Determining diffusion coefficients in inhomogeneous tissues using fluorescence recovery after photobleaching

Diffusion plays an important role in the transport of nutrients and signaling molecules in cartilaginous tissues. Diffusion coefficients can be measured by fluorescence recovery after photobleaching (FRAP). Available methods to analyze FRAP data, however, assume homogeneity in the environment of the bleached area and neglect geometrical restrictions to diffusion. Hence, diffusion coefficients in inhomogeneous materials, such as most biological tissues, cannot be assessed accurately. In this study, a new method for analyzing data from FRAP measurements has been developed, which is applicable to inhomogeneous tissues. It is based on a fitting procedure of the intensity recovery after photobleaching with a two-dimensional finite element analysis, which includes Fick's law for diffusion. The finite element analysis can account for distinctive diffusivity in predefined zones, which allows determining diffusion coefficients in inhomogeneous samples. The method is validated theoretically and experimentally in both homogeneous and inhomogeneous tissues and subsequently applied to the proliferation zone of the growth plate. Finally, the importance of accounting for inhomogeneities, for appropriate assessment of diffusivity in inhomogeneous tissues, is illustrated.     

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.     

Line FRAP with the confocal laser scanning microscope for diffusion measurements in small regions of 3-D samples

We present a truly quantitative fluorescence recovery after photobleaching (FRAP) model for use with the confocal laser scanning microscope based on the photobleaching of a long line segment. The line FRAP method is developed to complement the disk FRAP method reported before. Although being more subject to the influence of noise, the line FRAP model has the advantage of a smaller bleach region, thus allowing for faster and more localized measurements of the diffusion coefficient and mobile fraction. The line FRAP model is also very well suited to examine directly the influence of the bleaching power on the effective bleaching resolution. We present the outline of the mathematical derivation, leading to a final analytical expression to calculate the fluorescence recovery. We examine the influence of the confocal aperture and the bleaching power on the measured diffusion coefficient to find the optimal experimental conditions for the line FRAP method. This will be done for R-phycoerythrin and FITC-dextrans of various molecular weights. The ability of the line FRAP method to measure correctly absolute diffusion coefficients in three-dimensional samples will be evaluated as well. Finally we show the application of the method to the simultaneous measurement of free green fluorescent protein diffusion in the cytoplasm and nucleus of living A549 cells.     

Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery

Multiphoton fluorescence photobleaching recovery (MP-FPR) is a technique for measuring the three-dimensional (3D) mobility of fluorescent molecules with 3D spatial resolution of a few microns. A brief, intense flash of mode-locked laser light pulses excites fluorescent molecules via multiphoton excitation in an ellipsoidal focal volume and photobleaches a fraction. Because multiphoton excitation of fluorophores is intrinsically confined to the high-intensity focal volume of the illuminating beam, the bleached region is restricted to a known, three-dimensionally defined volume. Fluorescence in this focal volume is measured with multiphoton excitation, using the attenuated laser beam to measure fluorescence recovery as fresh unbleached dye diffuses in. The time course of the fluorescence recovery signal after photobleaching can be analyzed to determine the diffusion coefficient of the fluorescent species. The mathematical formulas used to fit MP-FPR recovery curves and the techniques needed to properly utilize them to acquire the diffusion coefficients of fluorescently labeled molecules within cells are presented here. MP-FPR is demonstrated on calcein in RBL-2H3 cells, using an anomalous subdiffusion model, as well as in aqueous solutions of wild-type green fluorescent protein, yielding a diffusion coefficient of 8.7 x 10(-7) cm(2)s(-1) in excellent agreement with the results of other techniques.     

