Joint Design of Protein Surface and Backbone Using a Diffusion Bridge Model

We appreciate the constructive feedback which helps us enhance the readability and quality of our paper. Below are our point-wise responses to the reviewer's concerns:

W1. The claim in the introduction that method addresses "Limited ability to generate diverse ..." while Table 1 shows their "Div" (diversity) metric decreased compared to baseline "PepFlow w/Bb" method.

Thank you for your insightful comment. Our claim: "Limited ability to generate diverse yet receptor-compatible surface configurations" . This refers to the diversity of 3D molecular surface geometries at the protein-protein interface. Table 1 metric: The "Div" metric measures structural diversity using TM-score between generated peptide backbones, which captures overall fold similarity rather than surface-specific diversity. Our method generates peptide surfaces de novo, enabling exploration of novel surface configurations that may not be captured by backbone TM-score diversity. A peptide can have similar overall fold (high TM-score) but present very different surface properties for receptor binding. In addition, since structures with TM-scores >0.5 maintain recognizable folds while allowing local surface variations, this constrained diversity may actually be more biologically relevant than purely maximizing structural dissimilarity.

W2. Compare against “PepFlow w/Bb+Seq+Ang.

Thank you for pointing this out. We include additional experiments comparing PepBridge variants directly against this method across all core metrics. Preliminary results show that “PepBridge w/Bb+Seq+Ang+Surf” achieves comparable or better performance in terms of stability, while "PepBridge w/Bb+Seq+surf" demonstrates better performance on affinity and diversity. We will include these results in Table 1 and expand the discussion accordingly.

W4. Add more textual descriptions to figures for clarity.

We appreciate this suggestion and have added more detailed textual explanations alongside key figures. The updated description for Figure 1 is: “In the top-down representation of a protein, the molecular surface plays a central role in mediating interactions between the peptide and its target receptor. Each residue is defined by its backbone and side-chain atoms, whose spatial positions are governed by a rotation $r$, a translation $m$, and torsional angles $\psi$ and ${\chi}$.”

w5 w6. Compare with recent methods PepHAR and PPFLOW in the main experiments.

Thank you for raising this point. We conducted additional experiments to benchmark our PepBridge variants against both methods. We will update Table 1 accordingly and expand the discussion.

Q1. How do different hyperparameters affect the final performance?

Thank you for this important question. We conducted experiments to assess how different sampling time steps affect the final performance metrics. The results indicate that longer sampling chains (e.g., 1000 steps) improve the quality of the generated structures. However, performance gains begin to plateau beyond 800 steps, and we observe a slight decrease in structural diversity, suggesting a trade-off between generation determinism and diversity. Based on this analysis, we selected 1000 steps as the default setting to achieve the best overall balance.

Q2. Although diversity metrics have been reported, there is a lack of in-depth analysis on how to find the optimal balance between diversity and quality.

PepBridge addresses the diversity-quality trade-off through flexible sampling strategies. We explored different sampling approaches to tune this balance: DDPM sampling: Incorporates Gaussian noise at each denoising step, yielding higher diversity but potentially lower structural quality. DDIM sampling: Uses deterministic denoising trajectories, producing higher-quality structures with reduced diversity. This flexibility allows us to prioritize either exploration (diversity) or optimization (quality) based on specific design objectives. Future work could develop adaptive sampling strategies that dynamically adjust noise levels based on intermediate quality metrics, potentially achieving both high diversity and quality simultaneously.

Q3. The method’s practical applicability, plans for experimental validation, and the reliability of its computational predictions.

Thank you for your insightful comment. While computational results show promising agreement with known structural and energetic metrics, experimental validation remains essential. We acknowledge these results represent necessary but not sufficient conditions for biological activity.

We are planning to develop a multi-tiered validation approach:

1. In silico validation:

Molecular dynamics (MD) simulations and free energy perturbation (FEP) calculations on top-ranked candidates to assess binding stability and thermodynamics under physiologically relevant conditions.

2. Biochemical assays:

Wet-lab binding assays (e.g., surface plasmon resonance, isothermal titration calorimetry) to measure actual binding affinities and kinetics.

3. Enhancing Computational Reliability:

To improve prediction reliability, we plan to develop PepBridge-derived scoring functions for virtual screening applications.

W3 Q4. Computational complexity and comparison of running time.

We appreciate the reviewer’s concern regarding the potential computational overhead introduced by our multi-modal diffusion framework. To address this, we compared PepBridge with several baseline methods in terms of training time, inference time per sample, GPU usage, and model size. The time cost is reported as the total time spent divided by the number of designed candidates. While multi-modal processing does introduce additional complexity relative to uni-modal approaches, our analysis shows that PepBridge achieves a good balance between computational efficiency and performance.

