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Authors: (1) JunJie Wee, Department of Mathematics, Michigan State University; (2) Jiahui Chen, Department of Mathematical Sciences, University of Arkansas; (3) Kelin Xia, Division of Mathematical Sciences, School of Physical and Mathematical Sciences Nanyang Technological University & xiakelin@ntu.edu.sg; (4)Guo-Wei Wei, Department of Mathematics, Michigan State University, Department of Biochemistry and Molecular Biology, Michigan State University, Department of Electrical and Computer Engineering, Michigan State University & xiakelin@ntu.edu.sg. Table of Links Abstract & Introduction Results Discussion Conclusion Materials and Methods Software and resources, Code and Data Availability Supporting Information, Acknowledgments & References Supporting Information Supporting Information is available for supplementary tables, figures, and methods. Acknowledgments This work was supported in part by NIH grants R01GM126189, R01AI164266, and R01AI146210, NSF grants DMS-2052983, DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, MSU Foundation, Bristol-Myers Squibb 65109, and Pfizer. It was supported in part by Nanyang Technological University Startup Grant M4081842.110, Singapore Ministry of Education Academic Research fund Tier 1 RG109/19 and Tier 2 MOE-T2EP20120-0013, MOE-T2EP20220- 0010, and MOE-T2EP20221-0003. References [1] Y. Qiu and G.-W. Wei, “Persistent spectral theory-guided protein engineering,” Nature Computational Science, vol. 3, no. 2, pp. 149–163, 2023. [2] R. Guerois, J. E. Nielsen, and L. Serrano, “Predicting changes in the stability of proteins and protein complexes: a study of more than 1000 mutations,” Journal of molecular biology, vol. 320, no. 2, pp. 369–387, 2002. [3] P. Sormanni, F. A. Aprile, and M. 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Ma, et al., “Biological structure and function emerge from scaling unsupervised learning to 250 million protein sequences,” Proceedings of the National Academy of Sciences, vol. 118, no. 15, p. e2016239118, 2021. [34] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” Advances in Neural Information Processing Systems, vol. 30, 2017. [35] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding,” in North American Chapter of the Association for Computational Linguistics, 2019. [36] J. Meier, R. Rao, R. Verkuil, J. Liu, T. Sercu, and A. Rives, “Language models enable zero-shot prediction of the effects of mutations on protein function,” in Advances in Neural Information Processing Systems (M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, eds.), vol. 34, pp. 29287–29303, Curran Associates, Inc., 2021. [37] J. Jumper, R. Evans, A. Pritzel, T. Green, M. Figurnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. Zˇ´ıdek, A. Potapenko, et al., “Highly accurate protein structure prediction with alphafold,” Nature, vol. 596, no. 7873, pp. 583–589, 2021. [38] J. Z. Xiang and B. Honig, “Jackal: A protein structure modeling package,” Columbia University and Howard Hughes Medical Institute, New York, 2002. [39] R. M. Rao, J. Liu, R. Verkuil, J. Meier, J. Canny, P. Abbeel, T. Sercu, and A. Rives, “MSA Transformer,” in International Conference on Machine Learning, pp. 8844–8856, PMLR, 2021. [40] P. Baldi, S. Brunak, Y. Chauvin, C. A. Andersen, and H. Nielsen, “Assessing the accuracy of prediction algorithms for classification: An Overview,” Bioinformatics, vol. 16, no. 5, pp. 412–424, 2000. [41] E. D. Levy, “A simple definition of structural regions in proteins and its use in analyzing interface evolution,” Journal of molecular biology, vol. 403, no. 4, pp. 660–670, 2010. [42] D. S. Goodsell, L. Autin, and A. J. Olson, “Illustrate: Software for biomolecular illustration,” Structure, vol. 27, no. 11, pp. 1716–1720, 2019. [43] Y. Hozumi, K. A. Tanemura, and G.-W. Wei, “Preprocessing of Single Cell RNA Sequencing Data Using Correlated Clustering and Projection,” Journal of Chemical Information and Modeling, 2023. [44] J. R. Munkres, Elements of algebraic topology. CRC Press, 2018. [45] A. J. Zomorodian, Topology for computing, vol. 16. Cambridge university press, 2005. [46] H. Edelsbrunner and J. Harer, Computational topology: an introduction. American Mathematical Soc., 2010. [47] K. Mischaikow and V. Nanda, “Morse theory for filtrations and efficient computation of persistent homology,” Discrete and Computational Geometry, vol. 50, no. 2, pp. 330–353, 2013. [48] D. Horak and J. Jost, “Spectra of combinatorial Laplace operators on simplicial complexes,” Advances in Mathematics, vol. 244, pp. 303–336, 2013. [49] B. Eckmann, “Harmonische funktionen und randwertaufgaben in einem komplex,” Commentarii Mathematici Helvetici, vol. 17, no. 1, pp. 240–255, 1944. [50] A. Zomorodian, “Topological data analysis,” Advances in applied and computational topology, vol. 70, pp. 1–39, 2012. [51] T. G. Project, GUDHI User and Reference Manual. GUDHI Editorial Board, 2015. [52] M. Mirdita, K. Schutze, Y. Moriwaki, L. Heo, S. Ovchinnikov, and M. Steinegger, “Colab- ¨ Fold: Making protein folding accessible to all,” Nature methods, vol. 19, no. 6, pp. 679– 682, 2022. [53] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine Learning in Python,” Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011. [54] T. J. Dolinsky, J. E. Nielsen, J. A. McCammon, and N. A. 