1932
0

Annual Review of Physical Chemistry

Volume 76, 2025

Reaction Coordinates Are Optimal Channels of Energy Flow

  • Ao Ma1 and Huiyu Li1
  • Center for Bioinformatics and Quantitative Biology, Richard and Loan Hill Department of Biomedical Engineering, The University of Illinois Chicago, Chicago, Illinois, USA; email: [email protected]
  • Vol. 76:153-179 (Volume publication date April 2025)
  • First published as a Review in Advance on February 04, 2025
  • Copyright © 2025 by the author(s).
    This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. See credit lines of images or other third-party material in this article for license information.

Abstract

Reaction coordinates (RCs) are the few essential coordinates of a protein that control its functional processes, such as allostery, enzymatic reaction, and conformational change. They are critical for understanding protein function and provide optimal enhanced sampling of protein conformational changes and states. Since the pioneering work in the late 1990s, identifying the correct and objectively provable RCs has been a central topic in molecular biophysics and chemical physics. This review summarizes the major advances in identifying RCs over the past 25 years, focusing on methods aimed at finding RCs that meet the rigorous committor criterion, widely accepted as the true RCs. Notably, the newly developed physics-based energy flow theory and generalized work functional method provide a general and rigorous approach for identifying true RCs, revealing their physical nature as the optimal channels of energy flow in biomolecules.

Keyword(s): dimension reduction, energy flow theory, enhanced sampling, generalized work functional, protein conformational change, reaction coordinate
Loading

Article metrics loading...

/content/journals/10.1146/annurev-physchem-082423-010652
2025-04-21
2025-06-19
Loading full text...

Full text loading...

/deliver/fulltext/physchem/76/1/annurev-physchem-082423-010652.html?itemId=/content/journals/10.1146/annurev-physchem-082423-010652&mimeType=html&fmt=ahah

