[Frontiers in Bioscience E3, 489-505, January 1, 2011]
Global analysis of autocorrelation functions and photon counting distributions
Victor V. Skakun1, Ruchira Engel2, Anatoli V. Digris1, Jan Willem Borst3, Antonie J.W.G. Visser3
1
TABLE OF CONTENTS
1. ABSTRACT
In fluorescence correlation spectroscopy (FCS) and photon counting histogram (PCH) analysis the same experimental fluorescence intensity fluctuations are used, 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 yielding the molecular brightness. Both FCS and PCH give information about the molecular concentration. Here we describe a global analysis protocol that simultaneously recovers relevant and common parameters in model functions of FCS and PCH from a single fluorescence fluctuation trace. The global analysis approach is described and tested with experimental fluorescence fluctuation data of enhanced green-fluorescent protein (eGFP) and dimeric eGFP (two eGFP molecules connected by a six amino acid long linker) in aqueous buffer. Brightness values and diffusion constants are recovered with good precision elucidating novel excited-state and motional properties of both proteins.
2. INTRODUCTION
In the last decades fluorescence correlation spectroscopy, originally introduced by Elson et al. (1-3) in the early 1970s, has become a widely used technique for studying various dynamic molecular processes (see many accounts in the 1990s (4-10)). It has found applications in measuring local concentrations, mobility coefficients, reaction rates and detection of intermolecular interactions in vitro and in vivo, excellently reviewed in this century(11-17). The sensitivity and non-invasive nature of this technique has made it one of the important techniques for studying molecular processes in cells and thus a useful tool for biochemists, biophysicists and biologists.
Fluorescence fluctuation methods are based on the detection of tiny, spontaneous fluctuations in fluorescence intensity caused due to deviations from thermal equilibrium in an open system. These fluctuations can arise e.g. due to diffusion of fluorescent molecules in and out of a well-defined observation volume generated by a focused laser beam. The intensity fluctuations can be monitored and autocorrelated over time as in fluorescence correlation spectroscopy (FCS) (13) or a distribution of fluorescence intensity amplitudes can be analyzed as in photon counting histogram (PCH) (18-22) or in fluorescence intensity distribution analysis (FIDA) (23-27).
Over time several developments in fluorescence fluctuation spectroscopy have taken place. Fluorescence cross- correlation spectroscopy (FCCS), reviewed in (13), is an established technique to monitor intracellular protein-protein interaction involving different measuring strategies (one-photon and two-photon excitation with dual-color detection, dual-color one-photon excitation and dual-color detection) (see, for instance, references (28-34)). Dual-color PCH analysis has been developed (35, 36) as well as 2D-FIDA (24, 25). Many technical improvements have been implemented in fluorescence fluctuation spectroscopy. Dual-focus FCS has been developed to remove experimental artifacts (37-39). Novel excitation strategies have been designed such as alternating laser excitation (40) or pulsed interleaved excitation (41) to remove cross-talk in dual channel measurements and modulated excitation to suppress triplet state buildup while keeping complete time range information in FCS (42). Fluorescence lifetime correlation spectroscopy has been developed to observe single dye diffusion in two distinct environments (43, 44). Scanning FCS has been implemented for precise measurements of diffusion coefficients (45). Total internal reflection FCS has been utilized to measure diffusion processes in the small confines of an evanescent field (46-49). Large progress has been made in the field of image correlation spectroscopy (reviewed by Kolin and Wiseman in 2007 (50)) and raster scanning image correlation spectroscopy (51-53).
Performing fluorescence fluctuation spectroscopy (FFS) experiments is relatively straightforward. But the theory for FFS is based on several assumptions, which do not always hold in a real experimental situation. For example, FFS analysis assumes that the experimental observation volume is Gaussian, which is not always true especially for in vivo measurements (54). Dead time and afterpulsing of the photodetectors may influence PCH analysis (55). There are a number of optical factors that influence FFS measurements such as cover-slide thickness, refractive index of the sample and optical saturation that should be taken into account for proper analysis (56, 57). In a confocal configuration for one-photon excitation a PCH model with a three-dimensional Gaussian observation volume profile does not adequately fit the experimental fluorescence fluctuation data (58, 59). In addition, there is a clear effect of bin time on PCH analysis, which influences the estimation of molecular concentration and brightness (25, 60). While it is crucial that the experimental setup must be optimized to close to ideal conditions, one should also be aware of all these factors in the data analysis procedure to avoid erroneous fits and misinterpretation of the experimental data.
