51
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Stachura S, Kneller GR. Communication: Probing anomalous diffusion in frequency space. J Chem Phys 2015; 143:191103. [DOI: 10.1063/1.4936129] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
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52
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Toppozini L, Roosen-Runge F, Bewley RI, Dalgliesh RM, Perring T, Seydel T, Glyde HR, García Sakai V, Rheinstädter MC. Anomalous and anisotropic nanoscale diffusion of hydration water molecules in fluid lipid membranes. SOFT MATTER 2015; 11:8354-8371. [PMID: 26338138 DOI: 10.1039/c5sm01713k] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We have studied nanoscale diffusion of membrane hydration water in fluid-phase lipid bilayers made of 1,2-dimyristoyl-3-phosphocholine (DMPC) using incoherent quasi-elastic neutron scattering. Dynamics were fit directly in the energy domain using the Fourier transform of a stretched exponential. By using large, 2-dimensional detectors, lateral motions of water molecules and motions perpendicular to the membranes could be studied simultaneously, resulting in 2-dimensional maps of relaxation time, τ, and stretching exponent, β. We present experimental evidence for anomalous (sub-diffusive) and anisotropic diffusion of membrane hydration water molecules over nanometer distances. By combining molecular dynamics and Brownian dynamics simulations, the potential microscopic origins for the anomaly and anisotropy of hydration water were investigated. Bulk water was found to show intrinsic sub-diffusive motion at time scales of several picoseconds, likely related to caging effects. In membrane hydration water, however, the anisotropy of confinement and local dynamical environments leads to an anisotropy of relaxation times and stretched exponents, indicative of anomalous dynamics.
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Affiliation(s)
- Laura Toppozini
- Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada.
| | | | | | | | - Toby Perring
- ISIS, Rutherford Appleton Laboratory, Didcot, UK
| | | | - Henry R Glyde
- Department of Physics and Astronomy, University of Delaware, Newark, Delaware, USA
| | | | - Maikel C Rheinstädter
- Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada.
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53
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Mardoukhi Y, Jeon JH, Metzler R. Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster. Phys Chem Chem Phys 2015; 17:30134-47. [PMID: 26503611 DOI: 10.1039/c5cp03548a] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
Abstract
We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law ∼T(-h) with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.
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Affiliation(s)
- Yousof Mardoukhi
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.
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54
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Abstract
The paper presents a rigorous derivation of the velocity autocorrelation function for an anomalously diffusing slow solute particle in a bath of fast solvent molecules. The result is obtained within the framework of the generalized Langevin equation and uses only scaling arguments and identities which are based on asymptotic analysis. It agrees with the velocity autocorrelation function of an anomalously diffusing Rayleigh particle whose dynamics is described by a fractional Ornstein-Uhlenbeck process in velocity space. A simple semi-analytical example illustrates under which conditions the latter model is appropriate.
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Affiliation(s)
- Gerald R Kneller
- Centre de Biophys. Moléculaire, CNRS, Rue Charles Sadron, 45071 Orléans, France; Synchrotron Soleil, L'Orme des Merisiers, 91192 Gif-sur-Yvette, France; and Université d'Orléans, Chateau de la Source-Av. du Parc Floral, 45067 Orléans, France
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55
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Sandev T, Chechkin AV, Korabel N, Kantz H, Sokolov IM, Metzler R. Distributed-order diffusion equations and multifractality: Models and solutions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042117. [PMID: 26565178 DOI: 10.1103/physreve.92.042117] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2015] [Indexed: 06/05/2023]
Abstract
We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.
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Affiliation(s)
- Trifce Sandev
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
- Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia
| | - Aleksei V Chechkin
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
- Akhiezer Institute for Theoretical Physics, Kharkov 61108, Ukraine
- Institute for Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
| | - Nickolay Korabel
- School of Mathematics, The University of Manchester, Manchester M60 1QD, United Kingdom
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - Igor M Sokolov
- Institute of Physics, Humboldt University Berlin, Newtonstrasse 15, D-12489 Berlin, Germany
| | - Ralf Metzler
- Institute for Physics and Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
- Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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56
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Bakalis E, Höfinger S, Venturini A, Zerbetto F. Crossover of two power laws in the anomalous diffusion of a two lipid membrane. J Chem Phys 2015; 142:215102. [DOI: 10.1063/1.4921891] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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57
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Safdari H, Chechkin AV, Jafari GR, Metzler R. Aging scaled Brownian motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042107. [PMID: 25974439 DOI: 10.1103/physreve.91.042107] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2015] [Indexed: 06/04/2023]
Abstract
Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.
