When One Plus One Does Not Equal Two: Fluorescence Anisotropy in Aggregates and Multiply Labeled Proteins

Abstract

The behavior of fluorescence anisotropy and polarization in systems with multiple dyes is well known. Homo-FRET and its consequent energy migration cause the fluorescence anisotropy to decrease as the number of like fluorophores within energy transfer distance increases. This behavior is well understood when all subunits within a cluster are saturated with fluorophores. However, incomplete labeling as might occur from a mixture of endogenous and labeled monomer units, incomplete saturation of binding sites, or photobleaching produces stochastic mixtures. Models in widespread and longstanding use that describe these mixtures apply an assumption of equal fluorescence efficiency for all sites first stated by Weber and Daniel in 1966. The assumption states that fluorophores have the same brightness when free in solution as they do in close proximity to each other in a cluster. The assumption simplifies descriptions of anisotropy trends as the fractional labeling of the cluster changes. However, fluorophores in close proximity often exhibit nonadditivity due to such things as self-quenching behavior or exciplex formation. Therefore, the anisotropy of stochastic mixtures of fluorophore clusters of a particular size will depend on the behavior of those fluorophores in clusters. We present analytical expressions for fractionally labeled clusters exhibiting a range of behaviors, and experimental results from two systems: an assembled tetrameric cluster of fluorescent proteins and stochastically labeled bovine serum albumin containing up to 24 fluorophores. The experimental results indicate that clustered species do not follow the assumption of equal fluorescence efficiency in the systems studied with clustered fluorophores showing reduced fluorescence intensity. Application of the assumption of equal fluorescence efficiency will underpredict anisotropy and consequently underestimate cluster size in these two cases. The theoretical results indicate that careful selection of the fractional labeling in strongly quenched systems will enhance opportunities to determine cluster sizes, making accessible larger clusters than are currently considered possible.

Introduction

Aggregation phenomena are important in understanding the behavior of a wide range of processes in living systems, such as receptor activation in membranes and subsequent downstream signaling (1); aggregation of CUL3-modified caspase-8 leading to cell death (2); the oligomerization of serotonin_1A receptors mediating inhibitory neurotransmission (3); and aggregation of β-amyloid peptide and homodimerization of amyloid precursor protein in the development of Alzheimer’s disease (4–6).

Fluorescence anisotropy is a convenient measure of aggregate formation due to its simplified labeling strategy. This experiment exploits the reduction of fluorescence anisotropy due to homo-fluorescence resonance energy transfer (FRET) among clustered fluorophores. These effects have been studied for nearly 50 years, beginning with the pioneering work of the Weber group, who first noted that assuming equal fluorescence efficiency of fluorophores when describing dye binding to a protein provides a useful simplification of the governing equations based on the binomial distribution (7). Yeow and Clayton rederived these expressions, extended them to other distributions, and applied them to understanding cluster size using a variety of experimental approaches (8). Both groups applied the assumption of equal fluorescence efficiency, which implies that the intensity of a solution of fluorophores would not be expected to change if the fluorophores were induced to assemble into dimers, tetramers, etc. This treatment has been widely applied, with examples including dissociation and reassociation of yeast enolase in the presence of KCl (9); the coordination number and geometric arrangement of fluorophore binding sites on CaATPase (10); rotational correlation time of the phosphorylation domain of CaATPase (11); determination of the average oligomeric state of phospholamban molecules (12); and calculation of the size of lysozyme oligomers in an anionic lipid membrane (13).

In the case of assemblies in which the fluorophores are far apart, the assumption likely holds. However, examination of the behavior of model systems indicates that interaction between fluorophores in molecular assemblies and clusters should be considered in more detail. In the literature, examples of fluorophores in close proximity resulting in either enhanced or decreased emission intensity have been reported. For example, DNA doubly labeled with pyrene on adjacent bases gave over six times the brightness of DNA with only a single label, and DNA labeled similarly with fluorescein resulted in a reduction in brightness (14). Although specific cases may exist, in general, the intensity of a cluster is not equal to the sum over the intensities of the individual molecules forming the cluster. Less is known about the impact of this behavior on anisotropy and the resulting interpretation of cluster size.