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Zhou, “Improving prediction of secondary structure, local backbone angles and solvent accessible surface area of proteins by iterative deep learning,” Scientific reports, vol. 5, no. 1, p. 11476, 2015. [60] C. Maria, J.-D. Boissonnat, M. Glisse, and M. Yvinec, “The GUDHI library: Simplicial complexes and persistent homology,” in Mathematical Software–ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings 4, pp. 167–174, Springer, 2014. This paper is under CC 4.0 license. available on arxiv Lead image by Cytonn Photography on Unsplash

Contributing Partners: NIH, NSF, NASA, and More Back TopLapGBT's Advancements

Authors: (1) JunJie Wee, Department of Mathematics, Michigan State University; (2) Jiahui Chen, Department of Mathematical Sciences, University of Arkansas; (3) Kelin Xia, Division of Mathematical Sciences, School of Physical and Mathematical Sciences Nanyang Technological University & xiakelin@ntu.edu.sg; (4)Guo-Wei Wei, Department of Mathematics, Michigan State University, Department of Biochemistry and Molecular Biology, Michigan State University, Department of Electrical and Computer Engineering, Michigan State University & xiakelin@ntu.edu.sg. Table of Links Abstract & Introduction Results Discussion Conclusion Materials and Methods Software and resources, Code and Data Availability Supporting Information, Acknowledgments & References 6 Software and resources Protein sequences are first preprocessed by AlphaFold 2 to generate wild type protein structures. In particular, 3D protein structures are generated from protein sequences using ColabFold [52]. Mutant proteins are generated from the Jackal software[38]. All TopLapGBT models are built using the sklearn machine learning library [53]. The hyperparameters for all the TopLapGBT are: n estimators = 20000, learning rate = 0.05, max depth = 7, subsample=0.4, min sample split = 3 and max features = sqrt. The PQR files, which contains the partial charge information of the proteins, are generated from the PDB2PQR software [54]. The PQR files for both the wild type proteins are generated with AMBER force field. The solvation energy and surface area information are calculated from the in-house online software package ESES [55] and MIBPB [56]. The pKa values are computed from the PROPKA software package [57]. The position-specific-scoring matrices (PSSM) are computed from the BLAST+ software [58] using the nr database. The secondary structure features and torsion angle sequence-based information are calculated from SPIDER [59]. The persistent Laplacian descriptors for both VR complexes and alpha complexes are calculated using the GUDHI software library [60]. All computational work in support of this research was performed using the resources from the National Super Computing Centre of Singapore (NSCC). Code and Data Availability The 3D protein structures and the TopLapGBT code can be found in https://github.com/ExpectozJJ/TopLapGBT. The source code for the R-S plot can be found at https://github.com/hozumiyu/RSI. This paper is under CC 4.0 license. available on arxiv

Software, Tools and Resources Utilized in TopLapGBT's Development

Authors: (1) JunJie Wee, Department of Mathematics, Michigan State University; (2) Jiahui Chen, Department of Mathematical Sciences, University of Arkansas; (3) Kelin Xia, Division of Mathematical Sciences, School of Physical and Mathematical Sciences Nanyang Technological University & xiakelin@ntu.edu.sg; (4)Guo-Wei Wei, 1Department of Mathematics, Michigan State University, Department of Biochemistry and Molecular Biology, Michigan State University, Department of Electrical and Computer Engineering, Michigan State University & xiakelin@ntu.edu.sg. Table of Links Abstract & Introduction Results Discussion Conclusion Materials and Methods Software and resources, Code and Data Availability Supporting Information, Acknowledgments & References 5 Materials and Methods In this section, we endeavor to elucidate key mathematical and computational foundations that are instrumental for the work presented in this study. Specifically, we delve into spectral graph theory, simplicial complex, and persistent Laplacian methods, highlighting their significance in capturing topological and spectral properties essential for the characterization of proteins. Additionally, we discuss machine learning and deep learning paradigms, focusing on their role in processing, analyzing, and interpreting these complex features, especially within the confines of test datasets and validation settings. 