@article{EliazarMetzlerReuveni2019, author = {Eliazar, Iddo and Metzler, Ralf and Reuveni, Shlomi}, title = {Poisson-process limit laws yield Gumbel max-min and min-max}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.022129}, pages = {12}, year = {2019}, abstract = {"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}, language = {en} } @article{EliazarMetzlerReuveni2019, author = {Eliazar, Iddo and Metzler, Ralf and Reuveni, Shlomi}, title = {Gumbel central limit theorem for max-min and min-max}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.020104}, pages = {6}, year = {2019}, abstract = {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.}, language = {en} } @article{JavanainenMartinezSearaMetzleretal.2017, author = {Javanainen, Matti and Martinez-Seara, Hector and Metzler, Ralf and Vattulainen, Ilpo}, title = {Diffusion of Integral Membrane Proteins in Protein-Rich Membranes}, series = {The journal of physical chemistry letters}, volume = {8}, journal = {The journal of physical chemistry letters}, publisher = {American Chemical Society}, address = {Washington}, issn = {1948-7185}, doi = {10.1021/acs.jpclett.7b01758}, pages = {4308 -- 4313}, year = {2017}, abstract = {The lateral diffusion of embedded proteins along lipid membranes in protein-poor conditions has been successfully described in terms of the Saffman-Delbruck (SD) model, which predicts that the protein diffusion coefficient D is weakly dependent on its radius R as D proportional to ln(1/R). However, instead of being protein-poor, native cell membranes are extremely crowded with proteins. On the basis of extensive molecular simulations, we here demonstrate that protein crowding of the membrane at physiological levels leads to deviations from the SD relation and to the emergence of a stronger Stokes-like dependence D proportional to 1/R. We propose that this 1/R law mainly arises due to geometrical factors: smaller proteins are able to avoid confinement effects much better than their larger counterparts. The results highlight that the lateral dynamics in the crowded setting found in native membranes is radically different from protein-poor conditions and plays a significant role in formation of functional multiprotein complexes.}, language = {en} } @article{KrapfMetzler2019, author = {Krapf, Diego and Metzler, Ralf}, title = {Strange interfacial molecular dynamics}, series = {Physics today}, volume = {72}, journal = {Physics today}, number = {9}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0031-9228}, doi = {10.1063/PT.3.4294}, pages = {48 -- 54}, year = {2019}, language = {en} } @article{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Transport in exclusion processes with one-step memory: density dependence and optimal acceleration}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {38}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab37e4}, pages = {19}, year = {2019}, abstract = {We study a lattice gas of persistent walkers, in which each site is occupied by at most one particle and the direction each particle attempts to move to depends on its last step. We analyse the mean squared displacement (MSD) of the particles as a function of the particle density and their persistence (the tendency to continue moving in the same direction). For positive persistence the MSD behaves as expected: it increases with the persistence and decreases with the density. However, for strong anti-persistence we find two different regimes, in which the dependence of the MSD on the density is non-monotonic. For very strong anti-persistence there is an optimal density at which the MSD reaches a maximum. In an intermediate regime, the MSD as a function of the density exhibits both a minimum and a maximum, a phenomenon which has not been observed before. We derive a mean-field theory which qualitatively explains this behaviour.}, language = {en} } @article{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Correlations and transport in exclusion processes with general finite memory}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab47fb}, pages = {31}, year = {2019}, language = {en} } @misc{JavanainenMartinezSearaMetzleretal.2017, author = {Javanainen, Matti and Martinez-Seara, Hector and Metzler, Ralf and Vattulainen, Ilpo Tapio}, title = {Diffusion of Proteins and Lipids in Protein-Rich Membranesa}, series = {Biophysical journal}, volume = {114}, journal = {Biophysical journal}, number = {3}, publisher = {Cell Press}, address = {Cambridge}, issn = {0006-3495}, doi = {10.1016/j.bpj.2017.11.3009}, pages = {551A -- 551A}, year = {2017}, language = {en} } @article{KrapfMarinariMetzleretal.2018, author = {Krapf, Diego and Marinari, Enzo and Metzler, Ralf and Oshanin, Gleb and Xu, Xinran and Squarcini, Alessio}, title = {Power spectral density of a single Brownian trajectory}, series = {New journal of physics : the open-access journal for physics}, volume = {20}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/aaa67c}, pages = {30}, year = {2018}, abstract = {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.