TY - GEN A1 - Gudowska-Nowak, Ewa A1 - Lindenberg, Katja A1 - Metzler, Ralf T1 - Preface: Marian Smoluchowski’s 1916 paper—a century of inspiration T2 - Journal of physics : A, Mathematical and theoretical Y1 - 2017 U6 - https://doi.org/10.1088/1751-8121/aa8529 SN - 1751-8113 SN - 1751-8121 VL - 50 IS - 38 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Krapf, Diego A1 - Marinari, Enzo A1 - Metzler, Ralf A1 - Oshanin, Gleb A1 - Xu, Xinran A1 - Squarcini, Alessio T1 - Power spectral density of a single Brownian trajectory BT - what one can and cannot learn from it JF - New journal of physics : the open-access journal for physics N2 - The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories. KW - power spectral density KW - single-trajectory analysis KW - probability density function KW - exact results Y1 - 2018 U6 - https://doi.org/10.1088/1367-2630/aaa67c SN - 1367-2630 VL - 20 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Ala-Nissila, Tapio A1 - Metzler, Ralf ED - 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. KW - solid-state nanopores KW - single-stranded-dna KW - posttranslational protein translocation KW - anomalous diffusion KW - monte-carlo KW - structured polynucleotides KW - dynamics simulation KW - equation approach KW - osmotic-pressure KW - membrane channel Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-76266 SN - 1744-683X VL - 45 IS - 10 SP - 9016 EP - 9037 PB - the Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Eliazar, Iddo A1 - Metzler, Ralf A1 - Reuveni, Shlomi T1 - Poisson-process limit laws yield Gumbel max-min and min-max JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - “A chain is only as strong as its weakest link” says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j)entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The min-max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i,j) matrix entry to be the time at which item j on copy i is lost; then, the min-max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit laws for the max-min and for the min-max. The limit laws hold whenever the entries' distribution has a density, and yield highly applicable approximation tools and design tools for the max-min and min-max of large random matrices. A brief of the results presented herein is given in: Gumbel central limit theorem for max-min and min-max Y1 - 2019 U6 - https://doi.org/10.1103/PhysRevE.100.022129 SN - 2470-0045 SN - 2470-0053 VL - 100 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Li, Yongge A1 - Mei, Ruoxing A1 - Xu, Yong A1 - Kurths, Jürgen A1 - Duan, Jinqiao A1 - Metzler, Ralf T1 - Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity JF - New Journal of Physics N2 - This work focuses on the dynamics of particles in a confined geometry with position-dependent diffusivity, where the confinement is modelled by a periodic channel consisting of unit cells connected by narrow passage ways. We consider three functional forms for the diffusivity, corresponding to the scenarios of a constant (D ₀), as well as a low (D ₘ) and a high (D d) mobility diffusion in cell centre of the longitudinally symmetric cells. Due to the interaction among the diffusivity, channel shape and external force, the system exhibits complex and interesting phenomena. By calculating the probability density function, mean velocity and mean first exit time with the Itô calculus form, we find that in the absence of external forces the diffusivity D d will redistribute particles near the channel wall, while the diffusivity D ₘ will trap them near the cell centre. The superposition of external forces will break their static distributions. Besides, our results demonstrate that for the diffusivity D d, a high dependence on the x coordinate (parallel with the central channel line) will improve the mean velocity of the particles. In contrast, for the diffusivity D ₘ, a weak dependence on the x coordinate will dramatically accelerate the moving speed. In addition, it shows that a large external force can weaken the influences of different diffusivities; inversely, for a small external force, the types of diffusivity affect significantly the particle dynamics. In practice, one can apply these results to achieve a prominent enhancement of the particle transport in two- or three-dimensional channels by modulating the local tracer diffusivity via an engineered gel of varying porosity or by adding a cold tube to cool down the diffusivity along the central line, which may be a relevant effect in engineering applications. Effects of different stochastic calculi in the evaluation of the underlying multiplicative stochastic equation for different physical scenarios are discussed. KW - diffusion KW - channel KW - space-dependent diffusivity Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/ab81b9 SN - 1367-2630 VL - 22 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - JOUR A1 - Vinod, Deepak A1 - Cherstvy, Andrey G. A1 - Wang, Wei A1 - Metzler, Ralf A1 - Sokolov, Igor M. T1 - Nonergodicity of reset geometric Brownian motion JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We derive. the ensemble-and time-averaged mean-squared displacements (MSD, TAMSD) for Poisson-reset geometric Brownian motion (GBM), in agreement with simulations. We find MSD and TAMSD saturation for frequent resetting, quantify the spread of TAMSDs via the ergodicity-breaking parameter and compute distributions of prices. General MSD-TAMSD nonequivalence proves reset GBM nonergodic. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.105.L012106 SN - 2470-0045 SN - 2470-0053 VL - 105 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Klett, Kolja A1 - Cherstvy, Andrey G. A1 - Shin, Jaeoh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Non-Gaussian, transiently anomalous, and ergodic self-diffusion of flexible dumbbells in crowded two-dimensional environments BT - coupled translational and rotational motions JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We employ Langevin-dynamics simulations to unveil non-Brownian and non-Gaussian center-of-mass self-diffusion of massive flexible dumbbell-shaped particles in crowded two-dimensional solutions. We study the intradumbbell dynamics of the relative motion of the two constituent elastically coupled disks. Our main focus is on effects of the crowding fraction phi and of the particle structure on the diffusion characteristics. We evaluate the time-averaged mean-squared displacement (TAMSD), the displacement probability-density function (PDF), and the displacement autocorrelation function (ACF) of the dimers. For the TAMSD at highly crowded conditions of dumbbells, e.g., we observe a transition from the short-time ballistic behavior, via an intermediate subdiffusive regime, to long-time Brownian-like spreading dynamics. The crowded system of dimers exhibits two distinct diffusion regimes distinguished by the scaling exponent of the TAMSD, the dependence of the diffusivity on phi, and the features of the displacement-ACF. We attribute these regimes to a crowding-induced transition from viscous to viscoelastic diffusion upon growing phi. We also analyze the relative motion in the dimers, finding that larger phi suppress their vibrations and yield strongly non-Gaussian PDFs of rotational displacements. For the diffusion coefficients D(phi) of translational and rotational motion of the dumbbells an exponential decay with phi for weak and a power-law variation D(phi) proportional to (phi - phi(star))(2.4) for strong crowding is found. A comparison of simulation results with theoretical predictions for D(phi) is discussed and some relevant experimental systems are overviewed. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.064603 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 6 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Thapa, Samudrajit A1 - Wagner, Caroline E. A1 - Metzler, Ralf T1 - Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels JF - Soft matter N2 - Native mucus is polymer-based soft-matter material of paramount biological importance. How non-Gaussian and non-ergodic is the diffusive spreading of pathogens in mucus? We study the passive, thermally driven motion of micron-sized tracers in hydrogels of mucins, the main polymeric component of mucus. We report the results of the Bayesian analysis for ranking several diffusion models for a set of tracer trajectories [C. E. Wagner et al., Biomacromolecules, 2017, 18, 3654]. The models with "diffusing diffusivity', fractional and standard Brownian motion are used. The likelihood functions and evidences of each model are computed, ranking the significance of each model for individual traces. We find that viscoelastic anomalous diffusion is often most probable, followed by Brownian motion, while the model with a diffusing diffusion coefficient is only realised rarely. Our analysis also clarifies the distribution of time-averaged displacements, correlations of scaling exponents and diffusion coefficients, and the degree of non-Gaussianity of displacements at varying pH levels. Weak ergodicity breaking is also quantified. We conclude that-consistent with the original study-diffusion of tracers in the mucin gels is most non-Gaussian and non-ergodic at low pH that corresponds to the most heterogeneous networks. Using the Bayesian approach with the nested-sampling algorithm, together with the quantitative analysis of multiple statistical measures, we report new insights into possible physical mechanisms of diffusion in mucin gels. Y1 - 2019 U6 - https://doi.org/10.1039/c8sm02096e SN - 1744-683X SN - 1744-6848 VL - 15 IS - 12 SP - 2526 EP - 2551 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Goychuk, Igor A. A1 - Kharchenko, Vasyl O. A1 - Metzler, Ralf T1 - Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion JF - Physical Chemistry Chemical Physics N2 - The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient. KW - royal soc chemistry KW - thomas graham house KW - science park KW - milton rd KW - cambridge cb4 0wf KW - cambs KW - england Y1 - 2014 SN - 1463-9076 IS - 16 SP - 16524 EP - 16535 PB - the Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Norregaard, Kamilla A1 - Metzler, Ralf A1 - Ritter, Christine M. A1 - Berg-Sorensen, Kirstine A1 - Oddershede, Lene Broeng T1 - Manipulation and Motion of Organelles and Single Molecules in Living Cells JF - Chemical reviews N2 - The biomolecule is among the most important building blocks of biological systems, and a full understanding of its function forms the scaffold for describing the mechanisms of higher order structures as organelles and cells. Force is a fundamental regulatory mechanism of biomolecular interactions driving many cellular processes. The forces on a molecular scale are exactly in the range that can be manipulated and probed with single molecule force spectroscopy. The natural environment of a biomolecule is inside a living cell, hence, this is the most relevant environment for probing their function. In vivo studies are, however, challenged by the complexity