To investigate this behavior, we studied the relationship between cluster formation and anisotropy using a combination of theoretical treatment, numerical simulations, and model systems containing multiple fluorophores. Model systems consisted of bovine serum albumin (BSA) labeled with fluorescein isothiocyanate (FITC) and a system of monomeric teal fluorescent protein (mTFP) monomer units that undergo template-directed assembly. BSA contains many sites for attaching FITC groups, and the mTFP system was tetrameric. The extent of homo-FRET-induced depolarization was measured and compared to conventional dye/protein (F/P) ratios to reveal the number of attached fluorophores. The results are applicable to extrinsically labeled molecules combining with native molecules in vivo and to experiments involving FPs that will both mix with a native protein fraction and have a fractional labeling component due to the balance of light and dark states.

Theory

When a fluorophore in close proximity to other molecules of the same type is excited with polarized light, its excitation energy will migrate to neighboring fluorophores. Theories describing the polarization behavior of fluorophores in clusters or under conditions of concentration quenching are well developed (15–18). These theories predict that clustered, randomly oriented fluorophores in close proximity will emit depolarized light and that the measured anisotropy therefore will be lower (19,20). As a consequence, homo-FRET-induced depolarization increases with the number of contributing fluorophores with computationally convenient forms when the interfluorophore distance (R) is <0.8 of the Förster radius (R₀) (18). This behavior has led to wide use of homo-FRET methods to study molecular self-assembly (21–23), quantify cluster sizes (24–26), and determine the extent of protein oligomerization in cells (27–29).

Homo-FRET is not the only process leading to changes in fluorescence anisotropy. Upon self-association, the anisotropy of the labeled proteins varies due to changes in rotational correlation time (30). Larger clusters rotate more slowly, producing an increase in fluorescence anisotropy. Nevertheless, due to the large size of proteins and their tendency to be hindered in vivo, homo-FRET effects dominate, and rotational diffusion can often be neglected. A variety of experiments have been reported in which proteins (subunits or clusters) were fractionally labeled (28,31) or fractionally photobleached (3,24,32).

Homo-FRET experiments provide evidence of both proximity and cluster size (25). However, labeling experiments often result in incomplete saturation of sites. This leads to mixtures consisting of a distribution of fluorophore cluster sizes, and the overall anisotropy is a weighted average of the anisotropies of the clusters within the distribution as predicted by the sum law of anisotropies. For example, Weber and Daniel (7) and Weber and Young (33) studied depolarization of 1-aniline-8-naphthalene sulfonate (ANS) bound to BSA as a function of increasing labeling frequency, and they were one of the first to study polarization of the fluorescence emission as a function of the average number of labeled ligand molecules attached to a protein. This created a mixture of species for which they assumed binomially distributed dyes on sites of equal binding affinity. They also assumed that a single transfer of the excitation energy was responsible for the depolarization of the fluorescence emission, which has since been treated in more detail (17,18).

The binomial theory implies that if there are $N$ binding sites on a protein (P) and $\overline{i}$ is the average number of labeled sites, then the fraction, $f_i$, of the protein that exists in the form of PX_i(0 ≤ i ≤ N), in which i fluorescent molecules or ligands (X) are bound to P, is given by the successive terms of the binomial distribution

$$f_i=\left(\begin{array}{c}N\\ i\end{array}\right){\left(\frac{\overline{i}}{N}\right)}^i{\left(1-\frac{\overline{i}}{N}\right)}^{N-i}.$$ (1)

Note that in this representation the overall fractional labeling, f, is given by $\overline{i}/N$. If the fraction associated with PX_i has a contribution of ${\varphi }_i$ to the total fluorescence intensity and its emission anisotropy is r_i, the observed emission anisotropy, r, of the ensemble, based on the sum law of anisotropies (7,8,15,34), is

$$r\left(N\right)=\sum _{i=1}^N{\varphi }_i{r}_i.$$ (2)

The value of r_i can be predicted using the considerations of Runnels and Scarlatta (18) and, when R < 0.8R₀, reduces to r₁/i. Weber and Daniels then applied an assumption of equal fluorescence efficiency of all sites, which gives ${\varphi }_i$ by the expression

$${\varphi }_i=\frac{i{f}_i}{\sum _{i=0}^{N}i{f}_i}.$$ (3)

The equal fluorescence efficiency assumption states that a species containing i fluorophores is weighted by i (e.g., a tetramer contributes four times as much per molecule as a monomer). Therefore, when Eq. 3 is expanded and Eq. 1 is substituted the number reduces from N to N − 1, yielding

$${\varphi }_i=\left(\begin{array}{l}N-1\\ i-1\end{array}\right){\left(f\right)}^{i-1}{\left(1-f\right)}^{\left(N-1\right)-\left(i-1\right)}.$$ (4)

