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Persistent spectral theory-guided protein engineering

Nature Computational Science volume 3, pages 149–163 (2023)Cite this article

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A preprint version of the article is available at bioRxiv.

Abstract

Although protein engineering, which iteratively optimizes protein fitness by screening the gigantic mutational space, is constrained by experimental capacity, various machine learning models have substantially expedited protein engineering. Three-dimensional protein structures promise further advantages, but their intricate geometric complexity hinders their applications in deep mutational screening. Persistent homology, an established algebraic topology tool for protein structural complexity reduction, fails to capture the homotopic shape evolution during filtration of given data. Here we introduce a Topology-offered Protein Fitness (TopFit) framework to complement protein sequence and structure embeddings. Equipped with an ensemble regression strategy, TopFit integrates the persistent spectral theory, which is a new topological Laplacian, and two auxiliary sequence embeddings to capture mutation-induced topological invariant, shape evolution and sequence disparity in the protein fitness landscape. The performance of TopFit is assessed by 34 benchmark datasets with 128,634 variants, involving a vast variety of protein structure acquisition modalities and training set size variations.

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Fig. 1: Conceptual diagram of the TopFit method.
Fig. 2: PST for topological persistence and homotopic shape evolution.
Fig. 3: Prediction from single embedding on fitness landscape measured by Spearman correlations.
Fig. 4: Comparisons between TopFit embeddings and other methods for fitness prediction.
Fig. 5: Comparisons between TopFit and other regression models for fitness predictions using Spearman correlation.

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Data availability

There are 34 DMS datasets with experimentally measured fitness used in this work including: 32 DeepSequence datasets8, the avGFP dataset41 and the GB1 dataset42. The original data sources of the 32 DeepSequence datasets are provided in Supplementary Data 1 and Supplementary Note 7. Structure data were obtained from PDB database22 and AF38, and the specific entry ID is provided in Supplementary Data 1. The data analyzed and generated in this work, including sequence-to-fitness datasets, optimized structure data, MSAs, fine-tune parameters for eUniRep models, predictions from evolutionary scores for individual mutations and sequence- and structure-based embeddings are available at https://github.com/WeilabMSU/TopFit (ref. 65) and our lab server https://weilab.math.msu.edu/Downloads/TopFit/. Source data for Figs. 3–5 and Extended Data Figs. 1, 3, 5–10 are available with this paper. Source data for Extended Data Figs. 2 and 4 are available in Supplementary Data 2.

Code availability

All source codes and models are publicly available at https://github.com/WeilabMSU/TopFit (ref. 65).

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Acknowledgements

This work was supported in part by NIH grants R01GM126189 and R01AI164266; NSF grants DMS-2052983, DMS-1761320 and IIS-1900473; NASA grant 80NSSC21M0023; Michigan Economic Development Corporation; MSU Foundation; Bristol-Myers Squibb 65109 and Pfizer. We thank C. Hsu and J. Listgarten for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, USA

    Yuchi Qiu & Guo-Wei Wei

  2. Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, USA

    Guo-Wei Wei

  3. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, USA

    Guo-Wei Wei

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Contributions

All authors conceived this work, and contributed to the original draft, review and editing. Y.Q. performed experiments and analyzed data. G.-W.W. provided supervision and resources and acquired funding.

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Correspondence to Guo-Wei Wei.

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Extended data

Extended Data Fig. 1 The average performance of various models over 34 datasets.

a,b, This is a supplement for Fig. 3a. a, Line plots show identical data with Fig. 3a with additional data for two TopFit strategies. b, Results are evaluated by NDCG. a,b, Ensemble regression is used, except ridge regression for Georgiev and one-hot embeddings. Absolute values of ρ were shown for evolutionary scores. The width of shade shows 95% confidence interval from n = 20 repeats. Evolutionary scores use absolute values for corresponding quantities.

Source data

Extended Data Fig. 2 Spearman correlation for various models on individual datasets.

This is a supplement for Fig. 3a. Each line plots show average ρ for each dataset over n = 20 repeats. Training data sizes are 24, 96, 168, and 240. The width of the shade shows 95% confidence interval.

