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We study the adsorption–desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition—demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces—are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye–Hückel approximation is often not feasible and the nonlinear Poisson–Boltzmann approach should be implemented. At low salt conditions, for instance, the electrostatic potential from the nonlinear Poisson–Boltzmann equation is smaller than the Debye–Hückel result, such that the required critical surface charge density for polyelectrolyte adsorption σc increases. The nonlinear relation between the surface charge density and electrostatic potential leads to a sharply increasing critical surface charge density with growing ionic strength, imposing an additional limit to the critical salt concentration above which no polyelectrolyte adsorption occurs at all. We contrast our simulations findings with the known scaling results for weak critical polyelectrolyte adsorption onto oppositely charged surfaces for the three standard geometries. Finally, we discuss some applications of our results for some physical–chemical and biophysical systems.

Under dilute in vitro conditions transcription factors rapidly locate their target sequence on DNA by using the facilitated diffusion mechanism. However, whether this strategy of alternating between three-dimensional bulk diffusion and one-dimensional sliding along the DNA contour is still beneficial in the crowded interior of cells is highly disputed. Here we use a simple model for the bacterial genome inside the cell and present a semi-analytical model for the in vivo target search of transcription factors within the facilitated diffusion framework. Without having to resort to extensive simulations we determine the mean search time of a lac repressor in a living E. coli cell by including parameters deduced from experimental measurements. The results agree very well with experimental findings, and thus the facilitated diffusion picture emerges as a quantitative approach to gene regulation in living bacteria cells. Furthermore we see that the search time is not very sensitive to the parameters characterizing the DNA configuration and that the cell seems to operate very close to optimal conditions for target localization. Local searches as implied by the colocalization mechanism are only found to mildly accelerate the mean search time within our model.

We study the ergodic properties of superdiffusive, spatiotemporally coupled Levy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement (delta x(2)) over bar around the ensemble value 3 - alpha (1 < alpha < 2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the average of (delta x(2)) over bar over an ensemble of trajectories as a function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.

We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a) > 0 after initiation of the process at t = 0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age ta of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed.

We comprehensively analyze the emergence of anomalous statistics in the context of the random relaxation ( RARE) model [Eliazar and Metzler, J. Chem. Phys. 137, 234106 ( 2012)], a recently introduced versatile model of random relaxations in random environments. The RARE model considers excitations scattered randomly across a metric space around a reaction center. The excitations react randomly with the center, the reaction rates depending on the excitations' distances from this center. Relaxation occurs upon the first reaction between an excitation and the center. Addressing both the relaxation time and the relaxation range, we explore when these random variables display anomalous statistics, namely, heavy tails at zero and at infinity that manifest, respectively, exceptionally high occurrence probabilities of very small and very large outliers. A cohesive set of closed-form analytic results is established, determining precisely when such anomalous statistics emerge.

Based on the space-fractional Fokker-Planck equation with a delta-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index alpha = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for alpha may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.

Low-dimensional, many-body systems are often characterized by ultraslow dynamics. We study a labelled particle in a generic system of identical particles with hard-core interactions in a strongly disordered environment. The disorder is manifested through intermittent motion with scale-free sticking times at the single particle level. While for a non-interacting particle we find anomalous diffusion of the power-law form < x(2)(t)> similar or equal to t(alpha) of the mean squared displacement with 0 < alpha < 1, we demonstrate here that the combination of the disordered environment with the many-body interactions leads to an ultraslow, logarithmic dynamics < x(2)(t)> similar or equal to log(1/2)t with a universal 1/2 exponent. Even when a characteristic sticking time exists but the fluctuations of sticking times diverge we observe the mean squared displacement < x(2)(t)> similar or equal to t(gamma) with 0 < gamma < 1/2, that is slower than the famed Harris law < x(2)(t)> similar or equal to t(1/2) without disorder. We rationalize the results in terms of a subordination to a counting process, in which each transition is dominated by the forward waiting time of an ageing continuous time process.