Fluorescence photobleaching recovery using total internal reflection interference fringes

Lateral diffusion measurements on cell membrane molecules, most commonly accomplished through fluorescence photobleaching recovery (FPR or FRAP), provide information on such molecules' size, environment, and participation in intermolecular interactions. However, difficulties arise in FPR measurements of lateral dynamics of materials, such as visible fluorescent protein (VFP) fusion proteins, where fluorescent intracellular species contribute to the fluorescence recovery signal and thus distort measurements intended to reflect surface molecules only. A new method helps eliminate these difficulties. In total internal reflection interference fringe FPR, interfering laser beams enter a 1.65-numercial aperture (NA) Olympus objective at the periphery of the back focal plane where the NA exceeds 1.38. This creates an extended interference pattern totally internally reflected at the coverslip-medium interface which excites fluorescence only from fluorescent molecules located where the cell contacts the coverslip. The large illuminated area interrogates many more membrane receptors than spot methods and hence obtains more diffusion information per measurement while rejecting virtually all interfering intracellular fluorescence. We report successful measurements of membrane dynamics of both VFP-containing and conventionally labeled molecules by this technique and compare them with results of other FPR methods.     

Developmental changes in the lateral diffusion of Leydig cell membranes measured by the FRAP method

A simple method for isolation and fluorescence labelling of Leydig cells (L-cells) from rat testes was developed. Lateral diffusion coefficients of both lipid and protein membrane fluorescent probes were measured by the method of fluorescence recovery after photobleaching (FRAP). Age-dependent changes in diffusibility of membrane lipids and proteins were discovered.     

Expanding the scope of quantitative FRAP analysis

In this study, new mathematical models were developed for analysis of fluorescence recovery after photobleaching (FRAP) data to account for features not represented in previous analysis: conical photobleaching geometry, spatial variations in binding of fluorescent molecules, and directed transport of fluorescent molecules. To facilitate computations in conical geometry, a fast computational method for calculation of fluorescence recovery is presented. Two approximations are presented to aid in FRAP analysis when binding varies spatially, one applying to cases of relatively fast diffusion and slow binding and the other to binding of molecules to small cellular structures. Numerical results show that using a model that represents the influential physical processes and that is formulated in the appropriate geometry can substantially improve the accuracy of FRAP calculations.     

Fringe pattern photobleaching, a new method for the measurement of transport coefficients of biological macromolecules.

The conventional method of studying mass transport in membranes by spot photobleaching and then following the recovery of fluorescence has disadvantages. Among them, the need for a high density of fluorescent molecules, the measurement of the beam profile, and a knowledge of the photobleaching processes are of a crucial importance. The application of a planar fringe pattern of light both for the bleaching and the monitoring of the fluorescent molecules solves these three major difficulties. Brownian diffusion coefficients and flow velocities can be measured independently and are averaged over the whole fringe pattern volume. These transport coefficients are explored over the wide range of experimentally accessible distances (from an interfringe spacing 0.5-50 micron). The quantification of the mobile and immobile components is further simplified by scanning the fringe pattern and detecting only a modulated fluorescence recovery signal. The fringe pattern photobleaching method is particularly adapted to the measurements of diffusion coefficients and flow velocity of membrane components, as well as of cytoplasmic proteins. The theoretical results and the test experiments with fluorescent bovine serum albumin are described.     

Simulations of (an)isotropic diffusion on curved biological surfaces

We present a computational particle method for the simulation of isotropic and anisotropic diffusion on curved biological surfaces that have been reconstructed from image data. The method is capable of handling surfaces of high curvature and complex shape, which are often encountered in biology. The method is validated on simple benchmark problems and is shown to be second-order accurate in space and time and of high parallel efficiency. It is applied to simulations of diffusion on the membrane of endoplasmic reticula (ER) in live cells. Diffusion simulations are conducted on geometries reconstructed from real ER samples and are compared to fluorescence recovery after photobleaching experiments in the same ER samples using the transmembrane protein tsO45-VSV-G, C-terminally tagged with green fluorescent protein. Such comparisons allow derivation of geometry-corrected molecular diffusion constants for membrane components from fluorescence recovery after photobleaching data. The results of the simulations indicate that the diffusion behavior of molecules in the ER membrane differs significantly from the volumetric diffusion of soluble molecules in the lumen of the same ER. The apparent speed of recovery differs by a factor of approximately 4, even when the molecular diffusion constants of the two molecules are identical. In addition, the specific shape of the membrane affects the recovery half-time, which is found to vary by a factor of approximately 2 in different ER samples.     

