Boltzmann-Aligned Inverse Folding Model as a Predictor of Mutational Effects on Protein-Protein Interactions

## Abstract

Abstract Predicting the change in binding free energy ( Δ ⁢ Δ ⁢ G Δ Δ 𝐺 \Delta\Delta G roman_Δ roman_Δ italic_G ) is crucial for understanding and modulating protein-protein interactions, which are critical in drug design.
Due to the scarcity of experimental Δ ⁢ Δ ⁢ G Δ Δ 𝐺 \Delta\Delta G roman_Δ roman_Δ italic_G data,
existing methods focus on pre-training, while neglecting the importance of alignment.
In this work, we propose the Boltzmann Alignment technique to transfer knowledge from pre-trained inverse folding models to Δ ⁢ Δ ⁢ G Δ Δ 𝐺 \Delta\Delta G roman_Δ roman_Δ italic_G prediction.
We begin by analyzing the thermodynamic definition of Δ ⁢ Δ ⁢ G Δ Δ 𝐺 \Delta\Delta G roman_Δ roman_Δ italic_G and introducing the Boltzmann distribution to connect energy with protein conformational distribution.
However, the protein conformational distribution is intractable; therefore, we employ Bayes’ theorem to circumvent direct estimation and instead utilize the log-likelihood provided by protein inverse folding models for Δ ⁢ Δ ⁢ G Δ Δ 𝐺 \Delta\Delta G roman_Δ roman_Δ italic_G estimation.
Compared to previous inverse folding-based methods, our method explicitly accounts for the unbound state of protein complex in the Δ ⁢ Δ ⁢ G Δ Δ 𝐺 \Delta\Delta G roman_Δ roman_Δ italic_G thermodynamic cycle, introducing a physical inductive bias and achieving both supervised and unsupervised state-of-the-art (SoTA) performance.
Experimental results on SKEMPI v2 indicate that our method achieves Spearman coefficients of 0.3201 (unsupervised) and 0.5134 (supervised) on SKEMPI v2, significantly surpassing the previously reported SoTA values of 0.2632 and 0.4324, respectively.
Futhermore, we demonstrate the capability of our method on binding
energy prediction, protein-protein docking and antibody optimization tasks.

## 1 Introduction

Protein–protein interactions (PPIs) are fundamental to the execution of diverse and essential biological functions across all organisms.
High-fidelity computational modeling of protein-protein interactions is indispensable. The interaction properties of protein-protein binding can be quantitatively characterized by binding free energy ($\Delta G$), the difference in Gibbs free energy between the bound state and the unbound state.
Predicting the change in binding free energy ($\Delta\Delta G$), also known as mutational effect, is crucial for modulating protein-protein interactions.
Accurate $\Delta\Delta G$ prediction facilitates the identification of mutations that enhance or diminish binding strength informs the efficient design of proteins,
thereby accelerating the development of therapeutic interventions and deepening our understanding of biological mechanisms.

Deep learning has demonstrated significant potential in protein modeling (Lin et al., ; Abramson et al., ), inspiring a paradigm shift in computational methods for predicting $\Delta\Delta G$. Traditional biophysics-based and statistics-based approaches (Park et al., ; Alford et al., ; Delgado et al., ) are increasingly being replaced by deep learning techniques (Shan et al., ; Liu et al., ).
Despite advancements, $\Delta\Delta G$ prediction is still hindered by the scarcity of annotated experimental data.
Consequently, pre-training on extensive unlabeled data has emerged as a prevalent strategy for $\Delta\Delta G$ prediction (Luo et al., ; Wu et al., ).
Recent studies observe that structure prediction models and inverse folding models implicitly capture the energy landscape (Bennett et al., ; Widatalla et al., ; Lu et al., ).
Although effective, these pre-training-based approaches simply employ supervised fine-tuning (SFT) and
neglect the importance of alignment.
Since SFT may result
in catastrophic forgetting of the general knowledge gained during unsupervised pre-training,
there remains an opportunity for better knowledge transfer.

In other biological tasks,
recent studies have adapted alignment techniques from large language models, such as direct preference optimization (DPO) (Rafailov et al., ), to integrate experimental fitness information into biological generative models (Zhou et al., ; Widatalla et al., ; Chennakesavalu et al., ; Liu et al., ).
However, directly adopting these alignment techniques for $\Delta\Delta G$ prediction is inadequate,
as they lack the physical inductive bias required for energy-related biological tasks.

In this work, we propose a technique named Boltzmann Alignment to transfer knowledge from pre-trained inverse folding models to $\Delta\Delta G$ prediction.
We first analyze the thermodynamic definition of $\Delta\Delta G$
and introduce the Boltzmann distribution to connect energy with protein conformational distribution,
thereby highlighting the potential of pre-trained probabilistic models.
However, the protein conformational distribution is intractable.
To address this,
we employ Bayes’ theorem to circumvent direct estimation and instead leverage the log-likelihood provided by protein inverse folding models for $\Delta\Delta G$ estimation.
Our derivation provides a rational perspective on the previously observed high correlation between binding energy and log-likelihood from inverse folding models (Bennett et al., ; Widatalla et al., ).
Compared to previous inverse folding-based methods, our method explicitly considers the unbound state of the protein complex, enabling fine-tuning of inverse folding models in a manner consistent with statistical thermodynamics.
Our contributions can be summarized as follows:

- •
We propose Boltzmann Alignment, which introduces physical inductive bias through Boltzmann distribution and thermodynamic cycle to transfer knowledge from pre-trained inverse folding models to $\Delta\Delta G$ prediction.
We not only provide an alignment technique, but also offer insight into the high correlation previously observed between binding energy and log-likelihood in inverse folding models.
- •
Experimental results on SKEMPI v2 indicate that our method achieves Spearman coefficients of 0.3201 (unsupervised) and 0.5134 (supervised) on SKEMPI v2, significantly exceeding the previously reported SoTA values of 0.2632 and 0.4324, respectively.
Further ablations demonstrate that our approach outperforms previous inverse folding-based methods and other alignment techniques.
Additionally, we showcase the broader applicability of our method in binding energy prediction, protein-protein docking, and antibody optimization.

## 2 Related Work

### 2.1 Alignment of Generative Models

The prevailing pre-training approach for generative models, which focuses on maximizing the likelihood of training data, often falls short in aligning with specific user preferences.
Consequently, there is growing interest in integrating task-specific information while maintaining the generative capabilities, especially in large language models (LLM). A commonly used alignment method is supervised fine-tuning (SFT), where models undergo additional training on curated
datasets with specific annotations, utilizing a straightforward loss function. However, SFT poses
the risk of overfitting to the fine-tuning dataset, which may result in catastrophic forgetting of the
general knowledge gained during unsupervised pre-training.
Much of the current alignment paradigm relies on reinforcement learning from human feedback (RLHF) (Ouyang et al., ).
Within this framework, human preferences provided as pairwise rankings are used to train a reward model, which can then optimize a policy using proximal policy optimization (PPO) (Schulman et al., ).
An alternative approach, known as direct preference optimization (DPO) (Rafailov et al., ),
serves as a plug-and-play method that maximizes the likelihood of preferences.
Numerous recent studies have expanded on the concepts of RLHF and DPO (An et al., ; Azar et al., ; Meng et al., ; Zhang et al., ).

