Single particle tracking of complex diffusion in membranes: simulation and detection of barrier, raft, and interaction phenomena

Single particle tracking is being used increasingly to follow the motion of membrane-associated receptors and lipids. Anomalous and complex diffusive behaviors are generally found in cell membranes. We developed computational algorithms to simulate particle trajectories and to detect complex diffusive behaviors in two dimensions, including confined and convective diffusion, intramembrane barrier and raft phenomena, and interparticle interactions. Little useful information regarding barrier, raft, and interaction effects were provided by standard computational procedures for identification of anomalous diffusion, including analysis of mean squared displacement, distributions of diffusion rates and range, and time evolution of particle position. New algorithms were developed and optimized to detect complex diffusive behaviors from simulated single particle trajectories. A barrier detection algorithm was developed on the basis of spatial averaging of particle positions in trajectories. A raft detection algorithm utilized spatially resolved diffusion coefficients and particle density functions. An interaction algorithm utilized interparticle distance distributions. The algorithms developed here are applicable to identify biologically important diffusive phenomena in cell membranes.     

Temporal correlation functions of concentration fluctuations: an anomalous case

We calculate, within the framework of the continuous time random walk (CTRW) model, multiparticle temporal correlation functions of concentration fluctuations (CCF) in systems that display anomalous subdiffusion. The subdiffusion stems from the nonstationary nature of the CTRW waiting times, which also lead to aging and ergodicity breaking. Due to aging, a system of diffusing particles tends to slow down as time progresses, and therefore, the temporal correlation functions strongly depend on the initial time of measurement. As a consequence, time averages of the CCF differ from ensemble averages, displaying therefore ergodicity breaking. We provide a simple example that demonstrates the difference between these two averages, a difference that might be amenable to experimental tests. We focus on the case of ensemble averaging and assume that the preparation time of the system coincides with the starting time of the measurement. Our analytical calculations are supported by computer simulations based on the CTRW model.     

Dynamics and conformational properties of Rouse polymers in random layered flows

Here we study the dynamics and the conformational properties of Rouse polymers in random layered flows. We obtain explicit expressions for the mean squared displacement of the center of mass of the polymer as a function of time and number of beads. The averages involve both the thermal noise and the realizations of the random flows. We find also an explicit dependence of the equilibrium value of the end-to-end vector on the number of beads and other dimensional parameters. Our results are compared to that for Rouse polymers immersed in non-random steady shear flows.     

Effective surface motion on a reactive cylinder of particles that perform intermittent bulk diffusion

In many biological and small scale technological applications particles may transiently bind to a cylindrical surface. In between two binding events the particles diffuse in the bulk, thus producing an effective translation on the cylindrical surface. We here derive the effective motion on the surface allowing for additional diffusion on the cylindrical surface itself. We find explicit solutions for the number of adsorbed particles at one given instant, the effective surface displacement, as well as the surface propagator. In particular sub- and superdiffusive regimes are found, as well as an effective stalling of diffusion visible as a plateau in the mean squared displacement. We also investigate the corresponding first passage problem.     

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times.     

Transient diffusion in a tube with dead ends

A particle diffusing in a tube with dead ends, from time to time enters a dead end, spends some time in the dead end, and then comes back to the tube. As a result, the particle spends in the tube only a part of the entire observation time that leads to slowdown of its diffusion along the tube. We study the transient diffusion in a tube with periodic identical dead ends formed by cavities of volume V(cav) connected to the tube by cylindrical channels of length L and radius a, which is assumed to be much smaller than the tube radius R and the distance l between neighboring dead ends. Assuming that the particle initial position is uniformly distributed over the tube, we analyze the monotonic decrease of the particle diffusion coefficient D(t) from its initial value D(0)=D, which characterizes diffusion in the tube without dead ends, to its asymptotic long-time value D(infinity)=D(eff)相似文献   

Calculation of the diffusion coefficient in a liquid based on the force covariance function

Methods of optimal use of statistical information on the Brownian motion of molecules in a liquid for calculating the diffusion coefficient are considered. Relationships between covariance functions for the displacement of a molecule, its velocity, and the force acting on the molecule are established. It is proposed the method of calculation of the diffusion coefficient estimates should be made using the force covariance function, which enables to reduce the time of simulation of the molecular motion. The proposed method is used for callculating diffusion coefficients for some systems.     

