Originally published as Biophys J. BioFAST on June 30, 2006.
doi:10.1529/biophysj.106.086181
Biophysical Journal 91:2687-2698 (2006)
© 2006 The Biophysical Society
Dual-Color Time-Integrated Fluorescence Cumulant Analysis
Bin Wu,
Yan Chen and
Joachim D. Müller
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota
Correspondence: Address reprint requests to Bin Wu, Tel.: 612-624-6045; E-mail: binwu{at}physics.umn.edu.
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ABSTRACT
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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.
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INTRODUCTION
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Fluorescence fluctuation spectroscopy (FFS) examines the fluctuating fluorescence signal from a small illumination volume <1 fl created by modern two-photon or confocal microscopy (1
,2) to characterize the behavior of fluorophores. Statistical analysis tools such as fluorescence correlation spectroscopy (3) and photon-counting histogram (PCH) or fluorescence intensity distribution analysis (4,5) are required to extract static and dynamic information from the stochastic fluorescence signal. Fluorescence correlation spectroscopy uses the correlation function to capture the temporal information of the physical process, while PCH uses the amplitude distribution of the fluctuations to characterize the concentration and brightness of each fluorescent species. Fluorescence intensity multiple distribution analysis (6) and photon arrival-time interval distribution (7) have been developed to take both temporal and amplitude information into account. A PCH theory that incorporates diffusion has also recently been described (8).
Moment analysis is an alternative technique for studying a fluctuating fluorescence signal and was originally developed in the late 80s and early 90s (9–12). Fluorescent cumulant analysis (FCA) (13) and time-integrated fluorescence cumulant analysis (TIFCA) (14) represent a further development of moment analysis. Cumulants are a special representation of moments that possess mathematical properties particularly suited for statistical analysis. For example, cumulants of independent random variables are additive. We previously discussed the advantages of using cumulants in analyzing fluorescence fluctuation data (13,14). 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. TIFCA generalizes the cumulant analysis to arbitrary sampling times, which makes it able to determine the dynamics as well as the concentration of fluorescent species. The exact theoretical treatment of TIFCA also allows the optimization of signal statistics in the analysis of FFS experiments.
In conventional FFS, all light is collected by a single detector. In most two-channel FFS experiments, the fluorescent signal is split by a dichroic mirror into two different detectors based on the color of the fluorophore. Two-channel FFS offers the possibility to resolve fluorophores according to their emission spectra, thus offering a method for detecting the association and dissociation between different species of biomolecules (15–17). The sensitivity of two-channel FFS in resolving species is dramatically improved over conventional single-channel FFS. In this article, we extend the theory of TIFCA to two-channel FFS experiments. A simple expression for the bivariate factorial cumulant of photon counts is derived for arbitrary binning times. Theoretical models are used to fit the experimental cumulants of the photon counts as a function of sampling time, which simultaneously determines the molecular brightness in each channel, the occupation number, and the diffusion time of each species from a single measurement. The statistical error of factorial cumulants is also derived and experimentally verified. The relative error of cumulants measures its statistical significance and provides weighting factors for data fitting.
Nonideal detector effects cause artifacts in the analysis of FFS data (18,19). These effects, if not accounted for, may lead to erroneous interpretation of the experimental data and therefore severely limit the practical use of the analysis technique. We develop in this article a theoretical model of nonideal detector effects on the factorial cumulants of photon counts and verify it experimentally.
The new technique is then applied to study EGFP and EYFP in living cells. We first measure EGFP and EYFP alone to demonstrate the validity of the theory in this challenging environment. The EGFP/EYFP pair exhibits strong spectral overlap, which poses a serious challenge for resolving species by FFS (17). Based on the parameters determined from the single-species measurements of EGFP and EYFP, we investigate the resolvability of the two proteins theoretically. Our results show a dramatic improvement of resolvability compared to conventional PCH analysis. Finally, we apply dual-color TIFCA to resolve for the first time binary mixtures of EGFP and EYFP over a wide concentration range from single measurements in living cells.
