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Thermally Robust Cooperative Function in Mechanical Networks

Ben Pisanty1, Jovana A. Kim1, Andrea J. Liu1,2, and Sidney R. Nagel3

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PRX Life 4, 033009 – Published 31 July, 2026

DOI: https://doi.org/10.1103/6n32-9vq7

Abstract

Elastic networks can be tuned to exhibit complex mechanical responses and have been extensively used to study protein allostery, where a localized strain regulates the conformation at a distant site. We show that cooperative binding, where one site enhances the other's ability to function, can be improved by tuning for bidirectional allostery: a symmetric coupling where the strain propagated between the two sites is independent of which site is perturbed. We identify a crossover temperature above which functionality breaks down due to thermal fluctuations. We introduce a modified tuning process to increase this crossover temperature, showing that function can be robust at biologically relevant temperatures if thermal stability is important.

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This article appears in the following collection:

PRX Life Collection on adaptation and learning

PRX Life presents a special collection on ‘Biological systems that learn’, curated by the organizers of the National Institute for Theory and Mathematics in Biology (NITMB) workshop together with early-career researchers. The papers in this collection illustrate how living systems acquire functions, showing it is useful to describe them as tunable matter with many individually adjustable degrees of freedom characterizing microscopic properties. Drawing inspiration from AI and non-biological systems that can acquire desired functions, the featured research highlights the timeliness of the topic and its potential to drive unifying frameworks for emergent collective function in living tunable matter.

Every article in this collection is subject to a rigorous peer review process, upholding the high standards applied to all papers in the journal. The PRX Life editorial team manages peer review and makes all editorial decisions.

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References (70)

