Non-Brownian diffusion in lipid membranes: Experiments and simulations.

The dynamics of constituents and the surface response of cellular membranes-also in connection to the binding of various particles and macromolecules to the membrane-are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder-through the addition of cholesterol or proteins-and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments-the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle-membrane binding energy may impact the dynamics and response of lipid membranes. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes.

We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x|(α) 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 |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α 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.

Numerical approach to unbiased and driven generalized elastic model.

From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent ß characterizing the growth of the mean squared displacement 〈(δh)(2)〉 of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments 〈∣δh∣(q)〉 with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe.

Aging and nonergodicity beyond the Khinchin theorem.

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 nonstationary 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.

How DNA coiling enhances target localization by proteins.

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.

Random time-scale invariant diffusion and transport coefficients.

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement delta2[over ] 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 delta2[over ] differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable delta2[over ]. 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.

Tight and loose shapes in flat entangled dense polymers.

We investigate the effects of topological constraints (entanglements) on two-dimensional polymer loops in the dense phase, and at the collapse transition (theta-point). Previous studies have shown that in the dilute phase the entangled region becomes tight, and is thus localised on a small portion of the polymer. We find that the entropic force favouring tightness is considerably weaker in dense polymers. While the simple figure-eight structure, created by a single crossing in the polymer loop, localises weakly, the trefoil knot and all other prime knots are loosely spread out over the entire chain. In both the dense and theta conditions, the uncontracted-knot configuration is the most likely shape within a scaling analysis. By contrast, a strongly localised figure-eight is the most likely shape for dilute prime knots. Our findings are compared to recent simulations.

The future is noisy: the role of spatial fluctuations in genetic switching.

A genetic switch may be realized by a certain operator sector on the DNA strand from which either genetic code, to the left or to the right of this operator sector, can be transcribed and the corresponding information processed. This switch is controlled by messenger molecules, i.e., they determine to which side the switch is flipped. Recently, it has been realized that noise plays an elementary role in genetic switching, and the effect of number fluctuations of the messenger molecules have been explored. Here we argue that the assumption of well stirredness taken in the previous models may not be sufficient to characterize the influence of noise: spatial fluctuations play a non-negligible part in cellular genetic switching processes.

Generation of unpredictable time series by a neural network.

A perceptron that "learns" the opposite of its own output is used to generate a time series. We analyze properties of the weight vector and the generated sequence, such as the cycle length and the probability distribution of generated sequences. A remarkable suppression of the autocorrelation function is explained, and connections to the Bernasconi model are discussed. If a continuous transfer function is used, the system displays chaotic and intermittent behavior, with the product of the learning rate and amplification as a control parameter.

Multichoice minority game.

The generalization of the problem of adaptive competition, known as the minority game, to the case of K possible choices for each player, is addressed, and applied to a system of interacting perceptrons with input and output units of a type of K-state Potts spins. An optimal solution of this minority game, as well as the dynamic evolution of the adaptive strategies of the players, are solved analytically for a general K and compared with numerical simulations.

Molecular switching with nonexponential relaxation patterns: a random walk approach.

The transition from an initial, locally stable configuration to a globally stable state in molecular switches is investigated in terms of a random walk model, effectively taking the reaction pathway through a potentially rugged energy landscape into account. Exponential and nonexponential scenarios are discussed and the implications on measurable quantities are explored.

Generalized chapman-kolmogorov equation: A unifying approach to the description of anomalous transport in external fields

The generalized Chapman-Kolmogorov equation [V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger, J. Stat. Phys. 9, 45 (1973)] is discussed. It is demonstrated that this equation unifies recently proposed kinetic equations of fractional order that describe anomalous transport in external fields, as well as continuous time random walks. The conditions under which the individual models can be established are discussed.

Subdiffusive transport close to thermal equilibrium: from the langevin equation to fractional diffusion

Subdiffusion in the presence or absence of an external force field is established on the basis of an extension of conventional Langevin dynamics to include long-tailed trapping events. It is demonstrated how the presence of the trapping events leads to the macroscopic observation of fractional diffusion, described by a fractional Klein-Kramers equation.

