excitation saturation in two-photon fluorescence correlation spectroscopy

Excitation saturation in two-photon fluorescencecorrelation spectroscopy

Keith Berland and Guoqing Shen

Fluorescence correlation spectroscopy �FCS� has become a powerful and sensitive research tool for thestudy of molecular dynamics at the single-molecule level. Because photophysical dynamics often dra-matically influence FCS measurements, the role of various photophysical processes in FCS measure-ments must be understood to accurately interpret FCS data. We describe the role of excitationsaturation in two-photon fluorescence correlation measurements. We introduce a physical model thatcharacterizes the effects of excitation saturation on the size and shape of the two-photon fluorescenceobservation volume and derive a new analytical expression for fluorescence correlation functions thatincludes the influence of saturation. With this model, we can accurately describe both the temporaldecay and the amplitude of measured fluorescence correlation functions over a wide range of illuminationpowers. © 2003 Optical Society of America

OCIS codes: 180.1790, 190.4180, 300.6410.

1. Introduction

Fluorescence correlation spectroscopy �FCS� is a pow-erful research tool for the study of molecular dynam-ics and interactions.1–9 FCS methods have alsoproven valuable for the characterization of the pho-tophysical properties of fluorescent molecules, suchas triplet-state kinetics, flicker of fluorescent pro-teins, and photobleaching kinetics.10–17 With FCS,statistical analysis is applied to fluorescence signalfluctuations measured from within a minute opensample volume. The volume is defined optically,typically with either confocal or two-photon micro-scopes. Fluorescence fluctuations can arise bothfrom the diffusion of molecules into and out of theobservation volume as well as from chemical reac-tions or photophysical mechanisms that can alter thepopulations of molecules within the volume. By an-alyzing the fluorescence fluctuations, one can recovervaluable information about the experimental systemincluding diffusion rates, molecular concentrations,the presence or absence of molecular interactions,chemical reaction rates and equilibrium constants, as

The authors are with the Department of Physics, Emory Uni-versity, 400 Dowman Drive, Atlanta, Georgia 30322-2430. Thee-mail address for K. Berland is [email protected].

Received 9 December 2002; revised manuscript received 30 May2003.

0003-6935�03�275566-11$15.00�0© 2003 Optical Society of America

5566 APPLIED OPTICS � Vol. 42, No. 27 � 20 September 2003

well as photophysical parameters. The recovery ofthis information is contingent on an accurate physi-cal model for the dynamic processes involved in aparticular system and the capability to calibrate thesample volume.

Although it is known that excitation saturation caninfluence two-photon fluorescence measurements, in-cluding two-photon FCS data,8 to our knowledge theeffects of excitation saturation have not previouslybeen well characterized in FCS theory and measure-ments. We demonstrate here that saturation cansignificantly influence both the amplitude G�0� andthe temporal decay of experimental fluorescence cor-relation functions. These effects are highlighted inFig. 1 that shows a series of two-photon FCS mea-surements for a single Rhodamine 6G �R6G� sampleat different average excitation powers. In the ab-sence of photophysical dynamics one would expecteach of these measured curves to be identical, whichthey clearly are not. Both photobleaching and exci-tation saturation play an important role in this re-sult. In this paper we discuss the physical basis ofthese observations, introduce modifications to FCStheory to include saturation effects, and demonstratethat this theory accurately models measured fluores-cence correlation functions over a wide range of exci-tation intensities. This new model of photophysicalprocesses in FCS measurements will be important foraccurate analysis of FCS data. Although excitationsaturation can be important in both one-photon con-focal and two-photon variations of FCS measure-

ments, we focus here on the role of saturation intwo-photon FCS.

2. Background

Fluorescence correlation measurements are essen-tially molecular counting experiments, in which themeasured fluorescence signals directly reflect thenumber of molecules occupying an optically definedobservation volume. By performing correlationanalysis on the observed fluorescence fluctuations,one can recover two types of information. First, theamplitude of the correlation functions reflects the av-erage number of molecules that occupy the measure-ment volume. In the case of a monodispersepopulation of molecules that experience Browniandiffusion but no other dynamic processes, the ampli-tude of the correlation function, typically specified asG�0�, is inversely proportional to the average numberof molecules within the measurement volume �G�0� �1�N�. Second, the temporal decay of the correlationfunction specifies the time scale for any observed dy-namic processes. When diffusion is the only rele-vant dynamic process, this time scale for thecorrelation function decay simply reflects the averagecrossing time for molecules diffusing through the vol-ume. The detailed nature of the decay reflects theshape of the observation volume. In a typical FCSinstrument, a laser focused through a high-numerical-aperture microscope objective lens servesas the excitation source, and the point-spread func-tion for the focused laser beam describes the spatialprofile of the excitation probability. For two-photonmicroscopes where there is no emission pinhole, thisprofile also completely specifies the measurementvolume.

