Venus-MAXWELL: Efficient Learning of Protein-Mutation Stability Landscapes using Protein Language Models
Authors: Yuanxi Yu
Fan Jiang
Xinzhu Ma
Liang Zhang
Bozitao Zhong
Wanli Ouyang
Guisheng Fan
Huiqun Yu
Liang Hong
Mingchen Li
Venue: NeurIPS 2025 poster ()
Keywords: Deep learning
Protein language model
Protein stability
Protein engineering
Primary area: machine_learning_for_sciences

Abstract: In-silico prediction of protein mutant stability, measured by the difference in Gibbs free energy change ($\Delta \Delta G$), is fundamental for protein engineering.
Current sequence-to-label methods typically employ two-stage pipelines: (i) encoding mutant sequences using neural networks (e.g., transformers), followed by (ii) the  $\Delta \Delta G$ regression from the latent representations.
Although these methods have demonstrated promising performance, their dependence on specialized neural network encoders significantly increases the complexity.
Additionally, the requirement to compute latent representations individually for each mutant sequence negatively impacts computational efficiency and poses the risk of overfitting.
This work proposes the Venus-MAXWELL framework, which reformulates mutation $\Delta \Delta G$ prediction as a sequence-to-landscape task.
In Venus-MAXWELL, mutations of a protein and their corresponding $\Delta \Delta G$ values are organized into a landscape matrix, allowing our framework to learn the $\Delta \Delta G$ landscape of a protein with a single forward and backward pass during training. To this end, we curated a new  $\Delta \Delta G$ benchmark dataset with strict controls on data leakage and redundancy to ensure robust evaluation.
Leveraging the zero-shot scoring capability of protein language models (PLMs), Venus-MAXWELL effectively utilizes the evolutionary patterns learned by PLMs during pre-training.
More importantly, Venus-MAXWELL is compatible with multiple protein language models.
For example, when integrated with the ESM-IF, Venus-MAXWELL achieves higher accuracy than ThermoMPNN with 10$\times$ faster in inference speed (despite having 50$\times$ more parameters than ThermoMPNN).
The training codes, model weights, and datasets are publicly available at https://github.com/ai4protein/Venus-MAXWELL.

Venus-MAXWELL: Efficient Learning of Protein-Mutation Stability Landscapes using Protein Language Models
Yuanxi Yu Fan Jiangl.2 Xinzhu Ma² Liang Zhang Bozitao Zhong 1.2
Wanli Ouyang
Guisheng Fan Huiqun Yu² Liang Hongl.2* Mingchen Li222,,44
Institute of Natural Sciences, Shanghai Jiao Tong University, China
2 Zhangjiang Institute for Advanced Study, Shanghai Jiao Tong University, China 3 Shanghai Artificial Intelligence Laboratory. China
School of Information Science and Engineering.
East China University of Science and Technology, China
Abstract
In-silico prediction of protein mutant stability, measured by the difference in Gibbs free energy change (AAG), is fundamental for protein engineering. Current sequence-to-label methods typically employ the two-stage pipeline: (i) encod- ing mutant sequences using neural networks (e.g., transformers), followed by (ii) the AAG regression from the latent representations. Although these methods have demonstrated promising performance, their dependence on specialized neu- ral network encoders significantly increases the complexity. Additionally, the requirement to individually compute latent representations for each mutant site negatively impacts computational efficiency and poses the risk of overfitting. This work proposes the Venus-M A AXWELLI framework, which reformulates mutation mutations
AAG prediction
snntencc-to-landssene task. In Benus-MAXWFIL protein and their corresponding AAG values are organized into landscape matrix, allowing our framework to learn the AAG' landscape of a protein with a single forward and backward pass during training. Besides, to facilitate fu- ture works, we also curated a large-scale AAG dataset with strict controls on data leakage and redundancy to ensure robust evaluation. Venus- MAxus-MAXWELL is compatible with multiple protein language models and enables these models for accurate and efficient AAG prediction. For example, when integrated with the ESM-IF, Venus-MAXWELJ achieves higher accuracy than ThermoMPNN with 10x faster in inference speed (despite having 50x more parameters than Ther- moMPNN). The training codes, model weights, and datasets publicly available at /tttps://gtthiiii com/ ai4protein wwwprroteeii//een--MAMMM. V
Introduction
Protein stability, determined by the change in Gibbs free energy (AG) of protein sequences between the native and unfolded states, plays critical role in various applications, ranging from therapeutic
protein design to industrial enzyme engineering 2 3 5 6.. In protein engincering, accurately predicting mutation-induced stability changes (AAG) is critical, and current methods for assessing stability across large mutant libraries remain prohibitively expensive and resource- ieoourceinttessvve, creating pressing need for efficient computational solutions. Consequently, developing accurate and efficient
"Corresponding authors: Mingchen Li (limc0sj tu. edu, en) and Liang Hong tu. edu. cn); TThese authors contributed equally to this work.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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computational tools for protein stability prediction is longstanding focus of research, spanning physics-based bassed and statistical methods 7 machine learning approaches [TO]
Deep learning, exemplified by AlphaFold2 [5]. has revolutionized protein-related tasks 16
18
with protein language models (PLMs), emerging powerful tools for capturing co-evolutionary patterns from vast sequence and structural datasets [22 23
19
20 21
24 25 26]. These models have shown powerful capabilities in wide-range downstream tasks, including stability prediction. Recent methods, such as Stability Oracle [33 and ThernoMPNN 14. leverage PLMs to achieve impressive performance AAG estimation. However, their reliance on complex architectures and massive computational requirements leads to prohibitively high training/inference costs, fundamen- tally limiting their scalability for large-scale protein engineering applications. These methods can be broadly categorized as sequence-to-labee approaches. Specifically. these methodologies involve two-stage process: first, a PL.M or a similar encoder model transforms a mutant sequence into a latent representation; subsequently, regression head predicts the AAG value from the representation.
The sequeencettolabbll paradigm, however, presents three main challenges: (i) Computational Inefficiency: This design requires computing a distinct latent vector for each mutant site (as shown in Figure M (A)), which is computationally expensive and heavily dependent on the encoder's design; (ii) Knowledge Misutilization: The regression head is often trained from scratch, failing to fully harness the rich evolutionary patterns PLMs learn during pre-training which are implicitly captured in the model's 's output likelihoods or logits. (iii) Architectural Rigidity: The reliance on a specific regression head design limits versatility and complicates the integration of different PLM types. These limitations highlight the need for generalizable and efficient approach for stability prediction.
(A)Traiitional mutation A&G prediction models:
Sequence-to labell
MECNL
Encoder
0.31 mml
(B) MAXWELL: Sequence-to pequeecc-to-landssaae landsaace
MAENL
Encoder
1.3 mol
MACNL
Encoder
(PLM)
Wild-type
A&G Landscape
Encoder
033 mol
seq.
Mutant seq.
Latent vectors
A4G
Figure 1: Comparison of Venus-MAXWELL and sequence-to-label models. (A): The sequence- to-label models compute a latent vector for each mutant site. (B): The Venus-MAXWELL outputs a landscape in one forward pass.
In this work, we introduce Vemus-MAXWELI (Matrix-wise landscape learning), an efficient sequence- to-landscape framework that reframes AAG prediction. Venus-MAXWELL is not merely faster model but fine-tuning paradigm that directly addresses the limitations of sequence-to-late approaches.
First, Venus- MANSSMMXXWLLL maximizes the utilization of pre-trained PLM knowledge. Our key insight is to formulate the framework as a differentiable, matrix-basee version of the PLM's native zero-shot scoring function. Many PLMs, pre-trained millions of sequences, AAG of mutant in a "zero-shot" manner by calculating the log- log-likeldoood difference between mutant and wild-type sequences (7 (28). Venus-MAXWELL Critically, its "initial performance" (before any fine-tuning) is mathematically equivalent to the underlying PLM's zero-shot prediction performance. Therefore, our fine-tuning process does not learn from scratch but starts from a significantly advanced foundation, directly enhancing the rich evolutionary information already captured by the PLM.
estimate the fitness like
designed to inherit this powerful capability.
Second, Menus-MAXWELL introduces a novel matrix-baseh fine-tuning methodology. Instead of adding a task-specific regression head, our method directly fine-tunes the PLM's output logits in matrix-wiid Miirix-wissee fashion (as depicted in Figure (B)), elegantly adapting the PLM for AAG estimation. This paradigm is highly versatile, serving universal fine-tuning framework compatible with diverse PLMs, including masked language models and inverse folding models.
Third, as direct benefit of this architecture, Venus-MAXWELJ is exceptionally computationally efficient. To predict all single-site mutations for a protein of 200 amino acids (3800 mutants), sequence-to-label model must exccute 3800 separate computations. In stark contrast, Venus-

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MAXWELL accomplishes all predictions in one computational pass using the wild-type sequence, enabling high-thne -trohhhhuuhhtt AAG prediction minimal cost. This capability particularly important for protein engineering workflows such as early-stage directed evolution, which often begin by comprehensively screening single-site mutations to identify stabilizing candidates. While complex multi-site mutant prediction is typically handled by specialized models (e.g., ECNET ProteinNPT 30 and PRIME(31), these models could offer multi-site mutant predictions based on training on single-site mutants. Venus-MAXWELS is thus positioned early-stage single-site predictions, which can then be used for experimental validation and as input prediction models, thereby accelerating the entire directed evolution
crucial tool to provide high-quality,
for downstream
pipeline.
Besides, we have observed that existing mutation AAG datasets often exhibit significant redundancy between training and test sets, as well as misalignment between mutant and wild-type sequences. To address these issues, we constructed high-quality dataset derived from public resources, which underwent rigorous de-duplication and redundancy reduction based on sequence alignment. As a result, we obtained strictly separated training and test sets, containing over 226K and 12K curated mutation entries, respectively, ensuring a fair evaluation of generalization. The difference between this dataset and those in previous studies [ lies in an intensive manual verification process to validate the sequence-mutation mutation consistency.
Utilizing this meticulously constructed AAG dataset for evaluation, we demonstrate that fine-tuning with our Cenus-MAXWELL framework significantly enhances the predictive capabilities of various PL.Ms. For example, when using ESM-IF as the base model, the fine-tuned ESM-IF not only outperforms the sequence-to-labe paradigm (which uses ESM-IF to generate features per mutant for regression) on the AAG prediction task but also surpasses ThermoMPNN, a current of-t-oo-thheart model designed explicitly for AAG prediction, while achieving over 10 times faster inference.
To summarize, our main contributions are as follows:
A matrix-based prquence-to-landseee framework:
offers new efficient and broadly applicable approach for fine-tuning diverse PLMs on A AG prediction task. This approach uniquely maximizes the utilization of pre-trained evolutionary knowledge to enable superior stability assessment with minimal computational overhead.
A qualified mutation AAG dataset:
We provide cleaned protein mutation AAG dataset. The sequences within the training and test sets have been processed to remove duplicates and reduce redundancy, ensuring that their sequence identity remains below 30%.
2
Related Work
Protein language models (PLMs). Similar to advances in natural language processing, PLMs have evolved into two dominant architectural paradigms: masked language models and auto-regressive
models. Masked language models, such as ESM series [22 24 25 32 33 employ bidirectional attention to predict masked residues, capturing rich contextual representations of protein sequences. Auto-regressive models, like ProGen2 [34] and Tranception [35], predict amino acids sequentially, excelling generative tasks. These pre-trained models have demonstrated success across various downstream applications, particularly in mutation effect prediction and protein design. Through fine-tuning, PLMs can effectively leverage their encoded evolutionary information to enhance protein fitness prediction, supporting directed evolution [29] 31] 36 (7). Moreover, PL.Ms often exhibit zero-shot capabilitics [22 24], but this ability is non-directional and not specifically tailored to stability prediction. Our work offers an efficient way to convert this non-directional prediction into
directional one by fine-tuning on AAG mutation datasets.
Physics-based methods for AAG prediction. Early efforts in protein stability prediction relied on physics-based methods or statistical analysis 7 g 99. Rosetta 77), widely adopted framework, employs Cartesian space sampling methods to predict stability changes based on high-resolution structural data. Similarly, FoldX g estimates AAG of mutants using an empirically derived energy function calibrated against experimental mutagenesis data. Such methods are often highly interpretable but are limited by computational complexity and input data precision, leading to lower accuracy and speed compared to contemporary deep learning methods, as we present in this work.
3

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Deep learning-baser methods for AAG prediction. Prior to the development of are-trained PLMs. several deep learning models have been designed to reduce the bias predicting stabilizing mutations The application of PI.M further advanced AAG prediction. For example, Stability Oracle integrates pretrained graph transformer model with data augmentation approach based on thermodynamic permutations to address the challenges of data bias and model generalization. Leveraging the knowledge overlap between sequence recovery and stability optimization tasks, ThermoMPNN [4] utilizes the embeddings from ProteinMPNN (38) for transfer learning, resulting in a fast and robust AAGprediction. Despite their success, these models often require complex architectures and extensive computational resources. Our approach, however, offers more general and efficient means of facilitating stability prediction with enhanced speed and accuracy, consequently improving its applicability range of directed evolution contexts.
10
1 12
3 Method
3.1 Preliminary
Protein Language Model (PLM). The PLM, whether it is GPT-style generative model or BERT- style de-noising model, can be viewed as a parameterized function fo that maps a protein sequence A (a), (2,...,,n)) an,) a probability distribution matrix E [=*
P = fe()),
(I)
where n is the sequence length and |A| is the size of the amino acid alphabet (typically 20 for standard amino acids). The element in the i-th row and j-th column of P is the probability that the i-th amino acid in the protein sequence is the j-th amino acid in the alphabet. Besides, PLMs can be extended by incorporating protein structural condition S, modifying the Equation 1 to P S (A, S). For simplicity, we omit the structural condition in the following, as it does not alter the framework.
PLM for mutation scoring. PLMs are typically trained large-scale protein databases, primarily composed of naturally occurring proteins, which exhibit higher stability due to evolutionary selection. This enables PLMs to predict mutation effects using "zero-shot" approach by comparing the probabilities of wild-type and mutant amino acids (24). For a mutation at position / from amino acid a, be the predicted zero-shot score is:
Score(a; + b,) = log fo(A) log fe(A)
(2)
where a higher score indicates a positive mutant. However, the score is often undirected (it could reflect activity, stability, or other properties). Notably, for inverse folding models, this specific log-likeldhood ratio has been mathematically derived to possess a free energy interpretation directly related to A&G 339 This provides strong theoretical foundation for its use. We therefore fine-tune the PLM using AAG data to enable it to explicitly predict mutation AAG.
3.2 AGG Landscape Construction
The AAG landscape of a given protein is a real-valued matrix Y E R where Yú, is the AAG value resulting from mutating the amino acid at position to the j-th amino acid. However, in practice, it is usually infeasible to experimentally measure AAG for all possible mutants, leading to sparse landscape. To track which entries have been experimentally measured, we define binary mask matrix M (0, 111,, where M.j 1 if i.j is available, and M.j 0 otherwise.
3.3 Framework Architecture
The overview of Venus-M AXWELI is shown in Figure 2 Specifically, by reformulating zero-shot mutation scoring (Equation 2 matrix scoring, Venus-MAXWALL enables PL.Ms to compute the mutation landscape in a single forward pass. Crucially, the process is differentiable, allowing PLMs to be fine-tuned on protein mutation AAG landscapes through backpropagation.
For protein sequence
(a1, ...,...: On Venus-MAXWELL computes the landscape as follows:
Sequence encoding. The sequence A is first tokenized into a one-hot encoding matrix X E (0, 1) 1)ax|AA, Ill, For each position i, the row X, is one-hot vector where the entry corresponding to the wild-type amino acid a; is set to 1, and all other entries are set to O.