Despite these advancements in LLM, developing effective methods to integrate experimental fitness information into biological generative models remains a crucial open question.
Park et al. () and Chennakesavalu et al. () explore alignment in the context of inverse molecular design, enhancing the desired properties of generated compounds against a drug target.
Liu et al. () fine-tunes retrosynthetic planning models with a preference-based loss to improve the quality of generated synthetic routes.
AbDPO (Zhou et al., ) introduces direct energy-based preference optimization to structure-sequence antibody co-design diffusion model, producing antibodies with low energy and high binding affinity. Mistani & Mysore () incorporates DPO into protein language models to generate peptides with desirable physicochemical properties that bind to a given target protein.
ProteinDPO (Widatalla et al., ) imbues structure-informed protein language model with biophysical information through DPO, enabling it to score stability and generate stable protein sequences. These efforts collectively underscore the potential of alignment techniques in advancing the performance of biological generative models.

### 2.2 Protein-Protein Interaction Modeling

Protein-protein interaction modeling has been extensively studied for decades, primarily concentrating on two key questions: where proteins interact, known as binding site prediction (Tubiana et al., ; Fang et al., ), and how they interact, addressed by protein-protein docking (Yan et al., ; Ketata et al., ).
Recently, AlphaFold3 (Abramson et al., ) reaches almost 80% success rate of predicting general protein complex structures, effectively tackling both the "where" and "how" of protein-protein interactions.
However, structure is not everything (Lowe, ); biomolecular interactions cannot be fully captured by static structures alone.
A comprehensive understanding requires appreciating the dynamic processes of association and dissociation between protein partners (Lu et al., ), which are quantified by measures such as the binding affinity (Kd) or binding free energy ($\Delta G$).
Predicting the change in binding free energy ($\Delta\Delta G$) is thus crucial for modulating protein-protein interactions; it enables the identification of mutations that enhance or diminish binding strength, which is essential for efficient protein design.
For example, in antibody discovery, accurately predicting $\Delta\Delta G$ is fundamental to antibody maturation.

A variety of methods have been developed to predict $\Delta\Delta G$.
Traditional approaches can be categorized into two main types: biophysical and statistical methods.
Biophysical methods (Park et al., ; Alford et al., ; Delgado et al., ) use energy calculations to model how proteins interact at the atomic level, but they often struggle with balancing speed and accuracy.
Statistical methods (Geng et al., ; Zhang et al., ) depend on feature engineering, utilizing descriptors that capture geometric, physical, and evolutionary characteristics of proteins.
Traditional approaches depend heavily on human expertise and struggle to accurately capture complex interactions between proteins.
Recently, deep learning-based approaches have emerged.
Despite advancements in end-to-end learning approaches (Shan et al., ),
$\Delta\Delta G$ prediction
is still limited by the lack of annotated experimental data.
As a result, pre-training on extensive unlabeled data has become a common strategy for improving predictions, involving various pre-training proxy tasks such as protein inverse folding (Yang et al., ), side-chain modeling (Luo et al., ; Liu et al., ; Mo et al., ), and masked modeling (Wu et al., ).
These in-depth explorations of pre-training proxy tasks not only establish a solid foundation but also highlight the urgent need for transferring acquired knowledge to accurately predict $\Delta\Delta G$.

## 3 Method

Based on the Boltzmann distribution and thermodynamic cycles, as illustrated on the right side of Figure 1, we first establish the connection between $\Delta\Delta G$ and protein sequence likelihoods, which we call Boltzmann Alignment (Sec. 3.1). Next, we present a method (Sec. 3.2) that integrates the inverse folding model into Boltzmann alignment. This method is named BA-Cycle and uses the inverse folding model to evaluate $\Delta\Delta G$ by predicting the likelihoods of protein sequences, as shown on the left side of Figure 1.
Upon BA-Cycle, we introduce a method named BA-DDG (Sec. 3.3) to fine-tune the inverse folding model using $\Delta\Delta G$-labeled data, introducing an inductive bias from statistical thermodynamics.

### 3.1 Boltzmann Alignment

To simplify the explanation, we introduce the case where the complex consists of two chains, chain A and chain B; however, the following content can be extended to cases with more chains. The binding free energy ($\Delta G$) of the complex is the difference between the Gibbs free energy of the bound state, denoted as $G_{\text{bnd}}$, and the Gibbs free energy of the unbound state, denoted as $G_{\text{unbnd}}$. Furthermore, using the Boltzmann distribution (Atkins et al., ), we can express the binding free energy ($\Delta G$) in terms of the probabilities of the two chains being in the bound state, $p_{\text{bnd}}$, and in the unbound state, $p_{\text{unbnd}}$ :

$$ $\Delta G=G_{\text{bnd}}-G_{\text{unbnd}}=-k_{\text{B}}T\cdot\left(\log p_{ \text{bnd}}-\log p_{\text{unbnd}}\right)$ (1) $$

where $k_{\text{B}}$ is the Boltzmann constant and $T$ is the thermodynamic temperature.
Specifically, the probabilities $p_{\text{bnd}}$ and $p_{\text{unbnd}}$ represent the probabilities of the protein complex structure being in the bound conformation ${\mathcal{X}}_{\text{bnd}}$ and the unbound conformation ${\mathcal{X}}_{\text{unbnd}}$, respectively, when the protein complex sequence ${\mathcal{S}}_{\text{AB}}$ is given.
Therefore, $p_{\text{bnd}}$ and $p_{\text{unbnd}}$ can be expressed as the conditional probabilities $p({\mathcal{X}}_{\text{bnd}}\mid{\mathcal{S}}_{\text{AB}})$ and $p({\mathcal{X}}_{\text{unbnd}}\mid{\mathcal{S}}_{\text{AB}})$, where we simplify and approximate ${\mathcal{X}}_{\text{bnd}}$ and ${\mathcal{X}}_{\text{unbnd}}$ as the backbone structure of the protein complex. Substituting these conditional probabilities into Eq.1 gives:

$$ $\Delta G=-k_{\text{B}}T\cdot\left(\log p({\mathcal{X}}_{\text{bnd}}\mid{ \mathcal{S}}_{\text{AB}})-\log p({\mathcal{X}}_{\text{unbnd}}\mid{\mathcal{S}} _{\text{AB}})\right)$ (2) $$

Estimating $p({\mathcal{X}}\mid{\mathcal{S}})$ poses several challenges. First, most current protein structure prediction models typically predict a single conformation and are not inherently probabilistic. Although these models provide uncertainty metrics such as pLDDT, pAE, and pTM (Abramson et al., ), and while these metrics have been observed to correlate with binding free energy (Lu et al., ), they do not offer a true probabilistic interpretation. Second, although recent efforts (Jing et al., ; ; Abramson et al., ) aim to introduce probabilistic models, specifically diffusion models, these neural networks learn to estimate the score $\nabla_{{\mathcal{X}}}\log p({\mathcal{X}}\mid{\mathcal{S}})$ rather than $p({\mathcal{X}}\mid{\mathcal{S}})$.
Moreover, due to the relatively low sample efficiency of diffusion models, directly estimating $p({\mathcal{X}}\mid{\mathcal{S}})$ remains intractable. Therefore, we use Bayes’ theorem to circumvent direct estimation:

$$ $\displaystyle p({\mathcal{X}}\mid{\mathcal{S}})$ $\displaystyle=\frac{p({\mathcal{S}}\mid{\mathcal{X}})\cdot p({\mathcal{X}})}{p ({\mathcal{S}})}$ (3) $$