Table: Comparison of Computation Time and Resources

Q5. The generation of peptides targeting receptors without obvious binding pockets.

Thank you for this important question about PepBridge's applicability to receptors lacking well-defined binding pockets. We conducted experiments using randomly sampled surface patches to quantify this effect.

These results demonstrate that PepBridge remains effective even when applied to less constrained or uncertain surface regions, maintaining high structural and biochemical performance.

For receptors without obvious binding pockets, we recommend a two-step approach:

Binding site prediction preprocessing: Users can integrate established binding site prediction tools to identify potential interaction regions before peptide generation

Multi-site exploration: When binding sites are uncertain, PepBridge can be applied to multiple predicted or suspected interaction regions, allowing exploration of different binding hypotheses.

Q6. The effectiveness of incorporating surface modeling.

We appreciate the reviewer’s question. We conducted ablation studies using PepBridge variants with and without surface representations. The results, shown in the table below, demonstrate that incorporating surface information leads to substantial improvements across multiple performance metrics.

We are grateful for your constructive feedback and thoughtful suggestions that helped us refine our work.

Thank you for responding to my points. Thank you for the additional experiments, but they have introduced additional questions.

W2: Overall, PepFlow seems to have more advantages, especially Div_stru and BSR, which seem to have a significant gap compared to others. According to Q6, the introduction of the surface significantly improved performance. I am curious about how PepFlow performs after introducing surface modeling.

Q1: In my opinion, as the time step gradually increases, the performance of the model should gradually stabilize. However, the results show that the performance of the model is still continuously improving. Moreover, according to the comparison results in Table 1, the magnitude of improvement is quite significant. I am curious about the upper limit.

Q3: Could the author provide more analysis and discussion on the generated results compared to the protein surface used in the real world?

Thank you for your insightful comment. From the results shown in our chart, when the number of inference time steps exceeds the number used during training, BSR and Stability show a slight increase followed by a small decrease, while Div_stru fluctuates within a reasonable range. The performance of the affinity metric declines. We intend to refine sampling granularity and scaling to better capture detailed performance patterns.

Ma et al. [1] investigated inference-time scaling for diffusion models, reporting performance metrics such as FID and IS on ImageNet, and CLIPScore and Aesthetic Score on DrawBench (Figure 1 in [1]). The results show that some metrics (e.g., Aesthetic Score) may decline when inference time exceeds the training regime, despite initial improvements. In our case, we hypothesize that increasing the inference time steps beyond the training regime can lead to over-smoothing and loss of fine structural details in the generated protein structures. This reduces binding affinity, even if stability remains relatively steady. While longer inference can provide more refined sampling in principle, excessive steps may degrade structural features, preventing a uniform performance plateau across all metrics. Although our study focuses on proteins in 3D space rather than images, the analogy with the image domain suggests that increased inference time is not beneficial across all quality measures. We believe that investigating inference time and inference-time scaling is an interesting direction for future work.

[1] Ma N, Tong S, Jia H, Hu H, Su YC, Zhang M, Yang X, Li Y, Jaakkola T, Jia X, Xie S. Inference-time scaling for diffusion models beyond scaling denoising steps.

We appreciate the reviewer’s thoughtful feedback and suggestions, which have helped us improve the clarity and completeness of our work. Below, we address each of the concerns raised.

Q1. I would be interested in seeing how consistent the results are, as the authors did not provide any standard deviation. In the caption of table 1 they state that they draw 40 candidates, but it is hard so say if this is enough for the results to be consistent. Also, did the authors run the inference with multiple different seeds?

Thank you for this insightful question. We indeed conducted inference using 5 different random seeds, generating 40 candidates per seed. The table below reports the standard deviation across all runs. These results demonstrate that the performance of our method is consistent across different random seeds.

Q2. How does the runtime of the different methods compare, and how does it scale with size? How many inference steps did have been used for the diffusion model and how does the number of steps impact the sample quality?

We appreciate the reviewer’s concern regarding the potential computational overhead introduced by our multi-modal diffusion framework. To address this, we compared PepBridge with several baseline methods in terms of training time, inference time per sample, GPU usage, and model size. The time cost is reported as the total time spent divided by the number of designed candidates.

While multi-modal processing does introduce additional complexity relative to uni-modal approaches, our analysis shows that PepBridge achieves a strong balance between computational efficiency and performance. The modest increase in resource usage is well-justified by the substantial gains in structural precision and interface quality, especially in biologically critical tasks such as therapeutic protein design.

