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Dual-Color Time-Integrated Fluorescence Cumulant Analysis

Bin Wu, Yan Chen, and Joachim D. Mu ¨ller

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

ABSTRACT

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.

We introduce dual-color time-integrated fluorescence cumulant analysis (TIFCA) to analyze fluorescence

INTRODUCTION

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 in-

formation 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 de-

veloped 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

inthelate80sandearly90s(9–12).Fluorescentcumulantanal-

ysis (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 occupa-

tion 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 con-

centration of fluorescent species. The exact theoretical treat-

ment 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 fluores-

cent 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 conven-

tional 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 simul-

taneously determines the molecular brightness in each

channel, the occupation number, and the diffusion time of

eachspeciesfromasinglemeasurement.Thestatisticalerrorof

factorial cumulants is also derived and experimentallyverified.

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.

Submitted March 30, 2006, and accepted for publication June 19, 2006.

Address reprint requests to Bin Wu, Tel.: 612-624-6045; E-mail: binwu@

physics.umn.edu.

? 2006 by the Biophysical Society

0006-3495/06/10/2687/12$2.00

doi: 10.1529/biophysj.106.086181

Biophysical Journal Volume 91October 20062687–2698 2687

Page 2

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 improve-

ment 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.

THEORY

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 detectingkAphotonsin channelA andkBphotonsinchannelB

is related to the probability distribution function P(WA,WB) of the integrated

light intensity WAand WB, according to Mandel’s formula (20),

ZN

3PðWA;WBÞdWAdWB;

where hAand hBare the detection efficiencies of the photon detectors and

T is the sampling time. For convenience, we set hA¼ hB¼ 1, which is

equivalent to measuring the intensity in counts per second (cps). As a

convention, we use km,nas a symbol for the (m,n)thcumulant and k[m,n]for

the (m,n)thfactorial 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 km,n(WA,WB) of the integrated intensity (WA, WB) is

determined experimentally by measuring the corresponding factorial

cumulant k[m,n](kA, kB) of the photon counts (kA,kB),

PðkA;kB;TÞ ¼

0

ZN

0

PoiðkA;hAWAÞPoiðkB;hBWBÞ

(1)

km;nðWA;WBÞ ¼ k½m;n?ðkA;kBÞ:

(2)

Next, we calculate the theoretical expression of km,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 PSFðr ~Þ. 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,

?

PSF3DG¼ exp ?2sðx21y2Þ

w2

?2sz2

w2

z

?

;

(3)

where s refers to an s-photon (s ¼ 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 r ~at

time t, the fluorescence intensity IQin channel Q (Q ¼ A, B) is given by

IQðtÞ ¼ lQPSFðr ~ðtÞÞ;

where lQ(measured in counts per second per molecule, i.e., cpsm) is the

brightness of fluorophores in channel Q. The integrated intensity WQwithin

the sampling time T is

ZT=2

With the above definition, the (m,n)thbivariate raw moment of (WA,WB) is

easily calculated as

ZT=2

PSFðr ~m1n;tm1nÞædt1???dtm1n:

The above integration has been solved and is expressed in terms of binning

functions Bm1n(T;td) in Wu and Mu ¨ller (14),

(4)

WQðTÞ ¼

?T=2

IQðtÞdt:

(5)

ÆWm

AWn

Bæ ¼ lm

Aln

B

?T=2

???

ZT=2

?T=2

ÆPSFðr ~1;t1Þ???

(6)

ÆWm

AWn

Bæ ¼ gm1nlm

Aln

B

VPSF

VBm1nðT;tdÞ;

(7)

where VPSFis the reference volume conventionally used in fluorescent

fluctuation experiments,

Z

and the coefficient gsis called the sthg-factor and defined as

R

According to Wu and Mu ¨ller (14), the binning function only depends on the

diffusion time td¼ 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)thbivariate cumulant kð1Þ

expressed as a series of terms of raw moments up to (m,n)thorder, where

kð1Þ

thermodynamic limit V / N, all terms except the (m,n)thraw moment

vanish (13),

VPSF¼

V

PSFðr ~Þdr ~:

(8)

gs¼

VðPSFðr ~ÞÞsdr ~

VPSF

:

(9)

m;nðWA;WBÞ of (WA,WB) can be

m;nðWA;WBÞ denotes the (m,n)thcumulant for a single molecule. In the

kð1Þ

m;nðWA;WBÞ ¼ ÆWm

AWn

Bæ:

(10)

