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Mechanobiology: protein refolding under force. Emerg Top Life Sci 2018; 2:687-699. [PMID: 33530665 DOI: 10.1042/etls20180044] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2018] [Revised: 09/19/2018] [Accepted: 09/19/2018] [Indexed: 02/03/2023]
Abstract
The application of direct force to a protein enables to probe wide regions of its energy surface through conformational transitions as unfolding, extending, recoiling, collapsing, and refolding. While unfolding under force typically displayed a two-state behavior, refolding under force, from highly extended unfolded states, displayed a more complex behavior. The first recording of protein refolding at a force quench step displayed an initial rapid elastic recoil, followed by a plateau phase at some extension, concluding with a collapse to a final state, at which refolding occurred. These findings stirred a lively discussion, which led to further experimental and theoretical investigation of this behavior. It was demonstrated that the polymeric chain of the unfolded protein is required to fully collapse to a globular conformation for the maturation of native structure. This behavior was modeled using one-dimensional free energy landscape over the end-to-end length reaction coordinate, the collective measured variable. However, at low forces, conformational space is not well captured by such models, and using two-dimensional energy surfaces provides further insight into the dynamics of this process. This work reviews the main concepts of protein refolding under constant force, which is essential for understanding how mechanotransducing proteins operate in vivo.
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Abstract
In a typical single-molecule force spectroscopy experiment, the ends of the
molecule of interest are connected by long polymer linkers to a pair of
mesoscopic beads trapped in the focus of two laser beams. At constant force
load, the total extension, i.e., the end-to-end distance of the molecule plus
linkers, is measured as a function of time. In the simplest systems, the
measured extension fluctuates about two values characteristic of folded and
unfolded states, with occasional transitions between them. We have recently
shown that molecular (un)folding rates can be recovered from such trajectories,
with a small linker correction, as long as the characteristic time of the bead
fluctuations is shorter than the residence time in the unfolded (folded) state.
Here, we show that accurate measurements of the molecular transition path times
require an even faster apparatus response. Transition paths, the trajectory
segments in which the molecule (un)folds, are properly resolved only if the
beads fluctuate more rapidly than the end-to-end distance of the molecule.
Therefore, over a wide regime, the measured rates may be meaningful but not the
transition path times. Analytic expressions for the measured mean transition
path times are obtained for systems diffusing anisotropically on a
two-dimensional free energy surface. The transition path times depend on the
properties both of the molecule and of the pulling device.
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Affiliation(s)
- Pilar Cossio
- Biophysics of Tropical Diseases Max Planck Tandem Group, University of Antioquia, Medellín, Colombia
| | - Gerhard Hummer
- Department of Theoretical Biophysics, Max Planck Institute of Biophysics, 60438 Frankfurt am Main, Germany
| | - Attila Szabo
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892-0520, USA
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Medina E, Satija R, Makarov DE. Transition Path Times in Non-Markovian Activated Rate Processes. J Phys Chem B 2018; 122:11400-11413. [DOI: 10.1021/acs.jpcb.8b07361] [Citation(s) in RCA: 26] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Makarov DE. Communication: Does force spectroscopy of biomolecules probe their intrinsic dynamic properties? J Chem Phys 2015; 141:241103. [PMID: 25554124 DOI: 10.1063/1.4904895] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/30/2023] Open
Abstract
In single-molecule pulling experiments, the molecule of interest is attached to a much larger object such as an atomic force microscope tip or a micrometer sized bead. The measured dynamics of molecular transitions is therefore affected by the hydrodynamic drag on the pulling instrument itself. By considering the transitions within the combined system (the molecule and the instrument), it is shown here that two distinct physical regimes exist: when the intrinsic stiffness of the molecule is greater than that of the linker connecting the molecule to the pulling setup then the pulling experiment probes the intrinsic dynamics of the molecule with only relatively small (and quantifiable) corrections resulting from the pulling setup. In contrast, when the stiffness of the linker exceeds that of the molecule, the molecular transition in question involves concerted motion of the molecule and the pulling setup and the hydrodynamic drag on the pulling instrument becomes the dominant source of friction along the molecular reaction coordinate. An analytical formula interpolating between these two cases is further derived. These results explain recent conflicting observations where some single-molecule pulling measurements report anomalously low diffusion coefficients along molecular reaction coordinates while others do not.
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Affiliation(s)
- Dmitrii E Makarov
- Department of Chemistry and Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, USA
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Berezhkovskii AM, Szabo A, Greives N, Zhou HX. Multidimensional reaction rate theory with anisotropic diffusion. J Chem Phys 2015; 141:204106. [PMID: 25429932 DOI: 10.1063/1.4902243] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023] Open
Abstract
An analytical expression is derived for the rate constant that describes diffusive transitions between two deep wells of a multidimensional potential. The expression, in contrast to the Kramers-Langer formula for the rate constant, is valid even when the diffusion is highly anisotropic. Our approach is based on a variational principle for the reactive flux and uses a trial function for the splitting probability or commitor. The theoretical result is validated by Brownian dynamics simulations.
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Affiliation(s)
- Alexander M Berezhkovskii
- Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology, National Institutes of Health, Bethesda, Maryland 20819, USA
| | - Attila Szabo
- Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20819, USA
| | - Nicholas Greives
- Department of Physics and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, USA
| | - Huan-Xiang Zhou
- Department of Physics and Institute of Molecular Biophysics, Florida State University, Tallahassee, Florida 32306, USA
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Abstract
In typical force spectroscopy experiments, a small biomolecule is attached to a soft polymer linker that is pulled with a relatively large bead or cantilever. At constant force, the total extension stochastically changes between two (or more) values, indicating that the biomolecule undergoes transitions between two (or several) conformational states. In this paper, we consider the influence of the dynamics of the linker and mesoscopic pulling device on the force-dependent rate of the conformational transition extracted from the time dependence of the total extension, and the distribution of rupture forces in force-clamp and force-ramp experiments, respectively. For these different experiments, we derive analytic expressions for the observables that account for the mechanical response and dynamics of the pulling device and linker. Possible artifacts arise when the characteristic times of the pulling device and linker become comparable to, or slower than, the lifetimes of the metastable conformational states, and when the highly anharmonic regime of stretched linkers is probed at high forces. We also revisit the problem of relating force-clamp and force-ramp experiments, and identify a linker and loading rate-dependent correction to the rates extracted from the latter. The theory provides a framework for both the design and the quantitative analysis of force spectroscopy experiments by highlighting, and correcting for, factors that complicate their interpretation.
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Nam GM, Makarov DE. Extracting intrinsic dynamic parameters of biomolecular folding from single-molecule force spectroscopy experiments. Protein Sci 2015; 25:123-34. [PMID: 26088347 DOI: 10.1002/pro.2727] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2015] [Revised: 06/17/2015] [Accepted: 06/17/2015] [Indexed: 11/12/2022]
Abstract
Single-molecule studies in which a mechanical force is transmitted to the molecule of interest and the molecular extension or position is monitored as a function of time are versatile tools for probing the dynamics of protein folding, stepping of molecular motors, and other biomolecular processes involving activated barrier crossing. One complication in interpreting such studies, however, is the fact that the typical size of a force probe (e.g., a dielectric bead in optical tweezers or the atomic force microscope tip/cantilever assembly) is much larger than the molecule itself, and so the observed molecular motion is affected by the hydrodynamic drag on the probe. This presents the experimenter with a nontrivial task of deconvolving the intrinsic molecular parameters, such as the intrinsic free energy barrier and the effective diffusion coefficient exhibited while crossing the barrier from the experimental signal. Here we focus on the dynamical aspect of this task and show how the intrinsic diffusion coefficient along the molecular reaction coordinate can be inferred from single-molecule measurements of the rates of biomolecular folding and unfolding. We show that the feasibility of accomplishing this task is strongly dependent on the relationship between the intrinsic molecular elasticity and that of the linker connecting the molecule to the force probe and identify the optimal range of instrumental parameters allowing determination of instrument-free molecular dynamics.
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Affiliation(s)
- Gi-Moon Nam
- Department of Chemistry, University of Texas at Austin, Austin, Texas, 78712
| | - Dmitrii E Makarov
- Department of Chemistry, University of Texas at Austin, Austin, Texas, 78712.,Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas, 78712
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9
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Abstract
Nonequilibrium pulling experiments provide detailed information about the thermodynamic and kinetic properties of molecules. We show that unperturbed free energy profiles as a function of molecular extension can be obtained rigorously from such experiments without using work-weighted position histograms. An inverse Weierstrass transform is used to relate the system free energy obtained from the Jarzynski equality directly to the underlying molecular free energy surface. An accurate approximation for the free energy surface is obtained by using the method of steepest descent to evaluate the inverse transform. The formalism is applied to simulated data obtained from a kinetic model of RNA folding, in which the dynamics consists of jumping between linker-dominated folded and unfolded free energy surfaces.
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Neher RA, Möbius W, Frey E, Gerland U. Optimal flexibility for conformational transitions in macromolecules. PHYSICAL REVIEW LETTERS 2007; 99:178101. [PMID: 17995372 DOI: 10.1103/physrevlett.99.178101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2007] [Indexed: 05/25/2023]
Abstract
Conformational transitions in macromolecular complexes often involve the reorientation of leverlike structures. Using a simple theoretical model, we show that the rate of such transitions is drastically enhanced if the lever is bendable, e.g., at a localized hinge. Surprisingly, the transition is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition rate is also least sensitive to the amount of "cargo" attached to the lever arm, which could be exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for multidimensional barrier crossing to configuration-dependent mobility matrices.
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Affiliation(s)
- Richard A Neher
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for Nanoscience (CeNS), LMU Munich, Theresienstrasse 37, 80333 Munich, Germany.
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Ignaczak A, Schmickler W. Effects of friction and asymmetric inner sphere reorganization energy on the electron transfer reaction rate—Two-dimensional simulations. Electrochim Acta 2007. [DOI: 10.1016/j.electacta.2006.08.032] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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12
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Pollak E, Talkner P. Reaction rate theory: what it was, where is it today, and where is it going? CHAOS (WOODBURY, N.Y.) 2005; 15:26116. [PMID: 16035918 DOI: 10.1063/1.1858782] [Citation(s) in RCA: 157] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hanggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hanggi, ibid. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even today, we do not know how to solve the real time quantum dynamics for systems with "many" degrees of freedom.
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Affiliation(s)
- Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel
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13
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Ignaczak A, Schmickler W. Simulations of adiabatic bond-breaking electron transfer reactions on metal electrodes. Chem Phys 2002. [DOI: 10.1016/s0301-0104(02)00385-3] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Bao JD. Multidimensional and memory effects on diffusion of a particle. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:061112. [PMID: 11415073 DOI: 10.1103/physreve.63.061112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2000] [Revised: 02/20/2001] [Indexed: 05/23/2023]
Abstract
The diffusion of an overdamped Brownian particle in the two-dimensional (2D) channel bounded periodically by a parabola is studied, where the particle is subject to an additive white or colored noise. The diffusion rate constant D* of the particle is evaluated by the quasi-2D approximation and the effective potential approach, and the theoretical result is compared with the Langevin simulation. The properties of the diffusion rate constant are stressed for weak and strong noise cases. It is shown that, in an entropy channel, the value of D* in units of Q decreases with increasing intensity of the colored noise. In the presence of energetic barriers, a nonmonotonic behavior of the reduced diffusion rate constant D*Q-1 as a function of the noise intensity is shown.
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Affiliation(s)
- J D Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
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15
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Barzykin AV, Frantsuzov PA. On the role of back reaction in the stochastic model of electron transfer. J Chem Phys 2001. [DOI: 10.1063/1.1329132] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Okada A. Qualitative Features of Electron Transfer Reaction for High Viscosity of Solvent and Low Activation Barrier. J Phys Chem A 2000. [DOI: 10.1021/jp0006328] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Akira Okada
- Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
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17
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Affiliation(s)
- Noam Agmon
- The Fritz Haber Research Center, Department of Physical Chemistry, The Hebrew University, Jerusalem 91904, Israel
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18
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Berezhkovskii A, Zitserman V, Yang DY, Lin S. Reversible chemical reactions in slowly relaxing environments: Kramers' turnover of the rate constant. Chem Phys 1998. [DOI: 10.1016/s0301-0104(98)00104-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Hershkovitz E, Pollak E. Multidimensional generalization of the Pollak–Grabert–Hänggi turnover theory for activated rate processes. J Chem Phys 1997. [DOI: 10.1063/1.473769] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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20
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Koper MTM. Temperature Dependence of the Transfer Coefficient of Simple Electrochemical Redox Reactions Due to Slow Solvent Dynamics. J Phys Chem B 1997. [DOI: 10.1021/jp9637993] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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22
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The steady-state Green's function method in unimolecular reactions. Generalization of the MFPT concept. I. Irreversible reactions. Chem Phys 1996. [DOI: 10.1016/0301-0104(95)00368-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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23
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Spirina OB, Cukier RI. Anisotropic dynamical effects on two‐dimensional potential energy surface reactions: Bond breaking electron transfer reactions. J Chem Phys 1996. [DOI: 10.1063/1.470850] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Berezhkovskii AM, Frishman AM, Pollak E. Variational transition state theory for multidimensional activated rate processes in the presence of anisotropic friction. J Chem Phys 1994. [DOI: 10.1063/1.467400] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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26
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Rate processes in dissipative systems: scaling in the canonical variational transition state theory. Chem Phys 1994. [DOI: 10.1016/0301-0104(94)00027-1] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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27
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Berezhkovskii AM, Dudko SA. Solvent dynamics influence on chemical reaction dynamics. J Chem Phys 1994. [DOI: 10.1063/1.467106] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Tannor DJ, Kohen D. Derivation of Kramers’ formula for condensed phase reaction rates using the method of reactive flux. J Chem Phys 1994. [DOI: 10.1063/1.467212] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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29
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Berezhkovskii A, Zitserman V. Multi-dimensional Kramers theory of the reaction rate with highly anisotropic friction. Energy diffusion for the fast coordinate versus overdamped regime for the slow coordinate. Chem Phys Lett 1993. [DOI: 10.1016/0009-2614(93)89347-k] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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30
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Berezhkovskii AM, Pollak E, Zitserman VY. Activated rate processes: Generalization of the Kramers–Grote–Hynes and Langer theories. J Chem Phys 1992. [DOI: 10.1063/1.463081] [Citation(s) in RCA: 95] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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