Extended Data Fig. 3 The frequency that an embedding is ranked as the best across 34 datasets using Spearman correlation.

a–d, This is a supplement for Fig. 3b including comparisons over different strategies. Histograms show the frequency that an embedding is ranked as the best across 34 datasets with 24, 96, 168 and 240 training data, respectively. For each dataset, the best embedding has average ρ over n=20 repeats within the 95% confidence interval of the embedding with the highest average ρ. Comparisons were performed for a sequence-based embeddings, b structure- and sequence-based embeddings, c structure-based embeddings, sequence-based embeddings, and evolutionary scores and d structure-based embeddings, sequence-based embeddings, evolutionary scores and two sets of TopFit (VAE+PST+ESM and VAE+PST+eUniRep). We showed and used absolute values Spearman correlation for evolutionary scores.

Source data

Extended Data Fig. 4 NDCG for various models on individual datasets.

This is an analog for Extended Data Figure 2 but using NDCG. Each line plots show average NDCG for each dataset over n = 20 repeats. Training data sizes are 24, 96, 168 and 240. The width of the shade shows 95% confidence interval.

Extended Data Fig. 5 The frequency that an embedding is ranked as the best across 34 datasets using NDCG.

a–d, This is an analog for Extended Data Figure 3 but measured by NDCG. Histograms show the frequency that an embedding is ranked as the best across 34 datasets with 24, 96, 168 and 240 training data, respectively. For each dataset, the best embedding has average NDCG over n = 20 repeats within the 95% confidence interval of the embedding with the highest average NDCG. Comparisons were performed for a sequence-based embeddings; b structure- and sequence-based embeddings; c structure-based embeddings, sequence-based embeddings and evolutionary scores and d structure-based embeddings, sequence-based embeddings, evolutionary scores and two sets of TopFit (VAE+PST+ESM and VAE+PST+eUniRep). We showed and used absolute values NDCG for evolutionary scores.

Source data

Extended Data Fig. 6 Relationships between quality of wild-type protein structure and PST performance.

a–b, This is a supplement for Fig. 3c. Boxplots show distribution of a percentages of coils for protein structure over 34 datasets and b third quartile (Q3) of B factors at alpha carbons over 26 X-ray datasets. Datasets were classified into two classes depending on whether PST embedding is the best embedding. Scatter plots show same data with boxplots but for individual datasets. One-sided Mann–Whitney U-test examines the statistical significance that two classes have different values. Boxplots display five-number summary where center line shows median, upper and lower limits of the box show upper and lower quartiles, and upper and lower whiskers show the maximum and the minimum by excluding “outliers” outside the interquartile range. In a, sample sizes for PST ranked as the best model are n = 21, n = 15, n = 18 and n = 19 for training data size 24, 96, 168 and 240, respectively. Sample sizes for PST not ranked as the best model are n = 13, n = 19, n = 16 and n = 15 for training data size 24, 96, 168 and 240, respectively. The p-values are 0.01, 3 × 10−5, 1 × 10−3 and 1 × 10−3 for training data size 24, 96, 168 and 240, respectively. In b, sample sizes for PST ranked as the best model are n = 8, n = 12, n = 14 and n = 15 for training data size 24, 96, 168 and 240, respectively. Sample sizes for PST not ranked as the best model are n = 18, n = 14, n = 12 and n = 11 for training data size 24, 96, 168 and 240, respectively. P values are 0.03, 0.02, 0.07 and 0.02 for training data size 24, 96, 168 and 240, respectively.

Source data

Extended Data Fig. 7 Model occurrence in ensemble regression.

This is a supplement for Fig. 3e to show model occurrence on individual datasets. For each repeat, the top N = 3 regressors were picked and counted. Histograms count the model occurrence over 20 repeats.

