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Time-Integrated Fluorescence Cumulant Analysis in Fluorescence Fluctuation Spectroscopy

School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota

Correspondence: Address reprint requests to Bin Wu, University of Minnesota, School of Physics and Astronomy, 116 Church St., SE, Minneapolis, MN 55455. Tel.: 612-624-6045; Fax: 612-624-4578; E-mail: binwu{at}physics.umn.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
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.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Biological systems are heterogeneous. Resolving such heterogeneous mixtures of molecules is an important and challenging problem. Fluorescence fluctuation spectroscopy (FFS) provides a promising tool for studying heterogeneous biological systems. The technique exploits fluorescence intensity fluctuations of molecules that pass through a very small optical observation volume. Each passage of a fluorescent molecule leads to a short burst of detected photons. Collectively, these diffusing molecules give rise to a stochastic fluorescence signal. Statistical analysis tools are required to extract information about the fluorescent molecules from the stochastic signal. The most widely used analysis technique, fluorescence correlation spectroscopy (FCS) (8), uses the autocorrelation function of the fluorescence signal. FCS resolves species based on the diffusion coefficient, which ultimately depends on molecular weight. Simulations, however, point out that FCS is limited to resolving species that differ in their diffusion time by at least a factor of five (10). Thus, many interesting cases, such as a monomer/dimer equilibrium, cannot be directly resolved by FCS.

Photon counting histogram (PCH) (3) and fluorescence intensity distribution analysis (5) are recently developed techniques that determine the molecular brightness and occupation number in the observation volume of the system from the experimental histogram of the photon counts. PCH complements FCS in that it distinguishes species by a difference in their molecular brightness instead of their molecular weight. PCH has been successfully applied to resolve binary dye mixtures (13) and probe protein interactions in living cells (2).

PCH is based on the analysis of the probability distribution function (pdf) of the photon counts. The probability distribution function and all of its moments contain, from a mathematical point of view, identical information (6). Moment analysis has been developed in the late 80s and early 90s to resolve heterogeneous bimolecular samples in FFS experiments (14,15,17,18). However, the potential of this approach has not been fully explored. A further development was the introduction of fluorescent cumulant analysis (FCA) (11). Cumulants are closely related to moments and have mathematical properties particularly suited for random variables. FCA uses simple analytical expressions that relate the factorial cumulants of the photon counts to the molecular brightness and occupation number in the observation volume. FCA successfully resolved binary mixtures according to brightness. We demonstrated that both FCA and PCH are equivalent techniques for analyzing FFS data.

PCH, FCS, and FCA theories explicitly or implicitly assume that the sampling time T is much smaller than the characteristic timescale of fluctuation in the fluorescence signal. However, few photons are collected during a short sampling period and the signal/noise ratio of the data is poor. Longer sampling times are needed to improve the signal, and the theory of PCH and FCA needs to be modified. Finding an exact solution for PCH at long sampling times has been difficult.

Cumulant analysis, in contrast to histogram analysis, allows an exact treatment of sampling times. In this article, we extend our previously developed FCA theory to arbitrary sampling times, which significantly increases the signal statistics of the technique. From a fluorescence intensity point of view, short sampling times capture the instantaneous fluorescence I, whereas longer sampling times measure the time-integrated fluorescence. To distinguish our previously developed FCA, which is only valid for short sampling times, we refer to our new theory as time-integrated fluorescence cumulant analysis (TIFCA).

Experimentally, we measure photon counts with a short sampling time. The photon counts are subsequently rebinned to obtain factorial cumulants of the photon counts for longer sampling times. Theoretical models are then fitted to experimental cumulants as a function of binning time. The fit determines simultaneously the molecular brightness, the occupation number, and the diffusion time of each species from a single measurement. Thus TIFCA is capable of distinguishing particles by both molecular brightness and diffusion time. In other words, the technique combines the advantages of both PCH and FCS.

In addition, we formulate and experimentally verify a mathematical model for the statistical error of factorial cumulants for arbitrary sampling times. The relative error of cumulants provides a measure of its statistical significance and provides weighing factors for data fitting. We also discuss the effect of binning time on the relative error of cumulants. Analysis of simple dye mixtures by FCA demonstrates that our theory successfully models the experimental data.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
We previously described factorial cumulants of photon counts in the limit of short sampling times (11). Here we expand the theory of cumulants to arbitrary binning times. We first introduce the binning function to capture the effect of binning time on the amplitude of cumulants, before deriving expressions of binning time-dependent cumulants. The variance of cumulants, which is needed to perform error analysis on experimental data, is discussed as well.