We have previously described the FCS data processor, which can be used to analyze FCS data (61). The FCS data processor provides the user with the flexibility of using different models and rigorous error analysis to judge the quality of the fit and decide on the model that best describes the data. By allowing the user to generate initial guesses, link and constrain fit parameters and fit several autocorrelation function (ACF) curves globally, it also improves the accuracy and speed of analysis. However, since both FCS and PCH/FIDA use the same experimental data it is lucrative to be able to analyze one set of data to get all the information about dynamics and brightness of the molecular species.
Here we describe new data analysis software, the FFS data processor that allows the complete analysis of FFS data. It incorporates all the features of the FCS data processor with additional possibilities of performing FIDA and PCH analyses on the experimental photon counting distributions (PCD). Starting from the recorded flow of photon arrival times (raw data) the FFS data processor allows the users to generate an ACF, PCD or other statistical characteristics of photon flows such as factorial cumulants (62). Several photon-counting distributions, generated at different bin times, can be analyzed globally to give robust values for the molecular brightness (25). It is also possible to perform global analyses of PCDs and ACFs simultaneously resulting in more accurate values of triplet-state and diffusion parameters, and also in robust, time-independent estimations of molecular brightness and number of molecules. The FFS data processor also implements various corrections for differences in shapes of confocal volumes, background fluorescence and for dynamic processes in intensity distribution analyses. The use of the software is demonstrated through fitting of experimental fluorescence fluctuation data obtained from monomeric and dimeric forms of enhanced green-fluorescent protein (eGFP) in aqueous buffer.
3. REVIEW OF THEORY
3.1. Fluorescence intensity distribution analysis
The theory of FIDA is based on the use of generation function (GF) concept. The photon counting generating function of the probability P(n) of detecting n photons emitted by a number of fluorescent molecules in a system at equilibrium in an open non-homogeneous observation volume V during a counting time interval (bin time) T can be written as (23):
, (1)
where i is an index of molecular species, Ci is the concentration, qi is the specific brightness of molecules (in counts per second per molecule), B(r) is the brightness profile function, which is the product of excitation intensity and detection efficiency, and l
is the mean background count rate of the detector. Brightness q is defined as 
, where I0 is the excitation intensity in the focus, 
and Q are molecular absorption cross section and fluorescence quantum yield, respectively, and κ is the detection efficiency of the confocal setup. It is assumed that the contribution of each single molecule to the recorded photon trace is independent and the emission intensity is constant during the counting time interval T.
Having the GF, it is easy to calculate the photon counting distribution, P(n), using inverse Fast Fourier Transform (FFT) of the characteristic function, which can be obtained from the GF by substituting x
by the complex exponent 
(23):
, 
, (2)
where m is the number of data points in the PCD.
The key advantage of the FIDA lies in the flexibility of a polynomial model for the brightness profile function allowing an approximation for a broad spectrum of confocal volume shapes (23, 25):
, 
, (3)
where a1, a2 are adjustable instrumental parameters. A0 is the volume scaling parameter and B0 is the brightness in the focus. B0 and A0 can be calculated from normalization equations:
(4)
Substituting Eq. 3 into Eqs. 4 gives:
(5)
Finally, one obtains:
, (6)
where 
, 
and
, 
. (7)
3.2. Photon counting histogram analysis
In PCH the total PCD from a number of molecules is calculated by successive convolutions of single-molecule PCDs (18):
, (8)
where p(1)(n,q) is a single-molecule PCD, Poi(n, η) denotes the Poisson distribution with the mean value η, B(r) is assumed to be Gaussian and Q is a constant taken such that the product QVeff is large enough to completely include the illuminated volume. The distribution P(n) of a number of molecules Neff in an open effective observation volume 
is the weighted average of 
convolved M times (18):
(9)
Finally, the PCD of a number of independent species is given by a convolution of the PCD of each species
. (10)
Correction for the background can be applied to PCH similarly as it is done in FIDA (23). Since the background signal has Poissonian statistics, it can be accounted for by an additional convolution with Poisson distribution with parameter λT
. (11)
Eqs. 8-9 often fail to fit one-photon excitation experimental data due to the large deviation of an actual brightness profile from an assumed 3D Gaussian approximation (23, 59). To improve the model, Perroud et al. (58) and Huang et al. (59) introduced additional fitting parameters Fk defined as the relative difference between the integral 
of the kth power of the actual brightness profile function B(r) (normalized to unity) and that of its 3D Gaussian approximation 
.
. (12)
Introducing Fk (Eq. 12) into the single-molecule PCD (Eq. 8) leads to (59, 63):
, (13)
where 
and qC is equal to:
. (14)
In most cases only the first-order correction (all Fk equal to zero except F1) is sufficient to get the best fit to the experimental data.
The polynomial approximation (Eq. 3) of the observation profile can be also applied to the PCH model. Eqs. 8 and 9 then take the form:
(15)
In this definition, results of PCH analysis with a polynomial profile are fully equivalent to results of FIDA.
3.3. Fluorescence correlation spectroscopy
The autocorrelation function describing j independent molecular species diffusing freely in a 3D-Gaussian shaped observation volume is (61):
, (16)
where 
, Ftrip and τtrip are, respectively, the fraction and the relaxation time of molecules in the triplet state, 
, wxy and wz are, respectively, the lateral and axial radii of the confocal detection volume, and τdiff is the lateral diffusion time, which is related to the diffusion coefficient Dtran via tdif = w
xy2/(4Dtran). In Eq. 16 it is assumed that each molecular species has the same triplet-state characteristics. For more details see comprehensive reviews (13, 16, 17).
3.4. Correction for dynamic processes in photon counting distribution analysis
Eqs. 1, 6, 8, 13 and 15 were derived under the assumption that the fluorescence intensity emitted by a molecule during bin time T is constant, which is valid for the limit of short bin times. The theory for an arbitrary bin time T was developed and successfully applied for analysis of PCDs (25, 26, 60, 64, 65). The most practical way to account for diffusion and other processes like triplet-state relaxation is to correct the brightness and number of molecules such that the first and second factorial cumulants of the PCD are exact (25, 60). It is just an approximation, because only the first two cumulants of PCD are exact, but it works well for a relatively wide range of experimental conditions (64). According to this theory one has to calculate the so-called binning correction factor (25, 60):
, (17)
where g(t) is a time dependent term of autocorrelation function in FCS
(18)
and to correct the brightness and the number of molecules in the following form:
(19)
where q(T) and N(T) are apparent parameters of the model dependent on bin time T and q0 and N0 are absolute values of brightness and concentration that are independent of T. In general, the binning correction factor can be calculated assuming two or even more diffusing components (such correction can be applied to a mixture of species with approximately equal brightness values but quite different hydrodynamic radii). For a model of multiple brightness components this correction has to be applied independently to each component. Triplet and diffusion characteristics can be either different or the same for each brightness component.
3.5. Relation between number of molecules and brightness in FCS, PCH and FIDA
The reference volume in PCH and FCS is defined as 
. The same definition of the reference volume is implicitly obtained in FIDA as can be seen from Eq. 7 and 
. Thus, the number of molecules NFIDA = Neff. However, 
, where 
. Correction of the brightness profile through Eq. 12 also influences the value of the brightness (see Eq. 14). Thus, neither FIDA nor PCH with brightness profile correction returns the true brightness q. The following relations must be taken into account (63):
, 
. (20)
For the sake of simplicity in the equations we have omitted the subscript 0 because these equations are valid for both q0, N0 and q(T) and N(T). While performing FCS and PCH/FIDA on the same data one usually likes to relate brightness calculated from the average intensity <I> to the brightness obtained from PCH/FIDA. In FIDA the expression for the first factorial cumulant K1 takes the following form (25):
. (21)
Similarly, for PCH analysis with brightness profile correction one obtains:
. (22)
Thus:
.