We study the parameter sensitivity of hetero-polymeric DNA within the purview of DNA breathing dynamics. The degree of correlation between the mean bubble size and the model parameters is estimated for this purpose for three different DNA sequences. The analysis leads us to a better understanding of the sequence dependent nature of the breathing dynamics of hetero-polymeric DNA. Out of the 14 model parameters for DNA stability in the statistical Poland-Scheraga approach, the hydrogen bond interaction epsilon(hb)(AT) for an AT base pair and the ring factor. turn out to be the most sensitive parameters. In addition, the stacking interaction epsilon(st)(TA-TA) for an TA-TA nearest neighbor pair of base-pairs is found to be the most sensitive one among all stacking interactions. Moreover, we also establish that the nature of stacking interaction has a deciding effect on the DNA breathing dynamics, not the number of times a particular stacking interaction appears in a sequence. We show that the sensitivity analysis can be used as an effective measure to guide a stochastic optimization technique to find the kinetic rate constants related to the dynamics as opposed to the case where the rate constants are measured using the conventional unbiased way of optimization.

Aging, the dependence of the dynamics of a physical process on the time t(a) since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first-passage dynamics. We find that for an aging continuous time random walk process, the scaling exponent of the density of first-passage times changes twice as the aging progresses and reveals an intermediate scaling regime. The first-passage dynamics depends on t(a) differently for intermediate and strong aging. Similar crossovers are obtained for the first-passage dynamics for a confined and driven particle. Comparison to the motion of an aged particle in the quenched trap model with a bias shows excellent agreement with our analytical findings. Our results demonstrate how first-passage measurements can be used to unravel the age t(a) of a physical system.

The rapid worldwide spread of severe viral infections, often involving novel mutations of viruses, poses major challenges to our health-care systems. This means that tools that can efficiently and specifically diagnose viruses are much needed. To be relevant for broad applications in local health-care centers, such tools should be relatively cheap and easy to use. In this paper, we discuss the biophysical potential for the macroscopic detection of viruses based on the induction of a mechanical stress in a bundle of prestretched DNA molecules upon binding of viruses to the DNA. We show that the affinity of the DNA to the charged virus surface induces a local melting of the double helix into two single-stranded DNA. This process effects a mechanical stress along the DNA chains leading to an overall contraction of the DNA. Our results suggest that when such DNA bundles are incorporated in a supporting matrix such as a responsive hydrogel, the presence of viruses may indeed lead to a significant, macroscopic mechanical deformation of the matrix. We discuss the biophysical basis for this effect and characterize the physical properties of the associated DNA melting transition. In particular, we reveal several scaling relations between the relevant physical parameters of the system. We promote this DNA-based assay as a possible tool for efficient and specific virus screening.

We study the parameter sensitivity of hetero-polymeric DNA within the purview of DNA breathing dynamics. The degree of correlation between the mean bubble size and the model parameters is estimated for this purpose for three different DNA sequences. The analysis leads us to a better understanding of the sequence dependent nature of the breathing dynamics of hetero-polymeric DNA. Out of the 14 model parameters for DNA stability in the statistical Poland-Scheraga approach, the hydrogen bond interaction epsilon(hb)(AT) for an AT base pair and the ring factor. turn out to be the most sensitive parameters. In addition, the stacking interaction epsilon(st)(TA-TA) for an TA-TA nearest neighbor pair of base-pairs is found to be the most sensitive one among all stacking interactions. Moreover, we also establish that the nature of stacking interaction has a deciding effect on the DNA breathing dynamics, not the number of times a particular stacking interaction appears in a sequence. We show that the sensitivity analysis can be used as an effective measure to guide a stochastic optimization technique to find the kinetic rate constants related to the dynamics as opposed to the case where the rate constants are measured using the conventional unbiased way of optimization. (c) 2014 AIP Publishing LLC.

