TY - JOUR
A1 - Schwarzl, Maria
A1 - Godec, Aljaz
A1 - Oshanin, Gleb
A1 - Metzler, Ralf
T1 - A single predator charging a herd of prey: effects of self volume and predator-prey decision-making
JF - Journal of physics : A, Mathematical and theoretical
N2 - 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.
KW - first passage process
KW - diffusion
KW - predator-prey model
Y1 - 2016
U6 - https://doi.org/10.1088/1751-8113/49/22/225601
SN - 1751-8113
SN - 1751-8121
VL - 49
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains
JF - New journal of physics
N2 - Westudy the mean first passage time (MFPT) to a reaction event on a specific site in a cylindrical geometry—characteristic, for instance, for bacterial cells, with a concentric inner cylinder representing the nuclear region of the bacterial cell. Asimilar problem emerges in the description of a diffusive search by a transcription factor protein for a specific binding region on a single strand of DNA.We develop a unified theoretical approach to study the underlying boundary value problem which is based on a self-consistent approximation of the mixed boundary condition. Our approach permits us to derive explicit, novel, closed-form expressions for the MFPT valid for a generic setting with an arbitrary relation between the system parameters.Weanalyse this general result in the asymptotic limits appropriate for the above-mentioned biophysical problems. Our investigation reveals the crucial role of the target aspect ratio and of the intrinsic reactivity of the binding region, which were disregarded in previous studies. Theoretical predictions are confirmed by numerical simulations.
KW - first passage time
KW - cylindrical geometry
KW - aspect ratio
KW - protein search
Y1 - 2017
U6 - https://doi.org/10.1088/1367-2630/aa8ed9
SN - 1367-2630
VL - 19
SP - 1
EP - 11
PB - IOP
CY - London
ER -
TY - JOUR
A1 - Mattos, Thiago G.
A1 - Mejia-Monasterio, Carlos
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - First passages in bounded domains When is the mean first passage time meaningful?
JF - Physical review : E, Statistical, nonlinear and soft matter physics
N2 - 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.
Y1 - 2012
U6 - https://doi.org/10.1103/PhysRevE.86.031143
SN - 1539-3755
VL - 86
IS - 3
PB - American Physical Society
CY - College Park
ER -
TY - JOUR
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains
JF - New journal of physics : the open-access journal for physics
N2 - 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.
KW - first passage time
KW - cylindrical geometry
KW - aspect ratio
KW - protein search
Y1 - 2017
U6 - https://doi.org/10.1088/1367-2630/aa8ed9
SN - 1367-2630
VL - 19
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Sposini, Vittoria
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Single-trajectory spectral analysis of scaled Brownian motion
JF - New Journal of Physics
N2 - 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.
KW - diffusion
KW - anomalous diffusion
KW - power spectral analysis
KW - single trajectory analysis
Y1 - 2019
U6 - https://doi.org/10.1088/1367-2630/ab2f52
SN - 1367-2630
VL - 21
PB - Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics
CY - Bad Honnef und London
ER -
TY - JOUR
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Dagdug, Leonardo
A1 - Berezhkovskii, Alexander M.
A1 - Skvortsov, Alexei T.
T1 - Trapping of diffusing particles by periodic absorbing rings on a cylindrical tube
JF - The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr
Y1 - 2019
U6 - https://doi.org/10.1063/1.5098390
SN - 0021-9606
SN - 1089-7690
VL - 150
IS - 20
PB - American Institute of Physics
CY - Melville
ER -
TY - JOUR
A1 - Grebenkov, Denis S
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Full distribution of first exit times in the narrow escape problem
JF - New Journal of Physics
N2 - In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small 'escape window' in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.
KW - narrow escape problem
KW - first-passage time distribution
KW - mean versus most probable reaction times
KW - mixed boundary conditions
Y1 - 2019
U6 - https://doi.org/10.1088/1367-2630/ab5de4
SN - 1367-2630
VL - 21
PB - Dt. Physikalische Ges.
CY - Bad Honnef
ER -
TY - JOUR
A1 - Krapf, Diego
A1 - Lukat, Nils
A1 - Marinari, Enzo
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Selhuber-Unkel, Christine
A1 - Squarcini, Alessio
A1 - Stadler, Lorenz
A1 - Weiss, Matthias
A1 - Xu, Xinran
T1 - Spectral Content of a Single Non-Brownian Trajectory
JF - Physical review : X, Expanding access
N2 - 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.