For a one-component model without background one obtains:
. (23)
Since the number of molecules in FCS does not depend on time, NFCS is equal to N0eff and
. (24)
3.6. Weighting of autocorrelation- and photon counting distribution functions
Weighting factors (66) of the ACF are calculated by the algorithm proposed by Wohland et al. (67). The intensity trace is subdivided into a number of non-overlapping sub-traces and the local autocorrelation function is calculated from each sub-trace. Then standard deviations for each point of the ACF are obtained from these local autocorrelation functions. Finally weighting factors of the ACF are calculated by dividing the obtained deviations by the square root of the number of sub-traces. This algorithm does not depend on the type of the ACF time scale (quasi-logarithmic or linear) and FCS model parameters (i.e. it does not require any prior knowledge about the explored system) and usually results in good estimations of standard deviations of the ACF. Weighting factors of the PCD are calculated as standard deviations of binomial distribution given by 
, where m is the total number of bins and 
(* denotes the experimental PCD). Because m is usually large, a binomial distribution is well approximated by a normal distribution and therefore the application of reduced χ2 criterion is justified.
4. MATERIALS AND METHODS
4.1. Samples
Monomeric and dimeric eGFP were purified according to a recently described procedure (68). The two eGFP molecules in the dimer were linked by six amino acids (GSGSGS). Purified monomeric and dimeric eGFP solutions were in 50 mM TRIS buffer (pH 8.0). For the measurements 200-�l solutions were added to an 8-chambered coverglass (Lab-Tek, Nalge Nunc International Corp., USA). Rhodamine 110 (R110) (Invitrogen, Breda, The Netherlands) in water was used for calibration measurements.
4.2. Instrumentation
The measurements were performed on the ConfoCor 2 - LSM 510 combination setup (Carl Zeiss, Jena, Germany) detailed in (31, 69, 70). eGFP was excited with the 488 nm line from an argon-ion laser (excitation intensity in the range 10-40 μ W) focused into the sample with a water immersion C-Apochromat 40x objective lens N.A. 1.2 (Zeiss). After passing through the main beam splitters HFT 488/633 the fluorescence was filtered with a band pass 505-550. The pinhole for confocal detection was set at 70 μm. The microscope was controlled by Zeiss AIM 3.2 software. Raw intensity fluctuation data consisting of up to 4 x 106 photons were collected from single measurements. The data collection time ranged between 30 s and 120 s.
5. RESULTS AND DISCUSSION
5. 1. General comments on global analysis
Dependence of model parameters in PCH on the bin time makes it difficult to recover the true time-independent brightness and number of molecules from just a single photon counting distribution. An attempt to fit PCD calculated at a very small bin time to avoid any influence of diffusion and triplet kinetics usually fails if the photon count rate is not high enough. The PCD calculated at such conditions has only a few data points, which makes it difficult to fit the data well. To get time-independent brightness and number of molecules from a fit of PCD calculated at higher bin time, knowledge of diffusion and triplet parameters is required (see Eqs. 17-19). FCS has been established as a robust method for estimation of triplet and diffusion parameters (71). Therefore it is advantageous to combine the features of both complementary FFS methods and to increase the information content available from a single measurement.
The fit of a set of PCDs calculated at different bin times in fluorescence intensity multiple distribution analysis (FIMDA) (25) or photon counting multiple histograms (PCMH) (60) allows recovering all parameters (including triplet and diffusion parameters) from a single analysis. However, both FIMDA and PCMH have a drawback. These methods use the correction in the form of an integral (see Eq. 17). It means that the same correction value B2(T) can be achieved for a different functional form of g(t); for example, one can set diffusion time to a very small value and compensate this by an appropriate triplet fraction and decay parameter values, and vice versa. Therefore after global analysis of several PCDs the fit can be satisfactory but results in erroneous values for diffusion, triplet and also for other parameters because of the presence of many local minima in the multi-dimensional space of fit parameters. As a consequence, the recovered parameters usually have much wider confidence intervals compared to those obtained by traditional FCS analysis.
Although good results can be obtained from a two-step analysis (fit of the ACF followed by a fit of the PCD while fixing diffusion and triplet parameters to values obtained in the first step), better results can be expected from global analysis of both ACF and PCD. Only one run of the fit is required to get estimations for all fit parameters in this case. The number of molecules appears in both FCS and PCH models and can be linked together. It already makes the global analysis of ACF and PCD more advantageous in comparison to a two-step analysis, because it allows reducing the number of fit parameters by one while increasing the total number of data points available for the analysis. The structure parameter, diffusion and triplet parameters in the FCS model and in the part correcting for triplet and diffusion kinetics in the PCH/FIDA model can also be linked together. Consequently, FCS will provide accurate and robust estimations for triplet and diffusion parameters thus minimizing the risk of obtaining erroneous parameter values. Vice versa PCH/FIDA will yield correct time-independent estimations for the brightness that helps to obtain correct concentrations in FCS, if the sample contains two (or more) brightness components (see Eq. 16). Factors γ2 and 
presented in the relation of qFIDA and qC with the true brightness (see Eq. 20) are independent of the number of components. It allows the use of global analysis of PCH, FIDA and FCS in any combination. In contrast to FIMDA and PCMH, global analysis of ACF and PCD allows extraction of time-independent values for brightness and concentration for molecular samples using just a single PCD and a single ACF. However, taking into account that PCD usually has a low number of data points, it is advisable to calculate several PCDs to increase the brightness information content of experimental data.