Table: Comparison of Computation Time and Resources

Q3. Although I am aware that diffusion bridges can be seen as a generalization of flow matching, I am curious to hear why the authors decided opted to use diffusion bridges instead of flow matching?

Thank you for this excellent question about our choice of diffusion bridges over flow matching. While both approaches can model transport between distributions, several factors made diffusion bridges more suitable for our protein surface generation task:

1. Stochastic exploration:

Protein-protein interactions inherently involve stochastic processes at the molecular level, including thermal fluctuations and conformational sampling. Diffusion bridges use stochastic differential equations (SDEs) that naturally incorporate noise during the generation process. This stochasticity is beneficial for exploring diverse protein surface configurations, as it allows the model to discover multiple viable binding modes rather than converging to deterministic solutions. Whereas the deterministic ordinary differential equations (ODEs) of flow matching may oversimplify the complex, probabilistic nature of surface complementarity between receptor and ligand.

2. Superior performance on structured biological data:

While flow matching has shown promise for image generation, diffusion models have demonstrated superior empirical performance on highly structured data like protein structures and molecular surfaces. Given that our 3D point cloud representations of protein surfaces exhibit complex geometric and biochemical constraints, the proven effectiveness of diffusion approaches on structured biological data was a decisive factor.

3. Flexibility in sampling:

The SDE framework allows us to easily switch between stochastic (DDPM) and deterministic (DDIM) sampling strategies, providing controllable trade-offs between diversity and quality.

Q4. It is not entirely clear to me how the authors encoded the surface structure. As I understood it, they use a point cloud for the surface, but I am not sure how many points do they use (one for each atom?) and if it could benefit from a more precise parameterization (i.e., more points).

We thank the reviewer for pointing out the need for clarification. We generate solvent-accessible surface representations using PyMol, which provides a triangulated molecular surface based on a probe radius that approximates both the Solvent-Accessible Surface Area (SASA) and Solvent-Excluded Surface (SES). From this surface, we sample a point cloud in which each point is annotated with 3D spatial coordinates and relevant physicochemical features, such as hydrophobicity and hydrogen bonding potential. To encode the surface of the receptor protein, we extract node-level features from the surface points and apply an MLP to obtain embeddings. Each surface node is represented by its 3D position ($surf_t$), hydrogen bonding potential ($surf_{hbond}$), and hydrophobicity score (${surf}_{hp}$). These features are concatenated and passed through an MLP to produce the surface node embeddings.

On average, we sample approximately three surface points per residue. This density was selected to balance surface resolution and computational efficiency. We also implement a subsampling strategy to avoid excessive surface points, which would significantly increase computational cost without substantial performance gains. While increasing the number of surface points could provide a finer geometric representation, our experiments showed diminishing returns in predictive performance beyond a certain point density. Exploring more adaptive or finer-grained surface parameterization remains an interesting direction for future work.

We sincerely thank you for your insightful comments, which have greatly enhanced the clarity and quality of our manuscript.

I thank the authors for their thorough response and additional clarifications.

I would like to follow up on a question regarding Q3. Did the authors consider other frameworks such as stochastic interpolants? It seems like it could also have been a good fit. Do they think that this would add any benefits? Also, the authors claim that diffusion models have shown superior performance. Could they provide any intuition on as to why?

We sincerely appreciate your valuable feedback and provide point-wise responses below.

Q1. There has been a long history to use surface complementarity to design ligands in traditional protein or small molecule design previous to deep learning era. I think authors should include this part of related works, and provide a discussion whether PepBridge can perform better than these methods.

You are absolutely correct that surface complementarity has been a fundamental principle in protein and small molecule design well before the deep learning era. We will add a section discussing classical approaches including:

Classical methods often relied on explicit modeling of geometric and physicochemical surface complementarity, such as shape matching or lock-and-key models. More recently, methods like MaSIF [1] represented protein surfaces as meshes and used predefined geometric and chemical descriptors to fingerprint surface patches. To reduce the high computational cost of mesh-based methods, dMaSIF [2] modeled molecular surfaces as point clouds, demonstrating competitive performance by assigning atom types and computing features directly from atomic positions.

In molecular generation, ShEPhERD [3] incorporates multiple surface-level features (e.g., shape, electrostatics, and pharmacophores) into a joint diffusion-denoising framework for 3D molecular graphs. DSR [4] models protein surfaces using implicit neural representations to capture dynamic structural features over time. Similarly, SurfPro [5] generates functional protein sequences conditioned on ground-truth surfaces and their biochemical properties. However, this reliance on known target surfaces limits its applicability in scenarios where accurate surface reconstruction is difficult or unavailable.

