arXiv subjects

Ralf Metzler

Publications and source records attributed to Ralf Metzler.

At least 181 records · Page 10Linked to original sources

arXiv2010

Quantitative analysis of single particle trajectories: mean maximal excursion method

An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions, i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean squared displacement we show that the mean maximal excursion analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the mean maximal excursion method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics, such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.

cond-mat.stat-mech↗

arXiv2010

Anomalous diffusion in correlated continuous time random walks

We demonstrate that continuous time random walks in which successive waiting times are correlated by Gaussian statistics lead to anomalous diffusion with mean squared displacement ~t^{2/3}. Long-ranged correlations of the waiting times with power-law exponent alpha (0 ~t^{alpha/(1+alpha)}. In contrast correlations in the jump lengths are shown to produce superdiffusion. We show that in both cases weak ergodicity breaking occurs. Our results are in excellent agreement with simulations.

cond-mat.stat-mech↗

arXiv2010

Fractional Brownian motion and generalized Langevin equation motion in confined geometries

Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time averaged mean squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume, and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data.

cond-mat.stat-mech↗

arXiv2009

Driven polymer translocation through nanopores: slow versus fast dynamics

We investigate the dynamics of polymer translocation through nanopores under external driving by 3D Langevin Dynamics simulations, focusing on the scaling of the average translocation time $τ$ versus the length of the polymer, $τ\sim N^α$. For slow translocation, i.e., under low driving force and/or high friction, we find $α\approx 1+ν\approx 1.588$ where $ν$ denotes the Flory exponent. In contrast, $α\approx 1.37$ is observed for fast translocation due to the highly deformed chain conformation on the trans side, reflecting a pronounced non-equilibrium situation. The dependence of the translocation time on the driving force is given by $τ\sim F^{-1}$ and $τ\sim F^{-0.80}$ for slow and fast translocation, respectively. These results clarify the controversy on the magnitude of the scaling exponent $α$ for driven translocation.

cond-mat.soft↗

arXiv2009

Fractional calculus approach to the statistical characterization of random variables and vectors

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.

cond-mat.stat-mech↗

arXiv2009

Polymer translocation out of confined environments

We consider the dynamics of polymer translocation out of confined environments. Analytic scaling arguments lead to the prediction that the translocation time scales like $τ\sim N^{β+ν_{2D}}R^{1+(1-ν_{2D})/ν}$ for translocation out of a planar confinement between two walls with separation $R$ into a 3D environment, and $τ\sim N^{β+1}R$ for translocation out of two strips with separation $R$ into a 2D environment. Here, $N$ is the chain length, $ν$ and $ν_{2D}$ are the Flory exponents in 3D and 2D, and $β$ is the scaling exponent of translocation velocity with $N$, whose value for the present choice of parameters is $β\approx 0.8$ based on Langevin dynamics simulations. These scaling exponents improve on earlier predictions.

cond-mat.soft↗

arXiv2009

Finding the optimum activation energy in DNA breathing dynamics: A Simulated Annealing approach

We demonstrate how the stochastic global optimization scheme of Simulated Annealing can be used to evaluate optimum parameters in the problem of DNA breathing dynamics. The breathing dynamics is followed in accordance with the stochastic Gillespie scheme with the denaturation zones in double stranded DNA studied as a single molecule time series. Simulated Annealing is used to find the optimum value of the activation energy for which the equilibrium bubble size distribution matches with a given value. It is demonstrated that the method overcomes even large noise in the input surrogate data.

cond-mat.stat-mech↗

arXiv2009

Bubble merging in breathing DNA as a vicious walker problem in opposite potentials

We investigate the coalescence of two DNA-bubbles initially located at weak domains and separated by a more stable barrier region in a designed construct of double-stranded DNA. In a continuum Fokker-Planck approach, the characteristic time for bubble coalescence and the corresponding distribution are derived, as well as the distribution of coalescence positions along the barrier. Below the melting temperature, we find a Kramers-type barrier crossing behavior, while at high temperatures, the bubble corners perform drift-diffusion towards coalescence. In the calculations, we map the bubble dynamics on the problem of two vicious walkers in opposite potentials. We also present a discrete master equation approach to the bubble coalescence problem. Numerical evaluation and stochastic simulation of the master equation show excellent agreement with the results from the continuum approach. Given that the coalesced state is thermodynamically stabilized against a state where only one or a few base pairs of the barrier region are re-established, it appears likely that this type of setup could be useful for the quantitative investigation of thermodynamic DNA stability data as well as the rate constants involved in the unzipping and zipping dynamics of DNA, in single molecule fluorescence experiments.

cond-mat.stat-mech↗

arXiv2008

Bulk-mediated surface diffusion on a cylinder: propagators and crossovers

We consider the effective surface motion of a particle that freely diffuses in the bulk and intermittently binds to that surface. From an exact approach we derive various regimes of the effective surface motion characterized by physical rates for binding/unbinding and the bulk diffusivity. We obtain a transient regime of superdiffusion and, in particular, a saturation regime characteristic for the cylindrical geometry. This saturation, however, in a finite system is not terminal but eventually turns over to normal surface diffusion. The first passage behavior of particles to the cylinder surface is derived. Consequences for actual systems are discussed.