The dynamics of single chains within a model polymer melt

Discontinuous molecular dynamics simulations are performed on a system containing 32 hard chains of length 192 at a volume fraction of phi = 0.45 to explore the idea that localized entanglements have a significant effect on the dynamics of the individual chains within an entangled polymer melt. Anomalous behavior can still be observed when studying the dynamics of the individual chains, although increased time averaging causes the anomalous relaxation-memory-release behavior that was observed previously in the system to smooth out. First, the individual chain mean squared displacements and apparent diffusion coefficients are calculated, and a wide distribution of diffusive behavior is found. Although the apparent diffusion coefficient curve averaged over all chains displays the predicted long-time diffusive behavior, the curves for the individual chains differ both qualitatively and quantitatively. They display superdiffusive, diffusive, and subdiffusive behavior, with the largest percentage of chains exhibiting superdiffusive behavior and the smallest percentage exhibiting the predicted diffusive behavior. Next, the individual chain end-to-end vector autocorrelation functions and relaxation times are determined, and a wide distribution of stress relaxation behavior is found. The times when the end-to-end vector autocorrelation functions relax completely span almost an order of magnitude in reduced time. For some chains, the end-to-end vector autocorrelation function relaxes smoothly toward zero similar to the system average; however, for other chains the relaxation is slowed greatly, indicating the presence of additional entanglements. Almost half of the chains exhibit the anomalous behavior in the end-to-end vector autocorrelation function. Finally, the dynamic properties are displayed for a single chain exhibiting anomalous relaxation-memory-release behavior, supporting the idea that the relaxation-memory-release behavior is a single-chain property.     

Coherent and diffusional motion of a chemisorbed atom

Experiments indicate that there are two extreme types of motion of an atom on a solid surface. One is characterized by an average velocity and has a mean square displacement proportion to the square of the time (we call this coherent). The other (called purely diffusional) is characterized by a diffusion coefficient and has a mean square displacement proportional to time. We present a simple stochastic model to explain the microscopic origin of these two extreme types of motion. In the case in which both types of motion coexist, the motion becomes diffusional for times longer than an intrinsic time depending on the intensity of the thermal fluctuations of the atom—lattice coupling.     

Supercooled dynamics of grain boundary particles in two-dimensional colloidal crystals

We experimentally investigate the dynamics of particles constituting grain boundaries in a two-dimensional colloidal crystal, using video-microscopy. A clear plateau in the mean square displacement of the grain boundary particles is found, followed by an upswing indicative of cage breaking. The van Hove correlation functions and the non-Gaussian parameter show that grain boundary particle dynamics are highly heterogeneous. Furthermore, we identified clusters of cooperatively moving particles and analyzed the time-dependence of the weight-averaged mean cluster size. We find good correlation between the behavior of the mean square displacement, and the time dependence of the non-Gaussian parameter and the cluster size, as also reported for various supercooled systems. Our results therefore provide experimental support for the similarity between particle dynamics in grain boundaries and in supercooled liquids as suggested by recent computer simulations.     

Aging correlation functions for blinking nanocrystals, and other on-off stochastic processes

Following recent experiments on power law blinking behavior of single nanocrystals, we calculate two-time intensity correlation functions I(t)I(t+t') for these systems. We use a simple two state (on and off) stochastic model to describe the dynamics. We classify possible behaviors of the correlation function and show that aging, e.g., dependence of the correlation function on age of process t, is obtained for classes of the on time and off time distributions relevant to experimental situation. Analytical asymptotic scaling behaviors of the intensity correlation in the double time t and t' domain are obtained. In the scaling limit I(t)I(t+t('))-->h(x), where four classes of behaviors are found: (i) finite averaged on and off times x=t' (standard behavior); (ii) on and off times with identical power law behaviors x=t/t' (case relevant for capped nanocrystals); (iii) exponential on times and power law off times x=tt' (case relevant for uncapped nanocrystals); (iv) for defected off time distribution we also find x=t+t'. Origin of aging behavior is explained based on simple diffusion model. We argue that the diffusion controlled reaction A+B <==>AB, when followed on a single particle level exhibits aging behavior.     