5.1 Persistent Laplacian characterization of proteins A simplicial complex is made up of a set of simplices and generalises beyond graph networks at higher dimensions [44, 45, 46, 47]. Every simplex is a finite set of vertices which can be interpreted as the atoms in a protein structure. Essentially, simplices can be a point (0-simplex), an edge (1-simplex), a triangle (2-simplex), a tetrahedron (3-simplex), or in higher dimensions, a p-simplex. In other words, a k-simplex σ k = {v0, v1, · · · , vk} is the convex hull formed by k + 1 affinely independent points v0, v1, · · · , vk as follows, Simplicial complex A geometric simplicial complex K is a finite set of geometric simplexes that satisfy two essential conditions. First, any face of a simplex from K is also in K. Second, the intersection of any two simplexes in K is either empty or shares faces. Commonly used methods to construct simplicial complexes are Cech complex, Vietoris-Rips complex, Alpha complex, Clique ˇ complex, Cubic complex, and Morse complex [44, 45, 46, 47]. In the case k = 0, then L0 = B1B⊤ 1 since ∂0 is a zero map Persistent Laplacian (PL) were first introduced by integrating graph Laplacian and multiscale filtration [25]. Analyzing the spectra of k-combinatorial Laplacian matrix allows both topological and geometric information (i.e. connectivity and robustness of simple graphs) to be obtained. However, this method is genuinely free of metrics or coordinates, which induced too little topological and geometric information that can be used to describe a single configuration. Persistent Laplacian Therefore, PL was extended to simplicial complexes. This allows a sequence of simplicial complexes from a filtration process to generate persistent Laplacian which is largely inspired by persistent homology and in earlier works in multiscale graphs. For the rest of this section, we introduce mainly on the construction of PL. First, a k-combinatorial Laplacian matrix is symmetric and positive semi-definite. Therefore, its eigenvalues are all real and non-negative. The multiplicity of zero spectra (also called harmonic spectra) reveals the topological information, and the geometric information will be preserved in the non-harmonic spectra. This nested sequence of simplicial complexes induces a family of chain complexes In order to illustrate the difference between PL and PH, Figure 5 describes a point cloud, basic simplices, a filtration process and the comparison between persistent Laplacian and persistent homology barcodes of 13 points. The filtration process in Figure 5(c) shows the different stages of a Rips filtration process for the 13 points. Figure 5(d) shows the persistent homology barcodes (in blue) and persistent non-harmonic spectra (in red). It can be seen that the nonharmonic spectra provides the additional homotopic shape evolution that is missing in persistent homology in the later part of the filtration process. 5.2 Persistent Laplacian descriptors where DE(·, ·) is the Euclidean distance between the two atoms and Loc(·) refers to the atom’s location which is either in the mutation site or in the rest of the protein. Here, we construct two types of simplicial complexes in our PL computation, such as Vietoris-Rips complex (VC) and Alpha complex (AC). Both complexes are used to characterize the first order interactions and higher order patterns respectively. To capture and characterize different types of atomatom interactions, we generate the PL based on different atom subsets by selecting one type of atom in the mutation site and one other atom type in the rest of the protein. Different types of atom-atom interactions characterize the different interactions in proteins. For example, interactions generated from carbon atoms are associated with hydrophobic interactions. Similarly, interactions between nitrogen and oxygen atoms correlate to hydrophilic interactions and/or hydrogen bonds. Both types of interactions are illustrated in Figure 6. Interactive PLs have the capability to unveil additional details about bonding interactions and offer a fresh and distinct representation of molecular interactions in proteins. For zero dimensions, we consider both the harmonic spectra and non-harmonic spectra information for each persistent Laplacian. Filtration using Rips complex with DI distance is used. The 0-dimensional PL features are generated from 0A to 6 ˚ A with 0.5 ˚ A gridsize. For the ˚ non-harmonic spectra information, we count the number of non-harmonic spectra and calculate seven statistical values of non-harmonic spectra such as sum, minimum, maximum, mean, standard deviation, variance and the sum of eigenvalues squared. This generates eight