A quantitative approach to analyze binding diffusion kinetics by confocal FRAP

Most of the important types of interactions that occur in cells can be characterized as binding-diffusion type processes, and can be quantified by kinetic rate constants such as diffusion coefficients (D) and binding rate constants (k_on and k_off). Confocal FRAP is a potentially important tool for the quantitative analysis of intracellular binding-diffusion kinetics, but how to dependably extract accurate kinetic constants from such analyses is still an open question. To this end, in this study, we developed what we believe is a new analytical model for confocal FRAP-based measurements of intracellular binding-diffusion processes, based on a closed-form equation of the FRAP formula for a spot photobleach geometry. This approach incorporates a binding diffusion model that allows for diffusion of both the unbound and bound species, and also compensates for binding diffusion that occurs during photobleaching, a critical consideration in confocal FRAP analysis. In addition, to address the problem of parametric multiplicity, we propose a scheme to reduce the number of fitting parameters in the effective diffusion subregime when D's for the bound and unbound species are known. We validate this method by measuring kinetic rate constants for the CAAX-mediated binding of Ras to membranes of the endoplasmic reticulum, obtaining binding constants of k_on ∼ 255/s and k_off ∼ 31/s.     

Challenges and artifacts in quantitative photobleaching experiments

Confocal fluorescence recovery after photobleaching (FRAP) is today the prevalent tool when studying the diffusional and kinetic properties of proteins in living cells. Obtaining quantitative data for diffusion coefficients via FRAP, however, is challenged by the fact that both bleaching and scanning take a finite time. Starting from an experimental case, it is shown by means of computer simulations that this intrinsic temporal limitation can lead to a gross underestimation of diffusion coefficients. Determining the binding kinetics of proteins to membranes with FRAP is further shown to be severely hampered by additional diffusional contributions, e.g. diffusion-limited binding. In some cases, the binding kinetics may even be masked entirely by diffusion. As current efforts to approach biological problems with biophysical models have to rely on experimentally determined model parameters, e.g. binding rates and diffusion constants, it is proposed that the accuracy in evaluating FRAP measurements can be improved by means of accompanying computer simulations.     

Investigation of binding mechanisms of nuclear proteins using confocal scanning laser microscopy and FRAP

Fluorescence recovery after photobleaching (FRAP) measurements offer an important tool towards analysing diffusion processes within living biological cells. A model is presented that aims to provide a rigorous theoretical framework from which binding information of proteins from FRAP data can be extracted. A single binding reaction is considered and a set of mathematical equations is introduced that incorporates the concentration of free proteins, vacant binding sites and bound complexes in addition to the on- and off-rates of the proteins. To allow a realistic FRAP model, characteristics of the instruments used to perform FRAP measurements are included in the equation. The proposed model has been designed to be applied to biological samples with a confocal scanning laser microscope (CSLM) equipped with the feature to bleach regions characterised by a radially Gaussian distributed profile. Binding information emerges from FRAP simulations considering the diffusion coefficient, radial extent of the bleached volume and bleach constant as parameters derived from experimental data. The proposed model leads to FRAP curves that depend on the on- and off-rates. Analytical expressions are used to define the boundaries of on- and off-rate parameter space in simplified cases when molecules can move on an infinite domain. A similar approach is ensued when movement is restricted in a compartment with a finite size. The theoretical model can be used in conjunction to experimental data acquired by CSLM to investigate the biophysical properties of proteins in living cells.     

An Improved Mathematical Approach for Determination of Molecular Kinetics in Living Cells with FRAP

The estimation of binding constants and diffusion coefficients of molecules that associate with insoluble molecular scaffolds inside living cells and nuclei has been facilitated by the use of Fluorescence Recovery after Photobleaching (FRAP) in conjunction with mathematical modeling. A critical feature unique to FRAP experiments that has been overlooked by past mathematical treatments is the existence of an `equilibrium constraint': local dynamic equilibrium is not disturbed because photobleaching does not functionally destroy molecules, and hence binding-unbinding proceeds at equilibrium rates. Here we describe an improved mathematical formulation under the equilibrium constraint which provides a more accurate estimate of molecular reaction kinetics within FRAP studies carried out in living cells. Due to incorporation of the equilibrium constraint, the original non-linear kinetic terms become linear allowing for analytical solution of the transport equations and greatly simplifying the estimation process. Based on mathematical modeling and scaling analysis, two experimental measures are identified that can be used to delineate the rate-limiting step. A comprehensive analysis of the interplay between binding-unbinding and diffusion, and its effect on the recovery curve, are presented. This work may help to bring clarity to the study of molecular dynamics within the structural complexity of living cells.     

