Communication: consistent picture of lateral subdiffusion in lipid bilayers: molecular dynamics simulation and exact results

This communication presents a molecular dynamics simulation study of a bilayer consisting of 128 dioleoyl-sn-glycero-3-phosphocholine molecules, which focusses on the center-of-mass diffusion of the lipid molecules parallel to the membrane plane. The analysis of the simulation results is performed within the framework of the generalized Langevin equation and leads to a consistent picture of subdiffusion. The mean square displacement of the lipid molecules evolves as ∝ t(α), with α between 0.5 and 0.6, and the fractional diffusion coefficient is close to the experimental value for a similar system obtained by fluorescence correlation spectroscopy. We show that the long-time tails of the lateral velocity autocorrelation function and the associated memory function agree well with exact results which have been recently derived by asymptotic analysis [G. Kneller, J. Chem. Phys. 134, 224106 (2011)]. In this context, we define characteristic time scales for these two quantities.     

Anomalous nonlinear dielectric and Kerr effect relaxation steady state responses in superimposed ac and dc electric fields

It is shown how the rotational diffusion model of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended to anomalous nonlinear dielectric relaxation and the dynamic Kerr effect by using a fractional kinetic equation. This fractional kinetic equation (obtained via a generalization of the noninertial kinetic equation of conventional rotational diffusion to fractional kinetics to include anomalous relaxation) is solved using matrix continued fractions yielding the complex nonlinear dielectric susceptibility and the Kerr function of an assembly of rigid dipolar particles acted on by external superimposed dc E0 and ac E1(t)=E1 cos omegat electric fields of arbitrary strengths. In the weak field limit, analytic equations for nonlinear response functions are also derived.     

Photonic superdiffusive motion in resonance radiation trapping

In this work we consider the relation between the jump length probability density function and the line shape function in resonance radiation trapping in atomic vapors. The two-sided jump length probability density function suitable for a unidimensional formulation of radiative transfer is also derived. As a side result, a procedure to obtain the Maxwell distribution of velocities from the Maxwell-Boltzmann distribution of speeds was obtained. General relations that give the asymptotic jump length behavior and the Levy flight parameter mu for any line shape are obtained. The results are applied to generalized Doppler, generalized Lorentz, and Voigt line shape functions. It is concluded that the lighter the tail of the line shape function, the less heavy the tail of the jump length probability density function, although this tail is always heavy, with mu < or =1.     

Photonic superdiffusive motion in resonance line radiation trapping Partial frequency redistribution effects

The relation between the jump length probability distribution function and the spectral line profile in resonance atomic radiation trapping is considered for partial frequency redistribution (PFR) between absorbed and reemitted radiation. The single line opacity distribution function [M. N. Berberan-Santos et al., J. Chem. Phys. 125, 174308 (2006)] is generalized for PFR and used to discuss several possible redistribution mechanisms (pure Doppler broadening; combined natural and Doppler broadening; and combined Doppler, natural, and collisional broadening). It is shown that there are two coexisting scales with a different behavior: the small scale is controlled by the intricate PFR details while the large scale is essentially given by the atom rest frame redistribution asymptotic. The pure Doppler and combined natural, Doppler, and collisional broadening are characterized by both small- and large-scale superdiffusive Levy flight behaviors while the combined natural and Doppler case has an anomalous small-scale behavior but a diffusive large-scale asymptotic. The common practice of assuming complete redistribution in core radiation and frequency coherence in the wings of the spectral distribution is incompatible with the breakdown of superdiffusion in combined natural and Doppler broadening conditions.     

Correlating anomalous diffusion with lipid bilayer membrane structure using single molecule tracking and atomic force microscopy

Anomalous diffusion has been observed abundantly in the plasma membrane of biological cells, but the underlying mechanisms are still unclear. In general, it has not been possible to directly image the obstacles to diffusion in membranes, which are thought to be skeleton bound proteins, protein aggregates, and lipid domains, so the dynamics of diffusing particles is used to deduce the obstacle characteristics. We present a supported lipid bilayer system in which we characterized the anomalous diffusion of lipid molecules using single molecule tracking, while at the same time imaging the obstacles to diffusion with atomic force microscopy. To explain our experimental results, we performed lattice Monte Carlo simulations of tracer diffusion in the presence of the experimentally determined obstacle configurations. We correlate the observed anomalous diffusion with obstacle area fraction, fractal dimension, and correlation length. To accurately measure an anomalous diffusion exponent, we derived an expression to account for the time-averaging inherent to all single molecule tracking experiments. We show that the length of the single molecule trajectories is critical to the determination of the anomalous diffusion exponent. We further discuss our results in the context of confinement models and the generating stochastic process.     