of the cell. In this review, we start with presenting relevant theoretical tools for analyzing single molecule data obtained in intracellular environments followed by a description of state-of-the art visualization techniques. The most commonly used force spectroscopy techniques, namely optical tweezers, magnetic tweezers, and atomic force microscopy, are described in detail, and their strength and limitations related to in vivo experiments are discussed. Finally, recent exciting discoveries within the field of in vivo manipulation and dynamics of single molecule and organelles are reviewed. Y1 - 2017 U6 - https://doi.org/10.1021/acs.chemrev.6b00638 SN - 0009-2665 SN - 1520-6890 VL - 117 IS - 5 SP - 4342 EP - 4375 PB - American Chemical Society CY - Washington ER - TY - JOUR A1 - Xu, Pengbo A1 - Zhou, Tian A1 - Metzler, Ralf A1 - Deng, Weihua T1 - Lévy walk dynamics in an external harmonic potential JF - Physical review : E, Statistical, nonlinear, and soft matter physics N2 - Levy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs-their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some longstanding puzzles around LWs. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.101.062127 SN - 2470-0045 SN - 2470-0053 SN - 1550-2376 SN - 1063-651X SN - 1539-3755 VL - 101 IS - 6 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Padash, Amin A1 - Aghion, Erez A1 - Schulz, Alexander A1 - Barkai, Eli A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf A1 - Kantz, Holger T1 - Local equilibrium properties of ultraslow diffusion in the Sinai model JF - New journal of physics N2 - We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 10(17) steps and thereby also study finite-time crossover phenomena. KW - Sinai diffusion KW - clustering KW - local equilibrium Y1 - 2022 U6 - https://doi.org/10.1088/1367-2630/ac7df8 SN - 1367-2630 VL - 24 IS - 7 PB - IOP Publishing CY - Bristol ER - TY - JOUR A1 - Capała, Karol A1 - Padash, Amin A1 - Chechkin, Aleksei V. A1 - Shokri, Babak A1 - Metzler, Ralf A1 - Dybiec, Bartłomiej T1 - Levy noise-driven escape from arctangent potential wells JF - Chaos : an interdisciplinary journal of nonlinear science N2 - The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Levy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Levy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Levy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point. Y1 - 2020 U6 - https://doi.org/10.1063/5.0021795 SN - 1054-1500 SN - 1089-7682 VL - 30 IS - 12 PB - American Institute of Physics CY - Woodbury, NY ER - TY - JOUR A1 - Thapa, Samudrajit A1 - Wyłomańska, Agnieszka A1 - Sikora, Grzegorz A1 - Wagner, Caroline E. A1 - Krapf, Diego A1 - Kantz, Holger A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories JF - New Journal of Physics N2 - Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations. KW - diffusion KW - anomalous diffusion KW - large-deviation statistic KW - time-averaged mean squared displacement KW - Chebyshev inequality Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/abd50e SN - 1367-2630 VL - 23 PB - Dt. Physikalische Ges. ; IOP CY - Bad Honnef ; London ER - TY - JOUR A1 - Ghosh, Surya K. A1 - Cherstvy, Andrey G. A1 - Petrov, Eugene P. A1 - Metzler, Ralf T1 - Interactions of rod-like particles on responsive elastic sheets JF - Soft matter N2 - What are the physical laws of the mutual interactions of objects bound to cell membranes, such as various membrane proteins or elongated virus particles? To rationalise this, we here investigate by extensive computer simulations mutual interactions of rod-like particles adsorbed on the surface of responsive elastic two-dimensional sheets. Specifically, we quantify sheet deformations as a response to adhesion of such filamentous particles. We demonstrate that tip-to-tip contacts of rods are favoured for relatively soft sheets, while side-by-side contacts are preferred for stiffer elastic substrates. These attractive orientation-dependent substrate-mediated interactions between the rod-like particles on responsive sheets can drive their aggregation and self-assembly. The optimal orientation of the membrane-bound rods is established via responding to the elastic energy profiles created around the particles. We unveil the phase diagramme of attractive–repulsive rod–rod interactions in the plane of their separation and mutual orientation. Applications of our results to other systems featuring membrane-associated particles are also discussed. Y1 - 2016 U6 - https://doi.org/10.1039/C6SM01522K SN - 1744-6848 SN - 1744-683X PB - RSC CY - London ER - TY - JOUR A1 - Xu, Pengbo A1 - Metzler, Ralf A1 - Wang, Wanli T1 - Infinite density and relaxation for Levy walks in an external potential BT - Hermite polynomial approach JF - Physical review N2 - Levy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(r ) = v0/./r. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.105.044118 SN - 2470-0045 SN - 2470-0053 VL - 105 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Wang, Wei A1 - Metzler, Ralf A1 - Sokolov, Igor M. T1 - Inertia triggers nonergodicity of fractional Brownian motion JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also dramatically alter the values of the ergodicity-breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped or massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying, e.g., the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.024115 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Sandev, Trifce A1 - Domazetoski, Viktor A1 - Kocarev, Ljupco A1 - Metzler, Ralf A1 - Chechkin, Aleksei T1 - Heterogeneous diffusion with stochastic resetting JF - Journal of physics : A, Mathematical and theoretical N2 - We study a heterogeneous diffusion process (HDP) with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyse the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like t (1/2) while the length scale xi(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the HDP with diffusion length increasing like t ( p/2) the length scale xi(t) grows like t ( p ). The obtained results are verified by numerical solutions of the corresponding Langevin equation. KW - heterogeneous diffusion KW - Fokker-Planck equation KW - Langevin equation KW - stochastic resetting KW - nonequilibrium stationary state KW - large deviation function Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac491c SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 7 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Xu, Yong A1 - Liu, Xuemei A1 - Li, Yongge A1 - Metzler, Ralf T1 - Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent a. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent alpha is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.062106 SN - 2470-0045 SN - 2470-0053 VL - 102 IS - 6 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Eliazar, Iddo A1 - Metzler, Ralf A1 - Reuveni, Shlomi T1 - Gumbel central limit theorem for max-min and min-max JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - The max-min and min-max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the max-min and min-max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit laws—akin to the central limit theorem—for the max-min and min-max of large random matrices. The limit laws intertwine random-matrix theory and extreme-value theory, couple the matrix dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix entries' distribution. Due to their generality and universality, as well as their practicality, these results are expected to have a host of applications in the physical sciences and beyond. Y1 - 2019 U6 - https://doi.org/10.1103/PhysRevE.100.020104 SN - 2470-0045 SN - 2470-0053 VL - 100 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Stojkoski, Viktor A1 - Sandev, Trifce A1 - Basnarkov, Lasko A1 - Kocarev, Ljupco A1 - Metzler, Ralf T1 - Generalised geometric Brownian motion BT - theory and applications to option pricing JF - Entropy N2 - Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness. KW - geometric Brownian motion KW - Fokker– Planck equation KW - Black– Scholes model KW - option pricing Y1 - 2020 U6 - https://doi.org/10.3390/e22121432 SN - 1099-4300 VL - 22 IS - 12 PB - MDPI CY - Basel ER - TY - GEN A1 - Metzler, Ralf T1 - Gaussianity Fair BT - the Riddle of Anomalous yet Non-Gaussian Diffusion T2 - Biophysical journal Y1 - 2017 U6 - https://doi.org/10.1016/j.bpj.2016.12.019 SN - 0006-3495 SN - 1542-0086 VL - 112 IS - 3 SP - 413 EP - 415 PB - Cell Press CY - Cambridge ER - TY - JOUR A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb T1 - Full distribution of first exit times in the narrow escape problem JF - New Journal of Physics N2 - In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small 'escape window' in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems. KW - narrow escape problem KW - first-passage time distribution KW - mean versus most probable reaction times KW - mixed boundary conditions Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab5de4 SN - 1367-2630 VL - 21 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - JOUR A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb T1 - From single-particle stochastic kinetics to macroscopic reaction rates BT - fastest first-passage time of N random walkers JF - New Journal of Physics N2 - We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N² for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells. KW - diffusion KW - first-passage KW - fastest first-passage time of N walkers Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/abb1de SN - 1367-2630 VL - 22 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - From continuous time random walks to the generalized diffusion equation JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases. KW - continuous time random walk (CTRW) KW - generalized diffusion equation KW - Mittag-Leffler functions KW - anomalous diffusion Y1 - 2018 U6 - https://doi.org/10.1515/fca-2018-0002 SN - 1311-0454 SN - 1314-2224 VL - 21 IS - 1 SP - 10 EP - 28 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Wang, Wei A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Thapa, Samudrajit A1 - Seno, Flavio A1 - Liu, Xianbin A1 - Metzler, Ralf T1 - Fractional Brownian motion with random diffusivity BT - emerging residual nonergodicity below the correlation time JF - Journal of physics : A, Mathematical and theoretical N2 - Numerous examples for a priori unexpected non-Gaussian behaviour for normal and anomalous diffusion have recently been reported in single-particle tracking experiments. Here, we address the case of non-Gaussian anomalous diffusion in terms of a random-diffusivity mechanism in the presence of power-law correlated fractional Gaussian noise. We study the ergodic properties of this model via examining the ensemble- and time-averaged mean-squared displacements as well as the ergodicity breaking parameter EB quantifying the trajectory-to-trajectory fluctuations of the latter. For long measurement times, interesting crossover behaviour is found as