Lateral diffusion in an archipelago. Dependence on tracer size.

In a pure fluid-phase lipid, the dependence of the lateral diffusion coefficient on the size of the diffusing particle may be obtained from the Saffman-Delbrück equation or the free-volume model. When diffusion is obstructed by immobile proteins or domains of gel-phase lipids, the obstacles yield an additional contribution to the size dependence. Here this contribution is examined using Monte Carlo calculations. For random point and hexagonal obstacles, the diffusion coefficient depends strongly on the size of the diffusing particle, but for fractal obstacles--cluster-cluster aggregates and multicenter diffusion-limited aggregates--the diffusion coefficient is independent of the size of the diffusing particle. The reason is that fractals have no characteristic length scale, so a tracer sees on average the same obstructions, regardless of its size. The fractal geometry of the excluded area for tracers of various sizes is examined. Percolation thresholds are evaluated for a variety of obstacles to determine how the threshold depends on tracer size and to compare the thresholds for compact and extended obstacles.     

Membrane lateral mobility obstructed by polymer-tethered lipids studied at the single molecule level

Obstructed long-range lateral diffusion of phospholipids (TRITC-DHPE) and membrane proteins (bacteriorhodopsin) in a planar polymer-tethered 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine bilayer is studied using wide-field single molecule fluorescence microscopy. The obstacles are well-controlled concentrations of hydrophobic lipid-mimicking dioctadecylamine moieties in the polymer-exposed monolayer of the model membrane. Diffusion of both types of tracer molecules is well described by a percolating system with different percolation thresholds for lipids and proteins. Data analysis using a free area model of obstructed lipid diffusion indicates that phospholipids and tethered lipids interact via hard-core repulsion. A comparison to Monte Carlo lattice calculations reveals that tethered lipids act as immobile obstacles, are randomly distributed, and do not self-assemble into large-scale aggregates for low to moderate tethering concentrations. A procedure is presented to identify anomalous subdiffusion from tracking data at a single time lag. From the analysis of the cumulative distribution function of the square displacements, it was found that TRITC-DHPE and W80i show normal diffusion at lower concentrations of tethered lipids and anomalous diffusion at higher ones. This study may help improve our understanding of how lipids and proteins in biomembranes may be obstructed by very small obstacles comprising only one or very few molecules.     

Lateral diffusion in an archipelago. The effect of mobile obstacles.

Lateral diffusion of mobile proteins and lipids (tracers) in a membrane is hindered by the presence of proteins (obstacles) in the membrane. If the obstacles are immobile, their effect may be described by percolation theory, which states that the long-range diffusion constant of the tracers goes to zero when the area fraction of obstacles is greater than the percolation threshold. If the obstacles are themselves mobile, the diffusion constant of the tracers depends on the area fraction of obstacles and the relative jump rate of tracers and obstacles. This paper presents Monte Carlo calculations of diffusion constants on square and triangular lattices as a function of the concentration of obstacles and the relative jump rate. The diffusion constant for particles of various sizes is also obtained. Calculated values of the concentration-dependent diffusion constant are compared with observed values for gramicidin and bacteriorhodopsin. The effect of the proteins as inert obstacles is significant, but other factors, such as protein-protein interactions and perturbation of lipid viscosity by proteins, are of comparable importance. Potential applications include the diffusion of proteins at high concentrations (such as rhodopsin in rod outer segments), the modulation of diffusion by release of membrane proteins from cytoskeletal attachment, and the diffusion of mobile redox carriers in mitochondria, chloroplasts, and endoplasmic reticulum.     

Influence of obstacles on lipid lateral diffusion: computer simulation of FRAP experiments and application to proteoliposomes and biomembranes

