Lateral diffusion in inhomogeneous membranes. Model membranes containing cholesterol.

The problem of lateral diffusion in inhomogeneous membranes is illustrated by a theoretical calculation of the lateral diffusion of a fluorescent lipid probe in binary mixtures of phosphatidylcholine and cholesterol under conditions of temperature and composition such that this lipid mixture consists of alternating parallel domains of fluid and solid lipid, having separations that are small compared with the distance scale employed in photobleaching experiments. The theoretical calculations clearly illustrate how inhomogeneities in membrane composition affecting the lateral motion of membrane components on a small (10-100 nm) distance scale can give complex diffusive responses in experiments such as fluorescence photobleaching that employ comparatively macroscopic distances (10-100 micrometers) for the measurement of diffusive recovery. The theoretical calculations exhibit the unusual dependence of the apparent lateral diffusion coefficient of a fluorescent lipid probe on lipid composition in binary mixtures of cholesterol and phosphatidylcholines as reported by Rubenstein et al. (1979, Proc. Natl. Acad. Sci. U.S.A., 76:15-18).     

Continuous photobleaching in vesicles and living cells: a measure of diffusion and compartmentation

We present a comprehensive and analytical treatment of continuous photobleaching in a compartment, under single photon excitation. In the very short time regime (t<0.1 ms), the diffusion does not play any role. After a transition (or short time regime), one enters in the long time regime (t>0.1-5 s), for which the diffusion and the photobleaching balance each other. In this long time regime, the diffusion is either fast (i.e., the photobleaching probability of a molecule diffusing through the laser beam is low) so that the photobleaching rate is independent of the diffusion constant and dependent only of the laser power, or the diffusion is slow (i.e., the photobleaching probability is high) and the photobleaching rate is mainly dependent on the diffusion constant. We illustrate our theory by using giant unilamellar vesicles ranging from approximately 10 to 100 microm in diameter, loaded with molecules of various diffusion constants (from 20 to 300 microm2/s) and various photobleaching cross sections, illuminated under laser powers between 3 and 100 microW. We also demonstrated that information about compartmentation can be obtained by this method in living cells expressing enhanced green fluorescent proteins or that were loaded with small FITC-dextrans. Our quantitative approach shows that molecules freely diffusing in a cellular compartment do experience a continuous photobleaching. We provide a generic theoretical framework that should be taken into account when studying, under confocal microscopy, molecular interactions, permeability, etc.     

Spatial distribution of competing populations

We consider spatial distributions of two competing and diffusing populations whose habitats are partly overlapping. As a model, certain reaction—diffusion equations are used in the finite and in the infinite regions with Dirichlet boundary conditions. On the assumption of extremely different diffusive rates of the two species, it is verified, by the use of singular perturbation techniques, that the slowly diffusing species can survive in some subregions, although the species with the greater diffusive rate rapidly occupies the region at the initial stage, and that coexistence of two populations is realized, reducing the effect of interspecific competition by spatial segregation. It will be also shown that the size of the region where the slowly moving population can survive exhibits a markedly qualitative change, depending on the values of some parameters, and that the population can extends the distribution infinitely, when the parameters satisfy a certain condition.     

Lateral diffusion measurement at high spatial resolution by scanning microphotolysis in a confocal microscope.

Fluorescence photobleaching methods have been widely used to study diffusion processes in the plasma membrane of single living cells and other membrane systems. Here we describe the application of a new photobleaching technique, scanning microphotolysis. Employing a recently developed extension module to a commercial confocal microscope, an intensive laser beam was switched on and off during scanning according to a user definable image mask. Thereby the location, geometry, and number of photolysed spots could be chosen arbitrarily, their size ranging from tens of micrometers down to the diffraction limit. Therewith we bleached circular areas on the surface of single living 3T3 cells labeled with the fluorescent lipid analog NBD-HPC. Subsequently, the fluorescence recovery process was observed using the attenuated laser beam for excitation. This yielded image stacks representing snapshots of the spatial distribution of fluorescent molecules. From these we computed the radial distribution functions of the photobleached dye molecules. The variance of these distributions is linearly related to the diffusion constant, time, and the mobile fraction of the diffusing species. Furthermore, we compared directly the theoretically expected and measured distribution functions, and could thus determine the diffusion coefficient from each single image. The results of these two new evaluation methods (D = 0.3 +/- 0.1 micron 2/s) agreed well with the outcome of conventional fluorescence recovery measurements. We show that by scanning microphotolysis information on dynamical processes such as diffusion of lipids or proteins can be acquired at the superior spatial resolution of a confocal laser scanning microscope.     

