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* Division of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637616; and Ludwig Institute for Cancer Research, Post Office Royal Melbourne Hospital, Melbourne, Australia
Correspondence: Address reprint requests to Dr. Andrew H. A. Clayton, Ludwig Institute for Cancer Research, PO Box 2008, Royal Melbourne Hospital, Melbourne, Australia. Tel.: 61-3-341-3155; Fax: 61-3-341-3104, E-mail: andrew.clayton{at}ludwig.edu.au.
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INTRODUCTION |
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–4). Because of the perceived importance of macromolecular oligomerization as a biological control mechanism, it is essential to perform experiments that can quantitatively assess the oligomerization states and extent of clustering on the nanometer scale (relevant to molecular contact).
The quantification of oligomerization on and inside cells is a difficult problem. Experimental strategies appropriate to cell and membrane preparations include measurement of protein translational and rotational diffusion (5) and fluorescence fluctuations in space (6) and/or time (7) by correlation spectroscopy/microscopy and intermolecular proximity between donor-labeled and acceptor-labeled proteins by fluorescence resonance energy transfer (8). In the latter situation, the characteristic nanometer distance scale of the energy transfer phenomenon renders it a particularly sensitive probe of protein association between fluorescently tagged proteins.
Homo-FRET, particularly energy migration FRET (9,10), is a simpler variant of energy transfer because it occurs between like chromophores and hence requires only one type of fluorescent label. It is manifested by the presence of depolarized fluorescence. Beginning with the first reported observation of concentration-dependent depolarization by Gaviola and Pringsheim (11), there exists a wide application of the phenomenon to photosynthesis, light-harvesting polymers, as well as synthetic and natural polychromophoric arrays (12). Much of the theoretical descriptions have focused on extracting information on interchromophore distance distributions from analysis of high-resolution time-resolved fluorescence anisotropy decay measurements (13). The descriptions result in complex decay functions that require data of high signal/noise ratio on samples of high purity. This circumstance is difficult to achieve in the complex milieu of the living cell and requires specialized instrumentation that is not readily available in most biological laboratories.
A simpler approach in terms of experimental implementation is to use steady-state anisotropy. Runnels and Scarlata have shown that under certain conditions the steady-state anisotropy is inversely proportional to the number of subunits in the oligomer (14). However, anisotropy as a single measure suffers from ambiguity of interpretation since factors other than energy transfer, such as excited state lifetime changes or changes in rotational motion, can hyper- or hypopolarize the emission. Several groups have reported cellular anisotropy measurements as a function of labeling (or photobleaching) as a qualitative probe of energy homotransfer. In the anisotropy enhancement after photobleaching experiment, first applied by the Mayor group (15), any depolarization caused by energy transfer is reversed by photodestruction of FRET acceptors (15). Similarly, mixing of labeled and unlabeled proteins should reverse interaction-dependent depolarization processes, as shown by Blackman et al. (16). In principle, these measurements allow information beyond confirmation of energy transfer and proximity to be obtained.
Our examination of the extent of energy migration as measured by fluorescence anisotropy as a function of fluorophore labeling (or photodepletion) gives valuable information on the actual oligomerization state of self-associating proteins. In the next section we present a general theoretical model for interpreting anisotropy data in terms of dilute solutions of oligomers and oligomerization distributions. The model predicts that the anisotropy as a function of labeling for an oligomer with N subunits is a polynomial of order N–1. We extend the formalism to account for the occurrence of interoligomer energy transfer and specifically treat interdimer energy transfer. This extension is needed because of the possible occurrence of depolarization between overexpressed oligomeric proteins at high density in cell membranes. In the Examples section we use our analytical methods to examine existing data from the literature on both intra- and interoligomer energy transfer between Band 3 dimers in solution and on membranes, intraoligomer energy transfer between triproximal GPI-proteins in rafts, and the consequences of submicrometer scale clustering of EGFR-eGFP on anisotropy enhancement data after photobleaching data.
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THEORY |
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,16). The exchange of labeled for unlabeled molecules would not be expected to alter the inherent motional properties of fluorophores, but it does decrease the probability for self-transfer of energy.
Dilute solution of homogeneous oligomers
First, we consider a single population of oligomers with N monomer subunits per oligomer. For a structure with N subunits, the fraction Fi that have i fluorescently labeled subunits when a fraction f of the subunits are labeled is, according to the binomial theorem (17,18),
 | (1) |
The total anisotropy of the oligomer population as a function of labeling (f) is given by the sum law of anisotropies (10), where the summation is from i = 1 to i = N:
 | (2) |
 | (3) |
 | (4) |
 | (5) |
 | (6) |
These expressions are obtained by noting that for an N-mer one has a polynomial of order N – 1 with the coefficients in the expansion (A1, A2, ... AN) derived from the (N – 1)th row of the Pascal's triangle.
