Originally published as Biophys J. BioFAST on September 2, 2005.
doi:10.1529/biophysj.105.067082
Biophysical Journal 89:3531-3547 (2005)
© 2005 The Biophysical Society
Molecular Brightness Determined from a Generalized Form of Mandel's Q-Parameter
Alvaro Sanchez-Andres,
Yan Chen and
Joachim D. Müller
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455
Correspondence: Address reprint requests to Joachim Mueller, University of Minnesota, School of Physics and Astronomy, 116 Church St. SE, Minneapolis, MN 55455. Tel.: 612-625-4369; Fax: 612-624-4578; E-mail: mueller{at}physics.umn.edu.
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ABSTRACT
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Mandel's Q-parameter, which is determined from the first two photon count moments, provides an alternative to PCH analysis for determining the brightness of fluorophores. Here, the definition of the Q-parameter is generalized to include correlations between photon counts that are separated by a time 
. We develop and experimentally verify a theory that takes the effects of dead time, afterpulsing, and the finite sampling time on the generalized parameter 
into account. 
which corresponds to the original Q-parameter, is severely affected by dead time and afterpulsing. for 
on the other hand, is quite robust with respect to nonideal detector effects. Thus, analysis of provides a robust method for determining the brightness of fluorophores. We extend the theory to a mixture of species, which is characterized by an apparent brightness. The brightness of EGFP in CV-1 cells is measured as a function of protein concentration to demonstrate the feasibility of analysis in cells. In addition, we monitor protein association of the ligand-binding domain of retinoid X receptor in the presence and absence of 9-cis-retinoic acid by analysis.
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INTRODUCTION
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Fluorescence fluctuation spectroscopy (FFS) derives information about biomolecules from statistical analysis of fluorescence intensity fluctuations. A number of different FFS techniques exist and provide different information about the sample. Fluorescence correlation spectroscopy (FCS) is the most widely used technique and derives information about the dynamic properties of the sample from the correlation in the signal fluctuations (1
,2). Other techniques, such as photon counting histogram (PCH) and cumulant analysis, target nondynamic properties of the sample (3,4). PCH analyzes the probability distribution function of the photon counts and determines the brightness of fluorescent molecules. The brightness of a fluorophore is given by the average number of photons emitted by one molecule over a specified time period. PCH analysis is useful for the study of particle aggregation and has been successfully applied to observe the oligomerization of proteins in living cells (5).
We briefly illustrate how brightness serves as a marker of the oligomeric state of a protein. A fluorescently labeled protein diffuses through the observation volume and produces a burst of detected photons. The average photon count rate of these bursts determines the molecular brightness of the labeled protein. If this protein associates to form a homodimer, the new complex will carry two fluorescent labels and produce on average twice as many photons as the monomeric protein. The molecular brightness of the dimer is therefore twice that of the monomer.
Protein oligomerization and aggregation are also measured by FCS, where changes in the diffusion coefficient induced by protein association are monitored by the autocorrelation function (for a review, see Thompson et al. (6)). Employing cross-correlation with dual-color detection provides a sensitive method for detecting protein interactions (7). Another approach uses the fluctuation amplitude 
of the autocorrelation function to detect changes in the aggregation state of proteins (8,9). The idea behind this method is that the effective number of diffusing particles decreases upon oligomerization with respect to the monomer concentration. This results in an increase of the amplitude of the autocorrelation of the fluorescence intensity, which is used as a marker for oligomerization. More sophisticated setups and analysis methods, such as scanning FCS (8,10) and higher-order FCS (11,12), have been employed as well.
The degree of oligomerization depends on protein concentration. To monitor oligomerization by brightness, we measure the brightness over a wide concentration range. Because fluorescence intensity is proportional to concentration, we measure at intensities where dead-time effects of the detector become significant (13). This nonideal detector effect results in an artificial decrease in the brightness and leads to erroneous interpretation of PCH experiments. We developed an improved PCH theory that corrects for dead-time and afterpulsing effects and accurately determines brightness over a wide range of intensities (14).