Now suppose there are Ntotalnoninteracting, diffusing molecules in the

volume V, such that in the thermodynamic limit V / N, the concentration

c ¼ Ntotal/V is kept constant. We define the average occupation number N in

the observation volume VPSF,

N ¼ cVPSF:

(11)

Since the fluorescence emitted by different noninteracting molecules is

statistically independent from each other, the cumulantof the total integrated

fluorescence intensities in the two channels is given by summing up the

corresponding cumulant of all individual molecules,

km;nðWA;WBÞ ¼ Ntotalkð1Þ

In the short sampling time limit T ? td, the binning function Bm1n(T;td) is

approximated by Tm1n(14). In this limit the bivariate cumulant simplifies to

m;n¼ Ngm1nlm

Aln

BBm1nðT;tdÞ: (12)

2688Wu et al.

Biophysical Journal 91(7) 2687–2698

Page 3

km;nðWA;WBÞ ¼ Ngm1nðlATÞmðlBTÞn:

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,

(13)

km;n¼ gm1n+

i

Nilm

Ailn

BiBm1nðT;tdiÞ:

(14)

Experimentally, photon counts are observed instead of integrated intensities.

Equation 2 allows us to measure the cumulants of the integrated intensities

km,n(WA, WB) by calculating the experimental estimates ˆ k½m;n?ðkA;kBÞ of the

factorial cumulant of the photon counts k[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 k[m,0]and k[0,n], which are essentially univariate in nature.

Thus we concentrate on cross-bivariate factorial cumulants. Since all cross

cumulantsare of order equalto or larger thantwo,the effect of correlationsis

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

factorialcumulants,wehave toderivethe formulasdirectly. Weillustratethe

process by deriving the variance of the first nontrivial cross-factorial

cumulant k[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,k[m,n]is a linear combinationof

ki,jwith i # m, j # n. For example, the cross-factorial cumulant k[2,1]is

given by

k½2;1?¼ k2;1? k1;1:

(15)

Next, each cumulant ki,jis replaced by its unbiased estimator k-statistic

ki,j(21) to construct the unbiased estimator of the factorial cumulant. In the

case of k[2,1], we get ˆ k½2;1?¼ k2;1? k1;1, where ˆ k½2;1?represents the unbiased

estimator of k[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 k1,1are

Var½k1;1? ¼1

nðk2

1;11k0;2k2;01k2;2Þ;

4k2

Var½k2;1? ¼1

n

1;1k2;012k0;2k2

2;015k2

2;114k2;0k2;21

4k1;2k3;014k1;1k3;11k0;2k4;01k4;2

Cov½k2;1;k1;1? ¼1

!

;

nð2k1;2k2;013k1;1k2;11k0;2k3;01k3;2Þ;

(16)

where n is the total number of data points. To obtain this result, we take the

large sample limit where n / N. The variance and covariance of ki,jis now

plugged into the variance of ˆ k½m;n?and results in the expression

Var½ˆ k½2;1?? ¼ Var½k2;1?1Var½k1;1?12Cov½k2;1;k1;1?:

Finally, the cumulants are transformed back to factorial cumulants by

MathStatica,

(17)

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 ˆ k½1;1?. The factorial cumulant k[1,1]and cumulant k1,1are equal by defi-

nition. The unbiased estimator for ˆ k½1;1?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 k1,1, k0,2, k2,0, and k2,2are transformed back into

factorial cumulants. Evaluating the expression results in

Var½ˆ k½1;1?? ¼1

nðk½1;1?1k2

3ðk½1;0?1k½2;0?Þ1k½2;1?1k½2;2?Þ:

½1;1?1k½1;2?1ðk½0;1?1k½0;2?Þ

(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 non-

paralyzable 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.

Var½ˆ k½2;1?? ¼1

n

4k½1;0?k½1;1?14k2

12k½1;2?k½2;0?12k½2;1?14k½1;0?k½2;1?116k½1;1?k½2;1?14k½2;0?k½2;1?15k2

2k½2;2?14k½1;0?k½2;2?14k½2;0?k½2;2?14k½1;1?k½3;0?14k½1;2?k½3;0?14k½3;1?1

4k½1;1?k½3;1?14k½3;2?1k½0;1?ð2k2

k½4;0?Þ1k½0;2?ð2k2

½1;1?14k½1;0?k2

½1;1?14k½1;0?k½1;2?112k½1;1?k½2;0?14k2

½1;1?k½2;0?1

½2;1?1

½1;0?12k½2;0?14k½1;0?k½2;0?12k2

½1;0?12k½2;0?14k½1;0?k½2;0?12k2

½2;0?14k½3;0?1

½2;0?14k½3;0?1k½4;0?Þ1k½4;1?1k½4;2?