Anomalous fluctuations in sliding motion of cytoskeletal filaments driven by molecular motors: model simulations

It has been found in in vitro experiments that cytoskeletal filaments driven by molecular motors show finite diffusion in sliding motion even in the long filament limit [Imafuku, Y. et al. Biophys. J. 1996, 70, 878-886. Noda, N. et al. Biophysics 2005, 1, 45-53]. This anomalous fluctuation can be evidence for cooperativity among the motors in action because fluctuation should be averaged out for a long filament if the action of each motor is independent. In order to understand the nature of the fluctuation in molecular motors, we perform numerical simulations and analyze velocity correlation in three existing models that are known to show some kind of cooperativity and/or large diffusion coefficient, i.e., the Sekimoto-Tawada model [Sekimoto, K.; Tawada, K. Phys. Rev. Lett. 1995, 75, 180], the Prost model [Prost, J. et al. Phys. Rev. Lett. 1994, 72, 2652], and the Duke model [Duke, T. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 2770]. It is shown that the Prost model and the Duke model do not give a finite diffusion in the long filament limit, in spite of the collective action of motors. On the other hand, the Sekimoto-Tawada model has been shown to give a diffusion coefficient that is independent of filament length, but it comes from the long time correlation whose time scale is proportional to filament length, and our simulations show that such a long correlation time conflicts with the experimental time scales. We conclude that none of the three models represent experimental findings. In order to explain the observed anomalous diffusion, we have to search for a mechanism that will allow both the amplitude and the time scale of the velocity correlation to be independent of the filament length.     

Simultaneous investigation of sedimentation and diffusion of a single colloidal particle near an interface

We describe here a new procedure for the simultaneous investigation of sedimentation and diffusion of a colloidal particle in close proximity to a solid, planar wall. The measurements were made using the optical technique of total internal reflection microscopy, coupled with optical radiation pressure, for dimensionless separation distances (gap width/radius of particle) ranging from 0.01 to 0.05. In this region, the hydrodynamic mobility and diffusion coefficient are substantially reduced below bulk values. The procedure involved measuring the mean and the variance of vertical displacements of a Brownian particle settling under gravity toward the plate. The spatially varying diffusion coefficient was calculated from the displacements at small times (where diffusive motion was dominant). The mobility relationship for motion normal to a flat plate was tested by measuring the average distance of travel versus time as the particle settled under the constant force of gravity. For the simple Newtonian fluid used here (aqueous salt solution), the magnitude of the diffusion coefficient and mobility, plus their dependence on separation distance, showed excellent agreement with predictions. This new technique could be of great value in measuring the mobility and diffusion coefficient for near-contact motion in more complex fluids for which the hydrodynamic correction factors are not known a priori, such as shear-thinning fluids.     

Water/hydrocarbon interfaces: effect of hydrocarbon branching on single-molecule relaxation

Water/hydrocarbon interfaces are studied using molecular dynamics simulations in order to understand the effect of hydrocarbon branching on the dynamics of the system at and away from the interface. A recently proposed procedure for studying the intrinsic structure of the interface in such systems is utilized, and dynamics are probed in the usual laboratory frame as well as the intrinsic frame. The use of these two frames of reference leads to insight into the effect of capillary waves at the interface on dynamics. The systems were partitioned into zones with a width of 5 A, and a number of quantities of dynamical relevance, namely, the residence times, mean squared displacements, the velocity auto correlation functions, and orientational time correlations for molecules of both phases, were calculated in the laboratory and intrinsic frames at and away from the interface. For the aqueous phase, translational motion is found to be (a) diffusive at long times and not anomalous as in proteins or micelles, (b) faster at the interface than in the bulk, and (c) faster upon reduction of the effect of capillary waves. The rotational motion of water is (a) more anisotropic at the interface than in the bulk and (b) dependent on the orientation of the covalent O-H bond with respect to the plane of the interface. The effect of hydrocarbon branching on aqueous dynamics was found to be small, a result similar to the effect on the interfacial water structure. The hydrocarbon phase shows a larger variation for all dynamical probes, a trend consistent with their interfacial structure.     

Microscopic Motion of Atoms in Simple Liquids at Equilibrium and with Shear Flow

Abstract

We investigate the microscopic mechanism of atomic motion and local stress relaxation in Lennard-Jones, LJ liquids using a new class of correlation functions that emphasise the interplay between an abitrary atom in the fluid and its surrounding shells of atoms. We use the linear momenta and stress tensor to characterise the time dependence of this interaction. We consider a series of correlation functions that give complementary information and build a picture of the single particle and small cluster motion. The central particle and first shell undergo a reversal in momentum at different times after the ‘collision’ of the central particle and its first shell of neighbours. This ‘phase difference’ becomes manifest in the subsequent dynamics probed by the new correlation functions. We also consider the effect of a non-newtonian shear flow on this local dynamical relaxation, using profile biased laminar flow equations of motion. In non-newtonian shear flow we find the momentum transfer between particle and cage to be less pronounced and occur over a wider time range.     

