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  • Open Access

Independent Training Trajectories Learn the Same Interaction Parameters in Coarse-Grained Protein Models

Marcos Lequerica-Mateos1, Jonathan Martin2, José N. Onuchic3,4,*, Faruck Morcos2,5,†, and Ivan Coluzza3,6,7,8,‡

  • *Contact author: jonuchic@rice.edu
  • †Contact author: faruckm@utdallas.edu
  • ‡Contact author: ic33@rice.edu
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PRX Life 4, 043010 – Published 4 November, 2026

DOI: https://doi.org/10.1103/dmcv-sx8q

Abstract

This work introduces a self-consistent training framework for learning residue–residue interaction parameters in matrix-based coarse-grained protein models. By combining direct coupling analysis with a physics-based design model, we construct a feedback loop in which the interaction matrix is iteratively updated from sequences generated by the model itself. We provide a mathematical foundation for this behavior by showing that, for a given iterative map satisfying explicit contact-locality and spectral assumptions, the training process is a contraction, guaranteeing convergence to a unique fixed point independent of the initial interaction matrix. This fixed-point property is general for coarse-grained models in which the residue-residue energy is represented by a distance-dependent term modulated by a type-dependent coupling matrix, while the learned interaction parameters remain specific to the underlying model physics. We then examine the empirical consequence of this framework in Caterpillar, a coarse-grained potential for folding and design. Independent training trajectories, defined by different target structures and stochastic sampling, converge to statistically equivalent interaction matrices. Each trajectory recovers the full interaction solution without requiring ensemble averaging, and the averaged trajectory reaches the same result. The resulting framework provides a link between sequence statistics and physically modeled interaction energies. Applied to protein design, it yields interaction matrices associated with improved structural fidelity and accurate contact prediction, while clarifying how learned sequence patterns relate to the interactions that stabilize protein structure.

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References (69)