KW - Biological Physics
KW - Interdisciplinary Physics
KW - Statistical Physics
Y1 - 2019
U6 - https://doi.org/10.1103/PhysRevX.9.011019
SN - 2160-3308
VL - 9
IS - 1
PB - American Physical Society
CY - College Park
ER -
TY - JOUR
A1 - Sposini, Vittoria
A1 - Grebenkov, Denis S
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Seno, Flavio
T1 - Universal spectral features of different classes of random-diffusivity processes
JF - New Journal of Physics
N2 - Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.
KW - diffusion
KW - power spectrum
KW - random diffusivity
KW - single trajectories
Y1 - 2020
U6 - https://doi.org/10.1088/1367-2630/ab9200
SN - 1367-2630
VL - 22
IS - 6
PB - Dt. Physikalische Ges.
CY - Bad Honnef
ER -
TY - JOUR
A1 - Grebenkov, Denis S.
A1 - Sposini, Vittoria
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Seno, Flavio
T1 - Exact distributions of the maximum and range of random diffusivity processes
JF - New Journal of Physics
N2 - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.
KW - random diffusivity
KW - extremal values
KW - maximum and range
KW - diffusion
KW - Brownian motion
Y1 - 2021
U6 - https://doi.org/10.1088/1367-2630/abd313
SN - 1367-2630
VL - 23
PB - Dt. Physikalische Ges.
CY - Bad Honnef
ER -
TY - JOUR
A1 - Grebenkov, Denis S
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - From single-particle stochastic kinetics to macroscopic reaction rates
BT - fastest first-passage time of N random walkers
JF - New Journal of Physics
N2 - We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N² for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.
KW - diffusion
KW - first-passage
KW - fastest first-passage time of N walkers
Y1 - 2020
U6 - https://doi.org/10.1088/1367-2630/abb1de
SN - 1367-2630
VL - 22
PB - Dt. Physikalische Ges.
CY - Bad Honnef
ER -
TY - JOUR
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control
JF - Communications Chemistry
N2 - Textbook concepts of diffusion-versus kinetic-control are well-defined for reaction-kinetics involving macroscopic concentrations of diffusive reactants that are adequately described by rate-constants—the inverse of the mean-first-passage-time to the reaction-event. In contradiction, an open important question is whether the mean-first-passage-time alone is a sufficient measure for biochemical reactions that involve nanomolar reactant concentrations. Here, using a simple yet generic, exactly solvable model we study the effect of diffusion and chemical reaction-limitations on the full reaction-time distribution. We show that it has a complex structure with four distinct regimes delineated by three characteristic time scales spanning a window of several decades. Consequently, the reaction-times are defocused: no unique time-scale characterises the reaction-process, diffusion- and kinetic-control can no longer be disentangled, and it is imperative to know the full reaction-time distribution. We introduce the concepts of geometry- and reaction-control, and also quantify each regime by calculating the corresponding reaction depth.
Y1 - 2018
U6 - https://doi.org/10.1038/s42004-018-0096-x
SN - 2399-3669
VL - 1
PB - Macmillan Publishers Limited
CY - London
ER -
TY - GEN
A1 - Grebenkov, Denis
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1159
KW - diffusion
KW - reaction cascade
KW - first passage time
Y1 - 2021
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-521942
SN - 1866-8372
ER -
TY - JOUR
A1 - Grebenkov, Denis
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes
JF - New Journal of Physics (NJP)
N2 - We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.
KW - diffusion
KW - reaction cascade
KW - first passage time
Y1 - 2021
U6 - https://doi.org/10.1088/1367-2630/ac1e42
SN - 1367-2630
VL - 23
PB - IOP - Institute of Physics Publishing
CY - Bristol
ER -
TY - GEN
A1 - Grebenkov, Denis S
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - From single-particle stochastic kinetics to macroscopic reaction rates
BT - fastest first-passage time of N random walkers
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N² for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1018
KW - diffusion
KW - first-passage
KW - fastest first-passage time of N walkers
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-484059
SN - 1866-8372
IS - 1018
ER -
TY - GEN
A1 - Grebenkov, Denis S.
