Maximum-entropy decomposition of fluorescence correlation spectroscopy data: application to liposome–human serum albumin association

Fluorescence correlation spectroscopy was used to measure the diffusion behavior of a mixture of DMPC or DMPC/DMPG liposomes with human serum albumin (HSA) and mesoporphyrin (MP), which was used as the fluorescent label for liposomes and HSA as well. For decomposing the fluorescence intensity autocorrelation function (ACF) into components corresponding to a liposome population, HSA and MP, we used a maximum entropy procedure that computes a distribution of diffusion times consistent with the ACF data. We found that a simple parametric non-linear fit with a discrete set of decay components did not converge to a stable parameter set. The distribution calculated with the maximum entropy method was stable and the average size of the particles calculated from the effective diffusion time was in good agreement with the data determined using the discrete-component fit.     

Separation of the rotational contribution in fluorescence correlation experiments

The theory of fluorescence correlation spectroscopy is reexamined with the aim of separating the contribution of rotational diffusion. Under constant excitation, fluorescence correlation experiments are characterized by three polarizations: one of the incident beam and two of the two photon detectors. A set of experiments of different polarizations is proposed for study. From the results of the experiments the isotropic factor of the fluorescence intensity correlation functions can be determined, which is independent of the rotational motion of the sample molecule. This function can be used to represent each fluorescence intensity correlation function as the product of the isotropic and the rotational factors. The theory is illustrated by an experiment in which rotational diffusion of porcine pancreatic lipase labeled with Texas Red was observed Texas Red is a label that allows precise fluorescence correlation experiments even in the nanosecond time range.     

The photon counting histogram in fluorescence fluctuation spectroscopy.

Fluorescence correlation spectroscopy (FCS) is generally used to obtain information about the number of fluorescent particles in a small volume and the diffusion coefficient from the autocorrelation function of the fluorescence signal. Here we demonstrate that photon counting histogram (PCH) analysis constitutes a novel tool for extracting quantities from fluorescence fluctuation data, i.e., the measured photon counts per molecule and the average number of molecules within the observation volume. The photon counting histogram of fluorescence fluctuation experiments, in which few molecules are present in the excitation volume, exhibits a super-Poissonian behavior. The additional broadening of the PCH compared to a Poisson distribution is due to fluorescence intensity fluctuations. For diffusing particles these intensity fluctuations are caused by an inhomogeneous excitation profile and the fluctuations in the number of particles in the observation volume. The quantitative relationship between the detected photon counts and the fluorescence intensity reaching the detector is given by Mandel's formula. Based on this equation and considering the fluorescence intensity distribution in the two-photon excitation volume, a theoretical expression for the PCH as a function of the number of molecules in the excitation volume is derived. For a single molecular species two parameters are sufficient to characterize the histogram completely, namely the average number of molecules within the observation volume and the detected photon counts per molecule per sampling time epsilon. The PCH for multiple molecular species, on the other hand, is generated by successively convoluting the photon counting distribution of each species with the others. The influence of the excitation profile upon the photon counting statistics for two relevant point spread functions (PSFs), the three-dimensional Gaussian PSF conventionally employed in confocal detection and the square of the Gaussian-Lorentzian PSF for two photon excitation, is explicitly treated. Measured photon counting distributions obtained with a two-photon excitation source agree, within experimental error with the theoretical PCHs calculated for the square of a Gaussian-Lorentzian beam profile. We demonstrate and discuss the influence of the average number of particles within the observation volume and the detected photon counts per molecule per sampling interval upon the super-Poissonian character of the photon counting distribution.     

Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution.

