PAPER: INTRODUCTIONInverse folding aims to predict amino acid sequences that fold into a given protein structure, and plays a fundamental role, for example, in the protein design pipeline of RFDiffusion (Watson et al., 2023). Recent deep learning approaches such as ESM-IF1 (Hsu et al., 2022) or ProteinMPNN (Dauparas et al., 2022) achieve remarkable accuracy in this task. However, instead of predicting a single ground truth sequence, it is often desirable to have a method that is able to generate a variety of different sequences with the desired fold, i.e., solving a one-to-may problem, see Fig. 2. This diversity could be leveraged for example by starting from a source sequence Sturmfels et al. (2022); Bryant et al. (2021) and taking different molecular environments into consideration (Krapp et al., 2023). Such an approach would expand the sequence design space while preserving structural consistency, enabling a larger pool of sequences for selection based on additional properties like thermostability, solubility, or toxicity. In drug discovery, for example, it would facilitate the generation of a large number of diverse candidates, allowing further selection optimized for properties such as bioavailability. Similarly, in biotechnology and enzyme engineering, it would facilitate the creation of enzymes with tailored properties, such as improved stability and activity under varying conditions. On the computational side instead, even after training, sampling from transformer-based architectures such as ESM-IF1 Hsu et al. (2022) or ProteinMPNN Dauparas et al. (2022) can be very expensive. This can severely limit the widespread use of such models, especially in virtual screening-like settings. In this work, we present an efficient method that is able to generate diverse protein sequences given a structure, including sequences far away from the native one (see Fig. 1). Recent architectures for inverse folding are based on encoder-decoder architectures, where a structure is encoded and a sequence decoded. During training, such models typically take into account only the native sequence of a given structure, maximizing its probability given the structure (Hsu et al., 2022;Dauparas et al., 2022).Figure 2: One-to-many nature of inverse folding. Top: Structure of 1KA0. Bottom: Some homologos sharing the fold.In our approach, we use the decoder to generate the complete set of parameters for a lightweight model that is sufficiently expressive to describe the sequence diversity of the multiple sequence alignment (MSA) of the native sequence. This architecture is trained endto-end to capture the broader probability distribution of sequences corresponding to a specific fold. We consider different choices for the lightweight model, and settle on the pairwise family (Figliuzzi et al., 2018), since it has been widely applied to protein sequence data Cocco et al. (2018a), proven to be good at generation (Russ et al., 2020) and demonstrated to capture information that enables fitness prediction (Poelwijk et al., 2017). A potential issue with pairwise models is that they typically have a number of parameters that is quadratic in the sequence length. We solve this issue by considering low-rank approximations, thus reducing the effective number of parameters to the same order of the sequence length. By doing so we drastically reduce the number of parameters, boosting training efficiency.Our model generates all of these parameters in a single forward pass, similar to previous research Li et al. (2023). Once this generation is done, the resulting pairwise model can be used for generating a large number of diverse sequences very efficiently leveraging CPUs on a standard machine, dramatically facilitating its use. We show that the models we generate are able to capture the diversity of the protein family better than other models and are able to find sequences far away from the natural sequence that are predicted to still fold into the same structure. We also show that this increased diversity translates into a more spread distribution in other properties, enabling selection of promising sequences from a larger pool. Codes to train the models and replicate some of the results can be found at the Potts Inverse Folding repository.model this distribution auto-regressively, using p(σ|X) = L i=1 p(σ i |σ i-1 , . . . , σ 1 , X), where L is the length, and train by minimizing the loss on the true sequence σ X , possibly after adding noise to the coordinates X (Hsu et al., 2022;Dauparas et al., 2022). Sampling amino acids auto-regressively requires a full forward pass through the neural network for every generated amino acid, making it very expensive to use in a virtual screening-like setting, where a large number of sequences are scanned for properties beyond folding into structure X. Moreover, minimizing the loss on the true sequence ignores the one-to-many property of inverse folding depicted in Figure 2, resulting in a distribution peaked around very few sequences (as we will show later) and not capturing other parts of sequence space that might be interesting for the problem at hand. We address both issues, reduced diversity and slow sampling, by having InvMSAFold output a simple and easy-to-sample-from model for fast inference and by training our architecture over multiple sequence alignments for any given input structure instead of a single sequence.
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REVIEW
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**Summary Of The Paper**

The paper presents *InvMSAFold*, a novel inverse folding framework that generates diverse protein sequences compatible with a target structure by training a lightweight pairwise model—specifically, a low-rank approximation of a Potts model—on multiple sequence alignments (MSAs) rather than relying solely on native sequences. The key innovation lies in the use of a two-step generation process: a deep neural network (built upon the ESM-IF1 encoder) generates the parameters of a pairwise model, which is then used for fast, efficient sampling of diverse sequences. The pairwise model incorporates low-rank couplings to reduce the number of parameters from $ O(L^2 q^2) $ to $ O(L q K) $, improving training efficiency and enabling CPU-based sampling. Two variants are introduced: *InvMSAFold-PW* using pseudo-log-likelihood training and *InvMSAFold-AR* using an autoregressive formulation. The model is evaluated on several fronts, including covariance reconstruction, sequence diversity, and predicted structural properties, with favorable results compared to state-of-the-art methods like ESM-IF1 and ProteinMPNN.

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**Strengths**

1. **Novel Integration of Low-Rank Pairwise Models with Deep Learning**: The paper effectively integrates a low-rank version of the Potts model—a well-established tool in statistical physics—with modern deep learning techniques. This fusion provides a computationally efficient way to model sequence diversity and offers insights into the relationship between structure and sequence.