These basic properties of correlation functions pro-vide the basis for a qualitative understanding of therole of both photobleaching and excitation saturationon FCS measurements �detailed quantitative models

are introduced in Section 3�. We first consider theeffects of photobleaching, in which a molecule tra-versing the observation volume is photochemicallydestroyed before exiting the volume. The disappear-ance of bleached molecules from the fluorescence sig-nal before the molecule actually leaves the volumeleads to shorter apparent molecular crossing timesand thus faster temporal decay of the correlationcurve. In addition, the average number of bleachedmolecules within the observation volume will reducethe apparent number of fluorescent molecules withinthe volume because bleached molecules can no longerbe observed by fluorescence measurements. Thisapparent reduction will lead to an increase in theamplitude of the correlation curve.

The role of excitation saturation is somewhat moresubtle. In the absence of saturation, two-photon flu-orescence excitation rates vary as the square of theoptical excitation flux.19,20 However, there is a limitto the maximum excitation rate of a fluorescent mol-ecule because once a molecule is excited it requires afinite amount of time to relax back to the ground statebefore it can be excited again. As the excitation rateapproaches this maximum threshold, additional in-creases in excitation levels will no longer cause cor-responding increases in molecular excitation ratesbecause of saturation, i.e., a depletion of the ground-state population. Because the excitation profile isnot uniform �specified by the point-spread function ofthe focused laser beam�, the onset of saturation willoccur at different excitation powers for different lo-cations within the laser-excited volume. For exam-ple, molecules directly at the center of the beam in thefocal plane will experience saturation prior to thosemolecules in the periphery of the beam. As the ex-citation flux is increased beyond the saturationthreshold, the molecular excitation rate will thus in-crease faster in the outer regions of the excitationvolume than in the central region of the focal volume.The net effect of this spatial disparity is an alterationof the size �increased� and spatial profile �wider andflatter� of the observation volume, as shown in section3. The effective increase in observation volumeleads to a larger average number of molecules occu-pying the volume and a corresponding decrease in theamplitude of the correlation function. The widervolume also increases the molecular crossing time,extending the decay time for the correlation curves.This saturation-modified observation volume isreadily observable by use of FCS measurements, anda quantitative description of this effect is our mainfocus in this paper. An accurate description of thisphenomenon is not only of fundamental interest, butis also useful for many specific applications of FCSmeasurements as we discussed in this paper.

3. Theory

We begin with a summary of the basic theory for FCSmeasurements and then introduce modifications toaccount for saturation. For simplicity we focus thetheory for samples with single, monodisperse molec-ular populations. Modifications for polydisperse

Fig. 1. FCS correlation curves for a single R6G sample in Trisbuffer �pH 8�. We observe substantial variations in both the am-plitude and the temporal relaxation of these FCS curves as theaverage excitation power is varied between 2.2 and 53 mW. Forvisual clarity, some of the FCS measurements used in the analysisare not shown on this graph.

20 September 2003 � Vol. 42, No. 27 � APPLIED OPTICS 5567

samples are straightforward and described else-where.5 In the absence of excitation saturation, thetwo-photon fluorescence signal is proportional to thesquare of the incident power, and the temporal fluc-tuations �F can be written as20,21:

�F�t� � F�t� � �F� � �C�r, t�I02S2�r�d3r . (1)

Here �F� is the time-averaged fluorescence signal,S�r� represents the spatial distribution of the focusedlaser, and �C�r, t� describes the fluctuations in localconcentration. I0 corresponds to the peak laser ex-citation flux; and the parameter accounts forsystem-dependent parameters such as the molecularabsorption cross section, fluorescence quantum yield,collection and detection efficiencies, as well as laser-pulse parameters. Fluorescence correlation func-tions are defined as

G�� F�r, t� �F�r�, t � ��

�F�2 . (2)

By using an appropriate model for the point-spreadfunction S�r�, one can easily calculate the correlationfunction associated with purely diffusive dynamics�no photophysics�. Here we use the three-dimensional Gaussian �3DG� model of the point-spread function given in a cylindrical coordinate by

S3DG�r, z� � exp� 2r2��02�exp� 2z2�z0

2� . (3)

The parameters �0 and z0 represent the beamwaist in the radial and axial directions, respectively.The 3DG model has been widely used for FCS theoryas the corresponding correlation functions can be re-duced to the following simple analytical form:

GD�� 2�2

�3�2�02z0�C� � 1

1 � ��D�

� � 11 � ��0�z0�

2��D��1�2

, (4)

where �D � �02�8D for two-photon excitation and D is

the molecular diffusion coefficient.22 The FCS ob-servation volume can be defined as S3DG

2 d3 r �in-tegration over all space� and results in an effectivevolume of V3DG � ��4�3�2 �0

2z0. From this expres-sion one can easily see that the time zero value of thecorrelation function is inversely proportional to theaverage number of molecules occupying the volume.Thus the time-independent prefactor �terms notfound inside the brackets� is also often written asG�0� � ��N. For the 3DG model, the value of thegeometric prefactor gamma is � � 1�22 � 0.35.