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THEORY
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Cumulants for arbitrary binning times
The derivation of dual-color cumulant analysis closely follows that of regular TIFCA (14). Since regular TIFCA theory only considers a single detection channel, we also refer to it as single-color or single-channel TIFCA. In the dual-color case, the fluorescence light is split into two different detectors with a dichroic mirror. A theory that describes the bivariate factorial cumulants of photon counts detected in two channels is needed. Here, we use the labels A and B to distinguish the two detection channels and the subscript Q refers to any one of the two channels. As a convention in this article, channel A always refers to the red channel and channel B to the blue channel. The probability distribution function P(kA,kB;T) of detecting kA photons in channel A and kB photons in channel B is related to the probability distribution function P(WA,WB) of the integrated light intensity WA and WB, according to Mandel's formula (20),
 | (1) |
where 
A and B are the detection efficiencies of the photon detectors and T is the sampling time. For convenience, we set A = B = 1, which is equivalent to measuring the intensity in counts per second (cps). As a convention, we use 
m,n as a symbol for the (m,n)th cumulant and [m,n] for the (m,n)th factorial cumulant. An up-carat (^) over a symbol indicates that it represents an experimentally measured physical quantity. Mandel's formula implies that the bivariate cumulant generating function of (WA,WB) equals the bivariate factorial cumulant generating function of (kA, kB). In other words, the cumulant m,n(WA,WB) of the integrated intensity (WA, WB) is determined experimentally by measuring the corresponding factorial cumulant [m,n](kA, kB) of the photon counts (kA,kB),
 | (2) |
Next, we calculate the theoretical expression of m,n(WA,WB) for arbitrary sampling times T. Assume that a single molecule is diffusing in a large, but closed volume V illuminated by focused laser light with a normalized point spread function (PSF) given by
.
The experimental PSF is approximated by a model function. Usually a three-dimensional Gaussian is used in the literature to describe the PSF of fluorescence fluctuation spectroscopy,
 | (3) |
where s refers to an s-photon (s = 1,2...) excitation experiment. The same PSF can be used for both detection channels since the fluorophores are coexcited by the same laser. When the molecule is located at position
at time t, the fluorescence intensity IQ in channel Q (Q = A, B) is given by
 | (4) |
where 
Q (measured in counts per second per molecule, i.e., cpsm) is the brightness of fluorophores in channel Q. The integrated intensity WQ within the sampling time T is
 | (5) |
With the above definition, the (m,n)th bivariate raw moment of (WA,WB) is easily calculated as
 | (6) |
The above integration has been solved and is expressed in terms of binning functions Bm+n(T;
d) in Wu and Müller (14),
 | (7) |
where VPSF is the reference volume conventionally used in fluorescent fluctuation experiments,
 | (8) |
and the coefficient 
s is called the sth -factor and defined as
 | (9) |
According to Wu and Müller (14), the binning function only depends on the diffusion time d = w2/4sD for a molecule with diffusion constant D in a focused laser beam with beam waist w. The binning function depends on the choice of PSF and has been previously determined for a three-dimensional Gaussian PSF (14). Because the excitation profile of both detection channels overlaps, only a single PSF is needed.
The (m,n)th bivariate cumulant 
of (WA,WB) can be expressed as a series of terms of raw moments up to (m,n)th order, where denotes the (m,n)th cumulant for a single molecule. In the thermodynamic limit V 
, all terms except the (m,n)th raw moment vanish (13),
 | (10) |
Now suppose there are Ntotal noninteracting, diffusing molecules in the volume V, such that in the thermodynamic limit V , the concentration c = Ntotal/V is kept constant. We define the average occupation number N in the observation volume VPSF,
 | (11) |
Since the fluorescence emitted by different noninteracting molecules is statistically independent from each other, the cumulant of the total integrated fluorescence intensities in the two channels is given by summing up the corresponding cumulant of all individual molecules,
 | (12) |
In the short sampling time limit T << d, the binning function Bm+n(T;d) is approximated by Tm+n (14). In this limit the bivariate cumulant simplifies to
 | (13) |
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,
 | (14) |
Experimentally, photon counts are observed instead of integrated intensities. Equation 2 allows us to measure the cumulants of the integrated intensities m,n(WA, WB) by calculating the experimental estimates
of the factorial cumulant of the photon counts [m,n] (kA, kB).
