Title: Accurately Predicting Protein Mutational Effects via a Hierarchical Many-Body Attention Network
Venue: NeurIPS 2025
Accepted: yes
Keywords: Protein Mutational Effects

Protein-Protein Interactions

Binding Free Energy

Hierarchical Many-Body Attention
Abstract: Predicting changes in binding free energy ($\Delta\Delta G$) is essential for understanding protein-protein interactions, which are critical in drug design and protein engineering. However, existing methods often rely on pre-trained knowledge and heuristic features, limiting their ability to accurately model complex mutation effects, particularly higher-order and many-body interactions.
To address these challenges, we propose H3-DDG, a Hypergraph-driven Hierarchical network to capture Higher-order many-body interactions across multiple scales. By introducing a hierarchical communication mechanism, H3-DDG effectively models both local and global mutational effects.
Experimental results demonstrate state-of-the-art performance on multiple benchmarks. On the SKEMPI v2 dataset, H3-DDG achieves a Pearson correlation of 0.75, improving multi-point mutations prediction by 12.10%. On the challenging BindingGYM dataset, it outperforms Prompt-DDG and BA-DDG by 62.61% and 34.26%, respectively.
Ablation and efficiency analyses demonstrate its robustness and scalability, while a case study on SARS-CoV-2 antibodies highlights its practical value in improving binding affinity for therapeutic design.

Accurately Predicting Protein Mutational Effects via a Hierarchical Many-Body Attention Network

Abstract

Predicting changes in binding free energy ( ∆∆ G ) is essential for understanding protein-protein interactions, which are critical in drug design and protein engineering. However, existing methods often rely on pre-trained knowledge and heuristic features, limiting their ability to accurately model complex mutation effects, particularly higher-order and many-body interactions. To address these challenges, we propose H3-DDG , a H ypergraph-driven H ierarchical network to capture H igherorder many-body interactions across multiple scales. By introducing a hierarchical communication mechanism, H3-DDG effectively models both local and global mutational effects. Experimental results demonstrate state-of-the-art performance on multiple benchmarks. On the SKEMPI v2 dataset, H3-DDG achieves a Pearson correlation of 0.75, improving multi-point mutations prediction by 12.10%. On the challenging BindingGYM dataset, it outperforms Prompt-DDG and BA-DDG by 62.61% and 34.26%, respectively. Ablation and efficiency analyses demonstrate its robustness and scalability, while a case study on SARS-CoV-2 antibodies highlights its practical value in improving binding affinity for therapeutic design.

1 Introduction

Protein-protein interactions (PPIs) [19, 9, 12, 32] are fundamental to numerous biological processes, driving key cellular functions such as signal transduction [21], immune response [35, 22], and metabolic regulation [40]. A precise understanding of how mutations alter binding free energy ( ∆∆ G ) in PPIs is critical for a wide range of applications, including drug design [2, 7, 20, 31], protein engineering [5], and elucidating the molecular basis of disease [33].

Binding free energy quantifies the thermodynamic stability of protein complexes and is inherently governed by the physical interactions between amino acids. These interactions span multiple spatial and structural scales, from pairwise atomic forces-such as hydrogen bonding and van der Waals interactions-to higher-order many-body effects, including hydrogen bond networks and π -π stacking interactions. Accurately modeling these intricate interactions is essential for predicting the functional and structural consequences of mutations [11, 13]. This becomes particularly challenging in multi-point mutation scenarios, where complex interdependencies between mutation sites often emerge, further complicating the prediction task.

Accurately predicting ∆∆ G remains a critical yet challenging task, as existing computational methods face inherent limitations. These methods can be categorized into structure-based approaches and inverse folding-based models. Structure-based approaches leverage well-designed training tasks, such as protein inverse folding [39], side-chain modeling [26, 23, 28], masked modeling [37], and data augmentation [38], to extract protein representations from structural data. However, they often

fall short in effectively transferring the learned structural knowledge to ∆∆ G prediction. This shortcoming arises from their limited capacity to explicitly model higher-order and many-body interactions, which are crucial for capturing the physicochemical impacts of mutations.

On the other hand, inverse folding-based models, such as ProteinMPNN-DDG [10] and BA-DDG [18], predict ∆∆ G by estimating likelihood differences between native and mutant sequences relative to stability. While these models perform well for single-point mutations, they rely on indirect proxies, such as Boltzmann-Alignment, to approximate binding free energy. This reliance limits their effectiveness in multi-point mutation scenarios, which involve intricate interdependencies among mutation sites and require a deeper understanding of protein energetics and many-body interaction dynamics. These limitations underscore the need for an approach capable of capturing higher-order interactions and adapting to the complexity of multi-point mutation scenarios.

In this work, we introduce H3-DDG , a H ypergraph-driven H ierarchical network to capture H igherorder many-body interactions across multiple scales for ∆∆ G predictions. By leveraging a manybody attention communication mechanism, H3-DDG effectively models higher-order and many-body interactions across multiple scales, enabling precise predictions of binding free energy changes, particularly in challenging multi-point mutation contexts.