The present paper describes a new experimental scheme for following diffusion and chemical reaction systems of fluorescently labeled molecules in the nanomolar concentration range by fluorescence correlation analysis. In the dual-color fluorescence cross-correlation spectroscopy provided here, the concentration and diffusion characteristics of two fluorescent species in solution as well as their reaction product can be followed in parallel. By using two differently labeled reaction partners, the selectivity to investigate the temporal evolution of reaction product is significantly increased compared to ordinary one-color fluorescence autocorrelation systems. Here we develop the theoretical and experimental basis for carrying out measurements in a confocal dual-beam fluorescence correlation spectroscopy setup and discuss conditions that are favorable for cross-correlation analysis. The measurement principle is explained for carrying out DNA-DNA renaturation kinetics with two differently labeled complementary strands. The concentration of the reaction product can be directly determined from the cross-correlation amplitude.     

Brownian dynamics simulations of fluorescence fluctuation spectroscopy

We have developed a program for the simulation of the fluorescence fluctuations as detected from highly diluted samples of (bio)molecules. The model is applied to translational diffusion and takes into account the hydrodynamic interactions. The solution concentration is kept constant by assuming periodic boundary conditions and spans here the range 0.5< C < 10 nM. We show that the fluorescence correlation functions can be accurately computed on systems of limited size (a few molecules per simulation box) by simulating for a total time approximately 100-300 times the diffusion relaxation time of the fluorescence autocorrelation function. The model is applied also to the simulation of the scanning fluorescence correlation spectroscopy (FCS) and of the photon counting histograms for the confocal collection configuration. Scanning FCS simulations of highly diluted samples (C approximately equals 0.5 nM) show anticorrelation effects in the autocorrelation functions of the fluorescence signal that are less evident for higher concentrations. We suggest here that this effect may be due to the non-uniform occupancy of the scanning area by the fluorophores.     

The standard deviation in fluorescence correlation spectroscopy

The standard deviation (SD) in fluorescence correlation spectroscopy (FCS) has been mostly neglected in applications. However, the knowledge of the correct SD is necessary for an accurate data evaluation, especially when fitting theoretical models to experimental data. In this work, an algorithm is presented that considers the essential features of FCS. It allows prediction of the performance of FCS measurements in various cases, which is important for finding optimal experimental conditions. The program calculates the SD of the experimental autocorrelation function online. This procedure leads to improved parameter estimation, compared to currently used theoretical approximations for the SD. Three methods for the calculation of the SD are presented and compared to earlier analytical solutions (D. E. Koppel. 1974. Phys. Rev. A. 10:1938-1945.), calculation directly from fluorescence intensity values, by averaging several FCS measurements, or by dividing one measurement into a set of shorter data packages. Although the averaging over several measurements yields accurate estimates for the SD, the other two methods are considerably less time consuming, can be run online, and yield comparable results.     

Molecular Brightness Characterization of EGFP In Vivo by Fluorescence Fluctuation Spectroscopy

We characterize the molecular properties of autofluorescence and transiently expressed EGFP in the nucleus and in the cytoplasm of HeLa cells by fluorescence correlation spectroscopy (FCS) and by photon counting histogram (PCH) analysis. PCH has been characterized and applied in vitro, but its potential for in vivo studies needs to be explored. Thus, this study mainly focuses on the characterization of PCH analysis in vivo. The strength of PCH lies in its ability to distinguish biomolecules by their molecular brightness value. Because the concept of molecular brightness is crucial for PCH analysis, we study the molecular brightness of EGFP and determine the statistical accuracy of its measurement under in vivo conditions. We started by characterizing the influence of autofluorescence on EGFP measurements. We found a molecular brightness of EGFP that is a factor of 10 higher than the brightness of the autofluorescence. Moment analysis demonstrates that the contribution of autofluorescence to fluorescence fluctuation experiments is negligible at EGFP concentrations of one protein per excitation volume. The molecular brightness of EGFP measured in the nucleus, the cytoplasm, and in vitro are identical and our study demonstrates that molecular brightness is a very stable and predictable quantity for cellular measurements. In addition to PCH, we also analyzed the autocorrelation function of EGFP. The diffusion coefficient of EGFP is a factor of 3 lower in vivo than compared to in vitro, and a simple diffusion process describes the autocorrelation function. We found that in the nucleus the fluorescence intensity is stable as a function of time, while measurements in the cytoplasm display fluorescence intensity drifts that complicate the data analysis. We introduce and discuss an analysis method that minimizes the influence of the intensity drifts on PCH analysis. This method allows us to recover the correct molecular brightness of EGFP even in the presence of drifts of the fluorescence intensity signal. We found the molecular brightness of EGFP to be a very robust parameter, and anticipate the use of PCH analysis for the study of oligomerization processes in vivo.     