Monte Carlo analysis of obstructed diffusion in three dimensions: application to molecular diffusion in organelles.

Molecular transport in the aqueous lumen of organelles involves diffusion in a confined compartment with complex geometry. Monte Carlo simulations of particle diffusion in three dimensions were carried out to evaluate the influence of organelle structure on diffusive transport and to relate experimental photobleaching data to intrinsic diffusion coefficients. Two organelle structures were modeled: a mitochondria-like long closed cylinder containing fixed luminal obstructions of variable number and size, and an endoplasmic reticulum-like network of interconnected cylinders of variable diameter and density. Trajectories were computed in each simulation for >10(5) particles, generally for >10(5) time steps. Computed time-dependent concentration profiles agreed quantitatively with analytical solutions of the diffusion equation for simple geometries. For mitochondria-like cylinders, significant slowing of diffusion required large or wide single obstacles, or multiple obstacles. In simulated spot photobleaching experiments, a approximately 25% decrease in apparent diffusive transport rate (defined by the time to 75% fluorescence recovery) was found for a single thin transverse obstacle occluding 93% of lumen area, a single 53%-occluding obstacle of width 16 lattice points (8% of cylinder length), 10 equally spaced 53% obstacles alternately occluding opposite halves of the cylinder lumen, or particle binding to walls (with mean residence time = 10 time steps). Recovery curve shape with obstacles showed long tails indicating anomalous diffusion. Simulations also demonstrated the utility of measurement of fluorescence depletion at a spot distant from the bleach zone. For a reticulum-like network, particle diffusive transport was mildly reduced from that in unobstructed three-dimensional space. In simulated photobleaching experiments, apparent diffusive transport was decreased by 39-60% in reticular structures in which 90-97% of space was occluded. These computations provide an approach to analyzing photobleaching data in terms of microscopic diffusive properties and support the paradigm that organellar barriers must be quite severe to seriously impede solute diffusion.     

Convolution-based one and two component FRAP analysis: theory and application

The method of fluorescence redistribution after photobleaching (FRAP) is increasingly receiving interest in biological applications as it is nowadays used not only to determine mobility parameters per se, but to investigate dynamic changes in the concentration or distribution of diffusing molecules. Here, we develop a new simple convolution-based approach to analyze FRAP data using the whole image information. This method does not require information about the timing and localization of the bleaching event but uses the first image acquired directly after photobleaching to calculate the intensity distributions, instead. Changes in pools of molecules with different velocities, which are monitored by applying repetitive FRAP experiments within a single cell, can be analyzed by means of a global model by assuming two global diffusion coefficients with changing portions. We validate the approach by simulation and show that translocation of the YFP-fused PH-domain of phospholipase Cδ1 can be quantitatively monitored by FRAP analysis in a time-resolved manner. The new FRAP data analysis procedure may be applied to investigate signal transduction pathways using biosensors that change their mobility. An altered mobility in response to the activation of signaling cascades may result either from an altered size of the biosensor, e.g. due to multimerization processes or from translocation of the sensor to an environment with different viscosity. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Fluorescence correlation spectroscopy with single-molecule sensitivity on cell and model membranes.

We report on the successful application of fluorescence correlation spectroscopy (FCS) to the analysis of single fluorescently labeled lipid analogue molecules diffusing laterally in lipid bilayers, as exemplified by time traces of fluorescence bursts of individual molecules entering and leaving the excitation area. FCS measurements performed on lipid probes in rat basophilic leukemia cell membranes showed deviations from two-dimensional Brownian motion with a single uniform diffusion constant. Giant unilamellar vesicles were employed as model systems to characterize diffusion of fluorescent lipid analogues in both homogeneous and mixed lipid phases with diffusion heterogeneity. Comparing the results of cell membrane diffusion with the findings on the model systems suggests possible explanations for the observations: (a) anomalous subdiffusion in which evanescent attractive interactions with disparate mobile molecules modifies the diffusion statistics; (b) alternatively, probe molecules are localized in microdomains of submicroscopic size, possibly in heterogeneous membrane phases.     