Literature Cited

  1. 1.
    Bryngelson JD, Onuchic JN, Socci ND, Wolynes PG. 1995.. Funnels, pathways, and the energy landscape of protein-folding—a synthesis. . Proteins 21::167–95
     [Crossref] [Google Scholar]
  2. 2.
    Dill KA, Chan HS. 1997.. From Levinthal to pathways to funnels. . Nat. Struct. Biol. 4::10–19
     [Crossref] [Google Scholar]
  3. 3.
    Onuchic JN, Luthey-Schulten Z, Wolynes PG. 1997.. Theory of protein folding: the energy landscape perspective. . Annu. Rev. Phys. Chem. 48::545–600
     [Crossref] [Google Scholar]
  4. 4.
    Jumper J, Evans R, Pritzel A, Green T, Figurnov M, et al. 2021.. Highly accurate protein structure prediction with AlphaFold. . Nature 596::583–89
     [Crossref] [Google Scholar]
  5. 5.
    Senior AW, Evans R, Jumper J, Kirkpatrick J, Sifre L, et al. 2020.. Improved protein structure prediction using potentials from deep learning. . Nature 577::706–10
     [Crossref] [Google Scholar]
  6. 6.
    Austin RH, Beeson KW, Eisenstein L, Frauenfelder H, Gunsalus IC. 1975.. Dynamics of ligand-binding to myoglobin. . Biochemistry 14::5355–73
     [Crossref] [Google Scholar]
  7. 7.
    Frauenfelder H, Sligar SG, Wolynes PG. 1991.. The energy landscapes and motions of proteins. . Science 254::1598–603
     [Crossref] [Google Scholar]
  8. 8.
    Henzler-Wildman K, Kern D. 2007.. Dynamic personalities of proteins. . Nature 450::964–72
     [Crossref] [Google Scholar]
  9. 9.
    Arrhenius S. 1889.. Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. . Z. Phys. Chem. 4::226–48
     [Crossref] [Google Scholar]
  10. 10.
    Hanggi P, Talkner P, Borkovec M. 1990.. Reaction-rate theory—50 years after Kramers. . Rev. Mod. Phys. 62::251–341
     [Crossref] [Google Scholar]
  11. 11.
    Berne BJ, Borkovec M, Straub JE. 1988.. Classical and modern methods in reaction-rate theory. . J. Phys. Chem. 92::3711–25
     [Crossref] [Google Scholar]
  12. 12.
    Chandler D. 1987.. Introduction to Modern Statistical Mechanics. New York:: Oxford Univ. Press
     [Google Scholar]
  13. 13.
    Grote RF, Hynes JT. 1980.. The stable states picture of chemical reactions. II. Rate constants for condensed and gas-phase reaction models. . J. Chem. Phys. 73::2715–32
     [Crossref] [Google Scholar]
  14. 14.
    Kassel LS. 1928.. Studies in homogeneous gas reactions I. . J. Phys. Chem. 32::225–42
     [Crossref] [Google Scholar]
  15. 15.
    Kramers HA. 1940.. Brownian motion in a field of force and the diffusion model of chemical reactions. . Physica 7::284–304
     [Crossref] [Google Scholar]
  16. 16.
    Langer JS. 1969.. Statistical theory of the decay of metastable states. . Ann. Phys. 54::258–75
     [Crossref] [Google Scholar]
  17. 17.
    Marcus RA. 1952.. Unimolecular dissociations and free radical recombination reactions. . J. Chem. Phys. 20::359–64
     [Crossref] [Google Scholar]
  18. 18.
    Rice OK, Ramsperger HC. 1927.. Theories of unimolecular gas reactions at low pressures. . J. Am. Chem. Soc. 49::1617–29
     [Crossref] [Google Scholar]
  19. 19.
    Wigner E. 1938.. The transition state method. . Trans. Faraday Soc. 34::29–41
     [Crossref] [Google Scholar]
  20. 20.
    Zwanzig R. 2001.. Nonequilibrium Statistical Mechanics. Oxford, UK:: Oxford Univ. Press.
     [Google Scholar]
  21. 21.
    Pechukas P. 1976.. Statistical approximations in collision theory. . In Dynamics of Molecular Collisions: Part B, ed. WH Miller , pp. 269–322. New York:: Plenum
     [Google Scholar]
  22. 22.
    Berezhkovskii A, Szabo A. 2005.. One-dimensional reaction coordinates for diffusive activated rate processes in many dimensions. . J. Chem. Phys. 122::014503
     [Crossref] [Google Scholar]
  23. 23.
    Berezhkovskii AM, Szabo A. 2013.. Diffusion along the splitting/commitment probability reaction coordinate. . J. Phys. Chem. B 117::13115–19
     [Crossref] [Google Scholar]