Experimental results demonstrate state-of-the-art performance across multiple benchmarks. On the SKEMPI v2 dataset [17], H3-DDG achieves a Pearson correlation of 0.75, improving multi-point mutations prediction by 12.10%. On the challenging BindingGYM dataset [24], it outperforms Prompt-DDG and BA-DDG by 62.61% and 34.26%, respectively. Notably, in ablation studies, the many-body attention mechanism proves to be critical, contributing a 5.4% improvement in prediction accuracy when incorporated. Additionally, the hypergraph-driven hierarchical design enables the model to effectively capture long-range dependencies, which are essential for accurately modeling complex mutation scenarios. A case study on SARS-CoV-2 antibodies further highlights H3-DDG's ability to predict binding affinity changes with high precision, demonstrating its practical utility in real-world applications. These results underscore the capability of H3-DDG to handle intricate mutational landscapes and its potential for broad applicability in protein engineering and drug design. The key contributions of this work are as follows:

We introduce a hierarchical communication mechanism to model local and global interactions, capturing higher-order effects like hydrogen bond networks and π -π stacking.

We propose a many-body attention network to explicitly model higher-order and many-body interactions, enabling robust predictions in complex mutational scenarios.

H3-DDG achieves state-of-the-art performance on multiple benchmark datasets, outperforming BA-DDG by 12.10% in multi-point mutation prediction on the SKEMPI v2 dataset and by 34.26% on the BindingGYM dataset.

2 Related Work

Efforts to predict the change in binding free energy ( ∆∆ G ) have resulted in a variety of computational approaches, broadly categorized into two main types: (1) methods leveraging structure-based training tasks and (2) methods utilizing inverse folding models for ∆∆ G prediction. While these approaches have shown promise, they also highlight significant limitations, particularly in capturing higher-order interactions and complex multi-point mutation effects [30, 32].

The first category focuses on designing training tasks that utilize structural information to enhance the modeling of protein energetics. These include techniques such as protein inverse folding [39], side-chain modeling [26, 23, 28], and masked modeling [37]. These approaches attempt to extract meaningful representations of protein structures by optimizing for tasks aligned with physical principles of molecular interactions. However, their reliance on pre-trained knowledge limits their ability to explicitly model higher-order and many-body interactions, which are critical for understanding the thermodynamic effects of mutations. This gap restricts their utility in directly predicting ∆∆ G , particularly in complex mutation scenarios.

The second category focuses on adapting inverse folding models to predict ∆∆ G [25]. Models such as ProteinMPNN-DDG [10] and BA-DDG [18] leverage the ability of inverse folding models to assess the likelihood of native and mutant amino acids within a structural context. These methods predict ∆∆ G by estimating the differences in likelihood scores between native and mutant sequences with

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their relative stability. However, likelihood-based scoring serves as an indirect proxy for binding free energy changes and often fails to capture explicit physical interactions critical to ∆∆ G prediction. Moreover, their performance degrades significantly in multi-point mutation scenarios, where the interdependencies between mutations require a nuanced understanding of higher-order effects.

To address these challenges, we introduces a hierarchical communication mechanism and a manybody attention network to explicitly model local and global interactions across multiple scales. These innovations capture higher-order and many-body effects, such as hydrogen bond networks and π -π stacking, while explicitly modeling mutation interdependencies in multi-point mutation scenarios.

3 Methodology

In this section, we introduce H3-DDG, a Hypergraph-driven, Hierarchical, and Higher-order interaction network designed to predict binding free energy changes ( ∆∆ G ) in protein-protein interactions (PPIs). The overall workflow is illustrated in Figure 1.

3.1 Preliminaries and Notations

We represent a protein graph as G = ( V, E ) , where each node i ∈ V represents a residue with feature vector h i , and each edge ( i, j ) ∈ E encodes interactions via feature vector e ij .

Building on ProteinMPNN-DDG [10] and BA-DDG [18], which link ∆∆ G prediction with inverse folding, we also fine-tune a pretrained inverse folding model (ProteinMPNN [6]) to tackle this task. H3-DDG extends the capabilities of the inverse folding model by introducing hierarchical graph construction and many-body higher-order attention, enabling it to effectively capture complex dependencies across multiple mutation sites and improve protein representation learning.

To predict binding free energy ∆ G , we compute amino acid probabilities and use negative loglikelihoods as a surrogate. For a wild-type (wt) complex and its mutant (mut) counterpart, the binding

free energy change is defined as:

3.2 Hierarchical Graph Construction for Multi-Scale Modeling

Residue-Level Graph. As described in Section 3.1, each residue i is represented by a feature vector h i , and each edge ( i, j ) encodes spatial or chemical interactions via feature vector e ij .

Following the approch of ProteinMPNN [6], residues are represented using key atoms: N, C α , C, O, and C β , To construct the amino acid-level graph. Edges are formed by identifying the topk nearest neighbors based on C α -C α distances. Each edge ( i, j ) incorporates geometric features using radial basis functions (RBFs) [4] over 25 atom pair distances (e.g., N-N, C-C). For a given distance d ij , the RBF is defined as:

where µ k are K predefined centers and σ is the bandwidth.