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Framework pipeline
[0.1)ami4]]
Encode
0
0
PLM
0
AA sequence (Wild-type protein)
Hadamard product
AA one-hot encodings (Protein sequence)
AA distribution (log scale)
Column sum
gexi
Element-wise subtraction
Replication
Mutation
AAG
M1T
mm
Encode
Joint loss
A2G
0.2 mel
Mutation landscape
Predicted mutation landscape
A&G of mutations
Masked (no experimental data)
Figure 2: Overview of Venus-MAXWELL. Venus-MAXWELL utilizes matrix-driven scoring to enable PLMs learn the mutation landscape of protein through single wrwaad-backkward computation process.
Probability prediction. The PLM computes log-probability matrix Q E Ra
Q log (P) log fo(A),
where fe is the parameterized PLM function, and the logarithm is applied element- wise. Each entry Q., represents the log-probability of amino acid ay at position
Wild-type probability extraction. To isolate the log-probabilities associated with the wild-type amino acids, compute the Hadamard product (element- wise multiplication) between the log- probability matrix Q and the one-hot encoding matrix X, yielding matrix Qu E R=
Qut Q X,
(4)
where the Hadamard product. Since Xi.j = only for the wild-type amino acid a; at position i, Qur retains log-probabilities at wild-type positions and zeros elsewhere.
Wild-type vector construction. The matrix Qui is then aggregated to produce compact representa- tion of the wild-type probabilities by summing Qut over the amino acid dimension:
(5)
where 1 A| IRAI a unity vector and v, is the log pog--rbbbbilty of the wild-type amino acid a,. Wild-type matrix replication. To enable comparison between wild-type and mutant probabilities, the wild-type log- log-probability vector v is replicated along the column dimension |A] times to form matrix V |a Specifically, form V through multiplication of o with the unity vector AI E IRI [AI,
WE =
1/
(6)
In this matrix, each row i contains the log-probability of the wild-type amino acid a, repeated across all columns j.
Mutation landscape computation. Finally, we subtract the replicated wild-type matrix " from the log- probabiiiyy matrix Q to obtain the mutation landscape matrix L E R
L Q Top wt
Each entry L,, represents the log log-pro probaabiiityy difference between the j-th amino acid and the wild- type amino acid at position i. To formalize this computation, we define function go as follows:
ge (A) L. Q Rep
(8)
= log fo (A) - ((log fo(A) o X). 1 T/A
(9)
5

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It is clear that ge is differentiable because the protein language model fo is differentiable. Addi- tionally, the mutation score equal the zero-shot scoring formula shown in Equation [ We also provide the algorithm and pseudo-code of Venus-MAXWELL, for training and landscape prediction in Appendix A. and Appendix A.2
3.3.1 Training objective
The training objective ensures the predicted AAG landscape closely aligns with the ground-truth AAG landscape. Given a protein sequence X, the experimentally measured mutation stability landscape E Rn*[A|, and a binary mask M E {0, (0xx]] indicating observed entries, the model predicts a landscape L (o
The training loss comprises two components: Ranking loss LRanking and mean squared error loss LMSE The first component encourages correct ranking of mutations by minimizing the negative Pearson correlation between the predicted and measured AAG values:
EN ( (Y Y)
Lanking Y, M,,)) =1
(10)
En,, Ma (L,, - L)3 Ei, (Y, YY
where T. and Y represent the mean predicted and ground-truth AAG values the masked positions. The second component, a mean squared error (MSE) loss, aligns absolute values:
MMSE K,M,L)
Σ M. L. 4d
(11)
Here, = MLP(L)- L by is the output of a two-layer multilayer perceptron (MLP), where (-)) denotes SELU activation function 40). This transformation adjusts for the scale mismatch between log- prgobbbbayyliyy scores and experimental AAG values. The parameters W1, W2, , bj, and ba are learnable. Specifically, the learnable weights have dimensions W1 RV and W E Px1, where V is the PLM's vocabulary size and D is the hidden dimension.
The total loss weighted sum of both components:
[())
Σ
LRanking (Y, M, M, + À LMSE (Y, M, go (A)),
(12)
M)DD
where À is a hyperparameter weighting factor that balances the two loss components. D is the training dataset, comprising multiple protein sequences annotated with sparse mutation landscape and corresponding binary mask. Each data point in 'D is tuple (A, Y, M). The training objective is to arg ming L()). Since go differential, we can utilize backpropagation
minimize the total loss: and gradient descent to optimize the parameters 6, which are initialized from the pre-trained model weights. The specific optimization details, hyperparameter selection, and implementation settings are described in Section 5.1 and Appendix A.3
4 Datasets Building
We collected large protein mutation AAG dataset, which includes 573 protein sequences and 239,408 mutations (average 418 mutations per protein) to evaluate the performance of Venus- MAXWELI. All data have been manually verified to ensure consistency between mutant and wild-type sequences, and duplicates were removed to maintain data integrity.
Test set. The test set is a compilation of mutation AAG data that we are currently able to collect, including P53 [77, Myoglobin [1]. SSym [17] S669 [4]) S8754 42), M1261 42] vb1432 (3) Duplicates within the datasets are addressed by selecting the AAG measurement with the highest absolute value. This was intended address the neutral bias and move the AAG predictions from a model away from 0. After removing duplicates, the test set contains 12,443 mutations across 308 proteins, namely 308 sparse mutation AAG landscape. This dataset, named TKs is organized by protein to enable per-protein prediction and evaluation, consistent with benchmarks like ProteinGym analysis, only proteins with at least five mutations were included.
Fireprotdb 44 and Thermomutdb 45
(27). To ensure sufficient data for meaningful per-protein

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23 and (2) sequence-only Training Set. The training set is derived from a large-scale dataset containing 272K protein mutation sequences, denoted as cDNA272K [6). This dataset is commonly used training dataset the task of protein stability prediction 13 To prevent potential data leakage issues, we used the MMSeqs2 tool [ to filter the training set, as illustrated in Figure (A). Specifically, we removed all sequences from the training set that share a sequence identity of 30% or higher with any sequence in the test set. This threshold 30% chosen to maintain a significant diversity of sequences, as it lies within the "ttilight zone", region where 95% of protein pairs have distinct structural folds [48). The resulting training set, Train226K, consists of 226K sequences with less than 30% sequence similarity to any sequences in the test set. All datasets are stored in a protein-specifii format facilitate model training and evaluation, and all sequences were folded using ColabFold 2.5 49 to generate structural input for PLMs if needed.
To further validate the separation between training and test sets, we performed pairwise global sequence alignment using the Needleman- Wunnshh algorithm [50] As shown in Figure 3 (B), the similarity distribution has a mean of 6.17% with over 88% of test-training pairs exhibiting less than 10% similarity. This low overlap confirms that our deduplication effectively reduces data leakage risks and enhances the model's ability to generalize to unseen sequences.
Experiments and Results
5.1 Experiments Setup
The ability of Venus-MAXWELLY to improve protein stability prediction (AAG) was systematically evaluated across wide range of PLMs. We strategically selected representative PLMs from two distinct categories: (1) structure-awar PLMs (ESM-IF and ProSST), which incorporate structural information either explicitly through hybrid architectures, with ESM-IF demonstrating superior zero-shot performance in stability prediction tasks [22] and ProSST representing an emerging paradigm that efficiently combines sequence and structural information
PLMMs (ESM-1b, ESM2, ESM-Iv), which are widely adopted in protein engineering applications and
serve pure sequence-based model representatives 24 51
52
33
53
Hyper-parameter settings. For all Venus-MAXWELL enhanced models, we utilized the Adam optimizer (54). All key hyperparameters were selected through rigorous 5-fold cross-validation procedure on the training set (Train226K) to prevent test set leakage. Based on this analysis, we set the initial learning rate to 5 x 10 the loss weighting factor À to 0.1, and the MLP hidden dimension D to V (the PLM's vocabulary size). The optimal number training epochs (approximately 7, determined via early stopping with patience of 5 epochs) validation process. The final models were then trained on the complete Train226K dataset for the determined optimal number of epochs. A detailed description of the cross-validation methodology,
also identified during this cross-
(A)
(B)
Needleman -unnahh- Sequence Similarity Distribution Between Test12K and Train226K
Training Set
Test Set
10000
Mean: 6.17%
cDNA272K
Ssym
8000
S669
Myo
MMseqs2
P53
S8754
M1261
Count
Thermo mutdb
4000
Idantity 30%
vb1
FP
432
2000
Deduplication
Train226K
Test12K
5
10
15
20
25
Similarity (%)
Figure 3: Dataset details. (A): Dataset construction process of Train226K and Test12K. (B): Global sequence alignment using Veenseeman-uusch confirmed the low sequence similarity between TK and Tra.n226K.
7

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hyperparameter search spaces, and full ablation results for each choice is provided in Appendix A.3 All experiments were conducted on PC with an NVIDIA RTX 4090 GPU.
Evaluation metrics. To rigorously assess model performance, we employed per-protein evaluation followed by averaging across the dataset. For each protein, calculated Pearson (pp) and Spearman (p+) correlation coefficients between predicted and experimental AAG values, capturing linear and monotonic relationships, respectively. For classification of stabilizing vs. destabilizing mutations (threshold at AAG = 0), we computed AUC-ROC and FI scores. This per-protein averaging approach prevents bias toward proteins with more mutations, ensuring balanced assessment across diverse protein structures 27 28
Baselines. We compared Venus-MAXWELI against: (1) ThermoMPNN, the current state-of- of-the-art neural network model [5 used with its publicly available checkpoin P Our training set is a subset of the training set of ThermoMPNN, there data leakage issues. (2) FoldX and Rosetta, two representative physics-based computational methods; and (3) Embedding transfer baselines using ProSST and ESM-IF, denoted "ProSST .LPP" and "ESM-IF MLP". The hyperparameter
settings of baseline models are shown in Appendix A.4
5.2 Model Performance
We evaluated mutations across 308 diverse proteins. As shown in Figure 4 (A), Venus-Mr MAWELL enhances zero- shot performance across multiple PLMs, achieving 0.143 (Wilcoxon signed-rank test, p < 0.01). With ESM-IF, Venus-MAXWELL reaches a Spearman correlation of 0.517, followed by ProSST at 0.479, highlighting the importance of structure- representations in protein stability prediction tasks [27 [8). This improvement stems from matrix- driven scoring (Equation 8). which maps sequences mutation landscapes in one forward pass.
performance on the Tcst12K dataset (Section 4 comprising 12,443
.ww.neeenaaness average Spearman correlation improvement of
(A)
0.6
(B)
0.6
0.517
Mean Spearann Score
0.517
0.5
0.5
0.427
Mean Speaman
0,410
0.4
0.374
0.369
0.358
0.384
0.4
0.368
0.302
0.3
0.268
0.251
0.263
0.3
0.234
0.2
0.2
Venus Venus-HArWELL HAXWELL (ESM-IF)
ESM-IF (Zero-shat)
Venus Venus- (ProSST)
ProSST (Zera-shat)
Venus MAXWELL (ESM-2)
ESM-2 Izero shot)
Venus MAXWELL (ESM-3) (ESM-Iv)
E5M-IV (Zero-shad
Venus NAXWELL (ESM-1) 1b)
ESM-18 14 iZero-shot)
Venus- MAXWELL (ESM-IF)
Vanus HAXWELL (ProSST)
ESM MLP
ProSS T MLP
TheroMPPN
Rosetta
FoldX
Figure 4: Performance comparison of Cenus-MAXWELL. (A): Comparison of Spearman cor- relation scores between zero-shot predictions and Venus-MAXWELJ enhanced predictions across different PLMs. The improvement from zero-shot to Venus-MAXWEL MAXWELL is statistically significant (Wilcoxon signed-rank test, p < 0.01). (B): Comparison with existing prediction methods. Venus- MAXWELL (ESM-IF) outperforms ThermoMPNN (Wilcoxon signed-rank test, < 0.01). All evaluations were performed the Test12K, with per-protein correlations averaged across entire set.
When compared to existing methods (Figure 4 (B)), embedding transfer methods struggle under the low similarity between Train226K and Te (Spearman: 0.302/0.234). The shysics-based Rosetta and FoldX method also demonstrates moderate performance. In contrast, the specialized architecture of ThermoMPNN delivers strong results (Spearman: 0.508). Notably, Venus-MAXWELL with ESM- with ProSST achieves comparable performance, without requiring architectural modifications to the base PLMs. To further validate these findings on established public benchmarks, we also benchmarked performance
IF surpasses ThermoMPNN (Spearman: 0.517, < 0.01) while
htttps://githuu..oom// Lawhheerrohhhheemp