Diffusional anisotropy in collagenous tissues: fluorescence imaging of continuous point photobleaching

Molecular transport in avascular collagenous tissues such as articular cartilage occurs primarily via diffusion. The presence of ordered structures in the extracellular matrix may influence the local transport of macromolecules, leading to anisotropic diffusion depending on the relative size of the molecule and that of extracellular matrix structures. Here we present what we believe is a novel photobleaching technique for measuring the anisotropic diffusivity of macromolecules in collagenous tissues. We hypothesized that macromolecular diffusion is anisotropic in collagenous tissues, depending on molecular size and the local organization of the collagen structure. A theoretical model and experimental protocol for fluorescence imaging of continuous point photobleaching was developed to measure diffusional anisotropy. Significant anisotropy was observed in highly ordered collagenous tissues such as ligament, with diffusivity ratios >2 along the fiber direction compared to the perpendicular direction. In less-ordered tissues such as articular cartilage, diffusional anisotropy was dependent on site in the tissue and size of the diffusing molecule. Anisotropic diffusion was also dependent on the size of the diffusing molecule, with greatest anisotropy observed for larger molecules. These findings suggest that diffusional transport of macromolecules is anisotropic in collagenous tissues, with higher rates of diffusion along primary orientation of collagen fibers.     

Diffusion and electrophoretic mobility of single-stranded RNA from molecular dynamics simulations

Hydrodynamic properties of small single-stranded RNA homopolymers with three and six nucleotides in free solution are determined from molecular dynamics simulations in explicit solvent. We find that the electrophoretic mobility increases with increasing RNA length, consistent with experiment. Diffusion coefficients of RNA, corrected for finite-size effects and solvent viscosity, agree well with those estimated from experiments and hydrodynamic calculations. The diffusion coefficients and electrophoretic mobilities satisfy a Nernst-Einstein relation in which the effective charge of RNA is reduced by the charge of transiently bound counterions. Fluctuations in the counterion atmosphere are shown to enhance the diffusive spread of RNA molecules drifting along the direction of the external electric field. As a consequence, apparent diffusion coefficients measured by capillary zone electrophoresis can be significantly larger than the actual values at certain experimental conditions.     

Permeability of composite chondrocyte-culture-millipore membranes to solutes of varying size and shape.

A model connective-tissue system was developed that is amenable to the determination of permeability coefficients of diffusing solutes. The system involves the culture of 13-day chick-embryo chondrocytes on a Millipore filter (HA:0.45 micron pore size) to form what is, in effect, a confluent, extremely thin cartilage slice of uniform thickness. These cultured chondrocyte membranes were used to measure the steady-state flux of radioactively labelled low-molecular-weight solutes and micro-ions. Similarly, the permeability coefficients of either radioactively labelled or enzymically active proteins across the membranes were determined. The membrane was found to have no marked effects on the diffusional behaviour of low-molecular-weight non-electrolytes (water, proline, ribose, glucose, sorbitol, raffinose). For micro-ions (Na+, SO42-, Cl-, glutamate, glucuronate,), the diffusive behaviour was found to be markedly affected by the ionic strength of the solvent used in a manner which was consistent with a Donnan distribution resulting from the immobilized proteoglycans. Globular proteins permeated the membrane at rates which decreased as the molecular size of the diffusing solute increased. The apparent diffusion rates of fibrinogen and of collagen through the membranes were greater than would be expected on the basis of their diffusion coefficients in free solution. Reasons for this behaviour are discussed.     

Analysis of lateral diffusion from a spherical cell surface to a tubular projection.

Cell surfaces are often heterogeneous with respect to the lateral distribution and mobility of membrane components. Because lateral mobility is related to membrane structure, measurement of a particular component's local diffusion coefficient within a distinct surface region provides useful information about the formation and maintenance of that region. Many structurally interesting cell surface features can be described as narrow tubular projections from the body of the cell. In a companion paper, we consider the thin "tethers" that can be mechanically drawn from the red blood cell membrane, and we measure the transport of fluorescent integral proteins from the surface of the cell body onto the tether. In this paper we present an analysis to describe the surface diffusion of membrane particles from a spherical shell onto a thin cylindrical process. Provision is made for different rates of diffusion within the two morphologically distinct regions. The relative role of each region in controlling the diffusive flux between regions is determined primarily by a single dimensionless parameter. This parameter incorporates the ratio of the two diffusion coefficients as well as the dimensions of each region. The analysis can be applied to a fluorescence photobleaching experiment in which the extended process is bleached. If the dimensions of the spherical cell body and the cylindrical extension are known, then the diffusion coefficients of both regions can be determined from the experimental fluorescence recovery curve.     