  1. J. Monod, J. Wyman, and J.-P. Changeux, On the nature of allosteric transitions: A plausible model, J. Mol. Biol. 12, 88 (1965).
  2. J.-P. Changeux, 50th anniversary of the word “allosteric,” Protein Sci. 20, 1119 (2011).
  3. C.-J. Tsai and R. Nussinov, A unified view of “how allostery works,” PLoS Comput. Biol. 10, e1003394 (2014).
  4. S. J. Wodak, E. Paci, N. V. Dokholyan, I. N. Berezovsky, A. Horovitz, J. Li, V. J. Hilser, I. Bahar, J. Karanicolas, G. Stock, P. Hamm, R. H. Stote, J. Eberhardt, Y. Chebaro, A. Dejaegere, M. Cecchini, J.-P. Changeux, P. G. Bolhuis, J. Vreede, P. Faccioli, et al., Allostery in its many disguises: From theory to applications, Structure 27, 566 (2019).
  5. M. M. Tirion, Large amplitude elastic motions in proteins from a single-parameter, atomic analysis, Phys. Rev. Lett. 77, 1905 (1996).
  6. I. Bahar, A. R. Atilgan, and B. Erman, Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential, Folding Des. 2, 173 (1997).
  7. T. Haliloglu, I. Bahar, and B. Erman, Gaussian dynamics of folded proteins, Phys. Rev. Lett. 79, 3090 (1997).
  8. K. Hinsen, Analysis of domain motions by approximate normal mode calculations, Proteins: Struct. Funct. Bioinf. 33, 417 (1998).
  9. A. J. Pak and G. A. Voth, Advances in coarse-grained modeling of macromolecular complexes, Curr. Opin. Struct. Biol. 52, 119 (2018), Special Issue: Cryo Electron Microscopy: The Impact of the Cryo-EM Revolution in Biology • Biophysical and Computational Methods - Part A.
  10. H. Flechsig and A. S. Mikhailov, Simple mechanics of protein machines, J. R. Soc. Interface 16, 20190244 (2019).
  11. M. R. Mitchell, T. Tlusty, and S. Leibler, Strain analysis of protein structures and low dimensionality of mechanical allosteric couplings, Proc. Natl. Acad. Sci. USA 113, E5847 (2016).
  12. H. Flechsig, Design of elastic networks with evolutionary optimized long-range communication as mechanical models of allosteric proteins, Biophys. J. 113, 558 (2017).
  13. J. W. Rocks, N. Pashine, I. Bischofberger, C. P. Goodrich, A. J. Liu, and S. R. Nagel, Designing allostery-inspired response in mechanical networks, Proc. Natl. Acad. Sci. USA 114, 2520 (2017).
  14. L. Yan, R. Ravasio, C. Brito, and M. Wyart, Architecture and coevolution of allosteric materials, Proc. Natl. Acad. Sci. USA 114, 2526 (2017).
  15. L. Yan, R. Ravasio, C. Brito, and M. Wyart, Principles for optimal cooperativity in allosteric materials, Biophys. J. 114, 2787 (2018).
  16. R. Ravasio, S. M. Flatt, L. Yan, S. Zamuner, C. Brito, and M. Wyart, Mechanics of allostery: Contrasting the induced fit and population shift scenarios, Biophys. J. 117, 1954 (2019).
  17. J. W. Rocks, H. Ronellenfitsch, A. J. Liu, S. R. Nagel, and E. Katifori, Limits of multifunctionality in tunable networks, Proc. Natl. Acad. Sci. USA 116, 2506 (2019).
  18. B. Bravi, R. Ravasio, C. Brito, and M. Wyart, Direct coupling analysis of epistasis in allosteric materials, PLoS Comput. Biol. 16, e1007630 (2020).
  19. E. Rouviere, R. Ranganathan, and O. Rivoire, Emergence of single- versus multi-state allostery, PRX Life 1, 023004 (2023).
  20. J. W. Rocks, E. Katifori, and A. J. Liu, Topological characterization of the continuum of allosteric response, arXiv:2401.13861.
  21. M. M. Rubin and J.-P. Changeux, On the nature of allosteric transitions: Implications of non-exclusive ligand binding, J. Mol. Biol. 21, 265 (1966).
  22. M. I. Stefan and N. Le Novère, Cooperative binding, PLoS Comput. Biol. 9, e1003106 (2013).
  23. D. Loutchko and H. Flechsig, Allosteric communication in molecular machines via information exchange: What can be learned from dynamical modeling, Biophys. Rev. 12, 443 (2020).
  24. M. Stern, A. J. Liu, and V. Balasubramanian, Physical effects of learning, Phys. Rev. E 109, 024311 (2024).