Dynamic scaling of polymer gels comprising nanoparticles

We present dynamic light scattering (DLS) measurements of soft poly(methyl-methacrylate) (PMMA) and polyacrylamide (PA) polymer gels prepared with trapped bodies (latex spheres or magnetic nanoparticles). We show that the anomalous diffusivity of the trapped particles can be analyzed in terms of a fractal Gaussian network gel model for the entire time range probed by DLS technique. This model is a generalization of the Rouse model for linear chains extended for structures with power law network connectivity scaling, which includes both percolating and uniform bulk gel limits. For a dilute dispersion of strongly scattering particles trapped in a gel, the scattered electric field correlation function at small wavevector ideally probes self-diffusion of gel portions imprisoning the particles. Our results show that the time-dependent diffusion coefficients calculated from the correlation functions change from a free diffusion regime at short times to an anomalous subdiffusive regime at long times (increasingly arrested displacement). The characteristic time of transition between these regimes depends on scattering vector as approximately q(-2), while the time decay power exponent tends to the value expected for a bulk network at small q. The diffusion curves for all scattering vectors and all samples were scaled to a single master curve.     

Generalized Kubo relations and conditions for anomalous diffusion: physical insights from a mathematical theorem

The paper describes an approach to anomalous diffusion within the framework of the generalized Langevin equation. Using a Tauberian theorem for Laplace transforms due to Hardy, Littlewood, and Karamata, generalized Kubo relations for the relevant transport coefficients are derived from the asymptotic form of the mean square displacement. In a second step conditions for anomalous diffusion are derived for the asymptotic forms of the velocity autocorrelation function and the associated memory function. Both spatially unconfined and confined diffusion processes are considered. The results are illustrated with semi-analytical examples.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

Molecular dynamics in the isothermal-isobaric ensemble: the requirement of a "shell" molecule. II. Simulation results

The results of a series of constant pressure and temperature molecular-dynamics (MD) simulation studies based on the rigorous shell particle formulation of the isothermal-isobaric (NpT) ensemble are presented. These MD simulations validate the newly proposed constant pressure equations of motion in which a "shell" particle is used to define uniquely the volume of the system [M. J. Uline and D. S. Corti, J. Chem. Phys. (to be published), preceding paper]. Ensemble averages obtained with the new MD NpT algorithm match the ensemble averages obtained using the previously derived shell particle Monte Carlo NpT method [D. S. Corti, Mol. Phys. 100, 1887 (2002)]. In addition, we also verify that the Hoover NpT MD algorithm [W. G. Hoover, Phys. Rev. A 31, 1695 (1985); 34, 2499 (1986)] generates the correct ensemble averages, though only when periodic boundary conditions are employed. The extension of the shell particle MD algorithm to multicomponent systems is also discussed, in which we show for equilibrium properties that the identity of the shell particle is completely arbitrary when periodic boundary conditions are applied. Self-diffusion coefficients determined with the shell particle equations of motion are also identical to those obtained in other ensembles. Finally, since the mass of the shell particle is known, the system itself, and not a piston of arbitrary mass, controls the time scales for internal pressure and volume fluctuations. We therefore consider the effects of the shell particle on the dynamics of the system. Overall, the shell particle MD algorithm is an effective simulation method for studying systems exposed to a constant external pressure and may provide an advantage over other existing constant pressure approaches when developing nonequilibrium MD methods.     

Single particle dynamics in ionic liquids of 1-alkyl-3-methylimidazolium cations

Ionic dynamics in room temperature molten salts (ionic liquids) containing 1-alkyl-3-methylimidazolium cations is investigated by molecular-dynamics simulations. Calculations were performed with united atom models, which were used in a previous detailed study of the equilibrium structure of ionic liquids [S. M. Urahata and M. C. C. Ribeiro, J. Chem. Phys. 120, 1855 (2004)]. The models were used in a systematic study of the dependency of several single particle time correlation functions on anion size (F-, Cl-, Br-, and PF6-) and alkyl chain length (1-methyl-, 1-ethyl-, 1-butyl-, and 1-octyl-). Despite of large mass and size of imidazolium cations, they exhibit larger mean-square displacement than anions. A further detailed picture of ionic motions is obtained by using appropriate projections of displacements along the plane or perpendicular to the plane of the imidazolium ring. A clear anisotropy in ionic displacement is revealed, the motion on the ring plane and almost perpendicular to the 1-alkyl chain being the less hindered one. Similar projections were performed on velocity correlation functions, whose spectra were used to relate short time ionic rattling with the corresponding long time diffusive regime. Time correlation functions of cation reorientation and dihedral angles of the alkyl chains are discussed, the latter decaying much faster than the former. A comparative physical picture of time scales for distinct dynamical processes in ionic liquids is provided.