@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{SchwarzlGodecOshaninetal.2016, author = {Schwarzl, Maria and Godec, Aljaz and Oshanin, Gleb and Metzler, Ralf}, title = {A single predator charging a herd of prey: effects of self volume and predator-prey decision-making}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {49}, journal = {Journal of physics : A, Mathematical and theoretical}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/49/22/225601}, pages = {19}, year = {2016}, abstract = {We study the degree of success of a single predator hunting a herd of prey on a two-dimensional square lattice landscape. We explicitly consider the self volume of the prey restraining their dynamics on the lattice. The movement of both predator and prey is chosen to include an intelligent, decision making step based on their respective sighting ranges, the radius in which they can detect the other species (prey cannot recognise each other besides the self volume interaction): after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase, respectively. We consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey. We find that the prey's sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight. We characterise the dynamics in terms of the mean distance between the predator and the nearest prey. It turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple Ornstein-Uhlenbeck picture. Reducing the many-body problem to a simple two-body problem by imagining predator and nearest prey to be connected by an effective Hookean bond, all features of the model such as prey density and sighting ranges merge into the effective binding constant.}, language = {en} } @article{GrebenkovMetzlerOshanin2017, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains}, series = {New journal of physics : the open-access journal for physics}, volume = {19}, 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/aa8ed9}, pages = {11}, year = {2017}, abstract = {We study 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. A similar 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. We analyse 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.}, language = {en} } @misc{GrebenkovMetzlerOshanin2017, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-403726}, pages = {11}, year = {2017}, abstract = {We study 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.}, language = {en} } @article{GrebenkovMetzlerOshanin2017, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains}, series = {New journal of physics}, volume = {19}, journal = {New journal of physics}, publisher = {IOP}, address = {London}, issn = {1367-2630}, doi = {10.1088/1367-2630/aa8ed9}, pages = {1 -- 11}, year = {2017}, abstract = {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.}, 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} } @article{GrebenkovMetzlerOshanin2018, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Towards a full quantitative description of single-molecule reaction kinetics in biological cells}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {20}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {24}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c8cp02043d}, pages = {16393 -- 16401}, year = {2018}, abstract = {The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.}, 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{SposiniMetzlerOshanin2019, author = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb}, title = {Single-trajectory spectral analysis of scaled Brownian motion}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab2f52}, 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} }