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.     

Lateral diffusion of molecules in two-component lipid bilayer: a Monte Carlo simulation study

Lateral diffusion of membrane components makes possible any in-plane membrane reaction and has a key role in signaling in cell membranes. In this report the equilibrium lateral diffusion of intrinsic molecules in an equimolar DMPC/DSPC mixture is simulated using a thoroughly tested two-state model of two-component phospholipid bilayers. The model has been successful in calculating the excess heat capacity function, the most frequent center-to-center distances between DSPC clusters, and the fractal dimensions of gel clusters (Sugar, I. P., Thompson, T. E., Biltonen, R. L. Biophys. J. 1999, 76, 2099-2110). In the gel/fluid mixed phase region, a diffusing intrinsic molecule may change its state from fluid to gel (or from gel to fluid) at any time. A common characterization of the diffusion of intrinsic molecules is given by the simulated average first-passage time curves. We find that these curves can be described as power functions containing two parameters, alpha and beta, except near the percolation threshold of gel/fluid or compositional clusters. We find also that the intrinsic molecules are involved in approximately normal diffusion, i.e., beta approximately 2 in the extreme gel and fluid phase regions, while in the gel/fluid and gel/gel mixed phase regions the diffusion is anomalous, i.e., beta not equal 2. In the mixed phase regions, when the initial local state of the diffusing molecule is not specified, each component is involved in sub-diffusion (beta > 2). In the gel/fluid mixed phase region molecules situated initially inside a fluid cluster are involved in sub-diffusion, but DMPC molecules situated initially inside a gel cluster are involved in super-diffusion (beta < 2). The possibility of anomalous diffusion in membranes apparently arises because the diffusing molecule visits a variety of different environments characterized by its relative proximity to various membrane components. The diffusion is actually anomalous when the components of the bilayer are nonrandomly distributed. The deviation from random distribution is strongly correlated with beta. Similar to the results of the NMR experiments, the calculated relative diffusion coefficient continuously decreases in the gel/fluid mixed phase region with decreasing temperature. In apparent contradiction, diffusion measured by fluorescence recovery after photobleaching (FRAP) demonstrates the existence of a threshold temperature, below which long-range diffusion of FRAP probe molecules is essentially blocked. This threshold temperature is highly correlated with the percolation temperature of gel clusters.     

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.     

Movement of proteins in an environment crowded by surfactant micelles: anomalous versus normal diffusion

Small proteins move in crowded cell compartments by anomalous diffusion. In many of them, e.g., the endoplasmic reticulum, the proteins move between lipid membranes in the aqueous lumen. Molecular crowding in vitro offers a systematic way to study anomalous and normal diffusion in a well controlled environment not accessible in vivo. We prepared a crowded environment in vitro consisting of hexaethylene glycol monododecyl ether (C(12)E(6)) nonionic surfactant and water and observed lysozyme diffusion between elongated micelles. We have fitted the data obtained in fluorescence correlation spectroscopy using an anomalous diffusion model and a two-component normal diffusion model. For a small concentration of surfactant (below 4 wt %) the data can be fitted by single-component normal diffusion. For larger concentrations the normal diffusion fit gave two components: one very slow and one fast. The amplitude of the slow component grows with C(12)E(6) concentration. The ratio of diffusion coefficients (slow to fast) is on the order of 0.1 for all concentrations of surfactant in the solution. The fast diffusion is due to free proteins while the slow one is due to the protein-micelle complexes. The protein-micelle interaction is weak since even in a highly concentrated solution (35% of C(12)E(6)) the amplitude of the slow mode is only 10%, despite the fact that the average distance between the micelles is the same as the size of the protein. The anomalous diffusion model gave the anomaly index (r(2)(t) approximately t(alpha)), alpha monotonically decreasing from alpha = 1 (at 4% surfactant) to alpha = 0.88 (at 37% surfactant). The fits for two-component normal diffusion and anomalous diffusion were of equally good quality, but the physical interpretation was only straightforward for the former.     