0

B

B

B

B

B

B

B

@

1

C

C

C

C

C

C

C

A

:

(18)

Dual-Color TIFCA2689

Biophysical Journal 91(7) 2687–2698

Page 4

The factorial-moment-generating function FkA;kBðu;vÞ (23) is an impor-

tant 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

GkA;kBðu;vÞ ¼

+

N

kA;kB¼0

PðkA;kBÞukAvkB:

(20)

The factorial-moment-generating function is given by

FkA;kBðu;vÞ ¼ GkA;kBð11u;11vÞ:

When FkA;kBðu;vÞ is expanded in a Taylor series, the coefficient of the

umvn/m!n! term describes the factorial moment m[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 ¼ A, B), to observe a fake count. In addition, we postulate that

different afterpulses are statistically independent from each other. Now

suppose there are kQreal counts detected in channel Q. The output counts k9Q

of the detector include afterpulses. We define a set of binary random

variables Xi

Q(i ¼ 1, ..., kQ) with a probability distribution

PQ;

1 ? PQ;

The definition of an afterpulse event leads to the following relation between

kQand k9Q,

(21)

PrðXi

QÞ ¼

Xi

Xi

Q¼ 1

Q¼ 0:

(

(22)

k9Q¼ kQ1 +

kQ

i¼1

Xi

Q:

(23)

The probability-generating function of (k9A,k9B) is given by

G9k9A;k9Bðu;vÞ ¼

+

N

k9A;k9B¼0

P9ðk9A;k9BÞuk9Avk9B;

(24)

where P9 (k9A,k9B) is the probability distribution of afterpulse-influenced

photon counts. With the help of a conditional probability distribution, we

can relate P9 (k9A,k9B) to the distribution of real photon counts P (kA,kB),

P9ðk9A;k9BÞ ¼ +

kA;kB

Pðk9A;k9BjkA;kBÞPðkA;kBÞ:

(25)

The conditional distribution can be written as the product of afterpulse

probabilities

Pðk9A;k9BjkA;kBÞ ¼

a

kA

i¼1

a

kB

j¼1

PrðXi

AÞPrðXj

BÞ;

(26)

sincedifferent afterpulses are statistically independent from each other. Thus

G9k9A;k9Bðu;vÞ is given by

N

PðkA;kBÞukAvkB

G9k9A;k9Bðu;vÞ ¼

+

kA;kB¼0

a

kA

i¼1

a

kB

j¼1

+

A;Xj

1

Xi

b¼0

uXi

AvXj

BPrðXi

AÞPrðXj

BÞ:

(27)

Notice that we plugged Eq. 23 into the exponent of u and v. Carrying out the

products and the second summation results in

G9k9A;k9Bðu;vÞ ¼

+

N

kA;kB¼0

3ðvð1 ? PB1vPBÞÞkB

¼ GkA;kBðuð1 ? PA1uPAÞ;vð1 ? PB1vPBÞÞ;

PðkA;kBÞðuð1 ? PA1uPAÞÞkA

(28)

where GkA;kBðu;vÞ 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,

F9k9A;k9Bðu;vÞ ¼ FkA;kBðð11uÞð11uPAÞ;ð11vÞð11vPBÞÞ:

(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,

k9½1;0?¼ k½1;0?ð11PAÞ;

k9½2;0?¼ k½2;0?ð11P2

k9½1;1?¼ k½1;1?ð11PAÞð11PBÞ:

To calculate the dead-time effect on factorial cumulants, we also consider

the factorial-moment-generating function. We define the dead-time param-

eter as dQðTÞ ¼ ty

in channel Q and T is the sampling time. We only consider the case where

dQ(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,

AÞ12PAðk½1;0?1k½2;0?Þ;

(30)

Q=T;ðQ ¼ A;BÞ, where ty

Qis the dead-time of the detector

P9ðkA;kBÞ ¼ PðkA;kBÞ1dAðkAðkA11ÞPðkA11;kBÞ

? kAðkA? 1ÞPðkA;kBÞÞ1dBðkBðkB11Þ

3PðkA;kB11Þ ? kBðkB? 1ÞPðkA;kBÞÞ:

(31)

The dead-time affected factorial moment generating function is

F9ðu;vÞ ¼

+

N

kA;kB¼0

P9ðkA;kBÞð11uÞkAð11vÞkB:

