Cumulant analysis in fluorescence fluctuation spectroscopy

A novel technique for the analysis of fluorescence fluctuation experiments is introduced. Fluorescence cumulant analysis (FCA) exploits the factorial cumulants of the photon counts and resolves heterogeneous samples based on differences in brightness. A simple analytical model connects the cumulants of the photon counts with the brightness epsilon and the number of molecules N in the optical observation volume for each fluorescent species. To provide the tools for a rigorous error analysis of FCA, expressions for the variance of factorial cumulants are developed and tested. We compare theory with experiment by analyzing dye mixtures and simple fluorophore solutions with FCA. A comparison of FCA with photon-counting histogram (PCH) analysis, a related technique, shows that both methods give identical results within experimental uncertainty. Both FCA and PCH are restricted to data sampling times that are short compared to the diffusion time of molecules through the observation volume of the instrument. But FCA theory, in contrast to PCH, can be extended to treat arbitrary sampling times. Here, we derive analytical expressions for the second factorial cumulant as a function of the sampling time and demonstrate that the theory successfully models fluorescence fluctuation data.     

Dual-color time-integrated fluorescence cumulant analysis

We introduce dual-color time-integrated fluorescence cumulant analysis (TIFCA) to analyze fluorescence fluctuation spectroscopy data. Dual-color TIFCA utilizes the bivariate cumulants of the integrated fluorescent intensity from two detection channels to extract the brightness in each channel, the occupation number, and the diffusion time of fluorophores simultaneously. Detecting the fluorescence in two detector channels introduces the possibility of differentiating fluorophores based on their fluorescence spectrum. We derive an analytical expression for the bivariate factorial cumulants of photon counts for arbitrary sampling times. The statistical accuracy of each cumulant is described by its variance, which we calculate by the moments-of-moments technique. A method that takes nonideal detector effects such as dead-time and afterpulsing into account is developed and experimentally verified. We perform dual-color TIFCA analysis on simple dye solutions and a mixture of dyes to characterize the performance and accuracy of our theory. We demonstrate the robustness of dual-color TIFCA by measuring fluorescent proteins over a wide concentration range inside cells. Finally we demonstrate the sensitivity of dual-color TIFCA by resolving EGFP/EYFP binary mixtures in living cells with a single measurement.     

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.     

Confidence Intervals for Concentration and Brightness from Fluorescence Fluctuation Measurements

The theory of photon count histogram (PCH) analysis describes the distribution of fluorescence fluctuation amplitudes due to populations of fluorophores diffusing through a focused laser beam and provides a rigorous framework through which the brightnesses and concentrations of the fluorophores can be determined. In practice, however, the brightnesses and concentrations of only a few components can be identified. Brightnesses and concentrations are determined by a nonlinear least-squares fit of a theoretical model to the experimental PCH derived from a record of fluorescence intensity fluctuations. The χ² hypersurface in the neighborhood of the optimum parameter set can have varying degrees of curvature, due to the intrinsic curvature of the model, the specific parameter values of the system under study, and the relative noise in the data. Because of this varying curvature, parameters estimated from the least-squares analysis have varying degrees of uncertainty associated with them. There are several methods for assigning confidence intervals to the parameters, but these methods have different efficacies for PCH data. Here, we evaluate several approaches to confidence interval estimation for PCH data, including asymptotic standard error, likelihood joint-confidence region, likelihood confidence intervals, skew-corrected and accelerated bootstrap (BCa), and Monte Carlo residual resampling methods. We study these with a model two-dimensional membrane system for simplicity, but the principles are applicable as well to fluorophores diffusing in three-dimensional solution. Using simulated fluorescence fluctuation data, we find the BCa method to be particularly well-suited for estimating confidence intervals in PCH analysis, and several other methods to be less so. Using the BCa method and additional simulated fluctuation data, we find that confidence intervals can be reduced dramatically for a specific non-Gaussian beam profile.     