Substituting this Bayes’ theorem equation into Eq.2 and eliminating the protein complex sequence probability $p({\mathcal{S}}_{\text{AB}})$ gives:

$$ $\displaystyle\Delta G$ $\displaystyle=-k_{\text{B}}T\cdot\left(\log\frac{p({\mathcal{S}}_{\text{AB}} \mid{\mathcal{X}}_{\text{bnd}})\cdot p({\mathcal{X}}_{\text{bnd}})}{p({ \mathcal{S}}_{\text{AB}})}-\log\frac{p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{ X}}_{\text{unbnd}})\cdot p({\mathcal{X}}_{\text{unbnd}})}{p({\mathcal{S}}_{ \text{AB}})}\right)$ (4) $\displaystyle=-k_{\text{B}}T\cdot\log\frac{p({\mathcal{S}}_{\text{AB}}\mid{ \mathcal{X}}_{\text{bnd}})\cdot p({\mathcal{X}}_{\text{bnd}})}{p({\mathcal{S}} _{\text{AB}}\mid{\mathcal{X}}_{\text{unbnd}})\cdot p({\mathcal{X}}_{\text{ unbnd}})}$ (5) $$

At this point, we have established the connection between the conditional probability of the protein sequence $p({\mathcal{S}}\mid{\mathcal{X}})$ and $\Delta G$. Next, based on this, we further analyze the connection between the conditional probability of the protein sequence $p({\mathcal{S}}\mid{\mathcal{X}})$ and $\Delta\Delta G$. Specifically, $\Delta\Delta G$ represents the difference between the $\Delta G$ of the wild type and the $\Delta G$ of the mutant type. The wild type and mutant type refer to the unmutated and mutated protein complexes, respectively. To distinguish between these two types, we denote them by adding “mut” and “wt” in the upper right corner of the notation; for example, $\Delta G^{\text{mut}}$ represents the $\Delta G$ of the mutant type. With this notation, we can express $\Delta\Delta G$ based on Eq.5 as follows:

$$ $\displaystyle\Delta\Delta G$ $\displaystyle=\Delta G^{\text{mut}}-\Delta G^{\text{wt}}$ (6) $\displaystyle=-k_{\text{B}}T\cdot\left(\log\frac{p({\mathcal{S}}_{\text{AB}}^{ \text{mut}}\mid{\mathcal{X}}_{\text{bnd}}^{\text{mut}})\cdot p({\mathcal{X}}_{ \text{bnd}}^{\text{mut}})}{p({\mathcal{S}}_{\text{AB}}^{\text{mut}}\mid{ \mathcal{X}}_{\text{unbnd}}^{\text{mut}})\cdot p({\mathcal{X}}_{\text{unbnd}}^ {\text{mut}})}-\log\frac{p({\mathcal{S}}_{\text{AB}}^{\text{wt}}\mid{\mathcal{ X}}_{\text{bnd}}^{\text{wt}})\cdot p({\mathcal{X}}_{\text{bnd}}^{\text{wt}})}{ p({\mathcal{S}}_{\text{AB}}^{\text{wt}}\mid{\mathcal{X}}_{\text{unbnd}}^{\text {wt}})\cdot p({\mathcal{X}}_{\text{unbnd}}^{\text{wt}})}\right)$ (7) $$

The backbone structure of the mutant type, ${\mathcal{X}}^{\text{mut}}$, is actually unknown. To address this issue, we introduce the assumption used in previous methods that the protein backbone structure remains unchanged before and after mutation. Therefore, we can approximate ${\mathcal{X}}_{\text{unbnd}}^{\text{wt}}$ to be equal to ${\mathcal{X}}_{\text{unbnd}}^{\text{mut}}$ and ${\mathcal{X}}_{\text{bnd}}^{\text{wt}}$ to be equal to ${\mathcal{X}}_{\text{bnd}}^{\text{mut}}$. Based on this assumption, we can eliminate the probability terms in Eq.7:

$$ $\Delta\Delta G=-k_{\text{B}}T\cdot\left(\log\frac{p({\mathcal{S}}_{\text{AB}}^ {\text{mut}}\mid{\mathcal{X}}_{\text{bnd}}^{\text{mut}})}{p({\mathcal{S}}_{ \text{AB}}^{\text{mut}}\mid{\mathcal{X}}_{\text{unbnd}}^{\text{mut}})}-\log \frac{p({\mathcal{S}}_{\text{AB}}^{\text{wt}}\mid{\mathcal{X}}_{\text{bnd}}^{ \text{wt}})}{p({\mathcal{S}}_{\text{AB}}^{\text{wt}}\mid{\mathcal{X}}_{\text{ unbnd}}^{\text{wt}})}\right)$ (8) $$

### 3.2 Probability Estimation

Eq.8 transforms $\Delta\Delta G$ into the likelihoods of protein sequences $p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}})$. Here, we demonstrate the method of estimating the sequence probabilities of the bound state, $p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}}_{\text{bnd}})$, and the unbound state, $p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}}_{\text{unbnd}})$, using the inverse folding model. Since we assume that the backbone structure remains unchanged before and after mutation, i.e., ${\mathcal{X}}^{\text{wt}}={\mathcal{X}}^{\text{mut}}$, the evaluation method for both the wild type and mutant type is identical, and no additional discussion is required.

The bound state. In our task, the backbone structure ${\mathcal{X}}_{\text{bnd}}$ of the bound state is usually known. Therefore, the inverse folding model can directly evaluate the probability $p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}}_{\text{bnd}})$. We denote the probability predicted by the inverse folding model as $p_{\theta}({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}}_{\text{bnd}})$, where $\theta$ is the model parameter.

The unbound state. The unbound state is typically not explicitly provided in known backbone structure; therefore, we propose an alternative probability estimation method. The unbound state approximately represents the scenario where chains A and B of the complex are relatively far apart, with minimal interactions between them. Thus, a reasonable estimation approach is to independently evaluate these two chains using the inverse folding model. We denote the structure and sequence of chain A as ${\mathcal{X}}_{\text{A}}$ and ${\mathcal{S}}_{\text{A}}$, and those of chain B as ${\mathcal{X}}_{\text{B}}$ and ${\mathcal{S}}_{\text{B}}$. The estimation method for $p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}}_{\text{unbnd}})$ can be expressed as:

$$ $p({\mathcal{S}}_{\text{AB}}\mid{\mathcal{X}}_{\text{unbnd}})\approx p_{\theta} ({\mathcal{S}}_{\text{A}}\mid{\mathcal{X}}_{\text{A}})\cdot p_{\theta}({ \mathcal{S}}_{\text{B}}\mid{\mathcal{X}}_{\text{B}})$ (9) $$

where $p_{\theta}({\mathcal{S}}_{\text{A}}\mid{\mathcal{X}}_{\text{A}})$ and $p_{\theta}({\mathcal{S}}_{\text{B}}\mid{\mathcal{X}}_{\text{B}})$ are the probabilities predicted by the inverse folding model.

Unsupervised estimation of $\Delta\Delta G$. Based on the estimation method described above, we can utilize a pretrained inverse folding model, ProteinMPNN (Dauparas et al., ), to achieve unsupervised evaluation of $\Delta\Delta G$. We name this approach BA-Cycle, which can be expressed as:

$$ $\widehat{\Delta\Delta G}=-k_{\text{B}}T\cdot\left(\log\frac{p_{\theta}({ \mathcal{S}}_{\text{AB}}^{\text{mut}}\mid{\mathcal{X}}_{\text{bnd}}^{\text{mut }})}{p_{\theta}({\mathcal{S}}_{\text{A}}^{\text{mut}}\mid{\mathcal{X}}_{\text{ A}}^{\text{mut}})\cdot p_{\theta}({\mathcal{S}}_{\text{B}}^{\text{mut}}\mid{ \mathcal{X}}_{\text{B}}^{\text{mut}})}-\log\frac{p_{\theta}({\mathcal{S}}_{ \text{AB}}^{\text{wt}}\mid{\mathcal{X}}_{\text{bnd}}^{\text{wt}})}{p_{\theta}( {\mathcal{S}}_{\text{A}}^{\text{wt}}\mid{\mathcal{X}}_{\text{A}}^{\text{wt}}) \cdot p_{\theta}({\mathcal{S}}_{\text{B}}^{\text{wt}}\mid{\mathcal{X}}_{\text{ B}}^{\text{wt}})}\right)$ (10) $$

Comparison with previous work.
Previous work (Bennett et al., ; Cagiada et al., ; Widatalla et al., ) attempts to use protein inverse folding models to predict $\Delta\Delta G$.
However, these studies do not explicitly account for the unbound state $p_{\text{unbnd}}$ in the thermodynamic cycle. We denote these methods as $\widehat{\Delta\Delta G}_{\text{Prev}}$, summarized as:

$$ $\widehat{\Delta\Delta G}_{\text{Prev}}=-k_{\text{B}}T\cdot\left(\log p_{\theta }({\mathcal{S}}_{\text{AB}}^{\text{mut}}\mid{\mathcal{X}}_{\text{bnd}}^{\text{ mut}})-\log p_{\theta}({\mathcal{S}}_{\text{AB}}^{\text{wt}}\mid{\mathcal{X}}_ {\text{bnd}}^{\text{wt}})\right)$ (11) $$

### 3.3 Supervision

We propose BA-DDG, a method that leverages $\Delta\Delta G$-labeled data to fine-tune BA-Cycle through Boltzmann Alignment. BA-DDG employs a forward process identical to that of BA-Cycle. During training, the parameters $\theta$ of the inverse folding model and $k_{\text{B}}T$ in Eq.10 are treated as learnable parameters that undergo optimization. The objective of BA-DDG is to minimize the discrepancy between the ground truth $\Delta\Delta G$ and the predicted $\widehat{\Delta\Delta G}$, while maintaining the distribution of the original pre-trained model, denoted as the reference model distribution $p_{\text{ref}}({\mathcal{S}}\mid{\mathcal{X}})$. The corresponding loss $\mathcal{L}_{\text{Boltzmann}}$ can be expressed as:

$$ $\mathcal{L}_{\text{Boltzmann}}=\underbrace{\|\Delta\Delta G-\widehat{\Delta \Delta G}\|}_{\text{minimize predicted error }}-\underbrace{\beta D_{\mathrm{ KL}}(p_{\theta}({\mathcal{S}}\mid{\mathcal{X}})\|p_{\text{ref }}({\mathcal{S}} \mid{\mathcal{X}}))}_{\text{distributional penalty }}$ (12) $$

where $\beta$ is a hyperparameter that controls the strength of the distributional penalty.

## 4 Experiments

In this section, we investigate three key questions through a series of experiments: (1) whether the physics-informed alignment technique we propose can achieve state-of-the-art (SoTA) accuracy on the SKEMPI v2 dataset under both unsupervised and supervised settings (Sec.4.2);
(2) whether the introduced thermodynamic cycle is indeed effective and has advantages over traditional SFT and DPO methods;
(3) whether predicted structures can effectively replace crystal structures as inputs in our method (Sec.4.3).
Before addressing these questions, we first introduce the benchmarks and baselines (Sec.4.1). Additionally, we explore potential applications of our method, taking binding energy prediction, protein-protein docking and antibody optimization as examples (Sec.4.4).

### 4.1 Benchmark

Dataset.
The SKEMPI v2 dataset (Jankauskaitė et al., ), the extensive annotated mutation dataset for 348 protein complexes, includes 7,085 amino acid mutations along with changes in thermodynamic parameters and kinetic rate constants.
Despite lacking structures of the mutated complexes, it serves as the most well-established benchmark for $\Delta\Delta G$ prediction.
To prevent data leakage, we follow Luo et al. () and Wu et al. (), first dividing the dataset into 3 folds by structure, ensuring each fold contains unique protein complexes. This division forms the basis of our 3-fold cross-validation process, where two folds are used for training and validation, and the third fold is reserved for testing.
This approach yields 3 different sets of parameters and ensures that every data point in SKEMPI v2 is tested once.

Evaluation metrics.
To thoroughly assess the performance of $\Delta\Delta G$ prediction, we utilize a total of 7 metrics. This includes 5 overall metrics: (1) Pearson correlation coefficient, (2) Spearman’s rank correlation coefficient, (3) minimized RMSE (root mean squared error), (4) minimized MAE (mean absolute error), and (5) AUROC (area under the receiver operating characteristic). To calculate AUROC, binary classification labels are assigned to mutations based on the sign of the $\Delta\Delta G$ predictions.
In practical settings, the correlation for individual protein complexes tends to be of greater importance. Consequently, we group mutations by their structural characteristics, calculate the Pearson and Spearman correlation coefficients for each group, and then report the average of these per-structure correlations as 2 additional metrics.

Baselines.
We compare our $\Delta\Delta G$ predictors, BA-Cycle and BA-DDG, with state-of-the-art unsupervised and supervised methods, respectively.
Unsupervised methods are categorized into three groups:
(1) traditional empirical energy functions such as Rosetta Cartesian $\Delta\Delta G$ (Alford et al., ) and FoldX (Delgado et al., );
(2) sequence/evolution-based approaches, exemplified by ESM-1v (Meier et al., ), Position-Specific Scoring Matrix (PSSM), MSA Transformer (Rao et al., ), and Tranception (Notin et al., );
(3) structure-informed pre-training-based methods not trained on $\Delta\Delta G$ labels, such as ESM-IF (Hsu et al., ), MIF-$\Delta$logits (Yang et al., ), RDE-Linear (Luo et al., ), and a model pre-trained to predict residue B-factors that are then used to predict $\Delta\Delta G$.
Supervised methods can be divided into 2 categories:
(1) end-to-end learning models, including DDGPred (Shan et al., ) and a model that utilizes a self-attention-based network (Jumper et al., ) as the encoder but employs an MLP to directly predict $\Delta\Delta G$ (End-to-End);
(2) structure-informed pre-training-based methods fine-tuned on $\Delta\Delta G$ labels, including MIF-Network (Yang et al., ), RDE-Network (Luo et al., ), DiffAffinity (Liu et al., ), Prompt-DDG (Wu et al., ), ProMIM (Mo et al., ), and
Surface-VQMAE (Wu & Li, ).

### 4.2 Main Results

Comparison with baselines.
According to Table 1, our BA-DDG outperforms all the baselines across all evaluation metrics. Notably, it demonstrates a significant improvement in per-structure correlations, highlighting its greater reliability for practical applications.
BA-Cycle achieves comparable performance to empirical energy functions and surpasses all unsupervised learning baselines.
Detailed results for single-point, multi-point, and all-point mutations are available in Table 6.

