Empower Structure-Based Molecule Optimization with Gradient Guided Bayesian Flow Networks

We sincerely appreciate the reviewer's thorough reading and insightful feedback, which have helped us identify areas for improved clarity and presentation. We shall address each point in our responses below, and we welcome further questions.

Q1: Explanation of BFN Fundamentals

We thank the reviewer for highlighting the need for improved clarity. We will enhance the paper's readability by providing a better overview and more detailed explanations of our guidance approach.

For experimental details of guidance, our gradient consists of pretrained MolCRAFT and plug-and-play energy functions. We describe the guided sampling in Algorithm 1 and Appendix D.1, and we will make them clearer.

Q2: Novelty

Thanks for raising this important question. While the work of Xue et al. establishes connections between BFNs and SDEs, our contribution goes beyond merely showing the theoretical feasibility. We derive a principled approach to gradient guidance specifically within the BFN framework, rather than reducing BFNs to SDEs and applying existing techniques.

Our contributions lie in both the methodology (deriving gradient guidance within BFNs and proposing a generalized sampling strategy) and in implementing and evaluating SBMO applications. (1) Methodologically, we show how gradient works through guided Bayesian update, offering a unique perspective as distinct from discretizing an SDE, and we believe contextualizing the guidance within BFN is a novel contribution. (2) Practically, the empirical results in Table 3 also show MolJO's distinct advantage compared to simply reducing BFN to SDE.

Q3: Limited Dataset Evaluation

We appreciate this concern. We add the evaluation on PoseBusters V2 test set (180 out of 384 complexes, after excluding those with sequence identity > 30% or with non-standard residues) as a held-out test. We also report the PoseBusters passing rate (PB-Valid), showing that MolJO's improvements are not dataset-specific or the result of overfitting.

Q4: Hyperparameter Choices and Potential Overfitting

As mentioned in Q3, the consistent performance on both datasets confirms that MolJO generalizes well beyond the specific choices for CrossDock. Moreover, they remain robust across reasonable ranges. For property predictors or training / sampling, we used the same architecture and the same BFN hyperparameters ($\beta_1, \sigma_1$) as MolCRAFT without specifically tuning for the task.

Q5: PoseCheck Metrics in Main Paper

We agree that the PoseCheck metrics should be presented in the main text rather than the Appendix for better accessibility, and we will revise our manuscript accordingly.

Though not directly incorporating strain energy as an objective, MolJO indeed improves the conformation quality as suggested by Figure 9, Appendix G, where our CDF is consistently above that of MolCRAFT. Furthermore, the evaluation on PoseBusters (see Q3) reveals notable improvements in PB-Valid.

Q6: Baseline Comparisons

We appreciate this excellent point. Our ablation studies in Section 5.4, Table 3 indeed provide direct comparisons, showing how each component contributes to performance improvements. Furthermore, we're expanding our comparison to include additional optimization baselines such as DecompDPO, where MolJO maintains competitive. Please refer to Q5 in Reviewer fSvs.

Q7: Relevance of Optimized Properties

We appreciate the reviewer's expertise in drug discovery. Our primary contribution is a general optimization framework whose efficacy can be validated through these in-silico metrics. Notably, we've observed that improvements correlate with enhanced key interactions and PB-Valid that are relevant to drug design.

Q8: Computational Overhead

Compared to MolCRAFT, MolJO requires approximately 2× or longer inference time in our updated study. The computational overhead depends on the complexity of the energy functions employed. MolCRAFT generally takes ~22s, previous MolJO with 9-layered energy proxies took ~146s, and we have experimented with 4-layered proxies that take ~45s. MolJO can be further accelerated with an efficient strategy where gradient is applied only at selected timesteps rather than at every step [2], which we leave for future work. We thank the reviewer for motivating this analysis, as it led to more efficient implementations.

[1] A Periodic Bayesian Flow for Material Generation.

[2] Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models.

We sincerely appreciate the reviewer's thorough reading and insightful feedback, which have helped us improve clarity and presentation. We shall address each point in our responses below, and we welcome further questions.

Questions

Q1: Similarities and Differences between Gaussian BFN and Guided Diffusion

Thank you for this insightful observation. We explain the fundamental differences and structural similarities between Gaussian BFN and guided diffusion as follows:

• Key Differences: Our approach guides in parameter space ($θ$) rather than in data space ($y$). In the continuous case, $θ$ exhibits lower input variance and therefore provides more informative signals for guiding the generative process toward the desired output. Guided diffusion typically steers the sampling process in the data space directly, while our BFN-based approach performs inference in the parameter space. Such parameter-space guidance arguably connects deeper to the final target properties due to lower input variance, which allows for more explicit Bayesian modeling of uncertainty and more reliable property optimization.
• Similarities: Both approaches use gradient information to steer the generative process toward desired outputs, operating for Gaussian distributed variables.

