ARTICLE

Maximum-entropy decomposition of fluorescence correlationspectroscopy data: application to liposome–human serumalbumin association

Received: 25 February 2003 / Revised: 25 April 2003 / Accepted: 25 April 2003 / Published online: 30 August 2003� EBSA 2003

Abstract Fluorescence correlation spectroscopy wasused to measure the diffusion behavior of a mixture ofDMPC or DMPC/DMPG liposomes with human serumalbumin (HSA) and mesoporphyrin (MP), which wasused as the fluorescent label for liposomes and HSA aswell. For decomposing the fluorescence intensity auto-correlation function (ACF) into components corre-sponding to a liposome population, HSA and MP, weused a maximum entropy procedure that computes adistribution of diffusion times consistent with the ACFdata. We found that a simple parametric non-linear fitwith a discrete set of decay components did not convergeto a stable parameter set. The distribution calculatedwith the maximum entropy method was stable and theaverage size of the particles calculated from the effectivediffusion time was in good agreement with the datadetermined using the discrete-component fit.

Keywords Fluorescence correlation spectroscopy ÆHuman serum albumin Æ Liposomes Æ Maximumentropy data analysis Æ Mesoporphyrin

Introduction

Fluorescence correlation spectroscopy (FCS) wasintroduced about 30 years ago (Elson and Magde 1974;Magde et al. 1972, 1974) as a method to analyze fluo-rescence intensity fluctuations from a small laser-illu-minated volume. The temporal behavior of thefluorescence intensity is characterized by its normalized

autocorrelation function (ACF). In a typical FCSexperiment, molecules free in solution move through theobservation volume by Brownian motion, generatingfluorescence fluctuations. Binding to a ligand slowsdown the molecules and increases the typical dwell timein the observation volume. The average number ofmolecules of free and bound species and the respectivediffusion coefficients can be recovered from the ACF.Numerous other sources of fluctuations can be studiedwith FCS (Chen et al. 2000 and references therein). Theimplementation of confocal microscopy for FCS (Qianand Elson 1991; Rigler et al. 1993) increases the sensi-tivity, improves the spatial resolution, and reduces theobservation volume well below 1 fL, corresponding toonly a few fluorescent molecules at nanomolar concen-trations. FCS is also becoming increasingly important inthe characterization of biological macromolecules andtheir interactions. Its high sensitivity, the low concen-trations used in the measurement, and the very smallobserved volume offer a very effective method to observeand characterize molecules inside living cells: observingthe behavior of the molecules in different parts of the cellcan yield information about the environment of thesemolecules.

The FCS experiments to which we applied themaximum entropy analysis here were originally plan-ned to characterize the interaction between humanserum albumin (HSA), mesoporphyrin, and two kindsof liposome compositions. In systems consisting ofHSA, differently charged small unilamellar vesicles(SUV), and mesoporphyrin (MP), steady-state andtime-resolved fluorescence measurements and differen-tial scanning calorimetry experiments had suggested aninteraction between HSA and liposomes (Bardos-Nagyet al. 1998; Galantai and Bardos-Nagy 2000; Galantaiet al. 2000). Before embarking on a detailed study ofthat interaction, which will be published separately, thispaper aims at studying the application of the maximumentropy method to FCS data containing a distributionof diffusing components. As a model we used neu-tral (1,2-dimyristoyl-sn-glycero-3-phosphatidylcholine,

Eur Biophys J (2004) 33: 59–67DOI 10.1007/s00249-003-0343-6

Karoly Modos Æ Rita Galantai Æ Iren Bardos-Nagy

Malte Wachsmuth Æ Katalin Toth Æ Judit FidyJorg Langowski

M. Wachsmuth Æ K. Toth Æ J. Langowski (&)Division of Biophysics of Macromolecules (H0500),Deutsches Krebsforschungszentrum,Im Neuenheimer Feld 280,69120 Heidelberg, GermanyE-mail: jl@dkfz-heidelberg.de