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Affiliation(s)
- Hadiseh Safdari
- Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
- Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Aleksei V Chechkin
- Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
- Max-Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
| | - Gholamreza R Jafari
- Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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58
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Yamamoto E, Akimoto T, Yasui M, Yasuoka K. Origin of 1/f noise in hydration dynamics on lipid membrane surfaces. Sci Rep 2015; 5:8876. [PMID: 25743377 PMCID: PMC4351557 DOI: 10.1038/srep08876] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2014] [Accepted: 02/09/2015] [Indexed: 12/15/2022] Open
Abstract
Water molecules on lipid membrane surfaces are known to contribute to membrane stability by connecting lipid molecules and acting as a water bridge. Although water structures and diffusivities near the membrane surfaces have been extensively studied, hydration dynamics on the surfaces has remained an open question. Here we investigate residence time statistics of water molecules on the surface of lipid membranes using all-atom molecular dynamics simulations. We show that hydration dynamics on the lipid membranes exhibits 1/f noise. Constructing a dichotomous process for the hydration dynamics, we find that residence times in each state follow a power-law with exponential cutoff and that the process can be regarded as a correlated renewal process where interoccurrence times are correlated. The results imply that the origin of the 1/f noise in hydration dynamics on the membrane surfaces is a combination of a power-law distribution with cutoff of interoccurrence times of switching events and a long-term correlation between the interoccurrence times. These results suggest that the 1/f noise attributed to the correlated renewal process may contribute to the stability of the hydration layers and lipid membranes.
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Affiliation(s)
- Eiji Yamamoto
- Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Takuma Akimoto
- Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Masato Yasui
- Department of Pharmacology, School of Medicine, Keio University, 35 Shinanomachi, Shinju-ku, Tokyo 160-8582, Japan
| | - Kenji Yasuoka
- Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
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59
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Reddy T, Shorthouse D, Parton DL, Jefferys E, Fowler PW, Chavent M, Baaden M, Sansom MSP. Nothing to sneeze at: a dynamic and integrative computational model of an influenza A virion. Structure 2015; 23:584-597. [PMID: 25703376 PMCID: PMC4353694 DOI: 10.1016/j.str.2014.12.019] [Citation(s) in RCA: 75] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2014] [Revised: 12/09/2014] [Accepted: 12/17/2014] [Indexed: 11/28/2022]
Abstract
The influenza virus is surrounded by an envelope composed of a lipid bilayer and integral membrane proteins. Understanding the structural dynamics of the membrane envelope provides biophysical insights into aspects of viral function, such as the wide-ranging survival times of the virion in different environments. We have combined experimental data from X-ray crystallography, nuclear magnetic resonance spectroscopy, cryo-electron microscopy, and lipidomics to build a model of the intact influenza A virion. This is the basis of microsecond-scale coarse-grained molecular dynamics simulations of the virion, providing simulations at different temperatures and with varying lipid compositions. The presence of the Forssman glycolipid alters a number of biophysical properties of the virion, resulting in reduced mobility of bilayer lipid and protein species. Reduced mobility in the virion membrane may confer physical robustness to changes in environmental conditions. Our simulations indicate that viral spike proteins do not aggregate and thus are competent for multivalent immunoglobulin G interactions.
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Affiliation(s)
- Tyler Reddy
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - David Shorthouse
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - Daniel L Parton
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - Elizabeth Jefferys
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - Philip W Fowler
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - Matthieu Chavent
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - Marc Baaden
- Institut de Biologie Physico-Chimique, Centre National de la Recherche Scientifique, UPR9080, Université Paris Diderot, Sorbonne Paris Cité, Paris, France
| | - Mark S P Sansom
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK.