These RBF ij features are concatenated with relative positional encodings PE ij , which include sequence offsets and chain identities. The final edge features e ij are computed as:

where W e is the learnable parameters and [ · ∥ · ] denotes concatenation. This graph effectively captures residue-level spatial and chemical interactions while preserving sequence context.

Mutation-Centered Hypergraph. Directly applying higher-order attention on amino acid-level graphs is computationally expensive, with complexity scaling as O ( N 3 ) or O ( N 4 ) , as demonstrated in [16, 30]. To address this challenge, we construct a mutation-centered hypergraph that reduces computational complexity while preserving critical local and global interactions.

We construct the hypergraph by clustering residues into hyperedges that capture higher-order spatial dependencies. Clustering is initialized at mutation sites to prioritize biologically salient regions, then expanded by iteratively adding residues with maximal Euclidean distance from existing centroids. Each residue is assigned to its nearest centroid based on C α coordinates. Hyperedge features h ( e ) are computed via mean-pooling over constituent residue embeddings h , yielding a compact representation that preserves structural topology and improves computational efficiency.

This mutation-centered hypergraph construction ensures that the influence of mutation sites is retained during graph compression and facilitates the efficient modeling of higher-order interactions. For further implementation details, refer to Appendix A.1.

Fine-Grained Mutation Subgraph. To capture the local effects of mutations, we extract finegrained subgraphs centered on each mutation site. These subgraphs include all residues with C α atoms within an 8Å radius of the mutation centroid in 3D space, preserving key structural and biochemical contexts. Node and edge features in these subgraphs, denoted as h ( m ) and e ( m ) , respectively, enable localized analysis of conformational changes and energetic impacts, providing a focused view of mutation-induced effects on binding affinity.

3.3 Many-Body Attention Network

In this section, we describe the many-body attention mechanisms applied at different hierarchical levels of the graph. These include pairwise (2-body) message passing over the residue-level graph, triplet (3-body) attention between hyperedges in the mutation-centered hypergraph, and quadruplet (4-body) attention within fine-grained mutation subgraphs, as illustrated in Figure 1(b).

2-body Message Passing in Residue-level Graph. We utilize a message passing mechanism to jointly update node features h i and edge features e ij in the residue-level graph (Section 3.2). Node-wise messages ∆ h i are computed by projecting concatenated node and edge features through

a multi-layer perceptron (MLP):

where [ · ∥ · ] denotes vector concatenation, ϕ is the GELU activation, and W m 1 , W m 2 , W m 3 are learnable projection matrices. The node features h i are updated via a residual edge with layer normalization and a position-wise feed-forward network (FFN):

Edge features e ij are updated using the updated node features through a similar MLP-based mechanism:

This mechanism enables the model to integrate residue-level spatial and chemical interactions efficiently across the residue-level graph.

3-body Attention Mechanism for Hyperedge Interactions. Building on the global structural context learned from 2-body message passing in the full-residue graph, we implement a 3-body attention mechanism to capture higher-order interactions between hyperedges in the mutation-centered hypergraph (Section 3.2). Here, each hyperedge is represented by its feature vector h ( e ) , capturing the residue cluster-level information.

To model pairwise relationships between hyperedges, we first construct a pairwise interaction tensor f ( e ) ij ∈ R N × N × D , where N is the number of hyperedges and D is the feature dimension. Here, i and j index hyperedges in the hypergraph. Each element f ( e ) ij is computed as:

where ϕ is a non-linear activation function (e.g., GELU) and ⊗ denotes the outer product.

An asymmetric attention mechanism is employed by linearly projecting f ( e ) ij through learnable weight matrices and biases to obtain the query vector q ij = W q f ( e ) ij + b q , key vector k jk = W k f ( e ) jk + b k , and value vector v jk = W v f ( e ) jk + b v . In addition, the mechanism introduces a bias term b ik and a gating vector g ik via separate linear projections for adaptive modulation. The triplet attention scores a ijk are then computed using a gated softmax function:

where Softmax ensures that the attention weights are normalized over hyperedge neighbors, and σ ( g ik ) applies a sigmoid activation to adaptively modulate the attention.

Each hyperedge aggregates attention-weighted messages from its spatial neighbors, and the aggregated features are fused with the corresponding full-residue node feature h r ( i ) via residual addition, followed by layer normalization:

Here, r ( i ) denotes the mapping from hyperedge index i to the corresponding residue index in the residue-level graph.

4-body Attention Mechanism around Mutation Sites. To capture higher-order interactions in the mutation environment, we extend the 3-body attention mechanism to 4-body attention on the Fine-Grained Mutation Subgraph (Section 3.2), where h ( m ) and e ( m ) denote the node and edge features of the mutation subgraph. This subgraph focuses on residues within an 8Å radius of the mutation site, enabling the model to learn localized interactions among residue quadruples. The

extension increases computational complexity by only a constant factor, preserving scalability while capturing more complex interactions. The full computational process is illustrated in Figure 1(c).