As noted above, the major influence of excitationsaturation is to limit the maximum fluorescence ex-citation rate. Thus the fluorescence signal from anyparticular molecule within the observation volumecannot exceed the corresponding fluorescence thresh-old. When the peak excitation rate is below thisthreshold level, the standard 3DG distribution accu-

rately describes the experimental excitation profileand the corresponding observation volume. As theillumination level increases, molecules at or near thecenter of the volume �which experience the highestexcitation flux� will begin to saturate. Further in-creases in the incident laser power will not increasethe fluorescence signal from molecules within the sat-uration region. The onset of saturation thus intro-duces a top-hat profile to the excitation profile. Aschematic diagram illustrating this effect is shown inFig. 2. We emphasize that there is no change in theprofile of the laser excitation with increasing illumi-nation levels, but rather in the profile of the effectivemolecular excitation rate.

In modeling this phenomenon, we aim to create anexpression for the top-hat profile that allows the eval-uation of fluorescence correlation functions that canreasonably be applied in curve-fitting routines. Weaccomplish this by constructing an expression com-posed solely of 3DG distribution functions that ap-proximates the top-hat profile, including the requiredthreshold behavior. The new effective excitationprofile is thus calculated when we subtract a second3DG profile from the standard illumination profile.When the width of the subtracted Gaussian distribu-tion is defined as the width of the original excitationprofile weighted by the relative amplitudes of theoriginal and subtracted distributions, the resultingexpression produces the desired top-hat distributionwith maximum amplitude corresponding to thethreshold level as shown below. One can choose toperform this subtraction on the linear distributionsS�r, z� representing the laser profile or on the squareddistributions S2�r, z� that characterize two-photon ex-citation probabilities because either distribution isstill Gaussian. For convenience, we thus definethreshold levels in terms of the squared intensitydistributions. This choice has negligible influenceon the ultimate effective excitation profile and elim-inates cross terms when we calculate the squared

Fig. 2. Schematic diagram for excitation saturation. As the mo-lecular excitation probability exceeds a threshold level, it cannot befurther increased. Saturation will be reached in the central re-gions of the focused laser before the tails. The net effect of satu-ration is thus a flatter top-hat profile for the excitation probability.


distributions. We use the following terms to de-scribe these distributions:

S02 �r, z�: squared 3DG distribution; S0 �r, z� repre-sents the laser excitation profile.

I02: the squared amplitude of the laser illumination,corresponding to the peak flux.

Ss2 �r, z�: squared 3DG distribution representingthe subtracted profile to achieve the top-hat distri-bution.

Is2: the amplitude of the subtracted illuminationcorresponding to the peak subtracted flux.

Ith2: the threshold excitation rate corresponding tothe maximum fluorescence excitation rate.

SE2 �r, z�: effective excitation profile after we ac-count for saturation.

To achieve a top-hat profile with a peak amplitudecorresponding to the threshold level, we first definethe subtracted amplitude as IS

2 � I02 Ith

2. Tocreate the top-hat profile, the ratio of the waists forthe laser excitation profile S0

2 �r, z� and the sub-tracted excitation profile Ss

2 �r, z� should match theamplitude ratio for these distributions. This rela-tionship is conveniently defined in terms of the di-mensionless parameter � with

� �IS

2

I02 �

�S2

�02 �

zS2

z02 �

I02 � Ith

2

I02 . (5)

This parameter has zero value when the excitationlevel matches the threshold level and approachesunity as the excitation flux greatly exceeds thethreshold. For clarity, we specify the definition ofSS �r, z� explicitly:

Ss�r, z� � exp� 2r2��s2�exp� 2z2�zs

2� . (6)

The resulting distribution that represents thesaturation-corrected fluorescence excitation profile isdefined as

SE2�r, z� �

I02S0

2�r, z� � Is2Ss

2�r, z�

Ith2 , for I0

2 � Ith2 .

(7)

This definition, along with the relative widths andamplitudes of the component incident and subtracteddistributions, is unique in that it limits the excitationrate to the threshold level without introducing oscil-lations in the spatial profile of SE

2 �r, z�. With thisdefinition, the effective fluorescence excitation ratecan then be written as Ith

2SE2 �r, z�. We note that an

analogous definition is suitable to treat saturation inone-photon confocal measurements, with the correc-tion defined in terms of linear intensities rather thanthe squared intensities. Figure 3 shows the radialdistribution for this newly defined effective excitationprofile at the focal plane �z � 0�. Surface plots forthe 3DG and saturation-corrected distributions arealso shown in Fig. 4. One can easily see that, forillumination levels above the threshold, the excita-

tion volume becomes larger with a flatter peak thanthe standard squared 3DG distribution.