Consequently, when Eq. 4 is applied to Eq. 2, the average anisotropy of the N species, PX_i, is:

$$r\left(N\right)=r_1+\sum _{i=1}^{N-1}{\left(-1\right)}^i\left(\begin{array}{l}N-1\\ i\end{array}\right){\left(f\right)}^i\times \left[\left(\begin{array}{l}i\\ 0\end{array}\right)r_1-\left(\begin{array}{l}i\\ 1\end{array}\right)r_2+\dots \pm \left(\begin{array}{l}i\\ i\end{array}\right)r_{i+1}\right].$$ (5)

The assumption of equal fluorescence efficiency of all sites allows the equations to be simplified to a reduced expression based on the binomial distribution of order (N − 1).

Yeow and Clayton (8) followed a similar course to provide a more general treatment and applied it to the estimation of protein cluster size distributions by linking the steady-state anisotropy and fractional labeling of the subunits within a protein cluster. The same discrete probability distribution is written to describe the distribution of the fraction of labeled monomers, f, between clusters, F_i:

$$F_i\left(i,f,N\right)=\frac{N!f^i{\left(1-f\right)}^{N-i}}{i!\left(N-i\right)!}.$$ (6)

The observed emission anisotropy of the ensemble, r, is then given by

$$r\left(f,N\right)=\frac{\sum _i^{N}i{F}_i{r}_i}{\sum _{i}i{F}_i},$$ (7)

and Eq. 7 is rewritten as

$$r\left(f,N\right)=A_1{f}^0{\left(1-f\right)}^{N-1}{r}_1+A_{2}f{\left(1-f\right)}^{N-2}{r}_2+\dots +A_N{f}^{N-1}{\left(1-f\right)}^0{r}_N,$$ (8)

where (A₁, A₂, … A_N ) are the elements of the (N − 1) row of Pascal’s triangle, and f is the fraction of labeled subunits. Using this model, a polynomial of order N − 1 describes an N-mer and the N-mers will have a range of 1–N labeled subunits.

Treatment for cases where equal fluorescence efficiency is not valid

Most dyes exhibit intensity quenching, enhancement, or a combination of the two (14,35–40). In these cases, a general but less convenient set of equations can be given by replacing Eq. 3 with one containing a term that accounts for the behavior of the fluorophore in a cluster of a particular size:

$${\varphi }_i=\frac{z_i{f}_i}{\sum _{i=0}^N{z}_i{f}_i},$$ (9)

where z_i is the relative intensity of the cluster PX_i, which contains i fluorophores. If z_i = {1, 2, 3, 4, …, N} (i.e., it follows the progression 1, 2, 3, 4, …, N for i = 1, 2, 3, 4, …, N), then the system follows the assumption of equal fluorescence intensity and reduces as described previously (7,8). Simple models can be used to investigate the general behavior of fluorophores that quench (z_i = {1, 1, 1, 1, …, N}) or enhance (z_i = {1, 4, 9, 16, …, N}) when in close proximity. This family of models can be conveniently represented as power laws in which z(i) = i^p. For example, the equal fluorescence efficiency model is z(i) = i¹; self-quenching might be modeled by z(i) = i⁰ and strong enhancement by z(i) = i². These models are simplistic relative to the wide range of known behaviors of clustered fluorophores, particularly self-quenching or enhanced fluorescence in excimers and exciplexes. They do, however, give clear information about the impact of quenching and enhancement on the predicted anisotropy of stochastic mixtures. In other cases, if the z_i follows a known functional progression z(i), this can be conveniently substituted into the equation. It is worth noting that z_i can take on any value, whereas in Eq. 3 and the simple power law models, the parameters are linked to the index, i. This definition focuses on the behavior of the cluster. A related parameter providing information about the behavior of individual fluorophores in the cluster can be defined by dividing z_i by i (i.e., ζ_i = z_i/i).