Single-particle tracking: effects of corrals.

Structural proteins of the membrane skeleton are thought to form "corrals" at the membrane surface, and these corrals may restrict lateral diffusion of membrane proteins. Recent experimental developments in single-particle tracking and laser trapping make it possible to examine the corral model in detail. Techniques to interpret these experiments are presented. First, escape times for a diffusing particle in a corral are obtained from Monte Carlo calculations and analytical solutions for various corral sizes, shapes, and escape probabilities, and reduced to a common curve. Second, the identification of corrals in tracking experiments is considered. The simplest way to identify corrals is by sight. If the walls are impermeable enough, a trajectory fills the corral before the diffusing particle escapes. The fraction of distinct sites visited before escape is calculated for corrals of various sizes, shapes, and escape probabilities, and reduced to a common curve. This fraction is also a measure of the probability that the diffusing species will react with another species in the corral before escaping. Finally, the effect of the sampling interval on the measurement of the short-range diffusion coefficient is examined.     

The fence and picket structure of the plasma membrane of live cells as revealed by single molecule techniques (Review)

Models of the organization of the plasma membrane of live cells as discovered through diffusion measurements of integral membrane molecules (transmembrane and GPI-anchored proteins, and lipid) at the single molecule level are discussed. Diffusion of transmembrane protein and, indeed, even lipid is anomalous in that the molecules tend to diffuse freely in limited size compartments, with infrequent intercompartment transitions. This average residency time in a compartment is dependent on the diffusing species and on its state of oligomerization, becoming completely confined to a single compartment upon sufficient oligomerization. This will be of great importance in determining cellular mechanisms for controlling the random diffusive motion of membrane molecules and in understanding signalling processes.     

Analysis of fluorophore diffusion by continuous distributions of diffusion coefficients: application to photobleaching measurements of multicomponent and anomalous diffusion.

Fluorescence recovery after photobleaching (FRAP) is widely used to measure fluorophore diffusion in artificial solutions and cellular compartments. Two new strategies to analyze FRAP data were investigated theoretically and applied to complex systems with anomalous diffusion or multiple diffusing species: 1) continuous distributions of diffusion coefficients, alpha(D), and 2) time-dependent diffusion coefficients, D(t). A regression procedure utilizing the maximum entropy method was developed to resolve alpha(D) from fluorescence recovery curves, F(t). The recovery of multi-component alpha(D) from simulated F(t) with random noise was demonstrated and limitations of the method were defined. Single narrow Gaussian alpha(D) were recovered for FRAP measurements of thin films of fluorescein and size-fractionated FITC-dextrans and Ficolls, and multi-component alpha(D) were recovered for defined fluorophore mixtures. Single Gaussian alpha(D) were also recovered for solute diffusion in viscous media containing high dextran concentrations. To identify anomalous diffusion from FRAP data, a theory was developed to compute F(t) and alpha(D) for anomalous diffusion models defined by arbitrary nonlinear mean-squared displacement <x2> versus time relations. Several characteristic alpha(D) profiles for anomalous diffusion were found, including broad alpha(D) for subdiffusion, and alpha(D) with negative amplitudes for superdiffusion. A method to deduce apparent D(t) from F(t) was also developed and shown to provide useful complementary information to alpha(D). alpha(D) and D(t) were determined from photobleaching measurements of systems with apparent anomalous subdiffusion (nonuniform solution layer) and superdiffusion (moving fluid layer). The results establish a practical strategy to characterize complex diffusive phenomena from photobleaching recovery measurements.     