One can recalculate the effective observation vol-ume for the new distribution, finding that

Vsat � SE2�r, z�d3r �

�3�2�02z0

8 �1 � �5�2

1 � � �� V3DG�1 � �5�2

1 � � � . (8)

The parameter �, defined above, ranges from zeroto one. The corresponding volume correction factorranges from unity at the threshold level to a factor of2.5 as the excitation level becomes large. This upperlimit, which is not realized for the actual volume,indicates that the saturation model begins to breakdown for excitation intensities several times thethreshold level as discussed below. Nevertheless,for the range of excitation levels of greatest interest,this model captures the key aspects of how saturationexpands the observation volume as well as influencesmeasured correlation functions. For this volume,

Fig. 3. Quantitative model for the saturation-corrected excitationprofiles. Shown are the squared profiles for the incident laser, thesubtracted profile to correct for saturation, and the resulting ef-fective excitation profile. The threshold excitation rate is alsoshown.

Fig. 4. Excitation saturation will increase the size of the obser-vation volume and flatten its shape. These surface plots show theprofiles of the excitation source S0

2 �k, z� and the saturation-corrected effective excitation profile SE

2 �r, z�.


the average fluorescence signal measured for excita-tion levels above the threshold will be

�F� � �C�Ith2Vsat � �C�I0

2V3DG�1 � �5�2� . (9)

With only Gaussian distributions in the expressionfor the saturation-altered measurement volume, thecalculation of corresponding correlation functions isstraightforward and yields the following result:

where � � �02�z0

2. It is useful to note that thisfunction reduces to the standard format of the corre-lation function �without saturation� when � � 0.The threshold value Ith is best determined when wecurve fit the measured FCS data. It can also bedetermined directly from measurements of the aver-age fluorescence intensity as a function of incidentpower, although this approach is less accurate asdiscussed below.

Fluorescence photobleaching is an important dy-namic process that can also be observed in FCS mea-surements.11,23,24 Although the details of thisprocess are not the focus of this paper, we must in-clude photobleaching in our model because substan-tial photobleaching is likely whenever one usessaturating illumination conditions. Photobleachingmodels have previously been introduced for FCS andtypically assume a constant photobleaching rate asthe molecule traverses the volume.11 With thissame assumption, the presence of photobleachingmodifies the theoretical form of the correlation curveby the multiplicative factor �1 B � B exp� kB��1 B��. Here kB is the photobleaching reactionrate, and B represents the average fraction of mole-cules within the observation volume that have beenphotobleached. When we include this factor, the fi-nal result for the correlation function that includesboth photobleaching terms and excitation saturationis

The bleaching rate kB and the bleached fraction Bare not truly independent parameters because oneexpects that a larger bleached fraction will be asso-

ciated with faster bleaching rates. To model thisdependency, we assume that molecules will bleachonly while within the optically defined observationvolume and that the time they are available forbleaching reactions is related to the average molecu-lar crossing time �D. We can then reasonably esti-mate the bleached fraction as

B �kB

kB � r��D. (12)

The variable r is an effective residency time fittingparameter that should be of the order of unity. Wedo not include any terms for triplet-state kinetics inour analysis as intersystem crossing is typically notobserved in two-photon FCS data.23

4. Materials and Methods

A. Sample Preparation

Rhodamine 6G was diluted in nanopure water �18.2M��cm�. We filtered the samples using 0.2-�m fil-ters, and we subsequently determined R6G concen-trations using absorption measurements. Sampleswere diluted to their final 32-nM concentration in100-mM Tris buffer �pH 8� and were loaded into mi-crobridge containers �Hampton Research, Lakeview,California� and sealed with coverslips. Prior to theiruse, the coverslips were submerged in a blocker ca-sein buffer for 10 min, rinsed with clean water, andair dried. This procedure helps minimize absorptionof dye molecules to the cover glass surfaces. We notethat a single sample was used for all FCS measure-ments shown.

B. Fluorescence Correlation Spectroscopy

FCS measurements were performed on a home-builtmicroscope based on designs that have been previ-ously described.21 The laser source was a mode-locked Tsunami Ti:sapphire laser pumped by a 5-W

Millennia V solid-state laser �Spectra-Physics, Moun-tain View, California�. The excitation wavelengthwas 780 nm, the laser-pulse repetition rate was 80

G�� 2�2

�C��3�2�02z0�1 � �5�2�2 � 1

�1 � ��D��1 � ��D�1�2 ��7�2

�1 � ��D��1 � ��D�1�2

�4�2�

�1 � 1�� 2��D��1 � 1�� 2��D�1�2� , (10)

G�� 2�2�1 � B � B exp� kB��1 � B�

�C��3�2 �02z0�1 � �5�2�2 � 1

�1 � ��D��1 � ��D�1�2 ��7�2

�1 � ��D��1 � ��D�1�2

�4�2�

�1 � 1�� 2��D��1 � 1�� 2��D�1�2� . (11)