We demonstrate the non-ergodicity of a simple Markovian stochastic process with space-dependent diffusion coefficient D(x). For power-law forms D(x) similar or equal to vertical bar x vertical bar(alpha), this process yields anomalous diffusion of the form < x(2)(t)> similar or equal to t(2/(2-alpha)). Interestingly, in both the sub- and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time-averaged mean-squared displacement <(delta(2)(Delta))over bar> remains linear in the lag time Delta and thus differs from the corresponding ensemble average < x(2)(t)>. We analyse the non-ergodic behaviour of this process in terms of the time-averaged mean- squared displacement (delta(2)) over bar and its random features, i.e. the statistical distribution of (delta(2)) over bar and the ergodicity breaking parameters. The heterogeneous diffusion model represents an alternative approach to non- ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.

The role of ergodicity in anomalous stochastic processes - analysis of single-particle trajectories
(2012)

Single-particle experiments produce time series x(t) of individual particle trajectories, frequently revealing anomalous diffusion behaviour. Typically, individual x(t) are evaluated in terms of time-averaged quantities instead of ensemble averages. Here we discuss the behaviour of the time-averaged mean squared displacement of different stochastic processes giving rise to anomalous diffusion. In particular, we pay attention to the ergodic properties of these processes, i.e. the (non)equivalence of time and ensemble averages.

Single-particle tracking has become a standard tool for the investigation of diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual trajectories. Here we study confined normal as well as anomalous diffusion, modeled by fractional Brownian motion and the fractional Langevin equation, and show that even for such ergodic systems time-averaged quantities behave differently from their ensemble-averaged counterparts, irrespective of how long the measurement time becomes. Knowledge of the exact behavior of time averages is therefore fundamental for the proper physical interpretation of measured time series, in particular, for extraction of the relaxation time scale from data.

We discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long-time properties, this process exhibits aging: events counted initially in a time interval [0, t] statistically strongly differ from those observed at later times [t(a,) t(a) + t]. The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous-time random walk. By this, we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking, along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.

It is generally believed that random search processes based on scale-free, Levy stable jump length distributions (Levy flights) optimize the search for sparse targets. Here we show that this popular search advantage is less universal than commonly assumed. We study the efficiency of a minimalist search model based on Levy flights in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space) based on two different optimization criteria with respect to minimal search time and search reliability (cumulative arrival probability). Although Levy flights turn out to be efficient search processes when the target is far from the starting point, or when relative to the starting point the target is upstream, we show that for close targets and for downstream target positioning regular Brownian motion turns out to be the advantageous search strategy. Contrary to claims that Levy flights with a critical exponent alpha = 1 are optimal for the search of sparse targets in different settings, based on our optimization parameters the optimal a may range in the entire interval (1, 2) and especially include Brownian motion as the overall most efficient search strategy.

Recent experiments reveal both passive subdiffusion of various nanoparticles and anomalous active transport of such particles by molecular motors in the molecularly crowded environment of living biological cells. Passive and active microrheology reveals that the origin of this anomalous dynamics is due to the viscoelasticity of the intracellular fluid. How do molecular motors perform in such a highly viscous, dissipative environment? Can we explain the observed co-existence of the anomalous transport of relatively large particles of 100 to 500 nm in size by kinesin motors with the normal transport of smaller particles by the same molecular motors? What is the efficiency of molecular motors in the anomalous transport regime? Here we answer these seemingly conflicting questions and consistently explain experimental findings in a generalization of the well-known continuous diffusion model for molecular motors with two conformational states in which viscoelastic effects are included.

Generalized facilitated diffusion model for DNA-binding proteins with search and recognition states
(2012)

Transcription factors (TFs) such as the lac repressor find their target sequence on DNA at remarkably high rates. In the established Berg-von Hippel model for this search process, the TF alternates between three-dimensional diffusion in the bulk solution and one-dimensional sliding along the DNA chain. To overcome the so-called speed-stability paradox, in similar models the TF was considered as being present in two conformations (search state and recognition state) between which it switches stochastically. Combining both the facilitated diffusion model and alternating states, we obtain a generalized model. We explicitly treat bulk excursions for rodlike chains arranged in parallel and consider a simplified model for coiled DNA. Compared to previously considered facilitated diffusion models, corresponding to limiting cases of our generalized model, we surprisingly find a reduced target search rate. Moreover, at optimal conditions there is no longer an equipartition between the time spent by the protein on and off the DNA chain.