The photon counting histogram in fluorescence fluctuation spectroscopy with non-ideal photodetectors

Fluorescence fluctuation spectroscopy utilizes the signal fluctuations of single molecules for studying biological processes. Information about the biological system is extracted from the raw data by statistical methods such as used in fluctuation correlation spectroscopy or photon counting histogram (PCH) analysis. Since detectors are never ideal, it is crucial to understand the influence of photodetectors on signal statistics to correctly interpret the experimental data. Here we focus on the effects of afterpulsing and detector dead-time on PCH statistics. We determine the dead-time and afterpulse probability for our detectors experimentally and show that afterpulsing can be neglected for most experiments. Dead-time effects on the PCH are concentration-dependent and become significant when more than one molecule is present in the excitation volume. We develop a new PCH theory that includes dead-time effects and verify it experimentally. Additionally, we derive a simple analytical expression that accurately predicts the effect of dead-time on the molecular brightness. Corrections for non-ideal detector effects extend the useful concentration range of PCH experiments and are crucial for the interpretation of titration and dilution experiments.     

High-pressure fluorescence correlation spectroscopy

We demonstrate that a novel high-pressure cell is suitable for fluorescence correlation spectroscopy (FCS). The pressure cell consists of a single fused silica microcapillary. The cylindrical shape of the capillary leads to refraction of the excitation light, which affects the point spread function of the system. We characterize the influence of these beam distortions by FCS and photon-counting histogram (PCH) analysis and identify the optimal position for fluorescence fluctuation experiments in the capillary. At this position within the capillary, FCS and photon-counting histogram experiments are described by the same equations as used in standard FCS experiments. We report the first experimental realization of fluorescence fluctuation spectroscopy under high pressure. A fluorescent dye was used as a model system for evaluating the properties of the capillary under pressure. The autocorrelation function and the photon count distribution were measured in the pressure range from 0 to 300 MPa. The fluctuation amplitude and the diffusion coefficient show a small pressure dependence. The changes of these parameters, which are on the order of 10%, are due to the pressure changes of the viscosity and the density of the aqueous medium.     

Dual-color photon-counting histogram

We report on the development of dual-color photon-counting histogram (PCH) analysis. Dual-color PCH is an extension of regular PCH and considers the photon counts received in two detection channels instead of one. Because each detection channel records a different color, dual-color PCH distinguishes fluorescent species not only by differences in their brightness, but also according to their color. The additional discrimination by color increases the sensitivity of PCH in resolving a mixture of species considerably. Most dual-color fluorescence fluctuation experiments are performed on fluorophores with overlapping emission spectra. This overlap results in spectral cross talk between the detector channels, which reduces resolvability. Here, we demonstrate that dual-color PCH is able to resolve binary dye mixtures in the presence of cross talk from a single measurement without any additional information about the sample. We discuss the effect of sampling time on the fit parameters of dual-color PCH. Differences between dual-color fluorescence correlation spectroscopy and dual-color PCH will also be addressed. We quantitatively resolve a mixture of the two fluorescent proteins CFP and YFP, which is challenging because of the strong spectral overlap of their emission spectra. Dichroic mirrors are needed to direct the light into the two detection channels. We quantify the influence of these filters on dual-color PCH analysis and determine the optimal transition wavelength of the dichroic mirror for the CFP-YFP pair.     

Statistical accuracy in fluorescence fluctuation experiments

The mathematical expression of the signal to noise ratio in fluorescence fluctuation experiments is derived for arbitrary sample profiles and for any mechanism of translational motion, and experimentally verified. The signal to noise ratio depends on the mean count rate per particle per dwell time, the mean number of particles per sample volume, time characteristics of the correlation function, sample profile characteristics, and the data collection time. Statistical accuracy of the third order moment of fluorescence intensity fluctuations is also studied. The optimum concentration for the third order moment analysis is about one particle per sample volume. Received: 13 February 1996 / Accepted: 20 September 1996     