Specifically, we construct two interaction tensors to encode localized multi-residue interactions as inputs to the 4-body attention mechanism:

Here, ϕ is a non-linear activation function (e.g., GELU), and ⊗ denotes the outer product. The tensors f ( m ) ij ∈ R N × N × D and f ( m ) jkl ∈ R N × E × D respectively encode node-node and node-edge interactions, where N and E are the number of residues and edges on the mutation subgraph, and D is the feature dimension. Indices i, j, k, l refer to nodes within the mutation subgraph.

Inspired by cross-attention, the tensors f ( m ) ij and f ( m ) jkl are projected into query, key, and value vectors:

where W q , W k , W v , W b , and W g are learnable projection matrices.

The quadruplet attention scores a ijkl are computed via a gated softmax function:

Each node in the mutation subgraph aggregates attention-weighted messages from its spatial neighbors, and the aggregated features are fused with the original node features via a residual connection, followed by dropout and layer normalization:

This 4-body attention mechanism enables the model to encode complex, localized interaction motifs around mutation sites, thereby improving its ability to predict binding free energy changes with structural and biophysical fidelity.

3.4 Prediction Module and Learning Objective

Following the multi-scale representation learning with many-body attention in Section 3.3, we obtain a residue-level representation h ∈ R N × d . Let s = { s 1 , . . . , s N } be the amino acid sequence. Inspired by autoregressive models like ProteinMPNN, we model P ( s i | s <i , h ) to compute the negative log-likelihoods E wt and E mut for wild-type and mutant sequences. As in RDE-Network [26] and BA-DDG [18], single-chain contributions are subtracted to isolate binding effects, yielding energy-like scores:

We minimize the mean squared error between ∆∆ G pred and ∆∆ G true over n training samples:

4 Experiments

4.1 Experimental Settings

Datasets. We used SKEMPI v2 [17], a benchmark with 7,085 mutations across 348 protein complexes, to evaluate ∆∆ G prediction. Following prior work [26, 37], we split the data into three non-overlapping folds by complex to avoid data leakage. Additionally, we evaluated on BindingGYM [24], the largest dataset for protein-protein interactions, with 508,962 curated entries and a high proportion of multi-point mutations. We used the hardest inter-assay split, focusing on the fold with the most multi-point mutations for testing (details in Appendix A.2).

Baselines. We compared H3-DDG against unsupervised and supervised methods. Unsupervised approaches include energy-based models (e.g., Rosetta [1], FoldX [8]), evolutionary sequence models (e.g., ESM-1v [27], Tranception [29]), and structure-guided pretrained models (e.g., ESM-IF [15], MIF-∆ logits [39]). Supervised methods include end-to-end architectures (e.g., DDGPred [34]) and pretraining-finetuning frameworks (e.g., MIF-Network [39], RDE-Network [26], DiffAffinity [23], Prompt-DDG [37], ProMIM [28], Surface-VQMAE [36], MSM-Mut [14], BA-DDG [18]).

Metrics. We use five metrics: Pearson, Spearman, RMSE, MAE, and AUROC. For SKEMPI, in addition to the global metrics, we also report per-structure metrics. For BindingGYM, metrics are reported per-DMS (deep mutational scanning). Detailed metric definitions and calculations are provided in Appendix A.3.

Implementation details. The hyperparameters and hardware details are provided in Appendix A.4. The code is available at https://github.com/biomed-AI/H3-DDG .

4.2 Results on SKEMPI v2 Dataset

As shown in Table 1, H3-DDG outperforms existing baseline approaches across all evaluation metrics in the all-, single-, and multi-point mutation scenarios. Notably, in practical applications where affinity modulation through multi-point amino acid mutations is particularly critical, H3-DDG demonstrates significant advantages in the multi-mutation prediction task: achieving a Spearman correlation coefficient of 0.7341, which is 10.39% higher than the second-best method; at the per-structure level, this metric reaches 0.5520, surpassing the second-best approach by 12.10%. Additional comparisons with more baseline methods are provided in Appendix B.2, Table 6.

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0.4448

0.4924

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0.4959

0.1763

0.4140

0.2056

0.3870

RDE-Linear

Difffinity

Prompt-DDG

BA-DDG

H3-DDG

RDE-Linear

DiffAffinity

Prompt-DDG

BA-DDG

H3-DDG

These results are enabled by our introduction of a hierarchical communication mechanism combined with a many-body attention network. This proposed framework effectively models both local and global interactions, capturing higher-order effects such as π -π stacking and explicitly addressing complex synergistic and many-body interactions in mutational scenarios. As further illustrated in Figure 2, the per-structure distributions of Pearson and Spearman correlation coefficients across five representative methods show that H3-DDG achieves significantly higher correlation values, with distributions concentrated in high-correlation regions, highlighting its robustness compared to other approaches.