Interacting neural networks.

Several scenarios of interacting neural networks which are trained either in an identical or in a competitive way are solved analytically. In the case of identical training each perceptron receives the output of its neighbor. The symmetry of the stationary state as well as the sensitivity to the used training algorithm are investigated. Two competitive perceptrons trained on mutually exclusive learning aims and a perceptron which is trained on the opposite of its own output are examined analytically. An ensemble of competitive perceptrons is used as decision-making algorithms in a model of a closed market (El Farol Bar problem or the Minority Game. In this game, a set of agents who have to make a binary decision is considered.); each network is trained on the history of minority decisions. This ensemble of perceptrons relaxes to a stationary state whose performance can be better than random.

From continuous time random walks to the fractional fokker-planck equation

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.

Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems.

Logarithmic oscillations superimposed on the temporal relaxation patterns of complex systems are considered from the standpoint of their hierarchical origin. We propose that a closer examination of experimental data should reveal logarithmic oscillations in systems that are characterized by a hierarchical structure of their dynamical degrees of freedom. On that footing, a new methodology of data analysis is proposed that may prove important for the dynamics of protein folding and of conformational fluctuations in proteins in which the relevant time scales of the dynamical evolution underlying the relaxation kinetics can be deduced from these oscillations.

Terson's syndrome in spontaneous subarachnoid hemorrhage: a prospective study in 60 consecutive patients.

Sixty consecutive patients with spontaneous subarachnoid hemorrhage (SAH) were prospectively studied by means of indirect funduscopy to address the question of incidence and prognostic implications of Terson's syndrome (TS) after SAH. Terson's syndrome was diagnosed in 10 (16.7%) of 60 patients and was associated with subarachnoid rebleeding in seven of 10. No correlation was found between anatomical localization of the ruptured aneurysm and TS laterality. Case fatality was nine (90%) of 10 in patients with TS compared to five (10%) of 50 in non-TS patients. It is concluded that TS is not infrequent (16.7%) in spontaneous SAH and has a poor prognosis, often heralding subarachnoid rebleeding.

Different concentrations of vasoactive intestinal polypeptide in aqueous humor of patients with proliferative vitreoretinopathy and cataract patients.

The pathophysiological events leading to cellular proliferation in proliferative vitreoretinopathy are largely unknown. An involvement of neuropeptides in that disease has recently been discussed, as substance P was found to be highly enriched in the intraocular fluid of patients with proliferative vitreoretinopathy. In the present study, aqueous humor was analyzed for another neuropeptide, vasoactive intestinal polypeptide. Radioimmunoassay revealed significantly increased levels of that polypeptide in the aqueous humor of patients with proliferative vitreoretinopathy as compared with cataract patients who served as controls. As vasoactive intestinal polypeptide contributes to the environment of the retinal pigment epithelial cell layer and induces proliferation of these cells in vitro, this peptide may be involved in the pathogenetic mechanisms leading to cellular proliferation in proliferative vitreoretinopathy.

Immunodiffusion and immunohistochemical investigations on the reactivity of oxide ceramic middle-ear implants.

Oxide ceramic materials (partial and total ossicular replacement prostheses) have been implants of preference for the reconstruction of the ossicular chain because of their excellent biocompatibility. The reaction on the surface of the implants takes place at three biodynamic levels according to the model of Stern's bilayer. We investigated the adsorption of proteins, which is determined by the cellular reaction and degradation to the surface using radial immunodiffusion and immunohistochemical methods. First, ceramic implants of aluminum oxide, hydroxyapatite, glass ceramic and zirconium oxide have individual actual (i.e. biological) surfaces. With a perthometer and the contact-free laser Focodyn method we determined each actual (i.e. biological) surface of the various ceramic implants mentioned above. Using radial immunodiffusion, the adsorption of albumin, glycoprotein, plasminogen, fibronectin, IgA, IgG and IgM shows characteristic rates of adsorption to the respective ceramic surfaces in correlating to the actual surface. A cross-check with fluorescent antibodies confirmed the protein adsorption. The individual surface adsorption of the proteins remains characteristic and is the basis for the recording of cellular reactions after implantation.