-C Bond
0.30
0.30
Reference
Reference Targestiff (50 0.251 DecompD (SD 0.24)
0.30
Reference
Reference
0.45)
0.25
MctCRAFT (SD 0.32)
0.25
Rocket2Hoi (SD
0.25
Density
0.20
0.20
Densiyy
0.20
Deastyy
0.20
0.15
0.15
0.15
0.15
0.10
0.10
0.10
0.10
0.05
0.05
0.05
0.05
0.00
0.80
1.6
0.00
1.2
1.4
1.6
0.00
1.2
1.4
Bond length (A)
Bond length (A)
Bond length (A)
Bond length (A)
(b) C:C Bond
0.30
0.30
0.30
0.30
All (SD 0.49)
largeltkff 0.421
0.25
MetCRAFT (SS 0.35)
0.25
Durs USD 0.24)
Pocket2Mol (SD 0.45)
DecompDiff (SD 0.351
Denydry
0.20
Demalyy
0.20
0.20-
Densty
0.20
0.15
0.15
0.15
0.35
0.10
0.10
0.05
0.05
0.05
0.05
0.00
0.00
0.00
0.00
1.6
1.2
1.4
1.2
Bond length (A)
Bond length (A)
Bond longth (A)
Bond longth (A)
(c) C-O Bond
Relerence
Seference lergetoiff
Reference MalCHAFF (SI 0.30)
Rellewnce
0.125
0.125
0.125
0.15
DecompDin 0.34)
0.100
Denaty
Dansit
Deneiy
Densilty
0.10
0.075
0.075
0.075
0.050
0.050
0.050
0.05
0.025
0.025
0.025
0.000
0.000
0.000
1.2
1.4
1.6
1.2
1.6
1.2
1.4
1.6
1.2
1.4
1.6
Bond lengtt (A)
Bond length (A)
Bond length (i)
Bond length (A)
(d) C-N Bond
Reference
Reference Targettin (SD 0.231
Refenence MolCRAFT (JSD 0.25)
Reference
0.25
Parbwi2Mol (SD c.08)
AR (JS 6.353
0.20
DecempDill (SD 0.31
0.20
0,20
Durs (J50 0.19)
Dunsiyy
Dersity
Dersity
0.15
Densty
0.115
0.15
0.10
0.10
0.30
0.05
0.05
0.05
0.05
0.00
0.00
0.00
1.2
1.4
1.6
1.2
1.4
1.6
1.2
1.4
1.6
1.2
1.4
1.6
Bond length (A)
Bond length (A)
Bond length
Bond length (A)
C:N Bond
Figure 14. 5 frequent bond length distribution of generated molecules compared with CrossDock reference molecules.
24

---

Piloting Structure-Base Based Drug Design via Modality-Specifn Optimal Schedule
Raference
Referance TorgatDitt UED 0.34
Reference
Reference Ours (SD 0.21)
0.0015
48 (J50 0.37)
0.0015
0.0015
NOICRAFT (SD 0.30)
0.0015
Pocket2Mal USD 0.38)
Decomp 0.34)
Density
0.0 0,0010
Dersity
0
Dersity
0.0010
Dersity
9.0010
0.0005
0.0005
0.0005
0.0005
0.0000
0.0000
0.0000
0,0000
110
130
140
100
110
130
100
120
130
140
11.0
120
140
Bond angle (*)
Bond angle (*)
Bond angle (*)
Bond angle (*)
(a) C-C-C Bond Angle
0.005
0.005
0.005
0.005
Reference Alt (ISa 0.57)
Reference Target IJS 0.28) DecompDlit (5D 0.25)
Reference MalCRATT QSD 0.17)
Reference Ours (SD 0.19)
0.004
Packet22Mll (150 0.48)
0.004
0.004
0.004
Denstyy
0.003
Density
0.003
Density
0.000
Densty
0.003
0.002
0.002
0.002
0.002
0.001
0.001
0.001
0.001
0.000
100
110
120
130
140
0.000
100
11.0
120
130
1440
0.000
100
110
120
130
140
0,000
100
110
120
130
140
Bond angle (7)
Bond angle (*)
Bond angle (*)
Bond angle (*)
(b) C:C:C Bond Angle
0.0025
Reference (JSO 4.4
0.0025
Referonce Target D 9.44] Decompbitf ISD 0.39)
0.0025
Reforence MuKIuAFT QSD 6.35)
0.0025
Reference Curs (SD 0.29)
0.0020
Pocket2MMl USD 0.48)
0.8020
0.0020
0,0020
Densly
0.0015
Density
0.0015
Density
0.0015
Denstyy
0.0015
0.0010
0.0010
0.0010
0.0010
0.0005
0.0005
0.0005
0,0005
0.0000
0.0000
0.0000
100
110
120
130
140
100
110
120
130
1400
100
110
120
130
140
100
110
120
130
140
Bond angle
Bond angle (*)
Bond angla (*)
Bond angle (?)