}, language = {en} } @misc{GudowskaNowakLindenbergMetzler2017, author = {Gudowska-Nowak, Ewa and Lindenberg, Katja and Metzler, Ralf}, title = {Preface: Marian Smoluchowski's 1916 paper—a century of inspiration}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {50}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {38}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aa8529}, pages = {8}, year = {2017}, language = {en} } @misc{Metzler2020, author = {Metzler, Ralf}, title = {Superstatistics and non-Gaussian diffusion}, series = {The European physical journal special topics}, volume = {229}, journal = {The European physical journal special topics}, number = {5}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2020-900210-x}, pages = {711 -- 728}, year = {2020}, abstract = {Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect "ensembles" of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed ("superstatistical") transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches.}, language = {en} } @article{PalyulinAlaNissilaMetzler2014, author = {Palyulin, Vladimir V. and Ala-Nissila, Tapio and Metzler, Ralf}, title = {Polymer translocation: the first two decades and the recent diversification}, series = {Soft matter}, volume = {45}, journal = {Soft matter}, number = {10}, editor = {Metzler, Ralf}, publisher = {the Royal Society of Chemistry}, address = {Cambridge}, issn = {1744-683X}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-76266}, pages = {9016 -- 9037}, year = {2014}, abstract = {Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous-infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis.}, language = {en} } @misc{PalyulinAlaNissilaMetzler2014, author = {Palyulin, Vladimir V. and Ala-Nissila, Tapio and Metzler, Ralf}, title = {Polymer translocation: the first two decades and the recent diversification}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-76287}, pages = {9016 -- 9037}, year = {2014}, abstract = {Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous-infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis.}, language = {en} } @article{MetzlerBauerRasmussenetal.2015, author = {Metzler, Ralf and Bauer, Maximilian and Rasmussen, Emil S. and Lomholt, Michael A.}, title = {Real sequence effects on the search dynamics of transcription factors on DNA}, series = {Scientific Reports}, volume = {5}, journal = {Scientific Reports}, number = {10072}, publisher = {Nature Publishing Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/srep10072}, year = {2015}, abstract = {Recent experiments show that transcription factors (TFs) indeed use the facilitated diffusion mechanism to locate their target sequences on DNA in living bacteria cells: TFs alternate between sliding motion along DNA and relocation events through the cytoplasm. From simulations and theoretical analysis we study the TF-sliding motion for a large section of the DNA-sequence of a common E. coli strain, based on the two-state TF-model with a fast-sliding search state and a recognition state enabling target detection. For the probability to detect the target before dissociating from DNA the TF-search times self-consistently depend heavily on whether or not an auxiliary operator (an accessible sequence similar to the main operator) is present in the genome section. Importantly, within our model the extent to which the interconversion rates between search and recognition states depend on the underlying nucleotide sequence is varied. A moderate dependence maximises the capability to distinguish between the main operator and similar sequences. Moreover, these auxiliary operators serve as starting points for DNA looping with the main operator, yielding a spectrum of target detection times spanning several orders of magnitude. Auxiliary operators are shown to act as funnels facilitating target detection by TFs.}, language = {en} } @misc{MetzlerBauerRasmussenetal.2015, author = {Metzler, Ralf and Bauer, Maximilian and Rasmussen, Emil S. and Lomholt, Michael A.}, title = {Real sequence effects on the search dynamics of transcription factors on DNA}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-79411}, year = {2015}, abstract = {Recent experiments show that transcription factors (TFs) indeed use the facilitated diffusion mechanism to locate their target sequences on DNA in living bacteria cells: TFs alternate between sliding motion along DNA and relocation events through the cytoplasm. From simulations and theoretical analysis we study the TF-sliding motion for a large section of the DNA-sequence of a common E. coli strain, based on the two-state TF-model with a fast-sliding search state and a recognition state enabling target detection. For the probability to detect the target before dissociating from DNA the TF-search times self-consistently depend heavily on whether or not an auxiliary operator (an accessible sequence similar to the main operator) is present in the genome section. Importantly, within our model the extent to which the interconversion rates between search and recognition states depend on the underlying nucleotide sequence is varied. A moderate dependence maximises the capability to distinguish between the main operator and similar sequences. Moreover, these auxiliary operators serve as starting points for DNA looping with the main operator, yielding a spectrum of target detection times spanning several orders of magnitude. Auxiliary operators are shown to act as funnels facilitating target detection by TFs.