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on specific, well-known subsets within Test12K (e.g., p53, s669, Myoglobin) against ThermoMPNN and Stability Oracle. As detailed in Appendix[A.. Cenus-MAXWERS consistently outperforms both SOTA models on these individual datasets, confirming its robust and state-of-the-art performance. These results demonstrate that Venus-MAXWEL. 's matrix-basee landscape approach effectively captures global stability patterns, outperforming both traditional regression methods and physics- based approaches by leveraging evolution pattern knowledge from pre-trained PLMs. For more details of comparison additional metrics, please refer to Table A4 in Appendix A.5
Furthermore, we investigated whether the inherent sparsity of the training landscapes adversely affects generalization. We conducted an ablation study by progressively removing random portions of the training data. The results show that Venus-MAA MAMusMMAXWLLLL performance is exceptionally robust, maintaining superior performance (Spearman: 0.501) even when 90% of the training mutants are removed. This demonstrates that the model learns generalizable stability principles from the diverse collection of proteins, rather than overfitting to sparse local landscapes. The full details of this sparsity analysis are also presented in Appendix A..5
To ensure that the superior performance of Venus-MAXWELL with ESM-IF and ProSST, both of which utilize structural information, does not stem from structural data leakage, we analyzed structural similarity between Train226K and Test 2K using TM-align [56). The average TM-score between training and test set structures is approximately 0.3, indicating low structural similarity. Furthermore, the weak correlation between prediction accuracy and structural similarity (see more details in Figure confirms Venus- MAXULLL 's robust generalization ability across diverse
A and Appendix (A.6) ( protein structures.
5.3 Prediction Speed Analysis
We evaluated Venus-MAXWELL inference efficiency in the context of directed evolution, where rapid screening of single-site mutants is crucial. Testing on single-site saturation mutagenesis of 10 random proteins from Test12K (averaged over five sampling runs), Venus-MAXWELL demonstrates remarkable speed improvements over baseline methods.
Table 1: Efficiency comparison of stability prediction methods.
Model
Trainable parameters
Prediction time (s)
Mutants per second
Venus- MAXWELL (ESM-IF)
142 M
2.11 + 0.04
20050 714
Cenus-MAXWELL (ProSST)
119 M
0.53 + 0.02
85471 + 3512 +
ESM-IF MLP
132 K
4744.76 + 335.9
ProSST + MLP
296 K
807.77 7.53
+ 2088 + 7
ThermoMPNN
2.68 M
20.27 + 1.81
Despite maintaining full model capacity (142 million parameters for Venus-MAXWELL (ESM-IF) and 119 million parameters for MODus-MXXWELL (ProSST)), Cenus-MAXWELL achieves exceptional
inference efficiency. As shown in Table I CenUs-MXXELLLEEMM per second, while ThermoMPNN (2,088 mutants/s) by over 10-fold and MLP-based baselines (9-51 mutants/s) by orders of magnitude. The efficiency improvement remained robust across sequence lengths (100-1000 residues), as detailed in Table Abble/Aaann and A8 Appendix (A7]] where we further analyze inference speed variations with sequence length, underscoring Venus-MAXWERS MAXWELL'S capability for parallel prediction of mutation landscapes. Notably, the training efficiency is similarly improved: Menus-MAXWELL (ESM- IF) completes one training epoch on Train226K in 41 seconds, over 200 times faster than embedding transfer methods, despite fine-tuning the entire PLM (see Figure A2 in Appendix A.8 for details). These efficiency gains, combined with the predictive accuracy reported in Section 5.2 highlight scalability for high-throughhu directed evolution.
predicts 20,050 mutants reaches 85,471 mutants per second, outperforming
5.4 Ablation Study
To dissect the contribution Venus-MAXWELLL'S components, we conducted ablation studies using ESM-IF as the base model, evaluating performance on Test12K (Table analysis reveals:

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Table 2: Ablation studies of Venus-MAXWELL (ESM-IF) on Test 12K datasets
Model
Ps
Pp
F1
AÚC
(ESM-IF)
0.517
0.542
0.427
0.678
MeDUS-MAXWELL (ESM-IF) (No LMSE)
0.506
0.528
0.330
0.638
VenuS-MAXWELL (ESM-IF) (No Lrnnking)
0.496
0.518
0.410
0.671
Venus-MAXWHLL (FSM-IF) (ESMFold)
0.492
0.519
0.389
0.662
Venus-MAXWELL (ESM-IF without pretrain)
0.202
0.234
0.382
0.501
ESM-IF + MLP
0.302
0.331
0.383
0.627
Zero-shot
0.375
0.376
0.218
0.575
Zero-shot (ESMFold)
0.354
0.360
0.217
0.574
ThermoMPNN
0.508
0.529
0.371
0.646
Ablation on training objective. Removing the MSE module ("No LMSE") maintained strong mutation ranking (Spearman: 0.506) but decreased classification performance (FI: 0.330, AUC: 0.638). Conversely, removing the ranking loss component ("No LRanking ") better preserved classification metrics (FI: 0.410, AUC: 0.671) while showing reduced correlation scores (Spearman: 0.496). Notably, both variants remained competitive with ThermoMPNN, demonstrating the robustness of our matrix-based approach even with partial architecture.
Ablation on structure quality. Using ESMFold-generated structures instead of ColabFold slightly reduced performance (51) As a result, if the structure is predicted using ESMFold instead of ColabFold, both our model (Venus-MAXWEL)
(ESMFold)) and the zero-shot method experienced slight decrease in scoring accuracy. Specifically, the Spearman correlation dropped from 0.517 to 0.492 for our model and from 0.375 to 0.354 for the zero-shot method, respectively.
Ablation on pre-training impact. Random initialization of ESM-IF severely degraded performance (Spearman: 0.202), demonstrating the crucial role of pre-trained weights in capturing evolutionary patterns. All models with pre- pre-trained weights outperformed both embedding transfer (Spearman: 0.302) and zero-shot baselines (0.375/0.354 for ISM-I))ESMFold), confirming the effectiveness of our matrix-based landscape approach.
Conclusion and Limitations
In this work, we introduced Venus-MAXWELL a universal and efficient framework that enhances protein stability prediction while preserving PLMs inherent capabilities. Through elegant ma- trix transformations, our approach enables rapid modeling of the complete mutation landscape, demonstrating superior performance in both ranking and classification metrics. Extensive evaluation demonstrates computational speedup three orders of magnitude, establishing a scalable paradigm for protein engineering and stability optimization.
While Venus-MAXWELL excels in stability prediction, its matrix-based approach holds some potential for broader applications. For example, the current matrix matrix-based implementation is inherently limited PLMs using single amino acids vocabulary tokens. excluding structure-aware tokenization schemes (e.g., SaProt (57)) or BPE tokenizer (58) (e.g., ProGPT (59)). Moreover, the framework focuses on single-site saturation mutations now, and extending to model combinatorial mutation effects poses a key challenge. Future work could address these limitations by developing adaptive tok- enization strategies, incorporating multi-sit mutation modeling, and evaluating Venus-MAr Monus-MAXWELLL on diverse protein function prediction tasks.
Acknowledgments and Disclosure of Funding
This work was supported by the National Key Research and Development Program of China (2024YFA09 1 Computational Biology Key Program of Shanghai Science and Technology Commission (231S1400600) Shanghai Jiao Tong University Scientific and Technological Innovation Funds X010000433,, and Science and Technology Innovation Key R&D Program of Chongqing (CSTB2022TIAD-STXOO0) CSTBBU22TIAD-STXOOO33
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References
[1] Sasha B Ebrahimi and Devleena Samanta. Engineering protein-based therapeutics through structural and chemical design. Nature Communications, 14(1):2411, 2023.
[2] Steven M Jay and Richard T Lee. Protein engineering for cardiovascular therapeutics: untapped potential for cardiac repair. Circulation research, 1 113(7):33-9443, 2013.
[3] Michaela Gebauer and Arne Skerra. Engineered protein scaffolds as next-generation therapeu- tics. Annual review of pharmacology and toxicology, 60(1):391-415, 2020.
[4] Elizabeth L Bell, William Finnigan, Scott P France, Anthony P Green, Martin A Hayes, Lorna J Hepworth, Sarah Lovelock, Haruka Nikuura, Sílvia Osuna, Elvira Romero, al. Biocatalysis. Nature Reviews Methods Primers, 1(1):1-21, 2021.
[5] Zhoutong Sun, Qian Liu, Ge Qu, Yan Feng, and Manfred T Reetz. Utility of b-factors in protein science: interpreting rigidity, flexibility, and internal motion and engineering thermostabillty. Chemical reviews, (993))6626- 1665, 2019.
[6] Fan Jiang, Jiahao Bian, Hao Liu, Song Li, Xue Bai, Lirong Zheng, Sha Jin, Zhuo Liu, Guang- Yu Yang. and Liang Hong. Creatinase: Using increased entropy to improve the activity and thermostability. The Journal of Physical Chemistry B, 127(12):2671-268) 2023.
[7] Rebecca Alford, Andrew Leaver-Fay, Jeliazko Jeliazkov, Matthew O' O'Maara, Frank DiMaio, Hahnbeom Park, Maxim V Shapovalov, P Douglas Renfrew, Vikram K Mulligan, Kalli Kappel, et al. The rosetta all alom energy function for macromolecular modeling and design. Journal of chemical theory and computation, 13(6):3031-300 2017.
[8] Joost Schymkowitz, Jesper Borg, Francois Stricher, Robby Nys, Frederic Rousseau, and Luis Serrano. The foldx web server: online force field. Nucleic acids research. 33(suppl 23:2382- W388, 2005.
[9] Shuangye Yin, Feng Ding, and Nikolay V Dokholyan. Eris: automated estimator of protein stability. Nature methods, 4(6):466-467, 2007.
[10] Bian Li, Yucheng T Yang. John A Capra, and Mark B Gerstein. Predicting changes in protein thermodynamic stability upon point mutation with deep 3d convolutional neural networks. PLoS computational biology, (6(11);e1008))
2020.
[11] Huali Cao, Jingxue Wang, Liping He, Yifei Qi, and John Z Zhang. Deepddg: predicting the stability change of protein point mutations using neural networks. Journal of chemical information and modeling, 59(4))150-1154, 2019.
[12] Lijun Quan, Qiang Lv, and Yang Zhang. Strum: structure-based prediction of protein stability changes upon single-point mutation. Bioinformatics, 32(19):296-2246, 2016.
[13] Daniel Diaz, Chengyue Gong, Jeffrey Ouyang-Zhang, James M Loy, Jordan Wells, David Yang, Andrew D Ellington, Alexandros G Dimakis, and Adam R Klivans. Stability oracle: a streeture-based graph-transformer framework for identifying stabilizing mutations. Nature Communications, 15(1):6170, 2024.
[14] Henry Dieckhaus, Michael Brocidiacono, Nicholas Z. Randolph, and Brian Kuhlman. Trans- fer learning to leverage larger datasets for improved prediction of protein stability changes. Proceedings of the National Academy of Sciences, 121(6):23148531211 2024.
[15] John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ron- neberger, Kathryn Tunyasuvunakool, Russ Bates, Augustin Žídek, Anna Potapenko, et al. Highly accurate protein structure prediction with alphafold. nature, 596(7873):583-58) 2021.
[16] Ziyi Zhou, Liang Zhang, Yuanxi Yu, Banghao Wu, Mingchen Li, Liang Hong, and Pan Tan. Enhancing efficiency of protein language models with minimal wet-lab data through few-shot learning. Nature Communications, 15(1):5566, 2024.

---

[17] Bian Li, Yucheng T Yang. John A Capra, and Mark B Gerstein. Predicting changes in protein thermodynamic stability upon point mutation with deep 3d convolutional neural networks. PLoS computational biology, 16(1 1 10080201 2020.
[18] Lasse M Blaabjerg, Maher M Kassem, Lydia L Good, Nicolas Jonsson, Matteo Cagiada, Kristoffer E Johansson, Wouter Boomsma, Amelic Stein, and Kresten Hindorff-Larsen Rapid protein stability prediction using deep learning representations. Elife, 12:e82593, 2023.
[19] Shuyu Wang, Hongzhou Tang, Peng Shan, Zhaoxia Wu, and Lei Zuo. Pros-gun: predicting effects of mutations on protein stability using graph neural networks. Computational Biology and Chemistry, 107:107952, 2023.
[20] Yuanxi Yu, Fan Jiang, Bozitao Zhong, Liang Hong. and Mingchen Li. Entropy-driven zero-shot deep learning model selection for viral proteins. Physical Review Research, 7(1):013229 2025.
[21] Liang Zhang, Kuan Luo, Ziyi Zhou, Yuanxi Yu, Fan Jiang, Banghao Wu, Mingchen Li, and Liang Hong. A deep retrieval-enhanced meta-learning framework for enzyme optimum ph prediction. Journal of Chemical Information and Modeling, 65(7):3761-3770, 2025.
[22] Chloe Hsu, Robert Verkuil, Jason Liu, Zeming Lin, Brian Hie, Tom Sercu, Adam Lerer, and Alexander Rives. Learning inverse folding from millions of predicted structures. In International conference machine learning, pages 8946-8970. PMLR, 2022.
[23] Mingchen Li, Yang Tan, Xinzhu Ma, Bozitao Zhong, Huiqun Yu, Ziyi Zhou, Wanli Ouyang, Bingxin Zhou, Pan Tan, and Liang Hong. ProSST: Protein language modeling with quantized structure and disentangled attention. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[24] Joshua Meier, Roshan Rao, Robert Verkuil, Jason Liu, Tom Sercu, and Alex Rives. Language models enable zero-shot prediction of the effects of mutations on protein function. Advances in neural information processing systems, 34:29287-229333 2021.
[25] Zeming Lin, Halil Akin, Roshan Rao, Brian Hie, Zhongkai Zhu, Wenting Lu, Nikita Smetanin, Robert Verkuil, Ori Kabeli, Yaniv Shmueli, al. Evolutionary- prediction of atomic-level protein structure with a language model. Science, 379(6637):1123-11 1130, 2023.
[26] Ahmed Elnaggar, Michael Heinzinger, Christian Dallago, Ghalia Rehawi, Yu Wang, Llion Jones, Tom Gibbs, Tamas Feher, Christoph Angerer, Martin Steinegger, et al. Prottrans: Toward understanding the language of life through self-supervised learning. IEEE transactions on pattern analysis and machine intelligence, 44(10):71 12-777777 2021.
[27] Pascal Notin, Aaron Kollasch, Daniel Ritter, Lood Van Nickerk, Steffanic Paul, Han Spinner, Nathan Rollins, Ada Shaw, Rose Orenbuch, Ruben Weitzman, et al. Proteingym: Large- scale benchmarks for protein fitness prediction and design. Advances in Neural Information Processing Systems,
2023.
[28] Liang Zhang, Hua Pang, Chenghao Zhang, Song Li, Yang Tan, Fan Jiang, Mingchen Li, Yuanxi Yu, Ziyi Zhou, Banghao Wu, al. Venusmuthub; A systematic evaluation of protein mutation effect predictors on scall-scale experimental data. Acta Pharmaceutica Sinica B, 2025.
[29] Yunan Luo, Guangde Jiang, Tianhao Yu, Yang Liu, Lam Vo, Hantian Ding, Yufeng Su, Wes- ley Wei Qian, Huimin Zhao, and Jian Peng. Ecnet is an evolutionary context-integrated deep learning framework for protein engineering. Nature communications, 12(1):5743, 2021.
[30] Pascal Notin, Ruben Weitzman, Debora Marks, and Yarin Gal. Proteinnpt: Improving pro- tein property prediction and design with non-parametric transformers. Advances in Neural Information Processing Systems, 00:33529-33563,, 2023.
[31] Fan Jiang, Mingchen Li, Jiajun Dong, Yuanxi Yu, Xinyu Sun, Banghao Wu, Jin Huang, Liqi Kang, Yufeng Pei, Liang Zhang, et al. A general temperature-uuided language model to design proteins of enhanced stability and activity. Science Advances, 10(48)ead:26411 2024. 2024.
12