  24. 24.
    Chandler D. 1978.. Statistical mechanics of isomerization dynamics in liquids and transition-state approximation. . J. Chem. Phys. 68::2959–70
     [Crossref] [Google Scholar]
  25. 25.
    Wigner E. 1932.. Crossing of potential thresholds in chemical reactions. . Z. Phys. Chem. B19::203–16
     [Crossref] [Google Scholar]
  26. 26.
    Eyring H. 1935.. The activated complex in chemical reactions. . J. Chem. Phys. 3::107–15
     [Crossref] [Google Scholar]
  27. 27.
    Berezhkovskii AM, Szabo A, Greives N, Zhou HX. 2014.. Multidimensional reaction rate theory with anisotropic diffusion. . J. Chem. Phys. 141::204106
     [Crossref] [Google Scholar]
  28. 28.
    Karplus M, Porter RN, Sharma RD. 1965.. Exchange reactions with activation energy. I. Simple barrier potential for (H, H2). . J. Chem. Phys. 43::3259–87
     [Crossref] [Google Scholar]
  29. 29.
    Dror RO, Green HF, Valant C, Borhani DW, Valcourt JR, et al. 2013.. Structural basis for modulation of a G-protein-coupled receptor by allosteric drugs. . Nature 503::295–99
     [Crossref] [Google Scholar]
  30. 30.
    Shan Y, Gnanasambandan K, Ungureanu D, Kim ET, Hammaren H, et al. 2014.. Molecular basis for pseudokinase-dependent autoinhibition of JAK2 tyrosine kinase. . Nat. Struct. Mol. Biol. 21::579–84
     [Crossref] [Google Scholar]
  31. 31.
    Bussi G, Laio A. 2020.. Using metadynamics to explore complex free-energy landscapes. . Nat. Rev. Phys. 2::200–12
     [Crossref] [Google Scholar]
  32. 32.
    Hénin J, Lelievre T, Shirts MR, Valsson O, Delemotte L. 2022.. Enhanced sampling methods for molecular dynamics simulations. . arXiv:2202.04164 [cond-mat.stat-mech]
  33. 33.
    Torrie GM, Valleau JP. 1977.. Non-physical sampling distributions in Monte-Carlo free-energy estimation: umbrella sampling. . J. Comput. Phys. 23::187–99
     [Crossref] [Google Scholar]
  34. 34.
    Barducci A, Bussi G, Parrinello M. 2008.. Well-tempered metadynamics: a smoothly converging and tunable free-energy method. . Phys. Rev. Lett. 100::020603
     [Crossref] [Google Scholar]
  35. 35.
    Laio A, Parrinello M. 2002.. Escaping free-energy minima. . PNAS 99::12562–66
     [Crossref] [Google Scholar]
  36. 36.
    Valsson O, Tiwary P, Parrinello M. 2016.. Enhancing important fluctuations: rare events and metadynamics from a conceptual viewpoint. . Annu. Rev. Phys. Chem. 67::159–84
     [Crossref] [Google Scholar]
  37. 37.
    Darve E, Pohorille A. 2001.. Calculating free energies using average force. . J. Chem. Phys. 115::9169–83
     [Crossref] [Google Scholar]
  38. 38.
    Darve E, Rodriguez-Gomez D, Pohorille A. 2008.. Adaptive biasing force method for scalar and vector free energy calculations. . J. Chem. Phys. 128::144120
     [Crossref] [Google Scholar]
  39. 39.
    Glielmo A, Husic BE, Rodriguez A, Clementi C, Noé F, Laio A. 2021.. Unsupervised learning methods for molecular simulation data. . Chem. Rev. 121::9722–58
     [Crossref] [Google Scholar]
  40. 40.
    Mehdi S, Smith Z, Herron L, Zou Z, Tiwary P. 2024.. Enhanced sampling with machine learning. . Annu. Rev. Phys. Chem. 75::347–70
     [Crossref] [Google Scholar]
  41. 41.
    Lv C, Zheng LQ, Yang W. 2012.. Generalized essential energy space random walks to more effectively accelerate solute sampling in aqueous environment. . J. Chem. Phys. 136::044103
     [Crossref] [Google Scholar]
  42. 42.
    Zheng L, Chen M, Yang W. 2009.. Simultaneous escaping of explicit and hidden free energy barriers: application of the orthogonal space random walk strategy in generalized ensemble based conformational sampling. . J. Chem. Phys. 130::234105
     [Crossref] [Google Scholar]
  43. 43.
    Zuckerman DM, Chong LT. 2017.. Weighted ensemble simulation: review of methodology, applications, and software. . Annu. Rev. Biophys. 46::43–57
     [Crossref] [Google Scholar]
  44. 44.
    Dickson A, Dinner AR. 2010.. Enhanced sampling of nonequilibrium steady states. . Annu. Rev. Phys. Chem. 61::441–59
     [Crossref] [Google Scholar]
  45. 45.
    Dickson A, Warmflash A, Dinner AR. 2009.. Nonequilibrium umbrella sampling in spaces of many order parameters. . J. Chem. Phys. 130::074104
     [Crossref] [Google Scholar]
  46. 46.
    Zhang BW, Jasnow D, Zuckerman DM. 2010.. The “weighted ensemble” path sampling method is statistically exact for a broad class of stochastic processes and binning procedures. . J. Chem. Phys. 132::054107
     [Crossref] [Google Scholar]
  47. 47.
    Faradjian AK, Elber R. 2004.. Computing time scales from reaction coordinates by milestoning. . J. Chem. Phys. 120::10880–89
     [Crossref] [Google Scholar]
  48. 48.
    Peng C, Zhang L, Head-Gordon T. 2010.. Instantaneous normal modes as an unforced reaction coordinate for protein conformational transitions. . Biophys. J. 98::2356–64
     [Crossref] [Google Scholar]
  49. 49.
    Levy RM, Srinivasan AR, Olson WK, McCammon JA. 1984.. Quasi-harmonic method for studying very low frequency modes in proteins. . Biopolymers 23::1099–112
     [Crossref] [Google Scholar]
  50. 50.
    Du R, Pande VS, Grosberg AY, Tanaka T, Shakhnovich ES. 1998.. On the transition coordinate for protein folding. . J. Chem. Phys. 108::334–50
     [Crossref] [Google Scholar]
  51. 51.
    Bolhuis PG, Chandler D, Dellago C, Geissler PL. 2002.. Transition path sampling: throwing ropes over rough mountain passes, in the dark. . Annu. Rev. Phys. Chem. 53::291–318
     [Crossref] [Google Scholar]
  52. 52.
    Onsager L. 1938.. Initial recombination of ions. . Phys. Rev. 54::554–57
     [Crossref] [Google Scholar]
  53. 53.
    van Kampen NG. 1978.. Escape and splitting probabilities in diffusive and non-diffusive Markov processes. . Prog. Theor. Phys. Suppl. 64::389–401
     [Crossref] [Google Scholar]
  54. 54.
    Pratt LR. 1986.. A statistical method for identifying transition states in high dimensional problems. . J. Chem. Phys. 85::5045–48
     [Crossref] [Google Scholar]
  55. 55.
    Ryter D. 1987.. On the eigenfunctions of the Fokker-Planck operator and of its adjoint. . Physica A 142::103–21
     [Crossref] [Google Scholar]
  56. 56.
    Li W, Ma A. 2014.. Recent developments in methods for identifying reaction coordinates. . Mol. Simul. 40::784–93
     [Crossref] [Google Scholar]
  57. 57.
    Ma A, Dinner AR. 2005.. Automatic method for identifying reaction coordinates in complex systems. . J. Phys. Chem. B 109::6769–79 This paper presented the first machine learning method for identifying tRCs in complex molecules.
     [Crossref] [Google Scholar]
  58. 58.
    Weinan E, Vanden-Eijnden E. 2010.. Transition-path theory and path-finding algorithms for the study of rare events. . Annu. Rev. Phys. Chem. 61::391–420
     [Crossref] [Google Scholar]
  59. 59.
    Lu J, Vanden-Eijnden E. 2014.. Exact dynamical coarse-graining without time-scale separation. . J. Chem. Phys. 141::044109
     [Crossref] [Google Scholar]
  60. 60.
    Krivov SV. 2018.. Protein folding free energy landscape along the committor - the optimal folding coordinate. . J. Chem. Theory Comput. 14::3418–27
     [Crossref] [Google Scholar]
  61. 61.
    Roux B. 2022.. Transition rate theory, spectral analysis, and reactive paths. . J. Chem. Phys. 156::134111
     [Crossref] [Google Scholar]
  62. 62.
    Ma A, Nag A, Dinner AR. 2006.. Dynamic coupling between coordinates in a model for biomolecular isomerization. . J. Chem. Phys. 124::144911
     [Crossref] [Google Scholar]
  63. 63.
    Bolhuis PG, Dellago C, Chandler D. 2000.. Reaction coordinates of biomolecular isomerization. . PNAS 97::5877–82
     [Crossref] [Google Scholar]
  64. 64.
    ten Wolde PR, Chandler D. 2002.. Drying-induced hydrophobic polymer collapse. . PNAS 99::6539–43
     [Crossref] [Google Scholar]
  65. 65.
    McCormick TA, Chandler D. 2003.. Grid-flux method for learning the solvent contribution to the mechanisms of reactions. . J. Phys. Chem. B 107::2796–801
     [Crossref] [Google Scholar]
  66. 66.
    Li W, Ma A. 2016.. Reaction mechanism and reaction coordinates from the viewpoint of energy flow. . J. Chem. Phys. 144::114103 This paper developed the energy flow theory and defined the exact potential energy flows.
     [Crossref] [Google Scholar]
  67. 67.
    Wu S, Li H, Ma A. 2022.. A rigorous method for identifying one-dimensional reaction coordinate in complex molecules. . J. Chem. Theory Comput. 18::2836–44 This paper developed the generalized work functional method.
     [Crossref] [Google Scholar]
  68. 68.
    Mori Y, Okazaki K, Mori T, Kim K, Matubayasi N. 2020.. Learning reaction coordinates via cross-entropy minimization: application to alanine dipeptide. . J. Chem. Phys. 153::054115
     [Crossref] [Google Scholar]
  69. 69.
    Hooft F, Perez de Alba Ortiz A, Ensing B. 2021.. Discovering collective variables of molecular transitions via genetic algorithms and neural networks. . J. Chem. Theory Comput. 17::2294–306
     [Crossref] [Google Scholar]
  70. 70.
    Wu S, Ma A. 2022.. Mechanism for the rare fluctuation that powers protein conformational change. . J. Chem. Phys. 156::05419 This paper uncovered a mechanism of energy activation that fundamentally differs from the Kramers picture.
     [Google Scholar]
  71. 71.
    Li H, Ma A. 2020.. Kinetic energy flows in activated dynamics of biomolecules. . J. Chem. Phys. 153::094109 This paper defined the exact kinetic energy flows through individual coordinates.
     [Crossref] [Google Scholar]
  72. 72.
    Lu C, Li X, Wu D, Zheng L, Yang W. 2016.. Predictive sampling of rare conformational events in aqueous solution: designing a generalized orthogonal space tempering method. . J. Chem. Theory Comput. 12::41–52
     [Crossref] [Google Scholar]
  73. 73.
    Peters B, Trout BL. 2006.. Obtaining reaction coordinates by likelihood maximization. . J. Chem. Phys. 125::054108
     [Crossref] [Google Scholar]
  74. 74.
    Antoniou D, Schwartz SD. 2009.. The stochastic separatrix and the reaction coordinate for complex systems. . J. Chem. Phys. 130::151103
     [Crossref] [Google Scholar]
  75. 75.
    Frassek M, Arjun A, Bolhuis PG. 2021.. An extended autoencoder model for reaction coordinate discovery in rare event molecular dynamics datasets. . J. Chem. Phys. 155::064103
     [Crossref] [Google Scholar]
  76. 76.
    Jung H, Covino R, Arjun A, Leitold C, Dellago C, et al. 2023.. Machine-guided path sampling to discover mechanisms of molecular self-organization. . Nat. Comput. Sci. 3::334–45
     [Crossref] [Google Scholar]
  77. 77.
    Mori T, Saito S. 2020.. Dissecting the dynamics during enzyme catalysis: a case study of Pin1 peptidyl-prolyl isomerase. . J. Chem. Theory Comput. 16::3396–407
     [Crossref] [Google Scholar]
  78. 78.
    Wang Y, Ribeiro JML, Tiwary P. 2019.. Past–future information bottleneck for sampling molecular reaction coordinate simultaneously with thermodynamics and kinetics. . Nat. Commun. 10::3573
     [Crossref] [Google Scholar]
  79. 79.
    Noé F, Nüske F. 2013.. A variational approach to modeling slow processes in stochastic dynamical systems. . Multiscale Model. Simul. 11::635–55
     [Crossref] [Google Scholar]
  80. 80.
    Nuske F, Keller BG, Perez-Hernandez G, Mey AS, Noé F. 2014.. Variational approach to molecular kinetics. . J. Chem. Theory Comput. 10::1739–52
     [Crossref] [Google Scholar]
  81. 81.
    Mardt A, Pasquali L, Wu H, Noé F. 2018.. VAMPnets for deep learning of molecular kinetics. . Nat. Commun. 9::5
     [Crossref] [Google Scholar]
  82. 82.
    Lechner W, Rogal J, Juraszek J, Ensing B, Bolhuis PG. 2010.. Nonlinear reaction coordinate analysis in the reweighted path ensemble. . J. Chem. Phys. 133::174110
     [Crossref] [Google Scholar]
  83. 83.
    Peters B. 2012.. Inertial likelihood maximization for reaction coordinates with high transmission coefficients. . Chem. Phys. Lett. 554::248–53
     [Crossref] [Google Scholar]
  84. 84.
    Peters B, Bolhuis PG, Mullen RG, Shea JE. 2013.. Reaction coordinates, one-dimensional Smoluchowski equations, and a test for dynamical self-consistency. . J. Chem. Phys. 138::054106
     [Crossref] [Google Scholar]
  85. 85.
    Antoniou D, Schwartz SD. 2011.. Reply to “Comment on ‘Toward identification of the reaction coordinate directly from the transition state ensemble using the kernel PCA method. ’.” J. Phys. Chem. B 115::12674–75
     [Crossref] [Google Scholar]
  86. 86.
    Antoniou D, Schwartz SD. 2011.. Toward identification of the reaction coordinate directly from the transition state ensemble using the kernel PCA method. . J. Phys. Chem. B 115::2465–69
     [Crossref] [Google Scholar]
  87. 87.
    Quaytman SL, Schwartz SD. 2007.. Reaction coordinate of an enzymatic reaction revealed by transition path sampling. . PNAS 104::12253–58
     [Crossref] [Google Scholar]
  88. 88.
    Jung H, Covino R, Hummer G. 2019.. Artificial intelligence assists discovery of reaction coordinates and mechanisms from molecular dynamics simulations. . arXiv:1901.04595 [physics.chem-ph]
  89. 89.
    Kikutsuji T, Mori Y, Okazaki K, Mori T, Kim K, Matubayasi N. 2022.. Explaining reaction coordinates of alanine dipeptide isomerization obtained from deep neural networks using Explainable Artificial Intelligence (XAI). . J. Chem. Phys. 156::154108
     [Crossref] [Google Scholar]
  90. 90.
    Mehdi S, Wang D, Pant S, Tiwary P. 2022.. Accelerating all-atom simulations and gaining mechanistic understanding of biophysical systems through state predictive information bottleneck. . J. Chem. Theory Comput. 18::3231–38
     [Crossref] [Google Scholar]
  91. 91.
    Ribeiro JML, Bravo P, Wang Y, Tiwary P. 2018.. Reweighted autoencoded variational Bayes for enhanced sampling (RAVE). . J. Chem. Phys. 149::072301
     [Crossref] [Google Scholar]
  92. 92.
    Wang D, Tiwary P. 2021.. State predictive information bottleneck. . J. Chem. Phys. 154::134111
     [Crossref] [Google Scholar]
  93. 93.
    Thiede EH, Giannakis D, Dinner AR, Weare J. 2019.. Galerkin approximation of dynamical quantities using trajectory data. . J. Chem. Phys. 150::244111
     [Crossref] [Google Scholar]
  94. 94.
    Swope WC, Pitera JW, Suits F. 2004.. Describing protein folding kinetics by molecular dynamics simulations. 1. Theory. . J. Phys. Chem. B 108::6571–81
     [Crossref] [Google Scholar]
  95. 95.
    Schütte C, Fischer A, Huisinga W, Deuflhard P. 1999.. A direct approach to conformational dynamics based on hybrid Monte Carlo. . J. Comput. Phys. 151::146–68
     [Crossref] [Google Scholar]
  96. 96.
    Pande VS, Beauchamp K, Bowman GR. 2010.. Everything you wanted to know about Markov State Models but were afraid to ask. . Methods 52::99–105
     [Crossref] [Google Scholar]
  97. 97.
    Chodera JD, Singhal N, Pande VS, Dill KA, Swope WC. 2007.. Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. . J. Chem. Phys. 126::155101
     [Crossref] [Google Scholar]
  98. 98.
    Molgedey L, Schuster HG. 1994.. Separation of a mixture of independent signals using time delayed correlations. . Phys. Rev. Lett. 72::3634–37
     [Crossref] [Google Scholar]
  99. 99.
    Naritomi Y, Fuchigami S. 2011.. Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: the case of domain motions. . J. Chem. Phys. 134::065101
     [Crossref] [Google Scholar]
  100. 100.
    Schwantes CR, Pande VS. 2013.. Improvements in Markov state model construction reveal many non-native interactions in the folding of NTL9. . J. Chem. Theory Comput. 9::2000–9
     [Crossref] [Google Scholar]
  101. 101.
    Perez-Hernandez G, Paul F, Giorgino T, De Fabritiis G, Noé F. 2013.. Identification of slow molecular order parameters for Markov model construction. . J. Chem. Phys. 139::015102
     [Crossref] [Google Scholar]
  102. 102.
    Wu H, Noé F. 2020.. Variational approach for learning Markov processes from time series data. . J. Nonlinear Sci. 30::23–66
     [Crossref] [Google Scholar]
  103. 103.
    Lorpaiboon C, Thiede EH, Webber RJ, Weare J, Dinner AR. 2020.. Integrated variational approach to conformational dynamics: a robust strategy for identifying eigenfunctions of dynamical operators. . J. Phys. Chem. B 124::9354–64
     [Crossref] [Google Scholar]
  104. 104.
    Webber RJ, Thiede EH, Dow D, Dinner AR, Weare J. 2021.. Error bounds for dynamical spectral estimation. . SIAM J. Math Data Sci. 3::225–52
     [Crossref] [Google Scholar]
  105. 105.
    Chen SJ, Hassan M, Jernigan RL, Jia K, Kihara D, et al. 2023.. Protein folds vs. protein folding: differing questions, different challenges. . PNAS 120::e2214423119
     [Crossref] [Google Scholar]
  106. 106.
    Truhlar DG, Garrett BC. 1980.. Variational transition-state theory. . Acc. Chem. Res. 13::440–48
     [Crossref] [Google Scholar]
  107. 107.
    Keck JC. 1967.. Variational theory of reaction rates. . Adv. Chem. Phys. 13::85–121
     [Crossref] [Google Scholar]
  108. 108.
    Mouaffac L, Palacio-Rodriguez K, Pietrucci F. 2023.. Optimal reaction coordinates and kinetic rates from the projected dynamics of transition paths. . J. Chem. Theory Comput. 19::5701–11
     [Crossref] [Google Scholar]
  109. 109.
    Berkowitz M, Morgan JD, McCammon JA, Northrup SH. 1983.. Diffusion-controlled reactions - a variational formula for the optimum reaction coordinate. . J. Chem. Phys. 79::5563–65
     [Crossref] [Google Scholar]
  110. 110.
    Huo SH, Straub JE. 1997.. The MaxFlux algorithm for calculating variationally optimized reaction paths for conformational transitions in many body systems at finite temperature. . J. Chem. Phys. 107::5000–6
     [Crossref] [Google Scholar]
  111. 111.
    Kirmizialtin S, Elber R. 2011.. Revisiting and computing reaction coordinates with directional milestoning. . J. Phys. Chem. A 115::6137–48
     [Crossref] [Google Scholar]
  112. 112.
    Elber R, Karplus M. 1987.. A method for determining reaction paths in large molecules: application to myoglobin. . Chem. Phys. Lett. 139::375–80
     [Crossref] [Google Scholar]
  113. 113.
    Henkelman G, Uberuaga BP, Jónsson H. 2000.. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. . J. Chem. Phys. 113::9901–4
     [Crossref] [Google Scholar]
  114. 114.
    Jonsson H, Mills G, Jacobsen KW. 1998.. Nudged elastic band method for finding minimum energy paths of transitions. . In Classical and Quantum Dynamics in Condensed Phase Simulations, ed. BJ Berne, G Ciccoti, DF Coker , pp. 385–404. Singapore:: World Scientific
     [Google Scholar]
  115. 115.
    Sheppard D, Terrell R, Henkelman G. 2008.. Optimization methods for finding minimum energy paths. . J. Chem. Phys. 128::134106
     [Crossref] [Google Scholar]
  116. 116.
    Weinan E, Vanden-Eijnden E. 2006.. Towards a theory of transition paths. . J. Stat. Phys. 123::503–23
     [Crossref] [Google Scholar]
  117. 117.
    Weinan E, Ren WQ, Vanden-Eijnden E. 2007.. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. . J. Chem. Phys. 126::164103
     [Crossref] [Google Scholar]
  118. 118.
    Maragliano L, Fischer A, Vanden-Eijnden E, Ciccotti G. 2006.. String method in collective variables: minimum free energy paths and isocommittor surfaces. . J. Chem. Phys. 125::24106
     [Crossref] [Google Scholar]
  119. 119.
    Weinan E, Ren WQ, Vanden-Eijnden E. 2002.. String method for the study of rare events. . Phys. Rev. B 66::052
     [Google Scholar]
  120. 120.
    Weinan E, Ren W, Vanden-Eijnden E. 2005.. Finite temperature string method for the study of rare events. . J. Phys. Chem. B 109::6688–93
     [Crossref] [Google Scholar]
  121. 121.
    Ren W, Vanden-Eijnden E, Maragakis P, Weinan E. 2005.. Transition pathways in complex systems: application of the finite-temperature string method to the alanine dipeptide. . J. Chem. Phys. 123::134109
     [Crossref] [Google Scholar]
  122. 122.
    Pan AC, Sezer D, Roux B. 2008.. Finding transition pathways using the string method with swarms of trajectories. . J. Phys. Chem. B 112::3432–40
     [Crossref] [Google Scholar]
  123. 123.
    Li H, Wu S, Ma A. 2022.. Origin of protein quake: energy waves conducted by a precise mechanical machine. . J. Chem. Theory Comput. 18::5692–702 This paper applied the GWF method and energy flow theory to energy relaxation in myoglobin.
     [Crossref] [Google Scholar]
  124. 124.
    Wu S, Li H, Ma A. 2022.. Exact reaction coordinates for flap opening in HIV-1 protease. . PNAS 119::e2214906119 This paper marks the first successful identification of tRCs for a large-scale protein conformational change.
     [Crossref] [Google Scholar]
  125. 125.
    Li H, Ma A. 2025.. Generalization of fluctuation-dissipation relation enables predictive and optimal enhanced sampling of protein conformational states and transitions. . Nat. Commun. 16::786 This paper uncovered that both energy relaxation and activation are controlled by the same tRCs.
     [Crossref] [Google Scholar]
  126. 126.
    Chaudhury S, Makarov DE. 2010.. A harmonic transition state approximation for the duration of reactive events in complex molecular rearrangements. . J. Chem. Phys. 133::034118
     [Crossref] [Google Scholar]
  127. 127.
    Zhang BW, Jasnow D, Zuckerman DM. 2007.. Transition-event durations in one-dimensional activated processes. . J. Chem. Phys. 126::074504
     [Crossref] [Google Scholar]
  128. 128.
    Manuel AP, Lambert J, Woodside MT. 2015.. Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories. . PNAS 112::7183–88
     [Crossref] [Google Scholar]
  129. 129.
    Neupane K, Foster DA, Dee DR, Yu H, Wang F, Woodside MT. 2016.. Direct observation of transition paths during the folding of proteins and nucleic acids. . Science 352::239–42
     [Crossref] [Google Scholar]
  130. 130.
    Stewman SF, Tsui KK, Ma A. 2020.. Dynamic instability from non-equilibrium structural transitions on the energy landscape of microtubule. . Cell Syst. 11::608–24.e9
     [Crossref] [Google Scholar]
  131. 131.
    Manuchehrfar F, Li H, Ma A, Liang J. 2022.. Reactive vortexes in a naturally activated process: non-diffusive rotational fluxes at transition state uncovered by persistent homology. . J. Phys. Chem. B 126::9297–308
     [Crossref] [Google Scholar]
  132. 132.
    Hummer G. 2004.. From transition paths to transition states and rate coefficients. . J. Chem. Phys. 120::516–23
     [Crossref] [Google Scholar]
  133. 133.
    Bozovic O, Jankovic B, Hamm P. 2020.. Sensing the allosteric force. . Nat. Commun. 11::5841
     [Crossref] [Google Scholar]
  134. 134.
    Gianni S, Jemth P. 2023.. Allostery frustrates the experimentalist. . J. Mol. Biol. 435::167934
     [Crossref] [Google Scholar]
  135. 135.
    Fischer E. 1894.. Einfluss der Configuration auf die Wirkung der Enzyme. . Berichte Dtsch. Chem. Ges. 27::2985–93
     [Crossref] [Google Scholar]
/content/journals/10.1146/annurev-physchem-082423-010652
Loading
Reaction Coordinates Are Optimal Channels of Energy Flow
Annual Review of Physical Chemistry 76, 153 (2025); https://doi.org/10.1146/annurev-physchem-082423-010652
/content/journals/10.1146/annurev-physchem-082423-010652
/content/journals/10.1146/annurev-physchem-082423-010652
Loading

Data & Media loading...

Most Cited Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error