In general, the anisotropy of an N-mer cluster with i labeled subunits, where a fraction, f, of the monomers are labeled, is given by

$$r\left(f,N\right)=\frac{\sum _{i=1}^N{A}_{i,N}{z}_i{f}^i{\left(1-f\right)}^{N-i}{r}_i}{\sum _{i=1}^N{A}_{i,N}{z}_i{f}^i{\left(1-f\right)}^{N-i}},$$ (10)

where, A_i,N is the i^th element of the N^th row of Pascal’s triangle indexed such that the first element, 1, is given by A_0,0, and r_i is the anisotropy of the cluster with i fluorophores. If measured values of r_i are available, these should be used. Otherwise, as in the case for Eq. 2, the considerations of Runnels and Scarlatta can be applied (18). The index begins at 1 due to the fact that the unlabeled fraction does not contribute directly to the anisotropy.

The challenge working with these expressions is recovering values of z_i in the absence of a known equation that predicts z_i values or the application of simplifying assumptions. One approach is to measure fluorescence intensity while titrating a fluorescent ligand, X, with a clustering agent or protein, P, over a range of fractional labeling. Total fluorophore concentration should be held constant while investigating over the range [P] = N[X] to [P] ≫> N[X]. Let I be normalized intensities of solutions at each fractional labeling (f) value. A system of equations can be set up that allow z_i to be estimated. In general,

$$I\left(f,N\right)=\sum _{i=1}^N{A}_{i,Ni}{z}_i{f}^i{\left(1-f\right)}^{N-i}.$$ (11)

This approach requires knowledge of f and N and assumes that each labeling site is independent of the others. Other approaches might be to construct the individual species forming a mixture (41) or to use synthetic biology approaches (42). Once z_i is known, anisotropy can be computed.

Materials and Methods

Model system

BSA was labeled with FITC according to previously reported methods (43). Briefly, BSA (Sigma-Aldrich, St. Louis, MO) dissolved in pH 9.0 carbonate-bicarbonate buffer (Sigma-Aldrich) reacted with FITC (Thermo-Fisher Scientific, Waltham, MA) in a darkened lab. The FITC was first dissolved in a few drops of dimethylformamide (DMF), then made up to volume in pH 9.0 carbonate-bicarbonate buffer. The mole ratio of FITC to BSA was varied over the range 0.016–70 by changing the amount of FITC added while keeping BSA constant. Samples were left on a shaker to react for 8 h at room temperature in the dark. Free unreacted dye was separated from the reaction mixture by repeated dialysis against 0.01 M buffer containing 0.02% (w/v) sodium azide solution to inhibit bacterial growth. Dialysis membranes (cutoff, 12–14,000 Da; Medicell International, London, United Kingdom) were prepared according to standard methods and immediately used. Dialysis was considered complete when free FITC fluorescence in the outer solution was no longer detectable. The extent of fluorophore conjugation with the protein in each reaction was monitored by absorbance (V530; Jasco, Tokyo, Japan) in the regions corresponding to protein and dyes (279 nm for BSA, 495 nm for FITC).

To verify the F/P ratio in FITC/BSA, the method of normalized integrated absorbance described by Matveeva et al. was used to correct for spectral shift on dye binding (44). These corrections are needed in systems with high labeling ratios due to changes in the absorbance characteristics in free and conjugated FITC. After correction, the F/P ratio in prepared BSA samples was computed according to standard methods (45). Errors in this system were evaluated by replicate measurements of the F/P ratio. With this method, reproducibility was excellent at low F/P, but less so at high values due to the high absorbance of the FITC moiety at the wavelength used to assess protein concentration. Error bars in this measurement represent twice the uncertainty in the replicate data sets, expressed as a percentage.

Fluorescence measurements

Fluorescence spectra and steady-state emission anisotropies of all BSA samples were measured in a fluorometer equipped with removable polarizers (Cary Eclipse; Varian, Palo Alto, CA). The excitation and emission bandwidths were both set to 5 nm. To avoid inconsistency in the polarizer settings across the set of measurements, the parallel orientation of all the samples was measured before moving the polarizers and making perpendicular measurements. BSA anisotropy measurements were measured at the emission maximum and are shown by vertical error bars representing ±2 SD for five repeated measurements of anisotropy.