2. **Comprehensive Evaluation Across Multiple Metrics**: The authors conduct extensive evaluations on covariance reconstruction, sequence diversity, and structural prediction. They provide detailed comparisons with established benchmarks like ESM-IF1 and ProteinMPNN, and demonstrate that *InvMSAFold* achieves superior performance in several aspects, particularly in capturing the broader sequence space and enabling faster sampling.

3. **Efficient Sampling Strategy**: The introduction of a lightweight model that can be sampled efficiently on standard CPUs represents a significant improvement in usability for virtual screening and other high-throughput applications. This addresses a real-world bottleneck in inverse folding pipelines.

4. **Clear Motivation and Practical Applications**: The paper clearly articulates the need for a one-to-many inverse folding solution and highlights the relevance of the method in drug discovery, biotechnology, and enzyme engineering. This contextual framing enhances the significance of the contribution.

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**Weaknesses**

1. **Insufficient Discussion of Biological Interpretability and Loss of Information Due to Low-Rank Approximation** (High Severity): The paper makes strong claims about the effectiveness of the low-rank approximation but provides limited insight into whether this approximation discards biologically meaningful couplings. Without evidence that the chosen rank $ K = 48 $ is sufficient to preserve essential interactions, the validity of the model remains questionable.

2. **Ambiguous Role of Noise Injection in Encoder Training** (Medium Severity): The addition of Gaussian noise to the ESM-IF1 encoder's embeddings is described briefly, but the rationale and impact of this technique on model performance are not thoroughly investigated. Understanding this aspect is crucial for replicating the results and assessing the robustness of the learned representations.

3. **Limited Justification for Using Pairwise Over Higher-Order Models** (Medium Severity): Although pairwise models are well-suited for certain tasks, they are known to lack the capacity to capture higher-order interactions, which may be critical for accurate sequence design. The paper does not adequately justify why pairwise models are preferred over more expressive alternatives, nor does it address the implications of this choice on the model's utility in real-world scenarios.

4. **Lack of Statistical Significance Analysis for Empirical Results** (Medium Severity): Several results, such as the KL divergence values presented in Table 1, are reported without accompanying measures of statistical significance (e.g., p-values, confidence intervals). This weakens the interpretation of the results and limits the ability to draw definitive conclusions about the superiority of *InvMSAFold* over baseline methods.

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**Questions For The Authors**

1. **How was the optimal rank $ K $ (set to 48 in the experiments) determined? Was cross-validation or ablation study conducted to verify that this value avoids significant information loss due to the low-rank approximation?**

2. **What is the theoretical basis for the coupling scaling in Equation 9, where couplings are divided by $ \max(i, j) $? Does this scaling alter the physical interpretation of the couplings, and how was its necessity validated?**

3. **Why is the full log-likelihood (Equation 2) not maximized directly for *InvMSAFold-PW*, despite its intractability? Could this decision introduce biases in the model's ability to represent rare variants in the MSA?**

4. **Are the KL divergence values reported in Table 1 statistically significant? Please provide p-values or confidence intervals for these comparisons.**

5. **Could you clarify the role of the Gaussian noise added to the ESM-IF1 encoder's embeddings during training? Specifically, what effect does this have on the learned representations, and how was the noise variance calibrated?**

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**Limitations Not Addressed By The Authors**

- **No Evaluation Against RFdiffusion**: Despite citing RFdiffusion in the introduction, the paper does not compare *InvMSAFold* against this recently proposed diffusion-based inverse folding method. This omission leaves open the question of how *InvMSAFold* performs relative to newer and arguably more powerful approaches.

- **No Investigation Into Generalization Beyond Homologous Sequences**: The model is trained exclusively on homologous MSAs derived from the CATH database. There is no analysis of how well it performs on de novo designs or structures with little or no homologous data available in public databases.

- **No Exploration of Higher-Order Interactions**: The reliance on pairwise models restricts the model’s ability to capture higher-order interactions that may be critical for accurate sequence design. No attempt is made to quantify the impact of this limitation or propose future directions to address it.

- **No Detailed Discussion of Ethical Implications of Sequence Diversity**: While the paper includes a brief ethical section, it lacks a deeper investigation into the potential risks associated with generating highly diverse sequences, particularly in the context of dual-use applications or unintended consequences.

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**Soundness:** 3 (Good)

While the paper presents a solid technical foundation and demonstrates empirical improvements over existing methods, it lacks depth in explaining the assumptions underlying the low-rank approximation and fails to fully justify the choice of pairwise models over more expressive alternatives. The absence of statistical significance analysis and comprehensive ablation studies also limits the soundness of the claims.

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**Contribution:** 3 (Good)

The paper contributes a novel and practically useful approach to inverse folding that improves upon existing methods in terms of efficiency and diversity of generated sequences. However, the novelty is somewhat constrained by the continued reliance on pairwise models, and the broader applicability of the method remains unclear.

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**Confidence:** 4 (Very Confident)

Despite the identified shortcomings, the paper presents a coherent and well-executed method with clear advantages in terms of speed and diversity of output. The technical details are largely explained, and the results are compelling enough to warrant further scrutiny.

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**Rating:** 8 (Accept)

The paper makes a valuable contribution to the field of inverse folding by introducing an efficient and diverse sequence generation method. Its strengths lie in the integration of deep learning with traditional statistical physics-inspired models and its practical benefits for high-throughput applications. However, the paper would benefit from addressing the listed weaknesses and providing more rigorous validation of its key assumptions. Overall, it deserves acceptance with the understanding that further refinements are needed.

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