Fluorescence Recovery After Photobleaching experiments were simulated using a computer approach in which a membrane lipid leaflet was mimicked using a triangular lattice obstructed with randomly distributed immobile and non-overlapping circular obstacles. Influence of the radius r and area fraction c of these obstacles and of the radius R of the observation area on the relative diffusion coefficient D ^* (Eq. (1)) and mobile fraction M was analyzed. A phenomenological equation relating D ^* to r and c was established. Fitting this equation to the FRAP data we obtained with the probe NBD-PC embedded in bacteriorhodopsin/egg-PC multilayers suggests that this transmembrane protein rigidifies the surrounding lipid phase over a distance of about 18 Å (two lipid layers) from the protein surface. In contrast, analysis of published diffusion constants obtained for lipids in the presence of gramicidin suggests that in terms of lateral diffusion, this relatively small polypeptide does not significantly affect the surrounding lipid phase. With respect to the mobile fraction M, and for point obstacles above the percolation threshold, an increase in R led to a decrease in M which can be associated with the existence of closed domains whose average size and diffusion properties can be determined. Adaptation of this model to the re-interpretation of the FRAP data obtained by Yechiel and Edidin (J Cell Biol (1987) 115:755–760) for the plasma membrane of human fibroblasts consistently leads to the suggestion that the lateral organization of this membrane would be of the confined type, with closed lipid domains of 0.5 µm² in area.Abbreviations and notations used BR bacteriorhodopsin - DMPC dimyristoylphosphatidylcholine - diOC18 dioctadecyloxatricarbocyanine - egf-PC egg-yolk phosphatidylcholine - NBD-PC 1-acyl2-[t2-[(7-nitro-2-1,3-benzoxadiazol-4-yl)amino]dodecanoyl]-sn-glycero-3-phosphocholine - MOPS 3-[N-morpholino]propane sulfonic acid - FRAP Fluoresence Recovery After photobleaching - D observed diffusion coefficient - D0 diffusion coefficient in the absence of obstacles - D ^* relative diffusion constant (Eq. 1) - M mobile fraction - c obstacle area fraction - r obstacle radius - R observation area radius - r _d diffusion area radius Correspondence to: A. Lopez     

To Hop or not to Hop: Exceptions in the FCS Diffusion Law

Diffusion obstacles in membranes have not been directly visualized because of fast membrane dynamics and the occurrence of subresolution molecular complexes. To understand the obstacle characteristics, mobility-based methods are often used as an indirect way of assessing the membrane structure. Molecular movement in biological plasma membranes is often characterized by anomalous diffusion, but the exact underlying mechanisms are still elusive. Imaging total internal reflection fluorescence correlation spectroscopy (ITIR-FCS) is a well-established mobility-based method that provides spatially resolved diffusion coefficient maps and is combined with FCS diffusion law analysis to examine subresolution membrane organization. In recent years, although FCS diffusion law analysis has been instrumental in providing new insights into the membrane structure below the optical diffraction limit, there are certain exceptions and anomalies that require further clarification. To this end, we correlate the membrane structural features imaged by atomic force microscopy (AFM) with the dynamics measured using ITIR-FCS. We perform ITIR-FCS measurements on supported lipid bilayers (SLBs) of various lipid compositions to characterize the anomalous diffusion of lipid molecules in distinct obstacle configurations, along with the high-resolution imaging of the membrane structures with AFM. Furthermore, we validate our experimental results by performing simulations on image grids with experimentally determined obstacle configurations. This study demonstrates that FCS diffusion law analysis is a powerful tool to determine membrane heterogeneities implied from dynamics measurements. Our results corroborate the commonly accepted interpretations of imaging FCS diffusion law analysis, and we show that exceptions happen when domains reach the percolation threshold in a biphasic membrane and a network of domains behaves rather like a meshwork, resulting in hop diffusion.     

Lateral diffusion and aggregation. A Monte Carlo study.

Aggregation in a lipid bilayer is modeled as cluster-cluster aggregation on a square lattice. In the model, clusters carry out a random walk on the lattice, with a diffusion coefficient inversely proportional to mass. On contact, they adhere with a prescribed probability, rigidly and irreversibly. Monte Carlo calculations show that, as expected, rotational diffusion of the aggregating species is highly sensitive to the initial stages of aggregation. Lateral diffusion of an inert tracer obstructed by the aggregate is a sensitive probe of the later stages of aggregation. Cluster-cluster aggregates are much more effective barriers to lateral diffusion of an inert tracer than the same area fraction of random point obstacles is, but random point obstacles are more effective barriers than the same area fraction of compact obstacles. The effectiveness of aggregates as obstacles is discussed in terms of particle-particle correlation functions and fractal dimensions. Results are applicable to aggregation of membrane proteins, and at least qualitatively to aggregation of gel-phase lipid during lateral phase separation.     

