TY - THES A1 - Krüsemann, Henning T1 - First passage phenomena and single-file motion in ageing continuous time random walks and quenched energy landscapes N2 - In der Physik gibt es viele Prozesse, die auf Grund ihrer Komplexität nicht durch physikalische Gleichungen beschrieben werden können, beispielsweise die Bewegung eines Staubkorns in der Luft. Durch die vielen Stöße mit Luftmolekülen führt es eine Zufallsbewegung aus, die so genannte Diffusion. Auch Moleküle in biologischen Zellen diffundieren, jedoch befinden sich in einer solchen Zelle im selben Volumen viel mehr oder viel größere Moleküle. Das beobachtete Teilchen stößt dementsprechend öfter mit anderen zusammen und die Diffusion wird langsamer, sie wird subdiffusiv. Mit der Zeit kann sich die Charakteristik der Subdiffusion ändern; dies wird als (mikroskopisches) Altern bezeichnet. Ich untersuche in der vorliegenden Arbeit zwei mathematische Modelle für eindimensionale Subdiffusion, einmal den continuous time random walk (CTRW) und einmal die Zufallsbewegung in einer eingefrorenen Energielandschaft (QEL=quenched energy landscape). Beide sind Sprungprozesse, das heißt, sie sind Abfolgen von räumlichen Sprüngen, die durch zufallsverteilte Wartezeiten getrennt sind. Die Wartezeiten in der QEL sind räumlich korrelliert, während sie im CTRW unkorrelliert sind. Ich untersuche in der vorliegenden Arbeit verschiedene statistische Größen in beiden Modellen. Zunächst untersuche ich den Einfluss des Alters und den Einfluss der Korrellationen einer QEL auf die Verteilung der Zeiten, die das diffundierendes Teilchen benötigt, um eine (räumliche) Schwelle zu überqueren. Ausserdem bestimme ich den Effekt des Alters auf Ströme von (sub)diffundierenden Partikeln, die sich auf eine absorbierende Barriere zubewegen. Zuletzt beschäftige ich mich mit der Diffusion einer eindimensionalen Anordnung von Teilchen in einer QEL, in der diese als harte Kugeln miteinander wechselwirken. Dabei vergleiche ich die gemeinsame Bewegung in einer QEL und als individuelle CTRWs miteinander über die Standartabweichung von der Startposition, für die ich das Mittel über mehrere QELs untersuche. Meine Arbeit setzt sich zusammen aus theoretischen Überlegungen und Berechnungen sowie der Simulation der Zufallsprozesse. Die Ergebnisse der Simulation und, soweit vorhanden, experimentelle Daten werden mit der Theorie verglichen. N2 - In the first part of my work I have investigated the ageing properties of the first passage time distributions in a one-dimensional subdiffusive continuous time random walk with power law distributed waiting times of the form $\psi(\tau) \sim \tau^{-1-\alpha}$ with $0<\alpha<1$ and $1<\alpha<2$. The age or ageing time $t_a$ is the time span from the start of the stochastic process to the start of the observation of this process (at $t=0$). I have calculated the results for a single target and two targets, also including the biased case, where the walker is driven towards the boundary by a constant force. I have furthermore refined the previously derived results for the non-ageing case and investigated the changes that occur when the walk is performed in a discrete quenched energy landscape, where the waiting times are fixed for every site. The results include the exact Laplace space densities and infinite (converging) series as exact results in the time space. The main results are the dominating long time power law behavior regimes, which depend on the ageing time. For the case of unbiased subdiffusion ($\alpha < 1$) in the presence of one target, I find three different dominant terms for ranges of $t$ separated by $t_a$ and another crossover time $t^{\star}$, which depends on $t_a$ as well as on the anomalous exponent $\alpha$ and the anomalous diffusion coefficient $K_{\alpha}$. In all three regimes ($t \ll t_a$, $t_a \ll t \ll t^{\star}$, $t \gg t^{\star}$) one finds power law decay with exponents depending on $\alpha$. The middle regime only exists for $t_a \ll t^{\star}$. The dominant terms in the first two regimes (ageing regimes) come from the probability distribution of the forward waiting time, the time one has to wait for the stochastic process to make the first step during the observation. When the observation time is larger than the second crossover time $t^{\star}$, the first passage time density does not show ageing and the non-ageing first passage time dominates. The power law exponents in the respective regimes are $-\alpha$ for strong ageing, $-1-\alpha$ in the intermediate regime, and $-1-\alpha/2$ in the final non-ageing regime. A similar split into three regimes can be found for $1<\alpha<2$, only with a different second crossover time $t^*$. In this regime the diffusion is normal but also age-dependent. For the diffusion in quenched energy landscapes one cannot detect ageing. The first passage time density shows a quenched power law $^\sim t^{-(1+2\alpha)/(1+\alpha)}$. For diffusion between two