Probing the structure of cytoplasm

We have used size-fractionated, fluorescent dextrans to probe the structure of the cytoplasmic ground substance of living Swiss 3T3 cells by fluorescence recovery after photobleaching and video image processing. The data indicate that the cytoplasm of living cells has a fluid phase viscosity four times greater than water and contains structural barriers that restrict free diffusion of dissolved macromolecules in a size-dependent manner. Assuming these structural barriers comprise a filamentous meshwork, the combined fluorescence recovery after photobleaching and imaging data suggest that the average pore size of the meshwork is in the range of 300 to 400 A, but may be as small as 200 A in some cytoplasmic domains.     

Reversible photobleaching of enhanced green fluorescent proteins

Color variants of green fluorescent protein (GFP) are increasingly used for multicolor imaging, fluorescence resonance energy transfer (FRET), and fluorescence recovery after photobleaching (FRAP). Here we show that experimental settings commonly used in these imaging experiments may induce an as yet uncharacterized reversible photobleaching of fluorescent proteins, which is more pronounced at acidic pH. Whereas the reversible photobleaching spectrum of eCFP corresponds to its absorption spectrum, reversible photobleaching spectra of yellow variants resemble absorption spectra of their protonated states. Fluorescence intensities recover spontaneously with time constants of 25-58 s. The recovery of eCFP can be further accelerated by illumination. The resulting steady-state fluorescence reflects a variable equilibrium between reversible photobleaching, spontaneous recovery, and light-induced recovery. These processes can cause significant artifacts in commonly applied imaging techniques, photobleach-based FRET determinations, and FRAP assays.     

Translational diffusion of bovine prothrombin fragment 1 weakly bound to supported planar membranes: measurement by total internal reflection with fluorescence pattern photobleaching recovery.

Previous work has shown that bovine prothrombin fragment 1 binds to substrate-supported planar membranes composed of phosphatidylcholine (PC) and phosphatidylserine (PS) in a Ca(2+)-specific manner. The apparent equilibrium dissociation constant is 1-15 microM, and the average membrane residency time is approximately 0.25 s-1. In the present work, fluorescence pattern photobleaching recovery with evanescent interference patterns (TIR-FPPR) has been used to measure the translational diffusion coefficients of the weakly bound fragment 1. The results show that the translational diffusion coefficients on fluid-like PS/PC planar membranes are on the order of 10(-9) cm2/s and are reduced when the fragment 1 surface density is increased. Control measurements were carried out for fragment 1 on solid-like PS/PC planar membranes. The dissociation kinetics were similar to those on fluid-like membranes, but protein translational mobility was not detected. TIR-FPPR was also used to measure the diffusion coefficient of the fluorescent lipid NBD-PC in fluid-like PS/PC planar membranes. In these measurements, the diffusion coefficient was approximately 10(-8) cm2/s, which is consistent with that measured by conventional fluorescence pattern photobleaching recovery. This work represents the first measurement of a translational diffusion coefficient for a protein weakly bound to a membrane surface.     

Accessing molecular dynamics in cells by fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) analyzes spontaneous fluctuations in the fluorescence emission of small molecular ensembles, thus providing information about a multitude of parameters, such as concentrations, molecular mobility and dynamics of fluorescently labeled molecules. Performed within diffraction-limited confocal volume elements, FCS provides an attractive alternative to photobleaching recovery methods for determining intracellular mobility parameters of very low quantities of fluorophores. Due to its high sensitivity sufficient for single molecule detection, the method is subject to certain artifact hazards that must be carefully controlled, such as photobleaching and intramolecular dynamics, which introduce fluorescence flickering. Furthermore, if molecular mobility is to be probed, nonspecific interactions of the labeling dye with cellular structures can introduce systematic errors. In cytosolic measurements, lipophilic dyes, such as certain rhodamines that bind to intracellular membranes, should be avoided. To study free diffusion, genetically encoded fluorescent labels such as green fluorescent protein (GFP) or DsRed are preferable since they are less likely to nonspecifically interact with cellular substructures.     