---

[32] Thomas Hayes, Roshan Rao, Halil Akin, Nicholas Sofroniew, Deniz Oktay, Zeming Lin, Robert Verkuil, Vincent Q Tran, Jonathan Deaton, Marius Wiggert, al. Simulating 500 million years of evolution with language model. Science, page eads0018, 2025.
[33] Alexander Rives, Joshua Meier, Tom Sercu, Siddharth Goyal, Zeming Lin, Jason Liu, Demi Guo, Myle Ott, C Lawrence Zitnick, Jerry Ma, et al. Biological structure and function emerge from scaling unsupervised learning to 250 million protein sequences. Proceedings of the National Academy of Sciences, 1 (18(15)=e20162) (011)3991111111111 18, 2021.
[34] Erik Nijkamp, Jeffrey A Ruffolo, Eli N Weinstein, Nikhil Naik, and Ali Madani. Progen2: exploring the boundaries of protein language models. Cell systems, (4(11))968-1 978, 2023.
[35] Pascal Notin, Mafalda Dias, Jonathan Frazer, Javier Marchena-Hurtade Aidan N Gomez, Debora Marks, and Yarin Gal. Tranception: protein fitness prediction with autoregressive transformers and inference-time retrieval. In International Conference on Machine Learning, pages (6990-17017, PMLR, 2022.
[36] Kaiyi Jiang, Zhaoqing Yan, Matteo Di Bernardo, Samantha R Sgrizzi, Lukas Villiger, Alisan Kayabolen, BJ Kim, Josephine K Carscadden, Masahiro Hiraizumi, Hiroshi Nishimasu, et al. Rapid in silico directed evolution by a protein language model with evolvepro. Science, page cadr6006, 2024.
[37] Mingchen Li, Liqi Kang, Yi Xiong, Yu Guang Wang, Guisheng Fan, Pan Tan, and Liang Hong. Sesnet: sequence-structure feature-inteerrated deep learning method for data-efficient protein engineering. Journal of Cheminformatics, 15(1):12, 2023.
[38] Justas Dauparas, Ivan Anishchenko, Nathaniel Bennett, Hua Bai, Robert J Ragotte, Lukas F Milles, Basile IM Wicky, Alexis Courbet, Rob J de Haas, Neville Bethel, et al. Robust deep learning-based protein sequence design using proteinmpnn. Science, 978(6615):49935) 2022.
[39] Jes Frellsen, Maher M Kassem, Tone Bengtsen, Lars Olsen, Kresten Lindorff-Larsed Jesper Ferkinghoff-Bae and Wouter Boomsma. Zero-shot protein stability prediction by inverse folding models: free energy interpretation. arXiv preprint arXiv:2506.05500 2025.
[40] Günter Klambauer, Thomas Unterthiner, Andreas Mayr, and Sepp Hochreiter. Self-normalizing neural networks. Advances in neural information processing systems, 30, 2017.
[41] Corrado Pancotti, Silvia Benevenuta, Giovanni Birolo. Virginia Alberini, Valeria Repetto. Tiziana Sanavia, Emidio Capriotti, and Piero Fariselli. Predicting protein stability changes upon single-point mutation: a thorough comparison of the available tools on a new dataset. Briefings in Bioinformatics, 23(2):bbab555, 2022.
[42] Yunxin Xu, Di Liu, and Haipeng Gong. Improving the prediction of protein stability changes upon mutations by geometric learning and a pre-training strategy. Nature Computational Science, pages 1-11, 2024.
[43] Yang Yang. Siddhaling Urolagin, Abhishek Niroula, Xuesong Ding, Bairong Shen, and Mauno Vihinen. Pon-tstab: protein variant stability predictor. importance of training data quality. International journal of molecular sciences, 19(4):1009 2018.
[44] Jan Stourac, Juraj Dubrava, Milos Musil, Jana Horackova, Jiri Damborsky, Stanislav Mazurenko, and David Bednar. Fireprotdb: database of manually curated protein stability data. Nucleic acids research, 49(D1):D319-D)
2021.
[45] Joicymara S Xavier, Thanh-Binh Nguyen, Malancha Karmarkar, Stephanie Portelli, Pâmela M Rezende, Joao PL. Velloso, David B Ascher, and Douglas EV Pires. Thermomutd:: ther- modynamic database for missense mutations. Nucleic acids research, 49(D1):D455-D47, 2021.
[46] Kotaro Tsuboyama, Justas Dauparas, Jonathan Chen, Elodie Laine, Yasser Mohseni Behbahani, Jonathan J Weinstein, Niall M Mangan, Sergey Ovchinnikov, and Gabriel J Rocklin. Mega-scale experimental analysis of protein folding stability biology and design. Nature, 620(7973):43-- 444, 2023.
13

---

[47] Martin Steinegger and Johannes Söding. Mmseqs2 enables sensitive protein sequence searching for the analysis of massive data sets. Nature biotechnology, 35 2017.
[48] Burkhard Rost. Twilight zone of protein sequence alignments. Protein engineering, 12(2):85-94,
1999.
[49] Milot Mirdita, Konstantin Schütze, Yoshitaka Moriwaki, Lim Heo, Sergey Ovchinnikov, and Martin Steinegger. Colabfold: making protein folding accessible to all. Nature methods, 19(6):679 08066799622222 2022.
[50] Vladimir Likic. The needleman-wunsch algorithm for sequence alignment. Lecture given the 7th Melbourne Bioinformatics Course, Bi021 Molecular Science and Biotechnology Institute, University of Melbourne, pages 1-46, 2008
[51] Zeming Lin, Halil Akin, Roshan Rao, Brian Hie, Zhongkai Zhu, Wenting Lu, Allan dos Santos Costa, Maryam Fazel-Zarandi, Tom Sercu, Sal Candido, et al. Language models of protein sequences the scale of evolution enable accurate structure prediction. BioRxiv, 2022:500902, 2022.
[52] Lin Tang. Large model predicts variant effects. Nature Methods, 20(10): 1448. 2023.
[53] Benjamin Livesey and Joseph A Marsh. Advancing variant effect prediction using protein language models. Nature genetics, 55(9):1426-1427, 2023.
[54] Diederik P Kingma. Adam: A method for stochastic optimization. arXiv preprint
arXiv:1412.6980, 2014.
[55] Antoni Beltran, Xiang'e er Jiang, Yue Shen, and Ben Lehner. Site-saturation mutagenesis of 500 human protein domains. Nature, pages 1-10, 2025.
[56] Yang Zhang and Jeffrey Skolnick. Tm-align: protein structure alignment algorithm based on the tm-score. Nucleic acids research, 33(7):2302-234) 220099 2005.
[57] Jin Su, Chenchen Han, Yuyang Zhou, Junjie Shan, Xibin Zhou, and Fajie Yuan. Saprot: Protein language modeling with structure-awes vocabulary. In The Twelfth International Conference on Learning Representations, 2024.
[58] Philip Gage. A new algorithm for data compression. The C Users Journal, 12(2):23-38, 1994.
[59] Noelia Ferruz, Steffen Schmidt, and Birte Höcker. Protgpt2 is a deep unsupervised language model for protein design. Nature communications, (4344, 2022.
[60] Jinrui Xu and Yang Zhang. How significant a protein structure similarity with tm-score= 0.5? Bioinformatics, 26(7):889-88) 8958999995-9999 2010.
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A Appendix
A.1 Algorithm
Algorithm Dotlines the training procedure for Venus-MAXWELL. The process iterates through the dataset, computing the predicted mutation landscape L, from the PLM's s prg-obbbaiilites for each protein A, The model parameters / are then updated using gradient descent based on a combined loss function that incorporates both ranking (CRanking) and absolute value (LMSE) objectives.
Algorithm I Venus-MAXWELL for mutation AAG landscape learning
Require: Dataset D (A,, Y, M.) where A, is the i-th wild-type protein sequence, Y, is the i-th mutation A AG landscape, and M, is the i-th binary mask matrix
Require: Protein language model fo with parameters f
Require: Alphabet size [A| of the protein language model
Require: Learning rate y/ number of epochs
1: 0g = 0
2: for to 7 do
3: for = 1 to N do
4:
Tokenize protein sequence into one-hot encoding matrix X,
5:
Compute nog-prbbability matrix Q; = log fo (A,) (Equation
6:
Compute scoring matrix L, log (A,)) (llog fo (A,) X, [(y) TAAl (Equation 8
7:
Compute loss L(0) = Lranking i, M,, L,) + X LMSE Y, M., L;)
8:
Update parameters: Og 0,-1 1) V a (Equation 10 and 11
9:
end for
end for
11: 0' = 0T
return for
Algorithm details the prediction process using a PLM fine-tuned by Venus-MAXWELL. Given protein sequence A, the model performs single forward pass to compute the tog-poobbbllity matrix and subsequently derives the mutation AAG landscape L based on Equation g
Algorithm 2 Venus-MAXWELL for mutation AAG landscape prediction
Require: Protein sequence A
Require: Protein language model fo with fine-tuned parameters 0'
1: Tokenize protein sequence into one-hot encoding matrix X
2: Compute probbbbbablity matrix Q, log fo A;) (Equation
3
3: Compute scoring matrix log fe (A) - ((log (A( o X 1AA) TAA (Equation 8
4: return Mutation AAG landscape L
A.2 Pseudo Code
Algorithm provides a Pytorch-like pseudocode implementation of the core training loop for Venus- MAXWELL. It illustrates how input sequences are processed, logits are computed, and transformed into the mutation landscape matrix, and how the ranking and MSE losses are calculated and combined for backpropagation and parameter updates.
21

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Algorithm 3 Pseudocode of Venus-MAXWELL training in a pytorch-like style.
If model protein languge model
for (seq, landsacpe, matrix) in loader:
Shape [seq_len] input_ids tokenizer
tensors"""t)
# Shape [enq_len, alphabet_size
one_hot = _.one_hot (input_ida ids, amm.claaseerallpaeet..ieze
# Shape [seg_len] logite model (input_ide)
Shape [seq_]em alphabet, _izee
logits logits. log_ log.sssaaa..oofmxxccccoo sof
Shape [seq_len, Saphabet_sizee [seq_ len, 1] [seq_len, alphabet_size) (broadcast)
logits logits (logits one_hot) mne_hot).eumddnn)) sum (eep_dim-True)
Shape mnum_mttanta, ]
predictions logita [mask) landacape [maak]
loss =1 Pearson (preditoott scores) mse_lose- MSE(predictions, scores)
loss rank_loss 0.1 mse_loss
loss. backward() optimizer step()
A.3 Hyperparameter Selection and Ablation Studies
To ensure a fair and robust selection of hyperparameters without test set leakage, we performed 5-fold cross-validation (CV) on the training dataset. The training set was split into five folds (Ti. T2, Ts. Ta, T.}. For each set of hyperparameters (e.g., a specific learning rate), we followed this procedure:
For i to 5:
Set the validation fold: D.al T,.
Set the internal training set: Dimin internal Dirain T;
Train the model on Diriin_itennt intemaal for a fixed number of epochs (e.g., epochs for the initial LR search).
Record the best Spearman correlation achieved on the validation fold D
Calculate the average Spearman p across all 5 folds.
We define the "optimal performance" as this 5-fold mean per-protein Spearman correlation P.. higher average score indicates better hyperparameter choice. After identifying the optimal hyperparameters (e.g., learning rate 5 x 10-"), we determined the average optimal training epochs (approximately 7) from the CV folds. Finally, trained single model the entire training dataset for 7 epochs and reported its performance on the held-out test set (Test12K). The test set was used exclusively for this final evaluation.
Ablation for Loss Weighting Factor
A balances the ranking loss (CRanking) and the MSE loss (CMSE). We conducted a sensitivity analysis by training Venus-MAXWELL (ESM-IF) with different values of X.
In our loss function (defined in Equation 12
As shown in Table Al the model achieved the best performance the test dataset when À 0.1. This suggests the framework performs optimally when the ranking loss serves as the primary training objective, while the MSE loss acts beneficial auxiliary objective to align the scale of the predictions. Based on this analysis, we set = 0.1 for all other experiments.
Ablation for Learning Rate (Ir)
22

---

Table A1: Ablation study on the weighting factor A. We report the average Spearman Ps and Pearson Pp correlation coefficients of Venus-MAXWELL (ESM-IF) on the Test12K dataset.
Weighting Factor (A)
Ps
Pp
0.0
0.506
0.528
0.1
0.517
0.542
0.2
0.514
0.540
0.4
0.504
0.538
0.8
0.494
0.523
0.486
0.519
Following the 5-fold CV methodology, we performed grid search for the optimal learning rate (Ir). The results are shown in Table A2 We selected 5 x 10 as the optimal learning rate, as demonstrated the best average performance across the validation sets.
Table A2: Performance of Venus-MAXWELL (ESM-IF) with different learning rates during 5-fold cross-validation. "Avg Spearman p" refers to the mean Spearman correlation across the 5 validation folds. Best performance is in bold.
Learning Rate (Ir)
Pa
x 10
0.184
10-3
0.278
5 X 10-1
0.416
x 10-1
0.499
10-5
0.517
X 10
0.501
X 10
0.508
10
0.492
Ablation for MLP Hidden Layer Dimension
The MLP (used in Equation II is a two-layer network where the learnable weights have dimensions W1 RR VxD and Wa RR pxl V is the PLM vocabulary size, and D is tunable hidden dimension. investigated the model's sensitivity to D, using ESM-IF (V = 35)) as the base model.
As shown in Table A3 the results indicate that the model achieves its best performance when the V (35). While performance relatively stable for up to 512, this
hidden dimension D analysis validates our choice of D = V, showing that increasing the MLP's complexity does not yield further benefits.
Table A3: Performance of VeNUS-MAXWELL (ESM-IF) with varying MI. hidden layer dimensions (D).
Hidden Dimension (D)
P
V (35)
0.517
64
0.506
128
0.516
256
0.515
512
0.505
1024
0.487
A.4 Evaluation Metrics and Embedding Transfer Baseline Models Configuration
For embedding transfer, features from frozen PLMs were fed into a three-layer MLP with SELU activations and 0.1 dropout, trained with Adam optimizer (learning rate 10-3 batch size 32). Following the similar training protocol as Venus-MAXWELL the optimal number of training epochs
23