A1 - Sposini, Vittoria
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Seno, Flavio
T1 - Exact distributions of the maximum and range of random diffusivity processes
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1142
KW - random diffusivity
KW - extremal values
KW - maximum and range
KW - diffusion
KW - Brownian motion
Y1 - 2021
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-503976
SN - 1866-8372
IS - 1142
ER -
TY - JOUR
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Towards a full quantitative description of single-molecule reaction kinetics in biological cells
JF - Physical chemistry, chemical physics : a journal of European Chemical Societies
N2 - 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.
Y1 - 2018
U6 - https://doi.org/10.1039/c8cp02043d
SN - 1463-9076
SN - 1463-9084
VL - 20
IS - 24
SP - 16393
EP - 16401
PB - Royal Society of Chemistry
CY - Cambridge
ER -
TY - GEN
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control
T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2 - Textbook concepts of diffusion-versus kinetic-control are well-defined for reaction-kinetics involving macroscopic concentrations of diffusive reactants that are adequately described by rate-constants—the inverse of the mean-first-passage-time to the reaction-event. In contradiction, an open important question is whether the mean-first-passage-time alone is a sufficient measure for biochemical reactions that involve nanomolar reactant concentrations. Here, using a simple yet generic, exactly solvable model we study the effect of diffusion and chemical reaction-limitations on the full reaction-time distribution. We show that it has a complex structure with four distinct regimes delineated by three characteristic time scales spanning a window of several decades. Consequently, the reaction-times are defocused: no unique time-scale characterises the reaction-process, diffusion- and kinetic-control can no longer be disentangled, and it is imperative to know the full reaction-time distribution. We introduce the concepts of geometry- and reaction-control, and also quantify each regime by calculating the corresponding reaction depth.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 527
Y1 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-422989
SN - 1866-8372
IS - 527
ER -
TY - GEN
A1 - Grebenkov, Denis S.
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains
N2 - 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.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 391
KW - aspect ratio
KW - cylindrical geometry
KW - first passage time
KW - protein search
Y1 - 2017
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-403726
ER -
TY - GEN
A1 - Sposini, Vittoria
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Single-trajectory spectral analysis of scaled Brownian motion
T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2 - 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.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 753
KW - diffusion
KW - anomalous diffusion
KW - power spectral analysis
KW - single trajectory analysis
Y1 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436522
SN - 1866-8372
IS - 753
ER -
TY - GEN
A1 - Grebenkov, Denis S
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
T1 - Full distribution of first exit times in the narrow escape problem
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small 'escape window' in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 810
KW - narrow escape problem
KW - first-passage time distribution
KW - mean versus most probable reaction times
KW - mixed boundary conditions
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-442883
SN - 1866-8372
IS - 810
ER -
TY - GEN
A1 - Sposini, Vittoria
A1 - Grebenkov, Denis S
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Seno, Flavio
T1 - Universal spectral features of different classes of random-diffusivity processes
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 999
KW - diffusion
KW - power spectrum
KW - random diffusivity
KW - single trajectories
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-476960
SN - 1866-8372
IS - 999
ER -
TY - GEN
A1 - Krapf, Diego
A1 - Marinari, Enzo
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Xu, Xinran
A1 - Squarcini, Alessio
T1 - Power spectral density of a single Brownian trajectory
BT - what one can and cannot learn from it
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 655
KW - power spectral density
KW - single-trajectory analysis
KW - probability density function
KW - exact results
Y1 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-424296
SN - 1866-8372
IS - 655
ER -
TY - JOUR
A1 - Krapf, Diego
A1 - Marinari, Enzo
A1 - Metzler, Ralf
A1 - Oshanin, Gleb
A1 - Xu, Xinran
A1 - Squarcini, Alessio
T1 - Power spectral density of a single Brownian trajectory
BT - what one can and cannot learn from it
JF - New journal of physics : the open-access journal for physics
N2 - The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.
KW - power spectral density
KW - single-trajectory analysis
KW - probability density function
KW - exact results
Y1 - 2018
U6 - https://doi.org/10.1088/1367-2630/aaa67c
SN - 1367-2630
VL - 20
PB - IOP Publ. Ltd.
CY - Bristol
ER -