target sites and the biased diffusion towards a target only two scaling regimes emerge, separated by the ageing time. In the ageing case $t \ll t_a$ the forward waiting time is again dominant with power law exponent $-\alpha$, while the non-ageing power law $-1-\alpha$ is found for all times $t \gg t_a$. An intermediate regime does not exist. The bias and the confinement have similar effects on the first passage time density. For quenched diffusion, the biased case is interesting, as the bias reduces correlations due to revisiting of the same waiting time. As a result, CTRW like behavior is observed, including ageing. Extensive computer simulations support my findings. The second part of my research was done on the subject of ageing Scher-Montroll transport, which is in parts closely related to the first passage densities. It explains the electrical current in an amorphous material. I have investigated the effect of the width of a given initial distribution of charge carriers on the transport coefficients as well as the ageing effect on the emerging power law regimes and a constant initial regime. While a spread out initial distribution has only little impact on the Scher-Montroll current, ageing alters the behavior drastically. Instead of the two classical power laws one finds four current regimes, up to three of which can appear in a single experiment. The dominant power laws differ for $t \ll t_a, t_c$, $t_a \ll t \ll t_c$, $t_c \ll t \ll t_a$, and $t \gg t_a,t_c$. Here, $t_c$ is the crossover time of the non-aged Scher-Montroll current. For strongly aged systems one can observe a constant current in the first regime while the others are dominated by decaying power laws with exponents $\alpha -1$, $-\alpha$, and $-1-\alpha$. The ageing regimes are the 1st and 3rd one, while the classical regimes are the 2nd and the 4th. I have verified the theory using numerical integration of the exact integrals and applied the new results to experimental data. In the third part I considered a single file of subdiffusing particles in an energy landscape. Every occupied site of the landscape acts as a boundary, from which a particle is immediately reflected to its previous site, if it tries to jump there. I have analysed the effects single-file diffusion a quenched landscape compared to an annealed landscape and I have related these results to the number of steps and related quantities. The diffusion changes from ultraslow logarithmic diffusion in the annealed or CTRW case to subdiffusion with an anomalous exponent $\alpha/(1+\alpha)$ in the quenched landscape. The behavior is caused by the forward waiting time, which changes drastically from the quenched to the annealed case. Single-file effects in the quenched landscape are even more complicated to consider in the ensemble average, since the diffusion in individual landscapes shows extremely diverse behavior. Extensive simulations support my theoretical arguments, which consider mainly the long time evolution of the mean square displacement of a bulk particle. KW - continuous time random walk KW - quenched energy landscape KW - first passage time KW - Scher-Montroll transport KW - single-file motion KW - Zufallsbewegung KW - Diffusion KW - eingefrorene Energielandschaft KW - Scher-Montroll Transport KW - Wartezeitverteilung Y1 - 2016 ER - TY - JOUR A1 - Godec, Aljaž A1 - Metzler, Ralf T1 - First passage time statistics for two-channel diffusion JF - Journal of physics : A, Mathematical and theoretical N2 - We present rigorous results for the mean first passage time and first passage time statistics for two-channel Markov additive diffusion in a 3-dimensional spherical domain. Inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes, which is shown to effect a non-trivial first passage behaviour. We also address the scenario of intermittent immobilisation. In both cases we prove that despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions the first passage statistics at long times do not display Poisson-like behaviour if none of the phases has a vanishing diffusion coefficient. This stands in stark contrast to the standard (one-channel) Markov diffusion counterpart. We also discuss the relevance of our results in the context of cellular signalling. KW - first passage time KW - Markov additive processes KW - Fokker-Planck equation KW - random search processes KW - coupled initial boundary value problem KW - cellular signalling KW - asymptotic analysis Y1 - 2017 U6 - https://doi.org/10.1088/1751-8121/aa5204 SN - 1751-8113 SN - 1751-8121 VL - 50 IS - 8 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Grebenkov, Denis S. 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 - 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 - 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 - 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 - GEN A1 - Grebenkov, Denis S. 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 -