Ligand-receptor kinetics measured by total internal reflection with fluorescence correlation spectroscopy

Total internal reflection excitation used in combination with fluorescence correlation spectroscopy (TIR-FCS) is a method for characterizing the dynamic behavior and absolute concentrations of fluorescent molecules near or at the interface of a planar substrate and a solution. In this work, we demonstrate for the first time the use of TIR-FCS for examining the interaction kinetics of fluorescent ligands in solution which specifically and reversibly associate with receptors in substrate-supported planar membranes. Fluorescence fluctuation autocorrelation functions were obtained for a fluorescently labeled IgG reversibly associating with the mouse receptor FcgammaRII, which was purified and reconstituted into substrate-supported planar membranes. Data were obtained as a function of the IgG solution concentration, the Fc receptor surface density, the observation area size, and the incident intensity. Best fits of the autocorrelation functions to appropriate theoretical forms gave measures of the average surface density of bound IgG, the local solution concentration of IgG, the kinetic rate constant for surface dissociation, and the rate of diffusion through the depth of the evanescent field. The average number of observed fluorescent molecules, both in solution and bound to the surface, scaled with the solution concentration of IgG, observation area size, and Fc receptor surface density as expected. The dissociation rate constant and rate of diffusion through the evanescent field agree with previous results, and all measured parameters were independent of the incident intensity.     

Fluorescence correlation spectroscopy close to a fluctuating membrane

Compartmentalization of the cytoplasm by membranes should have a strong influence on the diffusion of macromolecules inside a cell, and we have studied how this could be reflected in fluorescence correlation spectroscopy (FCS) experiments. We derived the autocorrelation function measured by FCS for fluorescent particles diffusing close to a soft membrane, and show it to be the sum of two contributions: short timescale correlations come from the diffusion of the particles (differing from free diffusion because of the presence of an obstacle), whereas long timescale correlations arise from fluctuations of the membrane itself (which create intensity fluctuations by modulating the number of detected particles). In the case of thermal fluctuations this second type of correlation depends on the elasticity of the membrane. To illustrate this calculation, we report the results of FCS experiments carried out close to a vesicle membrane. The measured autocorrelation functions display very distinctly the two expected contributions, and allow both to recover the diffusion coefficient of the fluorophore and to characterize the membrane fluctuations in term of a bending rigidity. Our results show that FCS measurements inside cells can lead to erroneous values of the diffusion coefficient if the influence of membranes is not recognized.     

Intramolecular Fluorescence Correlation Spectroscopy in a Feedback Tracking Microscope

We derive the statistics of the signals generated by shape fluctuations of large molecules studied by feedback tracking microscopy. We account for the influence of intramolecular dynamics on the response of the tracking system and derive a general expression for the fluorescence autocorrelation function that applies when those dynamics are linear. We show that in comparison to traditional fluorescence correlation spectroscopy, tracking provides enhanced sensitivity to translational diffusion, molecular size, heterogeneity, and long-timescale decays. We demonstrate our approach using a three-dimensional tracking microscope to study genomic λ-phage DNA molecules with various fluorescence label configurations.     

Time-integrated fluorescence cumulant analysis in fluorescence fluctuation spectroscopy

We introduce a new analysis technique for fluorescence fluctuation data. Time-integrated fluorescence cumulant analysis (TIFCA) extracts information from the cumulants of the integrated fluorescence intensity. TIFCA builds on our earlier FCA theory, but in contrast to FCA or photon counting histogram (PCH) analysis is valid for arbitrary sampling times. The motivation for long sampling times lies in the improvement of the signal/noise ratio of the data. Because FCA and PCH theory are not valid in this regime, we first derive a theoretical model of cumulant functions for arbitrary sampling times. TIFCA is the first exact theory that describes the effects of sampling time on fluorescence fluctuation experiments. We calculate factorial cumulants of the photon counts for various sampling times by rebinning of the original data. Fits of the data to models determine the brightness, the occupation number, and the diffusion time of each species. To provide the tools for a rigorous error analysis of TIFCA, expressions for the variance of cumulants are developed and tested. We demonstrate that over a limited range rebinning reduces the relative error of higher order cumulants, and therefore improves the signal/noise ratio. The first four cumulant functions are explicitly calculated and are applied to simple dye systems to test the validity of TIFCA and demonstrate its ability to resolve species.     

Stick-and-diffuse and caged diffusion: a comparison of two models of synaptic vesicle dynamics

Two models were recently proposed to enable us to understand the dynamics of synaptic vesicles in hippocampal neurons. In the caged diffusion model, the vesicles diffuse in small circular cages located randomly in the bouton, while in the stick-and-diffuse model the vesicles bind and release from a cellular cytomatrix. In this article, we obtain analytic expressions for the fluorescence correlation spectroscopy (FCS) autocorrelation function for the two models and test their predictions against our earlier FCS measurements of the vesicle dynamics. We find that the stick-and-diffuse model agrees much better with the experiment. We find also that, due to the slow dynamics of the vesicles, the finite experimental integration time has an important effect on the FCS autocorrelation function and demonstrate its effect for the different models. The two models of the dynamics are also relevant to other cellular environments where mobile species undergo slow diffusionlike motion in restricted spaces or bind and release from a stationary substrate.     

Continuous fluorescence microphotolysis and correlation spectroscopy using 4Pi microscopy

Continuous fluorescence microphotolysis (CFM) and fluorescence correlation spectroscopy (FCS) permit measurement of molecular mobility and association reactions in single living cells. CFM and FCS complement each other ideally and can be realized using identical equipment. So far, the spatial resolution of CFM and FCS was restricted by the resolution of the light microscope to the micrometer scale. However, cellular functions generally occur on the nanometer scale. Here, we develop the theoretical and computational framework for CFM and FCS experiments using 4Pi microscopy, which features an axial resolution of ∼100 nm. The framework, taking the actual 4Pi point spread function of the instrument into account, was validated by measurements on model systems, employing 4Pi conditions or normal confocal conditions together with either single- or two-photon excitation. In all cases experimental data could be well fitted by computed curves for expected diffusion coefficients, even when the signal/noise ratio was small due to the small number of fluorophores involved.     

Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion

Total internal reflection with fluorescence correlation spectroscopy (TIR-FCS) is a method for measuring the surface association/dissociation rates and absolute densities of fluorescent molecules at the interface of solution and a planar substrate. This method can also report the apparent diffusion coefficient and absolute concentration of fluorescent molecules very close to the surface. An expression for the fluorescence fluctuation autocorrelation function in the absence of contributions from diffusion through the evanescent wave, in solution, has been published previously (N. L. Thompson, T. P. Burghardt, and D. Axelrod. 1981, Biophys. J. 33:435-454). This work describes the nature of the TIR-FCS autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave contribute to the fluorescence fluctuations. The fluorescence fluctuation autocorrelation function depends in general on the kinetic association and dissociation rate constants, the surface site density, the concentration of fluorescent molecules in solution, the solution diffusion coefficient, and the depth of the evanescent field. Both general and approximate expressions are presented.     

Total internal reflection with fluorescence correlation spectroscopy: nonfluorescent competitors

Total internal reflection with fluorescence correlation spectroscopy is a method for measuring the surface association/dissociation rate constants and absolute densities of fluorescent molecules at the interface of a planar substrate and solution. This method can also report the apparent diffusion coefficient and absolute concentration of fluorescent molecules very close to the surface. Theoretical expressions for the fluorescence fluctuation autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave, in solution, contribute to the fluorescence fluctuations have been published previously. In the work described here, the nature of the autocorrelation function when both surface association/dissociation kinetics and diffusion through the evanescent wave contribute to the fluorescence fluctuations, and when fluorescent and nonfluorescent molecules compete for surface binding sites, is described. The autocorrelation function depends in general on the kinetic association and dissociation rate constants of the fluorescent and nonfluorescent molecules, the surface site density, the concentrations of fluorescent and nonfluorescent molecules in solution, the solution diffusion coefficients of the two chemical species, the depth of the evanescent field, and the size of the observed area on the surface. Both general and approximate expressions are presented.     

Imaging fluorescence correlation spectroscopy: nonuniform IgE distributions on planar membranes.

Fluorescence correlation spectroscopy is useful for detecting and characterizing molecular clusters that are smaller than or approximately equal to optical resolution in size. Here, we report the development of an approach in which the pixel-to-pixel fluorescence fluctuations from a single fluorescence image are spatially autocorrelated. In these measurements, tetramethylrhodamine-labeled, anti-trinitrophenyl IgE antibodies were specifically bound to substrate-supported planar membranes composed of trinitrophenyl-aminocaproyldipalmitoylphosphatidylethanolamine and dipalmitoylphosphatidylcholine. The antibody-coated membranes were illuminated with the evanescent field from a totally internally reflected laser beam, and the fluorescence arising from the IgE-coated membranes was recorded with a cooled CCD camera. The image was corrected for the elliptical Gaussian shape of the evanescent illumination after background subtraction. The spatial autocorrelation functions of the resulting images generated two useful parameters: the extrapolated initial values, which were related to the average cluster intensity and density; and the correlation distances, which were related to the average cluster size. These parameters varied with the IgE density, and unlabeled polyclonal anti-IgE enhanced the nonuniform IgE distributions. The autocorrelation functions calculated from images of planar membranes containing fluorescently labeled lipids rather than bound, labeled IgE demonstrated that the spatial nonuniformities were prominent only in the presence of IgE. Fluorescent beads were used to demonstrate the principles and the methods.     

Fluorescence Correlation Spectroscopy: The Case of Subdiffusion

The theory of fluorescence correlation spectroscopy is revisited here for the case of subdiffusing molecules. Subdiffusion is assumed to stem from a continuous-time random walk process with a fat-tailed distribution of waiting times and can therefore be formulated in terms of a fractional diffusion equation (FDE). The FDE plays the central role in developing the fluorescence correlation spectroscopy expressions, analogous to the role played by the simple diffusion equation for regular systems. Due to the nonstationary nature of the continuous-time random walk/FDE, some interesting properties emerge that are amenable to experimental verification and may help in discriminating among subdiffusion mechanisms. In particular, the current approach predicts 1), a strong dependence of correlation functions on the initial time (aging); 2), sensitivity of correlation functions to the averaging procedure, ensemble versus time averaging (ergodicity breaking); and 3), that the basic mean-squared displacement observable depends on how the mean is taken.     

Fluorescence correlation spectroscopy and Brownian rotational diffusion

A theroy relating rotational Brownian motion to the time autocorrelation function of the intensity of radiation from a fluorescent system composed of spherical rotors is presented. The calculation shows three relaxation times, two associated with the rotational diffusion, and the third associated with the natural decay of the fluorescence. The correlation function contains terms that relax independently of the fluorescence decay time, thus arbitrarily extending the time range over which rotational diffusion can be studied by fluorescence.