  1. B. I. Dahiyat and S. L. Mayo, De novo protein design: Fully automated sequence selection, Science 278, 82 (1997).
  2. B. Kuhlman, G. Dantas, G. C. Ireton, G. Varani, B. L. Stoddard, and D. Baker, Design of a novel globular protein fold with atomic-level accuracy, Science 302, 1364 (2003).
  3. G. L. Butterfoss and B. Kuhlman, Computer-based design of novel protein structures, Annu. Rev. Biophys. Biomol. Struct. 35, 49 (2006).
  4. P.-S. Huang, S. E. Boyken, and D. Baker, The coming of age of de novo protein design, Nature (London) 537, 320 (2016).
  5. E. Marcos and D. Silva, Essentials of de novo protein design: Methods and applications, WIREs Comput. Mol. Sci. 8, e1374 (2018).
  6. L. Sellés Vidal, M. Isalan, J. T. Heap, and R. Ledesma-Amaro, A primer to directed evolution: Current methodologies and future directions, RSC Chem. Biol. 4, 271 (2023).
  7. K. Yue and K. A. Dill, Inverse protein folding problem: Designing polymer sequences, Proc. Natl. Acad. Sci. USA 89, 4163 (1992).
  8. E. I. Shakhnovich and A. M. Gutin, Formation of unique structure in polypeptide chains: Theoretical investigation with the aid of a replica approach, Biophys. Chem. 34, 187 (1989).
  9. J. N. Onuchic and P. G. Wolynes, Theory of protein folding, Curr. Opin. Struct. Biol. 14, 70 (2004).
  10. W. G. Noid, Perspective: Coarse-grained models for biomolecular systems, J. Chem. Phys. 139, 90901 (2013).
  11. J. Jumper, R. Evans, A. Pritzel, T. Green, M. Figurnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. Žídek, A. Potapenko, A. Bridgland, C. Meyer, S. A. A. Kohl, A. J. Ballard, A. Cowie, B. Romera-Paredes, S. Nikolov, R. Jain, J. Adler, T. Back, et al., Highly accurate protein structure prediction with AlphaFold, Nature (London) 596, 583 (2021).
  12. N. Anand, R. R. Eguchi, I. I. Mathews, C. P. Perez, A. Derry, R. B. Altman, and P.-S. Huang, Protein sequence design with a learned potential, Nat. Commun. 13, 746 (2022).
  13. S. Chaudhury, S. Lyskov, and J. Gray, PyRosetta: A script-based interface for implementing molecular modeling algorithms using Rosetta, Bioinformatics 26, 689 (2010).
  14. T. Zheng, A. Rondina, G. Micklem, and P. Liò, Challenges and guidelines in deep generative protein design: Four case studies, in Proceedings of the ICML 2025 Workshop on Multi-modal Foundation Models and Large Language Models for Life Sciences, Vancouver, Canada (2025), arXiv:2411.18568.
  15. R. R. Stein, D. S. Marks, and C. Sander, Inferring pairwise interactions from biological data using maximum-entropy probability models, PLoS Comput. Biol. 11, e1004182 (2015).
  16. S. Cocco, C. Feinauer, M. Figliuzzi, R. Monasson, and M. Weigt, Inverse statistical physics of protein sequences: A key issues review, Rep. Prog. Phys. 81, 032601 (2018).
  17. R. M. Levy, A. Haldane, and W. F. Flynn, Potts Hamiltonian models of protein co-variation, free energy landscapes, and evolutionary fitness, Curr. Opin. Struct. Biol. 43, 55 (2017).
  18. C. Malbranke, D. Bikard, S. Cocco, R. Monasson, and J. Tubiana, Machine learning for evolutionary-based and physics-inspired protein design: Current and future synergies, Curr. Opin. Struct. Biol. 80, 102571 (2023).
  19. P. G. Wolynes, As simple as can be? Nat. Struct. Biol. 4, 871 (1997).
  20. J. Dauparas, I. Anishchenko, N. Bennett, H. Bai, R. J. Ragotte, L. F. Milles, B. I. M. Wicky, A. Courbet, R. J. de Haas, N. Bethel, P. J. Y. Leung, T. F. Huddy, S. Pellock, D. Tischer, F. Chan, B. Koepnick, H. Nguyen, A. Kang, B. Sankaran, A. K. Bera, et al., Others robust deep protein sequence design using ProteinMPNN, Science 378, 49 (2022).
  21. H. Khakzad, I. Igashov, A. Schneuing, C. Goverde, M. Bronstein, and B. Correia, A new age in protein design empowered by deep learning, Cell Syst. 14, 925 (2023).
  22. U. Göbel, C. Sander, R. Schneider, and A. Valencia, Correlated mutations and residue contacts in proteins, Proteins: Struct. Funct. Bioinf. 18, 309 (1994).
  23. M. Weigt, R. A. White, H. Szurmant, J. A. Hoch, and T. Hwa, Identification of direct residue contacts in protein–protein interaction by message passing, Proc. Natl. Acad. Sci. USA 106, 67 (2009).
  24. F. Morcos, A. Pagnani, B. Lunt, A. Bertolino, D. S. Marks, C. Sander, R. Zecchina, J. N. Onuchic, T. Hwa, and M. Weigt, Direct-coupling analysis of residue coevolution captures native contacts across many protein families, Proc. Natl. Acad. Sci. USA 108, E1293 (2011).