Results and Discussion

Theoretical predictions

To compare the conventional predictions (z(i) = i¹) to cases where z_i values follow progressions other than that expected from the assumption of equal fluorescence efficiency, the behavior of a trimer versus a tetramer and a 23-mer versus a 24-mer were investigated using simple power law models (Fig. 1). Although these models are simplistic, they illustrate general trends for cases where interactions between clustered fluorophores result in brightness changes. Under these conditions, as the fractional labeling increased, the predictions diverged with the quenched models consistently yielding higher anisotropy than either z(i) = i or z(i) = i². For the tetramer, the maximum difference in predicted anisotropy, Δr, between the two conditions (p = 0 and p = 1) was 0.048, with the models converging as f approached 0 and 1. This Δr is greater than what is expected going from a cluster with N = 3 to one with N = 4 in a saturated cluster (Δr = 0.033). The underprediction of anisotropy when the assumption of equal efficiency is applied to a system that exhibits strong self-quenching will result in an underprediction of cluster size. In a similar way, in cases where the clustering of fluorophores results in enhanced brightness, the conventional assumptions will overpredict anisotropy and overestimate cluster size.

Figure 1.

Predicted anisotropy for different fluorophore interaction terms (z_i) in stochastic mixtures in which each molecule within the mixture follows the rules of Runnels and Scarlatta and R < 0.8R₀. (a and b) In the graphs with N = 3 (dashed lines) and N = 4 (solid lines) (a) and N = 23 (dashed lines) and N = 24 (solid lines) (b), all models of fluorophore interaction converge in the limits of high (100%) and low (0%) fractional labeling. In both sets of data, the variations in anisotropy due to model assumptions exceed the difference expected due to changing the cluster size by one. The lowest anisotropy for each cluster size is given by 0.4/N in these simulations. The interaction models shown are for power laws (z(i) = i^p) with values of p indicated next to the corresponding curves. The curves labeled with 1 correspond to the equal fluorescence efficiency model. (c and d) Illustrations of the differences due to the cluster size changing by one. The quenched systems show greater resolving power based on anisotropy difference than do either the enhanced model or the equal fluorescence efficiency model. Note that the y axes scale with r₀ (e.g., using r₀ = 0.32 leads to a 20% reduction relative to r₀ = 0.4).

The simulations (Fig. 1) demonstrate a number of notable features. All fluorophore interaction models converge when f = 1. In the N = 24 system, when f > 0.60, the different interaction models are essentially indistinguishable. This might suggest that for best results f should be as large as possible to minimize fluorophore interaction effects on the outcome of an experiment. This intuition is incorrect if the purpose of the experiment is to discriminate cluster size. In both simulations, when p ≤ 1, the maximal Δr associated with a cluster size change of 1 did not occur when f = 1. For N changing from 3 to 4 and using the assumption of equal fluorescence efficiency (p = 1), the maximal Δr (0.035) was observed when f was near 0.5. In a quenched system (p = 0), the maximal Δr was 0.04 when f was near 0.6 (Fig. 1 c). In a similar way, when changing from N = 23 to N = 24 and p = 1, the maximal Δr (0.007) was observed when f was near 0.15. In a quenched system (p = 0), the maximal Δr was 0.01 when f was near 0.19 (Fig. 1 d). Although this Δr (0.01) remains challenging experimentally, it is within the capabilities of an instrument that can read to ±0.003 and is over an order of magnitude larger than the expected difference (0.0007) when f = 1. These results suggest that by carefully engineering fluorophore interaction and fractional labeling, larger clusters can be assessed using polarization methods.

To better understand the impact of fluorophore interaction on the anisotropy of mixtures of clusters, Δr was computed for p = 1 (equal fluorescence efficiency) versus p = 0 (quenched) for cluster sizes over the range N = 2–26. The models differed systematically, with the magnitude of Δr and the fractional labeling corresponding to the maximum difference changing with cluster size (Fig. 2). As the cluster size increased, the fractional labeling at maximum Δr approached 0 and the magnitude of Δr increased. For example, in a dimer, taking z_i = 1 yields an anisotropy that is 0.034 higher at a fractional labeling of 0.59. For a decamer, Δr is expected to be 0.055 higher at a fractional labeling of 0.146. Extrapolation of a reciprocal plot (not shown) indicates that the maximum Δr is 0.06 at the limit of infinite cluster size.

Figure 2.

Difference between p = 1 and p = 0 as a function of cluster size and fractional labeling. The predictions include clusters made up of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, and 26 units. The maximum difference increases with the cluster size but has an impact over a wider range of fractional labeling in the smaller clusters.

For all cases, the anisotropy predicted for the simple quenched model was lower than predicted using the assumption of equal fluorescence efficiency. Consequently, applying this assumption will systematically underestimate cluster sizes when fluorophores exhibit reduced brightness in clusters.

