- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Toolbox for quantifying memory in dynamics along reaction coordinates

Alessio Lapolla and Aljaž Godec*

  • Mathematical bioPhysics Group, Max Planck Institute for Biophysical Chemistry, Göttingen 37077, Germany

  • *agodec@mpibpc.mpg.de
PDF

Phys. Rev. Research 3, L022018 – Published 28 May, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L022018

Abstract

Memory effects in time series of experimental observables are ubiquitous, have important consequences for the interpretation of kinetic data, and may even affect the function of biomolecular nanomachines such as enzymes. Here we propose a set of complementary methods for quantifying conclusively the magnitude and duration of memory in a time series of a reaction coordinate. The toolbox is general, robust, easy to use, and does not rely on any underlying microscopic model. As a proof of concept we apply it to the analysis of memory in the dynamics of the end-to-end distance of the analytically solvable Rouse-polymer model, an experimental time series of extensions of a single DNA hairpin measured by optical tweezers, and the fraction of native contacts in a small protein probed by atomistic molecular dynamics simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (61)

  1. R. B. Best and G. Hummer, Reaction coordinates and rates from transition paths, Proc. Natl. Acad. Sci. USA 102, 6732 (2005).
  2. J. J. Portman, S. Takada, and P. G. Wolynes, Microscopic theory of protein folding rates. II. Local reaction coordinates and chain dynamics, J. Chem. Phys. 114, 5082 (2001).
  3. B. Peters, P. G. Bolhuis, R. G. Mullen, and J.-E. Shea, Reaction coordinates, one-dimensional Smoluchowski equations, and a test for dynamical self-consistency, J. Chem. Phys. 138, 054106 (2013).
  4. A. K. Faradjian and R. Elber, Computing time scales from reaction coordinates by milestoning, J. Chem. Phys. 120, 10880 (2004).
  5. A. Berezhkovskii and A. Szabo, One-dimensional reaction coordinates for diffusive activated rate processes in many dimensions, J. Chem. Phys. 122, 014503 (2005).
  6. G. Hummer, Position-dependent diffusion coefficients and free energies from Bayesian analysis of equilibrium and replica molecular dynamics simulations, New J. Phys. 7, 34 (2005).
  7. R. B. Best and G. Hummer, Coordinate-dependent diffusion in protein folding, Proc. Natl. Acad. Sci. USA 107, 1088 (2009).
  8. W. Zhang, C. Hartmann, and C. Schütte, Effective dynamics along given reaction coordinates, and reaction rate theory, Faraday Discuss. 195, 365 (2016).
  9. A. M. Berezhkovskii and D. E. Makarov, Communication: Coordinate-dependent diffusivity from single molecule trajectories, J. Chem. Phys. 147, 201102 (2017).
  10. B. J. Berne, M. Borkovec, and J. E. Straub, Classical and modern methods in reaction rate theory, J. Phys. Chem. 92, 3711 (1988).
  11. O. K. Dudko, G. Hummer, and A. Szabo, Theory, analysis, and interpretation of single-molecule force spectroscopy experiments, Proc. Natl. Acad. Sci. USA 105, 15755 (2008).
  12. K. Neupane, A. P. Manuel, and M. T. Woodside, Protein folding trajectories can be described quantitatively by one-dimensional diffusion over measured energy landscapes, Nat. Phys. 12, 700 (2016).
  13. K. Neupane, D. A. N. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, Direct observation of transition paths during the folding of proteins and nucleic acids, Science 352, 239 (2016).
  14. J. Gladrow, M. Ribezzi-Crivellari, F. Ritort, and U. F. Keyser, Experimental evidence of symmetry breaking of transition-path times, Nat. Commun. 10, 55 (2019).
  15. A. L. Thorneywork, J. Gladrow, Y. Qing, M. Rico-Pasto, F. Ritort, H. Bayley, A. B. Kolomeisky, and U. F. Keyser, Direct detection of molecular intermediates from first-passage times, Sci. Adv. 6, eaaz4642 (2020).
