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Discriminative Learning of Substitution Matrices and Gap Penalties for Pairwise Alignment of Biological Sequences

Authors Michał Aleksander Ciach , Elissavet Zacharopoulou , Michał Piotr Startek , Błażej Miasojedow , Panagiotis Alexiou

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Subject Classification

ACM Subject Classification
  • Applied computing → Molecular sequence analysis
Keywords
  • Sequence alignment
  • Substitution matrix
  • Logistic regression

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Abstract

Pairwise alignment scores are used to classify pairs of sequences in many areas of bioinformatics, including homology search, predicting interactions, or read mapping. The relative scores of different pairs strongly depend on the choice of a substitution matrix and gap penalties. However, current approaches for the estimation of these parameters typically describe patterns observed in a collection of ground-truth alignments instead of optimizing specifically for the task of classification. In this work, we present DiscrimAlign, a statistical model for discriminative learning of substitution matrices and gap penalties from a dataset of positive and negative pairs of unaligned DNA or amino acid sequences. The model links the alignment score of a sequence pair with the associated binary label through a logistic function and learns the parameters by likelihood maximization. We analyze theoretical properties of the model, derive and implement a learning procedure, study its performance in simulated experiments, and apply it to predict microRNA-target interactions. We show that sequence alignment with discriminative substitution matrices and gap penalties predicts the interactions comparably to black-box neural network classifiers while being more interpretable. An implementation of the model and reproducibility workflows are available at https://github.com/BioGeMT/DiscrimAlign.

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Michał Aleksander Ciach, Elissavet Zacharopoulou, Michał Piotr Startek, Błażej Miasojedow, and Panagiotis Alexiou. Discriminative Learning of Substitution Matrices and Gap Penalties for Pairwise Alignment of Biological Sequences. In 26th International Conference on Algorithms for Bioinformatics (WABI 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 390, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WABI.2026.17

Author Details

Michał Aleksander Ciach
  • Department of Applied Biomedical Science, Faculty of Health Sciences, University of Malta, Msida, Malta
Elissavet Zacharopoulou
  • Department of Applied Biomedical Science, Faculty of Health Sciences, University of Malta, Msida, Malta
Michał Piotr Startek
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Błażej Miasojedow
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Panagiotis Alexiou
  • Department of Applied Biomedical Science, Faculty of Health Sciences, University of Malta, Msida, Malta

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101244218. We acknowledge the BioGeMT Team (HORIZON-WIDERA-2022 Grant ID: 101086768).

Acknowledgements

We thank Mr Dave Galea for the help in running simulations and preparing figures.