function of the correlation time tau characterising the diffusivity dynamics. We unveil that at short lag times the EB parameter reaches a universal plateau. The corresponding residual value of EB is shown to depend only on tau and the trajectory length. The EB parameter at long lag times, however, follows the same power-law scaling as for fractional Brownian motion. We also determine a corresponding plateau at short lag times for the discrete representation of fractional Brownian motion, absent in the continuous-time formulation. These analytical predictions are in excellent agreement with results of computer simulations of the underlying stochastic processes. Our findings can help distinguishing and categorising certain nonergodic and non-Gaussian features of particle displacements, as observed in recent single-particle tracking experiments. KW - stochastic processes KW - anomalous diffusion KW - fractional Brownian motion KW - diffusing diffusivity KW - weak ergodicity breaking Y1 - 2020 U6 - https://doi.org/10.1088/1751-8121/aba467 SN - 1751-8113 SN - 1751-8121 VL - 53 IS - 47 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Guggenberger, Tobias A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions JF - Journal of physics : A, Mathematical and theoretical N2 - We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise (FGN) in a superharmonic external potential of the form U(x) proportional to x(2n) (n is an element of N). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential U(x). We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian Levy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by FGN always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled. KW - anomalous diffusion KW - Boltzmann distribution KW - non-Gaussian distribution Y1 - 2021 U6 - https://doi.org/10.1088/1751-8121/ac019b SN - 1751-8113 SN - 1751-8121 VL - 54 IS - 29 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Guggenberger, Tobias A1 - Pagnini, Gianni A1 - Vojta, Thomas A1 - Metzler, Ralf T1 - Fractional Brownian motion in a finite interval BT - correlations effect depletion or accretion zones of particles near boundaries JF - New Journal of Physics N2 - Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers. KW - anomalous diffusion KW - fractional Brownian motion KW - reflecting boundary conditions Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab075f SN - 1367-2630 VL - 21 PB - Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics CY - Bad Honnef und London ER - TY - JOUR A1 - Mardoukhi, Yousof A1 - Jeon, Jae-Hyung A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Fluctuations of random walks in critical random environments JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed. Y1 - 2018 U6 - https://doi.org/10.1039/c8cp03212b SN - 1463-9076 SN - 1463-9084 VL - 20 IS - 31 SP - 20427 EP - 20438 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Padash, Amin A1 - Chechkin, Aleksei V. A1 - Dybiec, Bartlomiej A1 - Pavlyukevich, Ilya A1 - Shokri, Babak A1 - Metzler, Ralf T1 - First-passage properties of asymmetric Levy flights JF - Journal of physics : A, Mathematical and theoretical N2 - Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times. KW - Levy flights KW - first-passage KW - search dynamics Y1 - 2019 U6 - https://doi.org/10.1088/1751-8121/ab493e SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 45 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Godec, Aljaž A1 - Metzler, Ralf T1 - First passage time statistics for two-channel diffusion JF - Journal of physics : A, Mathematical and theoretical N2 - We present rigorous results for the mean first passage time and first passage time statistics for two-channel Markov additive diffusion in a 3-dimensional spherical domain. Inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes, which is shown to effect a non-trivial first passage behaviour. We also address the scenario of intermittent immobilisation. In both cases we prove that despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions the first passage statistics at long times do not display Poisson-like behaviour if none of the phases has a vanishing diffusion coefficient. This stands in stark contrast to the standard (one-channel) Markov diffusion counterpart. We also discuss the relevance of our results in the context of cellular signalling. KW - first passage time KW - Markov additive processes KW - Fokker-Planck equation KW - random search processes KW - coupled initial boundary value problem KW - cellular signalling KW - asymptotic analysis Y1 - 2017 U6 - https://doi.org/10.1088/1751-8121/aa5204 SN - 1751-8113 SN - 1751-8121 VL - 50 IS - 8 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Sposini, Vittoria A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - First passage statistics for diffusing diffusivity JF - Journal of physics : A, Mathematical and theoretical N2 - A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law < r(2)(t)> similar or equal to Dt yet the distribution of particle displacements is strongly non-Gaussian. A central approach to describe this effect is the diffusing diffusivity (DD) model in which the diffusion coefficient itself is a stochastic quantity, mimicking heterogeneities of the environment encountered by the tracer particle on its path. We here quantify in terms of analytical and numerical approaches the first passage behaviour of the DD model. We observe significant modifications compared to Brownian-Gaussian diffusion, in particular that the DD model