---

was determined through five-fold cross-validation on the training set, with early stopping during validation (patience of 5 epochs based on Pearson correlation).
A.5
Model Performance Details
Table A4: Model performance across different metrics
Model Type
Model
Pa
Pp
F1
AÙC
ESM-1F (Zero-shot)
0.375
0.376
0.217
0.577
ProSST (Zero-shot)
0.369
0.370
0.372
0.560
PLM zero-shot
ESM-1b (Zero-shot)
0.263
0.278
0.281
0.524
ESM- (Zero-shot)
0.251
0.256
0.295
0.527
ESM-2 (Zero-shot)
0.268
0.284
0.289
0.533
ThermoMPNN
0.508
0.529
0.371
0.647
Baseline models
Rosetta
0.368
0.239
0.346
0.503
FoldX
0.410
0.431
0.322
0,581
ESM-IF + MLP
0.302
0.331
0.383
0.628
Embedding transfer
ProSST + MLP
0.234
0.263
0.360
0.608
Venus-MAXWELL (ESM-IF)
0.517
0.542
0.427
0.678
Venus-MAXWELL (ProSST)
0.479
0.502
0,405
0.604
Venus-MAXWELL (PLM)
Venus-MAXWELI (ESM-1b)
0.384
0.402
0.334
0.578
Venus-MAXWELI (ESM-1v)
(FSM-))
0.358
0.371
0.324
0.555
Venus-MAXWELL (ESM-2)
0.427
0.450
0.366
0.545
Table A4 provides detailed breakdown of the performance for all evaluated models across the four primary metrics: Mean Spearman correlation (p,), Mean Pearson correlation (pp), Mean F1 score (F1), and Mean Area Under the Curve (AUC). Models are categorized by type (PLM zero-shot, baseline SOTA model, and physical- based method, embedding transfer, and Venus-MAXWELLI to facilitate comparison within and across different approaches. The best performance for each metric highlighted bold. This comprehensive table allows for a granular assessment of each model's strengths and weaknesses on the 1K dataset.
In addition to this comprehensive performance comparison, further validated model specific, established public benchmarks. We clarify that Test12K is a curated meta-benchmark composed of these benchmarks (e.g., p53, s669). To directly address concerns about evaluation solely on our constructed dataset, Table A3 presents a detailed comparison against SOTA models on several individual subsets.
The results show that Venus-MAXWELL consistently outperforms both ThermoMPNN and Stability Oracle these well-known datasets. We deliberately excluded the popular ProteinGym benchmark, as its significant overlap with the training data of both our model and ThermoMPNN would lead to data leakage and invalidate the conclusions.
Table A5: Model Performance on Public Benchmarks (Mean per- per-protein Spearman P.).
Benchmark
Venus-MAXWELL (ESM-IF)
ThermoMMPNN
Stability Oracle
p53
0.751
0.702
0.727
s669
0.565
0.545
0.503
Myoglobin Ssym
0.724
0.620
0.669
0.602
0.516
0.567
Finally, to address concerns about the inherent sparsity of the training data and its potential to adversely affect generalization, we conducted an ablation study on training set sparsity. We evaluated Benus-MAXWELL (ESM-IF) after randomly removing p percent of the mutation entries from the Train226K dataset. Performance (Mean per-protein Spearman px) on the Test 2K set remains stable with extreme data removal.

---

As shown in Table A6 the framework demonstrates exceptional robustness. Performance remains remarkably stable, with a Spearman's correlation of 0.501 even when using only 10% (23K mutants) of the training data. A significant performance drop only occurs at the extreme of removing 95% of the data. This analysis confirms that our model learns generalizable stability principles from the diverse collection of proteins, rather than overfitting to sparse local landscapes.
Table A6: Effect of training set sparesity on model performance.
Data Removed (p)
Training Mutants
Ps
Pp
F1
AÚC
0%
226K
0.517
0.542
0.427
0.678
20%
181K
0.517
0.541
0.423
0.673
40%
136K
0.515
0.541
0.421
0.675
60%
90K
0,512
0.540
0.416
0.669
80%
45K
0.509
0.537
0.408
0.667
90%
23K
0.501
0.523
0.402
0.660
95%
11K
0.485
0.506
0.388
0.647
A.6
Generalization Ability Analysis Across Structual identity
In constructing the training and test sets, ensured low sequence similarity via MMseqs2 dedu- plication and Needleman- Wunschh validation (See Section + Since the top-performing Venus- MAXWELL enhanced models, ESM-IF and ProSST, leverage protein structure information. further assessed the structural similarity between the sets to evaluate the models' extrapolation capability to structurally novel proteins. We employed TM-align [56] compute TM-scores for each protein in Test12K against all structures Train226K, using the maximum score as the structural similarity metric.
As illustrated in Figure AT (A), the maximum TM-scores of the test proteins displayed a tightly clustered distribution, with the overall probability density peaking around a mean value of approx- imately 0.31. According to established conventions in structural biology, TM-scores below 0.5 indicate distinct folding topologies (60). This confirms the structural diversity between Train226K and Test12K, establishing a rigorous benchmark for extrapolation.
Further analysis of vealed only weak correlation (pp = 0.208, shown in Figure AT (B)). This finding indicates that the performance enhancement of structure-aware protein language models within the Venus- MAXWELL framework is independent of structural similarity to training samples, demonstrating strong generalization capabilities to unseen protein architectures.
(ESM-IF) performance relative to structural similarity re-
(A) Structure Similarity Distribution & Average Performance
(B)
Overall Correlation: r=0.208, p=2.38e
2.5
Density in Test12K
2.0
0.6
Peanson Correlation
0.75
0.50
1.5
0.4
Average Pearson
0.25
1.0
0.2
0.5
-0.25
-0.50
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Maximum TM-score
Maximum TM-score
Figure A1: Model generalization of Venus-MAXWELI (ESM-IF) against structural dissimilarity (A): Distribution of maximum TM-scores for Test12K proteins relative to Train226K. overlaid with average Pearson correlation per TM-score bin. Relatolaioos between maximum TM-score and prediction performance, showing minimal dependence on structural similarity to training proteins.
25

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A.7
Prediction Speed Scaling with Sequence Length
Table A7: Prediction time (s) across sequence length ranges for single-site mutagenesis
Model
0-200
200-400
400-600
600+
Venus-MAXWELL (ProSST)
0.30+1010
0.37
0.27=0.02
0.30:1.07
Venus-MAXWELL (ESM-IF)
1.33:0.10
1.65#0.10
1.78+0.13
2.19
ThermoMPNN
4.41:0.04
13.6410.01
20.6410.11
30.8910.13
ProSST + MLP
162.19+0.87 421.44+1.63
539.72+0067
844.52+3.68
1347.73=6683
RSM-IF+MLD
2846.02+1.13
6255.54+13.25
13554.11=7:27
Table A8: Mutants per second across different sequence length ranges for single-site mutagenesis
Model
0-200
200-400
400-600
600+
Venus-MAXWEI (ProSST)
30451--811
82336+-277736
1995
217882-+-3337
Venus-MAXWELL, (ESM-IF)
6513:+501
16887+9973
23993+1705
29425++2155
ThermoMPNN
1957=117
2038+t2
2058=11
2080
ProSST + MLP
53.22++0.29
51.51+00.06
50.29=0.22
47.67+#0.24
ESM-IF+ MLP
20.48.00008
9.77+00001
6.79=0.01
4.74
To assess the scalability of Venus-MAXWELL'S inference efficiency across diverse protein sizes, we evaluated its prediction speed for single-site saturation mutagenesis on proteins with different sequence length ranges from the Test12K dataset. For each sequence length range (0-200, 200-400, 400-600, and 600+ residues), we randomly selected five proteins, performed single-site saturation mutagenesis, and measured inference speed across all models. This process was repeated five times to compute the mean and standard deviation of prediction times and throughput.
As shown in Table A7 prediction times for all methods except Venus-MAXWELL (ProSST) increase with sequence length. For instance, ThermoMPNN's 's prediction time rises from 4.41 + 0.04 s (0-200 residues) 30.89 + 0.13 (600+ residues), and MLP- basedd baselines (ProSST-MLP and ESM- IF-MLP) exhibit even steeper increases, reaching up to 13,554.11 + 7.27 s for ESM-IF-MLP on sequences exceeding 600 residues. This trend is driven by factors: (1) longer sequences result in PLMS require significantly
a higher total number of single-site mutants, and (2) Transformer-bast more time to encode longer sequences due their quadratic complexity with respect sequence length.
In contrast, Venus-MAXWELL (ProSST) maintains rather stable prediction times (0.27-0.37 across all ranges), likely due to its remarkable high throughput, which remains well above the demands of current sequence lengths and mutation counts. Table A8 further validates this observation. The throughput of embedding transfer methods decreases with sequence length-fire from 53,22 0.29 to 47.67 = 0.24 mutants/s for ProSST MLP and from 20.48 = 0.08 to 4.74 + 0.00 mutants/s for ESM-IF + -efleeell their need to re-encode each mutant sequence, making them highly sensitive to sequence length. TPeN leveraging ProteinMPPN''s structure reduce encoding overhead, achieves a relatively stable throughput of approximately 2,000 mutants/s (1, (1.957-2,080 mutants/s), but this appears approach its practical upper limit.
Conversely, both throughput increases with sequence length, reaching pncers-of-magntuue higher performance. This scalability is likely attributable to
(ProSST) and MAXWELL (ESM-II) exhibit significant Section which efficiently handles the whole mutation landscapes.
optimized matrix processing architecture (see
A.8 Model Training Speed
This Figure A2 highlights the training efficiency of MAXWELL compared with other methods, using ESM-IF and ProSST as base models. Embedding transfer methods, which require encoding each mutant sequence during forward propagation, are significantly constrained in training speed. For FSSMIIF and ProSST, processing 100,000 mutant sequences takes over 2,000 seconds, with a single training epoch exceeding one hour. ThermoMPNN substantially reduces training time by encoding
26

---

Venus-MAXWELL & Embedding transfer Models
Venus-MAXXELL & ThemoMPNN
Venus-MAXWELLIESM-FF
Venus-MAWELLLEES--F
4000
Training Time (secondis)
ESM-F(MLP)
Training Time (seconas)
250
Venus=MAWWELLPPOSST
Venus- MAXXWELL/LLLL)OSSST
ThemoMPNN
3000
ProSST(MLP)
200
150
0k
20k
40k
60k
80k
100k
0k
20k
40k
60k
80k
100k
Number Mutants
Number Mutants
Figure A2: Training time analysis. (A): Comparison between Venus-MAXWELL models and their embedding transfer (MLP) counterparts for both ESM-IF and ProSST architectures. (B): Performance of Maxwell implementations contrasted with ThermoMPNN
only the wild-tpe sequence, processing 100,000 mutations in under 300 seconds. While the training efficiency of Venus-MAXWELLis further 20 times faster than ThermoMPNN, completing an epoch on the entire Train226K dataset in approximately 40 seconds.
27


