non-Gaussian Displacement Distributions Research Articles

Transport Dynamics of Water Molecules Confined between Lipid Membranes.

Water molecules confined between biological membranes exhibit a distinctive non-Gaussian displacement distribution, far different from that of bulk water. Here, we introduce a new transport equation for water molecules in the intermembrane space, quantitatively explaining molecular dynamics simulation results. We find that the unique transport dynamics of water molecules stems from the lateral diffusion coefficient fluctuation caused by their longitudinal motion in the direction perpendicular to the membranes. We also identify an interfacial region where water possesses distinct physical properties, which is unaffected by changes in the intermembrane separation.

Fickian Yet Non-Gaussian Nanoscopic Lipid Diffusion in the Raft-Mimetic Membrane.

Past experiments rationalized the observed dynamic heterogeneity and non-Gaussian diffusion in living cell membranes in terms of slow-active remodeling of the underlying cortical actin network. In this work, we demonstrate that the nanoscopic dynamic heterogeneity can also be explained via the lipid raft hypothesis, which postulates a phase separation between liquid-ordered (Lo) and liquid-disordered (Ld) nanodomains. Non-Gaussian displacement distribution is observed in the Lo domain for a long time, even when the mean square displacement becomes Fickian. This Fickian yet non-Gaussian diffusion is found particularly in the Lo/Ld interface consistent with the "diffusing diffusion" picture. A translational jump-diffusion model, previously employed to explain the diffusion-viscosity decoupling in supercooled water, is used here to quantitatively explain the long-term dynamic heterogeneity where a strong correlation between translational jump and non-Gaussian diffusion is observed. Therefore, this study proposes a novel approach to elucidate the dynamic heterogeneity and non-Gaussian diffusion in the cell membrane crucial for various cell membrane functionalities.

Universal Evolution of Fickian Non-Gaussian Diffusion in Two- and Three-Dimensional Glass-Forming Liquids.

Recent works show that glass-forming liquids display Fickian non-Gaussian Diffusion, with non-Gaussian displacement distributions persisting even at very long times, when linearity in the mean square displacement (Fickianity) has already been attained. Such non-Gaussian deviations temporarily exhibit distinctive exponential tails, with a decay length λ growing in time as a power-law. We herein carefully examine data from four different glass-forming systems with isotropic interactions, both in two and three dimensions, namely, three numerical models of molecular liquids and one experimentally investigated colloidal suspension. Drawing on the identification of a proper time range for reliable exponential fits, we find that a scaling law λ(t)∝tα, with α≃1/3, holds for all considered systems, independently from dimensionality. We further show that, for each system, data at different temperatures/concentration can be collapsed onto a master-curve, identifying a characteristic time for the disappearance of exponential tails and the recovery of Gaussianity. We find that such characteristic time is always related through a power-law to the onset time of Fickianity. The present findings suggest that FnGD in glass-formers may be characterized by a "universal" evolution of the distribution tails, independent from system dimensionality, at least for liquids with isotropic potential.

Non-Gaussian displacement distributions in models of heterogeneous active particle dynamics

We study the effect of randomly distributed diffusivities and speeds in two models for active particle dynamics with active and passive fluctuations. We demonstrate how non-Gaussian displacement distributions emerge in these models in the long time limit, including Cauchy-type and exponential (Laplace) shapes. Notably the asymptotic behaviours of such Cauchy shapes are universal and do not depend on the precise diffusivity distributions. Moreover, the resulting shapes of the displacement distributions with distributed diffusivities for the active models considered here are in striking contrast to passive diffusion models. For the active motion models our discussion points out the differences between active- and passive-noise. Specifically, we demonstrate that the case with active-noise is in nice agreement with measured data for the displacement distribution of social amoeba.

Heterogeneous run-and-tumble motion accounts for transient non-Gaussian super-diffusion in haematopoietic multi-potent progenitor cells.

Multi-potent progenitor (MPP) cells act as a key intermediary step between haematopoietic stem cells and the entirety of the mature blood cell system. Their eventual fate determination is thought to be achieved through migration in and out of spatially distinct niches. Here we first analyze statistically MPP cell trajectory data obtained from a series of long time-course 3D in vivo imaging experiments on irradiated mouse calvaria, and report that MPPs display transient super-diffusion with apparent non-Gaussian displacement distributions. Second, we explain these experimental findings using a run-and-tumble model of cell motion which incorporates the observed dynamical heterogeneity of the MPPs. Third, we use our model to extrapolate the dynamics to time-periods currently inaccessible experimentally, which enables us to quantitatively estimate the time and length scales at which super-diffusion transitions to Fickian diffusion. Our work sheds light on the potential importance of motility in early haematopoietic progenitor function.

Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile-immobile transport model with Poissonian switching.

We analyse mobile-immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching, we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Specifically, we obtain that, when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile.

An empirical method to characterize displacement distribution functions for anomalous and transient diffusion

We propose a practical empirical fitting function to characterize the non-Gaussian displacement distribution functions (DispD) often observed for heterogeneous diffusion problems. We first test this fitting function with the problem of a colloidal particle diffusing between two walls using Langevin Dynamics (LD) simulations of a raspberry particle coupled to a lattice Boltzmann (LB) fluid. We also test the function with a simple model of anomalous diffusion on a square lattice with obstacles. In both cases, the fitting parameters provide more physical information than just the Kurtosis (which is often the method used to quantify the degree of anomaly of the dynamics), including a length scale that marks where the tails of the DispD begin. In all cases, the fitting parameters smoothly converge to Gaussian values as the systems become less anomalous.

Non-Gaussian distribution of displacements for Lennard-Jones particles in equilibrium

Most mesoscale simulation methods assume Gaussian distributions of velocity-like quantities. These quantities are not true velocities, however, but rather time-averaged velocities or displacements of particles. We show that there is a large range of coarse-graining scales where the assumption of a Gaussian distribution of these displacements fails, and a more complex distribution is required to adequately express these distribution functions of displacements.

Particle Dynamics in Colloid-Polymer Mixtures with Different Polymer Architectures.

Nonadsorbing polymers are widely used as thickening agents for colloids. A quantitative description of the structure and dynamics of such colloid–polymer mixtures is crucial to reveal the mechanisms accounting for the desired mechanical properties. We use confocal microscopy to study colloids with three types of commonly used polymers with different architectures: linear, subgranular cross-linked, and branched microgels. All three thickeners give rise to heterogeneous colloidal dynamics, characterized by non-Gaussian displacement distributions. However, while the ensemble-averaged particle dynamics in these materials are very similar, the underlying individual particle dynamics are not. Linear polymers give rise to depletion attraction and the formation of colloidal gels, in which the majority of particles are immobilized, while a few weakly bound particles have much higher mobility. By contrast, the branched and cross-linked polymers thicken the continuous phase of the colloid, squeezing the particles into dense pockets, where the mobility is reduced and requires more cooperative rearrangements.

Unexpected crossovers in correlated random-diffusivity processes

The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by ‘viscoelastic’ anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of ‘viscoelastic yet non-Gaussian’ diffusion.

Gaussian Processes in Complex Media: New Vistas on Anomalous Diffusion

Normal or Brownian diffusion is historically identified by the linear growth in time of the variance and by a Gaussian shape of the displacement distribution. Processes departing from the at least one of the above conditions defines anomalous diffusion, thus a nonlinear growth in time of the variance and/or a non-Gaussian displacement distribution. Motivated by the idea that anomalous diffusion emerges from standard diffusion when it occurs in a complex medium, we discuss a number of anomalous diffusion models for strongly heterogeneous systems. These models are based on Gaussian processes and characterized by a population of scales, population that takes into account the medium heterogeneity. In particular, we discuss diffusion processes whose probability density function solves space- and time-fractional diffusion equations through a proper population of time-scales or a proper population of length-scales. The considered modelling approaches are: the continuous time random walk, the generalized grey Brownian motion, and the time-subordinated process. The results show that the same fractional diffusion follows from different populations when different Gaussian processes are considered. The different populations have the common feature of a large spreading in the scale values, related to power-law decay in the distribution of population itself. This suggests the key role of medium properties, embodied in the population of scales, in the determination of the proper stochastic process underlying the given heterogeneous medium.

A fluorogenic array for temporally unlimited single-molecule tracking.