One advantage of FCS lies in its insensitivity to the emission of molecules that are outside the focal point of the laser excitation beam along the optical axis (out-of-focus emission). The signal from these molecules is averaged in the calculation of the autocorrelation function and its contribution successfully canceled due to the chosen normalization. However, it is not true for PCH analysis and FIDA. The photon counting distribution keeps the information about each detected photon. It was shown (59, 60) that the first-order profile correction in PCH directly accounts for the out-of-focus emission. Taking into account the fact that PCH/FIDA is not sensitive to the difference in wxy and wz in contrast to FCS (59) one can conclude that the out-of-focus emission rather than the shape of the brightness profile itself has to be taken into account when performing global analysis of PCD and ACF.
5.2. Modification of the global χ2 criterion
Global analysis of experimental data obeying different functional forms may result in overestimation (or underestimation) of some model parameters, if appropriate weighting factors are not applied to each data point. This is especially important when the number of data points in these functions is quite different, for instance the ACF usually has 175 experimental data points versus only 15 data points in PCD. Such difference in number of data points leads to a significant difference in the number of degrees of freedom (calculated as number of data points minus number of fit parameters minus one) corresponding to each analyzed curve. Thus, the standard global χ2 criterion (72) becomes relatively insensitive to small deviations between the analyzed and best fit curves, for the curves with the lower degrees of freedom. To avoid this problem a modified χ2 criterion is proposed that takes into account the specific weight of each individual curve that participates in the global analysis. The equation of this modified χ2 is the following:
, (25)
where 
and 
are the measured value and the standard deviation of the ith experimental data point in the jth measured curve and 
is the theoretical value for this point; M is the number of globally analyzed curves; ni is the number of points in ith curve ; 
and 
is the number of free and linked parameters in ith model respectively, 
is the number of parameter groups (sets of linked parameters), 
, 
. As follows from Eq. 25 the contribution of each analyzed curve to the value of the global χ2 is made equal by dividing the sum in brackets by the corresponding number of degrees of freedom (thus obtaining the local χ2) and finally multiplying the total sum of the local χ2 by the average number of degrees of freedom 
. In case of equal number of degrees of freedom in each model (n1 = ...= nM; m1 = ...= mM) Eq. 25 reduces correctly to the standard global χ2 criterion (72):
. (26)
5.3. Monomeric and dimeric eGFPs as an experimental test system
Analysis of fluctuating fluorescence intensity data of monomeric eGFP has been performed in the FFS data processor by global analysis of one ACF and two PCD's calculated from raw data with bin times of 50 m s and 100 m s, respectively, linking all common parameters. The FCS model with brightness correction (Eq. 16) and the PCH model with second-order corrected Gaussian profile (Eqs. 9-11, 13; k = 1, 2) and triplet-diffusion correction (Eqs. 17-19) were used in the analysis. A complete algorithm of the PCH model calculation is presented in Figure 1. Steps, needed for calculation of a single-component PCH, are presented. For a two-component PCH one needs to perform these steps independently for each component and then to convolve the obtained arrays (Eq. 10).
At the beginning of the analysis only the ACF was fitted in order to get proper initial guesses for the parameters Ftrip, τtrip, τdiff and a. Then PCDs were added to the analysis and parameter groups were formed. The parameters N0eff, Ftrip, τtrip, τdiff and a were linked across all models (FCS and PCH) and F1, F2 and q0C only across the PCH model. The parameter Ginf was fixed to unity and the parameter λ was fixed to zero. Performing the preliminary analysis of ACF is advisable, because the FFS data processor does not calculate initial guesses for the correcting parameters Ftrip, τtrip, τdiff and a of the PCH model (see Eq. 18). Initial guesses for these parameters can not be obtained from a separate PCD. It was found that the global analysis of some PCDs (as well as some PCDs and ACF together) started from non-optimal initial guesses, are often trapped in local minima due to the integral form of the applied correction (see Eq. 17). Therefore, starting the global fit from almost optimal values of Ftrip, τtrip, τdiff, a prevents the analysis from trapping in local minima and increases its stability.