Q2: Regarding the Guided Term's Energy Function E

We thank the reviewer for suggesting this clarification. While we included this discussion in the Appendix, we agree it should be introduced earlier for clarity and will revise accordingly in revision.

The energy function forms part of a Boltzmann distribution over parameters $θ$. Although the target property is explicitly defined only at $t=0$, Bayesian inference defines the parameter space $θ$ for any timestep $i$. This allows us to associate property values with every $\theta_i$ throughout the generative process, enabling prediction of time-dependent properties during intermediate optimization stages.

We train a predictor $E(\theta_t, t)$ directly over the parameter space to estimate $\nabla_{\theta_t} p_E(\theta_t)$ given $\theta_t$ at different accuracy levels. This approach proves effective as it provides a consistent guidance signal throughout the sampling process.

Q3: Connection to Energy-Based Compositional Diffusion

We appreciate the connection to [6], which introduced how to interpret diffusion models from the perspective of energy-based model (EBM), and thus applies the additivity from EBM to diffusions.

In our approach, the energy function can be viewed as the negative log-likelihood estimated by a conditional model for given properties. In practice, our method resembles classifier guidance since we train property predictors to serve as the energy function. However, our framework doesn't require explicit "labels" as conditions for controllable generation.

Q4: Preservation of Key Interactions

We appreciate the reviewer's expertise in drug design and the important point raised about preserving key pharmacophoric interactions. We fully agree that in real-world drug optimization, certain critical molecular substructures forming intermolecular interactions (e.g., π-π stacking, hydrogen bonding) should be preserved rather than arbitrarily modified.

Our MolJO framework actually addresses this concern by design. The framework allows for precise control over which molecular substructures should be modified and which should be preserved, while our energy function can guide the redesign of remaining substructures. This better aligns with practical applications such as scaffold hopping.

Other Suggestions

Q5: Additional Related Work and Baselines

We thank the reviewer for suggesting these important methods for comparison. We agree that including these baselines will enhance the completeness of our evaluation. We commit to including comprehensive comparisons with these methods in our revision.

We have cited the results from CBGBench (DiffBP, D3FG) and borrowed the samples to calculate the median Vina affinities and Success Rate based on the statistics available for VoxBind. For DecompDPO, we directly cite the numbers from their paper. It can be seen that our MolJO maintains superiority in optimizing the overall properties, reflected by its highest Success Rate (51.3%).

Q6: CBGBench for Constraint Optimization

We appreciate the reviewer highlighting CBGBench. This is indeed an excellent work that sets up extensive experimental design for structure-based molecule optimization. We will add a thorough discussion of CBGBench, particularly in the context of constraint optimization tasks.

We sincerely thank the reviewer for the careful reading and insightful feedback that helps improve the clarity and completeness of our work.

Questions

Q1: Table Value Inconsistencies

We apologize for the confusion. The values in Table 1 and 3 were obtained from different runs. For Table 3 (ablation studies), we sampled 10 molecules per pocket instead of 100 per pocket. We will clarify this in our revision.

Q2: Fig (Table) 3 Clarification

Thanks for bringing up this issue that helps improve our clarity. All models use the same size specifications from DecompDiff. The differences result from some methods having failed molecule generations (e.g., invalid, not connected). We apologize for the confusion and will correct this labeling (from "Figure" to "Table") in our revision.

For guided SDE, we followed Xue et al. and perform guidance by modifying the conditional score following common practices.

Q3: "Me-better" Definition

Thank you for noting this oversight. "Me-better" refers to molecules with improved specific properties over existing compounds—a term borrowed from traditional drug discovery [1]. We will add the definition in the revision.

[1] Me-Better Drug Design Based on Nevirapine and Mechanism of Molecular Interactions with Y188C Mutant HIV-1 Reverse Transcriptase. https://pubmed.ncbi.nlm.nih.gov/36364174/

Q4: Effect of Joint Guidance and Backward Correction

Thank you for this insightful question. The ablation study is actually presented in Table 3, which directly addresses this question by isolating the contributions of each component. We apologize for the confusion caused by our initial abbreviation "w/ (guidance)" (MolJO-based) and "w/o (guidance)" (MolCRAFT-based) that can be misinterpreted as w/ or w/o (BC). We will improve the clarity in our revision.

As the reviewer suggested, we evaluated "w/ guidance, Vanilla" (row 4) that corresponds to MolJO at k=1 (guidance without BC), and "w/o guidance, BC" (row 3) that corresponds to MolCRAFT with BC. Table 3 shows that both components contribute to performance gains. Notably, the combined improvement exceeds the sum of individual improvements, suggesting these components work synergistically.

Q5: Top-of-N Comparison

Thanks for the great suggestion. We will add top-of-N comparisons for all methods in our revision, which generally shows what the "concentrated space" for desirable drug-like candidates looks like for generative models. Our method shows the best Success Rate (70.3%), indicating better optimization efficiency.