Variance of the bivariate factorial cumulants
The variance of an experimental quantity is an important measure of its statistical accuracy. In fluorescence cumulant analysis, the variance of the factorial cumulant is used as the weight in the nonlinear least-squares fit to the theoretical model. The variance is also a good indicator of how many statistically significant cumulants are present in the data. It is difficult to exactly calculate the variance of multivariate factorial cumulants of experimental photon counts with a finite number of correlated data points. We use a technique called moments-of-moments, which ignores the correlation in the data, to calculate the variance of the factorial cumulant. A detailed discussion of the limitations of this technique has been presented for the univariate case, where we also introduced correction terms that account for correlations (14). This approach is directly applicable to cumulants like [m,0] and [0,n], which are essentially univariate in nature. Thus we concentrate on cross-bivariate factorial cumulants. Since all cross cumulants are of order equal to or larger than two, the effect of correlations is negligible. The formulation we propose here also works for the univariate case, which constitutes a good validation of the method.
Because there is very little literature on the variance of multivariate factorial cumulants, we have to derive the formulas directly. We illustrate the process by deriving the variance of the first nontrivial cross-factorial cumulant [2,1]. First we express the factorial cumulant of photon counts in terms of cumulants of photon counts using the software MathStatica (MathStatica, Sydney, Australia). Generally, [m,n] is a linear combination of i,j with i 
m, j n. For example, the cross-factorial cumulant [2,1] is given by
 | (15) |
Next, each cumulant i,j is replaced by its unbiased estimator k-statistic ki,j (21) to construct the unbiased estimator of the factorial cumulant. In the case of [2,1], we get 
, where 
represents the unbiased estimator of [2,1]. The next task is to calculate the variance and covariance of ki,j. We apply the tensor representation technique of Kaplan (22). The algorithm in Kaplan's article is implemented in Mathematica (Wolfram Research, Champaign, IL). For example, the variance and covariance of k2,1 and k1,1 are
 | (16) |
where n is the total number of data points. To obtain this result, we take the large sample limit where n .
The variance and covariance of ki,j is now plugged into the variance of 
and results in the expression
 | (17) |
Finally, the cumulants are transformed back to factorial cumulants by MathStatica,
 | (18) |
Other variances of factorial cumulants are calculated in the same manner. We realize from this example that the resulting formulas are very long and cumbersome. However, all expressions are just simple polynomial and very easy to implement on computer. Here, for reference, we also list the variance of 
. The factorial cumulant [1,1] and cumulant 1,1 are equal by definition. The unbiased estimator for is thus given by the corresponding k-statistics k1,1. The variance Var[k1,1] is given by the first line in Eq. 16. Again the cumulants 1,1, 0,2, 2,0, and 2,2 are transformed back into factorial cumulants. Evaluating the expression results in
 | (19) |
Nonideal detector effect
Up to this point, the theory of factorial cumulants of the photon counts (Eq. 14) assumes that the photodetectors are ideal. Real detectors are never ideal and this needs to be taken into account in the theoretical description of photon count statistics. Particularly, dead-time and afterpulsing cause significant changes in the photon count statistics of PCH (18,19). An afterpulse is a fake pulse after the detection of a real photon count. Dead-time describes a period of time after the registration of a photon in which the detector is unable to generate photon signals. The dead-time of nonparalyzable detectors, such as an actively quenched avalanche photodiode (APD), is unaffected by photons reaching the detector during the dead-time. A detailed description of these nonideal effects on fluorescent fluctuation experiments, especially PCH analysis, can be found elsewhere (18,19). Here we discuss the effect of nonideal detectors on the factorial cumulants of the photon counts. In the following, primed quantities are used to represent physical quantities measured with a nonideal detector.
