Diffusion in macromolecular crowded media: Monte Carlo simulation of obstructed diffusion vs. FRAP experiments

The diffusion of tracer particles in 3D macromolecular crowded media has been studied using two methodologies, simulation and experimental, with the aim of comparing their results. First, the diffusion of a tracer in an obstructed 3D lattice with mobile and big size obstacles has been analyzed through a Monte Carlo (MC) simulation procedure. Secondly, fluorescence recovery after photobleaching (FRAP) experiments have been carried out to study the diffusion of a model protein (alpha-chymotrypsin) in in vitro crowded solution where two type of Dextran molecules are used as crowder agents. To facilitate the comparison, the relative size between the tracer and the crowder is the same in both studies. The results indicate a qualitative agreement between the diffusional behaviors observed in the two studies. The dependence of the anomalous diffusion exponent and the limiting diffusion coefficient on the obstacle size and excluded volume shows, in both cases, a similar tendency. The introduction of a reduced mobility parameter in the simulation model accounting for the short-range tracer?Cobstacle interactions allows obtaining a quantitative agreement between the limiting diffusion coefficient values yielded by both procedures. The simulation?Cexperiment quantitative agreement for the anomalous diffusion exponent requires further improvements. As far as we know, this is the first reported work where both techniques are used in parallel to study the diffusion in macromolecular crowded media.     

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.     

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.     

Relaxation dynamics of a linear molecule in a random static medium: a scaling analysis

We present extensive molecular dynamics simulations of the motion of a single linear rigid molecule in a two-dimensional random array of fixed overlapping disklike obstacles. The diffusion constants for the center of mass translation, D(CM), and for rotation, D(R), are calculated for a wide range of the molecular length, L, and the density of obstacles, rho. The obtained results follow a master curve Drho(micro) approximately (L(2)rho)(-nu) with an exponent micro=-3/4 and 1/4 for D(R) and D(CM), respectively, that can be deduced from simple scaling and kinematic arguments. The nontrivial positive exponent nu shows an abrupt crossover at L(2)rho=zeta(1). For D(CM) we find a second crossover at L(2)rho=zeta(2). The values of zeta(1) and zeta(2) correspond to the average minor and major axis of the elliptic holes that characterize the random configuration of the obstacles. A violation of the Stokes-Einstein-Debye relation is observed for L(2)rho>zeta(1), in analogy with the phenomenon of enhanced translational diffusion observed in supercooled liquids close to the glass transition temperature.     

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.     

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.     

Monte Carlo simulation on polymer translocation in crowded environment

The effect of crowded environment with static obstacles on the translocation of a three-dimensional self-avoiding polymer through a small pore is studied using dynamic Monte Carlo simulation. The translocation time τ is dependent on polymer-obstacle interaction and obstacle concentration. The influence of obstacles on the polymer translocation is explained qualitatively by the free energy landscape. There exists a special polymer-obstacle interaction at which the translocation time is roughly independent of the obstacle concentration at low obstacle concentration, and the strength of the special interaction is roughly independent of chain length N. Scaling relation τ ~ N(1.25) is observed for strong driving translocations. The diffusion property of polymer chain is also influenced by obstacles. Normal diffusion is only observed in dilute solution without obstacles or in a crowded environment with weak polymer-obstacle attraction. Otherwise, subdiffusion behavior of polymer is observed.     

Lateral diffusion of proteins in the plasma membrane: spatial tessellation and percolation theory

The obstructed diffusion of proteins in the plasma membrane is studied using computer simulation and an analysis using spatial tessellation and percolation theory. The membrane is modeled as a two-dimensional space with fixed hard disc obstacles, and the proteins are modeled as hard discs. The simulations show a transition from normal to anomalous diffusion as the area fraction, phim, of obstacles is increased and to confined diffusion for area fractions above the pecolation threshold, which occurs for phim=0.22. A Voronoi tessellation procedure is used to map the continuous space system onto an effective lattice model, with the connectivity of bonds determined from a geometric criterion. The estimate of the percolation threshold obtained from this lattice model is in excellent agreement with the simulation results, although the nature of the dynamics in the continuous space model is different from lattice models. At high obstacle area fractions (but below the percolation threshold), the primary mode of transport is a hopping motion between voids, consistent with experiment. The simulations and analysis emphasize the importance of structural correlations between obstacles.     

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.     

The effect of obstacle conductivity and electric field on effective mobility and dispersion in electrophoretic transport: a volume averaging approach

The method of volume averaging has been used to determine the effective electrophoretic mobility and dispersion coefficients for molecular transport of point-like solutes in a two-phase porous medium where the electrical conductivity and the diffusion and mobility coefficients may vary in both phases. The formal theory, derived in previous work, is numerically evaluated for cases where the obstacle phase has a large or small conductivity relative to the fluid phase and where the diffusion coefficient of the solute in the obstacle phase can be large or small relative to that in the fluid phase. In agreement with previous Monte Carlo methods, the effective electrophoretic mobility is not a function of media conductivity or electric field when the obstacles are impermeable to solute transport or have small diffusion solute diffusion coefficients. However, the dispersion coefficient is a strong function of electric field and varies with obstacle conductivity when diffusive transport is small in the obstacles relative to the fluid. In contrast, the effective electrophoretic mobility is a function of electric field when conductivity of the obstacles is much larger than the fluid and when the obstacles are very permeable to solute but have low electrical conductivity.     