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60
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Abstract
Modern single particle tracking techniques and many large scale simulations produce time series r(t) of the position of a tracer particle. Standardly these are evaluated in terms of the time averaged mean squared displacement. For ergodic processes such as Brownian motion, one can interpret the results of such an analysis in terms of the known theories for the corresponding ensemble averaged mean squared displacement, if only the measurement time is sufficiently long. In anomalous diffusion processes, that are widely observed over many orders of magnitude, the equivalence between (long) time and ensemble averages may be broken (weak ergodicity breaking). In such cases the time averages may no longer be interpreted in terms of ensemble theories. Here we collect some recent results on weakly non-ergodic systems with respect to the time averaged mean squared displacement and the inherent irreproducibility of individual measurements. We also address the phenomenon of ageing, the dependence of physical observables on the time span between initial preparation of the system and the start of the measurement.
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Affiliation(s)
- Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
- Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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61
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Ghosh SK, Cherstvy AG, Metzler R. Non-universal tracer diffusion in crowded media of non-inert obstacles. Phys Chem Chem Phys 2015; 17:1847-58. [DOI: 10.1039/c4cp03599b] [Citation(s) in RCA: 98] [Impact Index Per Article: 10.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
Abstract
For tracer motion in an array of attractive obstacles we observe transient, non-ergodic anomalous diffusion depending on the obstacle density.
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Affiliation(s)
- Surya K. Ghosh
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Andrey G. Cherstvy
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
- Department of Physics
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62
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Palyulin VV, Ala-Nissila T, Metzler R. Polymer translocation: the first two decades and the recent diversification. SOFT MATTER 2014; 10:9016-37. [PMID: 25301107 DOI: 10.1039/c4sm01819b] [Citation(s) in RCA: 139] [Impact Index Per Article: 13.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous-infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis.
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Affiliation(s)
- Vladimir V Palyulin
- Institute for Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany.
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63
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Cherstvy AG, Metzler R. Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012134. [PMID: 25122278 DOI: 10.1103/physreve.90.012134] [Citation(s) in RCA: 56] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2014] [Indexed: 06/03/2023]
Abstract
We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x|(α) of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P(x,t), we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.
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Affiliation(s)
- A G Cherstvy
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - R Metzler
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany and Department of Physics, Tampere University of Technology, 33101 Tampere, Finland
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64
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Soula H, Caré B, Beslon G, Berry H. Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium. Biophys J 2014; 105:2064-73. [PMID: 24209851 DOI: 10.1016/j.bpj.2013.07.023] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2013] [Revised: 07/04/2013] [Accepted: 07/16/2013] [Indexed: 01/06/2023] Open
Abstract
Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.
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Affiliation(s)
- Hédi Soula
- EPI Beagle, INRIA Rhône-Alpes, F-69603, Villeurbanne, France; Université de Lyon, Inserm UMR1060, CarMeN, F-69621 Villeurbanne, France.
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65
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Polanowski P, Sikorski A. Simulation of diffusion in a crowded environment. SOFT MATTER 2014; 10:3597-3607. [PMID: 24663121 DOI: 10.1039/c3sm52861h] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We performed extensive and systematic simulation studies of two-dimensional fluid motion in a complex crowded environment. In contrast to other studies we focused on cooperative phenomena that occurred if the motion of particles takes place in a dense crowded system, which can be considered as a crude model of a cellular membrane. Our main goal was to answer the following question: how do the fluid molecules move in an environment with a complex structure, taking into account the fact that motions of fluid molecules are highly correlated. The dynamic lattice liquid (DLL) model, which can work at the highest fluid density, was employed. Within the frame of the DLL model we considered cooperative motion of fluid particles in an environment that contained static obstacles. The dynamic properties of the system as a function of the concentration of obstacles were studied. The subdiffusive motion of particles was found in the crowded system. The influence of hydrodynamics on the motion was investigated via analysis of the displacement in closed cooperative loops. The simulation and the analysis emphasize the influence of the movement correlation between moving particles and obstacles.