may have a faster first passage dynamics. Moreover we find a universal crossover point of the survival probability independent of the initial condition. KW - diffusion KW - superstatistics KW - first passage Y1 - 2018 U6 - https://doi.org/10.1088/1751-8121/aaf6ff SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 4 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Mutothya, Nicholas Mwilu A1 - Xu, Yong A1 - Li, Yongge A1 - Metzler, Ralf A1 - Mutua, Nicholas Muthama T1 - First passage dynamics of stochastic motion in heterogeneous media driven by correlated white Gaussian and coloured non-Gaussian noises JF - Journal of physics. Complexity N2 - We study the first passage dynamics for a diffusing particle experiencing a spatially varying diffusion coefficient while driven by correlated additive Gaussian white noise and multiplicative coloured non-Gaussian noise. We consider three functional forms for position dependence of the diffusion coefficient: power-law, exponential, and logarithmic. The coloured non-Gaussian noise is distributed according to Tsallis' q-distribution. Tracks of the non-Markovian systems are numerically simulated by using the fourth-order Runge-Kutta algorithm and the first passage times (FPTs) are recorded. The FPT density is determined along with the mean FPT (MFPT). Effects of the noise intensity and self-correlation of the multiplicative noise, the intensity of the additive noise, the cross-correlation strength, and the non-extensivity parameter on the MFPT are discussed. KW - first passage KW - diffusion KW - non-Gaussian KW - correlated noise Y1 - 2021 U6 - https://doi.org/10.1088/2632-072X/ac35b5 SN - 2632-072X VL - 2 PB - IOP Publishing CY - Bristol ER - TY - JOUR A1 - Palyulin, Vladimir V A1 - Blackburn, George A1 - Lomholt, Michael A A1 - Watkins, Nicholas W A1 - Metzler, Ralf A1 - Klages, Rainer A1 - Chechkin, Aleksei V. T1 - First passage and first hitting times of Lévy flights and Lévy walks JF - New Journal of Physics N2 - For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms. KW - Lévy flights KW - Lévy walks KW - first-passage time KW - first-hitting time Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab41bb SN - 1367-2630 VL - 21 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Blackburn, George A1 - Lomholt, Michael A. A1 - Watkins, Nicholas W. A1 - Metzler, Ralf A1 - Klages, Rainer A1 - Chechkin, Aleksei V. T1 - First passage and first hitting times of Levy flights and Levy walks JF - New journal of physics : the open-access journal for physics N2 - For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms. KW - Levy flights KW - Levy walks KW - first-passage time KW - first-hitting time Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab41bb SN - 1367-2630 VL - 21 IS - 10 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Liu, Lin A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Facilitated Diffusion of Transcription Factor Proteins with Anomalous Bulk Diffusion JF - The journal of physical chemistry : B, Condensed matter, materials, surfaces, interfaces & biophysical chemistry N2 - What are the physical laws of the diffusive search of proteins for their specific binding sites on DNA in the presence of the macromolecular crowding in cells? We performed extensive computer simulations to elucidate the protein target search on DNA. The novel feature is the viscoelastic non-Brownian protein bulk diffusion recently observed experimentally. We examine the influence of the protein-DNA binding affinity and the anomalous diffusion exponent on the target search time. In all cases an optimal search time is found. The relative contribution of intermittent three-dimensional bulk diffusion and one-dimensional sliding of proteins along the DNA is quantified. Our results are discussed in the light of recent single molecule tracking experiments, aiming at a better understanding of the influence of anomalous kinetics of proteins on the facilitated diffusion mechanism. Y1 - 2017 U6 - https://doi.org/10.1021/acs.jpcb.6b12413 SN - 1520-6106 VL - 121 SP - 1284 EP - 1289 PB - American Chemical Society CY - Washington ER - TY - JOUR A1 - Scott, Shane A1 - Weiss, Matthias A1 - Selhuber-Unkel, Christine A1 - Barooji, Younes F. A1 - Sabri, Adal A1 - Erler, Janine T. A1 - Metzler, Ralf A1 - Oddershede, Lene B. T1 - Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods. Y1 - 2022 U6 - https://doi.org/10.1039/d2cp01384c SN - 1463-9076 SN - 1463-9084 VL - 25 IS - 3 SP - 1513 EP - 1537 PB - RSC Publ. CY - Cambridge ER - TY - JOUR A1 - Grebenkov, Denis S. A1 - Sposini, Vittoria A1 - Metzler, Ralf A1 - Oshanin, Gleb A1 - Seno, Flavio T1 - Exact distributions of the maximum and range of random diffusivity processes JF - New Journal of Physics N2 - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process. KW - random diffusivity KW - extremal values KW - maximum and range KW - diffusion KW - Brownian motion Y1 - 2021 U6 - https://doi.org/10.1088/1367-2630/abd313 SN - 1367-2630 VL - 23 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - JOUR A1 - Akimoto, Takuma A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevE.98.022105 SN - 2470-0045 SN - 2470-0053 VL - 98 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb T1 - Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains JF - New journal of physics N2 - Westudy the mean first passage time (MFPT) to a reaction event on a specific site in a cylindrical geometry—characteristic, for instance, for bacterial cells, with a concentric inner cylinder