Model system verification

Assembled fluorescent proteins along a DNA template (N = 4)

Recently, a system for template-directed assembly of monomeric teal fluorescent protein (mTFP) was described (42). This system consisted of mTFP fused to a short peptide nucleic acid (PNA) sequence and assembled along a complementary DNA strand with regular repeats. By titrating mTFP-PNA with DNA containing four repeats of the complementary sequence, the behavior of a tetrameric system over a range of fractional labeling values was studied. Intensity measurements indicate that the assumption of equal fluorescence intensity did not hold (Fig. 3). A solution with a 4:1 mTFP/DNA template was 1.82 times brighter than mTFP in solution, indicating some self-quenching. If the p = 1 assumption held, it would be expected to be four times brighter. The progression of intensities during the titration was well approximated by an empirical self-quenching model in which z(i) = 1.24ⁱ⁻¹ when normalized to the monomeric species. The equal fluorescence efficiency model did not give a reasonable prediction of the progression of intensities.

Figure 3.

Normalized fluorescence intensity in a tetrameric self-assembling system of fluorophores. The dashed line represents the behavior predicted according to the assumption of equal fluorescence intensity (z(i) = i¹), and the solid line is an empirical fit of z(i) = 1.24ⁱ⁻¹.

Using previously published data, the trajectory of anisotropy with f was compared (Fig. 4) to models including equal fluorescence efficiency and the empirical quenching model from Fig. 3. To analyze this existing data set, an assumption was made that only systems of adjacent mTFPs undergo homo-FRET and that in the absence of homo-FRET, the assumption of equal fluorescence efficiency holds. When computing the anisotropy weighted average (Scheme 1), the equal fluorescence efficiency model gives an r₂ consisting of 50.0% monomer and 50.0% dimer and an r₃ consisting of 16.7% monomer, 33.3% dimer, and 50.0% trimer. In a similar way, the quenching model gives an r₂ composed of 61.7% monomer and 38.3% dimer and an r₃ of 26.4% monomer, 32.8% dimer, and 40.7% trimer. Based on an F-test, the data reported for the anisotropy in this tetrameric system conform significantly better (p < 0.05) to the quenching model than to the equal fluorescence efficiency model. The latter model is presented for comparison purposes only. Except for purposes of illustration, based on the data in Fig. 3, there was no justification for invoking the equal fluorescence efficiency model.

Figure 4.

Anisotropy behavior of tetrameric template-directed assembly of mTFP-PNA by DNA. Previously reported data (black squares) (42) are shown in comparison to equal fluorescence efficiency (dashed line) and empirical quenching (solid line) models. Corrections have been made to account for forms in which fluorophores are located outside of 0.8 R/R₀ (see text for details).

Scheme 1.

Illustration of all possible species for a PX₄ (tetrameric) system using PNA-linked mTFPs assembled on a DNA template with four repeats. Sixteen (2⁴) possibilities exist, and when f = 0.5, the relative amounts of PX₀, PX₁, PX₂, PX₃, and PX₄ are 1:4:6:4:1. For the PX₃ subpopulation, there are two cases with three adjacent mTFPs and two cases in which two adjacent mTFPs are separated from an isolated mTFP by a gap of one repeat unit. Assuming that only adjacent mTFPs transfer energy, the anisotropy (r₃) of this ensemble will be a weighted average of the elements composing it. For example, in the case of equal fluorescence intensity, the relative contributions of the species r₁/3, r₁/2, and r₁ to the ensemble composing r₃ will be 6:4:2 (50%, 33.3%, and 16.7%), respectively. To see this figure in color, go online.

BSA labeled with FITC (N = 24)

To study the impact of fluorophore interaction on larger clusters, exhaustive labeling of BSA was studied. BSA has 60 lysine residues, with approximately half buried in the interior and additional residues in hydrophobic pockets, leaving a subset to react (46,47). Dye binding to BSA has been studied previously (37,44,48). Estimates of the F/P ratio for BSA under potentially saturating levels of FITC and other similarly reactive dyes vary widely in the literature and reflect the conditions used in individual laboratories (35,43,49). The maximal F/P ratio under our conditions was determined by varying the ratio of FITC to protein in the reaction mixture while keeping the amount of protein and the reaction time constant (Fig. 5). The saturation of FITC reactive sites on the surface of BSA was treated using a Langmuir-type binding model. F/P_max was found to be 24 FITC/BSA and the good correspondence to the Langmuir model suggests that all FITC reactive sites on BSA reacted independently. The F/P_max for the FITC/BSA system fell within the range of 15–25 reported for similar experiments with FITC (35,43) and similar amine reactive dyes (49). F/P_max = 24 was used subsequently to scale the fractional labeling of BSA.