The factorial-moment-generating function 
(23) is an important theoretical tool for treating nonideal photodetector effects of factorial cumulant. It is based on the probability-generating function (23), which for a bivariate distribution P(kA,kB) is defined as
 | (20) |
The factorial-moment-generating function is given by
 | (21) |
When is expanded in a Taylor series, the coefficient of the umvn/m!n! term describes the factorial moment µ[m,n] (23).
Now let us first treat the effect of afterpulses, which has an analytical solution. Assume that after detecting a real count, there is a probability of PQ, (Q = A, B), to observe a fake count. In addition, we postulate that different afterpulses are statistically independent from each other. Now suppose there are kQ real counts detected in channel Q. The output counts k'Q of the detector include afterpulses. We define a set of binary random variables 
(i = 1, ..., kQ) with a probability distribution
 | (22) |
The definition of an afterpulse event leads to the following relation between kQ and k'Q,
 | (23) |
The probability-generating function of (k'A,k'B) is given by
 | (24) |
where P' (k'A,k'B) is the probability distribution of afterpulse-influenced photon counts. With the help of a conditional probability distribution, we can relate P' (k'A,k'B) to the distribution of real photon counts P (kA,kB),
 | (25) |
The conditional distribution can be written as the product of afterpulse probabilities
P(k|<|^\prime|>|_|<|\mathrm|<|A|>||>|,k|<|^\prime|>|_|<|\mathrm|<|B|>||>||<|\vert|>|k_|<|\mathrm|<|A|>||>|,k_|<|\mathrm|<|B|>||>|)|<|=|>||<| _|<|\mathrm|<|i|>||<|=|>|1|>|^|<|\mathrm|<|k|>|_|<|\mathrm|<|A|>||>||>||>||<|_|<|\mathrm|<|j|>||<|=|>|1|>|^|<|\mathrm|<|k|>|_|<|\mathrm|<|B|>||>||>||>|\mathrm|<|Pr|>|(X_|<|\mathrm|<|A|>||>|^|<|\mathrm|<|i|>||>|)\mathrm|<|Pr|>|(X_|<|\mathrm|<|B|>||>|^|<|\mathrm|<|j|>||>|), | (26) |
since different afterpulses are statistically independent from each other. Thus 
is given by
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G|<|^\prime|>|_|<|\mathrm|<|k|<|^\prime|>||>|_|<|\mathrm|<|A|>||>|,\mathrm|<|k|<|^\prime|>||>|_|<|\mathrm|<|B|>||>||>|(u,v)|<|=|>||<|\sum_|<|\mathrm|<|k|>|_|<|\mathrm|<|A|>||>|,\mathrm|<|k|>|_|<|\mathrm|<|B|>||>||<|=|>|0|>|^|<||<|\infty|>||>||>|P(k_|<|\mathrm|<|A|>||>|,k_|<|\mathrm|<|B|>||>|)u^|<|\mathrm|<|k|>|_|<|\mathrm|<|A|>||>||>|v^|<|\mathrm|<|k|>|_|<|\mathrm|<|B|>||>||>|\\&&|<|_|<|\mathrm|<|i|>||<|=|>|1|>|^|<|\mathrm|<|k|>|_|<|\mathrm|<|A|>||>||>||>||<|_|<|\mathrm|<|j|>||<|=|>|1|>|^|<|\mathrm|<|k|>|_|<|\mathrm|<|B|>||>||>||>||<|\sum_|<|\mathrm|<|X|>|_|<|\mathrm|<|A|>||>|^|<|\mathrm|<|i|>||>|,\mathrm|<|X|>|_|<|\mathrm|<|b|>||>|^|<|\mathrm|<|j|>||>||<|=|>|0|>|^|<|1|>||>|u^|<|\mathrm|<|X|>|_|<|\mathrm|<|A|>||>|^|<|\mathrm|<|i|>||>||>|v^|<|\mathrm|<|X|>|_|<|\mathrm|<|B|>||>|^|<|\mathrm|<|j|>||>||>|\mathrm|<|Pr|>|(X_|<|\mathrm|<|A|>||>|^|<|\mathrm|<|i|>||>|)\mathrm|<|Pr|>|(X_|<|\mathrm|<|B|>||>|^|<|\mathrm|<|j|>||>|). | (27) |
Notice that we plugged Eq. 23 into the exponent of u and v. Carrying out the products and the second summation results in
 | (28) |
where 
is the probability-generating function for an ideal detector. Once the probability-generating function is known, the factorial-moment-generating function in the presence of afterpulsing is readily derived,
 | (29) |
Taking derivative on both sides of the above equation, we obtain the relation between the afterpulse-influenced factorial moments and the ideal factorial moments. Since factorial cumulants can be expressed in terms of factorial moments, we established a relationship between the afterpulse-influenced factorial cumulant and the ideal factorial cumulant. The above algorithm is programmed in Mathematica and provides a convenient method to express the afterpulse-influenced factorial cumulant in terms of ideal factorial cumulants. A few simple cases are listed below,