Improving Parameter Inference from FRAP Data: an Analysis Motivated by Pattern Formation in the <Emphasis Type="BoldItalic">Drosophila</Emphasis> Wing Disc

Fluorescence recovery after photobleaching (FRAP) is used to obtain quantitative information about molecular diffusion and binding kinetics at both cell and tissue levels of organization. FRAP models have been proposed to estimate the diffusion coefficients and binding kinetic parameters of species for a variety of biological systems and experimental settings. However, it is not clear what the connection among the diverse parameter estimates from different models of the same system is, whether the assumptions made in the model are appropriate, and what the qualities of the estimates are. Here we propose a new approach to investigate the discrepancies between parameters estimated from different models. We use a theoretical model to simulate the dynamics of a FRAP experiment and generate the data that are used in various recovery models to estimate the corresponding parameters. By postulating a recovery model identical to the theoretical model, we first establish that the appropriate choice of observation time can significantly improve the quality of estimates, especially when the diffusion and binding kinetics are not well balanced, in a sense made precise later. Secondly, we find that changing the balance between diffusion and binding kinetics by changing the size of the bleaching region, which gives rise to different FRAP curves, provides a priori knowledge of diffusion and binding kinetics, which is important for model formulation. We also show that the use of the spatial information in FRAP provides better parameter estimation. By varying the recovery model from a fixed theoretical model, we show that a simplified recovery model can adequately describe the FRAP process in some circumstances and establish the relationship between parameters in the theoretical model and those in the recovery model. We then analyze an example in which the data are generated with a model of intermediate complexity and the parameters are estimated using models of greater or less complexity, and show how sensitivity analysis can be used to improve FRAP model formulation. Lastly, we show how sophisticated global sensitivity analysis can be used to detect over-fitting when using a model that is too complex.     

Accurate quantification of diffusion and binding kinetics of non-integral membrane proteins by FRAP

Non-integral membrane proteins frequently act as transduction hubs in vital signaling pathways initiated at the plasma membrane (PM). Their biological activity depends on dynamic interactions with the PM, which are governed by their lateral and cytoplasmic diffusion and membrane binding/unbinding kinetics. Accurate quantification of the multiple kinetic parameters characterizing their membrane interaction dynamics has been challenging. Despite a fair number of approximate fitting functions for analyzing fluorescence recovery after photobleaching (FRAP) data, no approach was able to cope with the full diffusion-exchange problem. Here, we present an exact solution and matlab fitting programs for FRAP with a stationary Gaussian laser beam, allowing simultaneous determination of the membrane (un)binding rates and the diffusion coefficients. To reduce the number of fitting parameters, the cytoplasmic diffusion coefficient is determined separately. Notably, our equations include the dependence of the exchange kinetics on the distribution of the measured protein between the PM and the cytoplasm, enabling the derivation of both k(on) and k(off) without prior assumptions. After validating the fitting function by computer simulations, we confirm the applicability of our approach to live-cell data by monitoring the dynamics of GFP-N-Ras mutants under conditions with different contributions of lateral diffusion and exchange to the FRAP kinetics.     

A Generalization of Theory for Two-Dimensional Fluorescence Recovery after Photobleaching Applicable to Confocal Laser Scanning Microscopes

Fluorescence recovery after photobleaching (FRAP) using confocal laser scanning microscopes (confocal FRAP) has become a valuable technique for studying the diffusion of biomolecules in cells. However, two-dimensional confocal FRAP sometimes yields results that vary with experimental setups, such as different bleaching protocols and bleaching spot sizes. In addition, when confocal FRAP is used to measure diffusion coefficients (D) for fast diffusing molecules, it often yields D-values that are one or two orders-of-magnitude smaller than that predicted theoretically or measured by alternative methods such as fluorescence correlation spectroscopy. Recently, it was demonstrated that this underestimation of D can be corrected by taking diffusion during photobleaching into consideration. However, there is currently no consensus on confocal FRAP theory, and no efforts have been made to unify theories on conventional and confocal FRAP. To this end, we generalized conventional FRAP theory to incorporate diffusion during photobleaching so that analysis by conventional FRAP theory for a circular region of interest is easily applicable to confocal FRAP. Finally, we demonstrate the accuracy of these new (to our knowledge) formulae by measuring D for soluble enhanced green fluorescent protein in aqueous glycerol solution and in the cytoplasm and nucleus of COS7 cells.     

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.     