Generalized Einstein relations from a three-temperature theory of gaseous ion transport

Generalized Einstein relations between mobility and diffusion coefficients are derived within the framework of a recently developed three-temperature kinetic theory of gaseous ion transport. A previously assumed connection between diffusion and differential mobility is firmly established within low-order approximations of the kinetic theory. Equations are obtained for the approximate calculation of the unobservable parallel and perpendicular ion temperatures, and for two higher moments of the ion velocity distribution function that appear as correction terms in the generalized Einstein relations. The present theory is tested on inverse-power potentials and on several alkali ion-noble gas systems, and is compared to two previous semi-empirical generalized Einstein relations. Simple procedures are recommended for the estimation of the parameters and correction terms that occur, to permit approximate calculation of gaseous ion diffusion coefficients from mobility data without the need of a computer or specific knowledge of the ion-neutral interaction potential.     

Generalized Chudley-Elliott vibration-jump model in activated atom surface diffusion

Here the authors provide a generalized Chudley-Elliott expression for the activated atom surface diffusion which takes into account the coupling between both low-frequency vibrational motion (namely, the frustrated translational modes) and diffusion. This expression is derived within the Gaussian approximation framework for the intermediate scattering function at low coverage. Moreover, inelastic contributions (arising from creation and annihilation processes) to the full width at half maximum of the quasielastic peak are also obtained.     

The Kramers-Moyal expansion for electrochemical stochastic diffusion

It is proved that there is a general stochastic equation, according to which any random process in the transient mode can be presented by spatially homogeneous Kramers-Moyal expansion. In the electrochemical stochastic diffusion, an integral of the fluctuation component of electrode potential over the time plays the role of spatial coordinate. Based on these two facts, we derived a spatially homogeneous Kramers-Moyal expansion for the propagator of electrochemical stochastic diffusion. By using the limiting transition to long observation times, we obtained a time and spatially homogeneous asymptotic Kramers-Moyal expansion for the propagator of asymmetric non-Gaussian electrochemical stochastic diffusion. Under the conditions of Gaussian electrochemical noise, the asymptotic Kramers-Moyal expansion turns into the Einstein stochastic diffusion equation. The method of determining time and spatially homogeneous asymptotic Kramers-Moyal expansion for the propagator of asymmetric non-Gaussian electrochemical stochastic diffusion may be useful in the stochastic theory of slow electrochemical discharge and in the electrochemical noise diagnostics.     

A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes

This article reviewed anomalous diffusion phenomena coupled with facile and sluggish charge-transfer reactions at fractal interface. Firstly, the generalised diffusion equation (GDE) involving a fractional derivative which describes diffusion towards and from fractal interface was briefly introduced. And then, anomalous diffusion coupled with facile charge-transfer reaction at fractal interface, i.e., diffusion-controlled transfer process across fractal interface, was mathematically examined by the generalised Cottrell, Sand, Randles-Sevcik and Warburg equations theoretically derived from the analytical solutions to the GDE under the semi-infinite boundary condition. Finally, in order to provide a guideline in analysing anomalous diffusion coupled with sluggish charge-transfer reaction at fractal interface, i.e., non-diffusion-controlled transfer process across fractal interface, this review covered the recent researches into the effect of surface roughness on non-diffusion-controlled transfer process within the intercalation electrodes.     

Anomalous diffusion of particles with inertia in external potentials

Recently a new type of Kramers-Fokker-Planck equation has been proposed [Friedrich, R.; Jenko, F.; Baule, A.; Eule, S. Phys. Rev. Lett. 2006, 96, 230601] describing anomalous diffusion in external potentials. In the present paper, the explicit cases of a harmonic potential and a velocity-dependent damping are incorporated. Exact relations for moments for these cases are presented, and the asymptotic behavior for long times is discussed. Interestingly, the bounding potential and the additional damping by itself lead to a subdiffusive behavior. While acting together, the particle becomes localized for long periods of time.     