  25. M. Stern, M. Guzman, F. Martins, A. J. Liu, and V. Balasubramanian, Physical networks become what they learn, Phys. Rev. Lett. 134, 147402 (2025).
  26. C. P. Goodrich, A. J. Liu, and S. R. Nagel, The principle of independent bond-level response: Tuning by pruning to exploit disorder for global behavior, Phys. Rev. Lett. 114, 225501 (2015).
  27. N. Pashine, D. Hexner, A. J. Liu, and S. R. Nagel, Directed aging, memory, and nature's greed, Sci. Adv. 5, eaax4215 (2019).
  28. T. Tlusty, A. Libchaber, and J.-P. Eckmann, Physical model of the genotype-to-phenotype map of proteins, Phys. Rev. X 7, 021037 (2017).
  29. M. Guzman, F. Martins, M. Stern, and A. J. Liu, Microscopic imprints of learned solutions in tunable networks, Phys. Rev. X 15, 031056 (2025).
  30. W. Zheng, B. R. Brooks, and D. Thirumalai, Low-frequency normal modes that describe allosteric transitions in biological nanomachines are robust to sequence variations, Proc. Natl. Acad. Sci. USA 103, 7664 (2006).
  31. T. J. Glembo, D. W. Farrell, Z. N. Gerek, M. F. Thorpe, and S. B. Ozkan, Collective dynamics differentiates functional divergence in protein evolution, PLoS Comput. Biol. 8, e1002428 (2012).
  32. P. D. Townsend, T. L. Rodgers, E. Pohl, M. R. Wilson, T. C. B. McLeish, and M. J. Cann, Global low-frequency motions in protein allostery: CAP as a model system, Biophys. Rev. 7, 175 (2015).
  33. J.-P. Eckmann, J. Rougemont, and T. Tlusty, Colloquium: Proteins: The physics of amorphous evolving matter, Rev. Mod. Phys. 91, 031001 (2019).
  34. K. Husain and A. Murugan, Physical constraints on epistasis, Mol. Biol. Evol. 37, 2865 (2020).
  35. C. J. Russo, K. Husain, and A. Murugan, Soft modes as a predictive framework for low-dimensional biological systems across scales, Annu. Rev. Biophys. 54, 401 (2025).
  36. K. Henzler-Wildman and D. Kern, Dynamic personalities of proteins, Nature (London) 450, 964 (2007).
  37. , Thermally robust cooperative function in mechanical networks, Github, 2026, https://github.com/jandrejevic12/physical_learning.
  38. K. Gunasekaran, B. Ma, and R. Nussinov, Is allostery an intrinsic property of all dynamic proteins? Proteins: Struct. Funct. Bioinf. 57, 433 (2004).
  39. R. G. Smock and L. M. Gierasch, Sending signals dynamically, Science 324, 198 (2009).
  40. D. Kern and E. R. Zuiderweg, The role of dynamics in allosteric regulation, Curr. Opin. Struct. Biol. 13, 748 (2003).
  41. H. N. Motlagh, J. O. Wrabl, J. Li, and V. J. Hilser, The ensemble nature of allostery, Nature (London) 508, 331 (2014).
  42. M. Stern, D. Hexner, J. W. Rocks, and A. J. Liu, Supervised learning in physical networks: From machine learning to learning machines, Phys. Rev. X 11, 021045 (2021).
  43. M. Stern, S. Dillavou, M. Z. Miskin, D. J. Durian, and A. J. Liu, Physical learning beyond the quasistatic limit, Phys. Rev. Res. 4, L022037 (2022).
  44. K. Moritsugu and J. C. Smith, Coarse-grained biomolecular simulation with REACH: Realistic extension algorithm via covariance Hessian, Biophys. J. 93, 3460 (2007).
  45. E. Lyman, J. Pfaendtner, and G. A. Voth, Systematic multiscale parameterization of heterogeneous elastic network models of proteins, Biophys. J. 95, 4183 (2008).
  46. L. Yang, G. Song, and R. L. Jernigan, Protein elastic network models and the ranges of cooperativity, Proc. Natl. Acad. Sci. USA 106, 12347 (2009).
  47. D. Ming and M. E. Wall, Allostery in a coarse-grained model of protein dynamics, Phys. Rev. Lett. 95, 198103 (2005).
  48. I. Dubanevics and T. C. McLeish, Optimising elastic network models for protein dynamics and allostery: Spatial and modal cut-offs and backbone stiffness, J. Mol. Biol. 434, 167696 (2022), Special Issue: Allostery: From Mechanisms to Therapies.
  49. L. Zhu, D. Frenkel, and P. G. Bolhuis, Role of fluctuations in ligand binding cooperativity of membrane receptors, Phys. Rev. Lett. 106, 168103 (2011).