  25. D. S. Marks, L. J. Colwell, R. Sheridan, T. A. Hopf, A. Pagnani, R. Zecchina, and C. Sander, Protein 3D structure computed from evolutionary sequence variation, PLoS ONE 6, e28766 (2011).
  26. A. S. Lapedes, B. G. Giraud, and C. Jarzynski, Using sequence alignments to predict protein structure and stability with high accuracy, Los Alamos Report No. LA-UR-02-4481 (2002), arXiv:1207.2484.
  27. M. Ekeberg, C. Lövkvist, Y. Y. Lan, M. Weigt, and E. Aurell, Improved contact prediction in proteins: Using pseudolikelihoods to infer Potts models, Phys. Rev. E 87, 012707 (2013).
  28. H. Kamisetty, S. Ovchinnikov, and D. Baker, Assessing the utility of coevolution-based residue–residue contact predictions in a sequence- and structure-rich era, Proc. Natl. Acad. Sci. USA 110, 15674 (2013).
  29. H. Jacquin, A. Gilson, E. I. Shakhnovich, S. Cocco, and R. Monasson, Test and benchmark of statistical inverse methods for protein structure prediction, PLoS Comput. Biol. 12, e1004889 (2016).
  30. J. Martin, M. Lequerica Mateos, J. N. Onuchic, I. Coluzza, and F. Morcos, Machine learning in biological physics: From biomolecular prediction to design, Proc. Natl. Acad. Sci. USA 121, e2311807121 (2024).
  31. F. M. Ortega, F. Hossain, V. V. Volobouev, G. Meloni, H. Torabifard, and F. Morcos, Generative landscapes and dynamics to design functional multidomain artificial transmembrane transporters, ACS Cent. Sci. 11, 1452 (2025).
  32. S. W. Lockless and R. Ranganathan, Evolutionarily conserved pathways of energetic connectivity in protein families, Science 286, 295 (1999).
  33. N. Halabi, O. Rivoire, S. Leibler, and R. Ranganathan, Protein sectors: Evolutionary units of three-dimensional structure, Cell 138, 774 (2009).
  34. M. Figliuzzi, H. Jacquier, A. Schug, O. Tenaillon, and M. Weigt, Coevolutionary landscape inference and the context-dependence of mutations in beta-lactamase TEM-1, Mol. Biol. Evol. 33, 268 (2016).
  35. T. A. Hopf, J. B. Ingraham, F. J. Poelwijk, C. P. I. Schärfe, M. Springer, C. Sander, and D. S. Marks, Mutation effects predicted from sequence co-variation, Nat. Biotechnol. 35, 128 (2017).
  36. W. P. Russ, M. Figliuzzi, C. Stocker, P. Barrat-Charlaix, M. Socolich, P. Kast, D. Hilvert, R. Monasson, S. Cocco, M. Weigt, and R. Ranganathan, An evolution-based model for designing chorismate mutase enzymes, Science 369, 440 (2020).
  37. V. Tozzini, Coarse-grained models for proteins, Curr. Opin. Struct. Biol. 15, 144 (2005).
  38. C. Clementi, Coarse-grained models of protein folding: Toy models or predictive tools? Curr. Opin. Struct. Biol. 18, 10 (2008).
  39. S. Kmiecik, D. Gront, M. Kolinski, L. Wieteska, A. E. Dawid, and A. Kolinski, Coarse-grained protein models and their applications, Chem. Rev. 116, 7898 (2016).
  40. A. Davtyan, N. P. Schafer, W. Zheng, C. Clementi, P. G. Wolynes, and G. A. Papoian, AWSEM-MD: Protein structure prediction using coarse-grained physical potentials and bioinformatically based local structure biasing, J. Phys. Chem. B 116, 8494 (2012).
  41. J. K. Noel, F. Morcos, and J. N. Onuchic, Sequence co-evolutionary information is a natural partner to minimally-frustrated models of biomolecular dynamics, F1000Research 5, 106 (2016).
  42. I. Coluzza, Transferable coarse-grained potential for de novo protein folding and design, PLoS ONE 9, e112852 (2014).
  43. C. Cardelli, V. Bianco, L. Rovigatti, F. Nerattini, L. Tubiana, C. Dellago, and I. Coluzza, The role of directional interactions in the designability of generalized heteropolymers, Sci. Rep. 7, 4986 (2017).
  44. C. Cardelli, L. Tubiana, V. Bianco, F. Nerattini, C. Dellago, and I. Coluzza, Heteropolymer design and folding of arbitrary topologies reveals an unexpected role of alphabet size on the knot population, Macromolecules 51, 8346 (2018).
  45. C. Cardelli, F. Nerattini, L. Tubiana, V. Bianco, C. Dellago, F. Sciortino, and I. Coluzza, General methodology to identify the minimum alphabet size for heteropolymer design, Adv. Theory Simul. 2, 1900031 (2019).
  46. I. Coluzza, P. D. J. van Oostrum, B. Capone, E. Reimhult, and C. Dellago, Design and folding of colloidal patchy polymers, Soft Matter 9, 938 (2012)