 | (30) |
To calculate the dead-time effect on factorial cumulants, we also consider the factorial-moment-generating function. We define the dead-time parameter as 
, where 
is the dead-time of the detector in channel Q and T is the sampling time. We only consider the case where 
Q(T) is a small parameter. In this case the dead-time influenced probability-distribution function of photon counts can be expanded in a Taylor series of the dead-time parameters (19). We explicitly consider the first-order expansion,
 | (31) |
The dead-time affected factorial moment generating function is
 | (32) |
Plugging Eq. 31 into Eq. 32 and carrying out the summation, we obtain
 | (33) |
Expanding both sides of the above equation in a Taylor series and comparing the coefficients of u and v, we obtain a relation between dead-time influenced factorial moments and ideal factorial moments. As was done in the case of the afterpulse effect, the factorial moments are used to express the dead-time influenced factorial cumulant in terms of ideal factorial cumulants. A few examples are listed below,
 | (34) |
Higher-order expansions can be performed in a similar way and allow us to express the dead-time influenced moments in terms of ideal moments, but the results becomes more cumbersome and a simple analytical expression like Eq. 33 is not available. We explicitly treated the expansion up to the second-order and used it to correct for dead-time effects in the experimental data. Note that the derivation of the dead-time corrected cumulants is approximate. The maximum number of photon counts kMAX received during the sampling time T by a nonparalyzable detector with dead-time 
is kMAX = T/, but the theory sums photon counts from 0 to infinity. We have found that this approach describes fluorescence fluctuation data with 
k
0.1, which translates into an intensity of 2 x 106 cps for a dead-time of 50 ns (19,24).
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MATERIALS AND METHODS
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The instrument for the two-color fluorescence fluctuation experiments consists of a Zeiss Axiovert 200 microscope (Thornwood, NY) and a mode-locked Ti:Sapphire laser (Tsunami, Spectra-Physics, Mountain View, CA) pumped by an intracavity-doubled Nd:YVO4 laser (Millennia Vs, Spectra-Physics). 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 split into two channels. Photons counts are 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 two-channel 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 (Research Systems, 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.
pEGFP-C1 and pEYFP-C1 plasmids were obtained from Clontech (Mountain View, CA). COS cells were obtained from ATCC (Manassas, VA) and maintained in 10% fetal bovine serum (Hyclone Laboratories, Logan, UT) and DMEM media. Cells were subcultured into an eight-well cover-glass slide (Naglenunc International, Rochester, NY) and then transiently transfected using Polyfect (Qiagen, Valencia, CA) according to manufacturer's instructions. Before conducting measurements, the grow media was removed and replaced with Leibovitz L15 (Invitrogen, Carlsbad, CA). All in vivo experiments are performed at an excitation wavelength of 960 nm, and a 515-nm dichroic mirror is used to separate the fluorescence into two detection channels.