We study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated waiting times. In this model the current waiting time T-i is equal to the previous waiting time Ti-1 plus a small increment. Based on the associated coupled Langevin equations the force field is systematically introduced. We show that in a confining potential the relaxation dynamics follows power-law or stretched exponential pattern, depending on the model parameters. The process obeys a generalized Einstein-Stokes-Smoluchowski relation and observes the second Einstein relation. The stationary solution is of Boltzmann-Gibbs form. The case of an harmonic potential is discussed in some detail. We also show that the process exhibits aging and ergodicity breaking.

Combining the advection-diffusion equation approach with Monte Carlo simulations we study chaperone driven polymer translocation of a stiff polymer through a nanopore. We demonstrate that the probability density function of first passage times across the pore depends solely on the Peclet number, a dimensionless parameter comparing drift strength and diffusivity. Moreover it is shown that the characteristic exponent in the power-law dependence of the translocation time on the chain length, a function of the chaperone-polymer binding energy, the chaperone concentration, and the chain length, is also effectively determined by the Peclet number. We investigate the effect of the chaperone size on the translocation process. In particular, for large chaperone size, the translocation progress and the mean waiting time as function of the reaction coordinate exhibit pronounced sawtooth-shapes. The effects of a heterogeneous polymer sequence on the translocation dynamics is studied in terms of the translocation velocity, the probability distribution for the translocation progress, and the monomer waiting times. (C) 2011 American Institute of Physics.

We study the effects of ageing-the time delay between initiation of the physical process at t = 0 and start of observation at some time t(a) > 0-and spatial confinement on the properties of heterogeneous diffusion processes (HDPs) with deterministic power-law space-dependent diffusivities, D(x) = D-0 vertical bar x vertical bar(alpha). From analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the HDP we obtain striking similarities to ageing subdiffusive continuous time random walks with scale-free waiting time distributions. We also explore how both processes can be distinguished. For confined HDPs we study the long-time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement.

We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) < x(2)(t)> similar or equal to log(gamma)t. Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution psi(tau) similar or equal to 1/(tau log(1+gamma)tau) and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD.

Ageing single file motion
(2014)

During the life cycle of bacterial cells the non-mixing of the two ring-shaped daughter genomes is an important prerequisite for the cell division process. Mimicking the environments inside highly crowded biological cells, we study the dynamics and statistical behavior of two flexible ring polymers in the presence of cylindrical confinement and crowding molecules. From extensive computer simulations we determine the degree of ring-ring overlap and the number of inter-monomer contacts for varying volume fractions phi of crowders. We also examine the entropic demixing of polymer rings in the presence of mobile crowders and determine the characteristic times of the internal polymer dynamics. Effects of the ring length on ring-ring overlap are also analyzed. In particular, on systematic variation of the fraction of crowding molecules, a (1 - phi)-scaling is found for the ring-ring overlap length along the cylinder axis, and a non-monotonic dependence of the 3D ring-ring contact number with a maximum at phi approximate to 0.2 is obtained. Our results demonstrate that polymer rings are demixed and separated by particular entropy-favourable partitioning of crowders along the axis of the cylindrical simulation box. These findings help to rationalize the implications of macromolecular crowding for circular DNA molecules in confined spaces inside bacteria as well as in localized cellular compartments inside eukaryotic cells.

We consider the area coverage of radial Levy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Levy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Levy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Levy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Levy flights in a square area. Our findings are of interest for a general understanding of Levy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns.

