Intermittent search processes switch between local Brownian search events and ballistic relocation phases. We demonstrate analytically and numerically in one dimension that when relocation times are Lévy distributed, resulting in a Lévy walk dynamics, the search process significantly outperforms the previously investigated case of exponentially distributed relocation times: The resulting Lévy walks reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets. We also show that a searching agent that uses the Lévy strategy is much less sensitive to the target density, which would require considerably less adaptation by the searcher.

We study the search process of a target on a rapidly folding polymer (“DNA”) by an ensemble of particles (“proteins”), whose search combines 1D diffusion along the chain, Lévy type diffusion mediated by chain looping, and volume exchange. A rich behavior of the search process is obtained with respect to the physical parameters, in particular, for the optimal search.

Many genetic processes depend on proteins interacting with specific sequences on DNA. Despite the large excess of nonspecific DNA in the cell, proteins can locate their targets rapidly. After initial nonspecific binding, they are believed to find the target site by 1D diffusion (“sliding”) interspersed by 3D dissociation/reassociation, a process usually referred to as facilitated diffusion. The 3D events combine short intrasegmental “hops” along the DNA contour, intersegmental “jumps” between nearby DNA segments, and longer volume “excursions.” The impact of DNA conformation on the search pathway is, however, still unknown. Here, we show direct evidence that DNA coiling influences the specific association rate of EcoRV restriction enzymes. Using optical tweezers together with a fast buffer exchange system, we obtained association times of EcoRV on single DNA molecules as a function of DNA extension, separating intersegmental jumping from other search pathways. Depending on salt concentration, targeting rates almost double when the DNA conformation is changed from fully extended to a coiled configuration. Quantitative analysis by an extended facilitated diffusion model reveals that only a fraction of enzymes are ready to bind to DNA. Generalizing our results to the crowded environment of the cell we predict a major impact of intersegmental jumps on target localization speed on DNA.

When DNA-binding proteins search for their specific binding site on a DNA molecule they alternate between linear 1-dimensional diffusion along the DNA molecule, mediated by nonspecific binding, and 3-dimensional volume excursion events between successive dissociation from and rebinding to DNA. If the DNA molecule is kept in a straight configuration, for instance, by optical tweezers, these 3-dimensional excursions may be divided into long volume excursions and short hops along the DNA. These short hops correspond to immediate rebindings after dissociation such that a rebinding event to the DNA occurs at a site that is close to the site of the preceding dissociation. When the DNA molecule is allowed to coil up, immediate rebinding may also lead to so-called intersegmental jumps, i.e., immediate rebindings to a DNA segment that is far away from the unbinding site when measured in the chemical distance along the DNA, but close by in the embedding 3-dimensional space. This effect is made possible by DNA looping. The significance of intersegmental jumps was recently demonstrated in a single DNA optical tweezers setup. Here we present a theoretical approach in which we explicitly take the effect of DNA coiling into account. By including the spatial correlations of the short hops we demonstrate how the facilitated diffusion model can be extended to account for intersegmental jumping at varying DNA densities. It is also shown that our approach provides a quantitative interpretation of the experimentally measured enhancement of the target location by DNA-binding proteins.

We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with “diffusing diffusivity”—giving rise to widely-observed non-Gaussian displacement statistics—and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels.

In this study we derive a single-particle equation of motion, from first principles, starting out with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential. Using a harmonization technique, we show that the resulting dynamical equation belongs to the class of fractional Langevin equations, a stochastic framework which has been proposed in a large body of works as a means of describing anomalous dynamics. Our work sheds light on the fundamental assumptions of these phenomenological models and a relation derived by Kollmann.