The dual-color photon counting histogram with non-ideal photodetectors

Dual-color photon counting histogram (PCH) analysis utilizes the photon counts in two detection channels to distinguish species by differences in brightness and color. Here we modify the existing dual-color PCH theory, which assumes ideal detectors, to include the non-ideal nature of the detector. Specifically, we address the effects of deadtime and afterpulsing. Both effects modify the shape of the dual-color PCH and thus potentially lead to incorrect values for the brightness and number of molecules if an ideal model is assumed. We use the modified theory to predict the effects of detector non-idealities on dual-color PCH as a function of concentration and brightness. In addition, we introduce a method based on moment analysis to determine the error in brightness due to non-ideal detector effects. We verify our theory experimentally by measuring a dye solution as a function of concentration and brightness. We determine the deadtime and afterpulse probability of our detectors and show that both effects play an important role in the analysis of dual-color PCH experiments. We demonstrate that resolving a mixture of CFP and YFP requires taking non-ideal detector effects into account. These corrections are also crucial for cellular measurements, as shown for GFP and RFP in mammalian cells.     

Fluorescence fluctuation analysis for the study of interactions between oligonucleotides and polycationic polymers

The interactions between a cationic polymer, poly(2-dimethylamino)ethyl methacrylate (pDMAEMA), and negatively charged rhodamine-labeled 25-mer phosphodiester oligonucleotides (Rh-ONs) were studied by fluorescence fluctuation spectroscopy and other techniques. The composition of the pDMAEMA/Rh-ON complexes was investigated as a function of the charge ratio (+/-) by increasing the pDMAEMA concentration and keeping the Rh-ON concentration constant. We applied two different methods for analyzing the fluorescence fluctuation profiles of the pDMAEMA/Rh-ON complexes, which depended on their composition. First, we analyzed the data with the photon counting histogram (PCH) technique, which determines the molecular brightness and the concentration of fluorophores (Chen et al, 1999). A particular challenge for the data analysis is the occurrence of sudden fluorescence bursts in the fluorescence fluctuation profiles, which are linked to the appearance of multimolecular complexes (i. e. when several Rh-ONs were present in one complex). A quantitative interpretation of the analysis for the complexes remains challenging and is connected to the rarity of the fluorescent bursts, which do not provide sufficient data statistics. To specifically address the problem of the fluorescent bursts we employed a method described by Van Craenenbroeck et al. (1999). This method, applicable only when data were integrated over much longer time bins, allowed us to estimate the number of fluorescence bursts which could be considered as a relative measure of the amount of multimolecular complexes present. When monomolecular complexes were formed, i. e. at high values of the charge ratio, highly intense fluorescence peaks were not present and the interpretation of the PCH analysis was more straightforward. The molecular brightness of the species (epsilon), as revealed from PCH analysis, was greater than epsilon for the free Rh-ONs, indicating that the Rh-ONs were attached to pDMAEMA chains.     

On the analysis of high order moments of fluorescence fluctuations.

A simple, straightforward analysis to characterize the distribution of aggregate sizes in a reversible aggregation system at equilibrium is presented. The method, an extension of fluorescence correlation spectroscopy (FCS), is based on measurements of higher order moments of spontaneous fluctuations of fluorescence intensity emitted from a defined open region of the sample. These fluctuations indicate fluctuations of the numbers of the fluorescent molecules in the observation region. Shot noise resulting from the random character of fluorescence emission and from the photoelectric detection system is modeled as a Poisson distribution and is subtracted from the measured photon count fluctuation moments to yield the desired fluorescence fluctuation moments. This analysis can also be used to estimate the fraction of immobile fluorophores in FCS measurements.     

Simultaneous diffusion and brightness measurements and brightness profile visualization from single fluorescence fluctuation traces of GFP in living cells

Fluorescence correlation spectroscopy (FCS) and photon-counting histogram (PCH) analysis use the same experimental fluorescence intensity fluctuations, but each analytical method focuses on a different property of the signal. The time-dependent decay of the correlation of fluorescence fluctuations is measured in FCS yielding, for instance, molecular diffusion coefficients. The amplitude distribution of these fluctuations is calculated by PCH analysis yielding information about the molecular brightness of fluorescent species. Analysis of both FCS and PCH results in the molecular concentration of the sample. Using a previously described global analysis procedure we report here precise, simultaneous measurements of diffusion constants and brightness values from single fluorescence fluctuation traces of green-fluorescent protein (GFP, S65T) in the cytoplasm of Dictyostelium cells. The use of a polynomial profile in PCH analysis, describing the detected three-dimensional shape of the confocal volume, enabled us to obtain well fitting results for GFP in cells. We could visualize the polynomial profile and show its deviation from a Gaussian profile.     

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.     

Observation volumes and {gamma}-factors in two-photon fluorescence fluctuation spectroscopy

Fluorescence fluctuation spectroscopy has become an important measurement tool for investigating molecular dynamics, molecular interactions, and chemical kinetics in biological systems. Although the basic theory of fluctuation spectroscopy is well established, it is not widely recognized that saturation of the fluorescence excitation can dramatically alter the size and profile of the fluorescence observation volume from which fluorescence fluctuations are measured, even at relatively modest excitation levels. A precise model for these changes is needed for accurate analysis and interpretation of fluctuation spectroscopy data. We here introduce a combined analytical and computational approach to characterize the observation volume under saturating conditions and demonstrate how the variation in the volume is important in two-photon fluorescence correlation spectroscopy. We introduce a simple approach for analysis of fluorescence correlation spectroscopy data that can fully account for the effects of saturation, and demonstrate its success for characterizing the observed changes in both the amplitude and relaxation timescale of measured correlation curves. We also discuss how a quantitative model for the observed phenomena may be of broader importance in fluorescence fluctuation spectroscopy.     

A method improving the accuracy of fluorescence recovery after photobleaching analysis

Fluorescence recovery after photobleaching has been an established technique of quantifying the mobility of molecular species in cells and cell membranes for more than 30 years. However, under nonideal experimental conditions, the current methods of analysis still suffer from occasional problems; for example, when the signal/noise ratio is low, when there are temporal fluctuations in the illumination, or when there is bleaching during the recovery process. We here present a method of analysis that overcomes these problems, yielding accurate results even under nonideal experimental conditions. The method is based on circular averaging of each image, followed by spatial frequency analysis of the averaged radial data, and requires no prior knowledge of the shape of the bleached area. The method was validated using both simulated and experimental fluorescence recovery after photobleaching data, illustrating that the diffusion coefficient of a single diffusing component can be determined to within ∼1%, even for small signal levels (100 photon counts), and that at typical signal levels (5000 photon counts) a system with two diffusion coefficients can be analyzed with <10% error.     

Dual-color photon counting histogram analysis of mRFP1 and EGFP in living cells

We investigate the potential of dual-color photon counting histogram (PCH) analysis to resolve fluorescent protein mixtures directly inside cells. Because of their small spectral overlap, we have chosen to look at the fluorescent proteins EGFP and mRFP1. We experimentally demonstrate that dual-color PCH quantitatively resolves a mixture of EGFP and mRFP1 in cells from a single measurement. To mimic the effect of protein association, we constructed a fusion protein of EGFP and mRFP1 (denoted EGFP-mRFP1). Fluorescence resonant energy transfer within the fusion protein alters the dual-channel brightness of the fluorophores. We describe a model for fluorescence resonant energy transfer effects on the brightness and incorporate it into dual-color PCH analysis. The model is verified using fluorescence lifetime measurements. Dual-color PCH analysis demonstrated that not all of the expressed EGFP-mRFP1 fusion proteins contained a fluorescent mRFP1 molecule. Fluorescence lifetime and emission spectra measurements confirmed this surprising result. Additional experiments show that the missing fluorescent fraction of mRFP1 is consistent with a dark state population of mRFP1. We successfully resolved this mixture of fusion proteins with a single dual-color PCH measurement. These results highlight the potential of dual-color PCH to directly detect and quantify protein mixtures in living cells.     

Oligomer size of the serotonin 5-hydroxytryptamine 2C (5-HT2C) receptor revealed by fluorescence correlation spectroscopy with photon counting histogram analysis: evidence for homodimers without monomers or tetramers

Fluorescence correlation spectroscopy (FCS) and photon counting histogram (PCH) are techniques with single molecule sensitivity that are well suited for examining the biophysical properties of protein complexes in living cells. In the present study, FCS and PCH were applied to determine the diffusion coefficient and oligomeric size of G-protein-coupled receptors. FCS was used to record fluctuations in fluorescence intensity arising from fluorescence-tagged 5-hydroxytryptamine 2C (5-HT(2C)) receptors diffusing within the plasma membrane of HEK293 cells and rat hippocampal neurons. Autocorrelation analysis yielded diffusion coefficients ranging from 0.8 to 1.2 μm(2)/s for fluorescence-tagged receptors. Because the molecular brightness of a fluorescent protein is directly proportional to the number of fluorescent proteins traveling together within a protein complex, it can be used to determine the oligomeric size of the protein complex. FCS and PCH analysis of fluorescence-tagged 5-HT(2C) receptors provided molecular brightness values that were twice that of GFP and YFP monomeric controls, similar to a dimeric GFP control, and unaltered by 5-HT. Bimolecular fluorescence complementation of the N- and C-terminal halves of YFP attached to 5-HT(2C) receptors was observed in endoplasmic reticulum/Golgi and plasma membranes with a brightness equal to monomeric YFP. When GFP-tagged 5-HT(2C) receptors were co-expressed with a large excess of untagged, non-fluorescent 5-HT(2C) receptors, the molecular brightness was reduced by half. PCH analysis of the FCS data were best described by a one-component dimer model without monomers or tetramers. Therefore, it is concluded that 5-HT(2C) receptors freely diffusing within the plasma membrane are dimeric.     

Molecular brightness determined from a generalized form of Mandel's Q-parameter

Mandel's Q-parameter, which is determined from the first two photon count moments, provides an alternative to PCH analysis for determining the brightness of fluorophores. Here, the definition of the Q-parameter is generalized to include correlations between photon counts that are separated by a time tau. We develop and experimentally verify a theory that takes the effects of dead time, afterpulsing, and the finite sampling time on the generalized parameter Q(tau) into account. Q(0), which corresponds to the original Q-parameter, is severely affected by dead time and afterpulsing. Q(tau) for tau>0, on the other hand, is quite robust with respect to nonideal detector effects. Thus, analysis of Q(tau) provides a robust method for determining the brightness of fluorophores. We extend the theory to a mixture of species, which is characterized by an apparent brightness. The brightness of EGFP in CV-1 cells is measured as a function of protein concentration to demonstrate the feasibility of Q(tau) analysis in cells. In addition, we monitor protein association of the ligand-binding domain of retinoid X receptor in the presence and absence of 9-cis-retinoic acid by Q(tau) analysis.     

Fluorescence intensity multiple distributions analysis: concurrent determination of diffusion times and molecular brightness

Fluorescence correlation spectroscopy (FCS) has proven to be a powerful technique with single-molecule sensitivity. Recently, it has found a complement in the form of fluorescence intensity distribution analysis (FIDA). Here we introduce a fluorescence fluctuation method that combines the features of both techniques. It is based on the global analysis of a set of photon count number histograms, recorded with multiple widths of counting time intervals simultaneously. This fluorescence intensity multiple distributions analysis (FIMDA) distinguishes fluorescent species on the basis of both the specific molecular brightness and the translational diffusion time. The combined information, extracted from a single measurement, increases the readout effectively by one dimension and thus breaks the individual limits of FCS and FIDA. In this paper a theory is introduced that describes the dependence of photon count number distributions on diffusion coefficients. The theory is applied to a series of photon count number histograms corresponding to different widths of counting time intervals. Although the ability of the method to determine specific brightness values, diffusion times, and concentrations from mixtures is demonstrated on simulated data, its experimental utilization is shown by the determination of the binding constant of a protein-ligand interaction exemplifying its broad applicability in the life sciences.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.