We report the results of single tracer particle tracking by optical tweezers and video microscopy in micellar solutions. From careful analysis in terms of different stochastic models, we show that the polystyrene tracer beads of size 0.52-2.5 mu m after short-time normal diffusion turn over to perform anomalous diffusion of the form < r(2)(t)> similar or equal to t(alpha) with alpha approximate to 0.3. This free anomalous diffusion is ergodic and consistent with a description in terms of the generalized Langevin equation with a power-law memory kernel. With optical tweezers tracking, we unveil a power-law relaxation over several decades in time to the thermal plateau value under the confinement of the harmonic tweezer potential, as predicted previously (Phys. Rev. E 85 021147 (2012)). After the subdiffusive motion in the millisecond range, the motion becomes faster and turns either back to normal Brownian diffusion or to even faster superdiffusion, depending on the size of the tracer beads.

Lateral diffusion plays a crucial role in numerous processes that take place in cell membranes, yet it is quite poorly understood in native membranes characterized by, e.g., domain formation and large concentration of proteins. In this article, we use atomistic and coarse-grained simulations to consider how packing of membranes and crowding with proteins affect the lateral dynamics of lipids and membrane proteins. We find that both packing and protein crowding have a profound effect on lateral diffusion, slowing it down. Anomalous diffusion is observed to be an inherent property in both protein-free and protein-rich membranes, and the time scales of anomalous diffusion and the exponent associated with anomalous diffusion are found to strongly depend on packing and crowding. Crowding with proteins also has a striking effect on the decay rate of dynamical correlations associated with lateral single-particle motion, as the transition from anomalous to normal diffusion is found to take place at macroscopic time scales: while in protein-poor conditions normal diffusion is typically observed in hundreds of nanoseconds, in protein-rich conditions the onset of normal diffusion is tens of microseconds, and in the most crowded systems as large as milliseconds. The computational challenge which results from these time scales is not easy to deal with, not even in coarse-grained simulations. We also briefly discuss the physical limits of protein motion. Our results suggest that protein concentration is anything but constant in the plane of cell membranes. Instead, it is strongly dependent on proteins' preference for aggregation.

This paper introduces and analyses a general statistical model, termed the RAndom RElaxations (RARE) model, of random relaxation processes in disordered systems. The model considers excitations that are randomly scattered around a reaction center in a general embedding space. The model's input quantities are the spatial scattering statistics of the excitations around the reaction center, and the chemical reaction rates between the excitations and the reaction center as a function of their mutual distance. The framework of the RARE model is versatile and a detailed stochastic analysis of the random relaxation processes is established. Analytic results regarding the duration and the range of the random relaxation processes, as well as the model's thermodynamic limit, are obtained in closed form. In particular, the case of power-law inputs, which turn out to yield stretched exponential relaxation patterns and asymptotically Paretian relaxation ranges, is addressed in detail.

In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.

Generalized space-time fractional diffusion equation with composite fractional time derivative
(2012)

We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grunwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of the form delta(x). t-beta/Gamma(1-beta) (beta > 0).

Naturally occurring lipid granules diffuse in the cytoplasm and can be used as tracers to map out the viscoelastic landscape inside living cells. Using optical trapping and single particle tracking we found that lipid granules exhibit anomalous diffusion inside human umbilical vein endothelial cells. For these cells the exact diffusional pattern of a particular granule depends on the physiological state of the cell and on the localization of the granule within the cytoplasm. Granules located close to the actin rich periphery of the cell move less than those located towards to the center of the cell or within the nucleus. Also, granules in cells which are stressed by intense laser illumination or which have attached to a surface for a long period of time move in a more restricted fashion than those within healthy cells. For granules diffusing in healthy cells, in regions away from the cell periphery, occurrences of weak ergodicity breaking are observed, similar to the recent observations inside living fission yeast cells [1].