MHz, and the pulse width was �100 fs. A home-made confocal system is used for these measure-ments. The dichroic mirror �675DCSX, ChromaTechnology, Brattleboro, Vermont� reflects the ex-panded laser �5�� toward the objective where it isfocused in the sample �Olympus UApo 40� waternumerical aperture of 1.15, Melville, New York�.The laser power at the sample was varied between2.2 and 53 mW. We measured the power outside themicroscope using a Labmaster powermeter �Coher-ent, Auburn, California�, and the measured valueswere multiplied by a correction factor to account forknown losses that were due to transmission throughthe optical system. We adjusted the excitationpower by rotating a ��2 plate in the front of a linearpolarizer. We collected the fluorescence signalswith the objective lens and filtered these using ashort-pass E700SP �Chroma Technology, Brattle-boro, Vermont� to block scattered laser excitation.The fluorescence signal was split with a 50–50 beamsplitter and directed to two avalanche photodiodemodules �SPCM-AQR-16-FC, EG&E, Vaudreuil,Canada� with home-built fiber couplers and 100-�mfiber. The output of the photon-counting moduleswas sent to an ALV-6000 correlator �Langen, Germa-ny� to calculate cross-correlation functions. use ofcross-correlation measurements prevents afterpuls-ing of the avalanche photodiodes from influencing thecorrelation signal, and thus all time channels of thecross-correlation data are reliable. This would notbe true for autocorrelation measurements, where thefast time channels do show the influence of afterpuls-ing. We obtained each cross-correlation curve by av-eraging four independent measurements withrecording times of 10–120 s each, depending on theexciting intensity. Estimates of the standard devi-ation for the correlation data are also calculated forthis average.25 All experiments were performed atroom temperature ��25 °C�. Data analysis wascompleted with Igor Pro software �WaveMetrics Inc.,Lake Oswego, Oregon�.

5. Results and Discussions

We first demonstrate in Fig. 5 that the average flu-orescence intensity does indeed follow the predictedpower dependence according to Eq. �9� over a largerange of excitation powers. This includes both theinitial quadratic power dependence as well as thefirst several measured points beyond the saturationthreshold. This suggests that the saturation theoryintroduced above will be useful for quantitative anal-ysis of FCS data in the saturation region, as is ex-plicitly demonstrated below. On the other hand, themodel clearly does not fit the data well for the largestexcitation powers. This is predicted because the sat-uration model allows only for an overall saturationvolume increase of 2.5 times, whereas the actualmeasured volume continues to increase beyond thislevel. When we directly fit the measured fluores-cence signal to Eq. �9�, it is thus necessary to selectwhich points to include for curve-fitting purposes.When the three highest �power� points are excluded

for curve-fitting analysis, a saturation thresholdvalue of 10.5 mW is recovered.26 This is not a par-ticularly robust method to determine the threshold asit is dependent on the fitted data range and does notaccount for photobleaching effects. Nevertheless,this value agrees reasonably well with the value of 9mW recovered from the FCS global analysis as de-scribed below.

We performed a series of FCS measurements forthe r6G sample using different laser excitation levels,as shown in Fig. 1. With increasing power on thesample, we observe both faster relaxation of the cor-relation curves and decreasing correlation ampli-tudes. Each of these changes reflects the influenceof photophysics. As described above, photobleach-ing tends to increase the correlation amplitude anddecrease the correlation time. Saturation will de-crease the correlation amplitude and increase thecorrelation time. These two photophysical phenom-ena thus have competing effects on the measuredcorrelation curves. The saturation theory providesa quantitative model for understanding the influenceof saturation on FCS measurements.

To demonstrate the accuracy and advantages of thenewly introduced saturation model, we compare thefitting results for three different models including �1�pure diffusion �Eq. �4��, �2� photobleaching only �Eq.�11� with � � 0�, and �3� the full saturation model �Eq.

Fig. 5. Power dependence of the average fluorescence intensityfor R6G excited at 780 nm. The inset shows the same data plottedon a log–log scale. Vertical axes show both the total fluorescenceand the average fluorescence counts per molecule per second. Atlow power, the data follow the expected quadratic dependence onexcitation intensity �dotted curve�. It is clear that this quadraticrelationship does not hold for the higher excitation levels. Thedata were fit to the saturation model of Eq. �9� with the amplitudeand saturation threshold as parameters. We determine from FCSfitting that, for this sample and beam waist, saturation is reachedwith an average input power of 9 mW, corresponding to a peakphoton flux of 8 � 1029 photons�cm2�s. The saturation model fitsthe data nicely up to around three times the threshold value.Beyond this level, the saturation model breaks down because themodel volume stops increasing even though the actual measure-ment volume continues to increase, as discussed in the text.


�11��. Fits in which either saturation- or bleaching-only models are used are performed as a global anal-ysis, with common experimental parameters that donot change linked across the data set. Thus thebeam waist ��0�, concentration �C�, radial and axialwaist parameter ��, saturation threshold �Ith�, andbleaching residency time parameter �r� are eachglobal fit parameters. Photobleaching rates �kB�that are expected to vary with excitation power arethus left as free parameters and are not linked glo-bally. For all the analysis reported here we fix thediffusion coefficient at its known value of 3 � 10 6

cm2�s. The measured fluorescence as a function ofinput power makes clear that the saturation modelwill not work well for the three highest illuminationpowers. Global fits are thus performed excludingthese three curves, which are subsequently fit inde-pendently. Fitting to the pure diffusion model can-not be done globally as there are no parameters thatcan account for the variation between measurements.It is therefore clear that the pure diffusion modelcannot account for the measured observations.Nonetheless, results are shown for curve fits to indi-vidual FCS curves to facilitate comparison with theother models.