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Affiliation(s)
- Piotr Polanowski
- Department of Molecular Physics, Technical University of Łódź, 90-924 Łódź, Poland
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66
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Piskorz TK, Ochab-Marcinek A. A Universal Model of Restricted Diffusion for Fluorescence Correlation Spectroscopy. J Phys Chem B 2014; 118:4906-12. [DOI: 10.1021/jp502467u] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Affiliation(s)
- Tomasz K. Piskorz
- Department
of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Poland
| | - Anna Ochab-Marcinek
- Institute
of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka
44/52, 01-224 Warsaw, Poland
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67
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Origin of subdiffusion of water molecules on cell membrane surfaces. Sci Rep 2014; 4:4720. [PMID: 24739933 PMCID: PMC5380161 DOI: 10.1038/srep04720] [Citation(s) in RCA: 59] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2014] [Accepted: 03/31/2014] [Indexed: 12/22/2022] Open
Abstract
Water molecules play an important role in providing unique environments for biological reactions on cell membranes. It is widely believed that water molecules form bridges that connect lipid molecules and stabilize cell membranes. Using all-atom molecular dynamics simulations, we show that translational and rotational diffusion of water molecules on lipid membrane surfaces exhibit subdiffusion and aging. Moreover, we provide evidence that both divergent mean trapping time (continuous-time random walk) and long-correlated noise (fractional Brownian motion) contribute to this subdiffusion. These results suggest that subdiffusion on cell membranes causes the water retardation, an enhancement of cell membrane stability, and a higher reaction efficiency.
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68
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Cherstvy AG, Chechkin AV, Metzler R. Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity. SOFT MATTER 2014; 10:1591-1601. [PMID: 24652104 DOI: 10.1039/c3sm52846d] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells.
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Affiliation(s)
- Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.
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69
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Metzler R, Jeon JH, Cherstvy AG, Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 2014; 16:24128-64. [DOI: 10.1039/c4cp03465a] [Citation(s) in RCA: 1046] [Impact Index Per Article: 104.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
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Affiliation(s)
- Ralf Metzler
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
- Physics Department
- Tampere University of Technology
| | - Jae-Hyung Jeon
- Physics Department
- Tampere University of Technology
- Tampere, Finland
- Korean Institute for Advanced Study (KIAS)
- Seoul, Republic of Korea
| | - Andrey G. Cherstvy
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
| | - Eli Barkai
- Physics Department and Institute of Nanotechnology and Advanced Materials
- Bar-Ilan University
- Ramat Gan, Israel
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70
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Stachura S, Kneller GR. Anomalous lateral diffusion in lipid bilayers observed by molecular dynamics simulations with atomistic and coarse-grained force fields. MOLECULAR SIMULATION 2013. [DOI: 10.1080/08927022.2013.840902] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- Sławomir Stachura
- Centre de Biophysique Moléculaire, CNRS, Rue Charles Sadron, 45071, Orléans, France
- Synchrotron Soleil, L'Orme de Merisiers, 91192, Gif-sur-Yvette, France
| | - Gerald R. Kneller
- Centre de Biophysique Moléculaire, CNRS, Rue Charles Sadron, 45071, Orléans, France
- Synchrotron Soleil, L'Orme de Merisiers, 91192, Gif-sur-Yvette, France
- Université d'Orléans, Chateau de la Source-Av. du Parc Floral, 45067, Orléans, France
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71
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Kursawe J, Schulz J, Metzler R. Transient aging in fractional Brownian and Langevin-equation motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062124. [PMID: 24483403 DOI: 10.1103/physreve.88.062124] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2013] [Indexed: 06/03/2023]
Abstract
Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t=0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on t(a) is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.