K. Modos Æ R. Galantai Æ I. Bardos-Nagy Æ J. FidyDepartment of Biophysics and Radiation Biology,Semmelweis University, Puskin utca 9,1088 Budapest, Hungary

DMPC) and negatively charged [19:1 w/w% of DMPCand 1,2-dimyristoyl-sn-glycero-3-phosphatidylglycerol(DMPC/DMPG)] SUV liposomes. Since MP bindsreversibly to HSA (Ka=2.5· 107 M)1) and also to theliposomes [KL=(1.5–4)·104 M)1, depending on the lipidcomponents of the liposomes], we assumed that—at asufficient excess of HSA or liposomes—we could followtheir diffusion through the fluorescent signal of MP.Utilizing the size difference of HSA and liposomes, andthe distinct binding affinity of MP to them, the formationof the HSA–liposome complexes was observed byMP-labelled HSA.

Materials and methods

Chemicals

The samples were prepared in 10 mM sodium phosphate buffer atpH 7.4. The DMPG, DMPC, 1·crystallized and lyophilized HSA,and MP dihydrochloride were purchased from Sigma. All sol-vents [chloroform, methanol, ethanol, dimethylformamide(DMF)] were obtained from Merck. The chemical components inthe samples were of spectroscopic purity. Sephadex G100Superfine, used to purify HSA, was purchased from PharmaciaFine Chemicals.

Preparation of porphyrin solution

A stock solution of MP was prepared in DMF and kept in the darkto avoid photodamage. For the measurements, solutions wereprepared by diluting with pH 7.4 phosphate buffer and usedimmediately after preparation. The concentration of porphyrinsolutions was determined by using the Soret band absorbance ofthe dye. The molar extinction coefficient for MP dissolved in DMF,�399=1.55·105 M)1 cm)1, was determined by ourselves and used inlater calculations.

Preparation of HSA solution

HSA was purified by gel filtration (Sephadex G100 column equil-ibrated with 10 mM phosphate buffer at pH 7.4) and characterizedby SDS gel electrophoresis. The purified fractions were lyophilizedand stored at )20 �C. The concentration of HSA solutions wasdetermined spectrophotometrically (Reddi et al. 1981).

Preparation of liposomes

Stock solutions of DMPC/DMPG (19:1) and DMPC in chloro-form/methanol (9:1) were dried to a film using a stream of nitrogen.Lipids were hydrated with buffer by hand-shaking followed bysonication (MSE Desintegrator P 6/527) until no further decreaseof turbidity could be detected. The samples were centrifuged(Beckman J2-21 centrifuge, 19,000 rpm, 45 min) to eliminateremnants of multilamellar vesicles as well as metal particles fromthe sonicator. The phospholipid content of the liposome suspen-sions was quantified according to Rouser et al. (1969). For calcu-lations and graphic representations, the molar lipid concentrationwas used instead of the liposome concentration in all cases whenliposomes were used. For calculating molar liposome concentra-tions, the liposome size distributions were assumed to be the sameas in our previous work under these conditions (Galantai et al.2000).

FCS experiments

FCS measurements were performed in an apparatus of our ownconstruction (Langowski and Tewes 2000). It consists of a modulecontaining the confocal optics, which connects to the video port ofan inverted microscope (Olympus IX-70). The laser (Omnichrome20 mW Ar-Kr laser) is coupled into the module via fiber optics.The observation lens was an Olympus 60·/1.2W water immersiontype with cover slip correction. For the experiments described here,an excitation wavelength of 488 nm was selected via a filter and adichroic mirror in the FCS module. The fluorescence was detectedafter separation through a dichroic mirror (505DRLP) and filter(LC 635DF55 for porphyrin or LC 530DF30 for Alexa 488)(Omega Optical, Brattleboro, USA) by avalanche photon diodedetectors (Perkin-Elmer single-photon counting module SPCM-AQ141). Autocorrelation functions were collected from the photonstream by an ALV-5000 correlator card and ALV correlatorsoftware (ALV, Langen, Germany). The experiments weretypically performed at a laser power of 2 lW, corresponding toa power density (in a focus of 0.15 lm radius) of 2.8 kW/cm2.Autocorrelation functions were accumulated in five blocks of 60 seach.