statistical values for each of the nine atomic pairs. Therefore, the dimension of 0-dimensional PL features for a protein is 72. In total, the 0-dimensional PL-based feature size after concatenating features at different dimensions for wild type and mutant is 1872. For one or two dimensions, we perform the filtration using Alpha complex with the DE distance. The limited number of atoms in the local protein structure can create only a few high-dimensional simplexes, resulting in minimal alterations in shape. As a result, it suffice to consider features from only harmonic spectra of persistent Laplacians by coding the topological invariants for the high-dimensional interactions. Using GUDHI[51], the persistence of the harmonic spectra can be represented by persistent barcodes. The topological feature vectors are generated by computing the statistics of bar lengths, births and deaths. Bars shorter than 0.1A are excluded as they do not exhibit any clear physical meaning. The remaining bars are ˚ then used for computing the statistics: (1) sum, maximum and mean for lengths of bars; (2) minimum and maximum for the birth values of bars; (3) minimum and maximum for the death values of bars. Each set of point clouds leads to a seven-dimensional vector. These features are calculated on nine single atomic pairs and one heavy atom pair. The dimension of oneand two-dimensional PL feature vectors for a protein is 140. In total, the higher-dimensional PL-based feature size after concatenating features at different dimensions for wild type, mutant and their difference is 420. 5.3 Persistent Homology Persistent homology is part of the harmonic spectra of PL. The homology groups in PH illustrate the persistence of topological invariants, hence providing the harmonic spectral information in PL. The site- and element-specific PH features are generated in a similar way as compared to PL. Similar to PL, filtration construction is also employed to PH. For the zero dimension, the filtration parameter can be discretized into several equally spaced bins, namely [0, 0.5], (0.5, 1], · · · , (5.5, 6]A. The death value of the bars are summed in each bin resulting in 12 ˚ ×18 features. For each bin, we count the numbers of persistent bars, resulting in a nine-dimensional vector for each point cloud. Similarly, this is performed for each of the nine single atomic pairs. Hence, the dimension of PH features for a protein is 216. For one or two dimensions, the identical featurization from the statistics of persistent bars in PH is used. The PH embedding combines features at different dimensions as described above and concatenated for wild type, mutant and their difference, resulting in a 648-dimensional vector 5.4 Transformer Features Recently, we have seen significant advancements in modelling protein properties using largescale protein transformer models trained on hundreds of millions of sequences. These models, like ESM [33] (evolutionary scale modeling) and ProtTrans[34, 35], have demonstrated impressive performance. Moreover, hybrid fine-tuning approaches that leverage both local and global evolutionary data have proven to enhance these models even further. For instance, eUniRep is an improved LSTM-based UniRep model achieved through fine-tuning with knowledge extracted from local multiple sequence alignments (MSAs). Additionally, the ESM model can be fine-tuned using either downstream task data or local MSAs. In our research, we employed the ESM-1b transformer, a model that falls under the transformer architecture. This particular variant was trained on a dataset of 250 million sequences using a masked filling procedure and boasts an architecture comprising 34 layers with a whopping 650 million parameters. The ESM transformer’s primary role in our work was to generate sequence embeddings. At each layer of the ESM model, it encoded a sequence of length L into a matrix sized at 1,280×L, excluding the start and terminal tokens. For our study, we utilized the sequence representation derived from the final (34th) layer and computed the average along the sequence length axis, resulting in a 1,280-component vector. 5.5 Performance Metrics PPV and NPV assesses the true positive and true negative proportion of the predicted results for each solubility class. PPV and NPV are computed based on TP, TN, FP and FN which represents the true positive, true negative, false positive and false negative values for each solubility class. For each solubility class, PPV and NPV can be computed by: This paper is under CC 4.0 license. available on arxiv

Interrogating TopLabGBT's Computational Framework