Supplementary Materials

References

  1. Adelin Albert and John A Anderson. On the existence of maximum likelihood estimates in logistic regression models. Biometrika, 71(1):1-10, 1984. Google Scholar
  2. Stephen F Altschul. Amino acid substitution matrices from an information theoretic perspective. Journal of molecular biology, 219(3):555-565, 1991. Google Scholar
  3. Olivier Bastien, Sylvaine Roy, and Éric Maréchal. Construction of non-symmetric substitution matrices derived from proteomes with biased amino acid distributions. Comptes rendus. Biologies, 328(5):445-453, 2005. Google Scholar
  4. Amir Beck. First-order methods in optimization. SIAM, 2017. Google Scholar
  5. Kevin Brick and Elisabetta Pizzi. A novel series of compositionally biased substitution matrices for comparing plasmodium proteins. BMC bioinformatics, 9(1):236, 2008. URL: https://doi.org/10.1186/1471-2105-9-236.
  6. Celeste J Brown, Audra K Johnson, and Gary W Daughdrill. Comparing models of evolution for ordered and disordered proteins. Molecular biology and evolution, 27(3):609-621, 2010. Google Scholar
  7. Christiam Camacho, George Coulouris, Vahram Avagyan, Ning Ma, Jason Papadopoulos, Kevin Bealer, and Thomas L Madden. Blast+: architecture and applications. BMC bioinformatics, 10(1):421, 2009. URL: https://doi.org/10.1186/1471-2105-10-421.
  8. Michał Aleksander Ciach, Elissavet Zacharopoulou, Michał Piotr Startek, Błażej Miasojedow, and Panagiotis Alexiou. DiscrimAlign. Software, version 1.0., (visited on 2026-08-12). URL: https://github.com/BioGeMT/DiscrimAlign
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.27630
  9. Frank H Clarke. Generalized gradients and applications. Transactions of the American Mathematical Society, 205:247-262, 1975. Google Scholar
  10. Frank H Clarke, Yu S Ledyaev, Ronald J Stern, and RR Wolenski. Nonsmooth analysis and control theory. Springer, 1998. Google Scholar
  11. Peter JA Cock, Tiago Antao, Jeffrey T Chang, Brad A Chapman, Cymon J Cox, Andrew Dalke, Iddo Friedberg, Thomas Hamelryck, Frank Kauff, Bartek Wilczynski, et al. Biopython: freely available python tools for computational molecular biology and bioinformatics. Bioinformatics, 25(11):1422, 2009. Google Scholar
  12. Cuong Cao Dang, Vincent Lefort, Vinh Sy Le, Quang Si Le, and Olivier Gascuel. Replacementmatrix: a web server for maximum-likelihood estimation of amino acid replacement rate matrices. Bioinformatics, 27(19):2758-2760, 2011. URL: https://doi.org/10.1093/BIOINFORMATICS/BTR435.
  13. Aris Daniilidis and Dmitriy Drusvyatskiy. Pathological subgradient dynamics. SIAM Journal on Optimization, 30(2):1327-1338, 2020. URL: https://doi.org/10.1137/19M1298147.
  14. Damek Davis, Dmitriy Drusvyatskiy, Sham Kakade, and Jason D Lee. Stochastic subgradient method converges on tame functions. Foundations of computational mathematics, 20(1):119-154, 2020. URL: https://doi.org/10.1007/S10208-018-09409-5.
  15. Margaret Dayhoff, R Schwartz, and B Orcutt. A model of evolutionary change in proteins. Atlas of protein sequence and structure, 5:345-352, 1978. Google Scholar
  16. Richard Durbin, Sean R Eddy, Anders Krogh, and Graeme Mitchison. Biological sequence analysis: probabilistic models of proteins and nucleic acids. Cambridge university press, 1998. URL: https://doi.org/10.1017/CBO9780511790492.
  17. Robert C Edgar. Optimizing substitution matrix choice and gap parameters for sequence alignment. Bmc Bioinformatics, 10(1):396, 2009. URL: https://doi.org/10.1186/1471-2105-10-396.
  18. Robert C Edgar. Muscle5: High-accuracy alignment ensembles enable unbiased assessments of sequence homology and phylogeny. Nature communications, 13(1):6968, 2022. Google Scholar
  19. Martin C Frith. How sequence alignment scores correspond to probability models. Bioinformatics, 36(2):408-415, 2020. URL: https://doi.org/10.1093/BIOINFORMATICS/BTZ576.