These methods demonstrate the power of surface-based design. However, they typically assume strict geometric complementarity or require hand-crafted features and ground-truth surfaces. In contrast, PepBridge leverages denoising diffusion bridge models (DDBMs) to learn a data-driven mapping between receptor and ligand interfaces. This allows PepBridge to capture non-obvious, flexible, and non-complementary interactions that classical methods might miss—especially important in peptide binding, where induced fit and flexible recognition often play a key role.

We will update the manuscript to reflect this broader context and provide a more detailed comparison highlighting the strengths and limitations of surface-based methods relative to PepBridge.

[1] Deciphering interaction fingerprints from protein molecular surfaces using geometric deep learning. Nature Methods.

[2] Fast end-to-end learning on protein surfaces. dMaSIF CVPR 2021.

[3] ShEPhERD: Diffusing shape, electrostatics, and pharmacophores for bioisosteric drug design. ICLR 2025.

[4] DSR: Dynamical Surface Representation as Implicit Neural Networks for Protein. NeurIPS 2023.

[5] SurfPro: Functional Protein Design Based on Continuous Surface. ICML 2024.

Q2.1 How is the robustness of surface complementary methods against imperfect structure? The error of crystal structure is not significant in whole protein structure, but can be harmful when describing very local surface. If we use AlphaFold-like models to predict the receptor structure, the error could be even larger.

This is a very good question regarding the sensitivity of surface-based methods to structural inaccuracies. ! We agree that while global structural deviations may be minor (e.g., 0.5–2.0 Å RMSD), even small atomic-level perturbations can significantly affect local surface features such as curvature, pocket geometry, accessible surface area, and electrostatic potential. This challenge is especially relevant when using predicted structures from models like AlphaFold, where the overall fold is typically reliable, but side-chain placements and flexible loop regions may be less accurate.

To address this issue and improve the robustness of PepBridge, we are considering the following strategies:

1. Learned Error Tolerance:

The model could be trained on a diverse set of protein structures—including high-resolution crystal structures, NMR ensembles, and computational models—covering a broad range of structural qualities. This heterogeneity allows the model to learn representations that are inherently more tolerant to local inaccuracies and noise, improving its generalizability across different structure sources.

2. Ensemble-Based Modeling:

To mitigate the reliance on a single static structure, especially for predicted or low-resolution inputs, PepBridge can be extended to incorporate ensemble-based inputs. Using multiple conformations or perturbed variants (e.g., from molecular dynamics simulations or AlphaFold confidence-based sampling) can help account for uncertainty and enhance prediction stability.

3. Empirical Robustness Evaluation:

We are actively investigating robustness by evaluating PepBridge on (i) structures with synthetic coordinate perturbations, (ii) comparisons between AlphaFold-predicted structures and corresponding crystal structures; and (iii) NMR ensembles that capture native conformational variability. These experiments help quantify how sensitive predictions are to structural imperfections.

Additionally, we acknowledge that proteins are not rigid entities; they undergo constant fluctuations in physiological conditions. Capturing the dynamic nature of the surface—rather than relying solely on static snapshots—could further enhance modeling accuracy for protein–protein interactions. Future work will explore integrating dynamic surface representations derived from conformational ensembles or time-averaged properties.

Q2.2 Another problem is that the surface complementarity condition can be misleading sometimes. I think this could probably be a major advantage for learning-based methods, because they don't need to assume a very strict complementarity but can learn from data. Could authors help me understand how is PepBridge designed to address this issue?

Thank you for this insightful question about surface complementarity limitations. We fully agree that strict surface complementarity assumptions can be misleading and that learning-based methods offer significant advantages in this regard.

However, it is worth noting that several key innovations are specifically designed in PepBridge to address the limitations of rigid surface complementarity:

1. Learning-based Surface Relationship Modeling:

Rather than assuming strict geometric complementarity, PepBridge employs denoising diffusion bridge models (DDBMs) to learn the complex relationship between receptor and ligand surfaces from data. This allows the model to capture subtle, non-complementary interactions that are crucial for binding but would be missed by purely geometric approaches.

2. Integration of Biochemical Properties:

PepBridge goes beyond surface geometry by incorporating biochemical properties alongside geometric features. This enables the model to learn that effective binding often involves electrostatic interactions, hydrophobic patches, and other physicochemical factors that don't necessarily require perfect shape complementarity.