FCS theory and data analysis

Overview of FCS theory

The primary data obtained in an FCS measurement is the time-dependent fluorescence intensity, F(t), which is proportional to thenumber of particles in the observation volume at time t. Theautocorrelation function of F(t) contains all the relevant informa-tion relating to the diffusion of the fluorophores. The normalizedautocorrelation function G(s) is computed as:

G sð Þ ¼ F tð ÞF t þ sð Þh iF tð Þh i2

ð1Þ

For obtaining quantities such as diffusion coefficients, concen-trations, or reaction rate constants, one has to fit a theoreticalcorrelation function to the measured G(s) which is based on amodel that contains these quantities as free parameters. For asolution of a single fluorescent species with diffusion coefficient Dand molar concentration c and for Gaussian profiles for the exci-tation intensity and detection efficiency, G(s) evaluates to (Rigleret al. 1993):

G sð Þ ¼ 1

cVeff1þ 4Ds

w20

� ��1 1þ 4Ds

z20

!�1=2þ 1 ð2Þ

Here, Veff is the effective observation volume, which depends on thegeometry of the focus for excitation and emission, w0 and z0 are thehalf-widths of the focus in the x–y plane (the observation plane ofthe lens), and in the z-direction, respectively.

Veff, w0, and z0 can be measured independently by calibrationwith a solution of a fluorophore of known concentration and dif-fusion coefficient. If only relative changes are of interest, one canuse the average particle number, N=cVeff, and an effective diffu-sion time, sdiff=w0

2/4D, as parameters:

G sð Þ ¼ 1N

1þ s

sdiff

!�1 1þ s

sdiffj2

!�1=2þ 1 ð3Þ

where j (also called the structure factor) is the axial ratio of theobservation volume, z0/w0.

For a high-aperture lens (NA=1.2) at optimal alignment, jtypically ranges between 4 and 6. In a typical FCS experiment, onedetermines j on a monodisperse solution of a known fluorophoreand keeps its value fixed for the measurements of the unknownsample.

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In a mixture of molecules with different diffusion coefficients,the fluorescence intensity autocorrelation function is a sum of thecontributions of the individual species. The general form of G(s) fora mixture of m different fluorescent species with diffusion timessdiff,i is then given by:

G sð Þ ¼ 1

N

Xm

i¼1qigi s; sdiff;i� �

þ 1 ð4Þ

with:

gi s; sdiff;i� �

¼ 1þ s�sdiff;i

� ��11þ s

�sdiff;i � j2� �� ��1=2 ð5Þ

The qi are the relative amplitudes corresponding to molecules withdistinct diffusion coefficients; they are related to their concentra-tions ci by:

qi ¼/2

i ciPmi¼1

/2i ci

ð6Þ

where /i is the quantum yield of species i.Up to Eq. (4), only number fluctuations in the detection volume

were considered to contribute to the fluctuations of the lightintensity at the detector, under the simplifying assumption that anexcited fluorophore will emit a constant light flux. Because of thequantum nature of light and the photophysics of fluorescent mol-ecules, this is not the case. The most important effect that has to beconsidered is a transition of the excited molecule into the tripletstate. This will "interrupt" the stream of photons for approximatelythe triplet lifetime of the fluorophore and add another contributionto the autocorrelation function, which—in good approximation—is(Widengren et al. 1995):

G sð Þ ¼ 1þ be�ks� � 1

N

Xm

i¼1qigi s; sdiff;i� �

þ 1 ð7Þ

The amplitude of the triplet term b and its relaxation rate kincrease with the excitation light intensity up to a limit given bythe excitation, emission, and intersystem crossing probabilities ofthe fluorophore.