(c) C-C-O Bond Angle
0.0030
0.0030
Reference All (JSD 0.54)
Refeeencc Tergelibill (5.D 0.45)
Rederence HUKJUET QSD 0.45)
Reference Durs (SD 0.42)
0.0025
0.0025
0.0025
0.0025
Pocket22Ml USD 0.51)
Decompoitr USD 0.43)
Dersity
0.0020
Dersity
0.0020
Dersity
0.0020
Dersity
0.0020
0.0015
0.0015
0.0015
0.0010
0.0010
0.0010
0.0010
0.0005
0.0005
0.0005
0,0005
0.0000
0.0000
0.0000
0.0000
11.
120
130
140
100
110
130
100
120
130
140
110
120
130
140
angle
Bond angle
Bond angle (*)
Bond angle
(d) C:C:N Bond Angle
0.0020
Reference
0.0070
Refiemance Terget USD 0.40)
0.0020
Referance MOXRAFT (SD 0.3R)
0.0070
Reference
AB, (J52 0.45)
Ours (SD 0.33)
0.0015
USa 0.64)
0.0015
Decomptilf
0.0015
0.0015
Denstty
Density
Density
Densiy
0.0010
0.0010
0.0010
0.0010
0.0005
0.0005
0.0005
0.0005
0.0000
0.0000
0.0000
0.0000
110
140
100
130
100
120
130
120
140
Bond angle (*)
Bond angle (*)
Bond angle (*)
Bond angle (?)
(e) C-C-N Bond Angle
Figure/ 15. 5 frequent bond angle distribution of generated molecules compared with CrossDock reference molecules.
25

---

Piloting Prructure-Bascee Drug Design via Modality-Specifn Optimal Schedule
0.0005
4B (SD 0.381
Reference
0.00025
Refarence ce
TargetDit (500 0.31)
0.000125
Refarence
MUICRAFT 1SD 0.281
0.000125
Reference Ours USD 0:25)
Podket2Mal (SDD 0.32)
0.00020
DecompD (SD 0.35)
0.000100
0.000100
Density
0.0003
Dencity
0.00015
Denoity
0.000075
Dereity
0.000075
0.0002
0.00010
0.000050
0.000050
0.0001
0.00005
0.000025
0.000025
0.0000
0.00000
0.000000
0.000000
-100
1000
1.00
100
-100
100
100
Torsion angle (*)
Torsion angle (*)
Torsion angle (*)
Torsion angle (?)