3. Flexible Surface-Structure Coupling:

Our joint design framework enables dynamic adjustment between surface geometry and backbone structure through Shape-Frame Matching Networks (SFMN). This flexibility encourages the generation of binding interfaces that may not exhibit perfect complementarity but are biologically relevant and energetically favorable.

4. Multi-Step Refinement Process:

The multi-model diffusion approach allows for iterative refinement where initial surface predictions can be adjusted based on structural constraints and vice versa, leading to more realistic binding interfaces that balance complementarity with other binding determinants.

All above designs make PepBridge generate peptides that engage targets through diverse binding modes, including those involving conformational flexibility and partial complementarity - scenarios where traditional geometric complementarity assumptions would fail.

Q3. I think there should be more ablation study contributed to understand the performance of separate components. For example, how is the DDBM surface diffusion part and structure design part influence each other? I notice that the authors already provide some ablation study on the change of joint performance, but I wonder if it is possible to decompose the model totally.

We thank the reviewer for this insightful question. To better understand the contributions of individual components in our model, we performed ablation studies using several PepBridge variants with and without surface diffusion information. As shown in the table below, incorporating surface features improves performance across the evaluation metrics.

We would like to thank you again for your valuable comment and suggestions, which significantly improve our paper quality.

This answer is adequate and has addressed all my concerns. I especially appreciate your detailed ablation study. I will keep my scores unchanged.

Thank you for your thoughtful questions and for engaging deeply with our work. We respond to these suggestions point-by-point below:

Q1. As I understand, the model does not use receptor backbone or sequence information. Would adding these information further help?

Thank you for this important question. Our model does indeed incorporate both receptor backbone and sequence information. Specifically, for both receptor and ligand residues, we initialize node embeddings using multiple structural and sequence features: residue indices, atom coordinates, backbone dihedral angles, side-chain angles, and the diffusion timestep $t$. For edge embeddings between residue pairs, we incorporate residue-type pairs, relative sequence positions, pairwise distances, and relative orientations (details can be found in Appendix D).

Our denoising network architecture is built on Invariant Point Attention (IPA), which uses SE(3)-invariant attention mechanisms to effectively model interactions between the receptor and peptide while preserving the geometric relationships encoded in the backbone structure.

Our experimental results demonstrate that receptor backbone and sequence information are crucial contributors to model performance. We believe this comprehensive incorporation of structural and sequence features contributes significantly to the model's performance.

Q2. Could PepFlow benefit from adding surface context similarly? Also, can you elaborate on why the performance gain is modest compared to PepFlow despite richer modeling

Thank you for this insightful question regarding both the potential for surface context in PepFlow and our performance comparison.

Surface Context in PepFlow:

PepFlow could potentially benefit from incorporating surface context similar to our approach. Surface complementarity is fundamental to protein-peptide binding, and adding explicit surface representations could enhance PepFlow's ability to model binding site compatibility and geometric constraints.

Performance Comparison and Methodological Differences:

The key differences lie in the underlying mathematical frameworks: PepFlow uses flow matching with deterministic ordinary differential equations (ODEs) that learn direct velocity fields between distributions. In contrast, our diffusion bridge approach employs stochastic differential equations (SDEs) that incorporate controlled noise during generation. This stochasticity offers several advantages for peptide-receptor binding:

1. Exploration of binding diversity:

The stochastic process naturally explores multiple viable binding modes, which is crucial given that peptides often exhibit conformational flexibility at binding sites.

2. Surface complementarity modeling:

The noise inherent in SDEs helps model the probabilistic nature of surface-surface interactions, where slight variations in positioning can lead to different binding modes.

3. Sampling flexibility:

Our SDE framework allows switching between stochastic (DDPM) and deterministic (DDIM) sampling strategies, providing controllable trade-offs between diversity and binding mode exploration.

Future Directions: We plan to investigate combining surface context with flow matching formulations to explore the benefits of both approaches.

Q3. Computational complexity analysis, runtime comparison with baselines, and the impact of computational efficiency on practical applicability.

We appreciate the reviewer’s concern regarding the potential computational overhead introduced by our multi-modal diffusion framework. To address this, we compared PepBridge with several baseline methods in terms of training time, inference time per sample, GPU usage, and model size. The time cost is reported as the total time spent divided by the number of designed candidates.

While multi-modal processing does introduce additional complexity relative to uni-modal approaches, our analysis shows that PepBridge achieves a strong balance between computational efficiency and performance. The modest increase in resource usage is well-justified by the substantial gains in structural precision and interface quality, especially in biologically critical tasks such as therapeutic protein design.

Table: Comparison of Computation Time and Resources

We appreciate your thorough review and advice to make our study more readable.