Anomalous diffusion in a gel-fluid lipid environment: a combined solid-state NMR and obstructed random-walk perspective

Lateral diffusion is an essential process for the functioning of biological membranes. Solid-state nuclear magnetic resonance (NMR) is, a priori, a well-suited technique to study lateral diffusion within a heterogeneous environment such as the cell membrane. Moreover, restriction of lateral motions by lateral heterogeneities can be used as a means to characterize their geometry. The goal of this work is to understand the advantages and limitations of solid-state NMR exchange experiments in the study of obstructed lateral diffusion in model membranes. For this purpose, simulations of lateral diffusion on a sphere with varying numbers and sizes of immobile obstacles and different percolation properties were performed. From the results of these simulations, two-dimensional 31P NMR exchange maps and time-dependent autocorrelation functions were calculated. The results indicate that the technique is highly sensitive to percolation properties, total obstacle area, and, within certain limits, obstacle size. A practical example is shown, namely the study of the well-characterized DMPC-DSPC binary mixture. The comparison of experimental and simulated results yielded obstacle sizes in the range of hundreds of nanometers, therefore bridging the gap between previously published NMR and fluorescence recovery after photobleaching results. The method could also be applied to the study of membrane protein lateral diffusion in model membranes.     

Sieve tube unloading and post-phloem transport of fluorescent tracers and proteins injected into sieve tubes via severed aphid stylets

A variety of fluorescent tracers and proteins were injected via severed aphid stylets into the sieve tubes of wheat (Triticum aestivum L.) grains to evaluate the dimensions of plasmodesmal channels involved in sieve element/companion cell (SE/CC) unloading and post-phloem transport. In the post-phloem pathway, where diffusion is the predominant mode of transport, the largest molecule to show mobility was 16-kD dextran, with a Stokes radius of 2.6 nm. This suggests that the aqueous channels for cell-to-cell transport must be about 8 nm in diameter. Even the largest tracer injected into the sieve tubes, 400-kD fluorescein-labeled Ficoll with a Stokes radius of about 11 nm, was unloaded from the SE/CC complex. However, in contrast to smaller tracers (< or =3 kD, with a Stokes radius < or = 1.2 nm), the unloading of fluorescein-labeled Ficoll and other large molecules from the SE/CC complex showed an irregular, patchy distribution, with no further movement along the post-phloem pathway. Either the plasmodesmal channels involved in SE/CC unloading are exceptionally large (perhaps as much as 42 nm in diameter), with only a very small fraction of plasmodesmata being conductive, or the larger tracers damage the plasmodesmata in some way, enlarging smaller channels.     

Permeability properties of cell-to-cell channels: Kinetics of fluorescent tracer diffusion through a cell junction

Summary We have analyzed the intracellular and cell-to-cell diffusion kinetics of fluorescent tracers in theChironomus salivary gland. We use this analysis to investigate whether membrane potential-induced changes in junctional permeability are accompanied by changes in cell-to-cell channel selectivity. Tracers of different size and fluorescence wavelength were coinjected into a cell, and the fluorescence was monitored in this cell and an adjacent one. Rate constants,k _j , for cell-to-cell diffusion were derived by compartment model analysis, taking into account (i) cell-to-cell diffusion of the tracers; (ii) their loss from the cells; (iii) their binding (sequestration) to cytoplasmic components; and (iv) their relative mobility to cytoplasm, as determined separately on isolated cells. In cell pairs, we compared a tracer'sk _j with the electrical cell-to-cell conductance,g _j .At cell membrane resting potential, thek _j 's ranged 3.8–9.2×10^–3 sec^–1 for the small carboxyfluorescein (mol wt 376) to about 0.4×10^–3 sec^–1 for a large fluorescein-labeled sugar (mol wt 2327). Cell membrane depolarization reversibly reducedg _j andk _j for a large and a small tracer, all in the same proportion. This suggests that membrane potential controls the number of open channels, rather than their effective pore diameter or selectivity. From the inverse relation between tracer mean diameter and relativek _j we calculate an effective, permeation-limiting diameter of approximately 29 Å for the insect cell-to-cell channel. Intracellular diffusion was faster than cell-to-cell diffusion, and it was not solely dependent on tracer size. Rate constants for intracellular sequestration and loss through nonjunctional membrane were large enough to become rate-limiting for cell-to-cell tracer diffusion at low junctional permeabilities.     