(a) C-C-C-C Torsion Angle
0.00150
Reference
Reference Ta (SD
0.0020
Reference McICNAFT 0.36)
Reference
(SD 0.711
0.0020
0.00125
Paciet2Mol (SD 0.531
0.0015
Decompoft (15 0.24)
0.0015
Deniity
0.00100
Density
Denctty
Density
0.0015
0,0010
0.00075
0.0010
0.0010
0.00050
0.0005
0.0005
0.0005
0.00025
0.00000
0.0000
0,0000
0.0000
-100
100
-100
1000
-100
100
-100
1000
Torsion angle (*)
Torsion angle (*)
Torsion angle (*)
Torsion angle (*)
(b) C.C.C:C Torsion Angle
Reference AN 0.371
0.000150
Reference Targalbiff osa 0.34)
0.00012
Reference
0.00012
Reference Durs USD 0.35)
0.0008
MoICWAFT BSD 0.34
Podket2Mal (500 0.37)
0.060125
DecompDif (SD 0.34)
0.00010
0.00010
Dendtyy
0.0 0.0006
Denstty
0.000100
Density
0.00008
Density
0.00000
0.0004
0.000075
0.00006
0.00006
0.000050
0.00004
0.00004
0.0002
00000225
0.00002
0.00002
0.0000
0.000000
0.00000
0.00000
-100
0
100
-100
100
-100
100
-100
0
100
Torsion angle (*)
Torsion angle
Torsion angle (*)
Torsion angle (?)
(c) C-C-O-C Angle
Reforonce All (JSD 0.43)
Reforence Sangeitalf QSD 0.40)
Raference
Reference Durs US5 0.39)
0.00015
0.00015
0.00015
MelCRAFT (isa D.40)
0,00015
Podee/2Mal QSD 0.40)
DeconpDrl QSD 0.411
Dersity
0. 0.00010
Dersity
0.00010
Dersiby
0.00010
Density
0,00010
0.00005
0.00005
0.00005
0,00005
0.00000
0,00000
0.00000
0.00000
-100
100
100
100
-100
Torsion angle (*)
Torsion angle (*)
Torsion angle
angle (*)
(d) C-C-C-O Torsion Angle
0.0005
Reference
0,00030
Reference TargetDWff (SD Decompiil 0.39)
0.00030
Reference
0.00030
Rafarence
MIlCRAF 0.35)
0.0004
0.00025
0.00025
0.00025
Density
Density
0.00020
Dersity
0.00070
Dersity
0.00020
0.0003
0.00015
0.00015
0.00015
0.0002
0.00010
0,00010
0.00010
0.0001
0,00005
0.00005
0.00805
0.0000
0.00000
0.00000
0.00000
-100
100
-100
100
-100
Torsion angle (7)
Torsion angle (*)
Tersion angle (?)
Torsion angle (*)
(c) C-C-N-C Torsion Angle
Figure 16. 5 frequent torsion angle distribution of generated molecules compared with CrossDock reference molecules.
26


Reviews:
[{"review_id": "BsgANi4veu", "reviewer": "Reviewer_GKyX", "summary": "This paper presents a novel method for finding optimal noise scheduling in structure-based generative models. Typically, deep generative networks used for structure-based drug design generate molecules directly within the binding pocket of a protein, allowing the problem to be defined as 3D molecular graph generation conditioned on the protein binding pocket. Diffusion models that have been applied to this problem generally diffuse continuous atom positions and discrete graph topology separately. The authors of this paper contend that designing a joint noise scheduler may enhance generation results for multi-modal data, where the variational lower bound is not solely dependent on the endpoints, as is the case in unimodal scenarios. A dynamic programming algorithm is proposed to find the optimal scheduler on the discretized grid. The experiments demonstrate the advantages of the proposed model in conditional molecule generation and docking scenarios.\n\n## update after rebuttal\n\nThe Authors addressed all my comments. I decided to maintain my positive score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the thorough evaluation and valuable feedback. Below, we address each point raised to improve the clarity, reproducibility, and depth of our work.\n\n## Questions\n**Q1: Architectural Changes**\n\nThe backbone of MolPilot aligns with DecompDiff, where we only replaced the MLP in the q, k, v calculation with LinearNoBias to reduce GPU memory consumption. This decreases memory usage from 42177MB to 34699MB for batch size 8 in the first 500 steps, enabling more efficient training without compromising performance. We report this baseline trained by default loss in Figure 6.\n\n**Q2: Sensitivity to Optimal Path Choice**\n\nThank you for this insightful question. We conducted additional experiments interpolating between linear $t_l$ (coef $c=0$) and our optimal time-rescaling functions $t_o$ ($c=1$), by setting time functions $t=c\\cdot t_l + (1-c)t_o$. Our findings show a clear trend of improving performance as we move toward the optimal schedule on both datasets.