FCS data analysis

For analyzing FCS data from samples of biological macromolecules,Eq. (7) is often applied since it is known a priori that the systemcontains a small number of discrete components (for example, in aprotein–protein or protein–DNA interaction). Thus, FCS autocor-relation functions were first analyzed by QUICKFIT, a fitting pro-gram developed in our group under the application frameworkDelphi (Borland, Scotts Valley, USA) and running on a PC underWindows. The software is based on the Levenberg–Marquardtalgorithm for non-linear least-squares multi-parameter curve fitting(Press et al. 1986) and allows the automatic analysis of a set of ACFs,fitting the amplitude and decay time of up to three diffusing com-ponents and up to two non-fluorescent contributions, as well as thebaseline and the structure factor. The parameters describing the non-fluorescent contribution were then used in the further analysis by themaximum entropy procedure.

We shall demonstrate in the following that for biological sam-ples which are intrinsically polydisperse, such as liposomes, proteinaggregates, or intracellular fluorescent components, a discretecomponent fit often yields uninterpretable results because thenumber of parameters necessary for a good fit exceeds the infor-mation content of the experiment. In such a case, a distribution ofdecay functions should be fitted to the data, which can be char-acterized by a small number of independent parameters (such aswidth, area, and first moment).

In that case, the sum in Eq. (7) is approximated by an integralover the diffusion time sdiff, and the amplitudes of the components,qi, by the desired normalized distribution q(sdiff):

G sð Þ ¼ 1þ be�ks� � 1

N

Zsmax

smin

q sdiffð Þ g s; sdiffð Þdsdiff

0@

1Aþ 1 ð8Þ

The task is then to find the inversion of the integral transfor-mation in Eq. (8), obtaining the distribution q(sdiff) from a mea-sured noisy data set G(s). Inversion problems of this kind areknown to be ill-posed, that is, an infinite number of solutionscompatible with experimental (or numerical) noise exists, whichmay deviate infinitely from each other. The distribution q(sdiff) inEq. (8) is positive additive, i.e. negative values are not allowed andthe transformation of a sum of distributions is equal to the sum ofthe transformations of the individual components. For finding sucha distribution function that is physically ‘‘reasonable’’ and com-patible with the data, a variety of regularization methods exist.

The ‘‘maximum entropy’’ method (MEM) has been used tosolve this and other problems involving inverse integral transfor-mations (Gull and Daniell 1978; Skilling and Bryan 1984). Here,the entropy of the distribution q(sdiff) is defined as:

S ¼ �XN

i¼1q sdiff;i� �

ln q sdiff;i� �

ð9Þ

where N is the number of discrete values of sdiff through which thedistribution is determined. Given two solutions of equal merit (e.g.,in terms of their least-squares fits to the data), the solution with thelarger entropy extracts the most information out of the originaldata without making unreasonable assumptions about unavailableinformation.

MEM analysis maximizes the posterior probability of a par-ticular model [here the distribution function q(sdiff)], given a mea-sured data set [here G(s)], by maximizing the parameter Q=aS)L.Here, L is the likelihood of the fit, given by the sum of the squareddeviations of the model points from the data (M points), normal-ized by their variance:

L ¼ 1

2

XMj¼1

Gtheo sj� �� Gexp sj

� �� �2.r2

j ð10Þ

and a is a regularization parameter that controls the relativeinfluence of the entropy and likelihood on the fit. Higher valuesof a will lead to smoother distributions, while for small a valuesthe distribution will contain features that are due to random noisein the data. The correct choice of a can be determined by acriterion based on Bayesian probability theory, as shown byBryan (1990).

Several years ago, Bryan (1990) and Langowski and Bryan(1991) proposed an MEM procedure that was particularly welladapted to oversampled data, such as small-angle scattering data ormultiexponential decays obtained in fluorescence decay or dynamiclight scattering. In such data sets, the number of independent piecesof information in the data is much smaller than the number of datapoints. It was shown there that the MEM fitting procedure can begreatly accelerated for oversampled data by representing the dis-tribution function q(sdiff) by a set of linearly independent basisfunctions.