Figure 5.

Spectroscopically determined F/P ratio with increasing mole ratio of FITC during reaction. Data (black squares) were measured up to a mole ratio of 70. A Langmuir-type fit (solid line) indicated a maximum F/P ratio of 24.

Examination of the normalized intensity of a series of labeled BSA samples revealed a clear maximum followed by reduced intensity as f increased (Fig. 6). The equal fluorescence efficiency (z(i) = i¹) and simplified self-quenching (z(i) = i⁰) models were computed and compared to the data. At the highest F/P ratios, FITC was highly quenched, in agreement with previous reports (cf. Voss et al. (35)). It is notable that the progression of intensity does not follow any of the simple power law models (Fig. 6) and clearly does not conform to the assumption of equal fluorescence efficiency. There is an initial rise in intensity per BSA for f between 0 and 0.1. In this system, when f reaches 0.1, molecules with three or fewer FITCs account for 77% of the total and dominate the fluorescence. The fluorescence per BSA dropped over the range f = 0.1–0.5. For f > 0.5 (samples dominated by BSA molecules with >10 FITCs attached), the normalized intensity was essentially constant. Similar quenching-induced behavior that deviates from equal fluorescence efficiency has been observed in a wide range of fluorescent dyes, including: FITC (14,35,36,50,51), Cy3 (37), Cy5 (37,38), Cy7 (37), Alexa 488 (36), Alexa 532 (36), Alexa 546 (36), Alexa 594 (36), and Alexa 647 (38). Other dye systems that exhibit enhancement or more complex deviations from equal fluorescence efficiency include: pyrene (14,39), terphenyl (14), terthiophene (14), a wide range of polyaromatic hydrocarbon moieties (40,52), and other compounds (53). In a similar way, the high levels of quenching as f approaches 1 are consistent with previous data (35).

Figure 6.

Normalized fluorescence intensity of BSA solutions as fractional labeling is increased. The fluorescence was measured on a set of solutions with Abs₄₉₄ = 0.047 ± 0.04 and corrected for the amount of BSA present. The dashed line is based on the assumption of equal fluorescence intensity (z(i) = i¹). The dotted line is for a simple quenching model (z(i) = i⁰). The solid line is a semiempirical fit primarily to guide the eye.

The general trend in this data was ascribed to two processes: trap sites within the FITC population and energy transfer. The self-quenching of FITC is complex. It is known that proteins with many FITCs attached are heavily quenched, but these recover in close proximity to a metal surface (50,51). However, to our knowledge, this type of behavior in FITC-labeled proteins (and proteins labeled with many other dyes) has been reported for stochastic mixtures rather than for proteins labeled with a specific number of fluorophores (cf. (35,36–38,50,51)). As a result, the behavior of the molecules with specific numbers of fluorophores is not known.

In a similar way, energy transfer in this system is complex. In the FITC-BSA system, R₀ will shift to lower values as the number of FITCs attached increases due to the known spectral shifts observed in solution. Due to the random labeling there will always be a wide range of distances. BSA is well modeled in solution as a triangular prismatic shape with dimensions 8.4 × 8.4 × 8.4 × 3.2 nm (54). Simplifying this to a globular protein 8 nm in diameter (55) and assuming that the 24 dyes evenly distribute over the surface, doubly labeled BSA will contain molecules in which the inter-FITC distance is between 2.9 and 7.9 nm, which brackets R₀ for homo-FRET. Further, over the full range of possible labeled states, there are 2²⁴ species, and simulating all of these is a challenging problem. In general, as the number of labels increases, a concomitant increase in the density of energy transfer partners in close proximity occurs. Since some of those energy transfer partners will be trap sites, the brightness of the FITC-labeled BSA will decrease. Based on these prior reports and invoking trap sites and energy transfer, the general features in Fig. 6 can be rationalized, but a full quantitative description is beyond the scope of this study.

In the context of interpreting clustering using anisotropy, the extent to which the data do not follow the assumption of equal fluorescence efficiency is striking. Based on inspection of Fig. 6, z_i appears to follow a model in which z_i ≈ 1 due to self-quenching processes over a limited range (f < 0.1). For f > 0.1, the applicability of all of the simple models was limited.