\n\nFor CrossDock:\n\n| c | Vina Score Avg | Vina Score Med | Vina Min Avg | Vina Min Med | QED | SA | Connected | PB-Valid |\n|---|---|---|---|---|---|---|---|---|\n| 0 (Linear) | -6.63 | -6.94 | -7.06 | -7.06 | 0.55 | 0.76 | 0.95 | 0.95 |\n| 0.25 | -6.66 | -6.88 | -7.02 | -7.01 | 0.55 | 0.76 | 0.94 | 0.96 |\n| 0.5 | -6.70 | -6.82 | -7.04 | -6.94 | 0.55 | 0.76 | 0.95 | 0.97 |\n| 0.75 | -6.87 | -6.97 | -7.21 | -7.11 | 0.55 | 0.76 | 0.96 | 0.96 |\n| 1 (Optimal) | -6.92 | -7.02 | -7.23 | -7.18 | 0.55 | 0.75 | 0.96 | 0.95 |\n\nFor PoseBusters:\n\n| c | Vina Score Avg | Vina Score Med | Vina Min Avg | Vina Min Med | QED | SA | Connected | PB-Valid |\n|---|---|---|---|---|---|---|---|---|\n| 0 (Linear) | -7.35 | -7.38 | -7.64 | -7.48 | 0.48 | 0.73 | 0.94 | 0.79 |\n| 0.25 | -7.42 | -7.37 | -7.66 | -7.50 | 0.48 | 0.73 | 0.94 | 0.79 |\n| 0.50 | -7.20 | -7.36 | -7.56 | -7.51 | 0.48 | 0.73 | 0.95 | 0.78 |\n| 0.75 | -7.44 | -7.43 | -7.75 | -7.54 | 0.48 | 0.73 | 0.95 | 0.79 |\n| 1 (Optimal) | -7.52 | -7.52 | -7.79 | -7.65 | 0.49 | 0.73 | 0.95 | 0.80 |\n\n## Additional Clarifications\n**Q3: Repository and Reproducibility**\n\nWe sincerely apologize for the oversight regarding the code repository. We had uploaded our code to GitHub before the reviewing process, but failed to verify the auto-update mechanism for the anonymous repository. This unintentional mistake has been rectified, and we have now force-updated the repository with necessary code to ensure reproducibility.\n\n**Q4: Notation Clarifications**\n\nWe appreciate your careful reading that identified these notation inconsistencies. $\\alpha$ in Equation 2 is actually the discretized $\\alpha_i$ for time step $i$ in the sequence, derived from time-dependent function $\\alpha(t)$, not a constant.  $K$ is defined as the number of classes for one-hot encoding (with $K_h$ and $K_A$ denoting atom and bond types, respectively).  $\\mathbf{e}_\\mathbf{x}$ is kronecker function, i.e. the projection from a class index $\\mathbf{x}=j$ to a one-hot vector $\\in \\mathbb{R}^K$ with the $j$-th value equal to 1.  We will revise the manuscript to improve clarity as requested.\n\n**Q5: Molecular Docking Study and Baselines**\n\nWe thank the reviewer for this important point about training data differences. Following this suggestion, we trained a version of our model on PDBbind from scratch (120 epochs over 1.5 days) to enable a more direct comparison. \n\n|  | DiffDock (850 epochs) | Ours (120 epochs) |\n|---|---|---|\n| RMSD < 2Å (%) | 38.0 | 17.1 |\n| RMSD < 2Å & PB-Valid (%) | 12.7 | 15.4 |\n\nAlthough the comparison remains imperfect, the results are still informative, for our primary goal was to demonstrate that ours is the first with genuine docking capability that can repurpose an SBDD model for the docking task. This capability stems from our decoupled training strategy rather than architectural changes. We will add clarification to the manuscript.\n\n**Q6: Novelty of Generated Compounds**\n\nThank you for this valuable suggestion. We have calculated comprehensive metrics to assess the novelty. As shown in the table below, MolPilot achieves the highest combined metric (0.864) among all methods, demonstrating its ability to generate novel, valid, and unique compounds.\n\n||Uniq|Valid|Novel|Combined (U×V×N)|\n|---|---|---|---|---|\n|AR|0.924|0.385|1|0.356|\n|Pocket2Mol | 0.910 | 0.677 |1| 0.616 |\n|TargetDiff |1| 0.812 |1| 0.812 |\n|DecompDiff|0.899 | 0.829 |1| 0.745 |\n|MolCRAFT|0.841 | 0.995|0.983 | 0.823 |\n|Ours|0.898|0.974|0.987|0.864|\n\nIn addition to these metrics, we have calculated the Fréchet ChemNet Distance (FCD), which measures distribution similarity between generated molecules and training distribution (similar to FID for images, lower is better). MolPilot achieves competitive FCD scores that demonstrate its ability to generate novel compounds while faithfully capturing the training distribution.