Let the distribution be related to the measured data through alinear transformation:

d ¼ Tf ð11Þ

where d is an M-element vector containing the autocorrelationfunction G(si), f is an N-element vector containing the distributionq(sdiff,j), and T is a transformation matrix whose elements are givenin our case by the values of Eq. (5) for all pairs of (si, sdiff,j):

Tij ¼ g si; sdiff;j� �

ð12Þ

In previous work (Bryan 1990), a non-linear solution of thisproblem in the singular value space of T was described whichgreatly improved the convergence and stability of the fittingprocedure. In those papers, special solutions were described fordynamic light scattering data and Fourier transformation.

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In the case of FCS, the distribution in its discretized form isgiven by the coefficients qj, and the data vector is the measuredcorrelation function G(si). The transformation matrix elements aregiven by Eq. (12). Application of the maximum entropy algorithmdescribed by Bryan (1990), however, is complicated by the presenceof the (1+be)ks) term in Eq. (8), which in itself is s-dependent. Toovercome this difficulty, we have chosen to determine the param-eters b and k from a separate discrete fit first, before applying theMEM algorithm.

Results and discussion

The rationale for using MP as a non-covalent fluores-cent label for both HSA and liposomes is based on ourprevious work on the physicochemical properties of thissystem (Galantai et al. 2000). We found that the boundporphyrin could be easily detected in our FCS setup andgave a well-defined correlation signal for HSA, lipo-somes, and the HSA–liposome mixtures; on the otherhand, the free porphyrin dye is hardly detectable [whichis in agreement with the increase of fluorescence inten-sity of MP after binding HSA or liposomes (Bardos-Nagy et al. 1998)]. We therefore used Alexa 488 as astandard for a small dye molecule.

As a first experiment, we characterized the diffusionof MP-tagged liposomes and the MP–HSA complexunder conditions of a large molar excess of lipid orprotein over the label. The MP concentration was5·10)9 M, HSA was at 10)6 M, and the total lipidconcentration was 10)3 M. Two types of liposomes wereused, as described in the Methods section: one preparedfrom 100% DMPC, the other from a 19:1 w/w%DMPC/DMPG mixture. From the known size distri-bution of the liposomes under these preparation condi-tions and the known surface area per lipid molecule(Galantai et al. 2000), we estimate a liposome concen-tration of 2·10)8 M. The average diameter for this typeof preparation was 75 nm for the DMPC liposomes and50 nm for the DMPC/DMPG liposomes (Galantai et al.2000).

Figure 1 (lower panel) shows typical autocorrelationfunctions (ACF) of the basic components under theseconditions. Figure 1 (upper panel) gives the ACF ofAlexa 488 as a control for a monodisperse system. TheAlexa data are used to calibrate the s scale and tocalculate the diffusion constant of the particles.

We obtained stable values for the diffusion con-stant, the amplitude, and the non-fluorescent term ofAlexa 488 using a non-linear least-squares fit withdiscrete components. The translational diffusion con-stant of Alexa 488 was determined by comparing itsdiffusion time with that of rhodamine 6G, with knowndiffusion coefficient. Here, we measured sdiff=52 ls forAlexa 488 with excitation at 488 nm and detection at515–545 nm, and sdiff=70 ls for rhodamine 6G withexcitation at 568 nm and detection at 608–662 nm.Correcting for the wavelength-dependent dimensions ofthe focal volume and using Dt=2.8·10)10 m2 s)1 forrhodamine 6G (Rigler et al. 1993), this yields a value

of Dt=2.4·10)10 m2 s)1 at room temperature forAlexa 488. The decay curves for the HSA–MP mixturecould be fitted with one or two discrete fluorescentcomponents; however, the two-component analysis didnot result in a significantly improved fit, as judged bythe v2 value. Table 1 presents the results of the discretefit at 25 �C. The diffusion constant of HSA, as deter-mined from the one-component fit in Table 1, is ingood agreement with the value of 6.0·10)11 m2 s)1 asmeasured by Oncley et al. (1947) at room temperature,and with the value of 6.27·10)11 m2 s)1 determined byFerrer et al. (2001).