Additional Clarifications

Q6: Tables 1-2 Component

Thank you for highlighting this potential source of confusion. Yes, Tables 1-2 results include backward correction. To clarify, our contribution over TAGMol is two-fold: (1) We derived joint guidance over atom types and coordinates within the BFN framework. (2) We proposed the BC sampling algorithm that further improves optimization performance. As described in Q4, the results in Table 3 demonstrate that both contributions are significant. In our revised manuscript, we will make these distinctions clearer to avoid confusion.

Q7: Backward Correction Explanation & Novelty

We acknowledge that "correcting the past" is potentially confusing terminology. The correction applies to timesteps [n-k, n), while preserving parameters from [0, n-k).

As for its novelty, we develop a more flexible sampling strategy, establishing a sliding window approach that generalizes previous methods and explores a more nuanced control of variance. We appreciate the reviewer connecting this to variance reduction techniques, and we will investigate this connection further in our future work.

Q8: Error Bars

Thanks for the advice! We report the error bars as 95% confidence intervals for our main result in Table 1, and will add it to the Appendix.

Q9: Presentation & Typos

We thank the reviewer for pointing these out, and we will revise our manuscript as requested.

We sincerely appreciate the reviewer's thorough reading and insightful questions, which have helped us identify areas for improved clarity and presentation. We shall address each point in our responses below as well as our revised manuscript, and we welcome further questions.

Q1: Smoothness of Energy Landscapes

Thanks for raising the insightful question about the limitations of first-order Taylor expansion in our approach. We have followed established guided diffusion methodology [1] which assumes that with sufficiently small step sizes, the changes in the energy landscape between consecutive steps remain modest. We acknowledge that this approach inherently assumes the energy landscape is relatively smooth in the local region of expansion, which may not hold universally for complex energy functions.

As the reviewer correctly pointed out, higher-order terms could potentially provide more accurate gradient estimates, which is an active area of research in guided diffusion models [2]. However, higher-order methods introduce computational overhead and potential numerical instability during sampling, which we will leave for future work. While the empirical results demonstrate the practical utility of our approach despite this approximation, we plan to further develop adaptive guidance schemes that adjust based on estimated local curvature.

[1] Diffusion Models Beat GANs on Image Synthesis.

[2] Inner Classifier-Free Guidance and Its Taylor Expansion for Diffusion Models.

Q2: BFN Derivation Clarity

We will enhance our presentation in the revision with a more detailed, step-by-step explanation of the Bayesian Flow Network derivation.

Q3: Schematic Diagram of Backward Correction

Thank you for this suggestion. In Figure 1D, we have a schematic diagram illustrating the backward correction process, and we will continue to make it clearer in our revision.

Q4: Statistical Significance

We acknowledge the importance of statistical validation. We have conducted pairwise t-tests comparing our guided Backward Correction against both Vanilla and guided SDE. The results show statistically significant improvements (p<0.05), and we will add it to our revision.

Q5: Molecule Size Bias

The reviewer makes an excellent point regarding molecule size bias, which is a crucial issue for SBDD. We apologize for the confusion caused by Table 4, and we would like to clarify that rather than solving the size bias issue directly, our primary goal in size-controlled experiments was to demonstrate that our guidance approach effectively improves molecules across the size ranges, and consistently outperforms baselines across different molecular sizes.

For Table 4, we will improve our presentation by calculating optimal scores separately for each size (Ref & Large), which will better highlight that our improvements stem from enhanced molecular quality rather than size bias.

Q6: SE(3)-equivariance with Real-world Pockets

The reviewer correctly identifies a limitation for unknown protein pockets, where reference positions are required to clip and obtain the pocket region. We acknowledge this as a limitation of current SBDD methods and certainly worth exploration. For known protein pockets, we simply subtract the centroid to ensure the center of mass is at the origin.

Q7: RMSD Ratio Without Symmetry Correction

Thank you for this important observation. Here we followed the calculation from MolCRAFT for fair comparison with existing methods. This approach calculates symmetry-corrected RMSD between Vina Docked PDBQT and generated molecules when possible, but falls back to non-symmetry-corrected values in cases where symmetry correction cannot be applied. We appreciate the reviewer highlighting this important consideration for accurate assessment of molecular pose consistency, and we will clarify the detail in our revision and investigate the issue further in the future.

Q8: Strain Energy Higher Than Ref

Thank you for this important observation. Although our results are generally within reasonable ranges for computational generation, we agree there is room for improvement. Future work could incorporate additional chemical prior knowledge to further reduce strain energy. We appreciate the reviewer highlighting this point, as it identifies an important direction for continued refinement of our approach.

Q9: Missing Citation to GraphVF

We sincerely thank the reviewer for bringing GraphVF to our attention. This work indeed makes valuable contributions by encoding different modalities in latent space for joint coordinate-type optimization. We agree that the unified space represents a promising direction. We will update our manuscript to include this important reference and add relevant discussion on how it relates to our work.