\n\n|Method | AR | Pocket2Mol | TargetDiff | DecompDiff | MolCRAFT | Ours |\n|---|---|---|---|---|---|---|\n|FCD|23.5|23.0|21.0|18.1|14.2 | 12.9 |", "claims_and_evidence": "The claims in the paper are supported by the experimental results.", "methods_and_evaluation_criteria": "The methods make sense for the problem at hand, but I noticed two problems with the method description and reproducibility:\n1. The linked repository is empty, making this work irreproducible.\n2. Some potential abuses of notation can complicate the understanding of the methods section. For example, $\\alpha$ in Equation 2 is constant, and later it is a function of time. $K$ is not introduced near that equation, only earlier $K_h$ and $K_A$ are introduced. $\\mathbf{e}_\\mathbf{x}$ is not defined.\n\nThe benchmarks are carefully selected to showcase the effectiveness of the methods.", "theoretical_claims": "I read the proofs but did not check them thoroughly for correctness.", "experimental_designs_or_analyses": "The experimental design and analyses are sound. However, the molecular docking study should include more baselines because MolPilot was trained on more data (CrossDocked), and models such as DiffDock were trained on PDBbind. Additionally, examining the novelty of the generated compounds would be valuable, such as through measuring their distance from the training set.", "relation_to_broader_scientific_literature": "The results presented in this paper demonstrate that careful optimization of the noise schedule enhances the outcomes of structure-based molecule generation. The proposed theory could have a wider impact on other multi-modal generation scenarios in chemistry and beyond.", "essential_references_not_discussed": "The key references have been described.", "other_strengths_and_weaknesses": "The strengths and weaknesses have been described in the other sections.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. Are there any architectural changes that have improved the performance of MolPilot compared to other models like MolCRAFT and DecompDiff? The difference is quite significant, especially regarding the docking experiment. If there are any modifications in the architecture mentioned in Section A.3 compared to previous models, it would be advisable to include this model, trained similarly to the other models, as a baseline.\n2. Have you tested how sensitive the results are to the choice of the optimal path, e.g. by interpolating between linear and optimal time-rescaling functions?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WR35p39JiH", "reviewer": "Reviewer_8Pth", "summary": "The present paper is concerned about structure-based drug design (SBDD) using a Bayesian Flow Network (BFN). One of the issues when applying BFN to SBDD is that the model has to generate a molecular graph, which is a discrete object, as well as its 3D structure, which is a continuous object. A numerical example in Section 2 demonstrates that many of the related models favor 3D structure than 2D structure, and it leads to an unrealistic 3D structure that is incompatible with the 2D structure. Theoretically, when the data has both continuous and discrete parts, the variational lower bound depends on the entire noise schedule, which motivates the authors to find the best noise scheduling. The authors have successfully derived a dynamic programming based algorithm to obtain the best noise schedule, which is the main theoretical contribution of the paper.\n\nThe authors investigate the effectiveness of the proposed method by numerical experiments. One of the highlights is that the proposed method achieves the best PB-Valid score, which suggests that the generated 2D and 3D structures are consistent.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the thorough reading of our manuscript and the positive evaluation. We are pleased that the reviewer recognized the key challenges and contributions of our work on deriving optimal schedules for structure-based drug design, and we certainly welcome further discussion.