**Table 1: Mean results of 3-fold cross-validation on the SKEMPI v2 dataset. Bold and underline indicate the best metrics for methods trained and not trained on the $\Delta\Delta G$ labels, respectively.**
| Supervision | Method | Per-Structure | Overall |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Pearson $\uparrow$ | Spear. $\uparrow$ | Pearson $\uparrow$ | Spear. $\uparrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | AUROC $\uparrow$ |  |  |
| ✗ | Rosetta | 0.3284 | 0.2988 | 0.3113 | 0.3468 | 1.6173 | 1.1311 | 0.6562 |
| FoldX | 0.3789 | 0.3693 | 0.3120 | 0.4071 | 1.9080 | 1.3089 | 0.6582 |  |
| ESM-1v | 0.0073 | -0.0118 | 0.1921 | 0.1572 | 1.9609 | 1.3683 | 0.5414 |  |
| PSSM | 0.0826 | 0.0822 | 0.0159 | 0.0666 | 1.9978 | 1.3895 | 0.5260 |  |
| MSA Transformer | 0.1031 | 0.0868 | 0.1173 | 0.1313 | 1.9835 | 1.3816 | 0.5768 |  |
| Tranception | 0.1348 | 0.1236 | 0.1141 | 0.1402 | 2.0382 | 1.3883 | 0.5885 |  |
| B-factor | 0.2042 | 0.1686 | 0.2390 | 0.2625 | 2.0411 | 1.4402 | 0.6044 |  |
| ESM-IF | 0.2241 | 0.2019 | 0.3194 | 0.2806 | 1.8860 | 1.2857 | 0.5899 |  |
| MIF-$\Delta$logit | 0.1585 | 0.1166 | 0.2918 | 0.2192 | 1.9092 | 1.3301 | 0.5749 |  |
| RDE-Linear | 0.2903 | 0.2632 | 0.4185 | 0.3514 | 1.7832 | 1.2159 | 0.6059 |  |
|  | BA-Cycle | 0.3722 | 0.3201 | 0.4552 | 0.4097 | 1.8402 | 1.3026 | 0.6657 |
| ✓ | DDGPred | 0.3750 | 0.3407 | 0.6580 | 0.4687 | 1.4998 | 1.0821 | 0.6992 |
| End-to-End | 0.3873 | 0.3587 | 0.6373 | 0.4882 | 1.6198 | 1.1761 | 0.7172 |  |
| MIF-Network | 0.3965 | 0.3509 | 0.6523 | 0.5134 | 1.5932 | 1.1469 | 0.7329 |  |
| RDE-Network | 0.4448 | 0.4010 | 0.6447 | 0.5584 | 1.5799 | 1.1123 | 0.7454 |  |
| DiffAffinity | 0.4220 | 0.3970 | 0.6609 | 0.5560 | 1.5350 | 1.0930 | 0.7440 |  |
| Prompt-DDG | 0.4712 | 0.4257 | 0.6772 | 0.5910 | 1.5207 | 1.0770 | 0.7568 |  |
| ProMIM | 0.4640 | 0.4310 | 0.6720 | 0.5730 | 1.5160 | 1.0890 | 0.7600 |  |
| Surface-VQMAE | 0.4694 | 0.4324 | 0.6482 | 0.5611 | 1.5876 | 1.1271 | 0.7469 |  |
| BA-DDG | 0.5453 | 0.5134 | 0.7118 | 0.6346 | 1.4516 | 1.0151 | 0.7726 |  |

Visualization for correlation analysis.
We present scatter plots of experimental and predicted $\Delta\Delta G$ for three representative methods in Fig.2, along with their overall Pearson and Spearman correlation scores. Additionally, Fig.3 shows the distribution of per-structure Pearson and Spearman correlation scores, as well as the average results across all structures.
It is evident that BA-DDG outperforms other methods in both qualitative visualization and quantitative metrics.

Figure: Figure 2: Comparison of correlations between experimental $\Delta\Delta G$ and $\Delta\Delta G$ predicted by three representative methods.
Refer to caption: x1.png

Figure: Figure 3: Distributions of per-structure Pearson correlation scores and Spearman correlation scores for six representative methods.
Refer to caption: x2.png

### 4.3 Ablation Study

Ablation study on thermodynamic cycle.
As shown in Table 2, we evaluate the impact of thermodynamic cycle using 3-fold cross-validation on the SKEMPI v2 dataset.
We compare our training-free version BA-Cycle to the basic usage of protein inverse folding models, as demonstrated in previous studies (Bennett et al., ; Cagiada et al., ; Widatalla et al., ). As described in Eq.11, these studies do not account for thermodynamic cycle, as they consider only $p_{\text{bnd}}$ and neglect $p_{\text{unbnd}}$.
Disregarding thermodynamics significantly harms performance, as BA-Cycle outperforms ESM-IF and the original ProteinMPNN.

**Table 2: Ablation study on thermodynamic cycle. We report the mean results of 3-fold cross-validation on the SKEMPI v2 dataset. Bold indicate the best metrics.**
| Method | Per-Structure | Overall |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Pearson $\uparrow$ | Spear. $\uparrow$ | Pearson $\uparrow$ | Spear. $\uparrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | AUROC $\uparrow$ |  |
| ESM-IF | 0.2241 | 0.2019 | 0.3194 | 0.2806 | 1.8860 | 1.2857 | 0.5899 |
| ProteinMPNN | 0.2825 | 0.2741 | 0.2753 | 0.3112 | 1.9869 | 1.4048 | 0.6191 |
| BA-Cycle | 0.3722 | 0.3201 | 0.4552 | 0.4097 | 1.8402 | 1.3026 | 0.6657 |

Ablation study on supervision methods.
As presented in Table 3, we evaluate the impact of various supervision methods.
The training and inference settings are the same as in the main results.
The implementation details of other alignment techniques can be found in App.A.1.
Boltzmann supervision yields the best performance across
all evaluation metrics, compared to other alignment techniques like SFT and DPO. These results clearly demonstrate the effectiveness of our proposed methods.

**Table 3: Ablation study on supervision methods. We report the mean results of 3-fold cross-validation on the SKEMPI v2 dataset. Bold indicate the best metrics.**
| Supervison | Per-Structure | Overall |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Pearson $\uparrow$ | Spear. $\uparrow$ | Pearson $\uparrow$ | Spear. $\uparrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | AUROC $\uparrow$ |  |
| SFT | 0.5167 | 0.4769 | 0.7048 | 0.6240 | 1.4661 | 1.0346 | 0.7725 |
| DPO | 0.4212 | 0.3913 | 0.5397 | 0.4870 | 1.7399 | 1.2422 | 0.7036 |
| Boltzmann | 0.5453 | 0.5134 | 0.7118 | 0.6346 | 1.4516 | 1.0151 | 0.7726 |

Inference with predicted structures.
The SKEMPI v2 benchmark includes wild-type crystal structures, which are also used by previous state-of-the-art methods such as Prompt-DDG (Wu et al., ) and RDE (Luo et al., ). However, crystal structures of mutants are often unknown, and using wild-type structures for mutants can introduce errors, especially with multiple mutation sites. In practical applications, such as antibody design, only sequence information may be available without a template crystal structure. To assess the usability of $\Delta\Delta G$ prediction models in various practical scenarios, we investigate whether our fine-tuned ProteinMPNN relies on crystal structures as inputs to predict $\Delta\Delta G$, as presented in Table 4.
While the performance with AlphaFold3 predicted structures is slightly lower, the differences are minimal, indicating that our method does not strictly require crystal structures when a reasonably accurate predicted structure is available.

**Table 4: Comparison of inference with AlphaFold3 (Abramson et al., ) predicted structures and crystal structures. We report the mean results of 3-fold cross-validation on the common subset of Test Set 1 from SKEMPI as defined in SSIPe (Huang et al., ) and the SKEMPI dataset as employed in DSMBind (Jin et al., ). During the 3-fold cross-validation, models are trained on the SKEMPI v2 dataset using crystal structures as inputs and validated on the subset using AlphaFold3 predicted structures. Both the AlphaFold3 predicted structures and the dataset are provided by Lu et al. ().**
| Structure | Per-Structure | Overall |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Pearson $\uparrow$ | Spear. $\uparrow$ | Pearson $\uparrow$ | Spear. $\uparrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | AUROC $\uparrow$ |  |
| Crystal | 0.8057 | 0.7322 | 0.8584 | 0.7946 | 1.3948 | 0.9713 | 0.8437 |
| AlphaFold3 | 0.8017 | 0.7459 | 0.8545 | 0.7821 | 1.4122 | 0.9817 | 0.8314 |