The liposome data from Fig. 1 were also first ana-lyzed with the discrete-component model. Here wefound that the fit became very unstable: the fittedparameters exhibited large variations between differentmeasurements on the same sample and for differentstarting values of the fit parameters on the same mea-surements. The reasons for this were presumably boththe higher noise level of the primary data and the widthof the size distribution of the liposomes. Taken together,this resulted in data that could not be fitted with a smallnumber of discrete decay components, and did notcontain enough information to determine reliably theD values and amplitudes of a larger number of compo-nents. Thus, for these types of sample a methodthat yields a distribution of D values might be moreappropriate.

Our previously described maximum entropy algo-rithm (MEM) (Bryan 1990) is well adapted to fittingproblems such as the one described here, i.e. to dataconsisting of a sum of smooth decay functions. A spe-cific feature of FCS data, which makes the application ofMEM less straightforward, is the presence of the non-fluorescent decay part which constitutes the s-dependentmultiplicative term in Eq. (7). While such constantparameters can in principle be determined by a Bayesianprobability criterion or integrated out of the problem [aswas done for the baseline for dynamic light scatteringdata by Langowski and Bryan (1991)], this procedurecomplicates the fit and can lead to instabilities.We therefore used the discrete fit in a first step fordetermining the total amplitude of the correlationfunction, 1/N (where N is the number of particles in thedetection volume), and the amplitude b and relaxationrate k of the non-fluorescent part.

The result was used to calculate the elements of thetransformation matrix T:

Ti;j ¼1

N1þ be�ksj� � 1

1þ sj�sdiff;i

ð13Þ

where sj is the correlation time (discretized by the indexj) and sdiff,i is the diffusion time of the i-th component.For a measured data set d with dj=G(sj), the mostprobable distribution vector f with fi=q(sdiff,i) is thendetermined through maximum-entropy optimization inthe singular vector space of T, as described by Bryan(1990).

62

Figure 2 shows the distribution functions obtainedfrom applyingMEM to the Alexa and the HSA data. TheAlexa sample gives a monomodal distribution with the

median of the peak at 50 ls, in good agreement with thevalue from the discrete fit. For sdiff<1 ls, the distributioncurve increases again to a constant value of 0.006. This is atypical feature of themaximum-entropy fitting algorithm;for regions in which no information exists (i.e., only datafor correlation times>1 ls were used), themost probablevalue of the distribution function is not zero but theaverage over the region of sdiff where experimental dataexist. For amore thorough discussion of this subject in theframework of Bayesian statistics, see Bryan (1990).

The diffusion time distribution for the FCS data fromthe HSA–MP mixture shows two clearly separated

Fig. 1 (a) FCS autocorrelationfunctions of Alexa 488 used as astandard: measured data (opencircles) and maximum entropyfit (solid line). (b) Residuals ofthe fit. (c) FCS autocorrelationfunctions of the lipid andprotein components in thepresence of MP

Table 1 Diffusion constants of Alexa 488 and HSA at 25 �Cderived from one-component fits using the discrete non-linear fit-ting method. Errors given are standard errors of the mean derivedfrom three different fits

sdiff (ms) Dt (m2 s)1)

Alexa 488 0.052±0.0007 (2.4±0.03)·10)10

HSA 0.173±0.012 (7.2±0.5)·10)11

63

Fig. 2 FCS diffusion timedistributions calculated byMEM from Alexa 488 (a) andHSA in the presence of MP (b);(c) MEM fit to the FCSautocorrelation function fromthe HSA sample in (b) andresiduals (d)

64

components. The main peak has a median value ofsdiff=0.15 ms, corresponding to a diffusion coefficient of8·10)11 m2 s)1, in reasonable agreement with both theone-component fit and the literature value. The secondpeak, at sdiff=18 ls, probably originates from free MPand/or incomplete compensation of the non-fluorescentterm. Since the main HSA peak was separated from thisfeature by an order of magnitude, we did not considerthe presence of this peak an important problem in thefollowing analysis.

The distribution curves calculated from the MEManalysis on the liposome samples are shown in Fig. 3. Thepeaks represent relative values normalized to a total area(correlation amplitude) of 1; for comparison, they have tobe multiplied with the 1/N value determined separately.