In our analysis, we use fitting functions pro-grammed exactly as written in Eqs. �4� and �11�, andwe do not decouple the temporal decay from the over-all amplitude of the FCS curves. Changes in thebeam-waist fitting parameter �0 thus influence boththe temporal decay and the amplitude of FCS curves.This coupling of the amplitude and relaxation timesis integral to the FCS theory, but is often ignored.We find it important to account for both the relax-ation times and the amplitude of the measured FCScurves, and doing so serves as a useful tool for modeldiscrimination.

Another important feature of our fitting routines isthat the fitting functions are programmed with con-ditional statements in the function definitions.Whenever the measurement power for a given FCScurve exceeds the saturation threshold parameter,the saturation parameter alpha is calculated withinthe fitting function according to the definition of Eq.�5�. When the measurement power is less than thesaturation threshold parameter, the function simplyassigns alpha a zero value. This method of definingalpha within a single fitting function allows us to usethe saturation threshold as a fitting parameter and tofit all data sets, whether they were acquired above orbelow the saturation threshold, with a single fittingfunction. Thus the saturation threshold need not beknown a priori to fit the FCS data. This is a pow-erful advantage that allows us to perform global anal-ysis of the data, fitting all measured curvessimultaneously with the saturation threshold as acommon parameter.

Figure 6 presents a typical FCS curve for R6G inwater, acquired at 16-mW excitation, and the corre-sponding best fits to the saturation and pure diffusionmodels. Curve fitting this single curve with thephotobleaching-only model also produces a similar fit

quality. This result demonstrates how each modelappears to fit this data. In fact, with the exception ofthe two highest excitation powers, any single FCSmeasurement from the series shown in Fig. 1 can befit reasonably well with any of the three models dis-cussed here. However, the saturation theory is theonly model that both fits the data reasonably andrecovers consistent experimental parameters acrossthe data set, as shown below. It is in this respectthat the saturation model clearly demonstrates itssuperiority.

To illustrate this, we first consider the recoveredvalues for the beam waist �0 when we fit FCS datawith each model. The fitting results are shown inFig. 7, which plots the recovered beam waist as afunction of incident power for both the pure diffusionmodel and the saturation model. The beam waist �0is one of the most important parameters in FCS mea-surements. The actual beam waist, defined by thedimensions of the focused laser beam, does notchange when excitation power is altered. However,the apparent beam waist recovered from FCS mea-surements can vary substantially, as shown in Fig. 7for the pure diffusion model. We note that this datacannot be fit to the pure diffusion model without alsoallowing the concentration C to vary as a free param-eter, and the recovered values increase dramaticallyas shown in Fig. 8. These findings controvert theexperimental conditions for which both the molecularconcentration and the beam waist have fixed valuesfor all measurements. The recovered values thusclearly indicate that free diffusion is an inappropriatemodel for this data and that photophysical effectsmust be included in the analysis to recover accurateexperimental parameters.

In contrast, all the FCS curves �excluding the threehighest powers� can be fit with single values for boththe beam waist and the concentration. These re-sults are also shown in Figs. 7 and 8. Thus thesaturation model fits all the data; and when consis-tent parameters are recovered across the data set, itsuggests that this is in fact the correct model for the

Fig. 6. Autocorrelation curve for R6G excitation with 16-mW av-erage power, which is above the saturation threshold. Shownhere are best fits from the saturation theory and the pure diffusionmodel. Both models appear to fit the data well, yet comparison ofthe recovered parameters from the full data set highlights thesuperiority of the saturation model.


system. This consistency across the data set impliesthat both �0 and C can be treated as parameters withtrue physical meaning, which of course they are, andthat the recovered values can be used to evaluate thequality of a particular physical model to fit a givenFCS data set.

The parameters recovered from global analysis are0.53 �m for the beam waist and 1.5 molecules��m3

concentration. The agreement between this concen-tration value and the concentration recovered from

the pure diffusion model fits at low power where sat-uration and bleaching play a minor role gives addi-tional confidence in the validity of these results.The recovered concentration does not correspond pre-cisely to the value determined from absorption mea-surements, although it is reasonably close. Thedifference mainly reflects that the actual FCS obser-vation volume is not truly a 3DG distribution, andthus the gamma factor calculated for the 3DG modelis not precise. Generally, FCS instrumentationmust be calibrated when absolute concentration is tobe determined by FCS, and our findings will notchange this requirement. On the other hand, theresults shown here indicate that, without the satu-ration model, instruments need to be recalibrated forevery measurement power to be used and for everyfluorophore of interest. On the other hand, once thesaturation threshold is determined for a given fluoro-phore, the saturation model will support accuraterecovery of both the concentration and the beamwaist with only a single calibration point. This pro-vides a significant practical advantage as well asadded confidence in FCS results because the experi-mentally recovered parameters can be well correlatedwith experimental values �e.g., concentration andbeam waist� that are known from other measure-ments.