Diffusion current on a nonuniform electrode: Calculation with integral equations

Identity of mathematical problems concerning calculation of the distribution of reactants’ concentrations and the current near the surface of a nonuniform (strip) electrode and distribution of displacements and forces in the case of an elastic layer “antiplane” deformation caused by the punch action. Formulas for calculating the current at a strip electrode are derived for various ratios between the electrode width and the diffusion layer thickness by means of asymptotic methods designed for calculating problems of mechanical contact interactions. It is noted that calculations of the diffusion current for involved activity distributions at the electrode surface may benefit from asymptotic methods of mechanics of contact interactions.     

New insights into diffusion in 3D crowded media by Monte Carlo simulations: effect of size, mobility and spatial distribution of obstacles

Particle diffusion in crowded media was studied through Monte Carlo simulations in 3D obstructed lattices. Three particular aspects affecting the diffusion, not extensively treated in a three-dimensional geometry, were analysed: the relative particle-obstacle size, the relative particle-obstacle mobility and the way of having the obstacles distributed in the simulation space (randomly or uniformly). The results are interpreted in terms of the parameters that characterize the time dependence of the diffusion coefficient: the anomalous diffusion exponent (α), the crossover time from anomalous to normal diffusion regimes (τ) and the long time diffusion coefficient (D*). Simulation results indicate that there are a more anomalous diffusion (smaller α) and a lower long time diffusion coefficient (D*) when obstacle concentration increases, and that, for a given total excluded volume and immobile obstacles, the anomalous diffusion effect is less important for bigger size obstacles. However, for the case of mobile obstacles, this size effect is inverted yielding values that are in qualitatively good agreement with in vitro experiments of protein diffusion in crowded media. These results underline that the pattern of the spatial partitioning of the obstacle excluded volume is a factor to be considered together with the value of the excluded volume itself.     

Adsorption relaxation for nonionic surfactants under mixed barrier-diffusion and micellization-diffusion control

Relaxation processes of surfactant adsorption and surface tension, which are characterized by two specific relaxation times, are theoretically investigated. We are dealing with fluid interfaces and small initial deviations from equilibrium. For surfactant concentrations below the critical micellization concentration (CMC), we consider adsorption under mixed barrier-diffusion control. General analytical expressions are derived, which are convenient for both numerical computations and asymptotic analysis. Series expansions for the short- and long-time limit are derived. The results imply that the short-time asymptotics is controlled by the adsorption barrier, whereas the long-time asymptotics is always dominated by diffusion. Furthermore, for surfactant concentrations above the CMC, adsorption under mixed micellization-diffusion control is considered. Again, a general analytical expression is derived for the relaxation of surfactant adsorption and surface tension, whose long- and short-time asymptotics are deduced. The derived equations show that at the short times the relaxation is completely controlled by the diffusion, whereas the long-time asymptotics is affected by both demicellization and diffusion. The micellar effect is manifested as an exponential (rather than square-root) decay of the perturbation. The derived expressions are applied to process available experimental data for the nonionic surfactant Triton X-100 and to determine the respective demicellization rate constant.     

Experimental evidence of distance-dependent diffusion coefficients of a globular protein observed in polymer aqueous solution forming a network structure on nanometer scale

The distance dependence of the diffusion coefficient (DDDC) of a globular protein (cytochrome c) in aqueous hyaluronan (HA) solution, which is a model system for extracellular matrices (ECMs), was measured by a combination of three kinds of spectroscopic measurements of diffusion coefficients, the time and space samplings of which are different. The results of the three methods are plotted against the diffusion distance derived from the consideration of each experimental condition. Due to the characteristic morphology of HA with an effective mesh structure, the proteins showed two extreme diffusion modes: (1) short (<10 nm) diffusion with rare contact with polymer chains; (2) long (>100 nm) diffusion significantly disrupted by polymer chains showing an approximately 30% reduction in diffusion coefficient. The transition from the short diffusion to the long one occurs in a very narrow range (10-100 nm) of diffusion distance and this unique character of HA realizing anomalous diffusion should provide suitable environments for various bioactivities when involved in ECM.     