}, language = {en} } @article{XuZhouMetzleretal.2020, author = {Xu, Pengbo and Zhou, Tian and Metzler, Ralf and Deng, Weihua}, title = {L{\´e}vy walk dynamics in an external harmonic potential}, series = {Physical review : E, Statistical, nonlinear, and soft matter physics}, volume = {101}, journal = {Physical review : E, Statistical, nonlinear, and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.101.062127}, pages = {12}, year = {2020}, abstract = {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.}, language = {en} } @misc{MardoukhiJeonMetzler2015, author = {Mardoukhi, Yousof and Jeon, Jae-Hyung and Metzler, Ralf}, title = {Geometry controlled anomalous diffusion in random fractal geometries}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {980}, issn = {1866-8372}, doi = {10.25932/publishup-47486}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-474864}, pages = {30134 -- 30147}, year = {2015}, abstract = {We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law ∼T-h with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.}, language = {en} } @misc{GranadoAbadMetzleretal.2020, author = {Granado, Felipe Le Vot and Abad, Enrique and Metzler, Ralf and Yuste, Santos B.}, title = {Continuous time random walk in a velocity field}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1005}, issn = {1866-8372}, doi = {10.25932/publishup-47999}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-479997}, pages = {28}, year = {2020}, abstract = {We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a L{\´e}vy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of L{\´e}vy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.}, language = {en} } @misc{NorregaardMetzlerRitteretal.2017, author = {Norregaard, Kamilla and Metzler, Ralf and Ritter, Christine M. and Berg-Sorensen, Kirstine and Oddershede, Lene Broeng}, title = {Manipulation and Motion of Organelles and Single Molecules in Living Cells}, series = {Chemical reviews}, volume = {117}, journal = {Chemical reviews}, number = {5}, publisher = {American Chemical Society}, address = {Washington}, issn = {0009-2665}, doi = {10.1021/acs.chemrev.6b00638}, pages = {4342 -- 4375}, year = {2017}, abstract = {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.}, language = {en} } @misc{Metzler2017, author = {Metzler, Ralf}, title = {Gaussianity Fair}, series = {Biophysical journal}, volume = {112}, journal = {Biophysical journal}, number = {3}, publisher = {Cell Press}, address = {Cambridge}, issn = {0006-3495}, doi = {10.1016/j.bpj.2016.12.019}, pages = {413 -- 415}, year = {2017}, language = {en} } @misc{Metzler2017, author = {Metzler, Ralf}, title = {Anomalous Diffusion in Membranes and the Cytoplasm of Biological Cells}, series = {Biophysical journal}, volume = {112}, journal = {Biophysical journal}, number = {3}, publisher = {Cell Press}, address = {Cambridge}, issn = {0006-3495}, doi = {10.1016/j.bpj.2016.11.2577}, pages = {476A -- 476A}, year = {2017}, language = {en} } @article{GodecMetzler2017, author = {Godec, Aljaž and Metzler, Ralf}, title = {First passage time statistics for two-channel diffusion}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {50}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aa5204}, pages = {17}, year = {2017}, abstract = {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.}, language = {en} } @article{EstradaDelvenneHatanoetal.2018, author = {Estrada, Ernesto and Delvenne, Jean-Charles and Hatano, Naomichi and Mateos, Jose L. and Metzler, Ralf and Riascos, Alejandro P. and Schaub, Michael T.}, title = {Random multi-hopper model}, series = {Journal of Complex Networks}, volume = {6}, journal = {Journal of Complex Networks}, number = {3}, publisher = {Oxford Univ. Press}, address = {Oxford}, issn = {2051-1310}, doi = {10.1093/comnet/cnx043}, pages = {382 -- 403}, year = {2018}, abstract = {We develop a mathematical model considering a random walker with long-range hops on arbitrary graphs. The random multi-hopper can jump to any node of the graph from an initial position, with a probability that decays as a function of the shortest-path distance between the two nodes in the graph. We consider here two decaying functions in the form of Laplace and Mellin transforms of the shortest-path distances. We prove that when the parameters of these transforms approach zero asymptotically, the hitting time in the multi-hopper approaches the minimum possible value for a normal random walker. We show by computational experiments that the multi-hopper explores a graph with clusters or skewed degree distributions more efficiently than a normal random walker. We provide computational evidences of the advantages of the random multi-hopper model with respect to the normal random walk by studying deterministic, random and real-world networks.