A small peak at around 1 ls is present in most of thedata sets; this corresponds to the non-fluorescent com-ponent, which is not eliminated entirely using theparameters estimated from the discrete fit. Through asystematic variation in the amplitude b with subsequentrecalculation of the transformation matrix, this peakcould probably be suppressed even more. The peak foundat sdiff<0.1 ms is also present in the buffer and probablyoriginates in a slight contamination by traces of fluores-cent molecules other than MP, whose quantum efficiencyis rather low. The peaks corresponding to the liposomesare well separated from the small contaminant.

Given the sdiff value of the Alexa 488 reference and itsknown diffusion coefficient Dt corrected to 32 �C, the Dt

of liposomes and HSA may be computed from their sdiffvalues (Table 2). Assuming a spherical shape for theprotein and liposomes, we can also transform the diffu-sion coefficient into a hydrodynamic radius Rh using theStokes–Einstein relation Dt=kBT/6pgRh, with kB beingBoltzmann’s constant, T the temperature, and g the vis-cosity of the solvent. The results in Table 2 show that forAlexa 488 and HSA the diffusion coefficients obtainedfrom the discrete-component fit and from the MEMprocedure are in very good agreement. The hydrodynamic

radius of the liposomes is similar to the value determinedthrough dynamic light scattering (Galantai et al. 2000).

For analyzing the interaction between liposomes andHSA in the presence of MP, we then mixed liposomepreparations of both lipid compositions (DMPC andDMPC/DMPG) with a HSA–MP solution at the con-centrations given in Table 3. FCS data were then taken

Fig. 3 Diffusion time distributions for DMPC and DMPC/DMPGliposomes in the presence of MP, as obtained from MEM fits to theFCS autocorrelation functions

Table 2 Diffusion times (measured at 32 �C), computed diffusionconstants, and hydrodynamic radii for the Alexa standard and thedifferent protein and liposome components as obtained from theMEM distributions

sdiff (ms) Dt (m2 s)1) r (nm)

Alexa 488 0.05 2.5·10)10

HSA 0.15 9.7·10)11 3DMPC liposome 0.6 2.4·10)11 12DMPC/DMPG liposome 0.7 2.1·10)11 14

Table 3 Compositions of the liposome/HSA/MP samples used inthe experiments

Lipids (mol/L) HSA (mol/L) MP (mol/L)

A 8.3·10)5 5·10)7 5·10)9

B 1.7·10)4 5·10)7 5·10)9

C 3.3·10)4 5·10)7 5·10)9

D 8.3·10)4 5·10)7 5·10)9

Fig. 4 FCS autocorrelation functions of (a) DMPC–liposome–HSA–MP and (b) DMPC/DMPG–liposome–HSA–MP solutions.Curves A, B, C, D represent the different concentrations that aregiven in Table 3

65

of these samples; typical autocorrelation functions areshown in Fig. 4. For these data, usually three discretecomponents were needed to obtain a satisfactory fit,although in some cases two were sufficient. In Table 4we only show the sdiff parameter corresponding to themain liposome component.

MEM analysis applied to the same data gave the dif-fusion time distributions shown in Fig. 5. Table 5 showsthe corresponding diffusion constants and hydrodynamicradii obtained from the median values of the peaks.

Comparing the results from the discrete-component fitwith the MEM analysis, we see a clear concentrationdependence of the liposome diffusion time obtained in thediscrete analysis: for the higher lipid concentrations, sdiffincreases. MEM, on the other hand, gives a stable valuefor the diffusion time of the liposome component thatshows no trend with changing concentration.We suppose

that the reason for this difference lies in the fact that theactual diffusion time distributions exhibit rather broadfeatures, which makes a discrete analysis more uncertainand will artificially shift values of one fitted componentwhen the amplitude of another one changes, as is thecase here. We have one sharper peak around 1 ms that