An alternative to determining the brightness by PCH is moment analysis (11,15–17). Two approaches exist; the first directly calculates higher-order moments from the photon counts (17), whereas the other uses higher-order correlation functions to determine moments (11). Because moments and the probability distribution function used by PCH are mathematically equivalent, both methods provide the same information. Here we limit our discussion to the first two moments of the photon counts. They are sufficient to calculate the brightness of a single species. In the case of multiple species, the first two moments determine the apparent brightness of the sample, which represents an average brightness of all the species present in the solution (5). Moment analysis is attractive because it provides a very convenient and simple approach for computing the brightness. However, just as in the case of PCH, moment analysis suffers from dead time and afterpulsing of the detector. Equations that treat dead-time and afterpulsing effects on moment analysis have been introduced for the limit of short sampling times (14).
Moment analysis is based on Mandel's Q-parameter (18), which is defined in terms of the first two photon count moments. In this article we generalize Mandel's Q-parameter by including correlations between photon counts separated by a time . We develop the theory that connects the generalized Q-parameter to the brightness of fluorescent molecules for arbitrary sampling times and in the presence of detector dead time and afterpulsing. We also discuss the relationship between and the autocorrelation function. To test the theory we perform and analyze experiments using simple dye solutions.
The special case 
corresponds to the original definition of Mandel's Q-parameter. We extend the theory of Mandel's Q-parameter by including sampling time effects into the data analysis of 
Most importantly, we show that 
is in contrast to remarkably robust against nonideal detector effects and only requires minor corrections to account for dead time and afterpulsing. Thus, the generalized Q-parameter provides an attractive method for analyzing brightness and is in several aspects superior to traditional analysis of the Q-parameter. We extend the theory of to include multiple species and introduce an approximation that provides a quick and convenient correction for dead-time effects. The low brightness and large protein concentrations typically encountered in cellular measurements present a challenge for PCH and conventional moment analysis (5). analysis, on the other hand, provides a robust method for determining the brightness of fluorophores in cells. We demonstrate the feasibility of analysis of cell data by determining the brightness of EGFP and by monitoring the protein association of a nuclear receptor.
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THEORY
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Mandel's parameter and brightness
PCH analysis provides a framework for determining the brightness 
in the limit of short sampling times. The brightness characterizes the number of photons received per molecule for a sampling time T. It is proportional to
 | (1) |
in the short sampling time limit, where 
is the photon count rate of a single molecule (3). We previously treated both and as equivalent measures that determine the brightness of a molecule. However, depends explicitly on the sampling time and the simple relationship between and of Eq. 1 is not valid for long sampling times. The parameter , on the other hand, characterizes the instantaneous brightness of a molecule, which is independent of the sampling time. Thus, we focus on the brightness in this article.
Moment and cumulant analysis provide an alternative for determining the brightness (4,17). We assume in the following a single diffusing species with a photon count rate of . The brightness of a single species in the limit of short sampling times is readily determined from Mandel's Q-parameter (3,18)
 | (2) |
where 
is a shape factor that depends on the point spread function 
(8). The factors 
are defined by
 | (3) |
If the PSF is normalized, 
its volume 
corresponds to the observation volume typically employed in FCS experiments, and the brightness and the average number of fluorophores N in the observation volume is proportional to the average of the photon counts, 
(3).
Although calculation of the brightness from the photon count moments of Eq. 2 is fast and convenient, previous work has shown that this method suffers from dead-time and afterpulsing effects of the photodetectors and yields inaccurate values of the brightness even at relatively low concentrations (14). An algorithm based on a first-order Taylor expansion that takes nonideal detector effects for short sampling times into account has been described (14).