Other recovered parameters from the global fits tothe saturation model include an axial and radialbeam-waist parameter �z0��0� of 16 � 2, a bleachingparameter r � 2.5 � 0.1, and a saturation thresholdof Ith � 9 � 0.5 mW. These are all physically rea-sonable values. The actual beam profile is Gaussianin the radial direction and Lorentzian in the axialdirection. To model the axial profile as a Gaussian�i.e., 3DG�, its nominal width must be extended tocapture the long tail of the Lorentzian, which ex-plains the relatively large value for z0��0. Thebleaching time parameter r also satisfies the orderunity requirement stated along with its definition.The saturation parameter also seems physically rea-sonable. Assuming a two-photon absorption crosssection of 100 GM and 100-fs laser pulses, we calcu-late that approximately 10% of the dye molecules perpulse are excited at the 9-mW saturation threshold.This reflects only an approximate assessment of theexcited fraction because the cross section and pulsewidths are not known precisely. Nevertheless, thiscalculation suggests that the recovered parametervalue is reasonable as ground-state depletion effectsstart to become important at such excitation levels.We are investigating whether the saturation thresh-old has a more precise interpretation in terms of mo-lecular cross sections, but such discussion is outsidethe scope of this paper. We note that we specify thethreshold in terms of an average power to highlightthe relatively low average excitation flux where sat-uration becomes important. However, it should bepointed out explicitly that saturation will actuallyoccur on a per molecule basis. The peak photon flux,which corresponds to �1030 �photons�cm2��s for thesystem as described here, is the actual fundamental

Fig. 7. Beam waist �0, as a function of excitation power, recov-ered from FCS curves for R6G. If the pure diffusion model is usedto fit the data, the recovered beam waist varies greatly. Such fitsalso require that the concentration C be released as a free param-eter. For the saturation model, all data sets through 22-mW ex-citation are fit well with a single value for the beam waist andconstant values of all other fitting parameters except the photo-bleaching rates. These findings indicate that saturation is thepreferred model. We note that the three highest input powersshown in Fig. 1 �26, 40, and 53 mW—not shown here� can also befit with this constant value for the beam waist; but at these higherpowers, � must also be treated as a power-dependent fitting pa-rameter as discussed in the text.

Fig. 8. Concentration of R6G molecules obtained from fits to FCSmeasurements shown in Fig. 1. When applying the saturationtheory introduced above, we can fit all the data sets with a singlevalue for the molecular concentration, as one would expect shouldbe possible. In contrast, fitting to the pure diffusion model pro-duces a large increase in the recovered concentration as power isincreased. The capability to recover a single, and thus meaning-ful, value for the concentration is an important result for FCSapplications.


quantity of interest for saturation effects. Thus thesaturation threshold stated in terms of the averagepower also depends on the beam-waist parametersand the laser-pulse width and shape.

We also attempted to fit the FCS curves to aphotobleaching-only model, without the influence ofsaturation �alpha value of zero�. In performing theglobal analysis using this fitting model, including thelinkage between the bleaching rate and the bleachedfraction given by Eq. �12�, we recover parameters thatdo not fit the individual data sets, and the model thusfails. This finding could in fact be predicted becausethe bleaching-only model requires an increase in cor-relation amplitude with higher bleaching rates, andthe observed correlation amplitudes actually de-crease. To further explore whether thephotobleaching-only model can explain our observa-tions, we also performed the global fits without theconstraints of Eq. �12� just in case this constraintcauses the discrepancy. What we find is that, in thiscase, the theory can in fact fit the data. However,the fit results are not stable in that the bleaching rateparameter tends to become negative. Fits areachieved only when this parameter �kB� is con-strained to positive nonzero values, and it then as-sumes the value of the constraint and is thus not astable fit. Moreover, the recovered bleached frac-tions are quite large, averaging around 30% evenwhen the bleaching rate parameter kB is essentiallyzero. Although the relationship in Eq. �12� is notexact, it is a nonphysical result that the bleachedfractions are substantial even in the absence of pho-tobleaching. Thus, although the bleaching-onlymodel with uncorrelated bleaching rates andbleached fractions has enough free parameters to fitindividual data curves, the recovered parameters arenot physically reasonable and we thus discount theresults from the model. They are therefore notshown in the figures.