There exists compelling experimental evidence in numerous systems for logarithmically slow time evolution, yet its full theoretical understanding remains elusive. We here introduce and study a generic transition process in complex systems, based on nonrenewal, aging waiting times. Each state n of the system follows a local clock initiated at t = 0. The random time tau between clock ticks follows the waiting time density psi (tau). Transitions between states occur only at local clock ticks and are hence triggered by the local forward waiting time, rather than by psi (tau). For power-law forms psi (tau) similar or equal to tau(-1-alpha) (0 < alpha < 1) we obtain a logarithmic time evolution of the state number < n(t)> similar or equal to log(t/t(0)), while for alpha > 2 the process becomes normal in the sense that < n(t)> similar or equal to t. In the intermediate range 1 < alpha < 2 we find the power-law growth < n(t)> similar or equal to t(alpha-1). Our model provides a universal description for transition dynamics between aging and nonaging states.

Following recent discoveries of colocalization of downstream-regulating genes in living cells, the impact of the spatial distance between such genes on the kinetics of gene product formation is increasingly recognized. We here show from analytical and numerical analysis that the distance between a transcription factor (TF) gene and its target gene drastically affects the speed and reliability of transcriptional regulation in bacterial cells. For an explicit model system, we develop a general theory for the interactions between a TF and a transcription unit. The observed variations in regulation efficiency are linked to the magnitude of the variation of the TF concentration peaks as a function of the binding site distance from the signal source. Our results support the role of rapid binding site search for gene colocalization and emphasize the role of local concentration differences.

We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random-walk approach, we derive the diffusion equations for surface and bulk diffusion including the surface-bulk coupling. From these exact dynamic equations, we analytically obtain the propagator of the effective surface motion. This approach allows us to deduce a superdiffusive, Cauchy-type behavior on the surface, together with exact cutoffs limiting the Cauchy form. Moreover, we study the long-time dynamics for the surface motion.

Combining extensive molecular dynamics simulations of lipid bilayer systems of varying chemical compositions with single-trajectory analyses, we systematically elucidate the stochastic nature of the lipid motion. We observe subdiffusion over more than 4 orders of magnitude in time, clearly stretching into the submicrosecond domain. The lipid motion depends on the lipid chemistry, the lipid phase, and especially the presence of cholesterol. We demonstrate that fractional Langevin equation motion universally describes the lipid motion in all phases, including the gel phase, and in the presence of cholesterol. The results underline the relevance of anomalous diffusion in lipid bilayers and the strong effects of the membrane composition.

We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(omega) distribution of the random variable omega = tau(1)/(tau(1) + tau(2)), which is a measure for how similar the first passage times tau(1) and tau(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(omega) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(omega), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.

Macromolecular crowding in living biological cells effects subdiffusion of larger biomolecules such as proteins and enzymes. Mimicking this subdiffusion in terms of random walks on a critical percolation cluster, we here present a case study of EcoRV restriction enzymes involved in vital cellular defence. We show that due to its so far elusive propensity to an inactive state the enzyme avoids non-specific binding and remains well-distributed in the bulk cytoplasm of the cell. Despite the reduced volume exploration capability of subdiffusion processes, this mechanism guarantees a high efficiency of the enzyme. By variation of the non-specific binding constant and the bond occupation probability on the percolation network, we demonstrate that reduced nonspecific binding are beneficial for efficient subdiffusive enzyme activity even in relatively small bacteria cells. Our results corroborate a more local picture of cellular regulation.

In both eukaryotic and prokaryotic DNA sequences of 30-100 base-pairs rich in AT base-pairs have been identified at which the double helix preferentially unwinds. Such DNA unwinding elements are commonly associated with origins for DNA replication and transcription, and with chromosomal matrix attachment regions. Here we present a quantitative study of local DNA unwinding based on extensive single DNA plasmid imaging. We demonstrate that long-lived single-stranded denaturation bubbles exist in negatively supercoiled DNA, at the expense of partial twist release. Remarkably, we observe a linear relation between the degree of supercoiling and the bubble size, in excellent agreement with statistical modelling. Furthermore, we obtain the full distribution of bubble sizes and the opening probabilities at varying salt and temperature conditions. The results presented herein underline the important role of denaturation bubbles in negatively supercoiled DNA for biological processes such as transcription and replication initiation in vivo.