Binning function
Our starting point is Mandel's formula (9), which relates the probability distribution function P(W) (pdf) of the integrated light intensity W to the pdf p(k,T) of the photon counts k,

(1)

where Poi(k, x) is the Poisson distribution with a mean of x. The parameter describes the detection efficiency of the photo detector. For convenience, we set = 1 and assume a constant spatial intensity profile across the detector area A. As is customary, we will set the area of the detector to 1. With these conventions, the integrated intensity W is expressed in units of photon counts per sampling time. For a given fluorescent light intensity I, W is the integral of I over the sampling time T,

(2)

It is well known that the moment-generating function Q_w(s) of W is equal to the factorial moment-generating function of the photon counts k (19). The logarithm of a generating function defines its corresponding cumulant-generating function. Thus Mandel's formula implicitly states that the cumulant-generating function of W equals the factorial cumulant-generating function of k. In other words, the r^th factorial cumulants of the photon counts equals the r^th cumulants of the integrated intensity _r(W):

(3)

For deriving the binning function it is convenient to treat the special case of a single molecule in the entire sample volume. The result obtained for this special case is later generalized to the case of N particles in an open volume. Let us assume that a single molecule diffuses in a large, but closed volume V, illuminated by a laser with a normalized beam profile given by Without losing generality, let us assume that the point-spread function has a characteristic length w, and that the shape of the point-spread function is determined collectively by a set of parameters . For example, for a three-dimensional Gaussian beam profile in an s-photon (s = 1, 2...) excitation experiment,

(4)

the characteristic length is the beam waist w, and the shape parameter is given by the squared beam waist ratio According to definition (Eq. 2), the r^th moment of the integrated intensity,

(5)

is related to the r^th order correlation function g_r of the fluorescence intensity, which is defined as

(6)

Without losing generality, let us assume that the time is ordered: t₁ < t₂ < < t_r. For a stationary diffusion process, the correlation function is a function of the time differences only, ₂ = t₂ – t₁, ₃ = t₃ – t₂,, and _r = t_r – t_r–1. In Appendix A, we show that g_r can be expressed as

(7)

where is the molecular brightness measured in counts per second per molecule (cpsm) and V_PSF is the reference volume conventionally used in fluorescence fluctuation experiments,

(8)

The coefficient _r is defined as

(9)

and the normalized correlation function G_r is given by

(10)

where D is the diffusion coefficient. It is straightforward to show that G_r(0,0,,0;_d,) = 1 when ₂ = = _r = 0, since

(11)

where (x) is Dirac's -function.

Suppose the characteristic length of the point-spread function is w, which means if we define a normalized coordinate the point-spread function only depends on the shape parameters , but is independent of w. We change the integration variables and define a characteristic diffusion time By inspecting Eq. 10, we observe that G_r depends on only through and on the point-spread function only through the shape parameters . In other words, the normalized correlation function has the scaling property

(12)

If we plug Eq. 7 into the r^th moment of the integrated intensity (Eq. 5), we obtain

(13)

where B_r is called the r^th binning function, and is defined as

(14)

Expressing the binning function in the form of Eq. 14 uses the fact that the hyper-cubic integration volume of Eq. 5 can be divided into r! equivalent blocks, where within each region the times t₁,t₂,,t_r = 0 are time-ordered and the integration of the correlation function for each block is identical. We also change the integration range of t_i from to [0,T] using the stationary property of our correlation function. Now we change the integration variables t₁ to _i = t_i – t_i–1 for i > 1 and express the integration volume using the transformed coordinates. Because the integrand is independent of t₁, it is straightforward to integrate t₁ out. The binning function is now written in the form

(15)

We change the integration variables to x_i = _i/_d and arrive at the final form of the binning function, which is

(16)

By inspecting Eq. 16, we clearly see that the binning function also obeys a scaling law,

(17)

which is important in practical application. The calculation of binning functions requires high-dimensional integration, which cannot be solved analytically for arbitrary point-spread functions. In practice, we calculate the binning function numerically. Because of its scaling property, we only need to calculate the binning function for the special case of _d = 1. We construct a table of its function values for different T and . The scaling property (Eq. 17) allows us to calculate the binning function for an arbitrary diffusion time by interpolation of the tabulated values.

However, an analytical expression of the binning function is obtained in the limit of short binning times (T << _d). By inserting Eq. 11, which is valid in this limit, into Eq. 10, we obtain a normalized correlation function G_r of 1. This approximation is used to determine the analytical form of the binning function for short binning times,

(18)

To calculate binning functions for arbitrary binning times one has to choose a point-spread function. A commonly used point-spread function in fluorescence fluctuation experiment is given by a three-dimensional Gaussian (Eq. 4). We calculate the normalized correlation function up to the fourth order according to Eq. 10,

(19)

We plug these correlation functions into Eq. 16 and integrate numerically. An analytical expression for the second binning function B₂ (T;_d,r) has been constructed earlier (11). In Fig. 1, we plot B_r/T^r up to the fourth order for _d = 1 as a function of the binning time. We notice that for T << _d, the function B_r/T^r goes to 1, as predicted by Eq. 18. We also find that higher order binning functions decay faster than lower order ones.