The saturation model does recover physically rea-sonable values for the bleaching parameters from thedata fits, and these are shown in Fig. 9. To obtainthe values for the curves acquired at 26, 40, and 53mW, it was necessary to fit each curve individuallywith additional free parameters. We chose to re-lease � and to keep all other parameters fixed at theirglobally determined values. The recovered valuesfor the ratio of the axial to radial beam waists arethen 19, 30, and 37, respectively. Alternatively, onecan release the beam-waist parameter. Because wedo not have a model that is fully accurate for dataacquired at these powers well above the saturationthreshold, we have no real justification for selectingone method over the other. The recovered bleachingparameters are not particularly sensitive to thischoice, and we thus have some hope that these high-est values are at least approximately correct. Atpresent, it is difficult to interpret these results interms of various physical models of the photobleach-ing process because the parameters are recoveredfrom a simple constant bleaching rate model. Nev-ertheless, this model already provides reasonable

data fits and is sufficient to demonstrate how thecompeting processes of saturation and photobleach-ing influence FCS measurements.

We conclude with further consideration of the am-plitude of the measured FCS data. From the recov-ered photobleaching and saturation parameters, wecan calculate the expected contributions of these pho-tophysical processes to the correlation amplitude.These results are shown in Fig. 10 that shows �1� themeasured FCS amplitude; �2� the expected amplitudefor the same system with saturation only, i.e., in theabsence of photobleaching; and �3� the expected am-plitude given the photobleaching parameters with no

Fig. 9. Photobleaching rates kB and the average bleached fractionB are shown for R6G as a function of excitation flux. These valueswere recovered from global fits to the saturation model.

Fig. 10. Both saturation and photobleaching influence the ampli-tude of the measured correlation traces. To better illustrate therole of each process, we separated out their influence on the am-plitude. Shown are the measured FCS amplitude, the expectedcorrelation amplitude with saturation only in the absence of pho-tobleaching, and the expected correlation amplitude given the re-covered photobleaching parameters with no saturation. Thesevalues are calculated as described in the text.


saturation. The saturation-only curves are calcu-lated directly from Eq. �11� by use of the recoveredexperimental parameters including the thresholdvalue of 9 mW and zero values for the photobleachingparameters. The three highest power points are ad-justed for the enlarged volume with the values of �recovered from individual curve fitting as describedabove. The photobleaching-only values are calcu-lated similarly with the recovered photobleaching pa-rameters and with alpha set equal to zero. This plotillustrates how the correlation amplitude depends onboth photophysical processes; we observe that, al-though the correlation amplitude is relatively flatover much of the power range, this is a result of thecompetition between bleaching and saturation.Therefore, for different fluorophores under varied ex-perimental conditions, such a graph can conceivablylook quite different. This includes the possibilitythat the amplitude will either increase or decreasemore dramatically.

6. Conclusions

We have introduced a new model for excitation sat-uration in two-photon microscopy and calculated the-oretical expressions for the corresponding correlationfunctions. This saturation model can accurately de-scribe the observed fluorescence correlation spectra ofR6G, recovering physically reasonable values for fit-ting parameters over a wide range of excitation pow-ers. Pure diffusion and photobleaching-only modelscannot achieve this. The ability to accurately fitboth the temporal relaxation and the amplitude ofmeasured correlation curves is an important capabil-ity. FCS curves contain a tremendous amount ofdetailed information about molecular dynamics in aparticular experimental system, but this informationcan be recovered only when the data are fit to appro-priate physical models. FCS curves can often be fitwith a variety of physical models; thus analysis pro-cedures that can assist in the identification of correctphysical models are tremendously useful. Fittingboth the amplitude and the temporal relaxation ofFCS curves simultaneously provides important con-straints on the fitting models as shown here. Theseconstraints are lost when FCS data are fit to normal-ized curves. Moreover, the capability to constrainexperimental parameters that should not change�such as molecular concentration and the beamwaist� to single physically relevant values across datasets where other experimental parameters are variedprovides additional confidence that physically accu-rate models are being used to analyze FCS data.The saturation theory introduced in this paperachieves this capability. We note that the capabilityto recover consistent values for the molecular concen-tration with the saturation model has important im-plications for FCS measurements. With thesemodels, FCS instruments could be calibrated on anabsolute scale to recover physically meaningful re-sults for the measured concentration. This capabil-ity will be important to quantify the concentration ofmolecules of interest in various environments where

it is otherwise difficult to measure �such as withinliving cells�. It can also be important to quantifymolecular interactions without the need to observechanges in the aggregation state of the molecules.

In conclusion, we have shown that photophysicaleffects play an important role in two-photon FCSmeasurements, even at relatively low excitation flux.Data analysis routines thus must account for bothbleaching and saturation to recover accurate experi-mental information, and the models introduced hereprovide the tools necessary to do so.

This work was partially supported by the EmoryUniversity Research Fund.

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26. Our fitting routines are programmed with conditional state-ments in the fit function definitions such that the parameteralpha is assigned the value zero whenever the power is lessthan the saturation threshold parameter. This allows for si-multaneous fitting of the entire curve, both above and belowthe saturation threshold, with asingle fitting function.


excitation saturation in two-photon fluorescence correlation spectroscopy

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