Generalized Mandel's parameter Q()
We now introduce an alternative method for calculating the brightness . It utilizes the photon count correlation function,
 | (4) |
where 
for 
and 
for 
The function 
was introduced to subtract the shot noise term for The symbol 
is the number of photon counts registered in the sampling time interval 
and 
indicates averaging. The correlation between detected photons that are separated by a time of is given by 
The photon count correlation function 
is identical to the fluorescence intensity correlation function 
of FCS in the short sampling time limit. We now introduce a generalization of Mandel's Q-parameter by multiplying with 
 | (5) |
We develop in the following expressions that relate to the brightness . In addition, we consider the effect of sampling time T, detector dead time 
and afterpulsing on 
We will see in the following that and behave very differently, and it becomes necessary to treat each case separately. We use 
to refer to 
and 
to refer to 
Note that is equal to the traditional Q-parameter.
The statistics of the photoelectron counts is closely related to the statistics of the integrated intensity
 | (6) |
If the intensity 
does not vary significantly over the sampling time period T, Eq. 6 simplifies to
 | (7) |
The validity of Eq. 7 specifies the short sampling time limit. In other words, the short sampling time limit requires that the timescale of intensity fluctuations is much larger than the sampling time. For purely diffusing fluorophores, the characteristic timescale of fluctuations is given by the diffusion time 
Thus, the short sampling time limit is valid for sampling times that are much shorter than the diffusion time 
In this limit the photon count correlation function equals the fluorescence intensity correlation function,
 | (8) |
However, in the following we will mainly consider long sampling times where Eq. 8 is no longer valid. We later discuss the relationship between and the generalized Mandel's parameter
In the absence of dead time, the probability distribution function (pdf) of the integrated intensity 
is related to the pdf of the photon counts 
by Mandel's formula (19)
 | (9) |
where 
is the Poisson distribution with average x.
Notation
To be consistent with previous work (14), we label dead-time-affected variables with a prime and afterpulsing affected variables with a star. For example, we denote the ideal pdf of observing k photons during the sampling time T by 
whereas the afterpulsing and dead-time-affected pdf is referred to as 
Dead-time effect on the generalized Q-parameter
Dead-time influences the moments of the photon counts and therefore changes the Q-function
 | (10) |
Equation 10 has the same form as Eq. 5, but every moment is replaced by the dead-time-affected moment. In addition, we introduced a shorthand notation, where is written as 
We also assumed a stationary process, so that correlations only depend on the time difference . Dead time does not change the fact that photon detection is a doubly stochastic process, and the probability distribution functions of k and W are related by
 | (11) |
which generalized to a bivariate distribution function is given by
 | (12) |
In the absence of dead time the detection process of each photon is statistically independent from the detection process of others, which yields a Poissonian probability function 
(19). However, dead-time effects destroy the statistical independence of the detection process. After the detection of one photon event, no other can be detected for a period of time equal to the dead time. As a result, the dead-time-affected conditional probability 
is no longer Poissonian.
O'Donell (20) developed an analytical expression for using a Taylor expansion in the dead-time parameter 
for nonparalyzable detectors. The parameter is defined as the quotient of the dead time 
and the sampling time (
). The expression to first order in is
 | (13) |
The bivariate conditional probability 
of detecting 
photons given an integrated intensity of 
and of detecting 
photons a time later given an integrated intensity of 
is given by
 | (14) |
The detection of photons is essentially instantaneous, but dead time introduces a statistical dependence for times less than the dead time. Thus, as long as 
and for integrated intensities and that do not temporally overlap (
) Eq. 14 is valid. These conditions are always fulfilled in our experiments.