  20. Martin C Frith, Michiaki Hamada, and Paul Horton. Parameters for accurate genome alignment. BMC bioinformatics, 11(1):80, 2010. URL: https://doi.org/10.1186/1471-2105-11-80.
  21. Renganayaki Govindarajan, Biji Christopher Leela, and Achuthsankar S Nair. Rblosum performs better than corblosum with lesser error per query. BMC Research Notes, 11(1):328, 2018. Google Scholar
  22. Richard Grantham. Amino acid difference formula to help explain protein evolution. science, 185(4154):862-864, 1974. Google Scholar
  23. Jerrold R Griggs, Phil Hanlon, Andrew M Odlyzko, and Michael S Waterman. On the number of alignments of k sequences. Graphs and Combinatorics, 6(2):133-146, 1990. URL: https://doi.org/10.1007/BF01787724.
  24. Dan Gusfield and Paul Stelling. Parametric and inverse-parametric sequence alignment with xparal. In Methods in enzymology, volume 266, pages 481-494. Elsevier, 1996. Google Scholar
  25. Michiaki Hamada, Yukiteru Ono, Kiyoshi Asai, and Martin C Frith. Training alignment parameters for arbitrary sequencers with last-train. Bioinformatics, 33(6):926-928, 2017. URL: https://doi.org/10.1093/BIOINFORMATICS/BTW742.
  26. Vaclav Hejret, Nandan Mysore Varadarajan, Eva Klimentova, Katarina Gresova, Ilektra-Chara Giassa, Stepanka Vanacova, and Panagiotis Alexiou. Analysis of chimeric reads characterises the diverse targetome of ago2-mediated regulation. Scientific Reports, 13(1):22895, 2023. Google Scholar
  27. Steven Henikoff and Jorja G Henikoff. Amino acid substitution matrices from protein blocks. Proceedings of the national academy of sciences, 89(22):10915-10919, 1992. Google Scholar
  28. Martin Hess, Frank Keul, Michael Goesele, and Kay Hamacher. Addressing inaccuracies in blosum computation improves homology search performance. Bmc Bioinformatics, 17(1):189, 2016. URL: https://doi.org/10.1186/S12859-016-1060-3.
  29. Reiner Horst and Nguyen V Thoai. Dc programming: overview. Journal of Optimization Theory and Applications, 103(1):1-43, 1999. Google Scholar
  30. Hyeonseo Hwang, Hee Ryung Chang, and Daehyun Baek. Determinants of functional microrna targeting. Molecules and cells, 46(1):21-32, 2023. Google Scholar
  31. Sumaiya Islam and Robert J Pantazes. Developing similarity matrices for antibody-protein binding interactions. Plos one, 18(10):e0293606, 2023. Google Scholar
  32. Kejue Jia and Robert L Jernigan. New amino acid substitution matrix brings sequence alignments into agreement with structure matches. Proteins: Structure, Function, and Bioinformatics, 89(6):671-682, 2021. Google Scholar
  33. Thorsten Joachims, Tamara Galor, and Ron Elber. Learning to align sequences: A maximum-margin approach. In New algorithms for macromolecular simulation, pages 57-69. Springer, 2005. Google Scholar
  34. Kaisa Joki, Adil M Bagirov, Napsu Karmitsa, Marko M Makela, and Sona Taheri. Double bundle method for finding clarke stationary points in nonsmooth dc programming. SIAM Journal on Optimization, 28(2):1892-1919, 2018. URL: https://doi.org/10.1137/16M1115733.
  35. David T Jones, William R Taylor, and Janet M Thornton. The rapid generation of mutation data matrices from protein sequences. Bioinformatics, 8(3):275-282, 1992. URL: https://doi.org/10.1093/BIOINFORMATICS/8.3.275.
  36. Samuel Karlin and Stephen F Altschul. Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes. Proceedings of the National Academy of Sciences, 87(6):2264-2268, 1990. Google Scholar
  37. Kazutaka Katoh, John Rozewicki, and Kazunori D Yamada. Mafft online service: multiple sequence alignment, interactive sequence choice and visualization. Briefings in bioinformatics, 20(4):1160-1166, 2019. URL: https://doi.org/10.1093/BIB/BBX108.
  38. John Kececioglu and Eagu Kim. Simple and fast inverse alignment. In Annual International Conference on Research in Computational Molecular Biology, pages 441-455. Springer, 2006. URL: https://doi.org/10.1007/11732990_37.
  39. Frank Keul, Martin Hess, Michael Goesele, and Kay Hamacher. Pfasum: a substitution matrix from pfam structural alignments. BMC bioinformatics, 18(1):293, 2017. URL: https://doi.org/10.1186/S12859-017-1703-Z.
  40. Eagu Kim and John Kececioglu. Learning scoring schemes for sequence alignment from partial examples. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 5(4):546-556, 2008. URL: https://doi.org/10.1109/TCBB.2008.57.
  41. Satoshi Koide, Keisuke Kawano, and Takuro Kutsuna. Neural edit operations for biological sequences. Advances in Neural Information Processing Systems, 31, 2018. Google Scholar
  42. Carolin Kosiol and Nick Goldman. Different versions of the dayhoff rate matrix. Molecular biology and evolution, 22(2):193-199, 2005. Google Scholar
  43. Jacek Krol, Inga Loedige, and Witold Filipowicz. The widespread regulation of microrna biogenesis, function and decay. Nature reviews genetics, 11(9):597-610, 2010. Google Scholar
  44. Igor B Kuznetsov. Protein sequence alignment with family-specific amino acid similarity matrices. BMC Research Notes, 4(1):296, 2011. Google Scholar
  45. Si Quang Le and Olivier Gascuel. An improved general amino acid replacement matrix. Molecular biology and evolution, 25(7):1307-1320, 2008. Google Scholar
  46. Claire Lemaitre, Aurélien Barré, Christine Citti, Florence Tardy, François Thiaucourt, Pascal Sirand-Pugnet, and Patricia Thébault. A novel substitution matrix fitted to the compositional bias in mollicutes improves the prediction of homologous relationships. BMC bioinformatics, 12(1):457, 2011. URL: https://doi.org/10.1186/1471-2105-12-457.
  47. Sean E McGeary, Kathy S Lin, Charlie Y Shi, Thy M Pham, Namita Bisaria, Gina M Kelley, and David P Bartel. The biochemical basis of microrna targeting efficacy. Science, 366(6472):eaav1741, 2019. Google Scholar
  48. Uros Midic, A Keith Dunker, and Zoran Obradovic. Protein sequence alignment and structural disorder: a substitution matrix for an extended alphabet. In Proceedings of the KDD-09 Workshop on Statistical and Relational Learning in Bioinformatics, pages 27-31, 2009. URL: https://doi.org/10.1145/1562090.1562096.
  49. James T Morton, Charlie EM Strauss, Robert Blackwell, Daniel Berenberg, Vladimir Gligorijevic, and Richard Bonneau. Protein structural alignments from sequence. BioRxiv, pages 2020-11, 2020. Google Scholar
  50. Tobias Müller, Rainer Spang, and Martin Vingron. Estimating amino acid substitution models: a comparison of dayhoff’s estimator, the resolvent approach and a maximum likelihood method. Molecular biology and evolution, 19(1):8-13, 2002. Google Scholar
  51. Tobias Müller and Martin Vingron. Modeling amino acid replacement. Journal of Computational Biology, 7(6):761-776, 2000. URL: https://doi.org/10.1089/10665270050514918.
  52. VN Novoseletsky, GA Armeev, and KV Shaitan. Construction and analysis of amino acid substitution matrices for optimal alignment of microbial rhodopsin sequences. Moscow University Biological Sciences Bulletin, 74(1):21-25, 2019. Google Scholar
  53. Evgenii Ofitserov, Vasily Tsvetkov, and Vadim Nazarov. Soft edit distance for differentiable comparison of symbolic sequences. arXiv preprint arXiv:1904.12562, 2019. URL: https://arxiv.org/abs/1904.12562.
  54. Lior Pachter and Bernd Sturmfels. Algebraic statistics for computational biology, volume 13. Cambridge university press, 2005. Google Scholar
  55. William R Pearson. Selecting the right similarity-scoring matrix. Current protocols in bioinformatics, 43(1):3-5, 2013. Google Scholar
  56. 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, 12:2825-2830, 2011. URL: https://doi.org/10.5555/1953048.2078195.