4.3 Results on BindingGYM Dataset

To validate the robustness of H3-DDG, we evaluate it on the larger and more challenging BindingGYM dataset, demonstrating its scalability and generalizability. As shown in Table 2, H3-DDG outperforms all baseline methods across all metrics, achieving the highest Pearson correlation (0.3057), which surpasses the second-best method (BA-DDG) by 34.26%. Particularly in the ≥ 3 mutation scenario, it also achieves the best Spearman correlation (0.2755), exceeding the second-best method by 21.96%. Additionally, H3-DDG achieves the best AUROC across all mutations and demonstrates competitive RMSE. These results highlight H3-DDG's robustness, scalability, and effectiveness for accurately predicting protein-protein binding in complex mutational landscapes, especially in multi-point mutation scenarios.

While H3-DDG's RMSE under single-point mutations is slightly higher than the baseline methods (e.g., BA-DDG), this can be attributed to the BindingGYM dataset spanning different DMS experiments, where the absolute ∆∆ G values vary significantly across experiments. However, we focus more on the ranking and correlation within each DMS experiment, making the Pearson and Spearman metrics more critical for evaluation.

4.4 Ablation Study

Ablation Study on Pooling Types.

As shown in Table 3, H3-DDG outperforms other graph pooling methods like DiffPool [41] and MinCutPool [3], which rely on predicted assignment matrices and fail to predefine mutation sites as central nodes. By explicitly designating mutation sites as cluster centers, H3-DDG effectively captures long-range dependencies, enabling superior performance in modeling many-body interactions.

Ablation Study on Attention Mechanisms. Table 9 also demonstrates

the importance of many-body attention in ∆∆ G prediction. While 3-body attention improves performance by modeling local interactions around mutation sites, extending to 4-body attention further enhances accuracy by capturing more complex dependencies. These findings highlight many-body attention as a key factor in H3-DDG's strong performance and efficient modeling. Our model strikes a balance between efficiency and performance, influenced by two main factors: the number of hyperedges in the hypergraph and the number of edges in the 4-body attention. Notably, our method achieves a Pearson correlation of 0.7341, surpassing BA-DDG's 0.6650, with a relatively small efficiency trade-off. Additional details are provided in Appendix B.4 and B.5.

4.5 SARS-CoV-2 Antibody Optimization

Predicting ∆∆ G is essential for identifying affinity-enhancing mutations. We target five SARS-CoV2 neutralizing mutations [34] within the heavy-chain CDRs (26 residues, 494 single-point variants). Models are fine-tuned on SKEMPI v2.0 to rank mutations by predicted ∆∆ G , with lower values indicating stronger binding. Table 4 benchmarks performance against top baselines, highlighting mutations ranked in the top 10%. Only our model achieves an average rank below 10%, demonstrating strong generalization and practical utility in antibody design.

5 Conclusion

In this work, we introduce H3-DDG, a novel framework that leverages hierarchical communication mechanisms and many-body attention to tackle the challenges of protein-protein binding prediction. By explicitly modeling higher-order interactions and designating mutation sites as cluster centers,

H3-DDG captures complex dependencies and synergistic effects, particularly excelling in multi-point mutation scenarios. Evaluations on SKEMPI v2 and BindingGYM show that H3-DDG outperforms state-of-the-art methods across most metrics, demonstrating robust performance and scalability. While H3-DDG shows promise for protein design and affinity prediction, its performance on large datasets and diverse mutations needs further study. Integrating it with experimental workflows will be key to validating real-world applicability. Future work will tackle these challenges to expand its impact.

Acknowledgments and Disclosure of Funding

This study has been supported by the Guangdong S&T Program [2024B0101040005], the Guangdong S&T Program [2023B1111030002], the National Natural Science Foundation of China [62041209], the Natural Science Foundation of Shanghai [24ZR1440600], the China Postdoctoral Science Foundation [2025M771540, GZB20250391], the Guangdong Basic and Applied Basic Research Foundation [2025A1515060011], and the Lingang Laboratory [LGL-8888].

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A Additional Details

A.1 Graph Pooling Method of H3-DDG

To enable scalable learning on biomolecular graphs, H3-DDG employs a deterministic graph pooling strategy that compresses node representations while preserving spatial and functional context. Given residue-wise features h and their C α coordinates x ∈ R N × 3 , we form a reduced set of hyper-nodes via Farthest Point Sampling (FPS) with mutation-aware initialization. Specifically, mutation sites are first selected as initial cluster centroids c k , and additional centroids are iteratively sampled to maximize coverage under Euclidean distance.

Each residue is then assigned to the nearest cluster centroid based on the Euclidean distance between its C α coordinates and the centroids. Feature aggregation is performed within each cluster by computing the mean of the features of all residues assigned to that cluster:

where S k is the set of residues assigned to cluster k . This produces a set of K cluster-level representations { h k } K k =1 , with K dynamically determined as K = ⌊ L/R ⌋ , where L is the number of valid residues and R is a predefined reduction ratio.

Anchored to mutation sites and guided by spatial diversity, our pooling method preserves locality and biological relevance. It is non-parametric and efficient, suitable for large or variable-length proteins.