View larger version (16K): [in this window] [in a new window]   FIGURE 1  Binning function r(T) up to fourth order. The function r/Tr is graphed for as a function of binning time T for second order (solid line), third order (dashed line), and fourth order (dash-dotted line). The binning functions are calculated for a diffusion time d = 1 and a squared beam waist ratio r = 25.

(20)

Now we treat the case of more than one molecule in the closed volume V. The r^th cumulant for a large number of N_total noninteracting, diffusing molecules is given by the sum of the r^th cumulant over all molecules. If the molecules are identical, we obtain by using Eq. 13,

(21)

Fluctuation experiments measure fluorescence emerging from an open excitation volume, which is much smaller than the total sample volume V. We use the concentration c, which is given by N_total/V, to define the average occupation number N in the observation volume V_PSF:

(22)

Thus, the r^th cumulant of the integrated fluorescence intensity simplifies to

(23)

When the binning time is short, the binning function B_r is approximated by T^r (Eq. 18). Thus the formula for the cumulant (Eq. 23) reduces to the special case for short binning times (11),

(24)

We rewrite Eq. 23 with the help of Eq. 17 in the following form:

(25)

Equations 23 and 25 describe the r^th cumulant of the integrated fluorescence intensity of diffusing molecules with a diffusion time _d, brightness , and an occupation number of N. We have written the cumulant as a function of T to stress that, experimentally, we are interested in the binning time-dependence of the cumulant. Eq. 25 is useful for calculating the cumulant function _r(T), because every species is described by the same binning function, if the binning time is normalized by the diffusion time. The cumulants of a mixture of noninteracting fluorescent species are given by the sum of the cumulants of each individual species according to the additive property of cumulants for independent random variables,

(26)

Experimentally we observe photon counts instead of integrated intensities. Equation 3 states that the factorial cumulants of the photon counts are identical to the cumulants of the integrated fluorescence intensity. In other words, the cumulants _r(T) of the integrated fluorescence intensity are experimentally determined by calculating the factorial cumulants of the photon counts.

Variance of the factorial cumulants
It is important to know the experimental error of factorial cumulants for judging the quality of fits to theoretical models. In statistics, the error is determined by the variance. We use a technique called moments-of-moments (7, 11) to calculate the variance of factorial cumulants. For example, the variances of the first five factorial cumulants are given in Müller (11). In this technique, to calculate the variance of r^th order factorial cumulant factorial cumulants up to the 2r^th order are needed. Here, for convenience, we rewrite the variance of the factorial cumulants and in terms of factorial cumulants (the variance of higher order factorial cumulants is readily calculated symbolically using the statistical software MathStatica (MathStatica, Sydney, Australia)):

(27)

where n is the number of data points. There is a potential problem, however. It is well known that the higher order moments are notoriously difficult to estimate in experiment. The signal/noise ratio decreases rapidly with increasing order of the cumulant (11). It seems that we cannot determine experimentally the error of high order factorial cumulants accurately since it involves even higher order moments. Fortunately, in our case, the variance of the factorial cumulants is mainly determined by the lower order factorial cumulants. This can be understood as follows. In the short binning time limit, the r^th order factorial cumulant is proportional to T^r (11). Therefore, higher order factorial cumulants are much smaller than the lower order ones. For instance, is mainly determined by and only weakly depends on and For very long binning time, the photon counts converge to a Gaussian distribution, which is solely determined by the first two cumulants. In other words, compared to the first two factorial cumulants, higher order factorial cumulants are not important in determining the error of second order factorial cumulants. The same argument applies to the variance of higher order factorial cumulants. This suggests that we may define a truncated variance for the r^th (r > 1) order factorial cumulant, which neglects contributions of all factorial cumulants higher than r. We list the truncated variance of factorial cumulant up to third order,

(28)

Another assumption of the moments-of-moments technique is that the experimental data, which in our case are photon counts, are independent of each other. This assumption is not valid for fluctuation experiments. In fact, FCS uses the correlation between photon counts to determine transport properties of particles. We know that the correlation between photon counts decays to zero for very long binning times. However, it is still necessary to investigate its effect on the variance of the factorial cumulants for short and intermediate binning times. As we will show, the effect of correlation is negligible for estimating the variance of the higher order factorial cumulants, but has a strong influence on the first order.