Global analysis of eGFP monomer data was done either separately or in combination with R110 calibration data. In the latter case parameters F1, F2 and a were linked across both sets of data (6 curves in total) while parameters N0eff, Ftrip, τtrip and τdiff formed two different sets of parameter groups. Analysis, being performed on a typically current personal computer (Intel Core 2 Duo 2GHz, 2Gb RAM), lasted not more than a few seconds in all cases. Confidence intervals at the 67% confidence level were calculated by the asymptotic standard-errors method as described in (73). The true value of the brightness q0 was calculated via Eq. 20. Analysis results of monomeric eGFP are summarized in Table 1. Graphical results are shown in Figure 2. The last column in Table 1 represents global analysis results of eGFP and R110 together. Although the recovered parameters are nearly identical in both separate and combined analyses, two PCDs of calibration data were not fitted well in the combined analysis. Local χ2 criteria were 8.0 and 2.6 in comparison to 3.4 and 1.2 in the analysis of R110 only. It is a consequence of the high sensitivity of the PCH model to the value of correction parameters F1 and F2. Therefore it is advisable to link these correction parameters in the global analysis of several measurements only if they do not differ too much. It can be argued that the out-of-focus signal (accounted for by parameters F1, F2) shows small variations from measurement to measurement. Although parameters F1, F2 are descriptive parameters of the used setup (59), it is not advisable to fix them to any value, as even small deviations of these parameters from the best-fit value will affect the resulted PCD substantially. In contrast to the standard one the modified global χ2 criterion is sensitive to deviation of any curve from the best fit (see Table 1). We have listed all local χ2 criterion values in Table 1 to compare standard and modified χ2 criteria.
Measurements were performed on different days in order to examine whether brightness values can be related to different excitation intensities. Brightness values obtained from the calibration dye R110 enable comparing brightness values of eGFP measured on different days. Taking into account that the brightness value depends on the laser intensity applied to the sample, it can be recalculated accurately enough from one measurement to another (in the non saturated regime) by applying the formula 
. We obtained 
= 27422 cpsm (counts per second per molecule), which is very close to the value obtained from the analysis (see Table 1).
We have presented the theory of PCH with Gaussian (Eqs. 9-11, 13; k = 1, 2) and polynomial (Eqs. 10, 11, 15) profiles. Both PCH models as well as the FIDA model can be used in the global analysis together with the model for FCS (Eq. 16). To test the theory and to compare results obtained using either Gaussian or polynomial profiles we performed an analysis of fluorescence fluctuations from monomeric eGFP and dimeric eGFP. The PCH model with the polynomial profile was calculated by the algorithm shown in Figure 1 with some modifications. The initial guesses were generated assuming a polynomial approximation, Eq. 15 was used instead of Eq. 8. The upper integration limit in Eq. 15 was set to 50 and different values of Q were assigned (if(qT < 10) Q = 1, else if(qT < 5) Q = 6, else if(qT < 50) Q = 5, else Q = 40). Measurements were performed directly after each other to avoid artifacts related to different experimental conditions. Graphical results are shown in Figure 3. Recovered parameters and χ2 criteria values are summarized in Table 2. After inspection of the results we can conclude that PCH analyses with both brightness profile approximations fit the experimental data equally well. Parameter values and theoretical curves are nearly identical and indistinguishable proving that both approximations can be used in the global analysis. Therefore these approximations can be used together in the global analysis. The pathways to reach global minima (and therefore intermediate fit parameter values) can differ, but the final result does not depend much on the used approximation.
Inspection of a large number of analyzed data showed that PCH analysis with a polynomial profile gave better results than PCH analysis with a Gaussian-corrected profile. In some cases good results could only obtained using PCH analysis with a polynomial profile. On the other hand, application of PCH analysis with a Gaussian profile is advantageous in a number of cases, because the best fit is obtained just by using a first-order correction. This drastically minimizes the risk of overfitting the data. It is therefore advisable to first try the PCH analysis with a Gaussian profile. When the obtained fit results are not acceptable, PCH analysis with a polynomial profile should be applied.