A.2 Details of the Inter-Assay Split in BindingGYM

To evaluate generalization to unseen protein-protein interactions, we adopt the inter-assay split strategy from the BindingGYM dataset, following the approach of [24]. In this setting, assays are first clustered into five groups based on the sequences of their mutated proteins. Data from one cluster is held out for testing, while the remaining four are used for training. This split evaluates the models' ability to generalize to new assays, which holds significant practical significance.

We evaluate performance under three levels of mutational depth: ALL (all mutants), <3 (mutants with fewer than 3 mutations), and ≥ 3 (mutants with 3 or more mutations).

A.3 Details of metric definitions and calculations

For the SKEMPI v2 dataset, we use seven quantitative metrics, including five standard global criteria: Pearson and Spearman correlation coefficients, root mean square error (RMSE), mean absolute error (MAE), and area under the ROC curve (AUROC). To calculate AUROC, mutations are classified based on the sign of AUROC, mutations are classified based on the sign of ∆∆ G . In practice, correlations within individual protein complexes are often more relevant. To this end, following RDE-Network [26], we group mutations by protein structure, exclude groups with fewer than 10 mutations, and compute correlations separately for each structure. For the BindingGYM dataset, we use per-DMS (deep mutational scanning) metrics, including Pearson, Spearman, AUROC, and RMSE.

A.4 Training Details

A.4.1 Hyper-parameters

We used the Adam optimizer with a learning rate of 4e-4 and a batch size of 1, 2, depending on GPU memory and graph size. The model was trained for 20,000 iterations with 4 attention heads and a hidden dimension of 128. The number of hyperedges was selected from L/ 10 , L/ 6 , L/ 4 , and the number of edges in the 4-body attention module from 1 N , 2 N , 3 N , where L and N denote the numbers of residues and nodes, respectively. The pre-trained ProteinMPNN module used its default 3-layer configuration.

A.4.2 Hardware

Experiments were run on dual Xeon Gold 6248R CPUs and an RTX 4090 GPU under Ubuntu 22.04.

B Additional Results

B.1 H3-DDG Performance on Predicted Complex Structures

To evaluate robustness, we used AlphaFold 3 (AF3) to predict the complex structures for SKEMPI v2 and applied H3-DDG for prediction. While AF3 achieves high-quality structure prediction, its accuracy may still be lower than experimental structures. Importantly, H3-DDG showed only a slight decrease in performance on AF3-predicted structures, with a drop of 5.4% in Pearson correlation, demonstrating strong robustness and practical applicability.

B.2 Extended Baseline Comparisons on SKEMPI v2

Table 6 reports extended baseline results on SKEMPI v2, including additional methods to complement the main text comparisons. These include energy function-based, sequence-based, and unsupervised approaches, providing a broader context for evaluating performance across diverse methodological categories.

B.3 Ablation Results under Different Mutation Depths

Table 7 presents the full ablation results of different pooling strategies and many-body attention mechanisms across mutation types, including single-point, multi-point, and all-point mutations on SKEMPI v2. The results demonstrate that our proposed H3-DDG consistently outperforms baseline pooling methods (e.g., DiffPool [41] and MinCutPool [3]) under all mutation settings. Furthermore, incorporating 4-body attention around mutation sites yields the best performance across all metrics, validating the effectiveness of higher-order attention in capturing complex interaction patterns.

B.4 Ablation study on Edge Count in 4-body Attention

We investigate the impact of varying the number of edges in the 4-body attention mechanism around mutation sites on ∆∆ G prediction. The number of edges in the 4-body attention is directly proportional to the computational complexity. As shown in Table 8, where N denotes the number of nodes involved in the 4-body attention computation, selecting 2 · N edges yields a computational complexity of 2 · |N| 3 . The results indicate that increasing the number of edges leads to only a constant-factor increase in computational cost, yet yields sustained improvements in predictive performance. For example, the Pearson correlation, which was already high at 0.7352, further increases to 0.7501. These findings highlight that adding more edges in the many-body attention mechanism near mutation sites significantly enhances the model's capacity to capture complex manybody interactions, while maintaining manageable computational overhead, thereby driving further gains in prediction accuracy.

B.5 Efficiency Analysis

Our model balances efficiency and performance. As shown in Table 9, increasing hyperedges from L/ 10 to L/ 4 steadily improves Pearson correlation, reaching a peak of 0.7501 with L/ 4 hyperedges and 3 N 4-body edges. This setting maintains acceptable efficiency (4.34 it/s), only slightly slower than BA-DDG (6.25 it/s, Pearson: 0.7118). Similarly, increasing 4-body edges from 1 . 5 N to 3 N yields incremental gains, indicating denser subgraph modeling enhances accuracy at manageable cost. Additionally, as shown in Table 10, larger cutoff radii improve performance by capturing richer structural context, but gains diminish beyond 8 Å while cost rises. Thus, 8 Å is chosen as the optimal cutoff, offering the best accuracy-efficiency trade-off.

B.6 Scatter Plot of Prediction Results

In Figure 3, we present scatter plots comparing experimental and predicted ∆∆ G values for the three methods under multi-point mutation scenarios.