A consequence of Mandel's formula is that the factorial moments of the photon counts are identical to the moments of the integrated intensity (21), 
If we use this relationship and combine Eqs. 13 and 14 with Eqs. 11 and 12, we obtain a relation between the dead-time-affected moments of the photon counts and the ideal moments of W,
 | (15) |
where we used 
Next, we express the ordinary moments of W as cumulants of W (see Appendix A), where denotes the cumulant. Thus, Eq. 15 written in terms of integrated intensity cumulants is
 | (16) |
Inserting Eq. 16 into Eq. 10 and ignoring higher order terms in we arrive at an expression of the dead-time-affected Q-parameter for
 | (17) |
The introduction of cumulants in Eq. 17 is useful, because the integrated intensity cumulants are connected to properties of the sample,
 | (18) |
as derived in Appendix A. We introduced in Eq. 18 the normalized correlation function of the integrated intensity
 | (19) |
The function 
is closely related to the r-th order cumulant correlation function of the intensity,
 | (20) |
Note, that is normalized (
), because 
(4). This implies according to Eq. 19 that 
for short sampling times. The correlation function depends only on the shape of the point spread function and the physical process responsible for generating correlations. We assume throughout this article that the physical process is stationary, so that the correlation function depends on time differences only, 
We now use the stationary property to rewrite the integrated intensity cumulant of Eq. 18
 | (21) |
Inserting Eq. 21 into Eq. 17 allows us to finally arrive at an expression for the dead-time-affected Q-function for 
 | (22) |
This equation is used to connect the experimentally determined 
with the brightness . In the absence of dead time (
) Eq. 22 describes the ideal parameter
 | (23) |
The function 
describes the sampling time dependence of the Q-parameter. The value of the function 
tends to one in the limit of short sampling times. Thus, 
is identical to the original Q-parameter (Eq. 2) in the limit of short sampling times.
To derive an expression for Q at 
we start with Eq. 10 and repeat all of the above steps in the derivation of 
but evaluate the expressions for Because of the shot noise term in and the unique dead-time dependence of each moment, we arrive at a very different expression to describe 
As we later show, 
is significantly more sensitive to dead-time effects than 
We found that we need to include second-order terms in to describe experimental data accurately by 
whereas a first-order correction in is sufficient for We describe in Appendix B the derivation of an expression for to second order in . The result is given by
 | (24) |
with
 | (25) |
In the limit of short sampling times, and by only keeping the first-order terms in , we recover the dead-time correction of moment analysis as previously described (14). Equations 24 and 25 extend the previous theory to second order and include the effects of sampling time on Q.
To calculate we need a physical model that describes our fluctuation experiments. We consider the case of diffusing molecules and assume a three-dimensional Gaussian (3DG) PSF. The second to fourth order normalized intensity correlation functions 
are given by Qian (16),
 | (26) |
where 
is the average diffusion time through the observation volume and r is the squared ratio of the radial and axial beam waist. The correlation functions 
for a two-dimensional Gaussian (2DG) PSF are formally obtained from 
by taking 
To calculate the dead-time-affected Q-function 
we need to evaluate 
and 
according to Eq. 22. To calculate requires the evaluation of 
and 
In general this requires numerical integration, however, it is possible to derive analytical solutions for special cases. We first consider the function and transform the integral of Eq. 19 using the fact that the integrand is stationary (4,22)
 | (27) |
For diffusing particles with a two-dimensional Gaussian PSF an analytical solution of is easily derived,
 | (28) |
where we introduced the sampling factor 
and 
The functions 
and 
which are needed for the evaluation of and are evaluated numerically. We later discuss an approximation for 
which only depends on 
and therefore avoids the need for numerical integration.
Multiple species
Eqs. 22 and 24 describe the effect of dead time on the Q-function for a single species. It is straightforward to expand the theory to multiple species, because cumulant functions are additive for statistically independent variables (23),
 | (29) |
The subscript i characterizes parameters of the i-th species. We now explicitly derive an expression for 
for multiple species. Using Eqs. 17 and 29 we get
 | (30) |
Inserting Eq. 21 into above equation allows us to model for multiple species. However, it is not possible to determine individual brightnesses from the parameter 
Only a single brightness, which we refer to as apparent brightness, can be inferred. The apparent brightness 
is not a physical brightness, but represents the best average brightness of the mixture, and is defined by Mandel's Q-parameter, 
(15). The apparent number of molecules is determined from the average photon counts 
We now extend the concept of apparent brightness to analysis.