It is difficult to exactly calculate the variance of factorial cumulants of arbitrary order for correlated data. Here we illustrate the calculation of variance for the first two factorial cumulants in the context of fluorescence fluctuation experiments, following the procedure outlined by Qian (17). The first factorial cumulant is just the intensity, which is defined as

(29)

where k_i is the i^th photon counts and n is the total number of data points. The true variance Var_T of for correlated data is given by

(30)

where represents the ensemble average. The first term Var_M is the one determined by moments-of-moments. The second term Var_Corr contains the contribution due to correlations between photon counts. In Appendix B, we calculate the covariance or correlation in the photon counts k(t_i)k(t_j) for arbitrary binning times (Eq. 45). If we plug Eq. 45 into Var_corr, it is written as

(31)

The summation in Eq. 31 can be carried out analytically by rearranging the order of summation. The final result is very simple,

(32)

Eq. 32 is exact for all binning and data acquisition times. When the binning time is short, the summation in Eq. 30 can be approximated by transforming it into an integral (17). The correction derived by this approximation contains only the first term of Eq. 32.

The variance of the second factorial cumulant can be calculated similarly. In Appendix C, we work out the correction for the case of short binning times. Since higher order correlation functions decay much faster than the lower order ones, the leading correction comes from Eq. 53. However, it involves fourth order factorial cumulants, and as we argued above, the correction should be small.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Instrumentation
The experiments are performed on our homebuilt two-photon microscope. A mode-locked Ti:sapphire laser (Tsunami, Spectra-Physics, Mountain View, CA) pumped by an intracavity doubled Nd:YVO4 laser (Millennia Vs, Spectra-Physics, Mountain View, CA) serves as the two-photon excitation source. An excitation wavelength of 780 nm was used for all dye experiments. A 63X Plan-Apochromat oil immersion objective (NA = 1.4) is used to focus the laser and collect the fluorescence. The light passes through an optical filter and is detected with an avalanche photodiode (APD) (SPCM-AQ-14, Perkin-Elmer, Dumberry, Québec). The output of the APD, which produces TTL pulses, was directly connected to a data acquisition card (FLEX02, Correlator.com, Bridgewater, NJ). The recorded photon counts were stored and later analyzed with programs written for IDL version 5.4 (RSI, Boulder, CO). A program written in Fortran with a nonlinear least-squares optimization routine from the Port Library (available at http://www.netlib.org) is used to fit the theoretical model to the experimental cumulants.

Sample preparation
Alexa488 (Molecular Probes, Eugene, OR) and Rhodamine 6G (Acros Organics, Morris Plains, NJ) were dissolved in water with 0.02% (by volume) of NP-40 (Sigma, St. Louis, MO). The small amount of detergent was added to prevent Rhodamine 6G from absorbing to the surface of our sample holder.

Data analysis
We use the software MathStatica to derive formulas of factorial cumulants up to the 20th order and the variance of the factorial cumulants up to the 10th order by the technique of moments-of-moments (7). These formulas are implemented into an analysis program written in IDL to calculate the experimental factorial cumulants and their errors.

We rebin the data to determine the factorial cumulants for different sampling times. The procedure is performed as follows: We feed the recorded sequence of photon counts into software to calculate the experimental factorial cumulants of photon counts of sampling time T. To get cumulant for a sampling or binning time of 2T, we add neighboring photon counts together to get a new sequence of photon counts with binning time 2T. We apply the same software on the rebinned data to get the cumulants for a binning time 4T. This process is repeated to calculate the cumulants for binning times of specific integer multiples of T. By rebinning, we calculate the factorial cumulants over binning times that cover three orders of magnitude.

We fit the experimentally determined factorial cumulants k_[r] to theoretical cumulants _[r] determined by Eq. 26 with a nonlinear least-squares fitting program. The reduced ² of the fit is given by

(33)

The value of K is the total number of cumulants used in the fit and p is the number of free fitting parameters of the model.

	RESULTS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

 
Single dye experiment
To test the TIFCA theory for arbitrary binning times, we perform experiments on simple fluorescent dye solutions. Each species is characterized by three parameters: its molecular brightness , the diffusion time _d, and the average number of molecules N in V_PSF. This information can be obtained by fitting the first two cumulants (11). Here we fit the first four cumulants simultaneously, while allowing ₃ and ₄ to be determined experimentally as free parameters. Fig. 2 shows the first four factorial cumulants as a function of binning time for Rhodamine 6G. The data was taken with a sampling time of 5 µs with a total measurement time of 600 s. The reduced ² of the fit is 0.67 with a recovered brightness of = 23,000 cpsm, a diffusion time of _d = 41 µs, and with N = 0.72 molecules in the observation volume. The fitted factors are ₃ = 0.246 ± 0.004 and ₄ = 0.208 ± 0.012. Conventional FCS analysis of the fluctuation data yields a diffusion time of _d = 42 µs and N = 0.73 molecules. These parameters are, within experimental error, identical to the results of TIFCA.