In the dye experiments, Rhodamine 6G (Acros Organics, Morris Plains, NJ) and Rhodamine 110 (Molecular Probes, Eugene, OR) were dissolved in ethanol, and Alexa 488 (Molecular Probes) was dissolved in water. The stock solutions are diluted to appropriate concentrations for FFS experiments before each measurement. The dyes are excited at 780 nm. The Rhodamine 6G and Rhodamine 110 experiments employ a dichroic mirror with a transition wavelength of 544 nm and the Alexa 488 experiments use a 515-nm dichroic mirror.
We use the software MathStatica to derive formulas of factorial cumulants up to the eighth-order. The variance of the factorial cumulants is calculated up to the fourth-order by the technique of moments-of-moments with programs written in Mathematica. These formulas are implemented into an analysis program written in IDL (RSI, Boulder, CO) 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: The recorded sequence of photon counts is fed 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 algorithm on the rebinned data to get the cumulants for a binning time of 2T. 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 sampling times that cover three orders of magnitude.
We fit the experimentally determined factorial cumulants 
to theoretical cumulants [r,s] determined by Eq. 14 with a nonlinear least-squares fitting program. The reduced 
2 of the fit is given by
 | (35) |
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.
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RESULTS AND DISCUSSION
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Single dye experiment
To test the dual-color TIFCA theory we perform experiments on simple fluorescent dye solutions. Each species is characterized by four parameters, its molecular brightness in each channel (A and B), the diffusion time d, and the average occupation number N in VPSF. For simplicity, we define the order of a bivariate cumulant by the sum of its indices. For example, the factorial cumulant [i,j] is of (i + j)th order. We fit cumulants with order smaller or equal to four simultaneously. Fig. 1 shows these cumulants as a function of binning time for Rhodamine 6G. The data was taken with a sampling time of 20 µs and a total measurement time of 130 s. The reduced 2 of the fit is 0.85 with a recovered brightness of A = 34,000 cpsm, B = 9600 cpsm, a diffusion time of d = 44 µs, and an occupation number of N = 2.7.
TOP
ABSTRACT
INTRODUCTION
of the factorial cumulant 
is given by 
and characterizes the noise/signal ratio. The experimental variance of 
for the bivariate case. In Fig. 2 we plotted the relative errors of the first few important cross cumulants 
, and 
. Not surprisingly, all relative errors have minima with respect to the binning time. In other words, rebinning allows us to maximize the signal/noise ratio of factorial cumulants, just as in the single-color case. There is some small deviation in 
and 
between theory and experiments at very short and very long sampling times. This is caused by the large error in determining 
1, which indicates a large uncertainty in the experimental cumulant itself and leads to scattering of 
(with i + j = 2) are plotted in Fig. 3 as a function of sampling time T for a concentrated Alexa 488 sample. The dashed line is a fit to the ideal model, which neglects nonideal detector effect. In this case, the quantity 
(with i + j = 2) should decay monotonically since it is proportional to the normalized binning function B2/T2, which we have shown to be a monotonically decreasing function (14
and 
. This is explained by looking at Eq. 34. The dead-time effect will always decrease the value of the univariate cumulants. In the short sampling time-limit T << 
of EGFP measured in a living cell. A relative error <1 indicates that the cumulant is statistically significant. For this data, all third-order cumulants are statistically significant. Since the data contain nine significant cumulants, it is possible to directly resolve species in the living cell. Again, we find that rebinning helps to optimize the signal/noise ratio, which increases the sensitivity of our data analysis. 
is divided by Tr, where r = i + j is the order of the cumulant. The cumulants with the same order are plotted in the same panel. The best fit to a single species model is shown as lines. The fit determined 
,
, PAA = 0.002, and PAB = 0.003; and Instrument 2: 
, PAA = 0.007, and PAB = 0.0046.