The diffusion coefficient of the individual species within a mixture often differs less than a factor of two, and we approximate the individual normalized intensity correlation functions of second order 
by an averaged correlation function 
We define 
and by
 | (31) |
Note that the ideal 
equals 
which is consistent with our earlier definition for short sampling times, 
because for short sampling times 
With this definition, we write Eq. 30 in terms of the apparent brightness and apparent number of particles:
 | (32) |
We now introduce an approximation to express 
in terms of and
 | (33) |
We will later discuss the validity of this approximation. Equation 32 together with Eq. 33 allow us to write an expression for 
which is identical to the single species case (see Eq. 22, if one replaces the brightness and the number of molecules by their apparent parameters.
Afterpulsing
In addition to dead time, afterpulsing is another experimental artifact of the detector that affects PCH and moment analysis. An afterpulse constitutes a spurious photoelectron event that is triggered by the detection of a real event in the photodetector. The generation of afterpulses and its statistics has been studied in detail elsewhere (24,25). Its effects on PCH and moment analysis have also been characterized (14). The probability to observe an afterpulse at time t after a real event is characterized by a function 
The probability of observing an afterpulse decreases rapidly with increasing time t. Thus, 
for t greater than a characteristic time 
For avalanche photodiode (APD) detectors, as commonly used in FFS experiments, the probability essentially drops to zero for times greater than a few microseconds. Hence, if we use a sampling time that is larger than the characteristic time 
we may safely assume that all afterpulses detected during a sampling period are caused by the real events detected in the same sampling period. In other words, there is no cross talk between neighboring sampling periods in terms of afterpulsing. We calculated in Appendix C the effect of afterpulsing on the generalized Mandel's parameter for sampling times larger than The effect of afterpulsing on the Q-function is given by
 | (34) |
where 
is the integrated probability of 
over the sampling period (
)
 | (35) |
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MATERIALS AND METHODS
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Instrumentation
A mode-locked Ti:sapphire laser (Tsunami, Spectra Physics, Mountain View, CA) pumped by an intracavity doubled Nd:YVO4 laser (Spectra Physics) serves as source for two-photon excitation. The laser produces 100-fs pulses with a repetition frequency of 80 MHz (tunable between 700 and 1000 nm). The experiments were carried out using a Zeiss Axiovert 200 microscope (Thornwood, NY) with a 63x plan apochromat oil immersion objective (N.A. = 1.4). An excitation wavelength of 780 nm was used for the dye experiments, and a wavelength of 905 nm was used for the cell measurements. The power at the sample was determined by measuring the laser power directly after the objective. The excitation power was <3 mW for solution measurements, and was 0.25 mW for cell measurements. No photobleaching was detected for any of the samples measured. A dichroic filter (Chroma Technology, Brattleboro, VT) was used to separate the fluorescence from the excitation light. Photon counts were detected with an avalanche photodiode (SPCM-AQ-14, PerkinElmer, Vaudreuil, Quebec). The output of the APD, which produces TTL pulses, was directed to a data acquisition card (ISS, Champaign, IL). The card records the complete sequence of photoelectron counts to computer memory. The data shown were taken using sampling times between 10 and 200 
The data were analyzed using programs written for IDL version 5.4 (Research Systems, Boulder CA).
Sample preparation
Alexa488 was purchased from Molecular Probes (Eugene, OR) and dissolved in pure water. Initial concentrations of the stock solutions were determined from absorption measurements using the excitation coefficients provided by Molecular Probes. Samples for the FFS experiments were prepared by diluting the stock solution either in water or in a 60:40 (v/v) glycerol/water solution.
