TY - JOUR A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity JF - Soft matter N2 - 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. Y1 - 2014 U6 - https://doi.org/10.1039/c3sm52846d SN - 1744-683X SN - 1744-6848 VL - 10 IS - 10 SP - 1591 EP - 1601 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Bauer, Maximilian A1 - Godec, Aljaz A1 - Metzler, Ralf T1 - Diffusion of finite-size particles in two-dimensional channels with random wall configurations JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick-Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda Y1 - 2014 U6 - https://doi.org/10.1039/c3cp55160a SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 13 SP - 6118 EP - 6128 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - de Carvalho, Sidney J. A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. T1 - Critical adsorption of polyelectrolytes onto charged Janus nanospheres JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Based on extensive Monte Carlo simulations and analytical considerations we study the electrostatically driven adsorption of flexible polyelectrolyte chains onto charged Janus nanospheres. These net-neutral colloids are composed of two equally but oppositely charged hemispheres. The critical binding conditions for polyelectrolyte chains are analysed as function of the radius of the Janus particle and its surface charge density, as well as the salt concentration in the ambient solution. Specifically for the adsorption of finite-length polyelectrolyte chains onto Janus nanoparticles, we demonstrate that the critical adsorption conditions drastically differ when the size of the Janus particle or the screening length of the electrolyte are varied. We compare the scaling laws obtained for the adsorption-desorption threshold to the known results for uniformly charged spherical particles, observing significant disparities. We also contrast the changes to the polyelectrolyte chain conformations close to the surface of the Janus nanoparticles as compared to those for simple spherical particles. Finally, we discuss experimentally relevant physicochemical systems for which our simulations results may become important. In particular, we observe similar trends with polyelectrolyte complexation with oppositely but heterogeneously charged proteins. Y1 - 2014 U6 - https://doi.org/10.1039/c4cp02207f SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 29 SP - 15539 EP - 15550 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Jeon, Jae-Hyung A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is < x(2)(t) similar or equal to 2K(t)t with K(t) similar or equal to t(alpha-1) for 0 < alpha < 2. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely, we demonstrate that under confinement, the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments, in particular, under confinement inside cellular compartments or when optical tweezers tracking methods are used. Y1 - 2014 U6 - https://doi.org/10.1039/c4cp02019g SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 30 SP - 15811 EP - 15817 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. Y1 - 2014 U6 - https://doi.org/10.1039/c4cp03465a SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 44 SP - 24128 EP - 24164 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Ala-Nissila, Tapio A1 - Metzler, Ralf T1 - Polymer translocation: the first two decades and the recent diversification JF - Soft matter N2 - 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. Y1 - 2014 U6 - https://doi.org/10.1039/c4sm01819b SN - 1744-683X SN - 1744-6848 VL - 10 IS - 45 SP - 9016 EP - 9037 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Nezhadhaghighi, M. Ghasemi A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Numerical approach to unbiased and driven generalized elastic model JF - The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr N2 - From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent beta characterizing the growth of the mean squared displacement <(delta h)(2)> of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe. Y1 - 2014 U6 - https://doi.org/10.1063/1.4858425 SN - 0021-9606 SN - 1089-7690 VL - 140 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Metzler, Ralf T1 - Speeding up the first-passage for subdiffusion by introducing a finite potential barrier JF - Journal of physics : A, Mathematical and theoretical N2 - We show that for a subdiffusive continuous time random walk with scale-free waiting time distribution the first-passage dynamics on a finite interval can be optimized by introduction of a piecewise linear potential barrier. Analytical results for the survival probability and first-passage density based on the fractional Fokker-Planck equation are shown to agree well with Monte Carlo simulations results. As an application we discuss an improved design for efficient translocation of gradient copolymers compared to homopolymer translocation in a quasi-equilibrium approximation. KW - first passage KW - anomalous diffusion KW - potential landscape KW - polymer translocation Y1 - 2014 U6 - https://doi.org/10.1088/1751-8113/47/3/032002 SN - 1751-8113 SN - 1751-8121 VL - 47 IS - 3 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Tomovski, Zivorad T1 - Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise JF - Journal of mathematical physics N2 - We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion. Y1 - 2014 U6 - https://doi.org/10.1063/1.4863478 SN - 0022-2488 SN - 1089-7658 VL - 55 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Levy flights do not always optimize random blind search for sparse targets JF - Proceedings of the National Academy of Sciences of the United States of America N2 - It is generally believed that random search processes based on scale-free, Levy stable jump length distributions (Levy flights) optimize the search for sparse targets. Here we show that this popular search advantage is less universal than commonly assumed. We study the efficiency of a minimalist search model based on Levy flights in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space) based on two different optimization criteria with respect to minimal search time and search reliability (cumulative arrival probability). Although Levy flights turn out to be efficient search processes when the target is far from the starting point, or when relative to the starting point the target is upstream, we show that for close targets and for downstream target positioning regular Brownian motion turns out to be the advantageous search strategy. Contrary to claims that Levy flights with a critical exponent alpha = 1 are optimal for the search of sparse targets in different settings, based on our optimization parameters the optimal a may range in the entire interval (1, 2) and especially include Brownian motion as the overall most efficient search strategy. KW - search optimization KW - stochastic processes KW - Levy foraging hypothesis Y1 - 2014 U6 - https://doi.org/10.1073/pnas.1320424111 SN - 0027-8424 VL - 111 IS - 8 SP - 2931 EP - 2936 PB - National Acad. of Sciences CY - Washington ER -