We describe three optical tags, ArrayG, ArrayD and ArrayG/N, for intracellular tracking of single molecules over milliseconds to hours. ArrayG is a fluorogenic tag composed of a green fluorescent protein-nanobody array and monomeric wild-type green fluorescent protein binders that are initially dim but brighten ~26-fold on binding with the array. By balancing the rates of binder production, photobleaching and stochastic binder exchange, we achieve temporally unlimited tracking of single molecules. High-speed tracking of ArrayG-tagged kinesins and integrins for thousands of frames reveals novel dynamical features. Tracking of single histones at 0.5 Hz for >1 hour with the import competent ArrayG/N tag shows that chromosomal loci behave as Rouse polymers with visco-elastic memory and exhibit a non-Gaussian displacement distribution. ArrayD, based on a dihydrofolate reductase nanobody array and dihydrofolate reductase-fluorophore binder, enables dual-color imaging. The arrays combine brightness, fluorogenicity, fluorescence replenishment and extended fluorophore choice, opening new avenues for tracking single molecules in living cells.

Anomalous, non-Gaussian, viscoelastic, and age-dependent dynamics of histonelike nucleoid-structuring proteins in live Escherichia coli

The dynamics of a particular protein inside live E. coli bacteria displays unusual behavior, with diffusion coefficients that follow a power law and a non-Gaussian distribution of displacements. The authors also find that the rheological behavior of the cytoplasm differs from that of a homogeneous protein solution, and exhibits a liquid-glass transition.

A model of non-Gaussian diffusion in heterogeneous media

Recent progress in single-particle tracking has shown evidence of the non-Gaussian distribution of displacements in living cells, both near the cellular membrane and inside the cytoskeleton. Similar behavior has also been observed in granular materials, turbulent flows, gels and colloidal suspensions, suggesting that this is a general feature of diffusion in complex media. A possible interpretation of this phenomenon is that a tracer explores a medium with spatio-temporal fluctuations which result in local changes of diffusivity. We propose and investigate an ergodic, easily interpretable model, which implements the concept of diffusing diffusivity. Depending on the parameters, the distribution of displacements can be either flat or peaked at small displacements with an exponential tail at large displacements. We show that the distribution converges slowly to a Gaussian one. We calculate statistical properties, derive the asymptotic behavior and discuss some implications and extensions.

How Heterogeneous Are Trehalose/Glycerol Cryoprotectant Mixtures? A Combined Time-Resolved Fluorescence and Computer Simulation Investigation.

Heterogeneity and molecular motions in representative cryoprotectant mixtures made of trehalose and glycerol are investigated in the temperature range 298 ≤ T (K) ≤ 353, via time-resolved fluorescence Stokes shift and anisotropy measurements, and molecular dynamics simulations of four-point density-time correlations and H-bond relaxations. Mixtures containing 5 and 20 wt % of trehalose along with neat glycerol are studied. Viscosity coefficients for these systems lie in the range 0.30 < η (P) < 23. Measured solute (Coumarin 153) rotation and solvation times reveal a substantial departure from the hydrodynamic viscosity dependence, suggesting the strong microheterogeneous nature of these systems. Fluorescence anisotropy decays are highly nonexponential, reflecting a non-Markovian character of the medium friction. A complete missing of the Stokes shift dynamics in these systems at 298 K but partial detection of it at other higher temperatures (shift magnitude being ∼400-600 cm-1) indicates rigid solute environments. An amorphous solid-like feature emerges in the simulated radial distribution functions at these temperatures. Analyses of mean squared displacements reveal rattling-in-a-cage motion, non-Gaussian displacement distributions, and strong dynamic heterogeneity features. Simulated dynamic structure factors and four-point correlations hint, respectively, at very long α-relaxation and correlated time scales at 298 K. This explains the long solute rotation times (∼80-200 ns) measured at 298 K. Stretched exponential decay of the simulated H-bond relaxations with long time scales further highlights the strong temporal heterogeneity and slow dynamics inherent to these systems. In summary, this work provides the first insight into the molecular motions and interspecies interaction in a representative cryoprotectant mixture, and stimulates further study to investigate the interconnection between cryoprotection and dynamic heterogeneity.

Diffusing diffusivity: a model for anomalous, yet Brownian, diffusion.

Wang et al. [Proc. Natl. Acad. Sci. U.S.A. 106, 15160 (2009)] have found that in several systems the linear time dependence of the mean-square displacement (MSD) of diffusing colloidal particles, typical of normal diffusion, is accompanied by a non-Gaussian displacement distribution G(x,t), with roughly exponential tails at short times, a situation they termed “anomalous yet Brownian” diffusion. The diversity of systems in which this is observed calls for a generic model. We present such a model where there is diffusivity memory but no direction memory in the particle trajectory, and we show that it leads to both a linear MSD and a non-Gaussian G(x,t) at short times. In our model, the diffusivity is undergoing a (perhaps biased) random walk, hence the expression “diffusing diffusivity”. G(x,t) is predicted to be exactly exponential at short times if the distribution of diffusivities is itself exponential, but an exponential remains a good fit for a variety of diffusivity distributions. Moreover, our generic model can be modified to produce subdiffusion.

Constant gradient PFG sequence and automated cumulant analysis for quantifying dispersion in flow through porous media

This paper describes a new variant of established stimulated echo pulse sequences, and an analytical method for determining diffusion or dispersion coefficients for Gaussian or non-Gaussian displacement distributions. The unipolar displacement encoding PFGSTE sequence uses trapezoidal gradient pulses of equal amplitude g and equal ramp rates throughout while sampling positive and negative halves of q-space. Usefully, the equal gradient amplitudes and gradient ramp rates help to reduce the impact of experimental artefacts caused by residual amplifier transients, eddy currents, or ferromagnetic hysteresis in components of the NMR magnet. The pulse sequence was validated with measurements of diffusion in water and of dispersion in flow through a packing of spheres. The analytical method introduced here permits the robust determination of the variance of non-Gaussian, dispersive displacement distributions. The noise sensitivity of the analytical method is shown to be negligible, using a demonstration experiment with a non-Gaussian longitudinal displacement distribution, measured on flow through a packing of mono-sized spheres.

MODEL OF THE FIELD LINE RANDOM WALK EVOLUTION AND APPROACH TO ASYMPTOTIC DIFFUSION IN MAGNETIC TURBULENCE

The turbulent random walk of magnetic field lines plays an important role in the transport of plasmas and energetic particles in a wide variety of astrophysical situations, but most theoretical work has concentrated on determination of the asymptotic field line diffusion coefficient. Here we consider the evolution with distance of the field line random walk using a general ordinary differential equation (ODE), which for most cases of interest in astrophysics describes a transition from free streaming to asymptotic diffusion. By challenging theories of asymptotic diffusion to also describe the evolution, one gains insight on how accurately they describe the random walk process. Previous theoretical work has effectively involved closure of the ODE, often by assuming Corrsin's hypothesis and a Gaussian displacement distribution. Approaches that use quasilinear theory and prescribe the mean squared displacement Δx 2 according to free streaming (random ballistic decorrelation, RBD) or asymptotic diffusion (diffusive decorrelation, DD) can match computer simulation results, but only over specific parameter ranges, with no obvious marker of the range of validity. Here we make use of a unified description in which the ODE determines Δx 2 self-consistently, providing a natural transition between the assumptions of RBD and DD. We find that the minimum kurtosis of the displacement distribution provides a good indicator of whether the self-consistent ODE is applicable, i.e., inaccuracy of the self-consistent ODE is associated with non-Gaussian displacement distributions.

Decoupling of rotational and translational diffusion in supercooled colloidal fluids

We use confocal microscopy to directly observe 3D translational and rotational diffusion of tetrahedral clusters, which serve as tracers in colloidal supercooled fluids. We find that as the colloidal glass transition is approached, translational and rotational diffusion decouple from each other: Rotational diffusion remains inversely proportional to the growing viscosity whereas translational diffusion does not, decreasing by a much lesser extent. We quantify the rotational motion with two distinct methods, finding agreement between these methods, in contrast with recent simulation results. The decoupling coincides with the emergence of non-Gaussian displacement distributions for translation whereas rotational displacement distributions remain Gaussian. Ultimately, our work demonstrates that as the glass transition is approached, the sample can no longer be approximated as a continuum fluid when considering diffusion.

A cumulant analysis for non-Gaussian displacement distributions in Newtonian and non-Newtonian flows through porous media

We use displacement encoding pulsed field gradient (PFG) nuclear magnetic resonance to measure Fourier components S(q) of flow displacement distributions P(zeta) with mean displacement (zeta) for Newtonian and non-Newtonian flows through rocks and bead packs. Displacement distributions are non-Gaussian; hence, there are finite terms above second order in the cumulant expansion of ln(S(q)). We describe an algorithm for an optimal self-consistent cumulant analysis of data, which can be used to obtain the first three (central) moments of a non-Gaussian P(zeta), with error bars. The analysis is applied to Newtonian and non-Newtonian flows in rocks and beads. Flow with shear-thinning xanthan solution produces a 15.6+/-2.3% enhancement of the variance sigma(2) of displacement distributions when compared to flow experiments with water.