CV-1 cells were obtained from ATCC (Manassas, VA) and maintained in 10% fetal bovine serum (Hyclone Laboratories, Logan, UT) and EMEM media. EGFP-C1 and EGFP-RXRLBDß vectors were generated as described previously (5). Transfections were carried out by using transfectin (Bio-Rad, Hercules, CA) according to manufacturer's instructions. Cells were subcultured into eight-well coverglass chamber slides (Naglenunc International, Rochester, NY) 48 h before measurements. Before measurements, the growth media was exchanged to Leibovitz's L-15 medium (no phenol red) with 10% fetal bovine serum (Invitrogen, Carlsbad, CA); 9-cis retinoic acid (Sigma-Aldrich, St. Louis, MO) was added to the media at 300 nM concentration. FFS measurements were performed 5 min after the addition of ligand.
Data analysis
is directly determined from the photon count moments of the FFS data. The generalized Q-function 
is calculated from the raw data according to Eq. 10 for 
The dead time of the detector was determined by exposing it to light of 
10 kcps and observing the output signal with a digital oscilloscope (Tektronix TDS 3034, Wilsonville, OR). The dead time is determined by the shortest time interval between consecutive pulses. We found a value of 50 ± 1 ns, which agrees with the manufacturer's specification. The autocorrelation function of the FFS data was used to determine the diffusion time of the fluorophores.
Our goal is to determine the brightness from the experimentally measured dead-time-affected Q-value. However, the mathematical models for and depend on both the brightness and the number of molecules, 
Thus, to determine the brightness we need another experimental observable. This observable is the dead-time-affected average number of photon counts 
According to Eq. 15 
is given by
 | (36) |
We solve above equation for N,
 | (37) |
Inserting Eq. 37 into the formulas for 
we find an equation that only depends on the brightness and is solved numerically. The algorithms for data analysis were implemented into programs written in IDL language and used for data and error analysis. Errors in both and were determined experimentally by dividing each data set into segments of equal length, and the value of the Q-parameter was calculated for each segment. We determined the standard deviation and mean of the Q-parameters for data analysis.
The functions 
depend on the diffusion time 
which is determined from analysis of the autocorrelation function. We empirically found that the diffusion time is a robust parameter that is little affected by dead-time effects. The determination of the diffusion time from experimental data is reliable as long as we make sure that photobleaching is absent. We calibrated the observation volume 
by measuring an Alexa488 solution of known concentration c and determined N and from and according to Eqs. 22 and 37. The volume is determined by 
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RESULTS AND DISCUSSION
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Dead-time effects on Q and Q0
Let us first compare traditional moment analysis, which uses the parameter, with -correlation analysis. To simplify the comparison we neglect afterpulsing and undersampling, and concentrate on the effects of dead time only. In the absence of dead time we would measure the ideal 
value. Dead time leads to a biased value 
The relative deviation 
captures the bias introduced by deadtime. Let us evaluate for 
and We refer to the generalized Q-parameter at as 
Note that in the short sampling time limit 
In addition, 
for correlation times 
Because the times and 
are much less than all functions in Eqs. 22 and 25 are set equal to one, which results in very simple equations. Fig. 1 shows the dead time induced relative deviation 
together with 
for traditional moment analysis as a function of fluorescence intensity 
We calculated 
and in the limit of short sampling times according to Eqs. 22 and 24 for a brightness of = 10,000 cps, a sampling time T = 10 µs, and a dead time of 50 ns, which corresponds to a dead-time parameter of 0.005. These are values we typically encounter in actual experiments. The number of molecules N was varied, which translates into intensity as 
Fig. 1 shows the behavior of 
up to intensities of 
cps, which is close to the upper limit of most photon counting experiments. At low intensities the relative deviation is small for both, and 
because dead-time effects are negligible in this regime. Both Q-values decrease with increasing intensity due to dead time, but the of is much less than that of 
For example, an intensity of 300,000 cps leads to a dead-time-induced relative deviation of 100% for whereas experiences only a relative decrease of 5% at the same intensity.