Authors: (1) JunJie Wee, Department of Mathematics, Michigan State University; (2) Jiahui Chen, Department of Mathematical Sciences, University of Arkansas; (3) Kelin Xia, Division of Mathematical Sciences, School of Physical and Mathematical Sciences Nanyang Technological University & xiakelin@ntu.edu.sg; (4)Guo-Wei Wei, 1Department of Mathematics, Michigan State University, Department of Biochemistry and Molecular Biology, Michigan State University, Department of Electrical and Computer Engineering, Michigan State University & xiakelin@ntu.edu.sg. Table of Links Abstract & Introduction Results Discussion Conclusion Materials and Methods Software and resources, Code and Data Availability Supporting Information, Acknowledgments & References 4 Conclusion In the multifaceted quest to understand mutation-induced solubility changes, various scientific domains including quantum mechanics, molecular mechanics, biochemistry, biophysics, and molecular biology have made significant contributions. Despite these concerted efforts, state of-art models have limitations, as evidenced by their normalized CPR value of 0.656 even after employing feature selection methods. Persistent homology (PH) has emerged as a powerful tool for capturing the complexity of biomolecular structures and has achieved noteworthy success in drug discovery applications. However, its inability to capture the nuances of homotopic shape evolution, crucial for delineating molecular interactions in proteins, presents a critical shortcoming. Our study introduces TopLapGBT, a novel model that integrates persistent Laplacian (PL) features with pretrained transformer features, thereby bridging the gap in capturing both topology and homotopic shape evolution. This innovative fusion leads to significant advancements in classification performance. Specifically, TopLapGBT achieves normalized CPR and GC2 scores of 0.688 and 0.361, respectively, marking improvements of 4.88% and 15.71% over the state-of-the-art PON-Sol2. These findings are further corroborated by an independent blind test, where TopLapGBT continues to outperform existing models. In summary, our proposed TopLapGBT model not only achieves superior performance over existing state-of-the-art methods but also introduces a more nuanced approach for the classification of protein solubility changes upon mutation. These results underscore the transformative potential of integrating geometric and topological features with machine learning in advancing the field of molecular biology. This paper is under CC 4.0 license. available on arxiv

TopLapGBT: Bridging Gaps in Protein Solubility Prediction with Cutting-Edge Fusion

Authors: (1) JunJie Wee, Department of Mathematics, Michigan State University; (2) Jiahui Chen, Department of Mathematical Sciences, University of Arkansas; (3) Kelin Xia, Division of Mathematical Sciences, School of Physical and Mathematical Sciences Nanyang Technological University & xiakelin@ntu.edu.sg; (4)Guo-Wei Wei, 1Department of Mathematics, Michigan State University, Department of Biochemistry and Molecular Biology, Michigan State University, Department of Electrical and Computer Engineering, Michigan State University & xiakelin@ntu.edu.sg. Table of Links Abstract & Introduction Results Discussion Conclusion Materials and Methods Software and resources, Code and Data Availability Supporting Information, Acknowledgments & References 3 Discussion The performance of machine learning models generally relies on the nature of the input features. In our model, the PL-based features depend on one main element which is the quality of the protein structures from AlphaFold 2 (AF2). The quality of AF2 structures are crucial in determining the performance of TopLapGBT. Recently, AF2 structures have been reported to achieve comparable performance to nuclear magnetic resonance (NMR) structures while ensemble methods can be used to enhance the performance by combining multiple NMR structures [1]. This allows AF2 structures to serve as a practical substitute for experimental structural data. Although AF2 structures are not as reliable as X-ray structures, the fusion of sequencebased pre-trained transformer features and PL-based features provides robust featurization even for low quality AF2 structural data. PL elucidates the precise mutation geometry and topology, while sequence-based pre-trained transformer features capture evolutionary patterns from an extensive sequence library. This synergy holds significance and can be applied to a diverse range of other challenges in the field of biomolecular research. For the rest of this section, we analyze the model’s performance based on the region of the mutations and the type of mutations. We also discuss the performance of different feature types using the Residue-Similarity plots. Table 2: Performance of TopLapGBT with existing state-of-the-art models on the independent blind test classification. The negative solubility samples are denoted as ”-” whereas the positive solubility change samples are denoted as ”+”. The samples with no solubility change are denoted as ”N”. Performance metrics include the positive predicted values (PPV), negative predicted values (NPV), sensitivity, specificity, correct prediction ratio (CPR) and generalised correlation (GC2 ). PPV refers to the proportions of positive predictions for each solubility class while NPV refers to the proportions of negative predictions for each solubility class. CPR calculates the percentage of correctly classified samples while GC2 measures the correlation coefficient of the classification. All normalized metrics are also reported. For each metric, the first value is without normalization while the second one is with normalization. 3.1 Performance analysis based on different mutation regions To delve deeper into the model’s performance, we categorize mutation samples based on their structural regions: interior and surface, as depicted in Figure 2 pre- and post-mutations. These regions are defined by their relative accessible solvent area (rASA), using a cutoff value c. A residue at the mutation site is classified as buried or interior if its rASA falls below this cutoff. While the discrete nature of c initially raised concerns, given that amino acids have a continuous exposure profile, empirical analyses on databases from organisms like Escherichia coli, Saccharomyces cerevisiae, and Homo sapiens have shown that an optimal rASA cutoff of approximately 25% is effective for distinguishing between surface and interior residues [41]. In our analysis, we apply this framework to identify surface and interior residues in the solubility dataset. We observe that some mutation sites undergo a regional transition, moving from one structural domain to another, consequent to the mutation. To gain nuanced insights into TopLapGBT’s performance, we segment the results according to the mutation’s structural location within the protein. We present these segmentations as heatmap plots that delineate both mutation regions and amino acid types. Structural regions are defined based on relative accessible surface area (rASA) [41]. By categorizing residues as either interior or surface, we can examine the influence of continuous amino acid exposure on solubility change classification post-mutation. Figure 2(b) displays accuracy scores for four types of mutations: interior-interior, interior-surface, surface-interior, and surface-surface. TopLapGBT attains an average accuracy score of 0.770 across these categories. Extended data in Figure S1 further breaks down accuracy scores for all 20 distinct amino acids within each region-pair, revealing variations in residue-residue pair performance. 3.2 Performance analysis based on different mutation types Switching focus to mutation types, our model’s capability in classifying solubility changes also merits exploration across the 20 distinct amino acid types in the dataset. In addition to this, we subgroup amino acids as charged, polar, hydrophobic, or special case. Table S1 enumerates the sample counts for each mutation group pair. Figure 3(a) displays accuracy scores for each mutation group pair, while Figure 3(b) shows scores for each amino acid pair. Notably, the special-charged and special-polar groups register the highest accuracy, whereas the polarhydrophobic and polar-special groups underperform. One plausible reason could be the inherent complexity in accurately classifying mutations with non-negative solubility changes. It’s worth noting that PON-Sol2 employed a two-layer classifier to improve classification [11]. Our results indicate that TopLapGBT surpasses the performance of this two-layer system. 3.3 Feature analysis based on Residue-Similarity plots The Residue Similarity Index (RSI) serves as a potent metric for evaluating the efficacy of dimensionality reduction in both clustering and classification contexts [43]. RSI has proven its value in generating classification accuracy scores that align well with supervised methods in single-cell typing. When applied to our solubility change dataset, Residue-Similarity (R-S) plots can be constructed to scrutinize how the Residue Index (RI) and Similarity Index (SI) may indicate the quality of cluster separation. Figure 4 juxtaposes the R-S plots derived from TopLapGBT against those from various feature sets utilized in model training. Across all visualizations, samples manifest a range of classification outcomes—both correct and incorrect—for each true label. However, a noteworthy observation is that Transformer-pretrain and persistent Laplacian-based features demon strate superior clustering attributes compared to auxiliary features. The high RI and SI scores for auxiliary features cause these data points to cluster near the upper regions of their respective sections. Despite this, the integrative use of all three feature types in TopLapGBT results in appreciable clustering performance, corroborated by the CPR metrics obtained in 10-fold