These equations are general and make no assumptions about the energy transfer or rotational dynamics properties of the fluorophores. The only requirement is that an oligomer with one subunit labeled yields a different fluorescence anisotropy than an oligomer containing more than one fluorescently labeled subunit. Runnels and Scarlata have provided a means of determining the anisotropy of an oligomeric protein as a function of the number of labeled subunits (14) from structural data.
Fig. 1 illustrates simulations of the anisotropy as a function of labeling for a monomer, dimer, and tetramer under the situation that energy migration depolarizes the fluorescence to an extent that the anisotropy is decreased to r1/N, where N is the number of labeled monomers per oligomer. This choice of anisotropy corresponds to the limit of efficient energy hopping (14) between randomly oriented but rotationally fixed fluorophore sites. This circumstance is achieved at separations of less than 0.8 Ro, where Ro is the Förster distance (e.g., 
5 nm for eGFP). It is notable that the oligomerization state is reflected in the curvature of the anisotropy as a function of labeling plot, a consequence of Eq. 3–5.
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N. The mean anisotropy as a function of labeling is given by
 | (7) |
The influence of the Poisson distribution of oligomerization states is presented in Fig. 2 using the same model as in Fig. 1 for N = 1, 2, 3, and 4. The corresponding single population oligomerization model is plotted for comparison. For all N, the Poisson model gives anisotropy plots that show greater curvature than the homogeneous model. This is because in the distributed oligomer case, Eq. 7 receives weighting from states of higher oligomer number. For the Poisson simulations presented in Fig. 2, the fluorescence population distribution of oligomers contains modes at N and N + 1. Parenthetically we note improved agreement using a Poisson model with mean oligomerization 
N
= N – 0.5 with a homogeneous model with oligomerization state N. In this case both models have a single mode at N.
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 | (8) |
,18). The effect of monomer on the anisotropy enhancement curves is shown for the situation of a bimodal monomer-tetramer equilibrium (Fig. 3). The presence of the monomer rescales the anisotropy plot by a constant value but does not change its shape. This is an important property of the anisotropy enhancement method; only sources of depolarization that result from homo-FRET are affected by labeling/bleaching. Thus, oligomerization states of proteins that are in equilibrium between monomeric and associated (oligomeric or concentrated) forms can be investigated. Simulations with the Poisson-distributed model and rN = rm/N show that Eq. 8 always yields a minimum estimate of the mean oligomerization state.
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). The equation for 2D transfer depolarization between monomers is:
 | (9) |
is the fluorescence lifetime and ß is the normalized 2D concentration of monomers (= 1.354C/2Co, where C/Co is the average number of monomers per circular area with radius equal to the Förster distance 
For a typical Förster distance of 5 nm, a cell expressing 1 million receptors on the cell surface (typical cell surface area ranges 1000–10,000 µm2) will have a normalized 2D concentration of ß = 0.008–0.08 molcules/Förster area. Simulations presented below show the extent to which these effects are anticipated to influence the measured anisotropy.
Concentrated 2D solution of homogeneous oligomers
We consider the oligomers to be homogeneous and preformed. (The reader is referred to the article by Jovin et al. (20) for the 3D monomer-dimer equilibrium case as a function of concentration.) It is instructive to consider the situation of concentration depolarization of oligomers. This can be readily treated under the reasonable assumption that intraoligomer and interoligomer energy transfer modes of depolarization are independent (21).
The time-dependent anisotropy of the system involving both intraoligomer and interoligomer transfers is given by:
 | (10) |
The consequences of energy transfer between like chromophores on the fluorescence and anisotropy decays within a dimer has been discussed by Tanaka and Mataga (22). For a fixed dimer geometry the time-resolved anisotropy is described by a single exponential decaying to a finite value:
 | (11) |
Here, ro is the anisotropy in the absence of rotation or energy transfer, and r
is the anisotropy at a long time after the excitation (i.e., 200 ns or longer). The transfer correlation time, 
, is related to the rate of energy transfer (w) by = (2w)–1.
Combining the two terms and following similar arguments as discussed above gives the following anisotropy decay expression for randomly organized dimers as an explicit function of dimer concentration (ß) and fractional labeling (f):
 | (12) |
Equations describing higher-order oligomerization states can also be readily derived using appropriate time-dependent versions of Eqs. 3–6 for r(t)oligomer and noting that the effective concentration of labeled oligomers (i.e., bearing one or more labeled subunits) is given by ß(1 – (1 – f)N).
The steady-state anisotropy is obtained on integration with the intensity decay function I(t):
 | (13) |
Implicit in Eq. 13 is the independence of the intensity decay to concentration or labeling effects (i.e., the absence of concentration quenching).