Thermobifida fusca cellulases exhibit limited surface diffusion on bacterial micro‐crystalline cellulose

Elucidation of cellulase–cellulose interactions is key to modeling biomass deconstruction and in understanding the processes that lead to cellulase inactivation. Here, fluorescence recovery after photobleaching and single molecule tracking (SMT) experiments are used to assess the surface diffusion of Thermobifida fusca cellulases on bacterial micro‐crystalline cellulose. Our results show that cellulases exhibit limited surface diffusion when bound to crystalline cellulose and that a large fraction of the cellulases remain immobile even at temperatures optimal for catalysis. A comparison of our experimental results to Monte Carlo (MC) simulations, which use published diffusion coefficients to model cellulase displacements, shows that even those enzymes that are mobile on the cellulose surface exhibit significantly slower diffusive motions than previously reported. In addition, it is observed that the enzymes that show significant displacements exhibit complex, non‐steady surface motions, which suggest that cellulose–bound cellulases exist in molecular states with different diffusive characteristics. These results challenge the notion that cellulases can freely diffuse over cellulose surfaces without catalyzing bond cleavage. Biotechnol. Bioeng. 2013; 110: 47–56. © 2012 Wiley Periodicals, Inc.     

A new FRAP/FRAPa method for three-dimensional diffusion measurements based on multiphoton excitation microscopy

We present a new convenient method for quantitative three-dimensionally resolved diffusion measurements based on the photobleaching (FRAP) or photoactivation (FRAPa) of a disk-shaped area by the scanning laser beam of a multiphoton microscope. Contrary to previously reported spot-photobleaching protocols, this method has the advantage of full scalability of the size of the photobleached area and thus the range of diffusion coefficients, which can be measured conveniently. The method is compatible with low as well as high numerical aperture objective lenses, allowing us to perform quantitative diffusion measurements in three-dimensional extended samples as well as in very small volumes, such as cell nuclei. Furthermore, by photobleaching/photoactivating a large area, diffusion along the optical axis can be measured separately, which is convenient when studying anisotropic diffusion. First, we show the rigorous mathematical derivation of the model, leading to a closed-form formula describing the fluorescence recovery/redistribution phase. Next, the ability of the multiphoton FRAP method to correctly measure absolute diffusion coefficients is tested thoroughly on many test solutions of FITC-dextrans covering a wide range of diffusion coefficients. The same is done for the FRAPa method on a series of photoactivatable green fluorescent protein solutions with different viscosities. Finally, we apply the method to photoactivatable green fluorescent protein diffusing freely in the nucleus of living NIH-3T3 mouse embryo fibroblasts.     

Fluorescence Recovery after Merging a Droplet to Measure the Two-dimensional Diffusion of a Phospholipid Monolayer

We introduce a new method to measure the lateral diffusivity of a surfactant monolayer at the fluid-fluid interface, called fluorescence recovery after merging (FRAM). FRAM adopts the same principles as the fluorescence recovery after photobleaching (FRAP) technique, especially for measuring fluorescence recovery after bleaching a specific area, but FRAM uses a drop coalescence instead of photobleaching dye molecules to induce a chemical potential gradient of dye molecules. Our technique has several advantages over FRAP: it only requires a fluorescence microscope rather than a confocal microscope equipped with high power lasers; it is essentially free from the selection of fluorescence dyes; and it has far more freedom to define the measured diffusion area. Furthermore, FRAM potentially provides a route for studying the mixing or inter-diffusion of two different surfactants, when the monolayers at a surface of droplet and at a flat air/water interface are prepared with different species, independently.     

Resolving Cytosolic Diffusive States in Bacteria by Single-Molecule Tracking

The trajectory of a single protein in the cytosol of a living cell contains information about its molecular interactions in its native environment. However, it has remained challenging to accurately resolve and characterize the diffusive states that can manifest in the cytosol using analytical approaches based on simplifying assumptions. Here, we show that multiple intracellular diffusive states can be successfully resolved if sufficient single-molecule trajectory information is available to generate well-sampled distributions of experimental measurements and if experimental biases are taken into account during data analysis. To address the inherent experimental biases in camera-based and MINFLUX-based single-molecule tracking, we use an empirical data analysis framework based on Monte Carlo simulations of confined Brownian motion. This framework is general and adaptable to arbitrary cell geometries and data acquisition parameters employed in two-dimensional or three-dimensional single-molecule tracking. We show that, in addition to determining the diffusion coefficients and populations of prevalent diffusive states, the timescales of diffusive state switching can be determined by stepwise increasing the time window of averaging over subsequent single-molecule displacements. Time-averaged diffusion analysis of single-molecule tracking data may thus provide quantitative insights into binding and unbinding reactions among rapidly diffusing molecules that are integral for cellular functions.