Spatio-temporal anomalous diffusion in heterogeneous media by nuclear magnetic resonance

In this paper, we describe nuclear magnetic resonance measurements of water diffusion in highly confined and heterogeneous colloidal systems using an anomalous diffusion model. For the first time, temporal and spatial fractional exponents, α and μ, introduced within the framework of continuous time random walk, are simultaneously measured by pulsed gradient spin-echo NMR technique in samples of micro-beads dispersed in aqueous solution. In order to mimic media with low and high level of disorder, mono-dispersed and poly-dispersed samples are used. We find that the exponent α depends on the disorder degree of the system. Conversely, the exponent μ depends on both bead sizes and magnetic susceptibility differences within samples. The new procedure proposed here may be a useful tool to probe porous materials and microstructural features of biological tissue.     

Anomalous diffusion in supported lipid bilayers induced by oxide surface nanostructures

Hierarchic structure and anomalous diffusion on submicrometer scale were introduced into an artificial cell membrane, and the spatiotemporal dependence of lipid diffusion was visualized on nanostructured oxide surfaces. We observed the lipid diffusion in supported lipid bilayers (SLBs) on step-and-terrace TiO(2)(100) and amorphous SiO(2)/Si surfaces by single molecule tracking (SMT) method. The SMT at the time resolution of 500 μs to 30 ms achieved observation of the lipid diffusion over the spatial and temporal ranges of 100 nm/millisecond to 1 μm/second. The temporal dependence of the diffusion coefficient in the SLB on TiO(2)(100) showed that the crossover from anomalous diffusion to random diffusion occurred around 10 ms. The surface fine architecture on substrates will be applicable to induce hierarchic structures on the order of 100 nm or less, which correspond to the microcompartment size in vivo.     

Non-fickian diffusion in thin polymer films

Diffusion of penetrants through polymers often does not follow the standard Fickian model. Such anomalous behavior can cause difficulty when designing polymer networks for specific uses. One type of non-Fickian behavior that results is so-called case II diffusion, where Fickian-like fronts initially move like √t with a transition to a non-Fickian concentration profile and front speed for moderate time. A mathematical model is presented that replicates this behavior in thin polymer films, and an analysis is performed that yields relevant dimensionless groups for study. An unusual result is derived: In certain parameter ranges, the concentration profile can change concavity, reflecting Fickian behavior for short times and non-Fickian behavior for moderate times. Asymptotic and numerical results are then obtained to characterize the dependence of such relevant quantities as failure time, front speed, and mass transport on these dimensionless groups. This information can aid in the design of effective polymer protectant films. © 1996 John Wiley & Sons, Inc.     

Interpretation of diffusion coefficients in nanostructured materials from random walk numerical simulation

We make use of the numerical simulation random walk (RWNS) method to compute the "jump" diffusion coefficient of electrons in nanostructured materials via mean-square displacement. First, a summary of analytical results is given that relates the diffusion coefficient obtained from RWNS to those in the multiple-trapping (MT) and hopping models. Simulations are performed in a three-dimensional lattice of trap sites with energies distributed according to an exponential distribution and with a step-function distribution centered at the Fermi level. It is observed that once the stationary state is reached, the ensemble of particles follow Fermi-Dirac statistics with a well-defined Fermi level. In this stationary situation the diffusion coefficient obeys the theoretical predictions so that RWNS effectively reproduces the MT model. Mobilities can be also computed when an electrical bias is applied and they are observed to comply with the Einstein relation when compared with steady-state diffusion coefficients. The evolution of the system towards the stationary situation is also studied. When the diffusion coefficients are monitored along simulation time a transition from anomalous to trap-limited transport is observed. The nature of this transition is discussed in terms of the evolution of electron distribution and the Fermi level. All these results will facilitate the use of RW simulation and related methods to interpret steady-state as well as transient experimental techniques.     

Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.     

Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell

The electrical response of an electrolytic cell in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order is analyzed. The boundary conditions at the electrodes limiting the sample are described by an integro-differential equation governing the kinetic at the interface. The analysis is carried out by supposing that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies the Poisson's equation. The results cover a rich variety of scenarios, including the ones connected to anomalous diffusion.     