  50. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6n32-9vq7 for model validation, detailed parameter exploration, and protein analysis.
  51. J. R. Gissinger, I. Nikiforov, Y. Afshar, B. Waters, M.-k. Choi, D. S. Karls, A. Stukowski, W. Im, H. Heinz, A. Kohlmeyer, and E. B. Tadmor, Type label framework for bonded force fields in LAMMPS, J. Phys. Chem. B 128, 3282 (2024).
  52. M. Sheinman, C. P. Broedersz, and F. C. MacKintosh, Nonlinear effective-medium theory of disordered spring networks, Phys. Rev. E 85, 021801 (2012).
  53. T. Ku, P. Lu, C. Chan, T. Wang, S. Lai, P. Lyu, and N. Hsiao, Predicting melting temperature directly from protein sequences, Comput. Biol. Chem. 33, 445 (2009).
  54. H. P. Erickson, Size and shape of protein molecules at the nanometer level determined by sedimentation, gel filtration, and electron microscopy, Biol. Proced. Online 11, 32 (2009).
  55. G. Song, Bridging between material properties of proteins and the underlying molecular interactions, PLoS One 16, e0247147 (2021).
  56. H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov, and P. E. Bourne, The Protein Data Bank, Nucl. Acids Res. 28, 235 (2000).
  57. Y. Chow, R. Pietranico, and A. Mukerji, Studies of oxygen binding energy to hemoglobin molecule, Biochem. Biophys. Res. Commun. 66, 1424 (1975).
  58. R. E. Alcantara, C. Xu, T. G. Spiro, and V. Guallar, A quantum-chemical picture of hemoglobin affinity, Proc. Natl. Acad. Sci. USA 104, 18451 (2007).
  59. F. A. Lindemann, Calculation of molecular Eigen-frequencies, Technical Report N-84-35086; NASA-TM-77759, National Aeronautics and Space Administration, Washington, DC, x1984, https://www.osti.gov/biblio/6089125.
  60. F. H. Stillinger and D. K. Stillinger, Computational study of transition dynamics in 55‐atom clusters, J. Chem. Phys. 93, 6013 (1990).
  61. Y. Zhou, D. Vitkup, and M. Karplus, Native proteins are surface-molten solids: Application of the Lindemann criterion for the solid versus liquid state, J. Mol. Biol. 285, 1371 (1999).
  62. J. L. S. Eshel Faraggi, A. K. Dunker, and A. Kloczkowski, Comparing NMR and x-ray protein structure: Lindemann-like parameters and NMR disorder, J. Biomol. Struct. Dyn. 36, 2331 (2018).
  63. M. Katava, G. Stirnemann, M. Zanatta, S. Capaccioli, M. Pachetti, K. L. Ngai, F. Sterpone, and A. Paciaroni, Critical structural fluctuations of proteins upon thermal unfolding challenge the Lindemann criterion, Proc. Natl. Acad. Sci. USA 114, 9361 (2017).
  64. R. J. Hawkins and T. C. B. McLeish, Coarse-grained model of entropic allostery, Phys. Rev. Lett. 93, 098104 (2004).
  65. A. Cooper and D. T. F. Dryden, Allostery without conformational change: A plausible model, Eur. Biophys. J. 11, 103 (1984).
  66. N. Popovych, S. Sun, R. H. Ebright, and C. G. Kalodimos, Dynamically driven protein allostery, Nat. Struct. Mol. Biol. 13, 831 (2006).
  67. C.-J. Tsai, A. del Sol, and R. Nussinov, Allostery: Absence of a change in shape does not imply that allostery is not at play, J. Mol. Biol. 378, 1 (2008).
  68. N. M. Goodey and S. J. Benkovic, Allosteric regulation and catalysis emerge via a common route, Nat. Chem. Biol. 4, 474 (2008).
  69. T. Liu, S. T. Whitten, and V. J. Hilser, Functional residues serve a dominant role in mediating the cooperativity of the protein ensemble, Proc. Natl. Acad. Sci. USA 104, 4347 (2007).
  70. J. O. Wrabl, J. Gu, T. Liu, T. P. Schrank, S. T. Whitten, and V. J. Hilser, The role of protein conformational fluctuations in allostery, function, and evolution, Biophys. Chem. 159, 129 (2011).

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PRX Life 4, 033009– Published 31 July, 2026
  • Received 18 September 2025
  • Accepted 1 July 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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