A closed-form analytic expression for FRAP formula for the binding diffusion model

One of the most dominant methods cells use for a large class of cellular processes is reaction (or binding) diffusion kinetics, which are controlled by kinetic constants such as diffusion coefficients and on/off binding rate constants. Fluorescence recovery after photobleaching (FRAP) can be used to determine these kinetic constants in living cells. While an analytic expression for FRAP formulae for pure diffusion has been available for some time, an analytic FRAP formula for the binding diffusion model has not been reported yet. Here, we present an analytic FRAP formula for the binding diffusion model in an explicit form allowing for diffusion of the bound complex for either a uniform circle laser profile or a Gaussian laser profile.     

A mathematical model of actin filament turnover for fitting FRAP data

A novel mathematical model of the actin dynamics in living cells under steady-state conditions has been developed for fluorescence recovery after photobleaching (FRAP) experiments. As opposed to other FRAP fitting models, which use the average lifetime of actins in filaments and the actin turnover rate as fitting parameters, our model operates with unbiased actin association/dissociation rate constants and accounts for the filament length. The mathematical formalism is based on a system of stochastic differential equations. The derived equations were validated on synthetic theoretical data generated by a stochastic simulation algorithm adapted for the simulation of FRAP experiments. Consistent with experimental findings, the results of this work showed that (1) fluorescence recovery is a function of the average filament length, (2) the F-actin turnover and the FRAP are accelerated in the presence of actin nucleating proteins, (3) the FRAP curves may exhibit both a linear and non-linear behaviour depending on the parameters of actin polymerisation, and (4) our model resulted in more accurate parameter estimations of actin dynamics as compared with other FRAP fitting models. Additionally, we provide a computational tool that integrates the model and that can be used for interpretation of FRAP data on actin cytoskeleton.     

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.     

Behavior of a fluorescent analogue of calmodulin in living 3T3 cells

We have prepared and partially characterized a lissamine-rhodamine B fluorescent analogue of calmodulin, LRB-CM. The analogue had a dye/protein ratio of approximately 1.0 and contained no free dye or contaminating labeled proteins. LRB-CM was indistinguishable from native calmodulin upon SDS PAGE and in assays of phosphodiesterase and myosin light chain kinase. The emission spectrum of LRB-CM was insensitive to changes in pH, ionic strength, and temperature over the physiological range, but the apparent quantum yield was influenced somewhat by divalent cation concentration. LRB-CM injected into living Swiss 3T3 fibroblasts became associated with nitrobenzoxadiazole-phallacidin staining stress fibers in some interphase cells. LRB-CM and acetamidofluorescein-labeled actin co-injected into the same cell both became associated with fibers in some cells, but in most cases association of the two analogues with fibers was mutually exclusive. This suggests that calmodulin may differ from actin in the timing of incorporation into stress fibers or that we have distinguished distinct populations of stress fibers. We were able to detect no direct interaction of LRB-CM with actin by fluorescence photobleaching recovery (FRAP) of aqueous solutions. Interaction of LRB-CM with myosin light chain kinase also was not detected by FRAP. This suggests that the mean lifetime of the calmodulin-myosin light chain kinase complex is too short to affect the diffusion coefficient of calmodulin. We examined various fluorescent derivatives of proteins and dextrans as suitable control molecules for quantitative fluorescent analogue cytochemistry in living cells. Fluorescein isothiocyanate-dextrans were found to be preferable to all the proteins tested, since their mobilities in cytoplasm were inversely dependent on molecular size and there was no evidence of binding to intracellular components. In contrast, FRAP of LRB-CM in the cytoplasm of living 3T3 cells suggested that the analogue interacts with intracellular components with a range of affinities. The mobility of LRB-CM in the cytoplasm was sensitive to treatment of the cells with trifluoperazine, which suggests that at least some of the intracellular binding sites are specific for calmodulin in the calcium-bound form. FRAP of LRB-CM in the nuclei of living 3T3 cells indicated that the analogue was highly mobile within the nucleus but entered the nucleus from the cytoplasm much more slowly than fluorescein isothiocyanate-dextran of comparable molecular size and much more slowly than predicted from its mobility in cytoplasm.     

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.     