Fig. 5 Calculated diffusion timedistributions values of mixturesof (a) HSA, MP, and DMPC,and (b) DMPC/DMPGliposomes characterized by theireffective diffusion time (sdiff).The graphs are the average overtwo measurements fromindependent samples

Table 4 Average sdiff values for DMPC and DMPC/DMPG lipo-somes at 32 �C as determined from a discrete-component fit.Values are the average from two independent samples at differentlipid concentrations as given in Table 3

DMPC sdiff (ms) DMPC/DMPG sdiff (ms)

A 0.36±0.2 0.70±0.04B 0.56±0.1 0.87±0.04C 0.92±0.06 0.55±0.16D 1.05±0.21 1.00±0.14

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corresponds to the liposome and another wide peakaround 0.1 ms that is related to the diffusion of free HSA.

From the peak areas in Fig. 5 we can then charac-terize the relative contributions of the HSA and lipo-some components to the FCS data. The peak at fastdiffusion times corresponds to the error in the estimateof the triplet amplitude and relaxation rate, which in thisversion of the FCS program has to be done beforehandthrough a discrete-component fit (as described above).That peak, therefore, was not included in the calcula-tion. The relative areas of the two peaks correspondingto HSA and liposomes are shown in Table 6; we clearlysee that the increasing lipid concentration from A to D isreflected in the relative contribution of the liposomepeak. However, in all samples a substantial amount ofHSA is still detected.

It is worth noting that the observation of liposomesin a system containing the HSA–MP complex andunlabelled liposomes is in good agreement with ourprevious results that showed the binding of HSA to thisSUV (Galantai and Bardos-Nagy 2000; Galantai et al.2000; Bardos-Nagy et al. 2003). The fact that the lipo-some signal increases with increasing amount of lipidconcentrations indicates that the amount of HSA–lipo-some complex increases, so the saturation condition isnot fulfilled at the applied HSA/lipid concentrationratios. Again, the Stokes’ radius of the liposomes issimilar to the value from dynamic light scattering(Galantai et al. 2000).

In summary, we conclude that FCS data from aheterogeneous system like the one studied here can beanalyzed more stably using the MEM method than witha discrete fitting algorithm. A further improvementof the method is possible by including discrete non-

fluorescent terms into the parameters, which will beimplemented in a future version of the program.

A Windows-executable version of the MEM program(written in Borland Delphi/Pascal) is available uponrequest from the authors.

References

Bardos-Nagy I,GalantaiR,KaposiA,Fidy J (1998)Difference in thetransport of metal and free-base porphyrins. Steady-state andtime-resolved fluorescence studies. Int J Pharm 175:255–267

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Table 5 Measured effective diffusion times at 32 �C, computeddiffusion constants, and hydrodynamic radii for the liposome peakin the different mixtures of liposomes, HSA, and MP. A, B, C, andD correspond to the concentration values in Table 3

Sample sdiff (ms) Dt (m2 s)1) r (nm)

DMPC (A) 1±0.17 (1.4±0.24)·10)11 21±3.6DMPC (B) 0.7±0.26 (2.1±0.8)·10)11 14±5.2DMPC (C) 0.97±0.09 (1.5±0.14)·10)11 19±1.8DMPC (D) 1±0.15 (1.4±0.21)·10)11 21±3.2DMPC/DMPG (A) 0.79±0.15 (1.8±0.34)·10)11 16±3DMPC/DMPG (B) 0.95±0.06 (1.5±0.1)·10)11 19±1.2DMPC/DMPG (C) 0.75±0.12 (1.9±0.3)·10)11 15±2.4DMPC/DMPG (D) 0.81±0.08 (1.7·±0.17)·10)11 17±1.7

Table 6 Relative area under the HSA and liposome peaks in Fig. 5

A B C D

DMPC/HSAPeak 2 (HSA) 0.810 0.774 0.707 0.621Peak 3 (liposomes) 0.190 0.226 0.293 0.379

DMPC-DMPG/HSAPeak 2 (HSA) 0.802 0.749 0.670 0.537Peak 3 (liposomes) 0.198 0.251 0.330 0.463

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