cross-validation. To solidify the rationale behind adopting robust supervised classifiers like TopLapGBT, we contrast the R-S plots with UMAP visualizations (shown in Figure S2). It becomes evident that UMAP plots fail to form clusters that are as distinct as those observed in R-S plots, thereby reinforcing the need for a specialized approach to classify mutation samples effectively. The impetus for utilizing structure-based features stems from the multifaceted relationship that exists among protein sequence, structure, and solubility. Factors such as hydrophobicity, charge distribution, and intermolecular interactions contribute to the complexity of protein solubility. Traditional prediction methods, which often rely on empirical rules or rudimentary descriptors, fall short in capturing this intricate molecular interplay. By employing advanced mathematical techniques like persistent Laplacian (PL) coupled with machine learning algorithms, we can decipher the complex patterns and relationships embedded within protein sequences and structures. Persistent Laplacian, in particular, provides a robust mathematical representation that captures both the topological and homotopic evolution of protein structures. Furthermore, machine learning models rooted in advanced mathematics offer several advantages for classifying changes in protein solubility. These models are well-suited for handling high-dimensional and complex data sets, such as those involving protein sequences and structures. They are also capable of learning non-linear relationships and capturing nuanced dependencies that are often overlooked by traditional linear models. Importantly, these advanced models can adeptly manage class-imbalanced datasets, which are commonly encountered in protein solubility studies. This paper is under CC 4.0 license. available on arxiv

Unveiling Protein Solubility Patterns: Insights from TopLapGBT's Performance Analysis

Authors: (1) JunJie Wee, Department of Mathematics, Michigan State University; (2) Jiahui Chen, Department of Mathematical Sciences, University of Arkansas; (3) Kelin Xia, Division of Mathematical Sciences, School of Physical and Mathematical Sciences Nanyang Technological University & xiakelin@ntu.edu.sg; (4)Guo-Wei Wei, 1Department of Mathematics, Michigan State University, Department of Biochemistry and Molecular Biology, Michigan State University, Department of Electrical and Computer Engineering, Michigan State University & xiakelin@ntu.edu.sg. Table of Links Abstract & Introduction Results Discussion Conclusion Materials and Methods Software and resources, Code and Data Availability Supporting Information, Acknowledgments & References 2 Results 2.1 Overview of TopLapGBT TopLapGBT integrates both structure-based and sequence-based features, derived from protein structures and sequences respectively, into a unified model. Our architecture comprises three distinct embedding modules: persistent Laplacian-based embeddings, sequence-based embeddings, and auxiliary feature embeddings, all of which feed into an ensemble classifier as depicted in Figure 1. In the persistent Laplacian-based feature embedding module, we employ persistent Laplacian techniques to generate features that encapsulate the structural attributes of proteins both pre- and post-mutation. This approach is particularly effective in capturing the structural alterations induced by mutations within the localized neighborhoods of the mutation sites. Mathematically, the persistent Laplacian builds a sequence of simplicial complexes through a filtration process, thereby characterizing atom-atom interactions across multiple scales (details in the Methods section). In the sequence-based feature embedding module, a pre-trained transformer model generates latent feature vectors extracted from protein sequences. Specifically, the transformer model used here is a 650M-parameter protein language model, trained on a corpus of 250M protein sequences spanning multiple organisms [39]. Finally, the auxiliary feature embedding module incorporates a variety of attributes such as surface area, partial charge, pKa shifts, solvation free energy, and secondary structural information, synthesized from both protein sequences and structures. These three distinct sets of feature embeddings are subsequently concatenated to produce a comprehensive feature vector. This vector is then fed into a gradient-boosting tree classifier to categorize the mutation-induced samples. 