Fluorescence Correlation Spectroscopy Measurements of the Membrane Protein TetA in Escherichia coli Suggest Rapid Diffusion at Short Length Scales

Structural inhomogeneities in biomembranes can lead to complex diffusive behavior of membrane proteins that depend on the length or time scales that are probed. This effect is well studied in eukaryotic cells, but has been explored only recently in bacteria. Here we used fluorescence recovery after photobleaching (FRAP) and fluorescence correlation spectroscopy (FCS) to study diffusion of the membrane protein TetA-YFP in E. coli. We find that the diffusion constant determined from FRAP is comparable to other reports of inner membrane protein diffusion constants in E. coli. However, FCS, which probes diffusion on shorter length scales, gives a value that is almost two orders of magnitude higher and is comparable to lipid diffusion constants. These results suggest there is a population of TetA-YFP molecules in the membrane that move rapidly over short length scales (∼ 400 nm) but move significantly more slowly over the longer length scales probed by FRAP.     

Improving parameter estimation for cell surface FRAP data

Fluorescence Recovery After Photobleaching (FRAP) using the confocal laser scanning microscope has become a standard method used to determine the diffusion coefficient and mobile fraction of cell surface proteins. A common experimental approach is to bleach a stripe on the cell surface and fit the ensuing FRAP curve to a 1D diffusion model. This model is derived from the time course of recovery to an infinitely long stripe bleached on an infinite flat plane. This choice of model dictates the use of a long bleach stripe. We demonstrate that, in the case of a long bleach stripe, the finite extent of the cell leads to significant errors in parameter estimation. We further show that these errors are reduced when a relatively small stripe is bleached. Unfortunately, diffusion to such a region is fundamentally two dimensional and therefore applying the 1D model of diffusion leads to significant errors. We derive an equation suitable for fitting to FRAP data acquired from small bleach regions and analyze its accuracy using simulated data. We propose that the use of a small bleach region along with a two dimensional diffusion model is the ideal protocol for cell surface FRAP.     

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.     

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.     

Validation of Normalizations,Scaling, and Photofading Corrections for FRAP Data Analysis

Fluorescence Recovery After Photobleaching (FRAP) has been a versatile tool to study transport and reaction kinetics in live cells. Since the fluorescence data generated by fluorescence microscopy are in a relative scale, a wide variety of scalings and normalizations are used in quantitative FRAP analysis. Scaling and normalization are often required to account for inherent properties of diffusing biomolecules of interest or photochemical properties of the fluorescent tag such as mobile fraction or photofading during image acquisition. In some cases, scaling and normalization are also used for computational simplicity. However, to our best knowledge, the validity of those various forms of scaling and normalization has not been studied in a rigorous manner. In this study, we investigate the validity of various scalings and normalizations that have appeared in the literature to calculate mobile fractions and correct for photofading and assess their consistency with FRAP equations. As a test case, we consider linear or affine scaling of normal or anomalous diffusion FRAP equations in combination with scaling for immobile fractions. We also consider exponential scaling of either FRAP equations or FRAP data to correct for photofading. Using a combination of theoretical and experimental approaches, we show that compatible scaling schemes should be applied in the correct sequential order; otherwise, erroneous results may be obtained. We propose a hierarchical workflow to carry out FRAP data analysis and discuss the broader implications of our findings for FRAP data analysis using a variety of kinetic models.     

Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope

Confocal scanning laser microscopes (CSLMs) are equipped with the feature to photobleach user-defined regions. This makes them a handy tool to perform fluorescence recovery after photobleaching (FRAP) measurements. To allow quantification of such FRAP experiments, a three-dimensional model has been developed that describes the fluorescence recovery process for a disk-shaped geometry that is photobleached by the scanning beam of a CSLM. First the general mathematical basis is outlined describing the bleaching process for an arbitrary geometry bleached by a scanning laser beam. Next, these general expressions are applied to the bleaching by a CSLM of a disk-shaped geometry and an analytical solution is derived that describes three-dimensional fluorescence recovery in the bleached area as observed by the CSLM. The FRAP model is validated through both the Stokes-Einstein relation and the comparison of the measured diffusion coefficients with their theoretical estimates. Finally, the FRAP model is used to characterize the transport of FITC-dextrans through bulk three-dimensional biological materials: vitreous body isolated from bovine eyes, and lung sputum expectorated by cystic fibrosis patients. The decrease in the diffusion coefficient relative to its value in solution was dependent on the size of the FITC-dextrans in vitreous, whereas it was size-independent in cystic fibrosis sputum.