### 4.4 Application

Figure: Figure 4: Left: The Spearman correlation between predicted and true binding free energy ($\Delta G$) on the SAbDab test set. Right: Performance of selection methods for rigid protein-protein docking as the number of generative samples increases. "Perfect Selection" shows the best possible performance with an ideal selection method. "Diffdock-PP" refers to the confidence model proposed by Diffdock-PP (Ketata et al., ). "BA-DDG" involves using our method to estimate $\Delta G$ for selection. Performance is assessed by calculating the fraction of 13 antibody complex generation tasks on the DIPS test set (Townshend et al., ) that achieve a C-RMSD of less than 5Å.
Refer to caption: x3.png

#### 4.4.1 Binding Energy Prediction

Antibody design is vital for treating diseases such as cancer and infections.
Although we lack a comprehensive explanation for canceling out $p({\mathcal{X}}_{\text{bnd}})$ and $p({\mathcal{X}}_{\text{unbnd}})$ in Eq.5, BA-DDG can approximate $\Delta G$ using only inverse folding:

$$ $\displaystyle\widehat{\Delta G}$ $\displaystyle\approx-k_{\text{B}}T\cdot\left(\log p_{\theta}({\mathcal{S}}_{ \text{AB}}\mid{\mathcal{X}}_{\text{AB}})-\left(\log p_{\theta}({\mathcal{S}}_{ \text{A}}\mid{\mathcal{X}}_{\text{A}})+\log p_{\theta}({\mathcal{S}}_{\text{B} }\mid{\mathcal{X}}_{\text{B}})\right)\right)$ (13) $$

We benchmark antibody-antigen binding energy predictors, with the experimental setup proposed in DSMBind (Jin et al., ).
DSMBind is an unsupervised method trained on 3,416 antibody-antigen complexes from SAbDab (Schneider et al., ). SAbDab has over 3,500 non-redundant antibody-antigen complex structures, but only 566 of them have binding energy labels. Other methods involve pre-training on the entire binding energy dataset from SKEMPI, followed by fine-tuning on antibody-antigen data from SKEMPI. We evaluate these methods on a SAbDab subset of 566 complexes with binding energy labels. As depicted in Fig.4-Left, BA-DDG achieves a Spearman correlation of 0.385, outperforming other methods.

#### 4.4.2 Rigid Protein-Protein Docking

DiffDock-PP (Ketata et al., ) is a diffusion generative model specifically designed for rigid protein-protein docking. It learns to translate and rotate unbound protein structures into their bound conformations. The docking process comprises two primary steps. First, the diffusion model generates samples. Then, a pre-trained confidence model evaluates these samples to identify the one with the highest confidence score, which is used as the final docking result.
BA-DDG can serve as an approximate binding free energy ($\Delta G$) estimator and can be employed as a selection method in diffusion-based docking process. A lower $\Delta G$ indicates a more spontaneous and favorable transition from the unbound to the bound state, corresponding to a higher confidence score for the generative docking sample. By integrating BA-DDG into the diffusion-based docking process, we provide a robust method for selecting the most plausible docking conformation, as presented in Fig.4-Right.

#### 4.4.3 Antibody Optimization

Antibody optimization against SARS-CoV-2.
Beyond serving as an energy scoring function, BA-DDG also acts as a structure-based protein designer, similar to ProteinMPNN. Consequently, we are curious whether the ProteinMPNN fine-tuned for $\Delta\Delta G$ prediction tends to generate sequences with higher affinity.
Shan et al. () identify five single-point mutations in a human antibody targeting SARS-CoV-2 that significantly enhance its neutralization effectiveness. These mutations are selected from a pool of 494 possible single-point mutations within the heavy chain CDR region of the antibody.
We calculate the perplexity differences and preference probabilities of these 494 possible single-point mutations relative to the wild-type sequence.
Specifically, normalized per-amino acid perplexity is obtained by subtracting the average per-amino acid perplexity of all possible sequences from that of the mutant sequence. Preference probabilities are derived using the Bradley-Terry model (Bradley & Terry, ) applied to the predicted probabilities of wild-type and mutant sequences.
The results, as presented in the Table 5, show that the fine-tuned BA-DDG generally achieves lower normalized perplexity and higher preference probabilities compared to the original ProteinMPNN, indicating a tendency to generate sequences with potentially higher binding affinity and improved neutralization effectiveness. This suggests that fine-tuning for $\Delta\Delta G$ prediction can enhance the design of more effective antibody sequences.

**Table 5: Normalized per-amino acid perplexity and preference probabilities of the five favorable mutations sequence on the antibody against SARS-CoV-2 (Shan et al., ) by original and fine-tuned ProteinMPNN.**
| Metric | Method | TH31W | AH53F | NH57L | RH103M | LH104F | Average |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Perplexity $\downarrow$ | ProteinMPNN | 6.018 | 0.179 | 0.401 | 2.895 | 2.521 | 2.403 |
| BA-DDG | 1.979 | 0.429 | 0.809 | 2.416 | 0.992 | 1.325 |  |
| Preference $\uparrow$ | ProteinMPNN | 12.61% | 62.55% | 75.62% | 13.93% | 36.20% | 42.47% |
| BA-DDG | 37.24% | 68.65% | 56.51% | 25.10% | 47.63% | 47.03% |  |

## 5 Discussion and conclusion

We propose Boltzmann Alignment, a method that leverages physical inductive bias through the Boltzmann distribution and thermodynamic cycle to enhance $\Delta\Delta G$ prediction by transferring knowledge from pre-trained inverse folding models.
Experiments on SKEMPI v2 yield Spearman coefficients of 0.3201 (unsupervised) and 0.5134 (supervised), significantly surpassing previous state-of-the-art values of 0.2632 and 0.4324. Additionally, we demonstrate the broader applicability of our approach in binding energy prediction, protein-protein docking, and antibody optimization.

Our method has certain limitations that we aim to address in future work.
Firstly, our approach requires crystal structures or reliable predicted structures as input; in cases where mutant protein structures are unavailable, we assume that the mutant backbone structure is identical to that of the wild-type protein; we take parts of the complex backbone structure as the backbone structure of single proteins.
This reliance on structural data may limit the applicability and effectiveness of our method.
Moreover, our current model does not consider side-chain conformations, which may be essential for $\Delta\Delta G$ prediction.
Incorporating side-chain flexibility could enhance predictive accuracy and provide deeper insights into protein-protein interactions.

The positive societal impact is that this work can assist in drug design and virtual screening potentially. No negative societal impact is perceived.

## Appendix A Implementation Details

### A.1 Other Alignment Techniques

#### A.1.1 SFT

The training objective of trivial SFT is as follows:

$$ $\mathcal{L}_{\text{SFT}}=\underbrace{\|\Delta\Delta G-\widehat{\Delta\Delta G} _{\text{Prev}}\|}_{\text{minimize predicted error }}-\underbrace{\beta D_{ \mathrm{KL}}(p_{\theta}(S\mid X)\|p_{\text{ref }}(S\mid X))}_{\text{ distributional penalty }}$ (14) $$

#### A.1.2 DPO

We utilize weighted DPO algorithm proposed in Widatalla et al. (). The weighted DPO objective is simply minimization of the KL-divergence between the distribution parameterized by the generative model and numerical scores ($\Delta\Delta G$ labels).