View larger version (23K): [in this window] [in a new window]   FIGURE 2  FCA analysis of a Rhodamine 6G solution. The data are taken with a sampling time of 5 µs and a total data acquisition time of 10 min. The factorial cumulants are calculated for binning times from 10 µs to 640 µs. The first four cumulants divided by Tr–1 are shown in A–D as diamonds. The best fit to a single species model is shown as a solid line together with the normalized residuals (diamonds) for each cumulant in the lower panel with a dashed line to guide the eye. The fit determined a brightness of = 23,000 cpsm, a diffusion time of d = 41 µs and an average number of molecules of N = 0.72. In the fit 2 was fixed to which corresponds to a three-dimensional Gaussian PSF, whereas 3 and 4 are free parameters. The fit determined 3 = 0.245 and 4 = 0.208.

View larger version (17K): [in this window] [in a new window]   FIGURE 3  The relative error of the factorial cumulant by moments-of-moments analysis as a function of the reduced binning time T/d. The relative error is shown up to order four in A–D for the Rhodamine 6G sample shown in Fig. 2. We determined the error in three different ways. The diamonds represent the relative error calculated from the experimental data. We also calculated theoretical cumulants from the fit parameters according to Eq. 26 and determined their relative error using the moments-of-moments variance (solid line). In addition, we also determined the relative error using the truncated moments-of-moments variance (dashed line). The data were taken with a sampling time of 5 µs and a total data acquisition time of 131 s.

To confirm this, we also calculate the truncated variances of cumulants according to Eq. 28, which ignores all cumulants of order higher than r. The relative error of the factorial cumulants based on the truncated variance is shown in Fig. 3 for r > 1 as dashed lines, which is almost indistinguishable from the exact theoretical ones (solid lines). Both errors are indistinguishable at very short and very long binning times as discussed in Theory, above. The differences at intermediate binning times are small and are negligible for all practical purposes.

Fig. 3 also demonstrates that the relative error of the second, third, and fourth factorial cumulant has a minimum with respect to the binning time. In other words, rebinning allows us to maximize the signal/noise ratio of factorial cumulants. This increase in the signal/noise ratio was the main motivation to introduce rebinned factorial cumulants. The increase in signal/noise is especially important for higher order cumulants. For example, binning allows us to reduce the relative error of the fourth cumulant by a factor of 16 compared to short binning times.

The relative error of factorial cumulants as a function of binning time is shaped by two competing factors. The factorial cumulants for longer binning times are obtained by rebinning the original data. For each rebinning step, the sampling time increases, and the number of data points decreases. On the one hand, fewer data points lead to an increase in the variance and the relative error. On the other hand, the number of photons collected from a single molecule during time T increases for each rebinning step, which increases the signal/noise ratio. Which of the two factors dominates depends on the reduced binning time T/_d. For short binning times (T << _d), is, according to Eq. 24, proportional to T_r. The variance is dominated by the lowest power of T, which, according to the moments-of-moments technique, is given by T^r. In addition, the variance also increases, because of the reduction in data points, which is proportional to T. These two effects lead to a variance that is proportional to T^r+1. Thus, the relative error is of order As long as r > 1, rebinning reduces the relative error of for short binning times.

In the other extreme, for very long binning times (T >> _d) we can assume that there is a characteristic time T_c, above which the photon counts are statistically independent. Without losing generality, we set the binning time T = mT_c. According to the additive property of cumulants for independent random variables, In other words, the cumulant is proportional to the binning time, The variance of is proportional to the highest power of T, which is again T^r+1. So the relative error scales as Thus, for r > 1, the relative error will increase as a function of binning time. The experimentally observed dependence of the relative error of cumulants with in Fig. 3 is exactly as predicted by theory. For short binning times, the relative error decreases, reaches a minimum, and then increases at long binning times.

The moments-of-moments technique assumes statistically independent variables and ignores correlations between data. We now consider the effect of the correlation between photon counts on the variance of factorial cumulants. To do this, we need an experimental estimate of the true variance for data with embedded correlations. We calculate this experimental variance Var_E by dividing a long data set into a number of smaller segments. For each segment, we calculate the factorial cumulants, and determine the average and the variance of these factorial cumulants. For comparison, we also calculate the variance of the factorial cumulants for each segment by the moments-of-moments technique. We average all these variances to arrive at the estimated variance Var_M based on the moments-of-moments technique, which ignores correlations. Fig. 4 shows the ratio between these two variances. A ratio 1 shows that the two variances agree with one another, whereas a ratio different from 1 indicates the importance to include correlations in the determination of the variance. Fig. 4 shows that Var_M is considerably smaller than Var_E for the first factorial cumulant, but is very close to Var_E for higher order factorial cumulants.

View larger version (13K): [in this window] [in a new window]   FIGURE 4  Ratio of moments-of-moments variance and the experimental variance of the factorial cumulants obtained from the Rhodamine 6G sample. The experimental variance includes correlations between the data points, which are ignored by the moments-of-moments method. Only the variance of the first cumulant is clearly influenced by the presence of correlations, whereas the variance of the higher order cumulant is accurately described by the moments-of-moments technique.

View larger version (14K): [in this window] [in a new window]   FIGURE 5  Variance of of a Rodamine 6G sample. Diamonds represent the variance determined by moments-of-moments. Triangles are experimentally determined variances, which are obtained by dividing the long data set into small segments and calculating the variance from the statistics of the cumulant of each segment. The solid line is the variance calculated according to Eq. 32, which takes correlations between photon counts into account. (A) Variance of as a function of total data acquisition time. The variance decreases inversely proportional to TDAQ as expected. The moments-of-moments variance underestimates the experimental variance in The corrected variance describes the experimental variance accurately. (B) Variance of as a function of reduced binning time T/d. At short binning times, the moments-of-moments technique underestimates the variance of but approaches the experimental variance for long binning times, because correlations between photon counts in adjacent bins vanish. The corrected variance describes the experimental variance accurately for all binning times.

Binary dye mixture
Resolving a binary dye mixture requires the determination of six parameters: the molecular brightness _i, the diffusion time _di, and the number of molecules N_i of each species i. The diffusion time of small organic dyes is generally very similar. In other words, we mainly rely on the molecular brightness to resolve species. Therefore, at least four factorial cumulants are needed to resolve a binary dye mixture. We mixed stock solutions of Rhodamine 6G and Alexa 488 together and performed a fluorescence fluctuation measurement. Fig. 6 shows the first four experimental cumulants of the binary dye mixture. The fit of the experimental cumulants to a single-species model is shown as solid lines. The model and experimental data are not in agreement. The misfit is especially apparent for the third and fourth factorial cumulant. The reduced ² of the fit is 153, which is not surprising, because we expect that the single-species model fails to describe experimental data of a binary dye mixture. In the same figure, we also show the fit of the experimental cumulant to a two-species model, which is shown as dashed lines. The reduced ² for the two-species model is 2.35 and describes the experimental data within experimental error.

View larger version (26K): [in this window] [in a new window]   FIGURE 6  FCA analysis of a binary mixture of Rhodamine 6G and Alexa 488. The sample was measured for 30 min with a sampling time of 5 µs. The first four cumulants are plotted as (diamonds) in A–D with respect to binning time T. The fit to a single-species model is shown as solid line. The lower panel shows the normalized residuals of the fit as a solid line. The best fit to a two-species model is shown as a dashed line. The normalized residuals of the two-species fit are shown as a dashed line in the lower panel. The recovered parameters are R = 24,000 cpsm, dR = 40 µs, NR = 0.18 for Rhodamine, and A = 9300 cpsm, dA = 37 µs, NA = 0.53 for Alexa.

View larger version (9K): [in this window] [in a new window]   FIGURE 7  FCA analysis of a dilution experiment of a binary mixture of Rhodamine 6G and Alexa 488. Each data set was taken for 30 min with binning time of 5 µs. (A) The brightness of Rhodamine 6G (diamond) and Alexa 488 (square) are shown together with their average value (solid lines). (B) The number of molecules of both species recovered from the fit. The solid line is a fit of the data to a straight line. Its slope determines the concentration ratio of the mixture. We recover a composition of 22% of Rhodamine 6G and 78% of Alexa 488.

	DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION APPENDIX A APPENDIX B APPENDIX C ACKNOWLEDGEMENTS REFERENCES

Photon counting histogram and fluorescence intensity distribution analysis analyze the pdf of the photon counts. Both techniques are only applicable for short sampling times (T >> _d). Extending PCH theory to arbitrary binning times requires a path integral approach, which does not appear attractive for practical use. Cumulants provide an alternative approach to analyze fluctuation data. We previously demonstrated that PCH and FCA are equivalent techniques in the limit of short binning times (11). However, FCA has an advantage over PCH, because an exact extension of the technique to arbitrary binning times is feasible. We developed in this article the exact theory of time-integrated fluorescence cumulants, which experimentally describe the factorial cumulants of photon counts at arbitrary binning times. PCH analysis and FCA are sensitive to molecular brightness; FCS is sensitive to the diffusion time. TIFCA combines aspects of both PCH and FCS, because molecular brightness and diffusion time are determined simultaneously.