The influence of a concentrated solution of monomers on the anisotropy versus label plot is shown in Fig. 4 for the situation where there are 1 million molecules on the surface of a cell with surface area 1000 µm2 (ß = 0.075). The maximum extent of depolarization, compared with a dilute monomer solution model is 0.04. The corresponding plots for 1 million dimers on a cell with surface area 1000 µm2 is shown for comparison (with ro = 0.4, r = 0.2, w = 1 ns, = 4 ns) in the presence (ß = 0.075) and absence (ß = 0) of concentration depolarization. Concentration depolarization decreases the anisotropy at all label efficiencies compared with the dilute dimer situation but does not cause the anisotropy versus labeling curve to deviate substantially from a linear relationship. This is expected to be general whenever the intradimer energy migration is the dominant depolarization process. Thus, it is reasonable to expect that a significant curvature in the anisotropy versus labeling curve is a reflection of either 1), an atypically high local concentration of monomeric protein or 2), association at a level greater than dimer.
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and the steady-state anisotropy r is expressed in Eq. 14:
 | (14) |
–1 + –1), B = 2ß[1 – (1 – f)2] / t
, and C = –1.
In the absence of concentration depolarization (ß = 0), a linear relationship exists between r and f with a constant gradient, Gß=0 = [(ro – r) / (–1 + –1)] + r – ro. Fig. 5 shows the variation of the gradient (G) for the r versus f curve at various f values for ß = 0, 0.001, 0.01, and 0.1, and ro = 0.4, r = 0.2, w = 1 ns and = 4 ns. We note that for low dimer concentrations (i.e., ß = 0.001), the G values remain close to Gß=0 (= –0.133) regardless of the degree of fluorescence labeling (f). By way of orientation, a concentration of 10–100 µM corresponds to ß = 0.001–0.01 (for a fluorophore with Ro = 5 nm). Deviation from Gß=0 is observed when ß increases and is most significant for larger ß. As for the 2D concentration depolarization of dimers, close to linear anisotropy as a function of labeling/bleaching plots are anticipated over most of the parameter space when intradimer energy migration plays a small role.
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). In the latter application the authors compared the anisotropy of eosin-labeled tetrameric and dimeric reconstituted receptors to the anisotropy of native receptor in cell membrane suspensions and, together with detailed calculations, deduced that the receptor did not form high-order oligomers in native membranes. Homo-FRET was confirmed by several methods, including the observation of the red edge effect, rapid dynamic fluorescence depolarization, and an anisotropy that varied as a function of labeling efficiency. Fig. 6 shows a plot of the average steady-state anisotropy as a function of the degree of labeling for the eosin-labeled band 3 in purified dimers and in cells/membranes, adapted from Fig. 1 of the Blackman et al. study (16). A clear increase in the anisotropy is observed for both systems, as expected for molecules undergoing homo-FRET. As can be seen from Fig. 6, the data for the dimer conforms to the expected linear relationship for a dimer undergoing homo-FRET. We have overlaid on the experimental data a simulation using the parameters obtained from the frequency domain depolarization measurements (ro = 0.37, r = 0.31, w = 1 ns, = 3 ns, ß = 0). The agreement between the dynamic depolarization data and the steady-state anisotropy as a function of fluorophore labeling is excellent. The anisotropy enhancement observed in the cell membrane suspensions is also reasonably linear, albeit with a lower anisotropy at the high label stoichiometry. As for the dimers, we have simulated the expected anisotropy enhancement curve including the effect of concentration depolarization between dimers at a dimer density of 1 million dimers per cell (using identical parameters for the dimer simulation with a ß = 0.06). Thus, the observed data can be adequately simulated assuming only dimeric proteins. As discussed above, significant higher-order association would have yielded a nonlinear anisotropy versus labeling curve, in accord with conclusions reached by the authors based on separately prepared and reconstituted samples. We conclude that the present theory is adequate for the detection of dimeric proteins in cell membranes under conditions of intra- and interdimer homotransfer. This example highlights how the theory can be used to constrain models of association in cases where detailed information about intradimer emFRET dynamics is available.