### A.2 Model

In ProteinMPNN, part of the input protein sequence is to be designed, and the rest is fixed; decoding skips fixed regions but includes them in the context for designing the remaining positions during autoregressive decoding.
When estimating $\Delta\Delta G$ with ProteinMPNN, we designate the mutation sites $S_{\text{D}}$ to be designed while fixing the rest of the sequence $S\setminus S_{\text{D}}$.
Let $\pi=(s_{1},s_{2},\ldots,s_{n})$ represent a randomly sampled decoding order, a permutation of the mutation sites $S_{\text{D}}$.
The joint probability $p_{\theta}(S\mid X)$ can be expanded step by step into the product of model-predicted probabilities for each amino acid site using the conditional probability formula:

$$ $\displaystyle p_{\theta}(S\mid X)$ $\displaystyle=p_{\theta}(S_{\text{D}}\mid X,S\setminus S_{\text{D}})$ (15) $\displaystyle=p_{\theta}(S_{\text{D}}\setminus\{s_{1}\}\mid X,S\setminus S_{ \text{D}}\cup\{s_{1}\})\cdot p_{\theta}(\{s_{1}\}\mid X,S\setminus S_{\text{D}})$ (16) $\displaystyle=\ldots$ $$

### A.3 Training

#### A.3.1 Hyper-parameters

We train the model over 20,000 iterations using the Adam optimizer, with a learning rate set at 0.0001 and a batch size of 2.
The small batch size is because we need to include both the wild-type and mutant complexes along with their respective monomer structures in a single batch for each iteration.
The model is insensitive to the loss weight parameter $\beta$, so it is typically set to a low value, such as 0.001, during training.

For the inverse folding model, we utilized the pre-trained ProteinMPNN, which includes 3 MPNN layers. It considers the 30 nearest neighbors to construct edges and has an embedding dimension of 256.

#### A.3.2 Hardware

All our experiments are conducted on a computing cluster with CPUs of AMD
EPYC
7763 64-Core of 3.52GHz
and
a single NVIDIA
GeForce
RTX 4090 24GB GPU.

## Appendix B Additional results

We compare BA-DDG with several representative methods under single-point, multi-point, and all-point mutations on SKEMPI v2.
The results reported in Table 6 demonstrate that BA-DDG achieves the best overall performance for single-point mutations and the best per-structure performance for multi-point mutations. Notably, BA-DDG performs better on single-point mutations compared to multi-point mutations, which may be related to the autoregressive decoding of ProteinMPNN. In our BA-DDG, when dealing with multi-point mutations, the probability estimation for a new site depends on the estimation of the previous site.
This coupling might affect performance, presenting an opportunity for future improvements.

**Table 6: Performance comparison under single-, multi- and all-point mutation on SKEMPI v2, where bold denotes the best metric under each setting.**
| Method | Mutations | Per-Structure | Overall |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Pearson $\uparrow$ | Spear. $\uparrow$ | Pearson $\uparrow$ | Spear. $\uparrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | AUROC $\uparrow$ |  |  |
| Rosetta | all | 0.3284 | 0.2988 | 0.3113 | 0.3468 | 1.6173 | 1.1311 | 0.6562 |
| single | 0.3510 | 0.4180 | 0.3250 | 0.3670 | 1.1830 | 0.9870 | 0.6740 |  |
| multiple | 0.1910 | 0.0830 | 0.1990 | 0.2300 | 2.6580 | 2.0240 | 0.6210 |  |
| FoldX | all | 0.3789 | 0.3693 | 0.3120 | 0.4071 | 1.9080 | 1.3089 | 0.6582 |
| single | 0.3820 | 0.3600 | 0.3150 | 0.3610 | 1.6510 | 1.1460 | 0.6570 |  |
| multiple | 0.3330 | 0.3400 | 0.2560 | 0.4180 | 2.6080 | 1.9260 | 0.7040 |  |
| DDGPred | all | 0.3750 | 0.3407 | 0.6580 | 0.4687 | 1.4998 | 1.0821 | 0.6992 |
| single | 0.3711 | 0.3427 | 0.6515 | 0.4390 | 1.3285 | 0.9618 | 0.6858 |  |
| multiple | 0.3912 | 0.3896 | 0.5938 | 0.5150 | 2.1813 | 1.6699 | 0.7590 |  |
| End-to-End | all | 0.3873 | 0.3587 | 0.6373 | 0.4882 | 1.6198 | 1.1761 | 0.7172 |
| single | 0.3818 | 0.3426 | 0.6605 | 0.4594 | 1.3148 | 0.9569 | 0.7019 |  |
| multiple | 0.4178 | 0.4034 | 0.5858 | 0.4942 | 2.1971 | 1.7087 | 0.7532 |  |
| RDE-Network | all | 0.4448 | 0.4010 | 0.6447 | 0.5584 | 1.5799 | 1.1123 | 0.7454 |
| single | 0.4687 | 0.4333 | 0.6421 | 0.5271 | 1.3333 | 0.9392 | 0.7367 |  |
| multiple | 0.4233 | 0.3926 | 0.6288 | 0.5900 | 2.0980 | 1.5747 | 0.7749 |  |
| DiffAffinity | all | 0.4220 | 0.3970 | 0.6609 | 0.5560 | 1.5350 | 1.0930 | 0.7440 |
| single | 0.4290 | 0.4090 | 0.6720 | 0.5230 | 1.2880 | 0.9230 | 0.7330 |  |
| multiple | 0.4140 | 0.3870 | 0.6500 | 0.6020 | 2.0510 | 1.5400 | 0.7840 |  |
| Prompt-DDG | all | 0.4712 | 0.4257 | 0.6772 | 0.5910 | 1.5207 | 1.0770 | 0.7568 |
| single | 0.4736 | 0.4392 | 0.6596 | 0.5450 | 1.3072 | 0.9191 | 0.7355 |  |
| multiple | 0.4448 | 0.3961 | 0.6780 | 0.6433 | 1.9831 | 1.4837 | 0.8187 |  |
| ProMIM | all | 0.4640 | 0.4310 | 0.6720 | 0.5730 | 1.5160 | 1.0890 | 0.7600 |
| single | 0.4660 | 0.4390 | 0.6680 | 0.5340 | 1.2790 | 0.9240 | 0.7380 |  |
| multiple | 0.4580 | 0.4250 | 0.6660 | 0.6140 | 1.9630 | 1.4910 | 0.8250 |  |
| BA-DDG | all | 0.5453 | 0.5134 | 0.7118 | 0.6346 | 1.4516 | 1.0151 | 0.7726 |
| single | 0.5606 | 0.5192 | 0.7321 | 0.6157 | 1.1848 | 0.8409 | 0.7662 |  |
| multiple | 0.4924 | 0.4959 | 0.6650 | 0.6293 | 2.0151 | 1.4944 | 0.7875 |  |

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. In Section 4.1, Dividing the dataset into 3 folds by structure cannot strictly prevent data leakage, different protein complexes might have similar protein binding structures, can you elaborate more about the method you used to split data and prevent data leakage? 
Could you explain more about data partition process, why it can help to reduce data leakage process?

2. Could you explain more about data partition process, and why it can help to reduce data leakage process?

3. The SKEMPI dataset includes both single-point mutations and multiple-point mutations. Given the varying complexity, predicting binding affinity for single-point mutations may be less challenging and could potentially yield more accurate prediction results compared to multiple-point mutations. This raises the question: why not consider training separate models tailored for single and multiple mutation binding affinity prediction tasks, rather than using a single model for both?
soundness: 3
strengths: 1. It creatively used Boltzmann Alignment, which introduces physical inductive bias through Boltzmann distribution and thermodynamic cycle to link binding energy and log-likelihood in inverse folding models.