TOP
ABSTRACT
INTRODUCTION
for 
for 
for 
The relative deviation of both numerators due to dead time is shown as an inset of Fig. 1 as a function of intensity. The figure clearly demonstrates that the second factorial moment 
is significantly less affected by dead time than the second factorial moment 
Thus the primary reason for the robustness of 
for comparison. 
is shown as a function of the fluorescence intensity in Fig. 2. The value of 
albeit only slightly. We accounted for this bias while fitting the data to Eq. 22. The solid line is a description of the data by theory for a dead time of 
and a brightness of 
Our theory successfully describes the experimentally observed 
The value of 
as a function of intensity is shown in Fig. 3 A together with the best fit of the data to Eq. 22 for a dead time of 50 ns. The fit determined a brightness of 
and describes the data within experimental error. 
which results in strong undersampling with a sampling factor x = 2. We graph in Fig. 3 B 
as a function of intensity. The solid line represents the best fit of the data to Eq. 22 with a dead time of 50 ns. Again, theory and experiment agree with one another. Because brightness is a property of the dye we expect it to be independent of the sampling frequency. The fitted brightness of 
for a sampling time of 80 µs is in excellent agreement with the brightness determined for a sampling time of T = 40 µs, and provides an additional check of the theory. 
for 
for 
The cumulative afterpulsing probability of the APD used for this experiment was determined to be 0.007. In other words, correcting 
A rigorous analysis that takes sampling time effects into account arrives at 
and 
The approximation of setting 
for 
for 
and a brightness of 
kcps, which is in good agreement with the value of 8.7 kcps obtained by 
To test this idea we use Eq. 22 and write 
and 
The relative change of 
and 
The function 
depends on the brightness 
we first find an upper limit for 
In addition, higher order correlation functions decay faster than lower order ones. Thus, 
This allows us to define the function 
In other words, the function 
overestimates the true contribution of 
-factors of a 3D Gaussian PSF and the deadtime of our detector (
kcps. For in vivo experiments on fluorescent proteins the brightness 
by Eq. 33. To investigate this approximation further, we consider the case of a binary mixture of two species with brightnesses 
and 
present at concentrations 
and 
For these conditions the exact expression is
introduced by the approximation is thus given by
and the brightness ratio 
of the two species. 
analysis in cells to probe the oligomerization of proteins, we investigate the relative error of a monomer-dimer and a monomer-tetramer mixture. The brightness ratio r of the monomer/dimer system is two and that of the monomer/tetramer system is four. In Fig. 6 we plot the relative error introduced by the approximation for the two systems. The worst case introduces an error of 35% for the monomer/tetramer sample, and an error of 11% for the monomer/dimer mixture. It is easy to show that the maximum of the relative error grows with increasing brightness ratio and reaches a limiting value of 100%. 
Thus, in the case of multiple species the relative error introduced by approximating 
using an apparent brightness is usually less than a 35% bias of a term with a relative deviation of <1%, and therefore unnoticeable given the experimental uncertainty of the data. Hence, the expression for 
and for simplicity ignore undersampling effects. Next, we use Eqs. 22 and 24 to determine the brightness from 
but choose dead times of 49, 50, and 51 ns. This range of dead times is consistent with the experimental uncertainty. Fig. 7 shows the brightness recovered by 
and 
The inset shows the relative deviation of the numerators of 
(dotted line) as a function of the fluorescence intensity.
The parameter 
A diffusion time of 425 µs was determined by autocorrelation analysis. Because 
we fit the data to 
of 1.1 determines a brightness of 18.36 ± 0.04 kcps.
and fit to 
and 80 
respectively. (D) 
after correcting for afterpulsing with 
1.2 leads to a brightness of 8.84 ± 0.05 kcps.
due to the approximation of 
which depends on the brightness ratio and the fractional concentration. The solid and dashed lines represent the relative error introduced by a monomer-tetramer and a monomer-dimer mixture as a function of the fractional concentration of the monomer. We used a brightness ratio of two for the dimer/monomer case and a ratio of four for the tetramer/monomer example.