1
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Cherayil BJ. Internal friction as a factor in the anomalous chain length dependence of DNA transcriptional dynamics. J Chem Phys 2024; 160:014903. [PMID: 38165100 DOI: 10.1063/5.0184878] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2023] [Accepted: 12/05/2023] [Indexed: 01/03/2024] Open
Abstract
Recent experiments by Brückner et al. [Science 380, 1357 (2023)] have observed an anomalous chain length dependence of the time of near approach of widely separated pairs of genomic elements on transcriptionally active chromosomal DNA. In this paper, I suggest that the anomaly may have its roots in internal friction between neighboring segments on the DNA backbone. The basis for this proposal is a model of chain dynamics formulated in terms of a continuum scaled Brownian walk (sBw) of polymerization index N. The sBw is an extension of the simple Brownian walk model widely used in path integral calculations of polymer properties, differing from it in containing an additional parameter H (the Hurst index) that can be tuned to produce varying degrees of correlation between adjacent monomers. A calculation using the sBw of the mean time τc for chain closure predicts-under the Wilemski-Fixman approximation for diffusion-controlled reactions-that at early times, τc varies as the 2/3 power of N, in close agreement with the findings of the Brückner et al. study. Other scaling relations of that study, including those related to the probability of loop formation and the mean square displacements of terminal monomers, are also satisfactorily accounted for by the model.
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Affiliation(s)
- Binny J Cherayil
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, Karnataka, India
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2
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Tian X, Xu X, Chen Y, Chen J, Xu WS. Explicit analytical form for memory kernel in the generalized Langevin equation for end-to-end vector of Rouse chains. J Chem Phys 2022; 157:224901. [PMID: 36546812 DOI: 10.1063/5.0124925] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
The generalized Langevin equation (GLE) provides an attractive theoretical framework for investigating the dynamics of conformational fluctuations of polymeric systems. While the memory kernel is a central function in the GLE, explicit analytical forms for this function have been challenging to obtain, even for the simple models of polymer dynamics. Here, we achieve an explicit analytical expression for the memory kernel in the GLE for the end-to-end vector of Rouse chains in the overdamped limit. Our derivation takes advantage of the finding that the dynamics of the end-to-end vector of Rouse chains with both free ends are equivalent to those of Rouse chains with one free end and the other fixed. For the latter model, we first show that the equations of motion of the Rouse modes as well as their statistical properties can be obtained under the boundary conditions where the free end is held fixed temporarily. We then analytically solve the terms associated with intrachain interactions in the GLE. By formally comparing these terms with the GLE based on the Rouse modes, we obtain an explicit expression for the memory kernel, along with analytical forms for the potential field and the random colored noise force. Our analytical memory kernel is confirmed by numerical calculations in the Laplace space and is shown to yield asymptotic behaviors that are consistent with previous studies. Finally, we utilize our analytical result to simulate the cyclization dynamics of Rouse chains and discuss the scaling of the cyclization time with chain length.
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Affiliation(s)
- Xiaofei Tian
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
| | - Xiaolei Xu
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
| | - Ye Chen
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
| | - Jizhong Chen
- School of Chemical Engineering and Light Industry, Guangdong University of Technology, Guangzhou 510006, People's Republic of China
| | - Wen-Sheng Xu
- State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People's Republic of China
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3
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Thonnekottu D, Chatterjee D. CRISPR-Cas9 Genome Interrogation: A Facilitated Subdiffusive Target Search Strategy. J Phys Chem B 2020; 124:3271-3282. [PMID: 32212662 DOI: 10.1021/acs.jpcb.0c00086] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
The functional application of RNA-guided CRISPR-associated Cas9 protein, a bacterial immune system-based protein complex, via which in vivo, highly specific, and well-regulated, gene-editing processes are being monitored at an unprecedented level, has led to remarkable progress in genetic engineering and technology. The complicated in vivo process of genome interrogation followed by gene editing by the Cas9 complex was recently reported by Knight et al. (Science, 2015, 350, 823-826) using an elegant single-particle tracking method, aided by the two-photon fluorescence correlation spectroscopic technique. In contrast to the usually observed fast target-searching and protein-binding events in biophysical systems, an interesting slow genome-interrogation process by the RNA-guided CRISPR-Cas9 system through a crowded chromatin environment of a mammalian cell has been revealed in Knight et al.'s study. Motivated by this experiment, in this paper, we provide a generalized theoretical framework to capture this particular target-searching mechanism of the CRISPR-Cas9 protein complex. We show that an analysis on the basis of 3D subdiffusion under a cylindrical volume, created by several nonspecific off-target interactions from the DNA strands, can capture the essential details of the process. Moreover, on the basis of this model, we quantify the dynamics of this process and estimate the survival probability, first passage time, and the intensity correlation function of the tagged proteins of the experiment. The results from our theoretical predictions are found to be consistent with the experimental observations, and hence, seem to provide a plausible microscopic picture of the process.
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Affiliation(s)
- Diljith Thonnekottu
- Department of Physics, Indian Institute of Technology Palakkad, Palakkad, Kerala 678557, India
| | - Debarati Chatterjee
- Department of Chemistry, Indian Institute of Technology Palakkad, Palakkad, Kerala 678557, India
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4
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Kappler J, Noé F, Netz RR. Cyclization and Relaxation Dynamics of Finite-Length Collapsed Self-Avoiding Polymers. PHYSICAL REVIEW LETTERS 2019; 122:067801. [PMID: 30822085 DOI: 10.1103/physrevlett.122.067801] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2018] [Revised: 11/23/2018] [Indexed: 06/09/2023]
Abstract
We study the cyclization and relaxation dynamics of ideal as well as interacting polymers as a function of chain length N. For the cyclization time τ_{cyc} of ideal chains we recover the known scaling τ_{cyc}∼N^{2} for different backbone models, for a self-avoiding slightly collapsed chain we obtain from Langevin simulations and scaling theory a modified scaling τ_{cyc}∼N^{5/3}. The cyclization and relaxation dynamics of a finite-length collapsed chain scale differently; this unexpected dynamic multiscale behavior is rationalized by the crossover between swollen and collapsed chain behavior.
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Affiliation(s)
- Julian Kappler
- Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
| | - Frank Noé
- Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany
| | - Roland R Netz
- Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
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5
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Satija R, Makarov DE. Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding. J Phys Chem B 2019; 123:802-810. [PMID: 30648875 DOI: 10.1021/acs.jpcb.8b11137] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.
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6
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Wu YW, Yu HY. Adhesion of a polymer-grafted nanoparticle to cells explored using generalized Langevin dynamics. SOFT MATTER 2018; 14:9910-9922. [PMID: 30475366 DOI: 10.1039/c8sm01579a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
We model a polymer-grafted stealth nanoparticle (SNP) as a composite system consisting of a spherical core coated with a porous polymeric brush with end-ligands. Adjacent to target cells, the near-wall hydrodynamics, thermal fluctuations, and thermodynamic adhesive interactions simultaneously impact the transient motion of the SNP. Employing both the Langevin framework for the effective hard sphere dynamics and the coupled generalized Langevin framework for the nanoparticle-polymer dynamics, we comprehensively investigate the velocity and position temporal relaxations of the SNP in the absence and presence of end-to-end distance fluctuations for the tethered polymer. We demonstrate that polymer structural relaxations substantially impact the SNP adhesive dynamics, especially when the grafted polymer is more flexible. Moreover, a long-time tail with t-3/2 scaling due to polymer chain-length fluctuations is observed in the velocity autocorrelation for a bound SNP. Finally, the thermodynamic effects of membrane morphology on SNP adhesion are explored by modifying the membrane-mediated binding potential of mean force.
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Affiliation(s)
- Yu-Wen Wu
- Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan.
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7
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Zheng Y, Li P, Zhao N, Hou Z. Kinetics of molecular transitions with dynamic disorder in single-molecule pulling experiments. J Chem Phys 2013; 138:204102. [PMID: 23742449 DOI: 10.1063/1.4801331] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Macromolecular transitions are subject to large fluctuations of rate constant, termed as dynamic disorder. The individual or intrinsic transition rates and activation free energies can be extracted from single-molecule pulling experiments. Here we present a theoretical framework based on a generalized Langevin equation with fractional Gaussian noise and power-law memory kernel to study the kinetics of macromolecular transitions to address the effects of dynamic disorder on barrier-crossing kinetics under external pulling force. By using the Kramers' rate theory, we have calculated the fluctuating rate constant of molecular transition, as well as the experimentally accessible quantities such as the force-dependent mean lifetime, the rupture force distribution, and the speed-dependent mean rupture force. Particular attention is paid to the discrepancies between the kinetics with and without dynamic disorder. We demonstrate that these discrepancies show strong and nontrivial dependence on the external force or the pulling speed, as well as the barrier height of the potential of mean force. Our results suggest that dynamic disorder is an important factor that should be taken into account properly in accurate interpretations of single-molecule pulling experiments.
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Affiliation(s)
- Yue Zheng
- College of Chemistry, Sichuan University, Chengdu 610064, China
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8
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Dolgushev M, Blumen A. Dynamics of discrete semiflexible chains under dihedral constraints: analytic results. J Chem Phys 2013; 138:204902. [PMID: 23742511 DOI: 10.1063/1.4807058] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Here we consider the dynamics of semiflexible polymers subject both to angular and to dihedral constraints. We succeed in obtaining analytically the dynamical matrix of such systems by extending the formalism developed by Dolgushev and Blumen [J. Chem. Phys. 131, 044905 (2009)]. This leads to a set of Langevin equations whose eigenvalues determine many dynamical properties. Exemplarily, we display the mechanical relaxation loss moduli [G"(ω)] as a function of several, distinct sets of microscopic stiffness parameters; it turns out that such differences lead to macroscopically distinct patterns.
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Affiliation(s)
- Maxim Dolgushev
- Theoretical Polymer Physics, University of Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany.
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9
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Sharma R, Cherayil BJ. Subdiffusion in hair bundle dynamics: the role of protein conformational fluctuations. J Chem Phys 2012; 137:215102. [PMID: 23231261 DOI: 10.1063/1.4768902] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
The detection of sound signals in vertebrates involves a complex network of different mechano-sensory elements in the inner ear. An especially important element in this network is the hair bundle, an antenna-like array of stereocilia containing gated ion channels that operate under the control of one or more adaptation motors. Deflections of the hair bundle by sound vibrations or thermal fluctuations transiently open the ion channels, allowing the flow of ions through them, and producing an electrical signal in the process, eventually causing the sensation of hearing. Recent high frequency (0.1-10 kHz) measurements by Kozlov et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 2896 (2012)] of the power spectrum and the mean square displacement of the thermal fluctuations of the hair bundle suggest that in this regime the dynamics of the hair bundle are subdiffusive. This finding has been explained in terms of the simple Brownian motion of a filament connecting neighboring stereocilia (the tip link), which is modeled as a viscoelastic spring. In the present paper, the diffusive anomalies of the hair bundle are ascribed to tip link fluctuations that evolve by fractional Brownian motion, which originates in fractional Gaussian noise and is characterized by a power law memory. The predictions of this model for the power spectrum of the hair bundle and its mean square displacement are consistent with the experimental data and the known properties of the tip link.
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Affiliation(s)
- Rati Sharma
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India
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10
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Bhattacharyya P, Sharma R, Cherayil BJ. Confinement and viscoelastic effects on chain closure dynamics. J Chem Phys 2012; 136:234903. [DOI: 10.1063/1.4729041] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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11
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Anomalous Brownian motion discloses viscoelasticity in the ear's mechanoelectrical-transduction apparatus. Proc Natl Acad Sci U S A 2012; 109:2896-901. [PMID: 22328158 DOI: 10.1073/pnas.1121389109] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The ear detects sounds so faint that they produce only atomic-scale displacements in the mechanoelectrical transducer, yet thermal noise causes fluctuations larger by an order of magnitude. Explaining how hearing can operate when the magnitude of the noise greatly exceeds that of the signal requires an understanding both of the transducer's micromechanics and of the associated noise. Using microrheology, we characterize the statistics of this noise; exploiting the fluctuation-dissipation theorem, we determine the associated micromechanics. The statistics reveal unusual Brownian motion in which the mean square displacement increases as a fractional power of time, indicating that the mechanisms governing energy dissipation are related to those of energy storage. This anomalous scaling contradicts the canonical model of mechanoelectrical transduction, but the results can be explained if the micromechanics incorporates viscoelasticity, a salient characteristic of biopolymers. We amend the canonical model and demonstrate several consequences of viscoelasticity for sensory coding.
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12
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Chatterjee D, Cherayil BJ. The stretching of single poly-ubiquitin molecules: Static versus dynamic disorder in the non-exponential kinetics of chain unfolding. J Chem Phys 2011; 134:165104. [DOI: 10.1063/1.3582899] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
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13
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Neusius T, Daidone I, Sokolov IM, Smith JC. Configurational subdiffusion of peptides: a network study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:021902. [PMID: 21405858 DOI: 10.1103/physreve.83.021902] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2010] [Revised: 12/10/2010] [Indexed: 05/30/2023]
Abstract
Molecular dynamics (MD) simulation of linear peptides reveals configurational subdiffusion at equilibrium extending from 10⁻¹² to 10⁻⁸ s. Rouse chain and continuous-time random walk models of the subdiffusion are critically discussed. Network approaches to analyzing MD simulations are shown to reproduce the time dependence of the subdiffusive mean squared displacement, which is found to arise from the fractal-like geometry of the accessible volume in the configuration space. Convergence properties of the simulation pertaining to the subdiffusive dynamics are characterized and the effect on the subdiffusive properties of representing the solvent explicitly or implicitly is compared. Non-Markovianity and other factors limiting the range of applicability of the network models are examined.
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Affiliation(s)
- Thomas Neusius
- Computational Molecular Biophysics, Universität Heidelberg, Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany
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14
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Taloni A, Chechkin A, Klafter J. Correlations in a generalized elastic model: fractional Langevin equation approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:061104. [PMID: 21230641 DOI: 10.1103/physreve.82.061104] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2010] [Indexed: 05/30/2023]
Abstract
The generalized elastic model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, and growing interfaces. On the other hand a probe (tracer) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a fractional Langevin equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree with the corresponding quantities calculated from the GEM. Furthermore we show that the Fox H -function formalism appears to be very convenient to describe the correlation properties within the FLE approach.
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15
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Taloni A, Chechkin A, Klafter J. Generalized elastic model yields a fractional Langevin equation description. PHYSICAL REVIEW LETTERS 2010; 104:160602. [PMID: 20482037 DOI: 10.1103/physrevlett.104.160602] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2009] [Revised: 03/17/2010] [Indexed: 05/29/2023]
Abstract
Starting from a generalized elastic model which accounts for the stochastic motion of several physical systems such as membranes, (semi)flexible polymers, and fluctuating interfaces among others, we derive the fractional Langevin equation (FLE) for a probe particle in such systems, in the case of thermal initial conditions. We show that this FLE is the only one fulfilling the fluctuation-dissipation relation within a new family of fractional Brownian motion equations. The FLE for the time-dependent fluctuations of the donor-acceptor distance in a protein is shown to be recovered. When the system starts from nonthermal conditions, the corresponding FLE, which does not fulfill the fluctuation-dissipation relation, is derived.
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16
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Chatterjee D, Cherayil BJ. Anomalous reaction-diffusion as a model of nonexponential DNA escape kinetics. J Chem Phys 2010; 132:025103. [DOI: 10.1063/1.3290987] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
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17
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Chatterjee D, Cherayil BJ. Brownian particles in stationary and moving traps: the mean and variance of the heat distribution function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011118. [PMID: 19658664 DOI: 10.1103/physreve.80.011118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2009] [Revised: 06/02/2009] [Indexed: 05/28/2023]
Abstract
A recent theoretical model developed by Imparato [Phys. Rev. E 76, 050101(R) (2007)] of the experimentally measured heat and work effects produced by the thermal fluctuations of single micron-sized polystyrene beads in stationary and moving optical traps has proved to be quite successful in rationalizing the observed experimental data. The model, based on the overdamped Brownian dynamics of a particle in a harmonic potential that moves at a constant speed under a time-dependent force, is used to obtain an approximate expression for the distribution of the heat dissipated by the particle at long times. In this paper, we generalize the above model to consider particle dynamics in the presence of colored noise, without passing to the overdamped limit, as a way of modeling experimental situations in which the fluctuations of the medium exhibit long-lived temporal correlations, of the kind characteristic of polymeric solutions, for instance, or of similar viscoelastic fluids. Although we have not been able to find an expression for the heat distribution itself, we do obtain exact expressions for its mean and variance, both for the static and for the moving trap cases. These moments are valid for arbitrary times and they also hold in the inertial regime, but they reduce exactly to the results of Imparato in appropriate limits.
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Affiliation(s)
- Debarati Chatterjee
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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18
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Kneller GR, Calandrini V. Self-similar dynamics of proteins under hydrostatic pressure-Computer simulations and experiments. BIOCHIMICA ET BIOPHYSICA ACTA-PROTEINS AND PROTEOMICS 2009; 1804:56-62. [PMID: 19540369 DOI: 10.1016/j.bbapap.2009.05.007] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2009] [Revised: 05/05/2009] [Accepted: 05/29/2009] [Indexed: 10/20/2022]
Abstract
Different experimental techniques, such as kinetic studies of ligand binding and fluorescence correlation spectroscopy, have revealed that the diffusive, internal dynamics of proteins exhibits autosimilarity on the time scale from microseconds to hours. Computer simulations have demonstrated that this type of dynamics is already established on the much shorter nanosecond time scale, which is also covered by quasielastic neutron scattering experiments. The autosimilarity of protein dynamics is reflected in long-time memory effects in the underlying diffusion processes, which lead to a non-exponential decay of the observed time correlation functions. Fractional Brownian dynamics is an empirical model which is able to capture the essential aspects of internal protein dynamics. Here we give a brief introduction into the theory and show how the model can be used to interpret neutron scattering experiments and molecular dynamics simulation of proteins in solution under hydrostatic pressure.
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Affiliation(s)
- G R Kneller
- Centre de Biophysique Moléculaire, 45071 Orléans, France.
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19
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Chaudhury S, Chatterjee D, Cherayil BJ. The dynamics of single enzyme reactions: A reconsideration of Kramers' model for colored noise processes. J Chem Phys 2008; 129:075104. [DOI: 10.1063/1.2969767] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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20
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Huang L, Makarov DE. The rate constant of polymer reversal inside a pore. J Chem Phys 2008; 128:114903. [DOI: 10.1063/1.2890006] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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21
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Li CB, Yang H, Komatsuzaki T. Multiscale complex network of protein conformational fluctuations in single-molecule time series. Proc Natl Acad Sci U S A 2008; 105:536-41. [PMID: 18178627 PMCID: PMC2206571 DOI: 10.1073/pnas.0707378105] [Citation(s) in RCA: 121] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2007] [Indexed: 11/18/2022] Open
Abstract
Conformational dynamics of proteins can be interpreted as itinerant motions as the protein traverses from one state to another on a complex network in conformational space or, more generally, in state space. Here we present a scheme to extract a multiscale state space network (SSN) from a single-molecule time series. Analysis by this method enables us to lift degeneracy--different physical states having the same value for a measured observable--as much as possible. A state or node in the network is defined not by the value of the observable at each time but by a set of subsequences of the observable over time. The length of the subsequence can tell us the extent to which the memory of the system is able to predict the next state. As an illustration, we investigate the conformational fluctuation dynamics probed by single-molecule electron transfer (ET), detected on a photon-by-photon basis. We show that the topographical features of the SSNs depend on the time scale of observation; the longer the time scale, the simpler the underlying SSN becomes, leading to a transition of the dynamics from anomalous diffusion to normal Brownian diffusion.
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Affiliation(s)
- Chun-Biu Li
- *Nonlinear Sciences Laboratory, Department of Earth and Planetary Sciences, Faculty of Science, Kobe University, Nada, Kobe 657-8501, Japan
- Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan
| | - Haw Yang
- Department of Chemistry, University of California, Berkeley, CA 94720; and
- Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
| | - Tamiki Komatsuzaki
- *Nonlinear Sciences Laboratory, Department of Earth and Planetary Sciences, Faculty of Science, Kobe University, Nada, Kobe 657-8501, Japan
- Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan
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22
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Prakash MK, Marcus RA. Dielectric dispersion interpretation of single enzyme dynamic disorder, spectral diffusion, and radiative fluorescence lifetime. J Phys Chem B 2007; 112:399-404. [PMID: 17956086 DOI: 10.1021/jp0758869] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A formulation based on measurable dielectric dispersion of enzymes is developed to estimate fluctuations in electrostatic interaction energy on time scales as long as milliseconds to seconds at a local site in enzymes. Several single molecule experimental obsevations occur on this time scale, currently unreachable by real time computational trajectory simulations. We compare the experimental results on the autocorrelation function of the fluctuations of catalysis rate with the calculations using the dielectric dispersion formulation. We also discuss the autocorrelation functions of the fluorescence lifetime and of spectral diffusion. We use a previously derived relation between the observables and the electric field fluctuations and calculate the latter using dielectric dispersion data for the proteins and the Onsager regression hypothesis.
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Affiliation(s)
- Meher K Prakash
- Noyes Laboratory of Chemical Physics MC 127-72, California Institute of Technology, Pasadena, CA 91125, USA
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23
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Chaudhury S, Cherayil BJ. Modulation of electron transfer kinetics by protein conformational fluctuations during early-stage photosynthesis. J Chem Phys 2007; 127:145103. [DOI: 10.1063/1.2783845] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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24
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Chaudhury S, Cherayil BJ. Dynamic disorder in single-molecule Michaelis-Menten kinetics: The reaction-diffusion formalism in the Wilemski-Fixman approximation. J Chem Phys 2007; 127:105103. [PMID: 17867782 DOI: 10.1063/1.2768059] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Single-molecule equations for the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] mechanism of enzyme action are analyzed within the Wilemski-Fixman [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)] approximation after the effects of dynamic disorder--modeled by the anomalous diffusion of a particle in a harmonic well--are incorporated into the catalytic step of the reaction. The solution of the Michaelis-Menten equations is used to calculate the distribution of waiting times between successive catalytic turnovers in the enzyme beta-galactosidase. The calculated distribution is found to agree qualitatively with experimental results on this enzyme obtained at four different substrate concentrations. The calculations are also consistent with measurements of correlations in the fluctuations of the fluorescent light emitted during the course of catalysis, and with measurements of the concentration dependence of the randomness parameter.
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Affiliation(s)
- Srabanti Chaudhury
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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Dubbeldam JLA, Milchev A, Rostiashvili VG, Vilgis TA. Polymer translocation through a nanopore: a showcase of anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:010801. [PMID: 17677402 DOI: 10.1103/physreve.76.010801] [Citation(s) in RCA: 55] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2007] [Revised: 06/21/2007] [Indexed: 05/16/2023]
Abstract
The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s (i.e., the translocated number of segments at time t ) and shown to be governed by a universal exponent alpha=2(2nu+2-gamma(1), where nu is the Flory exponent and gamma(1) is the surface exponent. Remarkably, it turns out that the value of alpha is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0<s<N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments <s(t) and <s2(t)-<s(t)>2, which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature.
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Affiliation(s)
- J L A Dubbeldam
- Max Planck Institute for Polymer Research, 10 Ackermannweg, 55128 Mainz, Germany
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Half- and full-integer power law for distance fluctuations: Langevin dynamics in one- and two-dimensional systems. Chem Phys 2007. [DOI: 10.1016/j.chemphys.2006.10.012] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Xing J, Kim KS. Protein fluctuations and breakdown of time-scale separation in rate theories. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:061911. [PMID: 17280100 DOI: 10.1103/physreve.74.061911] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2006] [Revised: 10/03/2006] [Indexed: 05/13/2023]
Abstract
A long-time fluctuation correlation function with a power-law form has been observed in recent single-molecule experiments by the Xie group. By analyzing the dynamics of an elastic network model (ENM) under white noise, we show that the observed long-time memory kernel can be explained by the discrepancy between the experimentally measured coordinate (or the coordinate directly coupled to protein function) and the minimum energy path of the system. Consequently, the dynamics of the measured collective coordinate has contributions from degrees of freedoms with a broad distribution of time scales. Our study also implies that the widely used ENM Hamiltonian should be viewed as a coarse-grained model of a protein over a rugged energy landscape. Large effective drag coefficients are needed to describe protein dynamics with the ENM's.
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Affiliation(s)
- Jianhua Xing
- Chemistry and Material Science Directorate, University of California and Lawrence Livermore National Laboratory, Livermore, California 94550, USA.
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Chaudhury S, Cherayil BJ. Structural relaxation in complex liquids: Non-Markovian dynamics in a bistable potential. J Chem Phys 2006; 125:184505. [PMID: 17115763 DOI: 10.1063/1.2374887] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
The time correlation function C(t) identical with <x(0)x(t)> of the distance fluctuations of a particle moving in a bistable potential under the action of fractional Gaussian noise (fGn) is calculated from a Smoluchowski-type equation derived from a generalized Langevin equation (GLE). The time derivative of this function, dC(t)dt, is compared with data from optical Kerr effect measurements of liquid crystal dynamics in the vicinity of the isotropic-to-nematic transition, which are related to the time derivative of an orientational correlation function. A number of characteristic features of the experimental decay curves, including short and intermediate time power law behavior and long time exponential relaxation, are qualitatively reproduced by the analytical calculations, even though the latter do not explicitly treat orientational degrees of freedom. The GLE formalism with fGn was, in fact, originally proposed as a model of protein conformational fluctuations, so the present results suggest that it may also serve more generally as a model of structural relaxation in complex condensed phase media.
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Affiliation(s)
- Srabanti Chaudhury
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India
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Abstract
In this paper we investigate some general aspects of stochastic models of dynamic disorder. First, we reexamine the Zwanzig model for the kinetics of escape through a fluctuating hole. We show that this model is trivially connected to the canonical model of the broadening of the zero-phonon line (ZPL) in crystals. This provides a new perspective of the Wang-Wolynes expression for the rate of escape from a geometric bottleneck with non-Markovian Gaussian fluctuations. Motivated by recent single-molecule experiments, we examine more general examples of fluctuation processes from the perspective of cumulant expansions. Finally, we discuss recent single-molecule experiments probing enzyme turnover performed by Xie and co-workers.
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Affiliation(s)
- David R Reichman
- Department of Chemistry, Columbia University, 3000 Broadway, New York, New York, 10025, USA
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Chaudhury S, Cherayil BJ. Approximate first passage time distribution for barrier crossing in a double well under fractional Gaussian noise. J Chem Phys 2006; 125:114106. [PMID: 16999465 DOI: 10.1063/1.2354089] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The distribution of waiting times, f(t), between successive turnovers in the catalytic action of single molecules of the enzyme beta-galactosidase has recently been determined in closed form by Chaudhury and Cherayil [J. Chem. Phys. 125, 024904 (2006)] using a one-dimensional generalized Langevin equation (GLE) formalism in combination with Kramers' flux-over-population approach to barrier crossing dynamics. The present paper provides an alternative derivation of f(t) that eschews this approach, which is strictly applicable only under conditions of local equilibrium. In this alternative derivation, a double well potential is incorporated into the GLE, along with a colored noise term representing protein conformational fluctuations, and the resulting equation transformed approximately to a Smoluchowski-type equation. f(t) is identified with the first passage time distribution for a particle to reach the barrier top starting from an equilibrium distribution of initial points, and is determined from the solution of the above equation using local boundary conditions. The use of such boundary conditions is necessitated by the absence of definite information about the precise nature of the boundary conditions applicable to stochastic processes governed by non-Markovian dynamics. f(t) calculated in this way is found to have the same analytic structure as the distribution calculated by the flux-over-population method.
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Affiliation(s)
- Srabanti Chaudhury
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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Chaudhury S, Cherayil BJ. Complex chemical kinetics in single enzyme molecules: Kramers’s model with fractional Gaussian noise. J Chem Phys 2006; 125:24904. [PMID: 16848608 DOI: 10.1063/1.2209231] [Citation(s) in RCA: 57] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
A model of barrier crossing dynamics governed by fractional Gaussian noise and the generalized Langevin equation is used to study the reaction kinetics of single enzymes subject to conformational fluctuations. The direct application of Kramers's flux-over-population method to this model yields analytic expressions for the time-dependent transmission coefficient and the distribution of waiting times for barrier crossing. These expressions are found to reproduce the observed trends in recent simulations and experiments.
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Affiliation(s)
- Srabanti Chaudhury
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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Tang J, Lin SH. Distance versus energy fluctuations and electron transfer in single protein molecules. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:061108. [PMID: 16906810 DOI: 10.1103/physreve.73.061108] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2006] [Indexed: 05/11/2023]
Abstract
Stochastic nature due to distance and energy fluctuations of single protein molecules involved in electron-transfer (ET) reactions is studied. Distance fluctuations have been assumed previously for causing the slow fluctuations in the ET rates between a donor-acceptor pair constrained to a native protein. Although the observed t(-1/2) power law can be derived using Langevin dynamics with a simple chain model, some discrepancies exist. The friction coefficient and the Rouse segment time constant deduced from experimental data are several orders of magnitude too large, even though the extracted force constant is reasonable. Therefore, questions are raised about the distance-fluctuation mechanism and the activationless ET hypothesis. As an alternative mechanism, we considered fluctuations in activation energy and analyzed the data from two different single protein experiments to determine spectral distribution of energy fluctuations.
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Affiliation(s)
- Jau Tang
- Center for Ultrafast Science and Technology, California Institute of Technology, Pasadena, California 91125, USA.
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Tang J, Marcus RA. Chain dynamics and power-law distance fluctuations of single-molecule systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:022102. [PMID: 16605373 DOI: 10.1103/physreve.73.022102] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2005] [Indexed: 05/08/2023]
Abstract
Chain-dynamics-induced distance fluctuations between any two points in a finite chain with or without cross links are investigated. This model leads to three regimes of temporal behavior for distance autocorrelation: (i) initial flat time dependence, (ii) t(-alpha) power law, and (iii) long-time exponential decay. For an ideal Rouse chain with frequency-independent friction, alpha = 1/2. The span of the characteristic power-law behavior of a long chain could be reduced significantly with the presence of cross links.
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Affiliation(s)
- Jau Tang
- Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125, USA.
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Min W, Xie XS. Kramers model with a power-law friction kernel: dispersed kinetics and dynamic disorder of biochemical reactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:010902. [PMID: 16486113 DOI: 10.1103/physreve.73.010902] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2005] [Indexed: 05/06/2023]
Abstract
Kramers' model for the rate of chemical reaction is generalized to explain the phenomena of dispersed kinetics and dynamic disorder in biochemical reactions, by incorporating the newly observed power-law friction kernel into the generalized Langevin equation for a one-dimensional reaction ordinate. This new model accounts for time scale overlap between conformational and chemical dynamics, and quantitatively describes the multi-exponential kinetics and memory effects of fluctuating rate constants, which have been revealed by recent single-molecule experiments.
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Affiliation(s)
- Wei Min
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138
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Min W, English BP, Luo G, Cherayil BJ, Kou SC, Xie XS. Fluctuating enzymes: lessons from single-molecule studies. Acc Chem Res 2005; 38:923-31. [PMID: 16359164 DOI: 10.1021/ar040133f] [Citation(s) in RCA: 273] [Impact Index Per Article: 14.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Recent single-molecule enzymology measurements with improved statistics have demonstrated that a single enzyme molecule exhibits large temporal fluctuations of the turnover rate constant at a broad range of time scales (from 1 ms to 100 s). The rate constant fluctuations, termed as dynamic disorder, are associated with fluctuations of the protein conformations observed on the same time scales. We discuss the unique information extractable from these experiments and the reconciliation of these observations with ensemble-averaged Michaelis-Menten equation. A theoretical model based on the generalized Langevin equation (GLE) treatment of Kramers' barrier crossing problem for chemical reactions accounts naturally for the observation of dynamic disorder and highly dispersed kinetics.
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Affiliation(s)
- Wei Min
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA
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