Diffusion at the liquid-vapor interface

Recently, the intrinsic sampling method has been developed in order to obtain, from molecular simulations, the intrinsic structure of the liquid-vapor interface that is presupposed in the classical capillary wave theory. Our purpose here is to study dynamical processes at the liquid-vapor interface, since this method allows tracking down and analyzing the movement of surface molecules, thus providing, with great accuracy, dynamical information on molecules that are "at" the interface. We present results for the coefficients for diffusion parallel and perpendicular to the liquid-vapor interface of the Lennard-Jones fluid, as well as other time and length parameters that characterize the diffusion process in this system. We also obtain statistics of permanence and residence time. The generality of our results is tested by varying the system size and the temperature; for the latter case, an existing model for alkali metals is also considered. Our main conclusion is that, even if diffusion coefficients can still be computed, the turnover processes, by which molecules enter and leave the intrinsic surface, are as important as diffusion. For example, the typical time required for a molecule to traverse a molecular diameter is very similar to its residence time at the surface.     

Calculating the hopping times of confined fluids: two hard disks in a box

The dynamical transition between the anomalous single file diffusion of highly confined fluids and bulk normal diffusion can be described by a phenomenological model involving a particle hopping time tau(hop). We suggest a theoretical formalism that will be useful for the calculation of tau(hop) for a variety of systems and test it using a simple model consisting of two hard disks confined to a rectangular box with hard walls. In the case where the particles are moving diffusively, we find the hopping time diverges as a power law in the threshold region with an exponent of -(3/2). Under conditions where the particles move inertially, transition state theory predicts a power law behavior with an exponent of -2. Molecular dynamics simulations confirm the transition state theory result for inertial dynamics, while Brownian dynamics simulations suggest the scaling exponent is highly sensitive to the details of the algorithm.     

Some aspects of diffusion: fluctuations in reaction diffusion,and geometric influences in nonuniform media

Summary We deal with two diffusion problems: Space-integrated conserved entities characterizing very fast - diffusion - controlled reactions, such as time lags, etc. are universal. They are given by relationships which do not reflect the failure of the mean field hydrodynamic equations. We present another application which does not reflect this failure, for determining the surface flux via a diffusion controlled reaction producing a colored product. Another anomalous diffusion process we considered is transport through cellular materials whose cell sizes are highly nonuniform. We have analyzed the effects of extreme nonuniformity by considering fractal-like models of cellular solids. The diffusion current through these models can exhibit anomalous time-dependencies which are not predicted by the diffusion equation. In particular, it is shown that the initial diffusion current can be characterized by a power-law dependence on the time. Furthermore, the exponent of the power law is given in terms of the distribution of cell sizes in the fractal-like cellular solid.     

What can we learn from single molecule trajectories?

Diffusing membrane constituents are constantly exposed to a variety of forces that influence their stochastic path. Single molecule experiments allow for resolving trajectories at extremely high spatial and temporal accuracy, thereby offering insights into en route interactions of the tracer. In this review we discuss approaches to derive information about the underlying processes, based on single molecule tracking experiments. In particular, we focus on a new versatile way to analyze single molecule diffusion in the absence of a full analytical treatment. The method is based on comprehensive comparison of an experimental data set against the hypothetical outcome of multiple experiments performed on the computer. Since Monte Carlo simulations can be easily and rapidly performed even on state-of-the-art PCs, our method provides a simple way for testing various - even complicated - diffusion models. We describe the new method in detail, and show the applicability on two specific examples: firstly, kinetic rate constants can be derived for the transient interaction of mobile membrane proteins; secondly, residence time and corral size can be extracted for confined diffusion.     

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.     

Molecular Diffusion Measurement in Lipid Bilayers over Wide Concentration Ranges: A Comparative Study

Molecular diffusion in biological membranes is a determining factor in cell signaling and cell function. In the past few decades, three main fluorescence spectroscopy techniques have emerged that are capable of measuring molecular diffusion in artificial and biological membranes at very different concentration ranges and spatial resolutions. The widely used methods of fluorescence recovery after photobleaching (FRAP) and single‐particle tracking (SPT) can determine absolute diffusion coefficients at high (>100 μm^?2) and very low surface concentrations (single‐molecule level), respectively. Fluorescence correlation spectroscopy (FCS), on the other hand, is well‐suited for the intermediate concentration range of about 0.1–100 μm^?2. However, FCS in general requires calibration with a standard dye of known diffusion coefficient, and yields only relative measurements with respect to the calibration. A variant of FCS, z‐scan FCS, is calibration‐free for membrane measurements, but requires several experiments at different well‐controlled focusing positions. A recently established FCS method, electron‐multiplying charge‐coupled‐device‐based total internal reflection FCS (TIR‐FCS), referred to here as imaging TIR‐FCS (ITIR–FCS), is also independent of calibration standards, but to our knowledge no direct comparison between these different methods has been made. Herein, we seek to establish a comparison between FRAP, SPT, FCS, and ITIR–FCS by measuring the lateral diffusion coefficients in two model systems, namely, supported lipid bilayers and giant unilamellar vesicles.     