\n\n**Regarding Theoretical Contributions**\n\nWe appreciate the reviewer's acknowledgment of one of our main contributions - the dynamic programming algorithm for VLB-optimal noise scheduling when dealing with the twisted discrete-continuous probability path. We believe the theoretical foundation is sound and provides an important advancement for handling the challenging joint generation of molecular graphs (discrete) and their 3D structures (continuous).\n\n**Regarding Experimental Results**\n\nWe are particularly encouraged by the reviewer's recognition of our experimental design and results, especially highlighting our improvement in PB-Valid. As the reviewer correctly noted, this metric demonstrates our method's ability to generate consistent 2D and 3D molecular structures - addressing a key limitation in previous approaches that often favored 3D structure at the expense of 2D topology.\n\nWe thank the reviewer again for the supportive evaluation. We remain committed to further developing and refining our method to contribute to the field of structure-based drug design.", "claims_and_evidence": "Most of the claims in this paper are clear and well supported by both empirical and theoretical statements.", "methods_and_evaluation_criteria": "This paper uses a standard benchmark for deep learning-based SBDD, which makes sense to me.", "theoretical_claims": "I have followed the theoretical statements, but not followed their proofs.", "experimental_designs_or_analyses": "The experiments are well designed and are very informative for readers. I have no addition to the experimental protocol.", "relation_to_broader_scientific_literature": "This paper is related to drug discovery, and in this sense, is connected to the broader scientific literature.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "txKfYNUHXN", "reviewer": "Reviewer_PhsR", "summary": "This paper focuses on modeling twisted probability path of multimodalities in structure-based drug design.\nIt analyzes the theoretical link between noise schedules and VLB in multimodality probabilistic modeling and further proposes a VLB-optimal scheduling strategy to address this bottleneck.\n\nBy integrating the VLB-optimal scheduling with the BFN framework, the proposed MolPilot achieves state-of-the-art performance in de novo design and competitive performance in local docking, demonstrating the effect of the proposed VLB-optimal scheduling.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the careful reading and insightful feedback. Below, we address each question and concern to improve the clarity and completeness of our work, as well as demonstrate its generality.\n\n## Questions\n**Q1: Integration with More Frameworks**\n\n> Why not integrate the proposed VOS with more frameworks? It's necessary to demonstrate its practical applicability within various frameworks. \n\nThank you for this valuable suggestion. To demonstrate VOS's broader applicability, we integrated it with the diffusion-based framework TargetDiff. Specifically, we reimplement TargetDiff with our generalized training objective and train 140k steps following the default training configuration (TargetDiff*, with code provided), and then derive the test-time optimal schedule that resembles the shape in Figure 4B. \n\nThe results on CrossDock show that VOS successfully enhances conformation quality for diffusion models too, with generated poses achieving Vina Scores closer to Vina Min values, indicating near-optimal realistic poses.\n\n| Metric | PB-Valid | Vina Score Avg | Vina Score Med | Vina Min Avg | Vina Min Med | Vina Dock Avg | Vina Dock Med | scRMSD<2 | Energy Passed | Connected | QED | SA |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| TargetDiff | 50.5% | -5.47 | -6.30 | -6.64 | -6.83 | -7.80 | -7.91 | 37.1% | 69.8% | 90.4% | 0.48 | 0.58 |\n| TargetDiff* | 53.7% | -6.27 | -6.31 | -6.82 | -6.78 | -7.87 | -7.90 | 36.8% | 70.3% | 89.2% | 0.50 | 0.62 |\n| TargetDiff* + VOS | **58.1%** | **-6.46** | **-6.53** | **-7.04** | **-7.09** | **-8.04** | **-8.12** | **40.2%** | **73.2%** | **93.4%** | 0.49 | 0.59 |\n\n**Q2: CrossDock Test Set Consistency**\n\n> Is the ID CrossDock test set identical to those used in previous studies?