  16. A. Lapolla and A. Godec, Manifestations of projection-induced memory: General theory and the tilted single file, Front. Phys. 7, 182 (2019).
  17. D. Hartich and A. Godec, Emergent memory and kinetic hysteresis in strongly driven networks, arXiv:2011.04628.
  18. N. van Kampen, Remarks on non-Markov processes, Braz. J. Phys. 28, 90 (1998).
  19. S. S. Plotkin and P. G. Wolynes, Non-Markovian Configurational Diffusion and Reaction Coordinates for Protein Folding, Phys. Rev. Lett. 80, 5015 (1998).
  20. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, 2010).
  21. D. E. Makarov, Interplay of non-Markov and internal friction effects in the barrier crossing kinetics of biopolymers: Insights from an analytically solvable model, J. Chem. Phys. 138, 014102 (2013).
  22. M. Ozmaian and D. E. Makarov, Transition path dynamics in the binding of intrinsically disordered proteins: A simulation study, J. Chem. Phys. 151, 235101 (2019).
  23. H. Meyer, P. Pelagejcev, and T. Schilling, Non-Markovian out-of-equilibrium dynamics: A general numerical procedure to construct time-dependent memory kernels for coarse-grained observables, EPL (Europhys. Lett.) 128, 40001 (2020).
  24. E. Herrera-Delgado, J. Briscoe, and P. Sollich, Tractable nonlinear memory functions as a tool to capture and explain dynamical behaviors, Phys. Rev. Research 2, 043069 (2020).
  25. F. Müller, U. Basu, P. Sollich, and M. Krüger, Coarse-grained second-order response theory, Phys. Rev. Research 2, 043123 (2020).
  26. W. Min, G. Luo, B. J. Cherayil, S. C. Kou, and X. S. Xie, Observation of a Power-Law Memory Kernel for Fluctuations within a Single Protein Molecule, Phys. Rev. Lett. 94, 198302 (2005).
  27. A. Lapolla and A. Godec, Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches, Phys. Rev. Lett. 125, 110602 (2020).
  28. A. Lapolla and A. Godec, Bethesf: Efficient computation of the exact tagged-particle propagator in single-file systems via the Bethe eigenspectrum, Comput. Phys. Commun. 258, 107569 (2021).
  29. A. Lapolla and A. Godec, Single-file diffusion in a bi-stable potential: Signatures of memory in the barrier-crossing of a tagged-particle, J. Chem. Phys. 153, 194104 (2020).
  30. A. Lapolla and A. Godec, Unfolding tagged particle histories in single-file diffusion: exact single- and two-tag local times beyond large deviation theory, New J. Phys. 20, 113021 (2018).
  31. S. C. Kou and X. S. Xie, Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule, Phys. Rev. Lett. 93, 180603 (2004).
  32. T. Neusius, I. Daidone, I. M. Sokolov, and J. C. Smith, Subdiffusion in Peptides Originates from the Fractal-Like Structure of Configuration Space, Phys. Rev. Lett. 100, 188103 (2008).
  33. S. Pressé, J. Peterson, J. Lee, P. Elms, J. L. MacCallum, S. Marqusee, C. Bustamante, and K. Dill, Single molecule conformational memory extraction: P5ab rna hairpin, J. Phys. Chem. B 118, 6597 (2014).
  34. X. Hu, L. Hong, M. Dean Smith, T. Neusius, X. Cheng, and J. Smith, The dynamics of single protein molecules is non-equilibrium and self-similar over thirteen decades in time, Nat. Phys. 12, 171 (2015).
  35. A. K. Sangha and T. Keyes, Proteins Fold by Subdiffusion of the Order Parameter, J. Phys. Chem. B 113, 15886 (2009).
  36. S. M. Avdoshenko, A. Das, R. Satija, G. A. Papoian, and D. E. Makarov, Theoretical and computational validation of the Kuhn barrier friction mechanism in unfolded proteins, Sci. Rep. 7, 269 (2017).
  37. Y. Cote, P. Senet, P. Delarue, G. G. Maisuradze, and H. A. Scheraga, Anomalous diffusion and dynamical correlation between the side chains and the main chain of proteins in their native state, Proc. Natl. Acad. Sci. USA 109, 10346 (2012).
  38. I. Grossman-Haham, G. Rosenblum, T. Namani, and H. Hofmann, Slow domain reconfiguration causes power-law kinetics in a two-state enzyme, Proc. Natl. Acad. Sci. USA 115, 513 (2018).