Source data

Extended Data Fig. 8 Comparisons between TopFit and other methods for mutation effects prediction using Spearman correlation.

a,b, This is an analog for Fig. 4a-b, but TopFit combines VAE score, eUniRep embedding, and PST embedding. All supervised models use 240 labeled training data. Results are evaluated by Spearman correlation ρ. DeepSequence VAE takes the absolute value of ρ. The average ρ from n = 20 repeats is shown. All 34 datasets are categorized by their structure modality used: X-ray, nuclear magnetic resonance (NMR), AlphaFold (AF) and cryogenic electron microscopy (EM). a, Dot plots show results across 34 datasets. b, Dot plots show pairwise comparison between TopFit with one method at each plot. Medians of difference for average Spearman correlation Δρ across all datasets are shown. One-sided rank-sum test determines the statistical significance that TopFit has better performance than VAE score, eUniRep embedding and PST embedding with P values 3 × 10−7, 2 × 10−7 and 4 × 10−7, respectively.

Source data

Extended Data Fig. 9 Comparisons between TopFit with other methods. TopFit consists of VAE score, ESM embedding and PST embedding.

This is a supplement for Fig. 4b to include results with various numbers of training data. Average Spearman correlation from n = 20 repeats are shown, and all datasets are categorized by their structure modality used: X-ray, nuclear magnetic resonance (NMR), AlphaFold (AF) and cryogenic electron microscopy (EM). One-sided rank-sum test determines the statistical significance that TopFit has better performance than other strategies, except we use null hypothesis that TopFit has worse performance than VAE with 24 training data. The p-values are shown in the corresponding subfigures. They are 1. TopFit versus VAE: P = 4 × 10−6, 2 × 10−5 and 1 × 10−6; 2. TopFit versus ESM: P = 2 × 10−7, 2 × 10−7 and 2 × 10−7 and 3. TopFit versus PST: P = 2 × 10−7, 2 × 10−7 and 2 × 10−7 for training data size 24, 96 and 168, respectively.

Source data

Extended Data Fig. 10 Comparisons between TopFit with other methods. TopFit consists of VAE score, eUniRep embedding and PST embedding.

This is a supplement for Extended Data Figure 8b to include results with various numbers of training data. Average Spearman correlation from n = 20 repeats are shown, and all datasets are categorized by their structure modality used: X-ray, nuclear magnetic resonance (NMR), AlphaFold (AF) and cryogenic electron microscopy (EM). One-sided rank-sum test determines the statistical significance that TopFit has better performance than other strategies, except we use null hypothesis that TopFit has worse performance than VAE with 24 training data. The p-values are shown in the corresponding subfigures. They are 1. TopFit versus VAE: P = 3 × 10−6, 3 × 10−5 and 8 × 10−7; 2. TopFit versus eUniRep: P = 4 × 10−7, 2 × 10−7 and 2 × 10−7; and 3. TopFit versus PST: P = 3 × 10−7, 2 × 10−7 and 3 × 10−7 for training data size 24, 96 and 168, respectively.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–16, Tables 1–5 and Notes 1–7.

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Supplementary Data 1

Dataset information. List of datasets, including their UniProt ID, the structure data ID, reference, sequence region for mutations and so on.

Supplementary Data 2

Raw data for computational results. Each folder is named by the name of dataset. Please refer to Supplementary Data 1 for the correspondence between dataset naming convention used in this file and that in figures. Dataset with additional suffix “_AF” or “_NMR” indicate AF or NMR structure was used for computations. In each folder, results are saved in “.csv” with eight possible names for different tasks. “ridge.csv”, “ensemble.csv”, “evolutionary_scores.csv” are results from random train/test splits using ridge regression, ensemble regression and evolutionary scores, respectively. “unseen_ensemble.csv” and “unseen_evolutionary_scores.csv” are results from train/test splits for unseen mutational sites using ensemble regression and evolutionary scores, respectively. “ridge_n_mut_1.csv”, “ensemble_n_mut_1.csv”, “evolutionary_scores_n_mut_1.csv” are results from extrapolation task that single mutations are used to predict multiple mutations for ridge regression, ensemble regression and evolutionary scores, respectively. Explanations for column name are available in “README.txt”. It provides raw data for all figures, Extended Data figures and Supplementary figures generated in this work, excluding Fig. 5, which is available from its own source data. Statistical data for Extended Data Figs. 2 and 4 are available directly from this data.

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Qiu, Y., Wei, GW. Persistent spectral theory-guided protein engineering. Nat Comput Sci 3, 149–163 (2023). https://doi.org/10.1038/s43588-022-00394-y

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