\n\nYes, we confirm that we used the identical CrossDock test set as proposed in previous studies, ensuring fair and consistent comparison with existing methods.\n\n## Weaknesses\n**W1: Clarity of Implementation Details**\n\n> The description of the proposed MolPilot is not clear, lacking many implementation details.\n\nWe appreciate this feedback and will enhance the clarity of our description by adding more detailed description to Section 4, Algorithm 1, and Appendix A. We will move key implementation details from the Appendix to the main text where appropriate. Thanks again for the advice, and we will revise our manuscript accordingly.\n\n**W2: Essential References**\n\n> This paper ignores many essential references, limiting its impact and connection to existing research. For example, many essential references are not discussed in this paper, such as IPDiff [1], DecompOpt [2], IRDiff [3], DecompDPO [4], and BINDDM[5]. \n\nThank you for highlighting these important references. We agree that IPDiff, DecompOpt, IRDiff, DecompDPO, and BINDDM represent meaningful advancements in controllability and practicality by incorporating guidance signals (e.g. binding affinity, interactions) to enhance molecular properties of generated ligands.\n\nWe will expand our related work section to include thorough discussion of these approaches and their contributions. Additionally, we report the results of IPDiff and BindDM on CrossDock, where we calculated PB-Valid and other metrics based on the samples obtained by using the official code, showing that they indeed enhance affinities through binding signals from pretrained predictors, yet also with the issue of conformation plausibility. \n\n|  | PB-Valid | Vina Score Avg | Vina Score Med | Vina Min Avg | Vina Min Med | Vina Dock Avg | Vina Dock Med | scRMSD<2 | Energy Passed | Connected | QED | SA |\n|---|---|---|---|---|---|---|---|---|---|---|---|---|\n| IPDiff | 37.1% | -6.41 | -7.01 | -7.45 | -7.48 | -8.57 | -8.51 | 33.7% | 51.1% | 90.4% | 0.52 | 0.59 |\n| BindDM | 31.3% | -5.92 | -6.81 | -7.29 | -7.34 | -8.41 | -8.37 | 38.6% | 42.3% | 91.5% | 0.51 | 0.58 |\n\n**W3: Code Repository**\n\nWe sincerely apologize for the oversight regarding our code repository. We had uploaded our code to GitHub before the reviewing process began, but failed to verify the *auto-update* mechanism for the anonymous repository. This unintentional mistake has been rectified now, and we have force-updated the repository with all necessary code to ensure full reproducibility, including the newly added experiment with TargetDiff.\n\n**W4: Ethical Review Concerns**\n\nWe appreciate the attention to formatting requirements. The missing content was due to that we have commented out `printAffiliationsAndNotice{}`, yet it was not intended to create extra space. As noted in the ICML 2025 Peer Review FAQ (March 18, 2025 update), the Program Chairs have acknowledged that papers with this specific issue will not be desk-rejected *\"due to confusion in the LaTeX template.\"* We will restore this element in our revision to ensure full compliance with formatting requirements.\n\nWe thank the reviewer again for the constructive feedback that helps us improve our manuscript, and we welcome further discussion.", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "This paper proposes a VLB-optimal scheduling strategy to model twisted probability path of multimodalities, which makes sense for structure-based drug design.\n\nFurthermore, it introduces an OOD subset of PoseBusters to serve as a more reliable held-out test, which also makes sense for this problem.", "theoretical_claims": "I checked the correctness of the theoretical analysis between noise schedules and VLB in multi-modality probabilistic modeling.", "experimental_designs_or_analyses": "I checked the soundness/validity of the experimental designs and analyses.