(32)

Plugging Eq. 31 into Eq. 32 and carrying out the summation, we obtain

F9ðu;vÞ ¼ Fðu;vÞ ? dAuð11uÞ@2

? dBvð11vÞ@2

@u2Fðu;vÞ

@v2Fðu;vÞ:

(33)

Expandingbothsidesof the aboveequationina Taylorseries andcomparing

the coefficientsof uandv,weobtaina relationbetweendead-timeinfluenced

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,

k9½1;0?¼k½1;0??dAðk2

k9½2;0?¼k½2;0??dAð2k2

k9½1;1?¼k½1;1??dAð2k½1;0?k½1;1??k½2;1?Þ?dBð2k½0;1?k½1;1??k½1;2?Þ:

½1;0?1k½2;0?Þ;

½1;0?12k½2;0?14k½1;0?k½2;0?13k½3;0?Þ;

(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

2690Wu et al.

Biophysical Journal 91(7) 2687–2698

Page 5

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 kMAXreceived during

the sampling time T by a nonparalyzable detector with dead-time tyis

kMAX¼ T/ty, but the theory sums photon counts from 0 to infinity. We have

found that this approach describes fluorescence fluctuation data with Ækæd #

0.1, which translates into an intensity of 2 3 106cps for a dead-time of 50 ns

(19,24).

MATERIALS AND METHODS

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 633 Plan-Apochromat oil immersion objective (NA ¼ 1.4) is

used to focus the laser and collect the fluorescence. The light passes through

anopticalfilter andis split into two channels. Photonscounts are detectedwith

an avalanche photodiode (APD) (SPCM-AQ-14, Perkin-Elmer, Dumberry,

Que ´bec). 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 experi-

ments 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.

WeusethesoftwareMathStaticatoderiveformulasoffactorialcumulants

uptotheeighth-order.Thevarianceofthefactorialcumulantsiscalculatedup

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 sam-

plingtimes.Theprocedureisperformedasfollows:Therecordedsequenceof

photon counts is fed into software to calculate the experimental factorial

cumulants of photon counts of sampling time T. To get cumulant for a sam-

plingorbinningtimeof2T,weaddneighboringphotoncountstogethertoget

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

timesofspecificintegermultiplesofT.Byrebinningwecalculatethefactorial

cumulants over sampling times that cover three orders of magnitude.

We fit the experimentally determined factorial cumulants ˆ k½r;s?to theoret-

ical cumulants k[r,s]determined by Eq. 14 with a nonlinear least-squares

fitting program. The reduced x2of the fit is given by

x2¼

1

ðK ? pÞ+

T

+

r;s

r0;s0

ðˆ k½r;s?ðTÞ ? k½r;s?ðTÞÞ2

Var½ˆ k½r;s?ðTÞ?

:

(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.

RESULTS AND DISCUSSION

Single dye experiment

To test the dual-color TIFCA theory we perform experiments

on simple fluorescent dye solutions. Each species is charac-

terized by four parameters, its molecular brightness in each

channel (lAand lB), the diffusion time td, 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 k[i,j]is of (i 1 j)thorder. We

fit cumulants with order smaller or equal to four simulta-

neously. Fig. 1 shows these cumulants as a function of

binning time for Rhodamine 6G. The data was taken with a

samplingtimeof20ms anda total measurement timeof130 s.

The reduced x2of the fit is 0.85 with a recovered brightness

of lA¼ 34,000 cpsm, lB¼ 9600 cpsm, a diffusion time of

td¼ 44 ms, and an occupation number of N ¼ 2.7.

We now investigate the variance of the factorial cumulants

as a function of binning time. The relative error dˆ k½r?of the

factorial cumulant ˆ k½r?is given by dˆ k½r?¼

and characterizes the noise/signal ratio. The experimental

variance of ˆ k½r?is determined by the moments-of-moments

method from the experimental factorial cumulants ˆ k½r?. We

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Varðˆ k½r?Þ

p

=ˆ k½r?

FIGURE 1

The data were taken with a sampling time of 20 ms and a total acquisition

time of 130 s. Each cumulant ˆ k½i;j?is divided by Tr, where r ¼ i 1 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 lA¼ 34,000cpsm and lB¼ 9600cpsm, a diffusion time of td¼

44ms,andan occupationnumber of N ¼2.7 witha reducedx2valueof 0.85.

Dual-color cumulant analysis of a Rhodamine 6G solution.

Dual-Color TIFCA 2691

Biophysical Journal 91(7) 2687–2698