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 - http://dx.doi.org/10.1039/c4cp03465a
SN - 1463-9076 (print)
SN - 1463-9084 (online)
VL - 16
IS - 44
SP - 24128
EP - 24164
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 - http://dx.doi.org/10.1039/c4cp02019g
SN - 1463-9076 (print)
SN - 1463-9084 (online)
VL - 16
IS - 30
SP - 15811
EP - 15817
PB - Royal Society of Chemistry
CY - Cambridge
ER -
TY - JOUR
A1 - Mardoukhi, Yousof
A1 - Jeon, Jae-Hyung
A1 - Metzler, Ralf
T1 - Geometry controlled anomalous diffusion in random fractal geometries
BT - looking beyond the infinite cluster
JF - Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies
N2 - 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 BT� h with h o 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.
Y1 - 2015
U6 - http://dx.doi.org/10.1039/c5cp03548a
SN - 1439-7641 (online)
IS - 17
SP - 30134
EP - 30147
PB - Wiley-VCH Verl.
CY - Weinheim
ER -
TY - GEN
A1 - Mardoukhi, Yousof
A1 - Jeon, Jae-Hyung
A1 - Metzler, Ralf
T1 - Geometry controlled anomalous diffusion in random fractal geometries
BT - looking beyond the infinite cluster
N2 - 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 BT� h with h o 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.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 207
Y1 - 2015
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-85247
ER -
TY - JOUR
A1 - Metzler, Ralf
A1 - Jeon, Jae-Hyung
A1 - Cherstvy, Andrey G.
A1 - Barkai, Eli
T1 - Anomalous diffusion models and their properties
BT - non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking
JF - physical chemistry, chemical physics : PCCP
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.
KW - intermittent chaotic systems
KW - Fokker-Planck equations
KW - time random-walks
KW - fluorescence photobleaching recovery
KW - fluctuation-dissipation theorem
KW - fractional dynamics approach
KW - photon-counting statistics
KW - weak ergodicity breaking
KW - flight search patterns
KW - levy flights
Y1 - 2014
U6 - http://dx.doi.org/10.1039/c4cp03465a
SN - 1463-9076 (print), 1463-9084 (online)
VL - 2014
IS - 16
SP - 24128
EP - 24164
ER -
TY - GEN
A1 - Metzler, Ralf
A1 - Jeon, Jae-Hyung
A1 - Cherstvy, Andrey G.
A1 - Barkai, Eli
T1 - Anomalous diffusion models and their properties
BT - non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking
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.
T3 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 174
KW - Fokker-Planck equations
KW - flight search patterns
KW - fluctuation-dissipation theorem
KW - fluorescence photobleaching recovery
KW - fractional dynamics approach
KW - intermittent chaotic systems
KW - levy flights
KW - photon-counting statistics
KW - time random-walks
KW - weak ergodicity breaking
Y1 - 2014
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-74448
SP - 24128
EP - 24164
ER -
TY - JOUR
A1 - Adamcik, Jozef
A1 - Jeon, Jae-Hyung
A1 - Karczewski, Konrad J.
A1 - Metzler, Ralf
A1 - Dietler, Giovanni
T1 - Quantifying supercoiling-induced denaturation bubbles in DNA
JF - Soft matter
N2 - In both eukaryotic and prokaryotic DNA sequences of 30-100 base-pairs rich in AT base-pairs have been identified at which the double helix preferentially unwinds. Such DNA unwinding elements are commonly associated with origins for DNA replication and transcription, and with chromosomal matrix attachment regions. Here we present a quantitative study of local DNA unwinding based on extensive single DNA plasmid imaging. We demonstrate that long-lived single-stranded denaturation bubbles exist in negatively supercoiled DNA, at the expense of partial twist release. Remarkably, we observe a linear relation between the degree of supercoiling and the bubble size, in excellent agreement with statistical modelling. Furthermore, we obtain the full distribution of bubble sizes and the opening probabilities at varying salt and temperature conditions. The results presented herein underline the important role of denaturation bubbles in negatively supercoiled DNA for biological processes such as transcription and replication initiation in vivo.
Y1 - 2012
U6 - http://dx.doi.org/10.1039/c2sm26089a
SN - 1744-683X (print)
VL - 8
IS - 33
SP - 8651
EP - 8658
PB - Royal Society of Chemistry
CY - Cambridge
ER -
TY - JOUR
A1 - Javanainen, Matti
A1 - Hammaren, Henrik
A1 - Monticelli, Luca
A1 - Jeon, Jae-Hyung
A1 - Miettinen, Markus S.
A1 - Martinez-Seara, Hector
A1 - Metzler, Ralf
A1 - Vattulainen, Ilpo
T1 - Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes
JF - Faraday discussions
N2 - 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.
Y1 - 2013
U6 - http://dx.doi.org/10.1039/c2fd20085f
SN - 1359-6640 (print)
VL - 161
IS - 1
SP - 397
EP - 417
PB - Royal Society of Chemistry
CY - Cambridge
ER -
TY - JOUR
A1 - Jeon, Jae-Hyung
A1 - Leijnse, Natascha
A1 - Oddershede, Lene B.
A1 - Metzler, Ralf
T1 - Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions
JF - New journal of physics : the open-access journal for physics
N2 - We report the results of single tracer particle tracking by optical tweezers and video microscopy in micellar solutions. From careful analysis in terms of different stochastic models, we show that the polystyrene tracer beads of size 0.52-2.5 mu m after short-time normal diffusion turn over to perform anomalous diffusion of the form < r(2)(t)> similar or equal to t(alpha) with alpha approximate to 0.3. This free anomalous diffusion is ergodic and consistent with a description in terms of the generalized Langevin equation with a power-law memory kernel. With optical tweezers tracking, we unveil a power-law relaxation over several decades in time to the thermal plateau value under the confinement of the harmonic tweezer potential, as predicted previously (Phys. Rev. E 85 021147 (2012)). After the subdiffusive motion in the millisecond range, the motion becomes faster and turns either back to normal Brownian diffusion or to even faster superdiffusion, depending on the size of the tracer beads.
Y1 - 2013
U6 - http://dx.doi.org/10.1088/1367-2630/15/4/045011
SN - 1367-2630 (print)
VL - 15
IS - 4
PB - IOP Publ. Ltd.
CY - Bristol
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 : PCCP
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 ?x2(t)? [similar{,} equals] 2K(t)t with K(t) [similar{,} equals] t[small alpha]-1 for 0 < [small 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.
KW - single-particle tracking
KW - living cells
KW - random-walks
KW - subdiffusion
KW - dynamics
KW - nonergodicity
KW - coefficients
KW - transport
KW - membrane
KW - behavior
Y1 - 2014
U6 - http://dx.doi.org/10.1039/C4CP02019G
VL - 30
IS - 16
SP - 15811
EP - 15817
PB - The Royal Society of Chemistry
CY - Cambridge
ER -