RDE-Linear

Prompt-DDG

H3-DDG

Pearson=0.4583

Pearson=0.6780

Pearson =0.7341

10

Spearman=0.4247

Spearman=0.6433

Spearman=0.6913

G

Predicted

0

5

10

10

5

0

5

10

10

-5

0

5

10

10

-5

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Experimental DDG

Experimental DDG

Experimental DDG

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents H3-DDG, a hypergraph-driven hierarchical network designed to capture higher-order many-body interactions for predicting binding free energy changes ($\\Delta \\Delta G$) in protein-protein interactions. The method constructs three hierarchical graph representations: residue-level graphs, mutation-centered hypergraphs, and fine-grained mutation subgraphs, employing 2-body, 3-body, and 4-body attention mechanisms to model complex interactions across multiple scales. H3-DDG achieves state-of-the-art performance on SKEMPI v2 and BindingGYM datasets, with particularly strong results in challenging multi-point mutation scenarios, demonstrating a 12.10% improvement over BA-DDG in multi-point mutations on SKEMPI v2.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths**\n\n---\n\n**S1**. Predicting $\\Delta \\Delta G$ for protein-protein interactions is crucial for drug design and protein engineering. The authors clearly analyze the limitations of existing methods in handling higher-order and many-body interactions, particularly for multi-point mutations where the exponential mutation space poses significant challenges.\n\n**S2**. The paper is generally well-written with clear presentation of the hierarchical architecture. The model design is simple yet effective, with well-motivated many-body attention mechanisms that effectively capture both local and global interactions. The hierarchical design effectively balances computational complexity with biological interpretability.\n\n**S3**. Extensive experiments across multiple benchmarks (SKEMPI v2, BindingGYM) demonstrate strong performance improvements. Thorough ablation studies convincingly show the effectiveness of each component, and practical validation on SARS-CoV-2 antibody optimization demonstrates real-world applicability.\n\n**S4**. The method achieves significant improvements in the challenging multi-point mutation scenarios, addressing a key limitation of existing approaches that struggle with complex interdependencies between mutation sites.\n\n### **Weaknesses**\n\n---\n\n**W1**. While the authors mention \"manageable computational overhead,\" detailed runtime comparisons with baseline methods are missing from the main text. Given the hierarchical nature and higher-order attention mechanisms, comprehensive computational cost analysis is crucial for practical adoption.\n\n**W2**. The repeated residual connections in equations (5) and (6) are unconventional and may preserve more original features, but this design choice detailed justification.\n\n**W3**. The paper lacks comparison with very recent methods like Light-DDG [1], which reportedly achieves 89.7$\\times$ inference acceleration and 15.45\\% performance gains over previous SOTA methods.\n\n---\n\n**Reference**\n[1] Wu et al. A Simple yet Effective DDG Predictor is An Unsupervised Antibody Optimizer and Explainer. ICLR 2025.\n\nQUESTIONS:\n**Q1**. How does H3-DDG compare with the recently proposed Light-DDG method in terms of both performance and computational efficiency?\n\n**Q2**. Can the hierarchical embeddings and many-body interaction representations learned by H3-DDG be effectively transferred to other protein engineering tasks?\n\n**Q3**. How sensitive is the method's performance to the choice of clustering parameters? \n\n**Q4**. What is the relative contribution of each hierarchical level (2-body, 3-body, 4-body) to the final performance, and how does this vary across different types of mutations? \n\n**Q5**. Given the computational requirements of the hierarchical many-body attention, what are the practical limitations for real-time protein design applications?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a neural network architecture to estimate free energy differences based on the output probabilities of an inverse folding model. The main contribution is a many-body attention mechanism operating on various graph and hypergraph representations.\n\nSTRENGTHS AND WEAKNESSES:\nWhile the introduction starts rather clearly, the methods lack rigor and clarity. Many elements are not defined or detailed (eg which 25 atom pair distances exactly in L118? How is K defined in Eq 2? How are positional encodings exactly computed, how is chain identity factored in? What is the precise definition of the hypergraph and its hyperedges? How are hyperedges constructed, by merging any two adjacent edges? Are they directed? What are the dimensions of the weight matrices W in the attention? How are hyperedge features $f_{ij}$ projected, are they flattened? What are the hyperparameters of the clustering? How were they chosen? What is the metric used in Table 4? etc.) Unfortunately this lack in clarity makes it hard to follow their (rather complex) model setup, and prevents reproducibility.\n\nThe motivation is sensible and the adressed problem is very relevant. However the claim that the proposed network learns hydrogen bond networks or $\\pi - \\pi$ stacking remains unsubstantiated. \n\nWhile I appreciate the presence of ablation studies, I have difficulties understanding what was done in the pooling ablation. Which part of the model was replaced, what were the baselines? A bit more context would be good here. I also wonder whether the observed performance gains might stem from the added graph representations and not the attention mechanisms. An ablation of the attention while keeping the full graph, the clustered graph, and the mutation subgraph (modeling them with ProteinMPNN) seems to be a relevant experiment here.\n\nI further wonder about the effect of hyperparameters. How does performance depend on eg the hypergraph construction?