Finally, to compare these results with those available from fits of multiple PCDs calculated at different bin times, two additional analyses using FIMDA and PCMH approaches were performed. We selected measurements of monomeric eGFP that were obtained on the first day (see Table 1). Thirty PCD curves were globally (FIMDA) or sequentially (PCMH) analyzed and results are collected in Table 3 (graphical results are not shown). Although there is close agreement between most recovered parameters, unrealistic estimations were obtained for the triplet-state parameters. Confidence intervals of triplet and diffusion parameters are also wider as compared to global analysis of one ACF and two PCDs despite analysis of only three curves in the latter case.
5.4. Comparison of brightness values and diffusion times of monomeric and dimeric eGFPs
Since we have obtained brightness values and diffusion times with good precision, the recovered parameters for monomeric and dimeric eGFPs can be closely compared. The brightness values (q0) are 34480 cpsm for monomeric eGFP and 59340 cpsm for dimeric eGFP (Table 2). The ratio of the values 
. Thus, the brightness of the eGFP-dimer is less than twice that of the eGFP-monomer. This can be explained by the presence of resonance energy transfer in dimeric eGFP. It must be noted that the two eGFP molecules are linked together by only six amino acids. One of the excited eGFP molecules transfers energy to the other eGFP molecule in the ground state, but does this with less than 100% efficiency. This is demonstrated by time-resolved fluorescence anisotropy experiments, which will be published separately.
The diffusion times (τdiff) are 62.4 μs for monomeric eGFP and 95.1 μs for dimeric eGFP. The ratio of diffusion times is the inverse of the ratio of translation diffusion constants, therefore 
. eGFP-monomer has a molecular mass of 27 kDa and eGFP-dimer has twice this mass, 54 kDa. When we assume that both proteins have a spherical shape, then 
. This value is much higher than the obtained value of 0.656 and we conclude that the translational diffusion is not that for spherical (globular) proteins. When the power of 1/3 is replaced by a power of 1/2, we obtain the ratio of diffusion constants for rod-like macromolecules and the value is 0.707, which is closer to the experimental value of 0.656. eGFP-monomer and eGFP-dimer move like rod-like molecules, which is in excellent agreement with the results of recent FCS experiments on eGFP oligomers (77).
6. CONCLUSIONS
We have presented the theory of global analysis of autocorrelation functions and photon counting distributions. Although the theories of FCS and PCH are well known, several important points have to be taken into consideration when both theories are combined in a global analysis. From the PCH side there are two major points. First, a correction must be applied for out-of-focus emission, which can be done by either a correction of the Gaussian profile via Eq. 12, or using a polynomial profile. Second, PCH must be corrected for dynamic processes allowing time-independent estimation of brightness values and number of molecules. From the FCS side the correction on difference in brightness values must be applied when multiple fluorescent species are present. The last important point is to use the same reference volume in both FCS and PCH. The use of the effective volume Veff is preferable, because it allows linking the number of molecules (N) through all FCS, PCH and FIDA models.
We have described a global analysis protocol that simultaneously recovers the relevant parameters in model functions of FCS and PCH from a single fluorescence fluctuation trace allowing the linking of common parameters. The analysis yields more accurate values of triplet-state and diffusion parameters, but also robust, time-independent estimations of molecular brightness and number of molecules. We have used a new data analysis software, the FFS data processor, which allows complete analysis of experimental fluorescence fluctuation data. The use of the software is demonstrated through fitting of experimental data obtained from monomeric and dimeric forms of eGFP in aqueous buffer. Information on this software is available at www.sstcenter.com.
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Key Words: Fluorescence Fluctuation Spectroscopy, Fluorescence Correlation Spectroscopy, Photon Counting Histogram, Fluorescence Intensity Distribution Analysis, Global Analysis, Green Fluorescent Protein
Send correspondence to: Victor V. Skakun, Department of Systems Analysis, Radio Physics and Electronics Faculty, Belarusian State University, Minsk, 220050, Belarus, Tel: 37517 2789665, Fax: 37517 2789345, E-mail:skakun@sstcenter.com