Application of pulsed field gradient NMR with high gradient strength for studies of self-diffusion in lipid membranes on the nanoscale

This work demonstrates the feasibility of noninvasive studies of lipid self-diffusion in model lipid membranes on the nanoscale using proton pulsed field gradient (PFG) NMR spectroscopy with high (up to 35 T/m) gradient amplitudes. Application of high gradients affords for the use of sufficiently small diffusion times under the conditions when the width of the gradient pulses is much smaller than the diffusion time. As a result, PFG NMR studies of partially restricted or anomalous diffusion in lipid bilayers become possible over length scales as small as 100 nm. This length scale is important because it is comparable to the size of membrane domains, or lipid rafts, which are believed to exist in biomembranes. In this work, high-gradient PFG NMR has been applied to study lipid self-diffusion in three-component planar-supported multibilayers (1,2-dioleoyl- sn-glycerol-3-phosphocholine/sphingomyelin/cholesterol). The degree of lipid orientation in the bilayers was determined with (31)P NMR. A special insert was designed to mechanically align the multibilayer stack at the magic angle with respect to the direction of the constant magnetic field to address the detrimental effects of proton dipole-dipole interactions on the NMR signal. This insert is an alternative to the conventional method of magic angle orientation of lipid membranes, the goniometer probe, which is not compatible with commercial high-gradient coils because of the lack of space in the magnet bore. Macroscopic orientation of the multibilayer stacks using the insert was confirmed with (1)H NMR spectroscopic studies and the comparison of results obtained from identical experiments using a goniometer probe for orientation. Diffusion studies were carried out at three different constant magnetic field strengths ( B 0) over a range of temperatures and diffusion times. The measured diffusivities were found to be in agreement with the data obtained previously by techniques that are limited to much larger length scales of diffusion observation than high-gradient PFG NMR.     

Contribution of nonlocal interactions to DNA elasticity

A nonlocal harmonic elastic rod model is proposed to describe the elastic behavior of short DNA molecules. We show that the nonlocal interactions contribute to effective bending energy of the molecule and affect its apparent persistence length. It is also shown that the anomalous behavior which has been observed in all-atom molecular dynamic simulations [A. K. Mazur, Biophys. J. 134, 4507 (2006)] can be a consequence of both nonlocal interactions between DNA base pairs and the intrinsic curvature of DNA.     

Tracking single lipase molecules on a trimyristin substrate surface using quantum dots

The mobility of single lipase molecules has been analyzed using single molecule tracking on a trimyristin substrate surface. This was achieved by conjugating lipases to quantum dots and imaging on spin-coated trimyristin surfaces by means of confocal laser scanning microscopy. Image series of single lipase molecules were collected, and the diffusion coefficient was quantified by analyzing the mean square displacement of the calculated trajectories. During no-flow conditions, the lipase diffusion coefficient was (8.0+/-5.0)x10(-10) cm2/s. The trajectories had a "bead on a string" appearance, with the lipase molecule restricted in certain regions of the surface and then migrating to another region where the restricted diffusion continued. This gave rise to clusters in the trajectories. When a flow was applied to the system, the total distance and average step length between the clusters increased, but the restricted diffusion in the cluster regions was unaffected. This can be explained by the lipase operating in two different modes on the surface. In the cluster regions, the lipase is likely oriented with the active site toward the surface and hydrolyzes the substrate. Between these regions, a diffusion process is proposed where the lipase is in contact with the surface but affected by the external flow.     

Single molecule fluorescence microscopy investigations on heterogeneity of translational diffusion in thin polymer films

Translational diffusion of single perylene diimide molecules in 25 nm thin polymer films was investigated by single molecule widefield fluorescence microscopy. Spatial heterogeneities in single molecule motion were detected and analyzed by a new, quantitative method which draws a comparison of log-Gaussian fits of experimentally determined diffusion coefficient-distributions and diffusion coefficient-distributions from Monte Carlo random walk simulations. Heterogeneities could be observed close to the glass transition temperature, but disappear at ca. 1.1 × T(g). At higher temperatures, heterogeneities do not exist or they average out on the time and length scales of observation. The observed heterogeneities also explain why the dependency of diffusion coefficients on temperature does not follow Vogel-Fulcher-Tammann behavior.