}, language = {en} } @article{KosztolowiczMetzlerWąsiketal.2020, author = {Kosztolowicz, Tadeusz and Metzler, Ralf and Wąsik, Slawomir and Arabski, Michal}, title = {Modelling experimentally measured of ciprofloxacin antibiotic diffusion in Pseudomonas aeruginosa biofilm formed in artificial sputum medium}, series = {PLoS ONE}, volume = {15}, journal = {PLoS ONE}, publisher = {PLOS}, address = {San Francisco, California, US}, issn = {1932-6203}, doi = {10.1371/journal.pone.0243003}, pages = {14}, year = {2020}, abstract = {We study the experimentally measured ciprofloxacin antibiotic diffusion through a gel-like artificial sputum medium (ASM) mimicking physiological conditions typical for a cystic fibrosis layer, in which regions occupied by Pseudomonas aeruginosa bacteria are present. To quantify the antibiotic diffusion dynamics we employ a phenomenological model using a subdiffusion-absorption equation with a fractional time derivative. This effective equation describes molecular diffusion in a medium structured akin Thompson's plumpudding model; here the 'pudding' background represents the ASM and the 'plums' represent the bacterial biofilm. The pudding is a subdiffusion barrier for antibiotic molecules that can affect bacteria found in plums. For the experimental study we use an interferometric method to determine the time evolution of the amount of antibiotic that has diffused through the biofilm. The theoretical model shows that this function is qualitatively different depending on whether or not absorption of the antibiotic in the biofilm occurs. We show that the process can be divided into three successive stages: (1) only antibiotic subdiffusion with constant biofilm parameters, (2) subdiffusion and absorption of antibiotic molecules with variable biofilm transport parameters, (3) subdiffusion and absorption in the medium but the biofilm parameters are constant again. Stage 2 is interpreted as the appearance of an intensive defence build-up of bacteria against the action of the antibiotic, and in the stage 3 it is likely that the bacteria have been inactivated. Times at which stages change are determined from the experimentally obtained temporal evolution of the amount of antibiotic that has diffused through the ASM with bacteria. Our analysis shows good agreement between experimental and theoretical results and is consistent with the biologically expected biofilm response. We show that an experimental method to study the temporal evolution of the amount of a substance that has diffused through a biofilm is useful in studying the processes occurring in a biofilm. We also show that the complicated biological process of antibiotic diffusion in a biofilm can be described by a fractional subdiffusion-absorption equation with subdiffusion and absorption parameters that change over time.}, language = {en} } @article{GranadoAbadMetzleretal.2020, author = {Granado, Felipe Le Vot and Abad, Enrique and Metzler, Ralf and Yuste, Santos B.}, title = {Continuous time random walk in a velocity field}, series = {New Journal of Physics}, volume = {22}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab9ae2}, pages = {27}, year = {2020}, abstract = {We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a L{\´e}vy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of L{\´e}vy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.}, language = {en} } @article{Metzler2019, author = {Metzler, Ralf}, title = {Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {11}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab4988}, pages = {18}, year = {2019}, abstract = {Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in general have been addressed in Statistical Physics. In particular, there now exists a very large range of applications of stochastic processes in various disciplines. Here we provide a summary of some of the recent developments in the field of stochastic processes, highlighting both the experimental findings and theoretical frameworks.}, language = {en} } @article{KrapfLukatMarinarietal.2019, author = {Krapf, Diego and Lukat, Nils and Marinari, Enzo and Metzler, Ralf and Oshanin, Gleb and Selhuber-Unkel, Christine and Squarcini, Alessio and Stadler, Lorenz and Weiss, Matthias and Xu, Xinran}, title = {Spectral Content of a Single Non-Brownian Trajectory}, series = {Physical review : X, Expanding access}, volume = {9}, journal = {Physical review : X, Expanding access}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {2160-3308}, doi = {10.1103/PhysRevX.9.011019}, pages = {13}, year = {2019}, abstract = {Time-dependent processes are often analyzed using the power spectral density (PSD) calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble average. Frequently, the available experimental datasets are too small for such ensemble averages, and hence, it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from S(f, T), the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable parametrized by frequency f and observation time T, for a broad family of anomalous diffusions-fractional Brownian motion with Hurst index H-and derive exactly its probability density function. We show that S(f, T) is proportional-up to a random numerical factor whose universal distribution we determine-to the ensemble-averaged PSD. For subdiffusion (H < 1/2), we find that S(f, T) similar to A/f(2H+1) with random amplitude A. In sharp contrast, for superdiffusion (H > 1/2) S(f, T) similar to BT2H-1/f(2) with random amplitude B. Remarkably, for H > 1/2 the PSD exhibits the same frequency dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for H > 1/2 the PSD is ageing and is dependent on T. Our predictions for both sub-and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels and by extensive simulations.}, language = {en} } @misc{SposiniMetzlerOshanin2019, author = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb}, title = {Single-trajectory spectral analysis of scaled Brownian motion}, series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, number = {753}, issn = {1866-8372}, doi = {10.25932/publishup-43652}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436522}, pages = {16}, year = {2019}, abstract = {Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.}, language = {en} } @article{SarabadaniMetzlerAlaNissila2022, author = {Sarabadani, Jalal and Metzler, Ralf and Ala-Nissila, Tapio}, title = {Driven polymer translocation into a channel: Isoflux tension propagation theory and Langevin dynamics simulations}, series = {Physical Review Research}, volume = {4}, journal = {Physical Review Research}, edition = {3}, publisher = {American Physical Society}, address = {College Park, Maryland, USA}, issn = {2643-1564}, doi = {10.1103/PhysRevResearch.4.033003}, pages = {033003-1 -- 033003-14}, year = {2022}, abstract = {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.}, language = {en} } @article{SposiniKrapfMarinarietal.2022, author = {Sposini, Vittoria and Krapf, Diego and Marinari, Enzo and Sunyer, Raimon and Ritort, Felix and Taheri, Fereydoon and Selhuber-Unkel, Christine and Benelli, Rebecca and Weiss, Matthias and Metzler, Ralf and Oshanin, Gleb}, title = {Towards a robust criterion of anomalous diffusion}, series = {Communications Physics}, volume = {5}, journal = {Communications Physics}, publisher = {Springer Nature}, address = {London}, issn = {2399-3650}, doi = {10.1038/s42005-022-01079-8}, pages = {10}, year = {2022}, abstract = {Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.}, language = {en} } @article{SecklerMetzler2022, author = {Seckler, Henrik and Metzler, Ralf}, title = {Bayesian deep learning for error estimation in the analysis of anomalous diffusion}, series = {Nature Communnications}, volume = {13}, journal = {Nature Communnications}, publisher = {Nature Publishing Group UK}, address = {London}, issn = {2041-1723}, doi = {10.1038/s41467-022-34305-6}, pages = {13}, year = {2022}, abstract = {Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusionmodel and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a wellcalibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.}, language = {en} } @article{AwadMetzler2020, author = {Awad, Emad and Metzler, Ralf}, title = {Crossover dynamics from superdiffusion to subdiffusion}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {23}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {1}, publisher = {De Gruyter}, address = {Berlin ; Boston}, issn = {1311-0454}, doi = {10.1515/fca-2020-0003}, pages = {55 -- 102}, year = {2020}, abstract = {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.}, language = {en} } @misc{KosztolowiczMetzlerWąsiketal.2020, author = {Kosztolowicz, Tadeusz and Metzler, Ralf and Wąsik, Slawomir and Arabski, Michal}, title = {Modelling experimentally measured of ciprofloxacin antibiotic diffusion in Pseudomonas aeruginosa biofilm formed in artificial sputum medium}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1107}, issn = {1866-8372}, doi = {10.25932/publishup-49086}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-490866}, pages = {16}, year = {2020}, abstract = {We study the experimentally measured ciprofloxacin antibiotic diffusion through a gel-like artificial sputum medium (ASM) mimicking physiological conditions typical for a cystic fibrosis layer, in which regions occupied by Pseudomonas aeruginosa bacteria are present. To quantify the antibiotic diffusion dynamics we employ a phenomenological model using a subdiffusion-absorption equation with a fractional time derivative. This effective equation describes molecular diffusion in a medium structured akin Thompson's plumpudding model; here the 'pudding' background represents the ASM and the 'plums' represent the bacterial biofilm. The pudding is a subdiffusion barrier for antibiotic molecules that can affect bacteria found in plums. For the experimental study we use an interferometric method to determine the time evolution of the amount of antibiotic that has diffused through the biofilm. The theoretical model shows that this function is qualitatively different depending on whether or not absorption of the antibiotic in the biofilm occurs. We show that the process can be divided into three successive stages: (1) only antibiotic subdiffusion with constant biofilm parameters, (2) subdiffusion and absorption of antibiotic molecules with variable biofilm transport parameters, (3) subdiffusion and absorption in the medium but the biofilm parameters are constant again. Stage 2 is interpreted as the appearance of an intensive defence build-up of bacteria against the action of the antibiotic, and in the stage 3 it is likely that the bacteria have been inactivated. Times at which stages change are determined from the experimentally obtained temporal evolution of the amount of antibiotic that has diffused through the ASM with bacteria. Our analysis shows good agreement between experimental and theoretical results and is consistent with the biologically expected biofilm response. We show that an experimental method to study the temporal evolution of the amount of a substance that has diffused through a biofilm is useful in studying the processes occurring in a biofilm. We also show that the complicated biological process of antibiotic diffusion in a biofilm can be described by a fractional subdiffusion-absorption equation with subdiffusion and absorption parameters that change over time.}, language = {en} } @article{StojkoskiSandevBasnarkovetal.2020, author = {Stojkoski, Viktor and Sandev, Trifce and Basnarkov, Lasko and Kocarev, Ljupco and Metzler, Ralf}, title = {Generalised geometric Brownian motion}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {12}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22121432}, pages = {34}, year = {2020}, abstract = {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.}, language = {en} } @article{SecklerMetzler2022, author = {Seckler, Henrik and Metzler, Ralf}, title = {Bayesian deep learning for error estimation in the analysis of anomalous diffusion}, series = {Nature Communications}, volume = {13}, journal = {Nature Communications}, number = {1}, publisher = {Nature portfolio}, address = {Berlin}, issn = {2041-1723}, doi = {10.1038/s41467-022-34305-6}, pages = {13}, year = {2022}, abstract = {Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusion model and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a well-calibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.
Diffusive motions in complex environments such as living biological cells or soft matter systems can be analyzed with single-particle-tracking approaches, where accuracy of output may vary. The authors involve a machine-learning technique for decoding anomalous-diffusion data and provide an uncertainty estimate together with predicted output.}, language = {en} } @misc{XuZhouMetzleretal.2022, author = {Xu, Pengbo and Zhou, Tian and Metzler, Ralf and Deng, Weihua}, title = {Stochastic harmonic trapping of a L{\´e}vy walk: transport and first-passage dynamics under soft resetting strategies}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {1866-8372}, doi = {10.25932/publishup-56040}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-560402}, pages = {1 -- 28}, year = {2022}, abstract = {We introduce and study a L{\´e}vy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.}, language = {en} } @misc{SarabadaniMetzlerAlaNissila2022, author = {Sarabadani, Jalal and Metzler, Ralf and Ala-Nissila, Tapio}, title = {Driven polymer translocation into a channel: Isoflux tension propagation theory and Langevin dynamics simulations}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1292}, issn = {1866-8372}, doi = {10.25932/publishup-57438}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-574387}, pages = {033003-1 -- 033003-14}, year = {2022}, abstract = {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.}, language = {en} } @misc{VilkAghionAvgaretal.2022, author = {Vilk, Ohad and Aghion, Erez and Avgar, Tal and Beta, Carsten and Nagel, Oliver and Sabri, Adal and Sarfati, Raphael and Schwartz, Daniel K. and Weiß, Matthias and Krapf, Diego and Nathan, Ran and Metzler, Ralf and Assaf, Michael}, title = {Unravelling the origins of anomalous diffusion}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1303}, issn = {1866-8372}, doi = {10.25932/publishup-57764}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577643}, pages = {16}, year = {2022}, abstract = {Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. This method decomposes anomalous transport into three primary effects: long-range correlations ("Joseph effect"), fat-tailed probability density of increments ("Noah effect"), and nonstationarity ("Moses effect"). We show that such a decomposition of real-life data allows us to infer nontrivial behavioral predictions and to resolve open questions in the fields of single-particle tracking in living cells and movement ecology.}, language = {en} } @article{MattosMejiaMonasterioMetzleretal.2012, author = {Mattos, Thiago G. and Mejia-Monasterio, Carlos and Metzler, Ralf and Oshanin, Gleb}, title = {First passages in bounded domains When is the mean first passage time meaningful?}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {86}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.86.031143}, pages = {8}, year = {2012}, abstract = {We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(omega) distribution of the random variable omega = tau(1)/(tau(1) + tau(2)), which is a measure for how similar the first passage times tau(1) and tau(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(omega) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(omega), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.}, language = {en} } @article{HerrmannMetzlerEngbert2017, author = {Herrmann, Carl J. J. and Metzler, Ralf and Engbert, Ralf}, title = {A self-avoiding walk with neural delays as a model of fixational eye movements}, series = {Scientific reports}, volume = {7}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-017-13489-8}, pages = {17}, year = {2017}, abstract = {Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.}, language = {en} } @article{XuLiuLietal.2020, author = {Xu, Yong and Liu, Xuemei and Li, Yongge and Metzler, Ralf}, title = {Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.062106}, pages = {16}, year = {2020}, abstract = {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.}, language = {en} } @article{MutothyaXuLietal.2021, author = {Mutothya, Nicholas Mwilu and Xu, Yong and Li, Yongge and Metzler, Ralf}, title = {Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {54}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {29}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/abfba6}, pages = {31}, year = {2021}, abstract = {We study the stochastic motion of a test particle in a heterogeneous medium in terms of a position dependent diffusion coefficient mimicking measured deterministic diffusivity gradients in biological cells or the inherent heterogeneity of geophysical systems. Compared to previous studies we here investigate the effect of the interplay of anomalous diffusion effected by position dependent diffusion coefficients and coloured non-Gaussian noise. The latter is chosen to be distributed according to Tsallis' q-distribution, representing a popular example for a non-extensive statistic. We obtain the ensemble and time averaged mean squared displacements for this generalised process and establish its non-ergodic properties as well as analyse the non-Gaussian nature of the associated displacement distribution. We consider both non-stratified and stratified environments.}, language = {en} } @article{EliazarMetzler2013, author = {Eliazar, Iddo and Metzler, Ralf}, title = {Anomalous statistics of random relaxations in random environments}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.022141}, pages = {12}, year = {2013}, abstract = {We comprehensively analyze the emergence of anomalous statistics in the context of the random relaxation ( RARE) model [Eliazar and Metzler, J. Chem. Phys. 137, 234106 ( 2012)], a recently introduced versatile model of random relaxations in random environments. The RARE model considers excitations scattered randomly across a metric space around a reaction center. The excitations react randomly with the center, the reaction rates depending on the excitations' distances from this center. Relaxation occurs upon the first reaction between an excitation and the center. Addressing both the relaxation time and the relaxation range, we explore when these random variables display anomalous statistics, namely, heavy tails at zero and at infinity that manifest, respectively, exceptionally high occurrence probabilities of very small and very large outliers. A cohesive set of closed-form analytic results is established, determining precisely when such anomalous statistics emerge.}, language = {en} } @article{KursaweSchulzMetzler2013, author = {Kursawe, Jochen and Schulz, Johannes H. P. and Metzler, Ralf}, title = {Transient aging in fractional brownian and langevin-equation motion}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {88}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.88.062124}, pages = {13}, year = {2013}, abstract = {Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t = 0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on ta is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.}, language = {en} } @article{BauerMetzler2013, author = {Bauer, Maximilian and Metzler, Ralf}, title = {In vivo facilitated diffusion model}, series = {PLoS one}, volume = {8}, journal = {PLoS one}, number = {1}, publisher = {PLoS}, address = {San Fransisco}, issn = {1932-6203}, doi = {10.1371/journal.pone.0053956}, pages = {8}, year = {2013}, abstract = {Under dilute in vitro conditions transcription factors rapidly locate their target sequence on DNA by using the facilitated diffusion mechanism. However, whether this strategy of alternating between three-dimensional bulk diffusion and one-dimensional sliding along the DNA contour is still beneficial in the crowded interior of cells is highly disputed. Here we use a simple model for the bacterial genome inside the cell and present a semi-analytical model for the in vivo target search of transcription factors within the facilitated diffusion framework. Without having to resort to extensive simulations we determine the mean search time of a lac repressor in a living E. coli cell by including parameters deduced from experimental measurements. The results agree very well with experimental findings, and thus the facilitated diffusion picture emerges as a quantitative approach to gene regulation in living bacteria cells. Furthermore we see that the search time is not very sensitive to the parameters characterizing the DNA configuration and that the cell seems to operate very close to optimal conditions for target localization. Local searches as implied by the colocalization mechanism are only found to mildly accelerate the mean search time within our model.}, language = {en} } @article{GodecMetzler2013, author = {Godec, Aljaz and Metzler, Ralf}, title = {Finite-Time effects and ultraweak ergodicity breaking in superdiffusive dynamics}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.020603}, pages = {5}, year = {2013}, abstract = {We study the ergodic properties of superdiffusive, spatiotemporally coupled Levy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement (delta x(2)) over bar around the ensemble value 3 - alpha (1 < alpha < 2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the average of (delta x(2)) over bar over an ensemble of trajectories as a function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.}, language = {en} } @article{SchulzBarkaiMetzler2013, author = {Schulz, Johannes H. P. and Barkai, Eli and Metzler, Ralf}, title = {Aging effects and population splitting in single-particle trajectoryaverages}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.020602}, pages = {5}, year = {2013}, abstract = {We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a) > 0 after initiation of the process at t = 0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age ta of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed.}, language = {en} } @article{VahabiSchulzShokrietal.2013, author = {Vahabi, Mahsa and Schulz, Johannes H. P. and Shokri, Babak and Metzler, Ralf}, title = {Area coverage of radial Levy flights with periodic boundary conditions}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.042136}, pages = {10}, year = {2013}, abstract = {We consider the area coverage of radial Levy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Levy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Levy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Levy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Levy flights in a square area. Our findings are of interest for a general understanding of Levy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns.}, language = {en} } @article{JeonLeijnseOddershedeetal.2013, author = {Jeon, Jae-Hyung and Leijnse, Natascha and Oddershede, Lene B. and Metzler, Ralf}, title = {Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions}, series = {New journal of physics : the open-access journal for physics}, volume = {15}, journal = {New journal of physics : the open-access journal for physics}, number = {4}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/15/4/045011}, pages = {16}, year = {2013}, abstract = {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.}, language = {en} } @article{MaityBandyopadhyayChattopadhyayetal.2013, author = {Maity, Alok Kumar and Bandyopadhyay, Arnab and Chattopadhyay, Sudip and Chaudhuri, Jyotipratim Ray and Metzler, Ralf and Chaudhury, Pinaki and Banik, Suman K.}, title = {Quantification of noise in bifunctionality-induced post-translational modification}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {88}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.88.032716}, pages = {7}, year = {2013}, abstract = {We present a generic analytical scheme for the quantification of fluctuations due to bifunctionality-induced signal transduction within the members of a bacterial two-component system. The proposed model takes into account post-translational modifications in terms of elementary phosphotransfer kinetics. Sources of fluctuations due to autophosphorylation, kinase, and phosphatase activity of the sensor kinase have been considered in the model via Langevin equations, which are then solved within the framework of linear noise approximation. The resultant analytical expression of phosphorylated response regulators are then used to quantify the noise profile of biologically motivated single and branched pathways. Enhancement and reduction of noise in terms of extra phosphate outflux and influx, respectively, have been analyzed for the branched system. Furthermore, the role of fluctuations of the network output in the regulation of a promoter with random activation-deactivation dynamics has been analyzed.}, language = {en} } @article{JeonBarkaiMetzler2013, author = {Jeon, Jae-Hyung and Barkai, Eli and Metzler, Ralf}, title = {Noisy continuous time random walks}, series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, volume = {139}, journal = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, number = {12}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0021-9606}, doi = {10.1063/1.4816635}, pages = {15}, year = {2013}, abstract = {Experimental studies of the diffusion of biomolecules within biological cells are routinely confronted with multiple sources of stochasticity, whose identification renders the detailed data analysis of single molecule trajectories quite intricate. Here, we consider subdiffusive continuous time random walks that represent a seminal model for the anomalous diffusion of tracer particles in complex environments. This motion is characterized by multiple trapping events with infinite mean sojourn time. In real physical situations, however, instead of the full immobilization predicted by the continuous time random walk model, the motion of the tracer particle shows additional jiggling, for instance, due to thermal agitation of the environment. We here present and analyze in detail an extension of the continuous time random walk model. Superimposing the multiple trapping behavior with additive Gaussian noise of variable strength, we demonstrate that the resulting process exhibits a rich variety of apparent dynamic regimes. In particular, such noisy continuous time random walks may appear ergodic, while the bare continuous time random walk exhibits weak ergodicity breaking. Detailed knowledge of this behavior will be useful for the truthful physical analysis of experimentally observed subdiffusion.}, language = {en} }