There has been an approach described in the literature that extends histogram analysis to longer sampling times (16). Fluorescence intensity multiple distribution analysis studies the histogram of photon counts at different sampling times, and extracts diffusion and brightness information. However, the model is based on an approximation. It utilizes the first two cumulants to correct the brightness and number of molecules for different binning times, but ignores all higher order cumulants. Resolution of two species requires information derived from the third- and fourth-order cumulants. The effect of the approximation introduced by fluorescence intensity multiple distribution analysis on the resolution of species has not been investigated yet. We stress that TIFCA is currently the only theory that is exact for arbitrary binning times. Thus, TIFCA provides an excellent framework to study the effect of binning on PCH and related techniques.

Error analysis is needed to judge the quality of models in describing the experimental data. We therefore studied the variance of the factorial cumulants. We apply moments-of-moments to calculate the variance. We assume a large sample size in our formulation of the theory. This condition is easily met in FFS experiment since we acquire routinely millions of data points. The moments-of-moments technique also assumes statistical independence of the sampled data. This requirement is problematic since FFS data are correlated. For all experimental conditions we tested, we found that the moments-of-moments variance of is sufficiently accurate for r > 2. For r = 1, significantly underestimates the variance and correlation must be explicitly taken into account. We derived an expression to calculate the variance of for correlated data, which successfully describes our experiments. In practice, can be measured with extreme accuracy. In fact, the accuracy is high enough to detect the presence of other noise sources, such as long-term fluctuations in the excitation power of the laser. Such additional noise sources are not accounted for in our theory, which leads to an underestimate of the true variance by Eq. 32. For r = 2, there is a small bias in which usually can be safely ignored. We determined the leading-term correction due to correlations for short binning times. Since the bias will be maximal for short binning times, Eq. 53 allows us to calculate the largest bias introduced by the moments-of-moments technique. The leading-term correction contains fourth-order factorial cumulants and should be small, as we argued in Results, above. For very long binning times, the correlation between different bins dies away and becomes asymptotically unbiased. In practice, we found that the bias is <20% in all cases studied. To summarize, the moments-of-moments technique is suitable to calculate for but correction must be applied for

We investigated the behavior of the relative error of factorial cumulant as a function of binning time. The relative error is an ideal indicator for the importance of a factorial cumulant for data analysis. For example, means is statistically insignificant and not useful for data analysis. Since each cumulant contains independent information for a random variable, the number of statistically significant cumulants specifies whether the data is sufficient to resolve species. The theoretical model of the variance of factorial cumulants allows us to calculate the number of significant cumulants for various experimental conditions and is useful for judging the feasibility of resolving species by experiment (11). Rebinning reduces the number of data points, but more photons are collected from a single molecule during the bin time. These two competing factors shape the dependence of the relative error on binning time T. Theory predicts that the relative error scales as for short binning times and as for long binning times. Thus, for r > 1, rebinning reduces the relative error for short binning times, but leads to an increase at long binning times. We verified this behavior of the relative error in Fig. 3 experimentally, where the relative error decreases for short binning times, reaches a minimum, and then increases for long binning times.

Let us briefly compare the signal/noise characteristics of PCH and TIFCA. It is straightforward to calculate factorial cumulants from the histogram of photon counts. However, only a finite number of statistically significant cumulants are contained within an experimental histogram. To resolve two species by brightness requires four statistically significant cumulants (11). Let us assume we want to improve the signal/noise ratio of the fourth cumulant by a factor of 10. Because the relative error of the fourth cumulant scales as we require a 100-fold longer data acquisition time T_DAQ to achieve a 10-fold signal/noise improvement by PCH. TIFCA, in contrast to PCH, offers two ways to improve the signal/noise ratio of the experiment. First, just as in the case of PCH, longer data acquisition times reduce the relative error of cumulants. Second, rebinning of data can reduce the relative error, without need for longer data acquisition times. In the short sampling time limit (T << _d), the relative error of the r^th order cumulant scales as Thus rebinning by a factor of 5 reduces, in the short sampling limit, the relative error of the fourth cumulant by a factor of 10, whereas PCH requires a 100-times longer data acquisition time to achieve the same improvement. This example serves to illustrate the advantage of TIFCA over PCH.