\n\nThat said, the experimental results are good and add value to the field. But given the issues of clarity and reproducibility I unfortunately cannot recommend to accept at this stage. Since there is no revision, I leave it to the discretion of the AC whether to trust that these issues will be addressed.\n\nQUESTIONS:\nSee above, further:\n- What is the raionale to choose 3-body interaction on the pooled graph and 4-body interaction on the mutation subgraph? Why not both the same or the other way around?\n- Can you quantify the added value of using a clustered graph instead of the full graph? How much is runtime reduced by this? Would it be feasible to run a comparison of the model on an unclustered graph?\n- L186: How does 4-body attention only increase complexity by constant factor?\n- Is using a pretrained ProteinMPNN necessary? Do they add performance? Can the model be trained from scratch? How do you justify using the pretrained weights for a model that has significantly different architecture?\n\nLIMITATIONS:\nThe discussion of limitations is very brief with one sentence in the conclusion. I would recommend to expand on this, and also discuss runtime complexity and scaling with various hyperparameters.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces H3-DDG, a novel  hierarchical network designed to predict binding free energy changes (`ΔΔG`) in PPIs mutants. The proposed captures higher-order interactions that are crucial for understanding mutational effects. The authors demonstrate SOTA on SKEMPI v2 and BindingGYM benchmarks. The authors also present a case study on SARS-Cov-2 antibody design.\n\nSTRENGTHS AND WEAKNESSES:\n#### Strengths \n\n* The use of hypergraph representations with many body attention mechanisms (up to 4-body) is well-motivated. This directly addresses the limitation of existing methods that primarily model pairwise interactions, missing crucial higher-order effects like hydrogen bond networks and `π-π` stacking.\n\n* The performance improvements are substantial - 12.10% better than BA-DDG on multi-point mutations in SKEMPI v2, and 34.26% improvement on BindingGYM\n\n* The hierarchical graph construction with mutation centered hypergraphs cleverly reduces computational complexity from O(N^3) or O(N^4) to manageable levels while preserving critical interaction information needed for model performance \n\n#### Weaknesses\n* the paper lacks theoretical analysis of why the specific choices (e.g., 8A radius for mutation subgraphs, specific hyperedge construction) are optimal. Some ablation on these hyperparameters would strengthen the work. \n* Table 8 shows that H3-DDG is still ~30% slower than BA-DDG in training. For very large protein complexes or proteome-wide screens, this could be limiting.  \n* The evaluation is limited on SKEMPI v2 and BindingGYM. Testing on additional datasets (Abdesign etc) would better demonstrate generalizability claims. \nThe paper doesn't compare against some recent methods like ThermoMPNN or the latest co-folding based approaches for stability prediction.\n\nQUESTIONS:\n* Could this architecture be adapted for related tasks like protein-ligand binding affinity prediction or enzyme design?\n* Given the success of AF3 and similar methods (Chai-1/ Boltz) , how does H3-DDG perform on predicted structures versus experimental structures?\n*  How sensitive is the method to the clustering approach for hypergraph construction?\n\nLIMITATIONS:\n* The model is trained on SKEMPI v2, which has known biases towards certain protein families and mutation types. How does this affect performance on underrepresented protein classes in the training dataset (membrane proteins etc) \n\n* The 4-body attention is restricted to local mutation subgraphs, potentially missing long range allosteric effects. \n\n* The model assumes standard conditions but `ΔΔG` is highly dependent on temperature, pH, and ionic strength. \n\n* The paper doesn't discuss GPU memory requirements. The 4-body attention on mutation subgraphs likely requires substantial memory, limiting application to large complexes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces H3-DDG, a hypergraph-driven hierarchical network to predict higher-order many-body interactions across multiple scales. The authors compared the performance of their method with that of others and demonstrated improved performance, especially in cases with three or more mutations.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The topic is undoubtedly critical to assessing the implications of protein mutations. The methods are clearly described with detailed comparison results. \n\nWeaknesses: The proposed methods are ad-hoc, although the proposals are all sensible. It is unclear what the practical implications of the improvement in prediction are, e.g., how these will translate into better drug targets, etc. Additionally, a discrepancy exists between the performance improvement reported in the abstract and that reported in the tables, which lowers the confidence in the results presented in the paper. Therefore, we have several questions followed by the previous mentioned weakness:\n\n1. The results are not reliable and the authors should include more discussions for ablation studies and hyper-parameter tuning. Do the authors also adjust the hyper-parameters of other baseline to make a fair comparison?\n\n2. The authors should mention the criteria for selecting the k-body attention design, as it seems that there is no unified rule for selecting a suitable k across all different tasks, and thus the authors should spend more context to discuss the effect of k.\n\n3. For the anti-body optimization task, it seems that H3-DDG did not perform well for RH103M, while other baseline methods such as MSM-Mut shows stronger performances. Could the authors explain more about the performance difference and what is the preference of proposed method?\n\nQUESTIONS:\nCan the authors discuss how the improvement translates into better performance in downstream tasks? It seems that the overall improvement over BA-DDG is relatively minor.\n\nLIMITATIONS:\nThe authors discussed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}