Diffusion-influenced reversible geminate ABCD reaction in the presence of an external field

We investigate a diffusion-influenced ground-state reversible geminate ABCD reaction in the presence of a constant external field in one dimension. In the Laplace domain, we first obtain the nonreactive Green function from which the reactive Green function is derived. Analytic asymptotic expressions of the survival probability are obtained in the time domain for both short and long time regions. There exist four regimes for the equilibrium survival probability according to the signs of the field intensities a1 and a2 that reactant and product states feel, respectively. Analysis of the long-time asymptotic behavior of the survival probability shows two regimes depending on the sign of a parameter K( identical with a(2)(2)D(2) -a(2)(1)D(1)), where D(1) and D(2) are the relative diffusion constants of corresponding states, respectively. Combining these two results, we predict a total of eight regimes for the long-time asymptotic behavior of the survival probability. We find that the long-time asymptotic behavior of the deviation of the effective survival probability shows the t(-3/2) power law when m( identical with min {a(2)(1)D(1), a(2)(2)D(2)}) not equal 0, whereas it shows t(-1/2) power law when m = 0. When one of the fields is turned off, the long-time asymptotic behavior of the survival probability shows a kinetic transition as the sign of the remaining field changes.     

The geometry and spectra of hyperbolic manifolds

This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic spaceH ⁿ by discrete subgroups Г, i.e. ℳ =H ⁿ/Г. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green’s function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green’s function, and hence the eigenfunctions, near infinity. This information is used to construct anS-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between theS-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of theS-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group Г. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles. Research supported in part by NSF grant DMS93-07438     

Mixing rules for multicomponent mixture mass diffusion coefficients and thermal diffusion factors

Mixing rules are derived for mass diffusion coefficient and thermal diffusion factor matrices by developing compatibility conditions between the fluid mixture equations obtained from nonequilibrium thermodynamics and Grad's 13-moment kinetic theory. The mixing rules are shown to be in terms of the species mole fractions and binary processes. In particular, the thermal diffusion factors for binary mixtures obtained by the Chapman-Enskog expansion procedure are suitably generalized for many-component mixtures. Some practical aspects of the results are discussed including the utilization of these mixing rules for high pressure situations.     

Super- and sub-Poissonian photon statistics for single molecule spectroscopy

We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave laser field. The spectral diffusion is modeled based on a stochastic approach, in the spirit of the Anderson-Kubo line shape theory. Using a generating function formalism we solve the generalized optical Bloch equations and obtain an exact analytical formula for the line shape and Mandel's Q parameter. The line shape exhibits well-known behaviors, including motional narrowing when the stochastic modulation is fast and power broadening. The Mandel parameter, describing the line shape fluctuations, exhibits a transition from a quantum sub-Poissonian behavior in the fast modulation limit to a classical super-Poissonian behavior found in the slow modulation limit. Our result is applicable for weak and strong laser fields, namely, for arbitrary Rabi frequency. We show how to choose the Rabi frequency in such a way so that the quantum sub-Poissonian nature of the emission process becomes strongest. A lower bound on Q is found and simple limiting behaviors are investigated. A nontrivial behavior is obtained in the intermediate modulation limit, when the time scales for spectral diffusion and the lifetime of the excited state become similar. A comparison is made between our results and previous ones derived, based on the semiclassical generalized Wiener-Khintchine formula.     

Theory of chemical reactions in solids. Part II

The probability W is deduced for an elementary chemical reaction due to fluctuations in a solid. W is found as a function of the phonon coupling constant. It is shown that anomalous diffusion kinetics can occur.     

Semi-bottom-up coarse graining of water based on microscopic simulations

The generalized dissipative particle dynamics (DPD) equation derived from the generalized Langevin equation under Markovian approximations is used to simulate coarse-grained (CG) water cells. The mean force and the friction coefficients in the radial and transverse directions needed for DPD equation are obtained directly from the all atomistic molecular dynamics (AAMD) simulations. But the dissipative friction forces are overestimated in the Markovian approximation, which results in wrong dynamic properties for the CG water in the DPD simulations. To account for the non-Markovian dynamics, a rescaling factor is introduced to the friction coefficients. The value of the factor is estimated by matching the diffusivity of water. With this semi-bottom-up mapping method, the radial distribution function, the diffusion constant, and the viscosity of the coarse-grained water system computed with DPD simulations are all in good agreement with AAMD results. It bridges the microscopic level and mesoscopic level with consistent length and time scales.