However, practically, there is a limit to the improvement in signal/noise achievable by binning. Each binning step moves us away from the short sampling time regime, and the increase in signal/noise slows until the relative error of the cumulant reaches a minimum. Further rebinning decreases the signal/noise ratio. Despite this limitation, we demonstrated in Fig. 3 D that the relative error of the fourth cumulant improved by more than an order-of-magnitude by rebinning of the experimental data. The reduction of relative error by binning increases with the order r of the cumulant, because the relative error scales as for T << _d. This is confirmed by the data in Fig. 3. The improvement in signal/noise achieved for each cumulant increases with its order.

As we mentioned earlier, the signal/noise ratio of cumulants decreases rapidly with its order and experimental data only contain a finite number of cumulants with For example, the relative error of cumulant in Fig. 3 increases by approximately one order-of-magnitude for each successive cumulant. Thus, we only have four factorial cumulants with relative error <1 when the binning time is small. But the relative error of the fifth-order cumulant, is reduced to <1 by rebinning and reaches a minimum value of 0.1. We have not made use of the additional information provided by in this article, where we focus on introducing the technique. But it is straightforward to extend the technique to include higher order cumulants.

To simplify the discussion regarding our new technique, we acquired data for relatively long times to have four significant cumulants at all sampling times. However, much shorter sampling times are sufficient for resolving the binary mixture presented in Fig. 6 by TIFCA. If we only take the first 2 min of acquired data and perform TIFCA analysis, we arrive at a reduced ² of 13 for a single-species fit. A two-species fit determines _R = 23,000 cpsm, _dR = 39 µs, N_R = 0.22 for the Rhodamine, and _A = 7600 cpsm, _dA = 37 µs, N_A = 0.53 for Alexa, which is in excellent agreement with the result obtained for the complete data set. A fit of the PCH of the same data sampled at 10 µs to a single-species fit yields a reduced ² of 1.8. Thus, in contrast to TIFCA, PCH is unable to resolve this binary mixture.

The point-spread function (PSF) of the laser at the focal point of the objective is complicated (4). In FFS experiments, the PSF is conventionally modeled by a two-dimensional Gaussian, three-dimensional Gaussian, or a Gaussian Lorentzian function. These serve as approximations of the physical PSF. FFS is only able to distinguish between different PSF models if the signal statistics are sufficient. For example, with a bright dye sample, FCS can easily distinguish between a two-dimensional Gaussian and a three-dimensional Gaussian PSF by least-squares fitting. If two model functions lead within experimental error to identical fitting results, then choosing a PSF is simply a matter of convention and mathematical convenience. For example, the Gaussian-Lorentzian and the three-dimensional Gaussian model fit experimental autocorrelation curves equally well (12). The situation is similar for PCH analysis (1), where both the three-dimensional Gaussian and Gaussian Lorentzian PSF describe experimental histograms. TIFCA depend on the PSF model in two ways: the g factors and binning functions. The binning function determines the shape of the cumulant function and is determined by integration of correlation functions, which ultimately depend on the PSF. A two-dimensional Gaussian PSF is unable to fit experimental cumulants (data not shown), which is not surprising because the two-dimensional Gaussian also fails to fit the experimental autocorrelation function. In the current implementation of TIFCA, we chose a three-dimensional Gaussian model for calculation of the correlation functions. The computed binning functions based on a three-dimensional Gaussian PSF describe the shape of the experimental cumulants very well. The -factors are associated with the amplitude of the cumulant and characterize the shape of the PSF. According to definition (Eq. 9), ₁ is always 1. The second g-factor, ₂, has to be calculated from the PSF model, because TIFCA is unable to measure this parameter directly. We choose the three-dimensional Gaussian model and set ₂ to However, higher-order factors can be determined by experiment, because a single species is completely specified by its first two cumulant functions (11). This allows us to treat ₃ and ₄ as free parameters, which are determined from fitting a single-species model to the first four cumulant functions. We recovered values of ₃ = 0.251 and ₄ = 0.223, which differ from the values predicted for a three-dimensional Gaussian model. This discrepancy highlights that the model functions only approximately describe the physical PSF. Instead of constructing a better PSF model function, we choose to determine the higher-order factors experimentally. In other words, we perform a calibration experiment that determines ₃ and ₄. Once the g factors are calibrated, we fix them in the analysis of subsequent experiments. We believe that this calibration procedure adds flexibility for adapting TIFCA to other instruments with slightly different optical properties.

The diffusion coefficient determines the shape, and the brightness and number of molecules controls the amplitude of the cumulant function. Thus, TIFCA not only distinguishes species by brightness, but also by their diffusion coefficients. If the diffusion coefficients of two species differ substantially, then the presence of the two species is reflected in an altered shape of the cumulant functions. However, in this article we resolved a binary mixture based on brightness alone, because the diffusion coefficient of both fluorophores is essentially identical and the shape of the cumulant functions is determined by a single diffusion time. Resolving species purely by brightness is experimentally the more challenging case.