cond-mat.stat-mech↗

arXiv2008

Quantitative approach to facilitated diffusion with intersegmental jumping

We study the impact of DNA coiling on the search rate of proteins moving along the DNA contour interspersed by three-dimensional (3D) bulk excursions. When the DNA is coiled proteins performing short 3D hops along a DNA segment can be captured by foreign DNA segments that have looped back close to the original segment. These intersegmental jumps are shown to enhance the search rate for a specific site on the DNA by lowering the tendency to resample previously visited sites on the DNA. The model developed here offers a quantitative description of recent single molecule experiments on facilitated diffusion of restriction enzymes EcoRV.

cond-mat.stat-mech↗

arXiv2008

Denaturation transition of stretched DNA

We generalize the Poland-Scheraga model to consider DNA denaturation in the presence of an external stretching force. We demonstrate the existence of a force-induced DNA denaturation transition and obtain the temperature-force phase diagram. The transition is determined by the loop exponent $c$ for which we find the new value $c=4ν-1/2$ such that the transition is second order with $c=1.85<2$ in $d=3$. We show that a finite stretching force $F$ destabilizes DNA, corresponding to a lower melting temperature $T(F)$, in agreement with single-molecule DNA stretching experiments.

cond-mat.soft↗

arXiv2007

The advantage of Lévy strategies in intermittent search processes

Search strategies based on random walk processes with long-tailed jump length distributions (Levy walks) on the one hand and intermittent behavior switching between local search and ballistic relocation phases on the other, have been previously shown to be beneficial in stochastic target finding problems. We here study a combination of both mechanisms: an intermittent process with Levy distributed relocations. We demonstrate how Levy distributed relocations reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets.

cond-mat.stat-mech↗

arXiv2007

Dynamics of DNA-breathing: Weak noise analysis, finite time singularity, and mapping onto the quantum Coulomb problem

We study the dynamics of denaturation bubbles in double-stranded DNA on the basis of the Poland-Scheraga model. We show that long time distributions for the survival of DNA bubbles and the size autocorrelation function can be derived from an asymptotic weak noise approach. In particular, below the melting temperature the bubble closure corresponds to a noisy finite time singularity. We demonstrate that the associated Fokker-Planck equation is equivalent to a quantum Coulomb problem. Below the melting temperature the bubble lifetime is associated with the continuum of scattering states of the repulsive Coulomb potential; at the melting temperature the Coulomb potential vanishes and the underlying first exit dynamics exhibits a long time power law tail; above the melting temperature, corresponding to an attractive Coulomb potential, the long time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed.

cond-mat.stat-mech↗

arXiv2007

Lévy Statistics and Anomalous Transport: Lévy flights and Subdiffusion

Levy flights and subdiffusive processes and their properties are discussed. We derive the space- and time-fractional transport equations, and consider their solutions in external potentials. An extensive list of references is included.

cond-mat.stat-mech↗

arXiv2007

Leapover lengths and first passage time statistics for Lévy flights

Exact results for the first passage time and leapover statistics of symmetric and one-sided Levy flights (LFs) are derived. LFs with stable index alpha are shown to have leapover lengths, that are asymptotically power-law distributed with index alpha for one-sided LFs and, surprisingly, with index alpha/2 for symmetric LFs. The first passage time distribution scales like a power-law with index 1/2 as required by the Sparre Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.

cond-mat.stat-mech↗

arXiv2007

Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary

We derive the boundary condition for a subdiffusive particle interacting with a reactive boundary with finite reaction rate. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with identical exponent for both the unbinding times from the boundary to the bulk and the rebinding times from the bulk. This causes a weak ergodicity breaking: typically, an individual particle either stays bound or remains in the bulk for very long times. We discuss why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolar

cond-mat.stat-mech↗

arXiv2007

Manipulating single enzymes by an external harmonic force

We study a Michaelis-Menten reaction for a single two-state enzyme molecule, whose transition rates between the two conformations are modulated by an harmonically oscillating external force. In particular, we obtain a range of optimal driving frequencies for changing the conformation of the enzyme thereby controlling the enzymatic activity (i.e. product formation). This analysis demonstrates that it is, in principle, possible to obtain information about particular rates within the kinetic scheme.

q-bio.BM↗

arXiv2006

Barrier crossing driven by Levy noise: Universality and the Role of Noise Intensity

We study the barrier crossing of a particle driven by white symmetric Levy noise of index $α$ and intensity $DD for three different generic types of potentials: (a) a bistable potential; (b) a metastable potential; and (c) a truncated harmonic potential. For the low noise intensity regime we recover the previously proposed algebraic dependence on $D$ of the characteristic escape time, $T_{\mathrm{esc}}\simeq C(α)/D^{μ(α)}$, where $C(α)$ is a coefficient. It is shown that the exponent $μ(α)$ remains approximately constant, $μ\approx 1$ for $0<α<2$; at $α=2$ the power-law form of $T_{\mathrm{esc}}$ changes into the known exponential dependence on 1/D; it exhibits a divergence-like behavior as $α$ approaches 2. In this regime we observe a monotonous increase of the escape time $T_{\mathrm{esc}}$ with increasing $α$ (keeping the noise intensity $D$ constant). The probability density of the escape time decays exponentially. In addition, for low noise intensities the escape times correspond to barrier crossing by multiple Levy steps. For high noise intensities, the escape time curves collapse for all values of $α$. At intermediate noise intensities, the escape time exhibits non-monotonic dependence on the index $α$, while still retaining the exponential form of the escape time density.

cond-mat.stat-mech↗