  39. A. G. T. Pyo and M. T. Woodside, Memory effects in single-molecule force spectroscopy measurements of biomolecular folding, Phys. Chem. Chem. Phys. 21, 24527 (2019).
  40. H. P. Lu, L. Xun, and X. S. Xie, Single-molecule enzymatic dynamics, Science 282, 1877 (1998).
  41. B. P. English, W. Min, A. M. van Oijen, K. T. Lee, G. Luo, H. Sun, B. J. Cherayil, S. C. Kou, and X. S. Xie, Ever-fluctuating single enzyme molecules: Michaelis-menten equation revisited, Nat. Chem. Biol. 2, 87 (2005).
  42. A. M. Berezhkovskii and D. E. Makarov, Single-molecule test for Markovianity of the dynamics along a reaction coordinate, J. Phys. Chem. Lett. 9, 2190 (2018).
  43. S. Kullback and R. Leibler, On information and sufficiency, Ann. Math. Stat. 22, 79 (1951).
  44. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, 2nd ed. (Springer-Verlag, Berlin, 1985).
  45. W. Feller, Non-Markovian processes with the semigroup property, Ann. Math. Stat. 30, 1252 (1959).
  46. Y. Klimontovich, Ito, stratonovich and kinetic forms of stochastic equations, Physica A 163, 515 (1990).
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.3.L022018 for a discretization of the anti-Itô Langevin equation (5), exact results for the Rouse polymer, details about MD simulations, the fraction of native contacts, the estimation of the diffusion landscape D(q), and a description of the uncertainty quantification.
  48. A. Berezhkovskii and A. Szabo, Time scale separation leads to position-dependent diffusion along a slow coordinate, J. Chem. Phys. 135, 074108 (2011).
  49. S. Sunagawa and M. Doi, Theory of diffusion-controlled intrachain reactions of polymers, Polymer J. 7, 604 (1975).
  50. S. N. Majumdar, Persistence in nonequilibrium systems, Curr. Sci. 77, 370 (1999).
  51. A. J. Bray, S. N. Majumdar, and G. Schehr, Persistence and first-passage properties in nonequilibrium systems, Adv. Phys. 62, 225 (2013).
  52. A. J. Bray, B. Derrida, and C. Godréche, Non-trivial algebraic decay in a soluble model of coarsening, EPL (Europhys. Lett.) 27, 175 (1994).
  53. B. Derrida, V. Hakim, and V. Pasquier, Exact First-Passage Exponents of 1D Domain Growth: Relation to a Reaction-Diffusion Model, Phys. Rev. Lett. 75, 751 (1995).
  54. S. N. Majumdar and A. J. Bray, Persistence with Partial Survival, Phys. Rev. Lett. 81, 2626 (1998).
  55. S. N. Majumdar, A. J. Bray, S. J. Cornell, and C. Sire, Global Persistence Exponent for Nonequilibrium Critical Dynamics, Phys. Rev. Lett. 77, 3704 (1996).
  56. P. E. Rouse, A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers, J. Chem. Phys. 21, 1272 (1953).
  57. K. H. Ahn, J. L. Schrag, and S. J. Lee, Bead-spring chain model for the dynamics of dilute polymer solutions, J. Non-Newtonian Fluid Mech. 50, 349 (1993).
  58. K. Neupane, A. P. Manuel, J. Lambert, and M. T. Woodside, Transition-path probability as a test of reaction-coordinate quality reveals DNA hairpin folding is a one-dimensional diffusive process, J. Phys. Chem. Lett. 6, 1005 (2015).
  59. M. Jager, Y. Zhang, J. Bieschke, H. Nguyen, M. Dendle, M. E. Bowman, J. P. Noel, M. Gruebele, and J. W. Kelly, Structure-function-folding relationship in a WW domain, Proc. Natl. Acad. Sci. USA 103, 10648 (2006).
  60. K. Lindorff-Larsen, S. Piana, R. O. Dror, and D. E. Shaw, How fast-folding proteins fold, Science 334, 517 (2011).
  61. R. B. Best, G. Hummer, and W. A. Eaton, Native contacts determine protein folding mechanisms in atomistic simulations, Proc. Natl. Acad. Sci. USA 110, 17874 (2013).

Outline

Information

Phys. Rev. Research 3, L022018– Published 28 May, 2021
  • Received 11 January 2021
  • Accepted 13 May 2021
Crossmark - Check for updates button
Creative Commons Logo

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. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation