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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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2
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Kumar L, Nash A, Harms C, Planas-Iglesias J, Wright D, Klein-Seetharaman J, Sarkar SK. Allosteric Communications between Domains Modulate the Activity of Matrix Metalloprotease-1. Biophys J 2020; 119:360-374. [PMID: 32585130 PMCID: PMC7376139 DOI: 10.1016/j.bpj.2020.06.010] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Revised: 05/30/2020] [Accepted: 06/09/2020] [Indexed: 11/23/2022] Open
Abstract
An understanding of the structure-dynamics relationship is essential for understanding how a protein works. Prior research has shown that the activity of a protein correlates with intradomain dynamics occurring at picosecond to millisecond timescales. However, the correlation between interdomain dynamics and the function of a protein is poorly understood. Here, we show that communications between the catalytic and hemopexin domains of matrix metalloprotease-1 (MMP1) on type 1 collagen fibrils correlate with its activity. Using single-molecule Förster resonance energy transfer, we identified functionally relevant open conformations in which the two MMP1 domains are well separated, which were significantly absent for catalytically inactive point mutant (E219Q) of MMP1 and could be modulated by an inhibitor or an enhancer of activity. The observed relevance of open conformations resolves the debate about the roles of open and closed MMP1 structures in function. We fitted the histograms of single-molecule Förster resonance energy transfer values to a sum of two Gaussians and the autocorrelations to an exponential and power law. We used a two-state Poisson process to describe the dynamics and calculate the kinetic rates from the fit parameters. All-atom and coarse-grained simulations reproduced some of the experimental features and revealed substrate-dependent MMP1 dynamics. Our results suggest that an interdomain separation facilitates opening up the catalytic pocket so that the collagen chains come closer to the MMP1 active site. Coordination of functional conformations at different parts of MMP1 occurs via allosteric communications that can take place via interactions mediated by collagen even if the linker between the domains is absent. Modeling dynamics as a Poisson process enables connecting the picosecond timescales of molecular dynamics simulations with the millisecond timescales of single-molecule measurements. Water-soluble MMP1 interacting with water-insoluble collagen fibrils poses challenges for biochemical studies that the single-molecule tracking can overcome for other insoluble substrates. Interdomain communications are likely important for multidomain proteins.
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Affiliation(s)
- Lokender Kumar
- Department of Physics, Colorado School of Mines, Golden, Colorado
| | - Anthony Nash
- Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, United Kingdom
| | - Chase Harms
- Department of Physics, Colorado School of Mines, Golden, Colorado
| | - Joan Planas-Iglesias
- Warwick Medical School, University of Warwick, Coventry, United Kingdom; Loschmidt Laboratories, Department of Experimental Biology, Faculty of Science, Masaryk University, Brno, Czech Republic
| | - Derek Wright
- Department of Physics, Colorado School of Mines, Golden, Colorado
| | - Judith Klein-Seetharaman
- Warwick Medical School, University of Warwick, Coventry, United Kingdom; Department of Chemistry, Colorado School of Mines, Golden, Colorado
| | - Susanta K Sarkar
- Department of Physics, Colorado School of Mines, Golden, Colorado.
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3
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Cao X, Zhang B, Zhao N. Effective temperature scaled dynamics of a flexible polymer in an active bath. Mol Phys 2020. [DOI: 10.1080/00268976.2020.1730992] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
Affiliation(s)
- Xiuli Cao
- College of Chemistry, Sichuan University, Chengdu, China
| | - Bingjie Zhang
- College of Chemistry, Sichuan University, Chengdu, China
| | - Nanrong Zhao
- College of Chemistry, Sichuan University, Chengdu, China
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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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Affiliation(s)
- Yukun Bian
- College of Physical Science and Technology, Sichuan University, Chengdu, China
| | - Peng Li
- College of Physical Science and Technology, Sichuan University, Chengdu, China
| | - Nanrong Zhao
- College of Chemistry, Sichuan University, Chengdu, China
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7
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Ding H, Jiang H, Zhao N, Hou Z. Diffusion of a Rouse chain in porous media: A mode-coupling-theory study. Phys Rev E 2017; 95:012121. [PMID: 28208313 DOI: 10.1103/physreve.95.012121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2016] [Indexed: 11/07/2022]
Abstract
We use a kinetic mode-coupling theory (MCT) combining with generalized Langevin equation (GLE) to study the diffusion and conformational dynamics of a bead-spring Rouse chain (RC) dissolved in porous media. The media contains fluid particles and immobile matrix ones wherein the latter leads to the lack of translational invariance. The friction kernel ζ(t) used in the GLE can be obtained directly by adopting a simple density-functional approach in which the density correlators calculated by MCT equations of porous media serve as inputs. Due to cage effects generated by surrounding particles, ζ(t) shows a very long tail memory in the high volume fraction of fluid and matrix. It is found that the long-time center-of-mass diffusion constant D_{CM} of the RC decreases with the increment of volume fraction, influencing more strongly by the matrix particles than by the fluid ones. The auto-correlation function (ACF) of the end-to-end distance fluctuation can also be calculated theoretically based on GLE. Of particular interest is that the power-law region of ACF has a nearly fixed length in logarithmic scale when it shifts to longer time range, with increasing the volume fraction of media particles. Moreover, the effect of lack of translational invariance has been investigated by comparing the results between fluid-matrix and pure fluid cases under identical total volume fraction.
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Affiliation(s)
- Huai Ding
- Department of Chemical Physics & Hefei National Laboratory for Physical Sciences at Microscales, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Huijun Jiang
- Department of Chemical Physics & Hefei National Laboratory for Physical Sciences at Microscales, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Nanrong Zhao
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Zhonghuai Hou
- Department of Chemical Physics & Hefei National Laboratory for Physical Sciences at Microscales, University of Science and Technology of China, Hefei, Anhui 230026, China
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Bian Y, Wang Z, Chen A, Zhao N. Fluctuating bottleneck model studies on kinetics of DNA escape from α-hemolysin nanopores. J Chem Phys 2015; 143:184908. [PMID: 26567685 DOI: 10.1063/1.4935118] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
We have proposed a fluctuation bottleneck (FB) model to investigate the non-exponential kinetics of DNA escape from nanometer-scale pores. The basic idea is that the escape rate is proportional to the fluctuating cross-sectional area of DNA escape channel, the radius r of which undergoes a subdiffusion dynamics subjected to fractional Gaussian noise with power-law memory kernel. Such a FB model facilitates us to obtain the analytical result of the averaged survival probability as a function of time, which can be directly compared to experimental results. Particularly, we have applied our theory to address the escape kinetics of DNA through α-hemolysin nanopores. We find that our theoretical framework can reproduce the experimental results very well in the whole time range with quite reasonable estimation for the intrinsic parameters of the kinetics processes. We believe that FB model has caught some key features regarding the long time kinetics of DNA escape through a nanopore and it might provide a sound starting point to study much wider problems involving anomalous dynamics in confined fluctuating channels.
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Affiliation(s)
- Yukun Bian
- Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
| | - Zilin Wang
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Anpu Chen
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Nanrong Zhao
- College of Chemistry, Sichuan University, Chengdu 610064, China
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9
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Li P, Dong Y, Zhao N, Hou Z. Distance fluctuation of a single molecule in Lennard-Jones liquid based on generalized Langevin equation and mode coupling theory. J Chem Phys 2014. [DOI: 10.1063/1.4870824] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023] Open
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10
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Zheng Y, Bian Y, Zhao N, Hou Z. Stretching of single poly-ubiquitin molecules revisited: dynamic disorder in the non-exponential unfolding kinetics. J Chem Phys 2014; 140:125102. [PMID: 24697481 DOI: 10.1063/1.4869206] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Abstract
A theoretical framework based on a generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and a power-law memory kernel is presented to describe the non-exponential kinetics of the unfolding of a single poly-ubiquitin molecule under a constant force [T.-L. Kuo, S. Garcia-Manyes, J. Li, I. Barel, H. Lu, B. J. Berne, M. Urbakh, J. Klafter, and J. M. Fernández, Proc. Natl. Acad. Sci. U.S.A. 107, 11336 (2010)]. Such a GLE-fGn strategy is made on the basis that the pulling coordinate variable x undergoes subdiffusion, usually resulting from conformational fluctuations, over a one-dimensional force-modified free-energy surface U(x, F). By using the Kramers' rate theory, we have obtained analytical formulae for the time-dependent rate coefficient k(t, F), the survival probability S(t, F) as well as the waiting time distribution function f(t, F) as functions of time t and force F. We find that our results can fit the experimental data of f(t, F) perfectly in the whole time range with a power-law exponent γ = 1/2, the characteristic of typical anomalous subdiffusion. In addition, the fitting of the survival probabilities for different forces facilitates us to reach rather reasonable estimations for intrinsic properties of the system, such as the free-energy barrier and the distance between the native conformation and the transition state conformation along the reaction coordinate, which are in good agreements with molecular dynamics simulations in the literatures. Although static disorder has been implicated in the original work of Kuo et al., our work suggests a sound and plausible alternative interpretation for the non-exponential kinetics in the stretching of poly-ubiquitin molecules, associated with dynamic disorder.
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Affiliation(s)
- Yue Zheng
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Yukun Bian
- Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
| | - Nanrong Zhao
- College of Chemistry, Sichuan University, Chengdu 610064, China
| | - Zhonghuai Hou
- Hefei National Laboratory for Physical Sciences at the Microscale and Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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11
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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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12
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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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13
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Wang Y, Zhao N, Yan Y. Langevin dynamics of correlated subdiffusion and normal diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041142. [PMID: 22680452 DOI: 10.1103/physreve.85.041142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2011] [Revised: 11/29/2011] [Indexed: 06/01/2023]
Abstract
We analyze the dissipative dynamics of a particle governed by a two-dimensional generalized Langevin equation with coupled fractional Gaussian noise and white noise in its respective coordinates, assuming the lowest-order coupling form. Two situations are studied: In the first the particle is free from external force and in the second the particle is subject to a two-dimensional harmonic potential. We derive the general expressions for the mean values, variances, and velocity autocorrelation function and evaluate their temporal evolutions via the numerical Laplace inversion technique. Through the analytical results of the short-time and long-time behaviors, we also explicitly elucidate the effects of fluctuation correlation coupling and interoscillator coupling on the dynamic behaviors of the particle. It is shown that in both situations the couplings do not affect the short-time behavior of self-diffusions in each coordinate, and the subdiffusive and normal diffusive features of these processes resemble those in a one-dimensional system with fractional Gaussian noise and white noise, respectively. However, over a long time period, the fluctuation correlation extends the characteristic time scales for the self-diffusions of a free particle; while only the interoscillator coupling induces a retardation of the relaxation processes of a bounded particle toward equilibrium. Moreover, both couplings generate a cross diffusion, whose long-time approximation has two possible forms, the selection of which depends on the relevant time scales of self-diffusions in each coordinate.
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Affiliation(s)
- Yingxi Wang
- College of Chemistry, Sichuan University, Chengdu 610064, China
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14
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Goychuk I. Viscoelastic Subdiffusion: Generalized Langevin Equation Approach. ADVANCES IN CHEMICAL PHYSICS 2012. [DOI: 10.1002/9781118197714.ch5] [Citation(s) in RCA: 91] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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15
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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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16
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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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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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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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Cieplak M, Niewieczerzał S. Hydrodynamic interactions in protein folding. J Chem Phys 2009; 130:124906. [DOI: 10.1063/1.3050103] [Citation(s) in RCA: 24] [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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Nishioka H, Ueda N, Kakitani T. Correlation between square of electron tunneling matrix element and donor-acceptor distance in fluctuating protein media. Biophysics (Nagoya-shi) 2008; 4:19-28. [PMID: 27857572 PMCID: PMC5036608 DOI: 10.2142/biophysics.4.19] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2008] [Accepted: 09/17/2008] [Indexed: 12/01/2022] Open
Abstract
Correlation between fluctuations of the square of electron tunneling matrix element TDA2 and the donor-acceptor distance RDA in the electron transfer (ET) reaction from bacteriopheophytin anion to the primary quinone of the reaction center in the photosynthetic bacteria Rhodobacter sphaeroides is investigated by a combined study of molecular dynamics simulations of the protein conformation fluctuation and quantum chemical calculations. We adopted two kinds of RDA ; edge-to-edge distance REE and center-to-center distance RCC . The value of TDA2 distributed over more than 5 orders of magnitude and the fluctuation of the value of RDA distributed over more than 1.8 Å for the 106 instantaneous conformations of 1 ns simulation. We made analysis of the time-averaged correlation step by step as follows. We divide the 106 simulation data into 1000/t parts of small data set to obtain the averaged data points of <TDA2> t and <REE > t or <RCC > t . Plotting the 1000/t sets of log10 <TDA2> t as a function of <REE > t or <RCC > t , we made a principal coordinate analysis for these distributions. The slopes <βE > t and <βC > t of the primary axis are very large at small value of t and they are decreased considerably as t becomes large. The ellipticity for the distribution of <TDA2> tvs <REE > t which can be a measure for the degree of correlation became very small when t is large, while it does not hold for the distribution of <TDA2> tvs <RCC > t . These results indicate that only the correlation between <TDA2> t and <REE > t for large t satisfies the well-known linear relation ("Dutton law"), although the slope is larger than the original value 1.4 Å-1. Based on the present result, we examined the analysis of the dynamic disorder by means of the single-molecule spectroscopy by Xie and co-workers with use of the "Dutton law".
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Affiliation(s)
- Hirotaka Nishioka
- Graduate School of Environmental and Human Sciences, Meijo University, Tempaku-ku, Nagoya 468-8502, Japan
| | - Nobuharu Ueda
- Department of General Education, Faculty of Science and Technology, Meijo University, Tempaku-ku, Nagoya 468-8502, Japan
| | - Toshiaki Kakitani
- Graduate School of Environmental and Human Sciences, Meijo University, Tempaku-ku, Nagoya 468-8502, Japan; Department of General Education, Faculty of Science and Technology, Meijo University, Tempaku-ku, Nagoya 468-8502, Japan
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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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22
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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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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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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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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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27
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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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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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