Reviews:
[{"review_id": "t1RmERGQFw", "reviewer": "Reviewer_y7hf", "summary": "This paper proposes Matrix-wise landscape learning(MAXWELL), a novel framework for efficiently learning protein mutation stability landscapes. MAXWELL transforms the mutation prediction task from the traditional sequence-to-label approach to a sequence-to-landscape paradigm, leveraging protein language models to predict the ∆∆G of protein mutations in a matrix-driven manner. The framework enables the learning of an entire mutation landscape with just a single forward and backward propagation, significantly improving computational efficiency. Additionally, the authors constructed a large-scale protein mutation ∆∆G dataset with strict controls on data leakage and redundancy to ensure robust model evaluation. Experimental results demonstrate that MAXWELL outperforms existing state-of-the-art methods in both prediction accuracy and computational efficiency.", "questions": "1.In Equation (2), the order of probability calculations for the mutant and wild-type proteins appears inconsistent with the sequence described in the \"Method\" section. Could this be a typographical error?  \n\n2.Does the training set contain a sufficient number of mutant variants to ensure scalability across all mutation sites and amino acid types? If the MASK matrix is overly sparse, could this adversely affect the model’s generalization capability?  \n\n3.Did the authors attempt to combine MSE and ranking loss during MLP training to validate the effectiveness of introducing matrix transformations and mutation scoring?  \n\n4.MAXWELL fine-tunes all parameters of the PLM, whereas the MLP baseline freezes the PLM’s parameters. Could this discrepancy lead to performance differences, and if so, has this been quantified to isolate the contribution of the proposed architecture?", "limitations": "Yes, the authors have discussed limitations in the paper.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n1.MAXWELL employs a matrix-driven scoring approach, transforming mutation prediction to comprehensive landscape prediction. By integrating both ranking loss and MSE loss during training, it enables more holistic optimization of model performance.  \n\n2. Compared to traditional methods, MAXWELL demonstrates significant speed improvements in both training and inference, allowing efficient processing of large-scale mutation data. \n\n3. The framework is compatible with various PLMs, effectively leveraging evolutionary information from pre-trained models to enhance prediction accuracy.  \n\nWeaknesses\n\n1.MAXWELL primarily focuses on modeling single-point saturation mutations, and does not yet address the more complex multi-point mutation combinatorial effects, which limits its applicability in complex mutational scenarios.  \n\n2.The paper lacks a detailed theoretical explanation for why the probabilities of mutant and wild-type proteins can be used to predict mutation scores and ∆∆G values. Although a probability-based scoring formula is proposed, the underlying theoretical basis and assumptions are not sufficiently elaborated.  \n\n3.The experimental validation in this study was conducted only on the author-constructed Test12K dataset, without additional testing on publicly available benchmarks.\n\n4.The paper only compares the proposed framework with zero-shot and MLP baselines , omitting comparisons with other fine-tuning frameworks.", "was_revised": false, "final_justification": "", "rebuttal": "We would like to appreciate the reviewer for the evaluation and comments. We would appreciate it if you could let us know whether our response addresses your concerns.\n\n**Response to weakness 1:** We agree that MAXWELL is primarily designed for training on single-point mutations. **However, it is still capable of making predictions for multi-point mutations** using the feasible prediction formula: $$Score(S) = \\sum_{(t,j)}L_{t,j}$$\nwhere $S = \\{(t_1,j_1),(t_2,j_2),...(t_k,j_k)\\}$ represents a multi-point mutant constructed by $k$ single-point muttations and $L$ is the mutation landscape as defined in the paper.\n\nTable R1 presents MAXWELL's performance on the multi-point mutation subset of the public Mega-scale ΔΔG dataset and M1261 dataset. **The results confirm that MAXWELL effectively enhances multi-point mutation prediction capabilities,** demonstrating its practical utility in this regard.\n\n**Table R1. Multi-point mutation prediction performance of related models**\n|Model|Mega-scale|M1261|\n|---|---|----|  \n|ESM-IF| 0.317|0.336|  \n|MAXWELL (ESM-IF)|0.597|0.492|  \n\nWhile MAXWELL can handle multi-point mutation prediction tasks, our primary objective is to contribute a **high-performance single-point mutation prediction model**. This focus aligns with typical directed evolution workflows, which often begin with single-point mutations and then leverage experimental data to guide multi-point exploration.\nMAXWELL can provide initial stability predictions without experimental data. These recommendations can then be integrated with other models, such as ECNet and ProteinNPT, to guide the design of multi-point mutations, creating a powerful, iterative optimization cycle when combined with wet-lab validation.\n\n**Response to weakness 2:**  The idea that \"**probabilities of mutant and wild-type proteins can be used to predict mutation scores**\" is a well-validated conclusion in protein language models (PLMs). ProteinGym's benchmark quantifies the correlation between these PLM-derived probabilities and mutant phenotype, clearly demonstrating a **notable association**. This has become a key indicator for evaluating the performance of PLMs.\nIntuitively, pre-trained PLMs have learned the natural distribution of numerous protein sequences. Their calculated probabilities essentially reflect a mutant's likelihood of existing in nature—an inherent indicator of mutation stability, as unstable mutants rarely persist in natural contexts.\n\nTheoretically, prior research [1] has established that the predictive power of these probabilities for stability stems from an approximate mathematical relationship: **the change in thermodynamic stability (ΔΔG) correlates with the logarithmic ratio of probabilities derived from inverse folding models**, as expressed by:\n\n$$\\beta \\Delta \\Delta G_{a \\rightarrow a^{\\prime}} \\approx -\\ln \\frac{p_{\\theta}\\left(a^{\\prime} \\mid x_{a}\\right)}{p_{\\theta}\\left(a \\mid x_{a}\\right)} - \\ln \\frac{p_{\\theta}(a)}{p_{\\theta}\\left(a^{\\prime}\\right)}$$\n\nTo enhance clarity, we will provide a detailed theoretical explanation of this relationship at the beginning of the Methods section. We believe this will help readers better grasp the underlying mechanism.\n\n**Response to weakness 3**: Test12K is a curated meta-benchmark composed exclusively of established, publicly available benchmarks to ensure a comprehensive and challenging evaluation. To directly address your concern, we have tested the performance of MAXWELL on several public benchmarks, comparing it against current state-of-the-art models.\n\nAs shown in Table R2, MAXWELL consistently outperforms ThermoMPNN and Stability Oracle on well-known benchmarks including p53, s669, and Myoglobin, all of which are subsets of Test12K. Evaluation protocol is: for each dataset, we calculate the correlation score on a per-protein basis and then average these scores. A full evaluation against Stability Oracle on the complete Test12K was not possible, as the officially provided source code was non-functional.\n\nThe results demonstrate that MAXWELL achieves state-of-the-art performance on these public benchmarks. We deliberately excluded the popular ProteinGym benchmark from our evaluation. This exclusion is due to ProteinGym's overlap with the training data of both our model and the ThermoMPNN model, which would lead to data leakage and invalidate the conclusions.\n\n**Table R2. Model Performance on Public Benchmarks (Mean Per-Protein Spearman)**\n|| MAXWELL|ThermoMPNN|Stability Oracle|\n|---|---|---|---|\n|Test12K|0.517|0.508| - |\n|p53|0.751|0.702|0.727|\n|s669|0.565|0.545|0.503 |\n|Myoglobin|0.724|0.620|0.669|\n|Ssym|0.602|0.516|0.567|\n\n**Response to weakness 4:** Thanks for the valuable reminder. We have additionally compared MAXWELL with other fine-tuning methods (all based on the ESM-IF): likelihood-based fine-tuning and one-hot augmented fine-tuning.\n\nAs shown in Table R3, MAXWELL consistently achieves the optimal performance. It's worth noting that while these alternative methods can be adapted for this task, they were originally designed for intra-protein learning (predicting new mutations on a protein seen during training) rather than generalizing to entirely new proteins, which is the focus of our benchmark.\n\nWe will add these new comparative results to the baseline section of our revised paper to provide a more comprehensive evaluation.\n\n**Table R3. Performance comparison between MAXWELL and other fine-tuning frameworks**\n||Spearman|\n|---|---|\n|MAXWELL|0.517|\n|Likelihood-based fine-tuning|0.401|\n|One-hot augmented fine-tuning|0.442|\n\n**Response to question 1:**  Thank you for your reminder. There was indeed a typo in Equation 2, which has now been corrected to: \n\n$$\nScore(a_t \\to b_t) = \\log\\left(f_{\\theta}(\\boldsymbol{\\mathrm{A}})\\right)\\_{t, b_t} - \\log\\left(f_{\\theta}(\\boldsymbol{\\mathrm{A}})\\right)\\_{t, a_t}\n$$\n\n**Response to question 2:**  Thank you for this important question. We agree that data sparsity is a key consideration. Our training set, Train226K dataset, is substantial, containing over 226,000 mutation entries across 255 proteins, with an average of around 887 mutations per protein. While the mutation landscape for any single protein is sparse (due to the experimental infeasibility of measuring all possible mutations), the scale and diversity of the combined dataset allow the model to learn generalizable patterns of sequence-stability relationships. MAXWELL's strong performance on the Test12K dataset, which exhibits low sequence and structural similarity to the training set, clearly demonstrates its robust generalization, indicating it's not merely memorizing sparse landscapes.\n\nTo verify how sparsity affects model generalization, we conducted ablation experiments. Using ESM-IF as the base model, we randomly removed $p%$ of the training data from the training set, where $p=0$ represents the original performance. The results, presented in Table R4, show that our framework is exceptionally robust to data sparsity. Performance remains remarkably stable, with a Spearman's correlation of 0.501 even when using only 10% (23K mutants) of the training data. A significant performance drop only occurs at the extreme of removing 95% of the data. This demonstrates that our model learns generalizable stability principles from the diverse collection of proteins, rather than overfitting to the sparse landscape of any single protein.\n\n**Table R4. Effect of training set sparsity on model performance**\n|$p$|Traning mutants|Spearman|\n|---|---|---|\n|0%|226K|0.517|\n|20%|181K|0.517|\n|40%|136K|0.515|\n|60%|90K|0.512|\n|80%|45K|0.509|\n|90%|23K|0.501|\n|95%|11K|0.485|\n\n**Response to question 3:** Thank you for this insightful question. We performed the suggested experiment, and the results validated our framework's design. As shown in Table R5, when we added the ranking loss to the MLP's output, the model's performance degraded.\nWe hypothesize that primary evolutionary information resides within the language model's output head, and its score rankings correlate with protein mutation ΔΔG. This correlation makes ranking loss suitable for the language model's direct output. Conversely, the randomly initialized MLP lacks this inherent evolutionary information, rendering it unsuitable for ranking loss application. We will include these results in our ablation experiments.\n\n**Table R5. Impact of loss in MLP on model performance**\n|name|Spearman|\n|---|---|\n|Only use MSE loss during MLP training|0.517|\n|Combine MSE and ranking loss during MLP training|0.507|\n\n**Response to question 4:** We acknowledge this potential risk. We did not perform full training for all models due to prohibitive computational speed. While MAXWELL only requires a single computation per protein during full training, MLP baselines demand feature computation for each individual mutant sequence, making their full training exceptionally slow. We supplemented experiments on full training experiments for ESM-IF + MLP and LoRA and the training results using LoRA (configured with lora_r = 8 and lora_alpha=16), as presented in Table R6. (Note: Full training for all baselines was limited by computing power and time constraints.)\n\nThe results in Table R6 indicate that the performance of ESM-IF + MLP is slightly lower than ESM-IF (LoRA) + MLP but notably higher than ESM-IF (no freeze) + MLP. For the fully trained ESM-IF, its performance drops significantly after training. We speculate this decline is due to overfitting caused by an excessively high number of model parameters. We'll include these results in our ablation experiments.\n\n**Table R6. Training results for ESM-IF + MLP**\n|Model|Spearman|\n|---|---|\n|MAXWELL|0.517|\n|ESM-IF (freeze)+MLP|0.302|\n|ESM-IF (no freeze)+MLP|0.121|\n|ESM-IF (LoRA)+MLP|0.327|\n\n[1] Zero-shot protein stability prediction by inverse folding models: a free energy interpretation. arXiv, 2025."}, {"review_id": "1GURPWQxZt", "reviewer": "Reviewer_2Njy", "summary": "This paper introduces MAXWELL, a framework for predicting protein mutation stability. Instead of predicting individual mutation effects separately, MAXWELL predicts an entire mutation landscape matrix in a single forward pass. The authors demonstrate improved computational efficiency while maintaining competitive predictive performance compared to existing sequence-to-label methods.", "questions": "1. Experiment details: Can you provide more details on the hyperparameter tuning process? Specifically:\n* What was the search space for the learning rate grid search?\n* How was \"optimal performance\" defined during hyperparameter selection?\n* Can you provide ablation studies for the loss weighting factor λ?\n* What are the specific dimensions used in the MLP projection layer? How sensitive is performance to these choices?\n\n2. Can you detail the advantage of your method compared to the conventional approach beyond computational efficiency? Can you clarify the statement about \"equivalent to zero-shot prediction performance\"?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* The proposed method is simple yet novel and sound. This matrix-based approach allows the model to learn relationships between different mutations more effectively, as mutations at different positions can influence each other through the shared representation. Moreover, the reported test-time speedups (10× faster than ThermoMPNN, orders of magnitude faster than MLP baselines) make this approach valuable for some protein engineering applications.\n* The dataset construction is rigorous. The dataset curation process is well-documented and addresses a common issue in many papers of the field: test data leakage. The authors use mmseq to filter the sequences with >30% identity between train/test sets and then conduct further verifications using pairwise global sequence alignment.\n\nWeakness:\n* Several aspects of the experimental setup lack proper justification or transparency. For example, the weighting factor λ = 0.1 in the joint loss appears arbitrary and the MLP architecture details (dimensions of W₁, W₂) are not specific. Plus, It's unclear whether \"optimal performance\" in tuning learning rates refers to validation loss (and how is the validation set constructed?), or inadvertently test set performance, which would be problematic.\n* The introduction is not well-written and weakens the motivation for the project. Overall, the introduction fails to adequately motivate why learning mutation landscapes jointly is superior to independent predictions beyond computational efficiency. I am confused by the  \"initial performance of ΔΔG prediction is equivalent to the zero-shot prediction performance of the PLM\" (lines 62-64) since the model is also fine-tuned in the proposed method.", "was_revised": true, "final_justification": "The author's response has resolved my concerns about the rigor of the experimental design. I believe the proposed approach will benefit future work in this area. Hence I recommend acceptance.", "rebuttal": "Thank you for the insightful comments. We have addressed each of your concerns below and will incorporate these clarifications into the revised manuscript. We hope our responses resolve your concerns and are happy to discuss further if any points remain unclear.\n\n**Response to weakness 1:** \nTo answer your question thoroughly, we'll address it in points.\n\n**Point 1.1** “The weighting factor λ = 0.1 in the joint loss appears arbitrary.”\n\n**Response:** We agree the choice of hyperparameters requires justification. The value for λ was determined empirically through an ablation study, with the results shown in Table R1.