Several approaches to estimating the values of z_i were attempted. Unconstrained fitting of our results to Eqs. 10 and 11 is difficult due to the limited data, the smoothness of the functions involved, and the large number of similarly valued parameters. Computed distributions for f > 0.5 indicated that species with i ≥ 10 dominated these solutions. Since these solutions showed no change in normalized intensity, the z_i were assumed to be constant when i ≥ 10. Unconstrained fitting of the remaining parameters was still unsatisfactory. Neighboring z_is differed greatly. A semiempirical approach was adopted in which the z_i parameters were generated by a fourth-degree polynomial over the range i = 1–10, and were constant afterward. These conditions require the parameters to be smooth and reasonably continuous. The resulting fit can only be said to be better than existing assumptions. Although useful for guiding the eye, any parameters obtained are approximate. Further work is needed to develop robust estimates of these values for large aggregates and clusters.

The same models were applied to the prediction of measured fluorescence anisotropy (Fig. 7). Studies of the depolarization behavior of FITC/BSA as a function of labeling have been reported (43,55,56), but explicit comparisons to theoretical predictions have not, to our knowledge, appeared previously. Although our data cover a wider range of F/P than do those of previous studies, the measured anisotropy of the FITC/BSA preparations was consistent with previously reported trends showing lower measured anisotropy as F/P increases (43,55,56). The observed depolarization is due to FITC-FITC homo-FRET for which the R₀ is 4.4 nm (57). The anisotropy predicted based on the assumption that r_i = r₁/i in the equal fluorescence efficiency, simple quenching, and semi-empirical models was compared to the measured data. At low values of f, the observed anisotropy is somewhat lower than expected due to the limitations of the assumption (r_i = r₁/i). At more experimentally realistic fractional labeling (f > 0.05), this assumption appears to hold well. It is clear that the results do not correspond to an equal fluorescence intensity model, and based on Fig. 6, there would normally be no reason to invoke it. The simple self-quenching model gave a good prediction of the observed results and, in this instance, the greatly increased complexity of the semiempirical model yielded anisotropy predictions indistinguishable from the simple self-quenching model. The equal fluorescence intensity model underpredicted the anisotropy and if used interpretively would underestimate the number of clustered fluorophores.

Figure 7.

Fluorescence anisotropy of BSA solutions as fractional labeling is increased. The dashed line is the behavior of a system predicted according to the assumption of equal fluorescence intensity (z(i) = i¹). The dotted line represents the simple quenching model (z(i) = i⁰). The solid line is a semiempirical fit primarily to guide the eye.

Conclusion

Studies of aggregate formation using fluorescence anisotropy are becoming increasingly important in the investigation of a variety of biophysical phenomena. Existing widely applied assumptions lead to underestimation of both the predicted anisotropy and the cluster size when interpreting data from fluorophore systems that exhibit self-quenching. The theory presented here emphasizes systems restricted to randomly oriented fluorophores where the interfluorophore distance is <0.8 R/R₀. Under these conditions, anisotropy may be readily predicted using models that exhibit enhancement, equal fluorescence, or quenching. The theory, however, can be applied to other cases by invoking the more detailed expressions for r_i provided by Runnels and Scarlatta (18). For the systems studied, once the broad behavior is known, anisotropy can be readily predicted. Variations in anisotropy with f are more easily predicted than is intensity. As noted by earlier workers, the interpretation of anisotropy is greatly enhanced by the availability of complementary data (8). More detailed knowledge of the photophysical behavior of fluorophores in clusters will greatly enhance interpretation of cluster size using anisotropy. In particular, the theoretical considerations provided here indicate that discrimination of cluster size is improved by unsaturated subunits (e.g., f ≠ 1) and by fluorophores that quench. It is also likely that more detailed understanding of different types of clustering is needed; some types grow by adding subunits such that the distance to the next fluorophore remains nearly constant, whereas others remain constant in size, so that the density of transfer partners increases. The first of these is represented by the mTFP model system and the second by BSA.

We anticipate that these considerations will be essential both in engineering optimal fluorophores for the study of clustering and in future studies using anisotropy to probe receptor activation, plaque formation, and related processes underlying disease states. In particular, these results could allow for additional development of in vivo methods involving imaging and anisotropy (26,58–61).