Robust pore size analysis of filamentous networks from three-dimensional confocal microscopy

We describe a robust method for determining morphological properties of filamentous biopolymer networks, such as collagen or other connective tissue matrices, from confocal microscopy image stacks. Morphological properties including pore size distributions and percolation thresholds are important for transport processes, e.g., particle diffusion or cell migration through the extracellular matrix. The method is applied to fluorescently labeled fiber networks prepared from rat-tail tendon and calf-skin collagen, at concentrations of 1.2, 1.6, and 2.4 mg/ml. The collagen fibers form an entangled and branched network. The medial axes, or skeletons, representing the collagen fibers are extracted from the image stack by threshold intensity segmentation and distance-ordered homotopic thinning. The size of the fluid pores as defined by the radii of largest spheres that fit into the cavities between the collagen fibers is derived from Euclidean distance maps and maximal covering radius transforms of the fluid phase. The size of the largest sphere that can traverse the fluid phase between the collagen fibers across the entire probe, called the percolation threshold, was computed for both horizontal and vertical directions. We demonstrate that by representing the fibers as the medial axis the derived morphological network properties are both robust against changes of the value of the segmentation threshold intensity and robust to problems associated with the point-spread function of the imaging system. We also provide empirical support for a recent claim that the percolation threshold of a fiber network is close to the fiber diameter for which the Euler index of the networks becomes zero.     

Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study

Anomalous subdiffusion is hindered diffusion in which the mean-square displacement of a diffusing particle is proportional to some power of time less than one. Anomalous subdiffusion has been observed for a variety of lipids and proteins in the plasma membranes of a variety of cells. Fluorescence photobleaching recovery experiments with anomalous subdiffusion are simulated to see how to analyze the data. It is useful to fit the recovery curve with both the usual recovery equation and the anomalous one, and to judge the goodness of fit on log-log plots. The simulations show that the simplest approximate treatment of anomalous subdiffusion usually gives good results. Three models of anomalous subdiffusion are considered: obstruction, fractional Brownian motion, and the continuous-time random walk. The models differ significantly in their behavior at short times and in their noise level. For obstructed diffusion the approach to the percolation threshold is marked by a large increase in noise, a broadening of the distribution of diffusion coefficients and anomalous subdiffusion exponents, and the expected abrupt decrease in the mobile fraction. The extreme fluctuations in the recovery curves at and near the percolation threshold result from extreme fluctuations in the geometry of the percolation cluster.     

Mechanism and size cutoff for steric exclusion from actin-rich cytoplasmic domains.

Subdomains of the cytoplasmic volume in tissue culture cells exclude large tracer particles relative to small. Evidence suggests that exclusion of the large particles is due to molecular sieving by the dense meshwork of microfilaments found in these compartments, but exclusion as a result of the close apposition of the dorsal and ventral plasma membrane of the cell in these regions has not been ruled out conclusively. In principle, these two mechanisms can be distinguished by the dependence of exclusion on tracer particle size. By fluorescence ratio imaging we have measured the partition coefficient (P/PO) into excluding compartments for tracer particles ranging in radius from 1 to 41 nm. The decay of P/PO as a function of particle radius is better fitted by three molecular sieving models than by a slit pore model. The sieving models predict a percolation cutoff radius of the order of 50 nm for partitioning into excluding compartments.     

Dependence of plastoquinol diffusion on the shape, size, and density of integral thylakoid proteins

The diffusion of plastoquinol in the chloroplast thylakoid membrane is modelled using Monte Carlo techniques. The integral proteins are seen as obstacles to diffusion, and features of percolation theory emerge. Thus, the diffusion coefficient diminishes with increasing distance and there is a critical threshold of protein concentration, above which the long-range diffusion coefficient is zero. The area occupied by proteins in vivo is assessed and appears to be around this threshold, as determined from calculations assuming randomly distributed noninteracting proteins. Slight changes in the protein arrangement lead to pronounced changes in diffusion behaviour under such conditions. Mobility of the proteins increases the protein occupancy threshold, while boundary lipids impermeable to PQ diffusion decrease it. Further, the obstruction of plastoquinone/plastoquinol binding sites in a random arrangement is evaluated.     

ARTIFACTUAL LOCALIZATION OF FERRITIN IN THE CILIARY EPITHELIUM IN VITRO

The accumulation of ferritin by the ciliary epithelium of the adult albino rabbit has been studied by electron microscopy. The experiments have been carried out under in vitro conditions, such that any uptake observed should be the result of passive diffusion of the tracerparticles rather than the product of active metabolic processes. The cells were fixed in osmium tetroxide and embedded in Araldite. Ferritin was found localized in three areas: in rows of apparent vesicles, free in the cytoplasmic matrix, and in the basement membrane. Some of the conclusions reached are as follows. The appearance of tracer in rows of vesicles is not in itself an adequate demonstration of pinocytosis. The permeability of the plasma membrane is drastically increased by osmium tetroxide fixation, so that tracer particles are free to diffuse across the membrane and wander through the cytoplasm. These results indicate the serious danger of being misled by artifacts when colloidal particles are used as tracers.     