2. This work achieved SOTA and outperforms other SFT and DPO methods. The method can not only be used in binding energy prediction, but also be used in  protein-protein docking, and antibody optimization.
summary: This paper employs pre-trained inverse folding models based on the Boltzmann distribution and thermodynamic cycles to associate energy with log-likelihood from inverse folding models for predicting ΔΔG. It introduces both unsupervised and supervised inverse folding models, each achieving state-of-the-art (SoTA) accuracy on the SKEMPI v2 dataset. Additionally, the study demonstrates that the proposed thermodynamic cycle is effective and offers advantages over traditional SFT and DPO methods. Notably, the method does not strictly require crystal structures when a reasonably accurate predicted structure is available. Furthermore, the paper explores applications of this approach in binding affinity energy prediction, protein-protein docking, and antibody optimization.
weaknesses: 1. The code is not available,  it should be open-sourced upon publication.

2. Dividing the dataset into 3 folds by structure cannot strictly prevent data leakage, different protein complexes might have similar protein binding structures.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 4
questions: - In section 4.4.3 authors utilise dataset from Shen et al (2022) however focus only on point mutations. This dataset is, I think, much richer and contains double and triple mutants from multiple optimisation rounds. I believe it would be important to expand this section to show proposed model performance more extensively. This, I believe, has a potential to boost the impact and trust in the model in a typical antibody design scenario. 
- Following on sec 4.4.3 - preference probabilities are a good metric but they in my view they are a bit abstract. I think a simpler metric of e.g. how the method ranks particular mutation would be more interpretable (as they relate to simple consideration of how many designs one can screen in the wet lab before finding a particular hit). 
- In the typical design scenario, as the problem is hard and the reported performance of the method is not perfect, one can also consider the uncertainty of the prediction which is an important and frequently used feature in e.g. protein folding models. It could potentially strengthen the paper if authors could discuss and provide more information on how uncertainty estimation could be introduced to the model.
- In appendix B - authors write "We compare BA-DDG with several representative methods under single-point, multi-point, and
all-point mutations on SKEMPI v2". It's not clear to me what is the difference between multi-point and all-point mutations. Is the all-point a union of point- and multi-point mutations (I'd assume so by looking at the following Table) ? For clarity authors can consider to explain this more in the paper. 
- The discussion about computational (non-ML) methods for ddG estimation is scarce. Even though it's purely ML venue, I miss a bit more thorough discussion of how the proposed method relates to force-field methods utilising more nuanced thermodynamical cycles / approaches like MM-GBSA/PBSA, FEP, TI, etc.
soundness: 3
strengths: The paper is very well written and, I believe, easy to follow both for machine-learning and also, to large extent, for biology-oriented audiences thus extending its potential impact. The limitations and weaknesses are appropriately discussed. It focuses on relevant and important problem in protein science - predicting the energetic effect of mutation on the free energy of protein-protein complex formation. The contribution is clearly original, correctly references and builds upon several already existing contributions in the field (in particular - inverse folding models, direct preference optimisation). Most importantly, benchmark reports significant improvement across several investigated metrics on commonly-accepted dataset (SKEMPI) and described ablations and controls (folded structures) strengthen the confidence in the reported result.
summary: In this paper authors propose a novel, thermodynamics inspired, approach for predicting binding free energy change upon mutation (ddG) building, in particular, on the capabilities of protein inverse folding models. Validation of the method employs standard literature benchmark (SKEMPI v2 curating and integrating a vast number of experimental measurements of ddG across various types of proteins). Beyond that, authors demonstrate applicability of their method across several tasks (binding energy prediction, antibody optimisation and docking). The reported results, in general, significantly outperform a spectrum of modern methods presented in the relevant literature both in supervised and unsupervised context.
weaknesses: - Method is structure based, it requires at least experimentally resolved or confidently predicted complex structure,
- The representation of protein is rather coarse, only backbone. Side-chain atoms and solvent molecules are omitted yet they are obviously important drivers of protein-protein interactions.
- No representation of dynamics (structures are represented as static snapshots), which can affect the method performance for mutations inducing significant conformational changes.
- The described method seems to excel at point mutations and there's a trend towards decreased performance in more complex multiple mutations.
Majority of those weaknesses are highlighted and discussed by the authors in the appropriate sections across the paper.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: Are there any specific types of mutations or protein interactions where the authors have observed the method to be particularly effective or challenging? Sharing such insights could help guide future research directions.
soundness: 3
strengths: - The integration of Boltzmann distribution within the framework to enhance ∆∆G prediction is innovative and addresses a significant challenge in computational biology.
- The authors effectively showcase the versatility of their method in binding energy prediction, protein-protein docking, and antibody optimization, indicating its broad applicability.
- The paper is well-organized, and the results are presented clearly, making it easy for readers to follow the methodology and findings.
summary: The paper presents Boltzmann Alignment, a novel approach for predicting changes in protein-protein binding free energy due to mutations. It utilizes pre-trained inverse folding models and introduces physical inductive bias through the Boltzmann distribution.
weaknesses: I haven't identified any significant weaknesses.

---

confidence: 4
contribution: 4
initial_rating: 6
presentation: 3
questions: 1) Could the authors explain the reasons for the relatively low MAE reported in Table 2?

2) Could the authors clarify how they determined the unbound state approximation for independent chain probabilities (Equation 9)?  The paper provides some explanation on this point, but it lacks sufficient detail.

3) The transformation between energy and probability, along with the approximation of unbound state probabilities, partially explains the high correlation between the inverse folding task and ΔΔG prediction. However, within the neural network, the sophisticated design of model doesn’t appear to provide additional information gain. 

Given the strengths of this paper, I would consider raising my score if the authors can address my concerns.
soundness: 3
strengths: 1) The paper is well-written.

2) The approach achieves state-of-the-art results compared to baselines.

3) The paper effectively incorporates domain-specific priors, specifically leverages Boltzmann principles to connect energy with probability, along with the assumption that unbound states can be approximated as independent probabilities for each chain. Though this assumption is idealized, it still is an interesting approach that introduces a motivated alignment.

4) The paper demonstrates practical applicability on binding energy prediction, protein-protein docking and antibody optimization tasks.
summary: The paper proposes a novel and interesting method that leverages Boltzmann Alignment to establish a more sophisticated connection between inverse folding tasks and binding free energy (ΔΔG) prediction. Despite relying on somewhat idealized approximations, the method leverages a thermodynamic cycle and the principles of the Boltzmann distribution to incorporate both bound and unbound states of the protein complex, offering a physically motivated alignment technique that addresses limitations of traditional pre-trained inverse folding models.  Through evaluations on the selected SKEMPI v2 dataset, the paper demonstrates that the method achieves significant improvements over existing approaches in both supervised and unsupervised settings.  Overall, the paper provides a new perspective for integrating domain knowledge into alignment techniques.
weaknesses: 1) The method makes strong assumptions, particularly in treating the unbound state as independent probabilities for each chain, which oversimplifies the inter-chain interactions. Additionally, the predictions of inverse folding model’s do not fully align with the predicted probabilities (not the ground truth), introducing biases when single-chain probabilities are combined to approximate complex states. This approximation may result in significant deviations in some situations.

2) The method has only been validated on a limited dataset, which lacks sufficient evidence to demonstrate the generalizability of the approach and its underlying assumptions.

3) It seems that the paper lacks sufficient discussion on how improvements in the inverse folding model would translate to gains in this approach.  It’s also unclear which specific module in this sophisticated approach contributes most significantly to the information gain.

Meta-review: The paper considers protein-protein interactions and proposes Boltzmann Alignment to predict the change in binding free energy resulting from mutation. The approach make use of physical inductive bias through Boltzmann distribution and thermodynamic cycle to link pre-trained inverse folding models to binding free-energy prediction. The paper is a pleasure to read and accessible. The contributions are novel and the approach intuitive and interesting. The value of the proposed method achieves SOTA results on SKEMPI v2 benchmark and its broader applicability is also convincingly demonstrated for binding energy prediction, protein-protein docking, and antibody optimization.

Decision: Accept (Spotlight)