Complex Bio Materials -Prof. Ralf Metzler- Finland Distinguished Professor Theoretical Biological Physics & Nonequilibrium Processes
Research highlights
In vivo anomalous diffusion and weak ergodicity breaking of lipid granules by Jae-Hyung Jeon, Vincent Tejedor, Stas Burov, Eli Barkai, Christine Selhuber-Unkel, Kirstine Berg-Sørensen, Lene Oddershede, and Ralf Metzler Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion. [Phys. Rev. Lett. 106, 048103 (2011)]	Time averaged mean squared displacement of lipid granules, showing distinct features of ergodicity breaking.
Partial twist release causes the formation of one denaturation bubble in each DNA plasmid (arrows).	Supercoiling induces denaturation bubbles in circular DNA by Jae-Hyung Jeon, Jozef Adamczik, Giovanni Dietler, and Ralf Metzler We present a theoretical framework for the thermodynamic properties of supercoiling-induced denaturation bubbles in circular double-stranded DNA molecules. We explore how DNA supercoiling, ambient salt concentration, and sequence heterogeneity impact on the bubble occurrence. An analytical derivation of the probability distribution to find multiple bubbles is derived and the relevance for supercoiled DNA discussed. We show that in vivo sustained DNA bubbles are likely to occur due to partial twist release in regions rich in weaker AT base pairs. Single DNA plasmid imaging experiments clearly demonstrate the existence of bubbles in free solution. [Phys. Rev. Lett. 105, 208101 (2010)]
Ageing and non-ergodicity beyond the Khinchin theorem by Stas Burov, Ralf Metzler, and Eli Barkai The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and { suggest a possible generalization} of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics. [Proc. Natl. Acad. Sci. USA 107, 13228 (2010)]	Linear growth turning over to a shallower power-law increase of an ageing, subdiffusive random walk under confinement.
The scaling of the translocation time with the chain length exhibits different exponents for slow and fast driving of the translocation process.	Driven polymer translocation through nanopores: slow versus fast dynamics by Kaifu Luo, Tapio Ala-Nissilä, See-Chen Ying, and Ralf Metzler 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 versus the length of the polymer. For slow translocation (low driving force and/or high friction) we find the scaling exponent 1.588, while for fast translocation the pronouncedly non-equilibrium situation is reflected by a decrease to 1.37. Our results clarify a controversy on the magnitude of the dynamic scaling exponent for driven translocation. [Europhys. Lett. 88, 68006 (2009)]
Facilitated diffusion with DNA coiling by Michael Lomholt, Bram van den Broek, Svenja-Marei Kalisch, Gijs Wuite, and Ralf Metzler When DNA-binding proteins search for their specific binding site on a DNA molecule they alternate between linear one-dimensional diffusion along the DNA molecule, mediated by non-specific binding, and three-dimensional volume excursion events between successive dissociation from and rebinding to DNA. 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. [Proc. Natl. Acad. Sci. USA 106, 8204 (2009)]	Compared to a straight DNA configuration (a), coiling-induced higher local concentration of DNA (b) increases the search rate in facilitated diffusion processes.
At physiological salt conditions the search rate of restriction enzymes for their specific target on a DNA chain is maximal close to the fully relaxed, coiled DNA configuration.	How DNA coiling enhances target localization by proteins by Bram van den Broek, Michael Lomholt, Svenja-Marei Kalisch, Ralf Metzler, and Gijs Wuite Here, we show direct single DNA molecule 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. [Proc. Natl. Acad. Sci. USA 105, 15738 (2008)]
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