The membrane skeleton of erythrocytes. A percolation model.

The spectrin network on the cytoplasmic surface of the erythrocyte membrane is modeled as a triangular lattice of spectrin tetramers. This network obstructs lateral diffusion of proteins and provides mechanical reinforcement to the membrane. These effects are treated in a systematic and unified manner in terms of a percolation model. The diffusion coefficient is obtained as a function of the fraction of normal spectrin tetramers for both static and fluctuating barriers. The elasticity of the network is calculated as a function of the fraction of normal spectrin and the ratio of bending to stretching energies. For static barriers, elasticity and lateral diffusion are incompatible: if a network is connected enough to be elastic, it is connected enough to block long-range lateral diffusion. The elasticity and the force required for mechanical breakdown go to zero at the percolation threshold; experimental evidence suggests the existence of a stability threshold at or near the percolation threshold. The model is qualitatively applicable to other cells with membrane skeletons, such as epithelial cells, in which localization of membrane proteins is essential to differentiation.     

From microscopy data to in silico environments for in vivo-oriented simulations

In our previous study, we introduced a combination methodology of Fluorescence Correlation Spectroscopy (FCS) and Transmission Electron Microscopy (TEM), which is powerful to investigate the effect of intracellular environment to biochemical reaction processes. Now, we developed a reconstruction method of realistic simulation spaces based on our TEM images. Interactive raytracing visualization of this space allows the perception of the overall 3D structure, which is not directly accessible from 2D TEM images. Simulation results show that the diffusion in such generated structures strongly depends on image post-processing. Frayed structures corresponding to noisy images hinder the diffusion much stronger than smooth surfaces from denoised images. This means that the correct identification of noise or structure is significant to reconstruct appropriate reaction environment in silico in order to estimate realistic behaviors of reactants in vivo. Static structures lead to anomalous diffusion due to the partial confinement. In contrast, mobile crowding agents do not lead to anomalous diffusion at moderate crowding levels. By varying the mobility of these non-reactive obstacles (NRO), we estimated the relationship between NRO diffusion coefficient (D_nro) and the anomaly in the tracer diffusion (α). For D_nro=21.96 to 44.49 μm²/s, the simulation results match the anomaly obtained from FCS measurements. This range of the diffusion coefficient from simulations is compatible with the range of the diffusion coefficient of structural proteins in the cytoplasm. In addition, we investigated the relationship between the radius of NRO and anomalous diffusion coefficient of tracers by the comparison between different simulations. The radius of NRO has to be 58 nm when the polymer moves with the same diffusion speed as a reactant, which is close to the radius of functional protein complexes in a cell.     

Lateral diffusion in a mixture of mobile and immobile particles. A Monte Carlo study.

The lateral diffusion coefficient for mixtures of mobile and immobile particles is obtained from Monte Carlo calculations of random walks by mobile tracers in the presence of immobile obstacles on a triangular lattice. The diffusion coefficient of the mobile species is obtained as a function of the area fractions of mobile and immobile species. The results are applied to diffusion of band 3 in the erythrocyte membrane, and indicate that obstruction of diffusion of mobile band 3 by band 3 and glycophorin attached to the membrane skeleton is not sufficient to explain the observed diffusion coefficient.     

Obstructed diffusion in phase-separated supported lipid bilayers: a combined atomic force microscopy and fluorescence recovery after photobleaching approach

Proteins and other macromolecules are believed to hinder molecular lateral diffusion in cellular membranes. We have constructed a well-characterized model system to better understand how obstacles in lipid bilayers obstruct diffusion. Fluorescence recovery after photobleaching was used to measure the lateral diffusion coefficient in single supported bilayers composed of mixtures of 1,2-dilauroylphosphotidylcholine (DLPC) and 1,2-distearoylphosphotidylcholine (DSPC). Because these lipids are immiscible and phase separate at room temperature, a novel quenching technique allowed us to construct fluid DLPC bilayers containing small disk-shaped gel-phase DSPC domains that acted as obstacles to lateral diffusion. Our experimental setup enabled us to analyze the same samples with atomic force microscopy and exactly characterize the size, shape, and number of gel-phase domains before measuring the obstacle-dependent diffusion coefficient. Lateral obstructed diffusion was found to be dependent on obstacle area fraction, size, and geometry. Analysis of our results using a free area diffusion model shows the possibility of unexpected long-range ordering of fluid-phase lipids around the gel-phase obstacles. This lipid ordering has implications for lipid-mediated protein interactions in cellular membranes.