  57. Samantha Petti, Nicholas Bhattacharya, Roshan Rao, Justas Dauparas, Neil Thomas, Juannan Zhou, Alexander M Rush, Peter Koo, and Sergey Ovchinnikov. End-to-end learning of multiple sequence alignments with differentiable smith-waterman. Bioinformatics, 39(1):btac724, 2023. URL: https://doi.org/10.1093/BIOINFORMATICS/BTAC724.
  58. Valery Polyanovsky, Alexander Lifanov, Natalia Esipova, and Vladimir Tumanyan. The ranging of amino acids substitution matrices of various types in accordance with the alignment accuracy criterion. BMC bioinformatics, 21(Suppl 11):294, 2020. URL: https://doi.org/10.1186/S12859-020-03616-0.
  59. Predrag Radivojac, Zoran Obradovic, Celeste J Brown, and A Keith Dunker. Improving sequence alignments for intrinsically disordered proteins. In Biocomputing 2002, pages 589-600. World Scientific, 2001. Google Scholar
  60. Rodolfo Rios-Zertuche. Examples of pathological dynamics of the subgradient method for lipschitz path-differentiable functions. Mathematics of Operations Research, 47(4):3184-3206, 2022. URL: https://doi.org/10.1287/MOOR.2021.1241.
  61. Stephanie Sammut, Katarina Gresova, Dimosthenis Tzimotoudis, Eva Marsalkova, David Cechak, and Panagiotis Alexiou. mirbench: novel benchmark datasets for microrna binding site prediction that mitigate against prevalent microrna frequency class bias. Bioinformatics, 41(Supplement_1):i542-i551, 2025. URL: https://doi.org/10.1093/BIOINFORMATICS/BTAF233.
  62. Eric W Sayers, Jeffrey Beck, Evan E Bolton, J Rodney Brister, Jessica Chan, Ryan Connor, Michael Feldgarden, Anna M Fine, Kathryn Funk, Jinna Hoffman, et al. Database resources of the national center for biotechnology information in 2025. Nucleic acids research, 53(D1):D20-D29, 2025. Google Scholar
  63. David J States, Warren Gish, and Stephen F Altschul. Improved sensitivity of nucleic acid database searches using application-specific scoring matrices. Methods, 3(1):66-70, 1991. Google Scholar
  64. Martin Steinegger and Johannes Söding. Mmseqs2 enables sensitive protein sequence searching for the analysis of massive data sets. Nature biotechnology, 35(11):1026-1028, 2017. Google Scholar
  65. Mark P Styczynski, Kyle L Jensen, Isidore Rigoutsos, and Gregory Stephanopoulos. Blosum62 miscalculations improve search performance. Nature biotechnology, 26(3):274-275, 2008. Google Scholar
  66. Fangting Sun, David Fernández-Baca, and Wei Yu. Inverse parametric sequence alignment. Journal of Algorithms, 53(1):36-54, 2004. URL: https://doi.org/10.1016/J.JALGOR.2004.04.008.
  67. Amlan Talukder, Xiaoman Li, and Haiyan Hu. Position-wise binding preference is important for mirna target site prediction. Bioinformatics, 36(12):3680-3686, 2020. URL: https://doi.org/10.1093/BIOINFORMATICS/BTAA195.
  68. Octavian Teodorescu, Tamara Galor, Jaroslaw Pillardy, and Ron Elber. Enriching the sequence substitution matrix by structural information. Proteins: Structure, Function, and Bioinformatics, 54(1):41-48, 2004. Google Scholar
  69. Rakesh Trivedi and Hampapathalu Adimurthy Nagarajaram. Substitution scoring matrices for proteins-an overview. Protein Science, 29(11):2150-2163, 2020. Google Scholar
  70. Simon Whelan and Nick Goldman. A general empirical model of protein evolution derived from multiple protein families using a maximum-likelihood approach. Molecular biology and evolution, 18(5):691-699, 2001. Google Scholar
  71. Alex J Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function. Journal of the American Mathematical Society, 9(4):1051-1094, 1996. Google Scholar
  72. Kazunori Yamada and Kentaro Tomii. Revisiting amino acid substitution matrices for identifying distantly related proteins. Bioinformatics, 30(3):317-325, 2014. URL: https://doi.org/10.1093/BIOINFORMATICS/BTT694.
  73. Yin Yao and Martin C Frith. Improved dna-versus-protein homology search for protein fossils. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 20(3):1691-1699, 2022. URL: https://doi.org/10.1109/TCBB.2022.3177855.
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