representing the nuclear region of the bacterial cell. Asimilar problem emerges in the description of a diffusive search by a transcription factor protein for a specific binding region on a single strand of DNA.We develop a unified theoretical approach to study the underlying boundary value problem which is based on a self-consistent approximation of the mixed boundary condition. Our approach permits us to derive explicit, novel, closed-form expressions for the MFPT valid for a generic setting with an arbitrary relation between the system parameters.Weanalyse this general result in the asymptotic limits appropriate for the above-mentioned biophysical problems. Our investigation reveals the crucial role of the target aspect ratio and of the intrinsic reactivity of the binding region, which were disregarded in previous studies. Theoretical predictions are confirmed by numerical simulations. KW - first passage time KW - cylindrical geometry KW - aspect ratio KW - protein search Y1 - 2017 U6 - https://doi.org/10.1088/1367-2630/aa8ed9 SN - 1367-2630 VL - 19 SP - 1 EP - 11 PB - IOP CY - London ER - TY - JOUR A1 - Mejia-Monasterio, Carlos A1 - Metzler, Ralf A1 - Vollmer, Jürgen T1 - Editorial: anomalous transport BT - applications, mathematical perspectives, and big data JF - Frontiers in Physics KW - anomalous (or non-Fickian) diffusion KW - anomalous heat conduction KW - stochastic dynamics KW - molecular overcrowding KW - dynamical systems Y1 - 2020 U6 - https://doi.org/10.3389/fphy.2020.622417 SN - 2296-424X VL - 8 PB - Frontiers Media CY - Lausanne ER - TY - JOUR A1 - Sarabadani, Jalal A1 - Metzler, Ralf A1 - Ala-Nissila, Tapio T1 - Driven polymer translocation into a channel: Isoflux tension propagation theory and Langevin dynamics simulations JF - Physical Review Research N2 - Isoflux tension propagation (IFTP) theory and Langevin dynamics (LD) simulations are employed to study the dynamics of channel-driven polymer translocation in which a polymer translocates into a narrow channel and the monomers in the channel experience a driving force fc. In the high driving force limit, regardless of the channel width, IFTP theory predicts τ ∝ f βc for the translocation time, where β = −1 is the force scaling exponent. Moreover, LD data show that for a very narrow channel fitting only a single file of monomers, the entropic force due to the subchain inside the channel does not play a significant role in the translocation dynamics and the force exponent β = −1 regardless of the force magnitude. As the channel width increases the number of possible spatial configurations of the subchain inside the channel becomes significant and the resulting entropic force causes the force exponent to drop below unity. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevResearch.4.033003 SN - 2643-1564 VL - 4 SP - 033003-1 EP - 033003-14 PB - American Physical Society CY - College Park, Maryland, USA ET - 3 ER - TY - JOUR A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb T1 - Distribution of first-reaction times with target regions on boundaries of shell-like domains JF - New Journal of Physics (NJP) N2 - We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted 'onion-shell' geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes. KW - diffusion KW - first-passage time KW - first-reaction time KW - shell-like geometries KW - approximate methods Y1 - 2021 U6 - https://doi.org/10.1088/1367-2630/ac4282 SN - 1367-2630 VL - 2021 SP - 1 EP - 23 PB - IOP Publishing CY - London ET - 23 ER - TY - GEN A1 - Javanainen, Matti A1 - Martinez-Seara, Hector A1 - Metzler, Ralf A1 - Vattulainen, Ilpo Tapio T1 - Diffusion of Proteins and Lipids in Protein-Rich Membranesa T2 - Biophysical journal Y1 - 2018 U6 - https://doi.org/10.1016/j.bpj.2017.11.3009 SN - 0006-3495 SN - 1542-0086 VL - 114 IS - 3 SP - 551A EP - 551A PB - Cell Press CY - Cambridge ER - TY - JOUR A1 - Bauer, Maximilian A1 - Godec, Aljaž A1 - Metzler, Ralf T1 - Diffusion of finite-size particles in two-dimensional channels with random wall configurations JF - Physical chemistry, chemical physics : PCCP ; 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 [J. Chem. Phys., 2012, 137, 024107]. KW - anomalous diffusion KW - fractional dynamics KW - transport KW - nonergodicity KW - coefficient Y1 - 2014 U6 - https://doi.org/10.1039/C3CP55160A SN - 1463-9084 SN - 1463-9076 VL - 16 IS - 13 SP - 6118 EP - 6128 PB - RSC Publications CY - Cambridge ER - TY - JOUR A1 - Kosztolowicz, Tadeusz A1 - Metzler, Ralf T1 - Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We propose a model of antibiotic diffusion through a bacterial biofilm when diffusion and/or absorption barriers develop in the biofilm. The idea of this model is: We deduce details of the diffusion process in a medium in which direct experimental study is difficult, based on probing diffusion in external regions. Since a biofilm has a gel-like consistency, we suppose that subdiffusion of particles in the biofilm may occur. To describe this process we use a fractional subdiffusion-absorption equation with an adjustable anomalous diffusion exponent. The boundary conditions at the boundaries of the biofilm are derived by means of a particle random walk model on a discrete lattice leading to an expression involving a fractional time derivative. We show that the temporal evolution of the total amount of substance that has diffused through the biofilm explicitly depends on whether there is antibiotic absorption in the biofilm. This fact is used to experimentally check for antibiotic absorption in the biofilm and if subdiffusion and absorption parameters of the biofilm change over time. We propose a four-stage model of antibiotic diffusion in biofilm based on the following physical characteristics: whether there is absorption of the antibiotic in the biofilm and whether all biofilm parameters remain unchanged over time. The biological interpretation of the stages, in particular their relation with the bacterial defense mechanisms, is discussed. Theoretical results are compared with empirical results of ciprofloxacin diffusion through Pseudomonas aeruginosa biofilm, and ciprofloxacin and gentamicin diffusion through Proteus mirabilis biofilm. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.032408 SN - 2470-0045 SN - 2470-0053 VL - 102 IS - 3 PB - American Physical Society CY - Melville, NY ER - TY - JOUR A1 - Molina-Garcia, Daniel A1 - Sandev, Trifce A1 - Safdari, Hadiseh A1 - Pagnini, Gianni A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Crossover from anomalous to normal diffusion BT - truncated power-law noise correlations and applications to dynamics in lipid bilayers JF - New Journal of Physics N2 - Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive–diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive–diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion–diffusion and subdiffusion–subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling. KW - anomalous diffusion KW - truncated power-law correlated noise KW - lipid bilayer membrane dynamics Y1 - 2018 U6 - https://doi.org/10.1088/1367-2630/aae4b2 SN - 1367-2630 VL - 20 PB - IOP Publishing Ltd CY - London und Bad Honnef ER - TY - JOUR A1 - Awad, Emad A1 - Metzler, Ralf T1 - Crossover dynamics from superdiffusion to subdiffusion BT - models and solutions JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function. KW - Cattaneo equation KW - telegrapher's equation KW - crossover dynamics KW - fractional dynamic equations KW - anomalous diffusion KW - superdiffusion and KW - subdiffusion KW - Fox H-functions Y1 - 2020 U6 - https://doi.org/10.1515/fca-2020-0003 SN - 1311-0454 SN - 1314-2224 VL - 23 IS - 1 SP - 55 EP - 102 PB - De Gruyter CY - Berlin ; Boston ER - TY - JOUR A1 - de Carvalho, Sidney J. A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. T1 - Critical adsorption of polyelectrolytes onto planar and convex highly charged surfaces BT - the nonlinear Poisson–Boltzmann approach JF - New journal of physics : the open-access journal for physics N2 - We study the adsorption–desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition—demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces—are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye–Hückel approximation is often not feasible and the nonlinear Poisson–Boltzmann approach should be implemented. At low salt conditions, for instance, the electrostatic potential from the nonlinear Poisson–Boltzmann equation is smaller than the Debye–Hückel result, such that the required critical surface charge density for polyelectrolyte adsorption σc increases. The nonlinear relation between the surface charge density and electrostatic potential leads to a sharply increasing critical surface charge density with growing ionic strength, imposing an additional limit to the critical salt concentration above which no polyelectrolyte adsorption occurs at all. We contrast our simulations findings with the known scaling results for weak critical polyelectrolyte adsorption onto oppositely charged surfaces for the three standard geometries. Finally, we discuss some applications of our results for some physical–chemical and biophysical systems. KW - polyelectrolyte adsorption KW - electrostatic interactions KW - critical phenomena KW - Debye screening Y1 - 2016 U6 - https://doi.org/10.1088/1367-2630/18/8/083037 SN - 1367-2630 VL - 18 PB - IOP Publ. CY - London ER - TY - JOUR A1 - Caetano, Daniel L. Z. A1 - Carvalho, Sidney Jurado de A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. T1 - Critical adsorption of multiple polyelectrolytes onto a nanosphere BT - splitting the adsorption-desorption transition boundary JF - Interface : journal of the Royal Society N2 - Employing extensive Monte Carlo computer simulations, we investigate in detail the properties of multichain adsorption of charged flexible polyelectrolytes (PEs) onto oppositely charged spherical nanoparticles (SNPs). We quantify the conditions of critical adsorption-the phase-separation curve between the adsorbed and desorbed states of the PEs-as a function of the SNP surface-charge density and the concentration of added salt. We study the degree of fluctuations of the PE-SNP electrostatic binding energy, which we use to quantify the emergence of the phase subtransitions, including a series of partially adsorbed PE configurations. We demonstrate how the phase-separation adsorption-desorption boundary shifts and splits into multiple subtransitions at low-salt conditions, thereby generalizing and extending the results for critical adsorption of a single PE onto the SNP. The current findings are relevant for finite concentrations of PEs around the attracting SNP, such as the conditions for PE adsorption onto globular proteins carrying opposite electric charges. KW - nanoparticles KW - polyelectrolytes KW - electrostatics KW - critical adsorption KW - phase-transition boundary Y1 - 2020 U6 - https://doi.org/10.1098/rsif.2020.0199 SN - 1742-5689 SN - 1742-5662 VL - 17 IS - 167 PB - Royal Society CY - London ER -