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Affiliation(s)
- Jochen Kursawe
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG
| | - Johannes Schulz
- Physics Department, Technical University of Munich, 85747 Garching, Germany
| | - Ralf Metzler
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG and Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany and Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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72
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73
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Javanainen M, Hammaren H, Monticelli L, Jeon JH, Miettinen MS, Martinez-Seara H, Metzler R, Vattulainen I. Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes. Faraday Discuss 2013; 161:397-417; discussion 419-59. [PMID: 23805752 DOI: 10.1039/c2fd20085f] [Citation(s) in RCA: 133] [Impact Index Per Article: 12.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Lateral diffusion plays a crucial role in numerous processes that take place in cell membranes, yet it is quite poorly understood in native membranes characterized by, e.g., domain formation and large concentration of proteins. In this article, we use atomistic and coarse-grained simulations to consider how packing of membranes and crowding with proteins affect the lateral dynamics of lipids and membrane proteins. We find that both packing and protein crowding have a profound effect on lateral diffusion, slowing it down. Anomalous diffusion is observed to be an inherent property in both protein-free and protein-rich membranes, and the time scales of anomalous diffusion and the exponent associated with anomalous diffusion are found to strongly depend on packing and crowding. Crowding with proteins also has a striking effect on the decay rate of dynamical correlations associated with lateral single-particle motion, as the transition from anomalous to normal diffusion is found to take place at macroscopic time scales: while in protein-poor conditions normal diffusion is typically observed in hundreds of nanoseconds, in protein-rich conditions the onset of normal diffusion is tens of microseconds, and in the most crowded systems as large as milliseconds. The computational challenge which results from these time scales is not easy to deal with, not even in coarse-grained simulations. We also briefly discuss the physical limits of protein motion. Our results suggest that protein concentration is anything but constant in the plane of cell membranes. Instead, it is strongly dependent on proteins' preference for aggregation.
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Affiliation(s)
- Matti Javanainen
- Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland
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74
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Khoshnood A, Jalali MA. Anomalous diffusion of proteins in sheared lipid membranes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032705. [PMID: 24125292 DOI: 10.1103/physreve.88.032705] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2012] [Revised: 08/08/2013] [Indexed: 06/02/2023]
Abstract
We use coarse grained molecular dynamics simulations to investigate diffusion properties of sheared lipid membranes with embedded transmembrane proteins. In membranes without proteins, we find normal in-plane diffusion of lipids in all flow conditions. Protein embedded membranes behave quite differently: by imposing a simple shear flow and sliding the monolayers of the membrane over each other, the motion of protein clusters becomes strongly superdiffusive in the shear direction. In such a circumstance, the subdiffusion regime is predominant perpendicular to the flow. We show that superdiffusion is a result of accelerated chaotic motions of protein-lipid complexes within the membrane voids, which are generated by hydrophobic mismatch or the transport of lipids by proteins.
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Affiliation(s)
- Atefeh Khoshnood
- Computational Mechanics Laboratory, Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran and Center of Excellence in Design, Robotics and Automation, Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran
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75
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Plesnar E, Subczynski WK, Pasenkiewicz-Gierula M. Comparative computer simulation study of cholesterol in hydrated unary and binary lipid bilayers and in an anhydrous crystal. J Phys Chem B 2013; 117:8758-69. [PMID: 23848956 DOI: 10.1021/jp402839r] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Models created with molecular dynamics simulations are used to compare the organization and dynamics of cholesterol (Chol) molecules in three different environments: (1) a hydrated pure Chol bilayer that models the Chol bilayer domain, which is a pure Chol domain embedded in the bulk membrane; (2) a 2-palmitoyl-3-oleoyl-d-glycerol-1-phosphorylcholine bilayer saturated with cholesterol (POPC-Chol50) that models the bulk membrane; (3) a Chol crystal. The computer model of the hydrated pure Chol bilayer is stable on the microsecond time scale. Some structural characteristics of Chol molecules in the Chol bilayer are similar to those in the POPC-Chol50 bilayer (e.g., tilt of Chol rings and chains), while others are similar to those in Chol crystals (e.g., surface area per Chol, bilayer thickness). The key result of this study is that the Chol bilayer has, unexpectedly, a dynamic structure, with Chol mobility similar to that in the POPC-Chol50 bilayer though slower. This is the major difference compared to Chol crystals, where Chol molecules are immobile. Also, water accessibility to Chol-OH groups in the Chol bilayer is not limited. On average, each Chol molecule makes 2.3 hydrogen bonds with water in the Chol bilayer, compared with 1.7 hydrogen bonds in the POPC-Col50 bilayer.