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) in their study of homo-FRET in GPI-linked proteins. They distinguished concentration depolarization between randomly organized monomers from oligomerization by analysis of the dependence of the energy transfer on the concentration of protein on the membrane and suggested that the GPI-linked proteins were localized in submicrometer domains with as many as 70 proteins. Fig. 7 shows a plot of the average anisotropy as a function of the degree of photobleaching (taken from Fig. 3 of Varma and Mayor (15)). The dotted line indicates a best fit to a dimer model, and the solid line to a trimer model (r1 = 0.367, r2 = 0.245, r3 = 0.230). Unlike dimeric band 3, the GPI-linked protein plot shows distinct upward curvature in the anisotropy versus labeling (bleaching) plot consistent with an oligomerization state that is greater than a dimer. Application of the monomer–N-mer model gave a good fit to the data with a minimum degree of oligomerization of 35% (fractional fluorescence) and a minimum trimeric stoichiometry (N = 2.7). We also simulated the situation of concentration depolarization between dimers. Although most of the parameters in this model are not known from independent sources, we found that an effective density of C/C0 = 0.2 was needed to account for the observed curvature in the data (i.e., ro = 0.37, r = 0.33, w = 1 ns, = 4 ns, ß = 0.2). Taken together our simulations suggest that GPI-linked proteins are dimeric or trimeric and, in the former case, would need to be present at very high (local) concentrations in the cell membrane. Thus, the existence of large oligomeric structures on the nanometer scale appears to be excluded from the present analysis. The data are more consistent with the presence of submicrometer scale rafts containing randomly dispersed small oligomers (dimers or trimers). This is in line with more recent studies suggesting aggregate sizes up to four proteins per cluster (23), as discussed below.
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). They observed photobleaching enhancement in anisotropy that was independent of the expression level of the protein in the cells and provided evidence from time-resolved anisotropy measurements for a rapid dynamic depolarization caused by close (nanoscale) association of proteins. The former observation rules out significant contributions from interoligomer transfers and is consistent with their assignment of the homo-FRET to nanoclusters containing up to four proteins (23).
GFP-tagged proteins
Several groups have reported homo-FRET as a result of oligomerization or concentration depolarization of enhanced green fluorescent protein (eGFP). The favorable Förster distance for this protein and its high intrinsic anisotropy (even as a free monomeric protein in solution) make it a good tag for detecting protein oligomerization using polarization microscopy together with photobleaching.
Recently the Jovin laboratory (24) has presented the first comprehensive homo-FRET measurements of the EGF receptor-eGFP using dynamic depolarization imaging microscopy, confocal polarization microscopy, and flow cytometry. Homo-FRET was inferred from measuring the eGFP anisotropy as a function of eGFP labeling (by photobleaching in a confocal microscope) and as a function of total eGFP-EGF receptor concentration (by observing the natural cell-by-cell variations in eGFP-EGF receptor expression in a flow cytometer). Evidence for preassociation of the receptor was found, although no attempt was made to report an oligomerization state for the receptor. 3D structural data (25), biochemical cross-linking studies (26), and single-molecule imaging suggest that the EGF receptor is dimeric (27). On the other hand, image correlation spectroscopy (6), phosphorescence anisotropy decay measurements (28), and scanning near-field optical microscopy (on erbB2) (29) appear to detect a higher-order form of the receptor in clusters on the submicrometer scale.
An important question is whether the receptors are randomly organized as monomers, dimers, or high-order species within these submicrometer clusters. In CHO cells expressing EGFR-eGFP, anisotropy enhancement on photobleaching experiments revealed an anisotropy enhancement of 0.05 and a nonlinear enhancement curve (14). We have considered the following models for receptor organization: 1), randomly dispersed monomers within the clusters, 2), randomly dispersed dimers within the clusters, and 3), higher-order associations on the nanometer scale. To estimate the number density of receptors within these clusters we used the image correlation spectroscopy data of Petersen, who detected clustering of the EGF receptor on A431 cells with an average of 51 receptors (taking into account the 33% label efficiency) per cluster and an average cluster diameter of 0.4 µm (6). This corresponds to a receptor monomer density of C/C0 = 0.08 and represents an upper bound because A431 cells over expresses the EGF receptor at a level of 2–3 million receptors on the membrane, whereas the value in CHO cells may be lower. According to model 1, the depolarization caused by monomeric receptors is calculated to be 0.04, and the enhancement plot is linear (R2 = 0.99). Model 2 was simulated using a receptor dimer density of C/C0 = 0.04 and a broad parameter range (ro = 0.4, r = –0.6 to 0.375, w = 0.1–100 ns, = 2.5 ns, ß = 0.04) using the experimental constraint that r(f = 1) – r(f = 0) = 0.05. This model also predicts predominantly linear anisotropy enhancement plots (R2 = 0.98). We conclude that a random association of monomers or dimers within submicrometer clusters cannot account for the available data. Therefore Model 3, the involvement of higher-order oligomers, is the preferred model to account for the observed curvature in the anisotropy enhancement data. The latter observation is compatible with our recent experiments on EGFR-eGFP expressed at normal levels in BaF/3 cells (30). Image correlation microscopy showed that in the presence of EGF the average number of receptors per cluster was 3.7, and the density of clusters at the membrane was 19 ± 4 clusters/µm2. The density of clusters is too low to facilitate significant interoligomer energy transfer; however, the number of receptors within the clusters is consistent with higher-order nanoscale associations. This example stresses the value of the theory in providing model discrimination, particularly in cases where data from several sources is available.
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ACKNOWLEDGEMENTS |
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Submitted on October 13, 2006; accepted for publication December 27, 2006.
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