On the use of NMR pulsed field-gradient spectroscopy for the study of anomalous diffusion in fractal networks

Theoretical questions of NMR spin echo attenuation due to diffusion in fractal systems are examined. The mean-square displacement of anomalous diffusion depends sublinearly on time. Therefore, this stochastic process is not invariant against time translation. The correct adjustment of the time scale is a crucial prerequisite for the proper reflection of molecular migration by means of spin-echo experiments.     

Criteria for diffusion limitation in coordination polymerization

The following criteria are proposed to judge whether a coordination polymerization may be diffusion controlled or not: (1) If the number-average molecular weight and polydispersity of the polymer calculated from kinetic rate constants as a function of time agree with the experimental values, the polymerization is not diffusion controlled. (2) The polymerization may be diffusion controlled if the Thiele modulus, the ratio of the characteristic diffusion time to the characteristic reaction time, is much greater than unity; if it is much smaller than unity, the polymerization is reaction controlled. (3) If an initial linear dependence of rate of polymerization on catalyst concentration changes over to a square-root dependence, the polymerization may be diffusion limited. (4) The polymerization is likely to be diffusion limited if the instantaneous rate of polymerization is proportional to the rate of particle growth when the proportionality coefficient is the surface area of the particle. Criterion (1) is a necessary and sufficient condition as stated, as its converse is not true. All the other criteria are merely necessary but not sufficient conditions. The established Ziegler–Natta catalysts have activities too low to cause diffusion limitation; the Phillips catalyst system is likely to be diffusion limited. The polydispersity of polyolefins produced with Ziegler–Natta catalysts are not the consequence of diffusion control but are the characteristics of the catalysts in their kinetics of initiation, propagation, chain transfer, and termination.     

A new FTIR-ATR cell for drug diffusion studies

The drug diffusion of most compounds, particularly hydrophilic molecules through the skin is limited by the permeation of the outermost cell layers of the epidermis, the stratum corneum(SC). For this reason it is of interest to characterize drug diffusion processes through this skin layer. A new FTIR-ATR cell was developed for non-invasive real time measurements of drug diffusion. The diffusion of water through an artificial polyethyleneglycol-polydimethylsiloxane membrane was studied. Additionally the diffusion of urea in human SC was analyzed. Based on a mathematical model the diffusion coefficients were derived. We could reveal that this cell associates the advantages of the Franz diffusion cell and the FTIR-ATR spectroscopy as a new powerful method for determining drug diffusion through biological membranes.     

An experimental and theoretical analysis of molecular separations by diffusion through ultrathin nanoporous membranes

Diffusion based separations are essential for laboratory and clinical dialysis processes. New molecularly thin nanoporous membranes may improve the rate and quality of separations achievable by these processes. In this work we have performed protein and small molecule separations with 15 nm thick porous nanocrystalline silicon (pnc-Si) membranes and compared the results to 1- and 3- dimensional models of diffusion through ultrathin membranes. The models predict the amount of resistance contributed by the membrane by using pore characteristics obtained by direct inspection of pnc-Si membranes in transmission electron micrographs. The theoretical results indicate that molecularly thin membranes are expected to enable higher resolution separations at times before equilibrium compared to thicker membranes with the same pore diameters and porosities. We also explored the impact of experimental parameters such as porosity, pore distribution, diffusion time, and chamber size on the sieving characteristics. Experimental results are found to be in good agreement with the theory, and ultrathin membranes are shown to impart little overall resistance to the diffusion of molecules smaller than the physical pore size cutoff. The largest molecules tested experience more hindrance than expected from simulations indicating that factors not incorporated in the models, such as molecule shape, electrostatic repulsion, and adsorption to pore walls, are likely important.     

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.     