2.2 Performance of TopLapGBT on PON-Sol2 dataset In our study, we utilize the dataset employed by PON-Sol2 as detailed in [11]. The dataset is comprised of 6,328 mutation samples, originating from 77 distinct proteins. These samples are categorized into three labels: decrease in solubility, increase in solubility, and no change in solubility. Specifically, the dataset contains 3,136 samples demonstrating a decrease in solubility, 1,026 samples showing an increase, and 2,166 samples with no change. Notably, the dataset exhibits a class imbalance, with a ratio of 1 : 0.69 : 0.34, indicating a bias towards samples that exhibit a decrease in solubility. To assess the performance of our model, we initially carry out a random 10-fold cross-validation on the dataset. Subsequently, an independent blind test prediction is executed to provide further validation of the model’s efficacy. In Table 1, we present a comparative analysis of the performance of existing classifiers by PON-Sol [5] and PON-Sol2 [11] against our proposed model, TopLapGBT, using 10-fold cross-validation. It should be noted that PON-Sol2 incorporates feature selection techniques such as recursive feature elimination (RFE). To provide a robust assessment of TopLapGBT’s performance, we conduct 10 repeated runs, and the mean values of these runs are reported to account for any randomness in the model’s output. Performance evaluation of our model, TopLapGBT, is conducted using a range of metrics, including Positive Predictive Value (PPV), Negative Predictive Value (NPV), Sensitivity, Specificity, Correct Prediction Ratio (CPR), and Generalized Squared Correlation (GC2 ). PPV and NPV quantify the proportions of correct positive and negative predictions for each solubility class, respectively. Given that we are dealing with a K-class problem with three distinct solubility classes, CPR and GC2 are particularly relevant for providing a holistic view of the model’s performance [40]. Specifically, CPR measures the overall accuracy of the model, while GC2 quantifies the correlation coefficient of the classification, ranging from 0 to 1. Larger values for these metrics denote better performance. Importantly, due to the class imbalance in the number of mutation samples across the categories, all performance metrics are normalized to ensure a robust and reliable evaluation of the model’s efficacy (further details are elaborated in the Methods section). Table 1: Comparison of performance metrics between TopLapGBT and both single layer and double layer classifiers of PON-Sol2 in the 10-fold crossvalidation. The negative solubility samples are denoted as ”-” whereas the positive solubility change samples are denoted as ”+”. The samples with no solubility change are denoted as ”N”. Performance metrics include the positive predicted values (PPV), negative predicted values (NPV), sensitivity, specificity, correct prediction ratio (CPR) and generalised correlation (GC2 ). PPV refers to the proportions of positive predictions for each solubility class while NPV refers to the proportions of negative predictions for each solubility class. CPR calculates the percentage of correctly classified samples while GC2 measures the correlation coefficient of the classification. All normalized metrics are also reported. For each metric, the first value is without normalization while the second one is with normalization. The proposed model, TopLapGBT, demonstrates significant performance gains over existing featurization methods in PON-Sol2 across all evaluation metrics [11]. Specifically, normalized CPR and GC2 scores of TopLapGBT stand at 0.688 and 0.361, marking improvements of 4.88% and 15.71% over PON-Sol2, respectively. These gains underscore the merit of incorporating both structure-based and sequence-based features into the model. To elucidate the contribution of Persistent Laplacian (PL)-based features, we also present a comparative analysis with our TopGBT model in Table 1. The TopGBT model utilizes persistent homology-based embeddings alongside auxiliary and pre-trained transformer features. While TopGBT still outperforms all existing PON-Sol2 models, the incorporation of PL-based features in TopLapGBT leads to an incremental improvement of 1% and 2% in CPR and GC2 metrics, respectively. This validates our approach of leveraging Persistent Laplacian to comprehensively capture both the topological and homotopic nuances in the evolution of protein structures. 2.3 Performance of TopLapGBT on independent test set To robustly assess the performance of TopLapGBT, we subjected it to an independent test using the same dataset employed by PON-Sol2 [11]. In this validation, TopLapGBT consistently outperformed all five existing models, as evidenced in Table 2. Specifically, TopLapGBT registers a normalized CPR of 0.564 and a normalized GC2 of 0.185, surpassing PON-Sol2 by 3.49% and 17.83%, respectively. Relative to TopGBT, the inclusion of PL-based features in TopLapGBT yielded incremental gains in both CPR and GC2 metrics, thereby further substantiating the utility of Persistent Laplacian in capturing the homotopic shape evolution within protein structures. This paper is under CC 4.0 license. available on arxiv

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