  47. I. Coluzza, P. D. J. van Oostrum, B. Capone, E. Reimhult, and C. Dellago, Sequence controlled self-knotting colloidal patchy polymers, Phys. Rev. Lett. 110, 075501 (2013).
  48. I. Coluzza, A coarse-grained approach to protein design: Learning from design to understand folding, PLoS ONE 6, e20853 (2011).
  49. See Supplemental Material at http://link.aps.org/supplemental/10.1103/dmcv-sx8q for model and sampling details, mathematical derivations, and supplementary analyses of matrix convergence and protein folding. It also provides the detailed mathematical and computational framework underpinning the study, including the Caterpillar Hamiltonian, sequence-sampling protocol, projection of position-dependent mfDCA couplings onto the amino-acid interaction matrix, and the full Jacobian derivation. It states the contact-locality, spectral, and residual-localization assumptions used to obtain sparse-graph inverse-covariance decay, compact-globule sequence localization, and the resulting contraction bound. It also contains the matrix-alignment analysis, natural-sequence construction, folding free-energy landscapes, contact-map recovery, and DRMSD comparisons across refinement pathways, which includes Refs.  [61, 62, 63, 64, 65, 66, 67, 68, 69].
  50. M. Lequerica-Mateos, J. Martin, J. N. Onuchic, F. Morcos, and I. Coluzza, Designing proteins: Mimicking natural protein sequence heterogeneity, J. Chem. Phys. 161, 194102 (2024).
  51. Z. Bagci, R. L. Jernigan, and I. Bahar, Residue coordination in proteins conforms to the closest packing of spheres, Polymer 43, 451 (2002).
  52. A. R. Atilgan, P. Akan, and C. Baysal, Small-world communication of residues and significance for protein dynamics, Biophys. J. 86, 85 (2004).
  53. S. Jaffard, Propriétés des matrices ‘bien localisées’ près de leur diagonale et quelques applications, Ann. Inst. Henri Poincaré C Anal. Non Linéaire 7, 461 (1990).
  54. S. Demko, Inverses of band matrices and local convergence of spline projections, SIAM J. Numer. Anal. 14, 616 (1977).
  55. S. Demko, W. F. Moss, and P. W. Smith, Decay rates for inverses of band matrices, Math. Comput. 43, 491 (1984).
  56. M. Benzi and N. Razouk, Decay bounds and O(n) algorithms for approximating functions of sparse matrices, Electron. Trans. Numer. Anal. 28, 16 (2007).
  57. A. Y. Grosberg, S. K. Nechaev, and E. I. Shakhnovich, The role of topological constraints in the kinetics of collapse of macromolecules, J. Physique 49, 2095 (1988).
  58. T. G. Dewey, Protein structure and polymer collapse, J. Chem. Phys. 98, 2250 (1993).
  59. M. V. Imakaev, K. M. Tchourine, S. K. Nechaev, and L. A. Mirny, Effects of topological constraints on globular polymers, Soft Matter 11, 665 (2015).
  60. E. P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67, 325 (1958).
  61. T. E. Creighton, Proteins: Structures and Molecular Properties, 2nd ed. (W. H. Freeman, New York, 1993).
  62. A. Irbäck, F. Sjunnesson, and S. Wallin, Three-helix-bundle protein in a Ramachandran model, Proc. Natl. Acad. Sci. USA 97, 13614 (2000).
  63. S. Miyazawa and R. L. Jernigan, Estimation of effective interresidue contact energies from protein crystal structures: Quasi-chemical approximation, Macromolecules 18, 534 (1985).
  64. J. Kyte and R. F. Doolittle, A simple method for displaying the hydropathic character of a protein, J. Mol. Biol. 157, 105 (1982).
  65. I. Coluzza and D. Frenkel, Virtual-move parallel tempering, ChemPhysChem 6, 1779 (2005).
  66. G. C. Boulougouris and D. Frenkel, Novel Monte Carlo scheme for systems with short-ranged interactions, J. Chem. Phys. 122, 244106 (2005).
  67. G. C. Boulougouris and D. Frenkel, Monte Carlo sampling of a Markov web, J. Chem. Theory Comput. 1, 389 (2005).
  68. R. A. Fisher, Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population, Biometrika 10, 507 (1915).
  69. D. T. Jones, S. M. Kandathil, and A. Valencia, High precision in protein contact prediction using fully convolutional neural networks and minimal sequence features, Bioinformatics 34, 3308 (2018).

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PRX Life 4, 043010– Published 4 November, 2026
  • Received 15 May 2026
  • Accepted 14 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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