Intracellular macromolecular mobility measured by fluorescence recovery after photobleaching with confocal laser scanning microscopes

Fluorescence recovery after photobleaching (FRAP) is a widely used tool for estimating mobility parameters of fluorescently tagged molecules in cells. Despite the widespread use of confocal laser scanning microscopes (CLSMs) to perform photobleaching experiments, quantitative data analysis has been limited by lack of appropriate practical models. Here, we present a new approximate FRAP model for use on any standard CLSM. The main novelty of the method is that it takes into account diffusion of highly mobile molecules during the bleach phase. In fact, we show that by the time the first postbleach image is acquired in a CLSM a significant fluorescence recovery of fast-moving molecules has already taken place. The model was tested by generating simulated FRAP recovery curves for a wide range of diffusion coefficients and immobile fractions. The method was further validated by an experimental determination of the diffusion coefficient of fluorescent dextrans and green fluorescent protein. The new FRAP method was used to compare the mobility rates of fluorescent dextrans of 20, 40, 70, and 500 kDa in aqueous solution and in the nucleus of living HeLa cells. Diffusion coefficients were lower in the nucleoplasm, particularly for higher molecular weight dextrans. This is most likely caused by a sterical hindrance effect imposed by nuclear components. Decreasing the temperature from 37 to 22 degrees C reduces the dextran diffusion rates by approximately 30% in aqueous solution but has little effect on mobility in the nucleoplasm. This suggests that spatial constraints to diffusion of dextrans inside the nucleus are insensitive to temperature.     

From fixed to FRAP: measuring protein mobility and activity in living cells

Experiments with fluorescence recovery after photobleaching (FRAP) started 30 years ago to visualize the lateral mobility and dynamics of fluorescent proteins in living cells. Its popularity increased when non-invasive fluorescent tagging became possible with the green fluorescent protein (GFP). Many researchers use GFP to study the localization of fusion proteins in fixed or living cells, but the same fluorescent proteins can also be used to study protein mobility in living cells. Here we review the potential of FRAP to study protein dynamics and activity within a single living cell. These measurements can be made with most standard confocal laser-scanning microscopes equipped with photobleaching protocols.     

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).     

Simplified Equation to Extract Diffusion Coefficients from Confocal FRAP Data

Quantitative measurements of diffusion can provide important information about how proteins and lipids interact with their environment within the cell and the effective size of the diffusing species. Confocal fluorescence recovery after photobleaching (FRAP) is one of the most widely accessible approaches to measure protein and lipid diffusion in living cells. However, straightforward approaches to quantify confocal FRAP measurements in terms of absolute diffusion coefficients are currently lacking. Here, we report a simplified equation that can be used to extract diffusion coefficients from confocal FRAP data using the half time of recovery and effective bleach radius for a circular bleach region, and validate this equation for a series of fluorescently labeled soluble and membrane‐bound proteins and lipids. We show that using this approach, diffusion coefficients ranging over three orders of magnitude can be obtained from confocal FRAP measurements performed under standard imaging conditions, highlighting its broad applicability.     

Lateral diffusion of proteins in the periplasm of Escherichia coli.

We have introduced biologically active, fluorescently labeled maltose-binding protein into the periplasmic space of Escherichia coli and measured its lateral diffusion coefficient by the fluorescence photobleaching recovery method. Diffusion of this protein in the periplasm was found to be surprisingly low (lateral diffusion coefficient, 0.9 X 10(-10) cm2 s-1), about 1,000-fold lower than would be expected for diffusion in aqueous medium and almost 100-fold lower than for an equivalent-size protein in the cytoplasm. Galactose-binding protein, myoglobin, and cytochrome c were also introduced into the periplasm and had diffusion coefficients identical to that determined for the maltose-binding protein. For all proteins nearly 100% recovery of fluorescence was obtained after photobleaching, indicating that the periplasm is a single contiguous compartment surrounding the cell. These data have considerable implications for periplasmic structure and for the role of periplasmic proteins in transport and chemotaxis.