\n\nThe model achieved the best performance on the test datasets when λ=0.1. This result suggests that the framework performs optimally when the ranking loss serves as the primary training objective and the MSE loss acts as a beneficial auxiliary objective to align the scale of the predictions. We will add this ablation study to the appendix of our revised manuscript.\n**Table R1. Performance of MAXWELL (ESM-IF) for various λ values**\n|λ|Spearman|Pearson|\n|---|---|---|\n|0.0|0.506|0.528|\n|0.1|0.517|0.542|\n|0.2|0.514|0.540|\n|0.4|0.504|0.538|\n|0.8|0.494|0.523|\n|1.0|0.486|0.519|\n\n**Point 1.2** “The MLP architecture details (dimensions of $W_1$, $W_2$) are not specific.”\n\n**Response:** The MLP consists of two hidden layers. The dimensions are $$W_1 \\in R^{V \\times V}, W_2 \\in R^{V \\times 1}$$ where V represents the vocabulary size of the protein language models (PLMs) (e.g., 35 for ESM-IF and 25 for ProSST). A hidden activation function is applied between $W_1$ and $W_2$.\n\nThe MLP's operation is defined as:\n$$MLP(L) = SELU(LW_1 + b_1)W_2 + b_2$$\nWe will revise this section in the paper.\n\n**Point 1.3** “It's unclear whether \"optimal performance\" in tuning learning rates refers to validation loss (and how is the validation set constructed?), or inadvertently test set performance, which would be problematic.”\n\n**Response:** We selected hyperparameters based on 5-fold cross-validation. All validation sets were derived solely from the training set, ensuring no leakage of test data into the training or validation process. Our procedure for selecting the learning rate (lr) and training epochs is as follows:\n```\nTrainingSet = {T1, T2, T3, T4, T5}\nFor each lr in {1e-3, 1e-4, 1e-5, 1e-6,5e-3,5e-4,5e-5,5e-6}:\n    Initialize scoreset as empty list\n    For i from 1 to 5:\n        InternalTrainingSet = TrainingSet \\ {Ti}  \n        InternalValidationSet = Ti \n        Train model on InternalTrainingSet for 10 epochs:\n            For each epoch in 1..10:\n                Record Spearman correlation score sj on InternalValidationSet\n                Track the epoch with the best score\n        \n        Add the best score of this fold to scoreset\n    CurrentLrScore = Average(scoreset)\nEnd For\n```\nTable R2 illustrates the performance of different learning rates during 5-fold cross-validation. We selected 5e-5 as the optimal lr, as it demonstrated the best performance across the 5-fold cross-validation. The best Spearman correlations on the validation sets for this lr were achieved at epochs {7,7,6,8,7} respectively, averaging 7 epochs. Subsequently, we trained the final model on the entire training dataset for 7 epochs, without any intermediate evaluation during this phase. After training, the model's performance was directly evaluated and reported on the test set. Throughout this entire process, the test set (Test12K) was used exclusively for reporting the final scores and was not involved in any hyperparameter tuning, thereby preventing any data leakage. \n\nWe will supplement the details of our hyperparameter selection process in the Appendix.\n\n**Table R2. Performance of MAXWELL (ESM-IF) with different learning rates in 5-Fold cross-validation**\n|Lr|Avg Spearman|\n|---|---|\n|1e-3|0.278|\n|1e-4|0.499|\n|1e-5|0.492|\n|1e-6|0.501|\n|5e-3|0.184|\n|5e-4|0.416|\n|5e-5|0.517|\n|5e-6|0.508|\n\n**Response to weakness 2:**\nTo answer your question thoroughly, we'll address it in points.\n\n**Point 2.1** On Motivation Beyond Computational Efficiency\n\n**Response:** Our framework offers several key advantages:\n\n**Versatility with Protein Language Models:** As PLMs continue to advance, our model is designed as a universal fine-tuning framework, compatible with diverse protein language models (PLMs), including masked language models and inverse folding language models. This allows researchers to easily integrate the latest or most suitable PLM for their target protein without re-designing the architecture.\n\n**Novel Fine-tuning Methodology:** We introduce a novel matrix-based fine-tuning paradigm. Unlike existing approaches that often involve adding separate regression heads or utilizing non-matrix-based likelihood methods, our method directly and elegantly fine-tunes the PLM's output logits, offering a more streamlined way to adapt PLMs for specific tasks.\n\n**Maximized PLM Knowledge Utilization:** MAXWELL fully capitalizes on the knowledge embedded within PLMs. The insights gained during the unsupervised pre-training phase of PLMs are largely preserved and directly integrated. By inheriting the pre-trained PLM, MAXWELL starts from a significantly advanced foundation.\n\n**Superior Accuracy:** These advantages translate to superior empirical performance. MAXWELL (ESM-IF) outperforms the current state-of-the-art model, ThermoMPNN. Furthermore, ensembling our model with ThermoMPNN boosts the Spearman correlation to 0.548 on the Test12K dataset.\n\nWe will elaborate on these model advantages in the Introduction section.\n\n**Point 2.2** On Clarifying \"Initial Performance\"\n\n**Response:**  \nThe phrase \"initial performance\" refers to the predictive ability of our framework before any fine-tuning begins. This intrinsic capability is equivalent to the zero-shot ΔΔG prediction ability of the underlying PLM itself.\n\nMAXWELL is initialized with the parameters of a pretrained PLM. Our framework's architecture is designed to be a differentiable, matrix-based version of the PLM's standard zero-shot scoring function. PLMs, pre-trained on millions of protein sequences, can compute a likelihood for a given protein sequence. The difference in likelihood between a mutant sequence and its wild-type counterpart can then be used to estimate the mutation's effect and ΔΔG. This approach, which doesn't require training on mutation-specific data, is known as zero-shot prediction. MAXWELL inherits this powerful zero-shot capability and can further enhance it through fine-tuning.\n\nFrom a mathematical perspective, each element in the output matrix L from an unfine-tuned MAXWELL model is equivalent to   $log (Mutant∣Wildtype) − log (Wild-type∣Wildtype)$, which represents the PLM's zero-shot output for ΔΔG estimation. The relationship between ΔΔG and the logarithmic ratio has also been mathematical derived[1].\n \n**Response to question 1:** We primarily focused on optimizing the learning rate and number of epochs. The adjustment process relied on 5-fold cross-validation. This involved splitting the training set into five distinct folds. In each iteration, we used four of these folds for training and reserved the remaining one for validation. This process was repeated five times, and the average score from these five evaluations was used to determine the best hyperparameter combination.\n\nOnce the optimal hyperparameters were identified, we trained the final model on the entire training dataset using these tuned parameters. Afterward, the model's performance was evaluated on the test set. We strictly ensured that only the training set was used during the hyperparameter tuning process, thereby preventing any data leakage.\n\n**Specifically:**\n\n**Point 1.1** What was the search space for the learning rate grid search?\n\n**Response:** The lr search range was set to {1e-3,1e-4,1e-5,1e-6,5e-3,5e-4,5e-5,5e-6}.\n\n**Point 1.2** How was \"optimal performance\" defined during hyperparameter selection?\n\n**Response:** Optimal performance refers to the mean Spearman score achieved across the validation sets in a 5-fold cross-validation, given a specific set of hyperparameters. A higher mean Spearman score indicates a better hyperparameter combination.\n\n**Point 1.3** Can you provide ablation studies for the loss weighting factor λ?\n\n**Response:** Yes，we also conducted an ablation study on the loss weighting factor λ, with the results presented in Table R1. (This is in response to a weakness identified.)\n\n**Point 1.4** What are the specific dimensions used in the MLP projection layer? How sensitive is performance to these choices?\n\n**Response:**  The MLP consists of two hidden layers. The dimensions are $W_1 \\in R^{V \\times V}, W_2 \\in R^{V \\times 1}$, where V represents the vocabulary size of the protein language models (PLMs) (e.g., 35 for ESM-IF and 25 for ProSST). A hidden activation function is applied between W_1 and W_2.\nThe MLP's operation is defined as:\n$$MLP(L) = SELU(LW_1 + b_1)W_2 + b_2$$\n\nThe MLP's dimensions are designed to align with the protein language model's (PLM) vocabulary size. Here, V is a fixed vocabulary size, but we can consider adjusting the dimension D. In our paper, we set D=V.\n\nWe conducted an ablation study on the size of D, and the results are presented in Table R3. Taking ESM-IF as an example, we performed five sets of experiments with D={V, 64, 128, 256, 512}, while keeping other hyperparameters constant. The results in Table R3 indicate that the model's performance is relatively insensitive to the intermediate layer dimension. Specifically, the model achieved its best performance when the intermediate layer size was set to D=V.\n\n**Table R3. Model performance with varying hidden layer sizes**\n|D|Avg Spearman|\n|---|---|\n|V(35)|0.517|\n|64|0.506|\n|128|0.516|\n|256|0.515|\n|512|0.505|\n|1024|0.487|\n**Response to question 2:** Please refer to our response to Weakness 2.\n\n[1] Zero-shot protein stability prediction by inverse folding models: a free energy interpretation. arXiv, 2025."}, {"review_id": "fHq0NlHabG", "reviewer": "Reviewer_94jx", "summary": "This manuscript addresses the problem of predicting protein mutant stability across a sequence of interest. Existing frameworks model the relationship between each mutant sequence and its corresponding stability score for every mutant/score pair, resulting in prohibitively high training and inference costs.\n\nThe authors propose using a mutation landscape matrix constructed by taking the difference between the matrix derived from the wild-type sequence and the matrix obtained from a protein language model. In computational experiments, this matrix is shown to perform better than existing frameworks such as Rosetta, ThermoMPNN, and ESM-IF+MLP. Moreover, the effectiveness of the proposed approach is demonstrated by its fast prediction capability.", "questions": "This reviewer can understand that the proposed method is more efficient than existing methods, but cannot understand why it outperforms the existing methods in terms of prediction accuracy, since the existing methods explicitly learn from mutated sequences, whereas the proposed method learns them only implicitly through language models.", "limitations": "The authors addressed the limitations of their work.", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength\n- Demonstrated efficiency and performance", "was_revised": true, "final_justification": "The authors have addressed my concern regarding the reason for the improved performance over existing embedding methods. I find their explanation convincing and therefore assign a final score of +1.", "rebuttal": "Thank you for your valuable feedback to help us improve our paper. We detail our response below, and please kindly let us know if you have any further questions.\n\n**Response to the question.**\nWe thank you for this thoughtful question, highlighting the need to explain why MAXWELL can outperform methods that explicitly regress from the embeddings of mutated sequences.\n\nWhile it may seem counterintuitive that an implicit method outperforms an explicit one, the superior accuracy of MAXWELL stems from two key points:\n\n1. **MAXWELL leverages the zero-shot mutation prediction capabilities of pretrained protein language models, providing a stronger starting point than methods that regress directly from the embeddings of mutated sequences.** Protein language models (PLMs) are pre-trained on millions of naturally occurring proteins, learning statistical patterns that reflect evolutionary constraints -- unstable proteins are unlikely to appear in nature. Consequently, the probability ratio (or log-probability difference) between a mutant amino acid and the wild-type residue at a given position can serve as a proxy for evolutionary preference. Larger differences suggest mutations that are more evolutionarily plausible and potentially more stable. \n\n    Importantly, MAXWELL retains every component of the protein language model, including its output head, thereby preserving the model's full zero-shot capability. Its initial performance is therefore equivalent to the zero-shot performance of the PLM itself. In contrast, methods that regress from the embeddings of mutated sequences typically discard the original language model head and instead attach a randomly initialized MLP regressor. As a result, their initial predictions are nearly random, and they lose the valuable prior knowledge encoded in the pretrained output head. To validate the importance of the pretrained output head, we include an ablation study in which the language model head is randomly re-initialized. As shown in Table R1, this leads to a drop in performance, confirming the critical role of the pretrained head in enabling MAXWELL's accuracy. Additionally, as  demonstrated in our paper, if the entire PLM is randomly initialized, performance drops dramatically, falling even below the zero-shot baseline. Taken together, these results prove that MAXWELL’s primary advantage lies in its ability to effectively leverage the evolutionary information captured by the PLM during pre-training. Because MAXWELL is compatible with various PLMs, its performance is poised to improve as this foundational technology continues to advance.\n\n    Table R1. Performance of MAXWELL with and without the pretrained PLM\n  \n    | Model | Spearman |\n    |---|---|\n    | MAXWELL (ESM-IF) with pretrained head | 0.517 |\n    | MAXWELL (ESM-IF) with randomly initialized head | 0.432 |\n    | MAXWELL (ESM-IF) with randomly initialized whole PLM | 0.202 |\n    | ESM-IF Zero-shot | 0.375 |\n\n\n2. **MAXWELL utilizes a matrix-based fine-tuning algorithm that enables efficient training on protein mutation ΔΔG data** and supports direct landscape-level prediction for unseen proteins. In contrast, traditional methods regress from the embeddings of individual mutated sequences and produce a scalar prediction for each mutation. MAXWELL instead reformulates the task as a two-dimensional mutational landscape regression problem. This matrix formulation treats the set of single-point mutations as a structured landscape matrix, enabling the model to learn a protein's mutational landscape within a single forward pass. This design not only accelerates training but also improves generalization across diverse proteins. To assess the impact of the matrix formulation, we conducted an ablation study in which the model was fine-tuned using a conventional pointwise regression approach. Each individual mutation was scored using a mean squared error (MSE) loss in this setting, corresponding to likelihood-based fine-tuning without the matrix structure. As shown in Table R2, removing the matrix-based structure resulted in a clear drop in performance and slower convergence, with the Spearman correlation decreasing from 0.507 to 0.401.\n\n    **Table R2.** Performance of matrix-wise (MAXWELL) and pointwise regression on ΔΔG landscape prediction (Base model: ESM-IF).\n  \n    | Fine-tuning Method | Landscape Structure | Spearman | Training Speed |\n    |---|---|---|---|\n    | MAXWELL | 2D landscape | 0.507 | Fast |\n    | Point-wise, likelihood-based | 1D (single mutation) | 0.401 | Slow |\n\nWe will revise the main paper to explicitly clarify why MAXWELL outperforms methods that regress from mutated sequence embeddings."}, {"review_id": "z55ccNWiLs", "reviewer": "Reviewer_LNEY", "summary": "The paper introduces Maxwell: a framework to fine-tune a pre-trained model to predict the $\\Delta\\Delta G$ values for all single substitution mutations for a reference protein, moving from sequence-label to sequence-landscape modeling. An unmasked reference protein is fed through a pre-trained protein language model, which returns per position logits. The reference sequence logits are subtracted, and the resulting matrix is interpreted as a mutation landscape. The logits are via a composite loss function trained to approximate experimental $\\Delta\\Delta G$ values. The paper curates a representative stability dataset which is used for benchmarking, where the proposed model outperforms other models.", "questions": "1. L44-46: As you point out later, most PLMs are either encoder-only or decoder-only. To state that most methods fail to harness the learned evolutionary patterns due to them only using the encoder while ignoring the decoder is therefore not entirely accurate in my opinion. For encoder-only models, the language model heads are typically deliberately simple MLPs which map from the latent representation to logits.\n2. L65: I think that the claim that existing stability datasets exhibit significant redundancy, data leakage, and misalignment should be quantitatively supported. Presumably, this should be easy to show. I think a figure - perhaps just in the appendix - showing these existing issues would show the benefit of the introduced dataset.