Complex Bio Materials -Prof. Ralf Metzler- Theoretical Biological Physics & Nonequilibrium Processes
Current Research
Random Time-Scale Invariant Diffusion and Transport Coefficients by Yong He, Stas Burov, Ralf Metzler, and Eli Barkai Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable . Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed. [Phys. Rev. Lett. 101, 058101 (2008)] See the accompanying Viewpoint.	In a subdiffusive continuous time random walk process the time averaged mean squared displacement is a random variable depending on individual trajectories.

Lévy-intermittent strategies with α<2 reduce oversampling on all scales in contrast to exponential strategies.	Lévy strategies in intermittent search processes are advantageous by Michael A. Lomholt, Tal Koren, Ralf Metzler, and Joseph Klafter 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. [Proc. Natl. Acad. Sci. USA 105, 11055 (2008)]
Denaturation transition of stretched DNA by Andreas Hanke, Martha G. Ochoa, and Ralf Metzler 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. [Phys. Rev. Lett. 100, 018106 (2008)]	Force-extension curves at fixed temperature
The random walker starts at x=0 and after a number of jumps crosses the point x=d, overshooting it by a distance l.	Leapover lengths and first passage time statistics for Lévy flights by Tal Koren, Michael A. Lomholt, Aleksei Chechkin, Joseph Klafter, and Ralf Metzler Exact results for the first passage time and leapover statistics of symmetric and one-sided Lévy flights (LFs) are derived. LFs with stable index α are shown to have leapover lengths, that are asymptotically power-law distributed with index α for one-sided LFs and, surprisingly, with index α/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. [Phys. Rev. Lett. 99, 160602 (2007)]
Subdiffusion and Weak Ergodicity Breaking in the Presence of a Reactive Boundary by Michael A. Lomholt, Irwin M. Zaid, and Ralf Metzler The boundary condition for a subdiffusive particle interacting with a reactive boundary with a finite reaction rate is derived. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with an 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. It is discussed why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolar. [Phys. Rev. Lett. 98, 200603 (2007)]	Distribution of the time average of the probability to find the particle bound
Mean turnover time as function of external driving frequency	Manipulating Single Enzymes by an External Harmonic Force by Michael A. Lomholt, Michael Urbakh, Ralf Metzler, and Joseph Klafter 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. [Phys. Rev. Lett. 98 168302 (2007)]
DNA Bubble Dynamics as a Quantum Coulomb Problem by Hans C. Fogedby and Ralf Metzler We study the dynamics of denaturation bubbles in double-stranded DNA. Demonstrating that the associated Fokker-Planck equation is equivalent to a Coulomb problem, we derive expressions for the bubble survival distribution W(t). Below T_m, W(t) is associated with the continuum of scattering states of the repulsive Coulomb potential. At T_m, the Coulomb potential vanishes and W(t) assumes a power-law tail with nontrivial dynamic exponents: the critical exponent of the entropy loss factor may cause a finite mean lifetime. Above T_m (attractive potential), the long-time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed. [Phys. Rev. Lett. 98 070601 (2007)]
	Fractal Dimension and Localization of DNA Knots by Erika Ercolini, Francesco Valle, Jozef Adamcik, Guillaume Witz, Ralf Metzler, Paolo De Los Rios, Joaquim Roca, and Giovanni Dietler The scaling properties of DNA knots of different complexities were studied by atomic force microscope. Following two different protocols DNA knots are adsorbed onto a mica surface in regimes of (i) strong binding, that induces a kinetic trapping of the three-dimensional (3D) configuration, and of (ii) weak binding, that permits (partial) relaxation on the surface. In (i) the radius of gyration of the adsorbed DNA knot scales with the 3D Flory exponent ν≈0.60 within error. In (ii), we find ν≈0.66, a value between the 3D and 2D (ν=3/4) exponents. Evidence is also presented for the localization of knot crossings in 2D under weak adsorption conditions [Phys. Rev. Lett. 98 058102 (2007)] Left: Weakly adsorbed knots on treated mica, AFM images
Sequence Sensitivity of Breathing Dynamics in Heteropolymer DNA by Tobias Ambjörnsson, Suman K. Banik, Oleg Krichevsky, and Ralf Metzler We study the fluctuation dynamics of localized denaturation bubbles in heteropolymer DNA with a master equation and complementary stochastic simulation based on novel DNA stability data. A significant dependence of opening probability and waiting time between bubble events on the local DNA sequence is revealed and quantified for a biological sequence of the T7 bacteriophage. Quantitative agreement with data from fluorescence correlation spectroscopy is demonstrated. [Phys. Rev. Lett. 97 128105 (2006)]	Fluorescence autocorrelation with theoretical description
	Optimal Target Search on a Fast-Folding Polymer Chain with Volume Exchange by Michael A. Lomholt, Tobias Ambjörnsson, and Ralf Metzler 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. [Phys. Rev. Lett. 95 260603 (2005)] Left: Search modes in in vitro gene regulation