\nI think it's necessary to demonstrate the effect of the proposed VOS by integrating it with more frameworks.", "relation_to_broader_scientific_literature": "This paper aims to overcome the twisted probability path bottleneck of multimodalities, thus related to various tasks involving multimodal generative modeling.", "essential_references_not_discussed": "Many essential references are not discussed in this paper, such as IPDiff [1], DecompOpt [2], IRDiff [3], DecompDPO [4], and BINDDM[5]. \n\n[1] Huang Z, Yang L, Zhou X, et al. Protein-ligand interaction prior for binding-aware 3d molecule diffusion models[C]//The Twelfth International Conference on Learning Representations. 2024.\n\n[2] Zhou X, Cheng X, Yang Y, et al. DecompOpt: Controllable and Decomposed Diffusion Models for Structure-based Molecular Optimization[C]//The Twelfth International Conference on Learning Representations. 2024.\n\n[3] Huang Z, Yang L, Zhou X, et al. Interaction-based retrieval-augmented diffusion models for protein-specific 3d molecule generation[C]//Forty-first International Conference on Machine Learning. 2024.\n\n[4] Cheng X, Zhou X, Yang Y, et al. Decomposed direct preference optimization for structure-based drug design[J]. arXiv preprint arXiv:2407.13981, 2024.\n\n[5] Huang Z, Yang L, Zhang Z, et al. Binding-adaptive diffusion models for structure-based drug design[C]//Proceedings of the AAAI Conference on Artificial Intelligence. 2024, 38(11): 12671-12679.", "other_strengths_and_weaknesses": "**Strengths**:\n1. This paper provides the theoretical analysis between noise schedules and VLB in multimodality probabilistic modeling, showing the importance of designing optimal noise schedules for this twisted probability path.\n\n2. This paper introduces an OOD subset of PoseBusters to serve as a more reliable held-out test, addressing these potential problems with CrossDock evaluation.\n\n**Weakness**:\n1. The description of the proposed MolPilot is not clear, lacking many implementation details.\n2. This paper ignores many essential references, limiting its impact and connection to existing research.", "other_comments_or_suggestions": "It's optional for authors to decide whether to provide their code.\nHowever, providing an anonymous repository link but with an empty codebase is not respectful.", "questions_for_authors": "1. Why not integrate the proposed VOS with more frameworks? It's necessary to demonstrate its practical applicability within various frameworks.\n\n2. Is the ID CrossDock test set identical to those used in previous studies?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_flag": "Flag this paper for an ethics review.", "ethics_expertise_needed": ["Other expertise"]}}]

Author rebuttal:

Meta-review: The paper introduces a novel approach, VLB-Optimal Scheduling (VOS), for improving generative modeling in structure-based drug design.

The theoretical contribution is well-recognized, particularly in how noise schedules affect variational lower bounds in multimodal generative modeling.

The experimental results demonstrate strong performance, with a remarkable PoseBusters passing rate of 95.9% on CrossDock.

**Concerns**

- Initially, the linked repository lacked code, raising concerns about reproducibility.
- Reviewers identified inconsistencies in notation that could affect readability.
- One reviewer suggested that more baselines should be evaluated, particularly since the model was trained on different datasets than its competitors.
- One reviewer questioned why VOS was not tested across different frameworks to demonstrate broader applicability.

**Rebuttal & Discussions**

- The authors provided missing code in the repository to ensure reproducibility.
- Clarified notation inconsistencies by explaining definitions and revising documentation for readability.
- Conducted additional experiments to validate the sensitivity of their approach to optimal noise scheduling.
- Expanded comparisons to include other frameworks (e.g., integrating VOS into TargetDiff).

This is a borderline paper. I recommend accepting it if there is available space.
Decision: Accept (poster)