We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) similar to vertical bar x vertical bar(alpha) of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P(x, t), we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus vertical bar alpha vertical bar of the scaling exponent. The fluctuations appear to diverge when the critical value alpha = 2 is approached, while for even larger alpha the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.

We study the elastic deformations in a cross-linked polymer network film triggered by the binding of submicron particles with a sticky surface, mimicking the interactions of viral pathogens with thin films of stimulus-responsive polymeric materials such as hydrogels. From extensive Langevin Dynamics simulations we quantify how far the network deformations propagate depending on the elasticity parameters of the network and the adhesion strength of the particles. We examine the dynamics of the collective area shrinkage of the network and obtain some simple relations for the associated characteristic decay lengths. A detailed analysis elucidates how the elastic energy of the network is distributed between stretching and compression modes in response to the particle binding. We also examine the force-distance curves of the repulsion or attraction interactions for a pair of sticky particles in the polymer network film as a function of the particle-particle separation. The results of this computational study provide new insight into collective phenomena in soft polymer network films and may, in particular, be applied to applications for visual detection of pathogens such as viruses via a macroscopic response of thin films of cross-linked hydrogels. (C) 2014 AIP Publishing LLC.

Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion
(2014)

The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient.

In various biological systems and small scale technological applications particles transiently bind to a cylindrical surface. Upon unbinding the particles diffuse in the vicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise to an effective surface translation, for which we here derive and discuss the dynamic equations, including additional surface diffusion. We discuss the time evolution of the number of surface-bound particles, the effective surface mean squared displacement, and the surface propagator. In particular, we observe sub- and superdiffusive regimes. A plateau of the surface mean-squared displacement reflects a stalling of the surface diffusion at longer times. Finally, the corresponding first passage problem for the cylindrical geometry is analysed.

We consider diffusion processes with a spatially varying diffusivity giving rise to anomalous diffusion. Such heterogeneous diffusion processes are analysed for the cases of exponential, power-law, and logarithmic dependencies of the diffusion coefficient on the particle position. Combining analytical approaches with stochastic simulations, we show that the functional form of the space-dependent diffusion coefficient and the initial conditions of the diffusing particles are vital for their statistical and ergodic properties. In all three cases a weak ergodicity breaking between the time and ensemble averaged mean squared displacements is observed. We also demonstrate a population splitting of the time averaged traces into fast and slow diffusers for the case of exponential variation of the diffusivity as well as a particle trapping in the case of the logarithmic diffusivity. Our analysis is complemented by the quantitative study of the space coverage, the diffusive spreading of the probability density, as well as the survival probability.

Velocity and displacement correlation functions for fractional generalized Langevin equations
(2012)

We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.

We consider the mean first-passage time of a random walker moving in a potential landscape on a finite interval, the starting and end points being at different potentials. From analytical calculations and Monte Carlo simulations we demonstrate that the mean first-passage time for a piecewise linear curve between these two points is minimized by the introduction of a potential barrier. Due to thermal fluctuations, this barrier may be crossed. It turns out that the corresponding expense for this activation is less severe than the gain from an increased slope towards the end point. In particular, the resulting mean first-passage time is shorter than for a linear potential drop between the two points.

Based on extensive Brownian dynamics simulations we study the thermal motion of a tracer bead in a cross-linked, flexible gel in the limit when the tracer particle size is comparable to or even larger than the equilibrium mesh size of the gel. The analysis of long individual trajectories of the tracer demonstrates the existence of pronounced transient anomalous diffusion. From the time averaged mean squared displacement and the time averaged van Hove correlation functions we elucidate the many-body origin of the non-Brownian tracer bead dynamics. Our results shed new light onto the ongoing debate over the physical origin of steric tracer interactions with structured environments.

We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion.