non-Brownian Diffusion Research Articles

Exact Solutions of Systems of Nonlinear Time-Space Fractional Partial Differential Equations Using an Iterative Method

Abstract Fractional partial differential equations are useful tools to describe transportation, anomalous, and non-Brownian diffusion. In the present paper, we propose the Daftardar-Gejji and Jafari method along with its error analysis for solving systems of nonlinear time–space fractional partial differential equations (PDEs). Moreover, we solve a variety of nontrivial time–space fractional systems of PDEs. The obtained solutions either occur in exact form or in the form of a series, which converges to a closed form. The proposed method is free from linearization and discretization and does not include any tedious calculations. Moreover, it is easily employable using the computer algebra system such as Mathematica, Maple, etc.

Nobias: analyzing anomalous diffusion in single-molecule tracks with nonparametric Bayesian inference

Single-molecule imaging techniques and single-particle tracking (SPT) of molecules inside living cells investigate biomolecular dynamics at nanometer spatial resolution and millisecond temporal resolution. However, the complexity of living cells requires rich statistical approaches to deal with heterogeneous dynamics and complex diffusion models. To achieve these goals, we have developed a new SPT analysis framework: NOBIAS (Nonparametric Bayesian Inference for Anomalous Diffusion in Single-Molecule Tracking), which combines nonparametric Bayesian statistics and recurrent neural network (RNN) methods to deal with SPT datasets that contain heterogeneous dynamics with a mixture of unknown number of diffusive states, even within a single trajectory, asymmetrical motion of target biomolecules, and non-Brownian anomalous diffusion.

Non-Brownian Interfacial Diffusion: Flying, Hopping, and Crawling

In theoretical work beginning in the 1990s, O’Shaughnessy and co-workers predicted that for lateral diffusion at a solid-liquid interface, a key step involves desorption, excursion through the liqu...

Measuring Hindered Diffusion Dynamics in Live Cell Plasma Membranes with Confocal and Super-Resolution Imaging

The cellular plasma membrane is a highly heterogeneous structure organised on nano-scales and displays a crucial interaction platform for proteins, lipids and soluble ligands. Investigating the molecular membrane organisation by measuring diffusion dynamics offers a better understanding of its biological function. Fluorescence correlation spectroscopy (FCS) is one of the prominent tools to elucidate these dynamics in living cells but can only report on the dynamics at one given spatial position at a time. Using scanning fluorescence correlation spectroscopy (sFCS), we obtain a multitude of FCS measurements at different spatial locations. Here, we present a statistical analysis pipeline for sFCS data which allows for the accurate determination of the diffusion dynamics and the differentiation of free (Brownian) from hindered (non-Brownian) diffusion modes. We show free diffusion for phospholipids in model membranes and cells but reveal hindered diffusion of sphingolipids and GPI-anchored proteins in cells. Notably, these measurements can be performed using standard fluorescent dyes or proteins on a conventional confocal laser scanning microscope. To further investigate the dynamics on single cell level, we combine sFCS with stimulated emission depletion (STED) microscopy and by alternating conventional and super-resolved excitation we introduce line interleaved excitation scanning STED-FCS (LIESS-FCS). With LIESS-FCS the diffusion modes can be directly determined at multiple spots within the cellular plasma membrane providing detailed insights into organisation and function. Overall we are presenting a novel toolkit to investigate nano-scale molecular diffusion dynamics for shedding a new light on membrane organisation and heterogeneity.

Membrane Diffusion Occurs by Continuous-Time Random Walk Sustained by Vesicular Trafficking

Diffusion in cellular membranes is regulated by processes that occur over a range of spatial and temporal scales. These processes include membrane fluidity, interprotein and interlipid interactions, interactions with membrane microdomains, interactions with the underlying cytoskeleton, and cellular processes that result in net membrane movement. The complex, non-Brownian diffusion that results from these processes has been difficult to characterize, and moreover, the impact of factors such as membrane recycling on membrane diffusion remains largely unexplored. We have used a careful statistical analysis of single-particle tracking data of the single-pass plasma membrane protein CD93 to show that the diffusion of this protein is well described by a continuous-time random walk in parallel with an aging process mediated by membrane corrals. The overall result is an evolution in the diffusion of CD93: proteins initially diffuse freely on the cell surface but over time become increasingly trapped within diffusion-limiting membrane corrals. Stable populations of freely diffusing and corralled CD93 are maintained by an endocytic/exocytic process in which corralled CD93 is selectively endocytosed, whereas freely diffusing CD93 is replenished by exocytosis of newly synthesized and recycled CD93. This trafficking not only maintained CD93 diffusivity but also maintained the heterogeneous distribution of CD93 in the plasma membrane. These results provide insight into the nature of the biological and biophysical processes that can lead to significantly non-Brownian diffusion of membrane proteins and demonstrate that ongoing membrane recycling is critical to maintaining steady-state diffusion and distribution of proteins in the plasma membrane.

Hydrodynamic diffusion in active microrheology of non-colloidal suspensions: the role of interparticle forces

Hydrodynamic diffusion in the absence of Brownian motion is studied via active microrheology in the ‘pure-hydrodynamic’ limit, with a view towards elucidating the transition from colloidal microrheology to the non-colloidal limit, falling-ball rheometry. The phenomenon of non-Brownian force-induced diffusion in falling-ball rheometry is strictly hydrodynamic in nature; in contrast, analogous force-induced diffusion in colloids is deeply connected to the presence of a diffusive boundary layer even when Brownian motion is very weak compared with the external force driving the ‘probe’ particle. To connect these two limits, we derive an expression for the force-induced diffusion in active microrheology of hydrodynamically interacting particles via the Smoluchowski equation, where thermal fluctuations play no role. While it is well known that the microstructure is spherically symmetric about the probe in this limit, fluctuations in the microstructure need not be – and indeed lead to a diffusive spread of the probe trajectory. The force-induced diffusion is anisotropic, with components along and transverse to the line of external force. The latter is identically zero owing to the fore–aft symmetry of pair trajectories in Stokes flow. In a naïve first approach, the vanishing relative hydrodynamic mobility at contact between the probe and an interacting bath particle was assumed to eliminate all physical contribution from interparticle forces, whereby advection alone drove structural evolution in pair density and microstructural fluctuations. With such an approach, longitudinal force-induced diffusion vanishes in the absence of Brownian motion, a result that contradicts well-known experimental measurements of such diffusion in falling-ball rheometry. To resolve this contradiction, the probe–bath-particle interaction at contact was carefully modelled via an excluded annulus. We find that interparticle forces play a crucial role in encounters between particles in the hydrodynamic limit – as they must, to balance the advective flux. Accounting for this force results in a longitudinal force-induced diffusion $D_{\Vert }=1.26aU_{S}{\it\phi}$, where $a$ is the probe size, $U_{S}$ is the Stokes velocity and ${\it\phi}$ is the volume fraction of bath particles, in excellent qualitative and quantitative agreement with experimental measurements in, and theoretical predictions for, macroscopic falling-ball rheometry. This new model thus provides a continuous connection between micro- and macroscale rheology, as well as providing important insight into the role of interparticle forces for diffusion and rheology even in the limit of pure hydrodynamics: interparticle forces give rise to non-Newtonian rheology in strongly forced suspensions. A connection is made between the flow-induced diffusivity and the intrinsic hydrodynamic microviscosity which recovers a precise balance between fluctuation and dissipation in far from equilibrium suspensions; that is, diffusion and drag arise from a common microstructural origin even far from equilibrium.

Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions

We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction ϕ of the crowders and the crowder–crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational—but not rotational—diffusivity of the particles D is a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law with the area fraction ϕ occupied by the crowders and the critical value Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.

Pore Formation in a Membrane Submitted to High Voltages. Influence of the Membrane Vicosity

We present here a new model of pore formation based on physical considerations of membrane energy. The idea is to revisit the pore expansion theory, as described by Weaver, Chizmadzhev et al. [4] in the 90's. We first recover the curvature-driven closure of the pore as obtain by Kroeger et al. [2], thanks to Langevin equation written on the pore area instead of the pore raldius. Then we consider the influence of the membrane viscosity around the region where pores have already appeared. This changes in membrane viscosity have been reported in the experiments of Tsai et al. [3] Around the pore region, the mobility of the lipids is constraint, which makes the membrane viscosity decrease locally. Then, once a critical pore radius is reached, it costs more to enlarge the pore, than to create another pore elsewhere. This has also an influence on the number of pores, which are hardly created in the high viscocity regions. Our model avoids an important drawback of the previsous models: the pore radius and the pore density cannot grow infinitely unlike the model of Krassowska, Neu et al [1]. After the derivation of the equations, we will present numerical simulations that corroborate quantitatively the experimental data obtained by patch-clamp experiments.[1] K. DeBruin et al. Modelling electroporation in a single cell. Biophysical Journal, 77, 1999.[2] Jens H Kroeger, et al.Curvature-driven pore growth in charged membranes during charge-pulse and voltage-clamp experiments. Biophysical journal, 96(3), 2009.[3] J. Tsai, et al. Non-brownian diffusion of membrane molecules in nanopatterned supported lipid bilayers. Nano Letters, 8(2), 2008.[4] J.C. Weaver and Y.A. Chimazdzhev. Theory of electroporation: A review. Bioelectrochemistry and Bioenergetics, 41, 1996.

A Toolbox of Fluorescence Microscopic Approaches Reveals Dynamics and Assembly of a Membrane-Associated Protein

A Toolbox of Fluorescence Microscopic Approaches Reveals Dynamics and Assembly of a Membrane-Associated Protein

Poly-ethylene glycol induced super-diffusivity in lipid bilayer membranes

Fluorescence lifetime correlation spectroscopy (FLCS) has been used to probe the influence of PEG-8000 on the fluidity of fluorescently labeled planar supported lipid bilayers on ozone plasma treated glass. The lipid membrane compositions examined were; DOPC, DOPC/DOPS (80/20 mol/mol) and DMPC, with and without cholesterol. The lateral diffusion coefficients (D) for supported lipid bilayer films of these layers without cholesterol were 7.9 ± 0.2, 7.9 ± 0.4 and 5.5 ± 0.1 μm2 s−1 respectively. The high fluidity reflected the super-hydrophilicity (contact angle of 0) of the ozone treated plasma glass substrate. Using DOPE conjugated Atto 655 as a probe, exposure of the lipid bilayer to a 30% wt/wt aqueous solution of PEG, followed by washing, dramatically increased the diffusion coefficients of the probe within the film. For example, the diffusion coefficient for the DOPC bilayer increases by nearly an order of magnitude to 51.4 ± 2.6 μm2 s−1. The autocorrelation curves for DOPC/DOPS (80/20 mol/mol) and DMPC bilayers required a two-component model for adequate fit of their behaviour yielding both fast and slow components of the diffusion. In all cases, when hydrophilic DOPE–Atto 655 was used as the probe, treatment of the lipid bilayer with PEG resulted in non-Brownian diffusion. Importantly, the observed diffusion behavior observed depends on the identity of probe. In contrast, when a hydrophobic probe (DOPE–NapthBodipy) was employed PEG showed relatively little impact on the observed diffusion rates. This was attributed to orientation of the reporter probe in the lipid bilayer and its aqueous interface. Specifically, DOPE–Atto 655 is believed to associate strongly with PEG mesh at the aqueous interface of the lipid bilayer, its diffusion strongly influenced by the structure in this region, whereas DOPE–NapthBodipy remains in the interior of the bilayer where it is relatively uninfluenced by PEG.

Lattice models of nonequilibrium bacterial dynamics

We study a model of self-propelled particles exhibiting run-and-tumble dynamics on alattice. This non-Brownian diffusion is characterized by a random walk with a finitepersistence length between changes of direction and is inspired by the motion of bacteriasuch as E. coli. By defining a class of models with multiple species of particles andtransmutation between species we can recreate such dynamics. These models admit exactanalytical results whilst also forming a counterpart to previous continuum models ofrun-and-tumble dynamics. We solve the externally driven non-interacting andzero-range versions of the model exactly and utilize a field-theoretic approach toderive the continuum fluctuating hydrodynamics for more general interactions. Wemake contact with prior approaches to run-and-tumble dynamics off lattice anddetermine the steady state and linear stability for a class of crowding interactions,where the jump rate decreases as density increases. In addition to its interestfrom the perspective of nonequilibrium statistical mechanics, this lattice modelconstitutes an efficient tool to simulate a class of interacting run-and-tumble modelsrelevant to bacterial motion, so long as certain conditions (that we derive) aremet.

Protein–DNA interactions in high speed AFM: single molecule diffusion analysis of human RAD54

High-speed AFM (atomic force microscopy also called scanning force microscopy) provides nanometre spatial resolution and sub-second temporal resolution images of individual molecules. We exploit these features to study diffusion and motor activity of the RAD54 DNA repair factor. Human RAD54 functions at critical steps in recombinational-DNA repair. It is a member of the Swi2/Snf2 family of chromatin remodelers that translocate on DNA using ATP hydrolysis. A detailed single molecular description of DNA-protein interactions shows intermediate states and distribution of variable states, usually hidden by ensemble averaging. We measured the motion of individual proteins using single-particle tracking and observed that random walks were affected by imaging-buffer composition. Non-Brownian diffusion events were characterized in the presence and in the absence of nucleotide cofactors. Double-stranded DNA immobilized on the surface functioned as a trap reducing Brownian motion. Distinct short range slides and hops on DNA were visualized by high-speed AFM. These short-range interactions were usually inaccessible by other methods based on optical resolution. RAD54 monomers displayed a diffusive behavior unrelated to the motor activity.

Non-Brownian Diffusion of Membrane Molecules in Nanopatterned Supported Lipid Bilayers

Molecules associated with the outer surface of living cells exhibit complex, non-Brownian patterns of diffusion. In this report, supported lipid bilayers were patterned with nanoscale barriers to capture key aspects of this anomalous diffusion in a controllable format. First, long-range diffusion coefficients of membrane-associated molecules were significantly reduced by the presence of the barriers, while short-range diffusion was unaffected. Second, this modulation was more pronounced for large molecular complexes than for individual lipids. Surprisingly, the quantitative effect of these barriers on long-range lipid diffusion could be accurately simulated using a simple, continuum-based model of diffusion on a nanostructured surface; we thus describe a metamaterial that captures the properties of the outer membrane of living cells.

Detection of Non-Brownian Diffusion in the Cell Membrane in Single Molecule Tracking

Molecules undergo non-Brownian diffusion in the plasma membrane, but the mechanism behind this anomalous diffusion is controversial. To characterize the anomalous diffusion in the complex system of the plasma membrane and to understand its underlying mechanism, single-molecule/particle methods that allow researchers to avoid ensemble averaging have turned out to be highly effective. However, the intrinsic problems of time-averaging (resolution) and the frequency of the observations have not been explored. These would not matter for the observations of simple Brownian particles, but they do strongly affect the observation of molecules undergoing anomalous diffusion. We examined these effects on the apparent motion of molecules undergoing simple, totally confined, or hop diffusion, using Monte Carlo simulations of particles undergoing short-term confined diffusion within a compartment and long-term hop diffusion between these compartments, explicitly including the effects of time-averaging during a single frame of the camera (exposure time) and the frequency of observations (frame rate). The intricate relationships of these time-related experimental parameters with the intrinsic diffusion parameters have been clarified, which indicated that by systematically varying the frame time and rate, the anomalous diffusion can be clearly detected and characterized. Based on these results, single-particle tracking of transferrin receptor in the plasma membrane of live PtK2 cells were carried out, varying the frame time between 0.025 and 33ms (0.03–40kHz), which revealed the hop diffusion of the receptor between 47-nm (average) compartments with an average residency time of 1.7ms, with the aid of single fluorescent-molecule video imaging.

Anomalous coalescence from a nonlinear Schroedinger equation with a quintic term: interpretation through Thompson's approach

Inspired by models for A+ A→ A(0) reactions with non-Brownian diffusion, we suggest a possible analytical explanation for the phenomena of anomalous coalescence of bubbles found in one-dimension (1d) by Josserand and Rica through numerical work [Phys. Rev. Letters 78 (1997) 1215]. The explanation firstly requires an exponent γ, which is sometimes used to describe anomalous diffusion. Here it displays an explicit dependence on the dimensionality ( γ= γ( d)=4/ d for d⩽2). So we have d c =2, coinciding with the upper critical dimension of A+ A→ A(0) reactions (Mod. Phys. Lett. B 13 (1999) 829; Mod. Phys. Lett. B 15(26) (2001) 1205) with Brownian diffusion condition ( γ=2). Thus anomalous coalescence emerges, only below the critical dimension ( d<2). We show that the typical size of the structures (bubbles) grows as R( t)∼ t 1/4 in 1d. An alternative explanation could also be thought as a diffusion constant D which depends on the average concentration (〈 n〉), namely D= D 0〈 n〉 α . It is introduced into an effective action for A+ A→ A(0) reactions. Therefore we are also able to reproduce the anomalous behavior for n( t) and R( t) in 1d, being α=0 for d⩾2 (mean field behavior) and α=2(2− d)/ d 2 for d⩽2.

Low-frequency structural plasma turbulence in the L-2M stellarator

Experiments on the L-2M stellarator have shown the occurrence of steady-state low-frequency strong structural (LFSS) turbulence throughout the entire plasma column. A key feature of LFSS turbulence is the presence of stochastic plasma structures. It is shown that different types of LFSS turbulence are correlated throughout the entire plasma volume. Stable non-Gaussian probability density distributions of all of the fluctuating plasma parameters are measured. The characteristic spatial and time scales of LFSS turbulence, which is responsible for non-Brownian diffusion in plasma, are determined.

Asymmetric transport and non-Gaussian statistics of passive scalars in vortices in shear

Transport of passive scalars in a chain of vortices in a shear layer is studied using a model motivated by the quasigeostrophic equation, and a discrete map model. Surrounding the vortices there is a stochastic layer where particles alternate chaotically between being trapped in the vortices, and moving following the shear flow. Transport in the stochastic layer is asymmetric: Mixing between the vortices and the up-stream flow is, in general, different from mixing between the vortices and the down-stream flow. We use the Melnikov method to study this asymmetry, and to construct a generalized separatrix map model for asymmetric transport. The statistics of the passive scalar is non-Gaussian. In particular, there is anomalous advection, and anomalous (non-Brownian) diffusion. Thus, transport in this system cannot be described by an advection-diffusion equation with an effective diffusivity. The probability density function (PDF) of particle displacements δx, P(δx,t), is asymmetric and broader than Gaussian. At large times, P relaxes to a self-similar limit distribution of the form t−γ/2f(X/tγ/2), where X≡δx−〈δx〉, f is a scaling function, and γ is the anomalous diffusion exponent. As a result, the moments scale as 〈Xn〉∼tnγ/2. We present a systematic study of the dependence of the mean, the variance, the skewness, and the flatness, on the parameters controlling the asymmetry of the flow. The PDFs of the duration of flight (motion following the shear flow) events, and vortex trapping events, exhibit algebraic decay. In some cases, the flights correspond to Lévy flights. The results of the model are compared with recent experiments on chaotic advection and Lévy flights in a rotating annulus.

250-GHz EPR of nitroxides in the slow-motional regime: models of rotational diffusion

A 250-GHz electron paramagnetic resonance (EPR) study of the slow rotational diffusion of two spin probes in toluene, viz., perdeuterated 2,2,6,6-tetramethyl-4-piperidone (PDT) and 3-doxylcholestane (CSL) is presented. EPR spectra were obtained in the slow-motional and near-rigid limit regions, which corresponds to rotational correlation times 10 -10 >τ R >10 -6 s. These two probes differ significantly in size and shape, permitting a detailed exploration of the sensitivity of 250-GHz EPR to different aspects of the molecular dynamics such as rotational anisotropy and non-Brownian diffusion. Nonlinear least-squares fitting based on full stochastic Liouville calculations provides a sensitive means for discriminating amongst motional models

Quantitation of non-einstein diffusion behavior of water in biological tissues by proton MR diffusion imaging: Synthetic image calculations

The non-Einstein diffusion behavior of water in a model biological tissue system, intact duck embryos, has been investigated by the use of an in vivo proton pulsed-gradient spin-echo (PGSE) MR imaging technique. Multipleframe MR images of the intact duck embryos and control solution (0.5 mM CuSO 4 doped water) were acquired systematically at different diffusion times and strengths of the diffusion-sensitizing magnetic field gradients of the PGSE sequence. These raw images were then used to generate various dynamic (self-diffusion coefficient) and structural (fractal, residual attenuation, and compartment fraction) diffusion parameter maps of water in the imaging objects on the basis of different Einstein and higher order (non-Brownian, Residual, and 2-compartment) diffusion models. The self-diffusion coefficients of the body tissues of the embryos obtained from all diffusion models were significantly lower than those of the surrounding embryonic fluid. The structural diffusion parameter maps obtained from the higher order diffusion models revealed that water molecules exhibited either non-Brownian, restricted, or compartmentalized diffusion behavior in the embryonic tissues, but Einstein or Brownian diffusion behavior in the embryonic fluid and control solution. The diffusion parameter maps, both dynamic and structural, were found to provide much better contrasts than the conventional relaxation time ( T 1, T 2, and biexponential T 2) maps in separating the tissues from the surrounding embryonic fluid in the duck embryos. The mathematical models and procedures for generating the dynamic and structural diffusion parameter maps are also presented in this paper.