\n3. L129: Phrasing: “primarily composed of mainly consisting of”\n4. Figure 2: I think the figure is slightly misleading w.r.t. the one-hot encodings as it shows these being fed directly into the PLM which would be odd. From the appendix, I can see that this is not actually the case as the input sequence is tokenized in two ways: regular tokenization for the PLM and one hot for the PLM output. I think this should be clarified in L148-150.\n5. Loss function. I am generally confused by the loss function, particularly the inconsistent use of the MLP and the motivation to optimize the Pearson correlation.\n    1. Regarding equation 10, the Pearson correlation coefficient captures the linear correlation between the predicted and ground truth values. In that case, it is not actually a ranking, right? I think labeling the loss as a ranking loss is incorrect and misleading. Have you explored using ranking losses? \n    2. L186: I am confused about the role of the MLP. The aim of the main architecture is to predict $\\Delta\\Delta G$ values, yet an MLP is used in the *loss* function to map from a logit-like space to $\\Delta\\Delta G$ space to compute the MSE loss. This necessarily means that the model outputs are not actually $\\Delta\\Delta G$s. Why is that? Wouldn’t it make more sense to have the MLP as a component after the element-wise subtraction? \n    3. In a similar vein, why is the MLP only used for the MSE loss and not the Pearson loss? \n    4. Is the MLP in the MSE loss trained jointly with the rest of the architecture or is it trained separately? \n6. Section 5.2: How are the zero-shot scores computed? For zero-shot scoring using sequence-only PLMs, a position would typically be masked and the logits of the masked position conditioned on non-masked residues would typically serves as the fitness proxy. Is this how the values are computed? Or are full sequence (pseudo-) likelihood computed? If so, how? If the sequences are non-masked, does this not result in very peaked distributions, and wouldn’t it benefit encoder-only models to be scored using masking?\n7. L190: The weighing factor of the losses is 0.1. Why this values? Did you experiment with others?\n8. Figure 4: In the related works, many $\\Delta\\Delta G$ predictors are referenced. I think it could be beneficial and fairly straightforward to include more of these in the results to add more granularity.\n9. Is Maxwell consistently better than the other baselines? Or are there instances where it is outperformed? I think an analysis or discussion of some qualitative results would be interesting to see, such as potential failures modes or instances where Maxwell is superior.\n10. I have some questions regarding the role of Maxwell for directed evolution, which is proposed as a use case for the model. Since Maxwell cannot handle multiple mutations, what does DE with Maxwell actually look like? At each step, would a single mutation be suggested, the resulting variant would then be used as input for the next round? ESM-IF1 which is used as PLM is structure conditioned. This presumably means that the structure would change over time, particularly if stability is optimized. Wouldn’t structure prediction be a potential bottleneck which would overrule the inference-speed gains of the proposed architecture? I think some more motivation for how Maxwell is used in a DE pipeline would help show the utility of the model.\n11. You mention on lines 330-331 that Maxwell can potentially be used for diverse protein function prediction. How would this work? Would it be using the same framework but just substituting the stability data with other datasets?", "limitations": "I think that the limitations of the framework should be discussed slightly more. This relates to the mentioned weaknesses and raised questions. E.g., when does the model fail? How sensitive is it to the structure quality? How can be used for DE if it cannot handle multiple mutations?", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1. Well-motivated problem to move from sequence-label to sequence-landscape mappings. I appreciate the efficiency comparisons.\n2. Simple idea of learning multiple $\\Delta\\Delta G$ values via regression on a full backbone.\n3. Beats ThermoMPNN across metrics.\n4. Well-written paper with clear notation.\n5. Rigorous handling of datasets and curation of novel stability dataset.\n\nWeaknesses\n\n1. The novelty of the approach and the methodological contributions are limited. Perhaps this is outweighed by the increased utility of the model given its rapid inference. I am however missing clear cases where Maxwell would have a substantial impact. Directed evolution is used as an example, but it is unclear to me exactly how this would work given the lack of multi-mutant predictions.\n2. Confusing ad-hoc loss function where the Pearson correlation between experimental labels and model “logits” is maximized while the MSE between the labels and MLP-transformed logits is minimized. I do not understand the use of the MLP. If the role of the MLP is to map from logits to $\\Delta\\Delta G$s, why is this not a core component of the model? See related questions. Perhaps I’m misunderstanding something.\n3. I am similarly missing a qualitative analysis of when the model performs well or poorly. Perhaps the model is just consistently better across proteins and dataset sizes, but it would nonetheless be illustrative to see.\n4. No code in the supplementary materials for reproducibility.\n\nI will consider raising my score if the weaknesses - particularly 1 and 2 - are adequately and convincingly addressed.", "was_revised": true, "final_justification": "I am pleased with the authors’ rebuttal which has addressed my raised weaknesses, questions, and limitations. I particularly appreciate the authors' acknowledgement that their model isn't necessarily ideal for multi-mutant modeling despite its capacity for it. The justification for redundancy in existing datasets, the detailed analyses for diverse proteins, the inclusion of additional baselines and experiments as well as the clarifying explanations has, in my opinion, strengthened the quality and utility of the paper and its methods and I therefore think the paper is of interest to the community and should be presented at this year's NeurIPS.", "rebuttal": "Thank you for carefully reviewing our manuscript and providing constructive feedback. Below we respond to your questions and concerns. We would appreciate it if you could let us know whether your concerns are addressed by our response.\n\n**Response to weakness 1.**\n\n(1) *Applications*. Directed evolution (DE) often begins with single-point mutations. MAXWELL can identify stable single mutations of the target protein for experimental validation, which can then be combined into multi-point mutants. The ΔΔG of multi-point mutants can be predicted by existing \"train-on-single-and-predict-on-multi\" models such as ECNet [1] or ProteinNPT [2]. Unlike these methods, MAXWELL requires no prior mutational data, making it useful in early-stage DE.\n\n(2) *Multi-Point Mutations Prediction*. Although MAXWELL isn't trained on multi-point mutants, it is able to predict their stability via additive scoring: $Score(M) = \\sum_{i=1}^{n} Score(m_i)$,  where $Score(m_i)$ is the individual predicted effect of each single-point mutation. On the multi-point mutations from public Mega-scale ΔΔG dataset and M1261 dataset, MAXWELL significantly improves prediction accuracy over the PLM baseline, increasing Spearman correlation from 0.317 and 0.272 to 0.597 and 0.446(Table R1).\n\nTable R1. Multi-point mutant prediction performance\n\n|Model|Mega-scale|M1261|\n|---|---|----|  \n|ESM-IF| 0.317|0.336|  \n|MAXWELL (ESM-IF)|0.597|0.492|  \n|ThermoMPNN|Not Supported|\n\nNote that MAXWELL and other ΔΔG models (ThermoMPNN, StabilityOracle) primarily target single-point mutants, because:\n\n* DE workflows prioritize reliable single-mutant prediction.\n* Accurate multi-point predictions usually need prior single-mutant data, often unavailable.\n* Multi-point datasets are typically smaller, restricting model effectiveness.\n\nWe hope this clarifies MAXWELL's design principles.\n\n**Response to weakness 2.**\n\nMLP serves as an auxiliary component rather than the core predictor for several reasons:\n\n(1) The MLP is randomly initialized and lacks pretraining, whereas the PLM output head is pretrained and estimates ΔΔG via log-probability differences between mutant and wild-type residues.\n\n(2) The MLP outputs ΔΔG values, but ranking is more relevant in practice. Rankings from the language model head are more accurate. For instance, MAXWELL (ESM-IF) trained only with MSE loss achieves a Spearman correlation of 0.492, lower than the language-model head trained solely with ranking loss (0.506 vs. 0.492, Table 2 in the main text).\n\n(3) The MLP mainly aids training by providing a ΔΔG-based loss. It improves PLM head performance (e.g., Spearman corr. from 0.506 to 0.517, Table 2 in main text).\n\n**Response to weakness 3.**\n\nWe performed a qualitative analysis to understand MAXWELL's behavior across proteins. We found a strong correlation (ρ = 0.685, R² = 0.469) between MAXWELL's performance and the PLM's zero-shot accuracy, confirming that our method effectively leverages pretrained knowledge. Correlations with protein length (ρ = -0.378, R² = 0.058) and dataset size (ρ = 0.307, R² = 0.022) were weak, suggesting robustness to these factors. Additionally, we examined structure quality using AlphaFold pLDDT scores and observed only a weak correlation with performance (ρ = 0.264, R² = 0.070), suggesting that MAXWELL is relatively insensitive to moderate structure noise. The slight size trend is driven by a few large datasets (>200 mutants) where MAXWELL performs consistently well. These findings will be included in the revision.\n\n**Response to weakness 4.**\nWe apologize for not including the code in the supplementary materials. Code and data will be released in the next version.\n\n---\n**Response to question 1.**\nThough simple, PLM output heads are pretrained to map representations to logits, enabling zero-shot ΔΔG prediction. To validate their importance, we re-initialized the head randomly. Table R4 shows performance drops sharply, confirming its critical role.\n\nTable R4. Performance of MAXWELL with and without the pretrained head (Spearman corr. on Test12K).\n|Model|Spearman|\n|---|---|\n|MAXWELL (ESM-IF) with pretrained head|0.517|\n|MAXWELL (ESM-IF) with randomly re-initialized head|0.432|\n\n**Response to question 2.**\nWe quantified the occurrences of duplicates and misalignments in publicly available stability datasets, including Myoglobin, S669, S8754, M1261, vb1432, Fireprotdb, and Thermomutdb. The results are presented in Table R5.\n\nTable R5. Summary of Data Issues in Public Stability Datasets\n|Dataset|Seq. Mismatch|Missing PDB ID|Missing Type|Duplicate Rows|Duplicate Mutant|\n|---|---|---|---|---|---|\n|Myoglobin|-|-|-|-|21|\n|S669|333|-|-|-|-|\n|S8754|-|506|-|-|3912|\n|M1261|-|-|-|-|447|\n|vb1432|-|131|-|-|4|\n|Fireprotdb|-|34|19|39600|5890|\n|Thermomutdb|-|373|-|-|2425|\n\n**Response to question 3.**\nWe apologize for the typo and will correct it.\n\n**Response to question 4.**\nThank you for pointing this out. The PLM input is indeed an integer token sequence. We will update the figure.\n\n**Response to question 5.**\nIn short, the MLP is only used for auxiliary tasks. Pearson correlation serves as an alternative ranking loss.\n\n*5.1* Pearson correlation serves as a simple yet effective ranking loss. We tested other ranking losses and observed similar performance (see Table R6).\n\nTable R6. Performance of different ranking losses.\n|Loss|Spearman|\n|---|---|\n|Pearson|0.517|\n|ListMLE|0.512|\n|Soft Spearman|0.514|\n\n*5.2* MAXWELL aims to rank ΔΔG values rather than predict their exact ΔΔG values, as ranking is more important in practice. The MLP is auxiliary (see response to Weakness 1). Applying the MLP after the element-wise operation yields similar results (Spearman: 0.514).\n\n*5.3* Adding Pearson loss to the MLP branch had negative effect (Spearman: 0.507).\n\n*5.4* They are trained jointly. Using only MSE loss reduces performance, and MAXWELL (ESM-IF) drops to Spearman corr. 0.496.\n\n**Response to question 6.**\nFor zero-shot scoring, we input the full, unmasked protein sequence into a BERT-style PLM, which outputs amino acid probability distributions at each position. The log-probability difference between the mutant and wild-type residues serves as a fitness proxy. We do not apply masking because (1) BERT-style models can handle unmasked inputs due to their pretraining (including random replacements and unchanged tokens), and (2) masking each position individually greatly increases computation and breaks the ability to compute the full mutational landscape in a single forward pass. And, in practice, the performance difference is negligible. For example, with ESM2, Spearman correlation is 0.273 (with masking) vs. 0.268 (without).\n\n**Response to question 7.**\nWe tested various λ values (Table R7), and λ = 0.1 yields the best performance.\n\nTable R7. MAXWELL (ESM-IF) Performance Across λ\n|λ|Spearman|Pearson|\n|---|---|---|\n|0.0|0.506|0.528|\n|0.1|0.517|0.542|\n|0.2|0.514|0.540|\n|0.4|0.504|0.538|\n|0.8|0.494|0.523|\n|1.0|0.486|0.519|\n\n**Response to question 8.**\nWe have evaluated additional models. Results are shown in Table R8:\n\nTable R8: Performance of Additional Baseline Models.\n|Model|Spearman|Pearson|\n|---|---|---|\n|FoldX|0.410|0.431|\n|ProGen2|0.187|0.191|\n|Tranception|0.244|0.254|\n\nUnfortunately, we were unable to reproduce StabilityOracle. The official code could not be executed.\n\n**Response to question 9.**\nWhile MAXWELL outperforms baselines on average, it underperforms ThermoMPNN on 143 of 308 proteins (~46%). These often overlap with proteins where the language model's zero-shot performance is low (114/143 below average), highlighting the importance of the PLM's intrinsic capability. Fortunately, our method is general and can improve as language models advance.\n\n**Response to question 10.**\nWe appreciate your valuable suggestion.\n(1) MAXWELL's role in DE is discussed in our response to Weakness 1.  \n(2) While MAXWELL focuses on single-point mutation prediction, it does not directly address iterative design or multi-site optimization. Such tasks are better suited for models like ECNet [1] or ProteinNPT [2] that integrate modeling with experimental feedback. MAXWELL can be trained with experimental data, and we plan to explore this in future work.  \n(3) Although ESM-IF is structure-aware, its mutant scoring only requires the wild-type sequence and structure, not mutant structures. So it introduces no computational bottleneck.\nWe will clarify MAXWELL's role in directed evolution more explicitly in the revised Introduction. Thank you again for the insightful feedback.\n\n**Response to question 11.**\nYes, your intuition is correct. Theoretically, our framework is task-agnostic and can be adapted for other functional prediction tasks by substituting the training data. We chose ΔΔG as our primary focus because it is currently the ideal use case, benefiting from both the availability of large-scale public data and a theoretical link to PLM log-likelihoods [3]. Extending this framework to other tasks is a future direction, contingent on the curation of suitable datasets and potential loss function adjustments.\n\n**Response to limitations**  \nWe appreciate your suggestion to clarify the limitations. As discussed in Response to Weakness 3, MAXWELL's performance primarily depends on PLM prior quality and is relatively robust to structure quality. Its role in directed evolution is addressed in Response to Weakness 1. For more advanced DE tasks such as iterative design or combinatorial optimization, we plan to explore these directions in future work. We will make these points clear in the revised manuscript.\n\n[1] ECNet is an evolutionary context-integrated deep learning framework for protein engineering. Nature Communications, 2021.\n\n[2] ProteinNPT: Improving Protein Property Prediction and Design with Non-Parametric Transformers. NeurIPS, 2023.\n\n[3] Zero-shot protein stability prediction by inverse folding models: a free energy interpretation. arXiv, 2025."}]

Author rebuttal:

Meta-review: This paper introduces MAXWELL, a new method for prediction of protein stability. Unique features of the model include the ability to predict an entire landscape of mutations with a single forward pass, which provides considerable computational speedups compared to existing approaches. The authors also present a new ddg benchmark set with better control of data leakage, and report state-of-the-art results. The reviewers raised some initial concerns regarding the original submission: the lack of theoretical foundation, the lack of testing on established benchmarks and relevant baselines, lack of justification of the experimental setup, and lack of convincing arguments for impact on the community. The authors provided an elaborate rebuttal, which was received positively, and all reviewers who responded to the rebuttal raised their score to a 5 - recommending acceptance of the paper.
Decision: Accept (poster)