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Affiliation(s)
- Elzbieta Plesnar
- Department of Computational Biophysics and Bioinformatics, Faculty of Biochemistry, Biophysics and Biotechnology, Jagiellonian University, Krakow, Poland
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76
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Goose JE, Sansom MSP. Reduced lateral mobility of lipids and proteins in crowded membranes. PLoS Comput Biol 2013; 9:e1003033. [PMID: 23592975 PMCID: PMC3623704 DOI: 10.1371/journal.pcbi.1003033] [Citation(s) in RCA: 124] [Impact Index Per Article: 11.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2012] [Accepted: 02/28/2013] [Indexed: 12/31/2022] Open
Abstract
Coarse-grained molecular dynamics simulations of the E. coli outer membrane proteins FhuA, LamB, NanC, OmpA and OmpF in a POPE/POPG (3∶1) bilayer were performed to characterise the diffusive nature of each component of the membrane. At small observation times (<10 ns) particle vibrations dominate phospholipid diffusion elevating the calculated values from the longer time-scale bulk value (>50 ns) of 8.5×10−7 cm2 s−1. The phospholipid diffusion around each protein was found to vary based on distance from protein. An asymmetry in the diffusion of annular lipids in the inner and outer leaflets was observed and correlated with an asymmetry in charged residues in the vicinity of the inner and outer leaflet head-groups. Protein rotational and translational diffusion were also found to vary with observation time and were inversely correlated with the radius of gyration of the protein in the plane of the bilayer. As the concentration of protein within the bilayer was increased, the overall mobility of the membrane decreased reflected in reduced lipid diffusion coefficients for both lipid and protein components. The increase in protein concentration also resulted in a decrease in the anomalous diffusion exponent α of the lipid. Formation of extended clusters and networks of proteins led to compartmentalisation of lipids in extreme cases. Biological membranes are selective barriers which control the entry/exit of molecules to/from the interior of a cell. They are composed of a lipid bilayer in which are embedded many membrane proteins. Whilst the individual components of membranes are relatively well characterised, the lateral organization and dynamics of the membrane remain less well understood. The lateral mobility of constituent membrane species affects many processes, including how quickly proteins complexes form and protein recruitment occurs, how quickly lipids can be modified/lysed, and the formation of disordered and ordered microdomains. Biological membranes can contain as much as 50% protein. The dynamics of these crowded environments differ greatly from the sparsely populated membranes often studied in silico or in vitro. We use molecular dynamics computer simulations to quantify how mobility within the membrane decreases as the protein concentration increases. We calculate a baseline diffusion of both lipids and selected bacterial outer membrane proteins in the simplest of systems, namely a single protein in a large lipid bilayer patch. In this case diffusion can be correlated with the size of the protein. We observe how proteins affect the mobility of adjacent lipids. As the protein concentration within our systems increases we show that diffusion of both the proteins and lipids is reduced.
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Affiliation(s)
- Joseph E. Goose
- Department of Biochemistry, University of Oxford, Oxford, United Kingdom
| | - Mark S. P. Sansom
- Department of Biochemistry, University of Oxford, Oxford, United Kingdom
- * E-mail:
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77
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Cherstvy AG, Metzler R. Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes. Phys Chem Chem Phys 2013; 15:20220-35. [DOI: 10.1039/c3cp53056f] [Citation(s) in RCA: 97] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023]
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78
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Jeon JH, Monne HMS, Javanainen M, Metzler R. Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. PHYSICAL REVIEW LETTERS 2012; 109:188103. [PMID: 23215336 DOI: 10.1103/physrevlett.109.188103] [Citation(s) in RCA: 153] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2012] [Indexed: 05/08/2023]
Abstract
Combining extensive molecular dynamics simulations of lipid bilayer systems of varying chemical compositions with single-trajectory analyses, we systematically elucidate the stochastic nature of the lipid motion. We observe subdiffusion over more than 4 orders of magnitude in time, clearly stretching into the submicrosecond domain. The lipid motion depends on the lipid chemistry, the lipid phase, and especially the presence of cholesterol. We demonstrate that fractional Langevin equation motion universally describes the lipid motion in all phases, including the gel phase, and in the presence of cholesterol. The results underline the relevance of anomalous diffusion in lipid bilayers and the strong effects of the membrane composition.
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Affiliation(s)
- Jae-Hyung Jeon
- Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
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79
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Ward AB, Guvench O, Hills RD. Coarse grain lipid-protein molecular interactions and diffusion with MsbA flippase. Proteins 2012; 80:2178-90. [DOI: 10.1002/prot.24108] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/04/2012] [Revised: 04/10/2012] [Accepted: 04/25/2012] [Indexed: 12/27/2022]
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