The structural integrity of the endoplasmic reticulum, and its possible regulation by inositol 1,3,4,5-tetrakisphosphate

The endoplasmic reticulum (ER) is a dynamic organelle thought to consist of a single interconnected network of membranes. Using fluorescence recovery after photobleaching (FRAP) of HEK-293 cells dually transfected with soluble fluorescent proteins targeted to the ER (GFP) and mitochondria (DsRed), we have confirmed this continuity, which contrasts that of the mitochondria, which behave as a population of discrete organelles. The degree of ER integrity (interconnected versus fragmented) has been suggested to be regulated in some cells by inositol 1,3,4,5-tetrakisphosphate (Ins(1,3,4,5)P(4)). In HEK-293 and freshly isolated murine lacrimal acinar cells, we manipulated ER structure by disrupting cellular Ca(2+) homeostasis with the Ca(2+) ionophore ionomycin, and by permeabilisation of the plasma membrane, protocols known to cause ER fragmentation. However, we were subsequently unable to detect by FRAP any significant effect of Ins(1,3,4,5)P(4) on ER integrity.     

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.     

Characterization of arabinoxylan/cellulose nanocrystals gels to investigate fluorescent probes mobility in bioinspired models of plant secondary cell wall

Biomass from lignocellulose (LC) is a highly complex network of cellulose, hemicellulose, and lignin, which is considered to be a sustainable source of fuels, chemicals and materials. To achieve an environmental friendly and efficient LC upgrading, a better understanding of the LC architecture is necessary. We have devised some LC bioinspired model systems, based on arabinoxylan gels, in which mobility of dextrans and BSA grafted with FITC has been studied by FRAP. Our results indicate that the probes diffusion is more influenced by their hydrodynamic radius than by the gel mesh size. The addition of some cellulose nanocrystals (CNCs) decreases polymer chain mobility and has low effect on the probes diffusion, suggesting that the gels are better organized in the presence of CNCs, as shown by rheological measurements and scanning electronic microscopy observations. This demonstrates that the FRAP analysis can be a powerful tool to screen the architecture of LC model systems.     

Measurement of the lateral mobility of cell surface components in single living cells by fluorescence recovery after photobleaching

The use of fluorescence recovery after photobleaching (FRAP) techniques to monitor the lateral mobility of plant lectin-receptor complexes on the surface of single, living mammalian cells is described in detail. FRAP measurements indicate that over 75% of the wheat germ agglutinin receptor (WGA-receptor) complexes on the surface of human embryo fibroblasts are mobile. These WGA-receptor complexes diffuse laterally (as opposed to flow) on the cell surface with a diffusion coefficient in the range of 2 × 10^?11 to 2 × 10^?10 cm²/sec. Both the percentage of mobile WGA-receptor complexes and the mean diffusion coefficient of these complexes are higher than that obtained from earlier FRAP measurements of the mobility of concanavalin A-receptor (Con A-receptor) complexes in a variety of cell types. The possible reasons for the differing mobilities of WGA and Con A receptors are discussed.     

Improved FRAP Measurements on Biofilms

We expand the standard fluorescence recovery after photobleaching (FRAP) model introduced by Axelrod et al. in 1976. Our goal is to capture some of the following common artifacts observed in the fluorescence measurements obtained with a confocal laser scanning microscope in biofilms: 1) linear drift, 2) exponential decrease (due to bleaching during the measurements), 3) stochastic Gaussian noise, and 4) uncertainty in the exact time point of the onset of fluorescence recovery. To fit the resulting stochastic model to data from FRAP measurements and to estimate all unknown model parameters, we apply a suitably adapted Metropolis-Hastings algorithm. In this way, a more accurate estimation of the diffusion coefficient of the fluorophore is achieved. The method was tested on data obtained from FRAP measurements on a cultivated biofilm.     

FKBP12-rapamycin-associated protein (FRAP) autophosphorylates at serine 2481 under translationally repressive conditions

The FKBP12-rapamycin associated protein (FRAP, also RAFT, mTOR) belongs to a family of phosphatidylinositol kinase-related kinases. These kinases mediate cellular responses to stresses such as DNA damage and nutrient deprivation in a variety of eukaryotes from yeast to humans. FRAP regulates G(1) cell cycle progression and translation initiation in part by controlling the phosphorylation states of a number of translational and cell cycle regulators. Although FRAP is known to be phosphorylated in vivo and to phosphorylate several proteins (including itself) in vitro, FRAP's phosphorylation sites and substrate specificity are unknown. We report here the identification of a FRAP autophosphorylation site. This site, Ser-2481, is located in a hydrophobic region near the conserved carboxyl-terminal FRAP tail. We demonstrate that the COOH-terminal tail is required for FRAP kinase activity and for signaling to the translational regulator p70(s6k) (ribosomal subunit S6 kinase). Phosphorylation of wild-type but not kinase-inactive FRAP occurs at Ser-2481 in vivo, suggesting that Ser-2481 phosphorylation is a marker of FRAP autokinase activity in cells. FRAP autophosphorylation is blocked completely by wortmannin treatment but not by rapamycin treatment, amino acid deprivation, or serum withdrawal, treatments that lead to acute dephosphorylation of eIF4E-binding protein (4E-BP1) and p70(s6k). Ser-2481 phosphorylation increases slightly upon c-Akt/PKB activation and dramatically upon calyculin A treatment of T-cells. These results suggest that FRAP-responsive dephosphorylation of 4E-BP1 and p70(s6k) occurs through a mechanism other than inhibition of intrinsic FRAP kinase activity.     

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.     

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.     

Spindle microtubule dynamics: modulation by metabolic inhibitors

Recent experiments have shown that spindle microtubules are exceedingly dynamic. Measurements of fluorescence recovery after photobleaching (FRAP), in cells previously microinjected with fluorescent tubulin, provide quantitative information concerning the rate of turnover, or exchange, of tubulin subunits with the population of microtubules in living cells at steady state. In an effort to elucidate the pathways and factors that regulate tubulin exchange with microtubules in living cells, we have investigated the energy requirements for tubulin turnover as measured by FRAP. Spindle morphology was not detectably altered in cells incubated with 5 mM sodium azide and 1 mM 2-deoxyglucose (Az/DOG) for 5 minutes, as assayed by polarized light microscopy and antitubulin immunofluorescence. In FRAP experiments on these ATP-depleted cells, the average rate of recovery and the average percent of bleached fluorescence recovered were reduced to 37% and 30% of controls, respectively. When the inhibitors were removed, cells continued through mitosis, and rapid FRAP was restored. In the presence of azide and glucose, the rate of recovery and percent of fluorescence recovered were only slightly reduced, demonstrating that energy production via glycolysis can support microtubule turnover. Longer incubations with Az/DOG altered the microtubule organization in mitotic cells: astral microtubules lengthened and spindle fibers shortened. Furthermore, both astral and spindle microtubules became resistant to nocodazole-induced disassembly under these conditions. Together these observations indicate that microtubule dynamics require ATP and suggest a relationship between microtubule organization and turnover.     

Mobility of the IsiA chlorophyll-binding protein in cyanobacterial thylakoid membranes

We are using fluorescence recovery after photobleaching (FRAP) to probe the dynamics of thylakoid membranes in vivo in cells of the cyanobacterium Synechococcus sp. PCC7942. We have shown previously that the light-harvesting phycobilisomes diffuse quite rapidly on the thylakoid membrane surface. However, the photosystem II core complexes appear completely immobile. This raises the possibility that all of the membrane integral protein complexes in the thylakoid membrane are locked into a rather rigid array. Alternatively, it is possible that photosystem II is specifically anchored in the membrane, with other membrane proteins able to diffuse around it. We have now resolved this question by studying the diffusion of a second integral membrane protein, the IsiA chlorophyll-binding protein. IsiA is induced under iron starvation and some other stress conditions. In iron-stressed cyanobacterial cells, a high proportion of chlorophyll fluorescence comes from IsiA. This makes it straightforward to examine the diffusion of IsiA by FRAP. We find that the complex is mobile with a mean diffusion coefficient of approximately 3 x 10(-11) cm(2) s(-1). Thus it is clear that some thylakoid membrane proteins are mobile and that there must be a specific anchor that prevents photosystem II diffusion. We discuss the implications for the structure and function of the cyanobacterial thylakoid membrane.     

荧光漂白后恢复技术及其在活细胞分子机制研究中的应用

荧光漂白后恢复（FRAP）是一项利用荧光探针研究活体细胞中各类分子迁移特性的技术。简要介绍了FRAP技术的原理和具体实施要求,列出了动态比和扩散系数的计算公式,并例举了近几年FRAP技术在细胞分子机制研究中的应用。