Anomalous penetrant diffusion as a probe of the local structure in a blend of poly(ethylene oxide) and poly(methyl methacrylate)

Pulse field gradient (PFG) NMR measurements have been made to study the diffusion of diethyl ether in blends of poly(ethylene oxide) (PEO) and poly(methyl methacrylate) (PMMA). The blends have 10–30 wt % PEO, a composition range within which these materials are amorphous glasses. The diffusion of diethyl ether through the blends is quite rapid, with diffusion constants in the range of 10^?7 to 10^?8 cm²/s. In PFG NMR experiments, the apparent diffusion constant depends on the timescale over which diffusion is observed. The values decrease to a plateau as the time increases, this being the signature of tortuous diffusion. Tortuous diffusion is usually observed in heterogeneous systems in which there are regions that support fast diffusion and regions that support slow diffusion or act as barriers. In these blends, PEO is known to undergo rapid segmental motion typical of a rubbery state well below the glass transition, whereas the segmental motion of PMMA is slower by many orders of magnitude. Mobile PEO provides a pathway for the diffusion of structurally similar diethyl ether, whereas solid‐like PMMA acts as a barrier. The size of the domains can be estimated either from a lattice model or from equations for tortuous diffusion. Micrometer sizes are indicated that are unexpectedly large, given the size of the polymer chains and the size of the concentration fluctuations, both of which are thought to be in the tens of nanometers. The lattice model and the equations for tortuous diffusion assume a random dispersion of impenetrable or less penetrable objects. This may not be the appropriate morphology for the diffusion pathway. Recently, large sizes have been indicated by PFG NMR experiments, in which a penetrant is thought to diffuse in a curvilinear fashion. In these blends, the pathway for diethyl ether is along the PEO backbone. A plot of the logarithm of the mean‐square displacement versus the logarithm of time has a slope of about 0.6, close to the value of 0.5 for pure curvilinear diffusion. Exponents with values in this range can also be associated with diffusion in a fractal space, which, in this situation, still consists of mobile PEO. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1053–1067, 2004     

Heterogeneous and anomalous diffusion inside lipid tubules

Self-assembled lipid tubules with crystalline bilayer walls are promising candidates for controlled drug delivery vehicles on the basis of their ability to release preloaded biological molecules in a sustained manner. While a previous study has shown that the release rate of protein molecules from lipid tubules depends on the associated molecular mass, suggesting that the pertinent diffusion follows the well-known Stokes-Einstein relationship, only a few attempts have been made toward investigating the details of molecular diffusion in the tubule interior. Herein, we have characterized the diffusion rates of several molecules encapsulated in lipid tubules formed by 1,2-bis(10,12-tricosadiynoyl)-sn-glycero-3-phosphocholine (DC8,9PC) using the techniques of fluorescence recovery after photobleaching (FRAP) and fluorescence correlation spectroscopy (FCS). Our results show that the mobility of these molecules depends not only on their positions in the DC8,9PC tubules but also on their respective concentrations. While the former indicates that the interior of the DC8,9PC tubules is heterogeneous in terms of diffusion, the latter further highlights the possibility of engineering specific conditions for achieving sustained release of a "drug molecule" over a targeted period of time. In addition, our FCS results indicate that the molecular diffusions inside the crystalline bilayer walls of the DC8,9PC tubules strongly deviate from the normal, stochastic processes, with features characterizing not only anomalous subdiffusions but also motions that are superdiffusive in nature.     

Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function

We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B → C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t(-1) to the anomalous t(-1/4) behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.     

Three-dimensional Brownian diffusion of rod-like macromolecules in the presence of randomly distributed spherical obstacles: molecular dynamics simulation

Brownian diffusion of rod-like polymers in the presence of randomly distributed spherical obstacles is studied using molecular dynamics simulations. It is observed that dependence of the reduced diffusion coefficient of these macromolecules on the available volume fraction can be described reasonably by a power law function. Despite the case of obstructed diffusion of flexible polymers in which reduced diffusion coefficient has a weak dependence on the polymer length, this dependence is noticeably strong in the case of rod-like polymers. Diffusion of these macromolecules in the presence of obstacles is observed that is anomalous at short time scales and normal at long times. Duration time of the anomalous diffusion regime is found that increases very rapidly with increasing both the polymer length and the obstructed volume fraction. Dynamics of diffusion of these polymers is observed that crosses over from Rouse to reptation type with increasing the density of obstacles.