Utility of FRET in studies of membrane protein oligomerization: The concept of the effective dissociation constant

Abstract

The activity of many membrane receptors is controlled through their lateral association into dimers or higher-order oligomers. Although Förster resonance energy transfer (FRET) measurements have been used extensively to characterize the stability of receptor dimers, the utility of FRET in studies of larger oligomers has been limited. Here we introduce an effective equilibrium dissociation constant that can be extracted from FRET measurements for EphA2, a receptor tyrosine kinase (RTK) known to form active oligomers of heterogeneous distributions in response to its ligand ephrinA1-Fc. The newly introduced effective equilibrium dissociation constant has a well-defined physical meaning and biological significance. It denotes the receptor concentration for which half of the receptors are monomeric and inactive, and the other half are associated into oligomers and are active, irrespective of the exact oligomer size. This work introduces a new dimension to the utility of FRET in studies of membrane receptor association and signaling in the plasma membrane.

Significance

Förster resonance energy transfer (FRET) has been extensively used to study the dimerization of membrane proteins in the plasma membrane. However, previous work has shown that FRET experiments cannot discern the exact oligomer size when it is larger than a dimer. Here we show that we can extract an effective equilibrium constant from FRET datasets, even when the exact oligomer size is unknown. The implication of this work is that FRET, along with fluorescence fluctuation techniques that directly measure the oligomer size, can be a very powerful tool in studies of membrane protein dynamics and oligomerization in the plasma membrane.

Introduction

Membrane proteins are abundant in eukaryotes and account for 20%–30% of the open reading frames (1). They play key roles in cell signaling, cell adhesion, recognition, motility, energy production, and transport of nutrients (2,3,4,5,6,7,8). Just as in the case of soluble proteins, the function of membrane proteins is often regulated through their homointeractions or through heterointeractions with partner proteins (9,10,11,12,13). While soluble proteins are typically studied in purified form using well-established quantitative methods, membrane proteins are fickle and easily lose their activity once extracted out of the native membranes they reside in (14). Thus, often the only viable option is to study them in the context of the complex membrane milieu (15,16,17,18,19,20,21). Because of such restrictions, knowledge of the folding, structure, and function of membrane proteins has been slow to emerge.

Studies of the second-largest class of membrane receptors, the receptor tyrosine kinases (RTKs), exemplify challenges and limitations in membrane protein research. RTKs are single-pass transmembrane proteins that control cell growth, differentiation, motility, and metabolism (22,23). They play profound roles in human development and are strongly implicated in disease. Their N-terminal extracellular (EC) regions, composed of characteristic arrays of structural domains, bind the activating ligands (24). They have single transmembrane helices and intracellular kinase domains. Despite their simple architecture and their significance for human health, there are currently no high-resolution structures of any of the 58 full-length RTKs, and there is no detailed mechanistic understanding of their activation (25). What is well known is that the function of RTKs is regulated through their self-association in the membrane. RTK monomers are inactive, whereas RTK dimers/oligomers are active, as kinases in close proximity cross-phosphorylate each other by acting as both enzymes and substrates (12,26,27,28,29). This cross-phosphorylation stimulates catalytic activity, resulting in the phosphorylation of cytoplasmic substrates and downstream signaling (12,30,31,32).

For many years, the activation of RTKs was believed to follow the simple model of ligand-induced dimerization (24,33). Later, it was shown that many RTKs dimerize even in the absence of ligand and that ligand binding stabilizes these dimers (13,34). Dissociation constants in the absence and in the presence of high saturating ligand concentrations have been measured using Förster resonance energy transfer (FRET). In these experiments, RTK expressions are varied over a wide range, and RTK concentrations in the plasma membrane are measured in hundreds of cells, along with FRET efficiencies, yielding dimerization curves and dissociation constants (46).

Recent work has revealed that some RTKs signal as oligomers that are larger than dimers (35,36,37,38). In such cases, interpretation of FRET data is more challenging (39). Here, we introduce the concept of the “effective dissociation constant,” which allows us to understand and predict the self-association, and therefore the activities, of RTKs even when their association state is poorly defined. We assess the utility of the concept using the receptor EphA2, which plays an important role in cell guidance during development and has been implicated in many cancers. EphA2 oligomerization has been characterized using fluorescence fluctuation methods such as Pulsed Interleaved Excitation Fluorescence Cross-Correlation Spectroscopy (PIE-FCCS), Number and Brightness (N&B), and Fluorescence Intensity Fluctuation (FIF) Spectroscopy (20,35,40,41). These methods can directly report on the distribution of oligomer sizes but have had limited use in determining oligomer stability (42,43). Prior fluorescence fluctuation studies have revealed that the size distributions of the ligand-bound EphA2 oligomers is heterogeneous, with an average oligomer size of ∼4 (44). In this work we measure FRET for EphA2 in the presence of its ligand ephrinA1-Fc, and we calculate the effective dissociation constant.

Materials and methods

Sample preparation

The EphA2 plasmid in the pcDNA3.1(+) vector was cloned in prior work (45). The plasmid encodes for human EphA2 tagged at the C terminus with a fluorescent protein (either eYFP or mTurquoise) via a 15-amino-acid GGS₅ linker (45).

HEK293T cells were purchased from American Type Culture Collection (Manassas, VA, USA). The cells were cultured in Dulbecco’s modified Eagle’s medium (Gibco, #31600034) supplemented with 10% fetal bovine serum (HyClone, #SH30070.03), 20 mM D-glucose and 18 mM sodium bicarbonate at 37°C in a 5% CO₂ environment. Twenty-four hours before transfection, cells were seeded in 35-mm glass coverslip, collagen-coated Petri dishes (MatTek, P35GCOL-1.5-14-C) at a density of 2.5 × 10⁵ cells per dish to reach ∼70% confluency at the day of the experiment. For transfection, Lipofectamine 3000 (Invitrogen, #L3000008) was used according to the manufacturer’s protocol. Single transfections were performed using 1–3 μg of plasmid DNA. Co-transfections were performed with 1–4 μg of total plasmid DNA in a 1:3 donor:acceptor ratio. Twelve hours after transfection, the cells were rinsed twice with phenol-red-free, serum-free starvation medium and then serum starved for at least 12 h. The starvation medium was supplemented with 0.1% BSA to coat the wall of the dishes.

FRET imaging and data analysis

Before imaging, HEK 293T cells were subjected to reversible osmotic stress by replacing the serum-free medium with a 37°C, 1:9 serum-free medium:deionized H₂O, 25 mM HEPES solution. In cells, the plasma membrane is normally highly ruffled and its topology in microscope images is virtually unknown (46). The reversible osmotic stress eliminates these wrinkles and allows the conversion of effective 3D protein concentrations into 2D receptor concentrations (46). The swelling medium was supplemented with 50 nM dimeric ephrinA1-Fc (R&D Systems, #602-A1-200). The cells were allowed to equilibrate for 10 min at room temperature. Images of cells were acquired using a two-photon microscope equipped with the OptiMiS spectral imaging system (Aurora Spectral Technologies). Two scans were performed for each cell– a scan at ${\lambda }_1=$ 840 nm in which the donor fluorophore (mTurquoise) is primarily excited and a scan at ${\lambda }_2=$ 960 nm where the acceptor fluorophore (eYFP) is primarily excited. These two scans are referred to as the FRET and acceptor scan, respectively. Each scan produces an image of 300 × 440 pixels, where every pixel contains a full emission spectrum in the range of 420–620 nm. Each dish was imaged for up to 2 h.

Spectra for each pixel acquired at ${\lambda }_1=$ 840 nm and ${\lambda }_2=$ 960 nm were unmixed into donor and acceptor components as described (46). Single transfections of either donor- or acceptor-tagged EphA2 were used as controls to acquire emission spectra of the donor only and the acceptor only and serve as a basis for the unmixing. These control spectra were averaged over many pixels and smoothed over the emission wavelengths (46) to produce the basis spectra $F_{basis}^{\mathrm{D}}$ and $F_{basis}^{\mathrm{A}}$ for the donor and acceptor. The FRET emission spectra, ${F\left(\lambda \right)}_{{\lambda }_1,{\lambda }_2}^{pixel}$, are assumed to be a linear sum of three contributions: the fluorescence of the donor in the presence of the acceptor $F^{\text{DA}}\left(\lambda \right)$ (with the spectral features of $F_{basis}^{\mathrm{D}}$), the fluorescence of the acceptor in the presence of the donor $F^{\text{AD}}\left(\lambda \right)$ (with the spectral features of $F_{basis}^{\mathrm{A}}$), and a background contribution. The unmixing of ${F\left(\lambda \right)}_{{\lambda }_1,{\lambda }_2}^{pixel}$ per each pixel was performed using linear least-squares optimization in MATLAB, as described in detail in (46). The unmixed spectra are then integrated to yield the cumulative fluorescence intensities.

Solution standards of soluble mTurquoise and eYFP were imaged to calculate concentrations from the integrated fluorescence intensities. Four standards for each fluorphore with known concentrations were imaged at both 840 and 960 nm, and a line of slope i was fit to the integrated intensity versus concentration data for every pixel of the image (46). The four slopes, $i_{\mathrm{D},\lambda 1}$, $i_{\mathrm{D},\lambda 2}$, $i_{\mathrm{A},\lambda 1}$, and $i_{\mathrm{D},\lambda 2}$, are calculated for every pixel, producing a fit for each fluorophore at both wavelengths.

From these experiments, we calculate 1) the donor concentration [D], 2) the acceptor concentration [A], and 3) the FRET efficiencies in the plasma membrane of each individual cell using the following equations:

$$E_{app}=1-\frac{F_{{\lambda }_1}^{\text{DA}}}{F_{{\lambda }_1}^{\mathrm{D}}}$$ (1)

$$\left[\mathrm{D}\right]=\frac{F_{{\lambda }_1}^{\mathrm{D}}}{i_{\mathrm{D},{\lambda }_1}}=\frac{F_{{\lambda }_1}^{\text{DA}}}{i_{\mathrm{D},{\lambda }_1}}+\frac{Q^{\mathrm{D}}}{Q^{\mathrm{A}}}\left(F_{{\lambda }_1}^{\text{AD}}-\frac{i_{\mathrm{A},{\lambda }_1}}{i_{\mathrm{A},{\lambda }_2}}{F}_{{\lambda }_2}^{\mathrm{A}}\right)$$ (2)

$$\left[\mathrm{A}\right]=\frac{F_{{\lambda }_2}^{\mathrm{A}}}{i_{\mathrm{A},{\lambda }_2}}=\frac{1}{i_{\mathrm{A}},{\lambda }_2}\left(F_{{\lambda }_2}^{\text{AD}}-\frac{i_{\mathrm{D},{\lambda }_2}}{i_{\mathrm{D},{\lambda }_1}}{F}_{{\lambda }_1}^{AD}\right)·{\left(1-\frac{i_{\mathrm{A},{\lambda }_1}}{i_{\mathrm{A},{\lambda }_2}}\frac{i_{\mathrm{D},{\lambda }_2}}{i_{\mathrm{D},{\lambda }_1}}\right)}^{-1}$$ (3)

In these equations, $E_{app}$ is the measured FRET efficiency, $F_{{\lambda }_{\mathrm{1,2}}}^{\text{DA}}$ is the total fluorescence of the donor or acceptor in the absence of FRET for excitation at ${\lambda }_1$ or ${\lambda }_2$, $F_{{\lambda }_1}^{\text{DA}}$ is the measured fluorescence of the donor in the presence of acceptors, and $F_{{\lambda }_2}^{\text{DA}}$ is the measured fluorescence of the acceptor, which is augmented due to FRET. $Q^{\mathrm{D}}$ and $Q^{\mathrm{A}}$ are the quantum yields of the donor and acceptor, respectively. To determine [A] and [D] as two-dimensional concentrations in the membrane, the pixel-level fluorescence intensities for $F^{\mathrm{D}}$, $F^{\mathrm{A}}$, and $F^{\text{AD}}$ are integrated over a membrane region (Reg) chosen by the researcher. The average integrated fluorescence per unit length of the membrane is the ratio of the integrated fluorescence of the region, $F_{{\lambda }_1,Reg}^{\text{DA}}$, $F_{{\lambda }_1,Reg}^{\mathrm{D}}$, or $F_{{\lambda }_2,Reg}^{\mathrm{A}}$, and the arc length of the region. The effective 2D concentration are calculated from the 3D concentration by multiplying the mean integrated fluorescence by the pixel width (46).

The measured FRET, $E_{app}$, has two contributions: $E_{oligomer}$, due to receptor dimerization/oligomerization, and $E_{prox}$, the “proximity FRET,” which occurs when a donor and an acceptor are randomly within 100 Å of each other in the absence of specific interactions (39,47), due to the confinement of the fluorophores to the two-dimensional membrane. The contribution due to oligomerization, $E_{oligomer}\text{,}$ can be calculated from the measured FRET efficiency, $E_{app}$, according to (39):

$$E_{oligomer}=\frac{E_{app}-E_{prox}}{1-2E_{prox}+E_{app}{E}_{prox}}$$ (4)

$E_{oligomer}$ can be written as (48):

$$E_{oligomer}=\frac{f_{oligomer}}{x_{\mathrm{D}}}\sum _{k=1}^{n-1}\frac{k\left(n-k\right)\tilde{E}}{1+\left(n-k-1\right)\tilde{E}}\left(\begin{array}{c}n\\ k\end{array}\right)x_{\mathrm{D}}^k{x}_{\mathrm{A}}^{n-k}$$ (5)

where $n$ is the oligomer order, $x_{\mathrm{D}}\&x_{\mathrm{A}}$ are the fraction of donors and acceptors, and $\tilde{E}$ is the “intrinsic FRET,” which depends on the distance between the fluorophores in the oligomer. $f_{oligomer}$ is the oligomeric fraction, which is a function of the total concentration $\left[\mathrm{T}\right]$ and depends on the dissociation constant, $K_D^{oligomer}$ (solid lines in Fig. 1). The dependence of $f_{oligomer}$ on $K_D^{oligomer}$ has been derived previously (46). Briefly, for the derivation we first write $f_{oligomer}$ as

$$f_{oligomer}=1-f_{monomer}=1-\frac{\left[m\right]}{\left[\mathrm{T}\right]}$$ (6)

where [m] is the monomer concentration. Next, we write the mass balance equation as

$$n\left[oligomer\right]+\left[m\right]=\left[\mathrm{T}\right]$$ (7)

Figure 1.

Determination of an effective dissociation constant from FRET data. (A) Single-cell FRET efficiencies for EphA2-mTuquoise (donor) and EphA2-YFP (acceptor). (B) EphA2-mTuquoise concentrations versus EphA2-YFP concentrations in single cells. (C) Fits to oligomer models. Each solid line is the oligomeric fraction curve for the best-fit $K_D^{oligomer}$, where “oligomer” denotes dimer (black), trimer (red), tetramer (blue), pentamer (magenta), and hexamer (green). The symbols are the binned experimental oligomeric fractions, with their standard errors, determined from the FRET efficiencies and the best-fit $\tilde{E}$. Different colors correspond to different oligomer orders, $n$. (D) Comparison of EphA2 effective dissociation constants calculated for different oligomerization models. Shown are calculated values with their standard errors. By ANOVA, there is no statistical significance between the values. To see this figure in color, go online.

Using Eq. 23 Below, Eq. 7 can be written as

$$\frac{n{\left[m\right]}^n}{K_D^{oligomer}}+\left[m\right]=\left[\mathrm{T}\right]$$ (8)

Equation 8 depends on the oligomer order $n$ and cannot be solved analytically for all values of $n$. However, it can be solved numerically as a function of any [T], $K_D^{oligomer}$, and $n$. Specifically, a root finding MATLAB function is used to solve for $\left[m\right]$ and the real positive root is taken:

$$\left[m\right]=\left[m\right]\left(\left[\mathrm{T}\right],K_D^{oligomer},n\right)$$ (9)

By combining Eqs. 5, 6, and 9, we obtain

$$E_{oligomer}=\frac{\left[\mathrm{T}\right]-\left[m\right]\left(\left[\mathrm{T}\right],K_D^{oligomer},n\right)}{\left[\mathrm{T}\right]x_{\mathrm{D}}}\sum _{k=1}^{n-1}\frac{k\left(n-k\right)\tilde{E}}{1+\left(n-k-1\right)\tilde{E}}\left(\begin{array}{c}n\\ k\end{array}\right)x_{\mathrm{D}}^k{x}_{\mathrm{A}}^{n-k}$$ (10)

Equation 10 is used to fit the data for any oligomerization model (any given $n$). $x_{\mathrm{A}}$, $x_{\mathrm{D}}$, $\left[\mathrm{T}\right]$, and $E_{oligomer}$ (after correction for proximity; see below) are measured in the experiment. $K_D^{oligomer}$ and $\tilde{E}$ are the unknowns, and their best-fit values are determined in the fit.

In the case of a dimer$,n=2$, Eq. 8 has an analytical solution:

$$\left[m\right]=\frac{K_D^{dimer}\left(-1+\sqrt{1+\frac{8\left[\mathrm{T}\right]}{K_D^{dimer}}}\right)}{4}$$ (11)

In this case, Eq. 10 is reduced to:

$$E_{dimer}=\frac{x_{\mathrm{A}}\tilde{E}}{\left[\mathrm{T}\right]}\left(\left[\mathrm{T}\right]-\frac{K_D^{dimer}}{4}\left(1-\sqrt{1+\frac{8\left[\mathrm{T}\right]}{K_D^{dimer}}}\right)\right)$$ (12)

Another layer of complexity in FRET data fitting is due to the fact that proximity FRET, $E_{prox}$, depends on the oligomer order, $n$, $K_D^{oligomer}$, and $\tilde{E}$. We thus use a two-step computational approach, verified in (39). Briefly, we utilize a computationally derived library of proximity FRET efficiencies calculated over a grid of association constants, $K_D^{oligomer}$, and $\tilde{E}$ values. In the first step, this proximity FRET library was used to perform a gridded search for the best-fit $K_D^{oligomer}$ and $\tilde{E}$ from the library (39). Because small changes in these parameters have negligible effects on the proximity FRET contribution, the proximity contribution is fixed in the second step and E_oligomer is calculated using Eq. 4. $K_D^{oligomer}$ and $\tilde{E}$ are varied in a MATLAB non-linear least-squares algorithm to determine the best-fit values of $K_D^{oligomer}$ and $\tilde{E}$ and their 68% confidence intervals.

Results

Theory: The effective dissociation constant

The dissociation constant describing protein-ligand interactions

Equilibrium constants describe the relative abundances of reactants and products for a reaction at equilibrium and report on which are favored. One such reaction is the binding of a ligand, L, to a protein, P, to form the complex, LP.

$$L+P\rightleftharpoons LP$$ (13)

The association equilibrium constants $K_A$ is defined as:

$$K_A=\frac{\left[LP\right]}{\left[L\right]\left[P\right]}$$ (14)

where $\left[L\right]$ denotes the free ligand concentration, $\left[P\right]$ is the free protein concentration, and $\left[LP\right]$ is the concentration of the complex. The dissociation constant, $K_D$, is the inverse of $K_A$:

$$K_D=\frac{\left[L\right]\left[P\right]}{\left[LP\right]}$$ (15)

The units for each constant are important to note, with $K_A$ being inverse concentration and $K_D$ being concentration. Of the two equilibrium constants, the usage of $K_D$ is preferred because $K_D$ has units of concentration, and its value can be compared to the reactants concentrations. At low reactant concentrations, the free reactants are favored. At high reactant concentrations, higher than $K_D$, the products are favored.

The dissociation constant has another feature, namely if $\left[LP\right]=\left[P\right]\text{,}$ then $K_D=\left[L\right]$. Thus, $K_D$ is the free ligand concentration at which the fraction of bound protein is 0.5 (50%). Often, $K_D$ is approximated as the total ligand concentration for which the bound protein fraction is 0.5. This approximation is made when the free ligand concentration is unknown, but is not always valid, depending on the experimental design.

Dissociation constants describing receptor self-association

The simplest case of homointeractions is dimerization, described by the following reaction scheme:

$$m+m\rightleftharpoons dimer$$ (16)

where $m$ denotes the monomer. The dissociation constant is

$$K_D^{dimer}=\frac{{\left[m\right]}^2}{\left[dimer\right]}$$ (17)

Brackets indicate the concentrations of monomers and dimers. $K_D$ in Eq. 17 has similar biological significance as in Eq. 15 and has the same units, concentration. However, since the interactions occur in the two-dimensional membrane, the units are receptors per unit area. The fraction of dimeric receptors is given by:

$$f_{dimer}=\frac{2\left[dimer\right]}{\left[\mathrm{T}\right]}$$ (18)

Where [T] is the total receptor concentration, $\left[m\right]+2\left[dimer\right]$. Substitution of Eq. 17 into Eq. 18 yields

$$f_{dimer}=\frac{2{\left[m\right]}^2}{K_D^{dimer}\left[\mathrm{T}\right]}$$ (19)

Now we write Eq. 19 for the specific total receptor concentration, $\left[{\mathrm{T}}^{\ast }\right]$, for which 50% of the receptors are dimeric and 50% are monomeric. For this particular concentration, $\left[{\mathrm{T}}^{\ast }\right]=2\left[m^{\ast }\right]$ and $f_{dimer}=f_{monomer}=0.5$. At this particular concentration,

$$f_{dimer}=\frac{1}{2}=\frac{\left[{\mathrm{T}}^{\ast }\right]}{2K_D^{dimer}}$$ (20)

Therefore,

$$K_D^{dimer}=\left[{\mathrm{T}}^{\ast }\right]$$ (21)

Equation 21 is the condition under which Eq. 20 is satisfied. We thus see that, when half of the receptors are in the dimeric state, the total receptor concentration, $\left[{\mathrm{T}}^{\ast }\right]$, is equal to the dissociation constant, $K_D^{dimer}$. Unlike in the case of ligand binding to a protein discussed above, there are no assumptions, and this relation is always exact.

The meaning of the dissociation constant becomes more nebulous if the receptors associate into higher-order oligomers. A monomer-oligomer model, given by the following reaction scheme, can be used to demonstrate this.

$$n\left[m\right]\rightleftharpoons \left[oligomer\right]$$ (22)

where $n$ is the oligomer order. The dissociation constant for this reaction can be defined as

$$K_D^{oligomer}=\frac{{\left[m\right]}^n}{\left[oligomer\right]}$$ (23)

The units of $K_D$ in Eq. 23 are not receptors per unit area but instead (receptors/μm²)ⁿ⁻¹. Thus, the physical-chemical meaning of $K_D^{oligomer}$ is not immediately obvious. In an analogy to the dimer case, we can write the oligomeric fraction as

$$f_{oligomer}=\frac{n\left[oligomer\right]}{\left[\mathrm{T}\right]}$$ (24)

We solve Eq. 23 for the oligomer concentration and we substitute it into Eq. 24 to obtain

$$f_{oligomer}=\frac{n{\left[m\right]}^n}{K_D^{oligomer}\left[\mathrm{T}\right]}$$ (25)

Now we write Eq. 25 for the specific receptor concentration, $\left[{\mathrm{T}}^{\ast }\right]$, when $\left[{\mathrm{T}}^{\ast }\right]=2\left[m^{\ast }\right]$ and $f_{oligomer}=f_{monomer}=0.5$. This is the concentration at which half of the receptors are associated into oligomers and half are not. For that particular concentration, the following equation holds:

$$\frac{1}{2}=\frac{n{\left[{\mathrm{T}}^{\ast }\right]}^{n-1}}{2^n{K}_D^{oligomer}}$$ (26)

Now we solve Eq. 26 for $\left[{\mathrm{T}}^{\ast }\right]$ and we define the effective dissociation constant as $K_{D_{eff}}^{oligomer}=\left[{\mathrm{T}}^{\ast }\right]$.

$$K_{D_{eff}}^{oligomer}=\left[{\mathrm{T}}^{\ast }\right]=2{\left(\frac{K_D^{oligomer}}{n}\right)}^{\frac{1}{n-1}}$$ (27)

$K_{D_{eff}}^{oligomer}$ is thus the concentration for which 50% of the receptors are associated into oligomers and 50% are monomeric. It has units of receptor concentration. Note that, if we solve Eq. 27 for a dimer $\left(n=2\right)$, then we recover

$$K_D^{dimer}=K_{D_{eff}}^{dimer}$$ (28)

FRET studies of EphA2 self-association in the plasma membrane

We used a quantitative FRET technique termed fully quantified spectral imaging FRET (FSI-FRET) (46), and we acquired a dataset for full-length EphA2 in the presence of the ligand ephrinA1-Fc. In these experiments, the fluorescent proteins eYFP or mTurquoise (a FRET pair) were attached to the C terminus of the full-length EphA2 with a flexible (GGS)₅ linker. The images, captured with a spectrally resolved two-photon microscope, were analyzed to calculate 1) the apparent FRET efficiencies $E_{app}$; 2) the donor concentration, EphA2-mTurq; and 3) the acceptor concentration EphA2-YFP in small areas of the plasma membrane of each cell (46). The data from 197 individual cells are shown in Fig. 1. In Fig. 1 A, we show the measured FRET efficiency, $E_{app}$, as a function of EphA2-YFP concentration. We see that FRET increases as a function of concentration, in accordance with the law of mass action. Fig. 1 B shows the donor, EphA2-mTurq, concentration versus the acceptor, EphA2-YFP, concentration. Expression of the receptors varies over a significant concentration range since the receptors are introduced via transient transfection, yielding a binding curve.

To analyze FRET data and to determine dissociation constants, we follow the established FSI-FRET protocol (46). Specifically, we fit predictions for models of association from a dimer to a hexamer to the FRET data using Eq. 10 for $n=$ 2–6. There are two unknown parameters in each fit (for an oligomer, or specific n): the dissociation constant $K_D^{oligomer}$ and $\tilde{E}$, a structural parameter that depends on the relative distances and the dynamics of the fluorophores in the oligomer. We vary the two unknown parameters, $K_D^{oligomer}$ and $\tilde{E}$, and determine their best-fit values, while also accounting for proximity FRET as discussed in section "Materials and methods.” In Fig. 1, we show the theoretical oligomerization curves (oligomeric fraction, $f_{oligomer}$, versus total EphA2 concentration, constructed for the best-fit $K_D^{oligomer}$ values). We also show the binned experimental oligomeric fractions, calculated from the measured FRET efficiencies for the best-fit $\tilde{E}$ values, and their SEs. Different colors correspond to different oligomer orders, $n$.

In Table 1, we present the best-fit $K_D^{oligomer}$ values for all oligomer models, which all have different units. Using the best-fit $K_D^{oligomer}$ values in Table 1, we calculate effective dissociation constants using Eq. 27. They are also shown in Table 1 and they all have the same units, EphA2 concentration in the membrane. By ANOVA, we find that they are not statistically different from each other. Thus, we find that all models used to fit the EphA2 FRET data yield the same effective dissociation constant.

Table 1.

Effective dissociation constants calculated for different oligomerization models along with the fit parameters of $K_D^{oligomer}$ and $\tilde{E}$

Oligomer model	$K_D^{oligomer},\tilde{E}$	$K_{D_{eff}}$ (rec/μm2)
Dimer	180 $\pm$ 30 (rec/μm2), 0.53 $\pm$ 0.02	180 $\pm$ 30
Trimer	40,000 $\pm$ 10,000 (rec2/μm4), 0.348 $\pm$ 0.006	240 $\pm$ 30
Tetramer	1.1 $\pm$ 0.4 $\times$ 107 (rec3/μm6), 0.261 $\pm$ 0.009	280 $\pm$ 30
Pentamer	2 $\pm$ 1 $\times$ 109 (rec4/μm8), 0.204 $\pm$ 0.008	290 $\pm$ 30
Hexamer	2 $\pm$ 2 $\times$ 1011 (rec5/μm10), 0.159 $\pm$ 0.006	220 $\pm$ 40

In Table 1, we also show the best-fit $\tilde{E}$ values. The best-fit $\tilde{E}$ for the dimer, 0.53, is straightforward to interpret, as it depends on the average distance between the fluorescent proteins in the dimer. Assuming free rotation of the fluorescent proteins and a value for the Förster radius of 54.5 Å for the mTurquoise/YFP pair, we calculate an average distance of ∼53 Å between the fluorescence proteins in the dimer.

Discussion

Challenges in the interpretation of FRET data in studies of membrane receptor association arise because the measured FRET efficiency does not only depend on the abundance of dimers/oligomers. It also depends on the distance between the fluorescent proteins in the dimer/oligomer. Thus, the oligomerization curves in Fig. 1 C can be only constructed once the structural contribution is accounted for and deconvoluted, by the determination of the best-fit Ẽ from a two-parameter fit of an oligomerization model to the FRET data. Furthermore, the contribution of proximity FRET, which depends on the oligomer size, oligomer stability, and Ẽ, also needs to be also accounted for in the fit. This is accomplished following a two-step protocol, discussed in detail in prior work (39,47) and summarized in section “materials and methods,” for each model of association. As a result of this procedure, a dissociation constant is calculated, with units depending on the association model. Additional challenges in data interpretation exist, as previous work has shown that FRET experiments cannot discern the exact oligomer size (39). At best, they can differentiate between a dimer and a higher-order oligomer, but a trimer cannot be distinguished from a tetramer or a higher-order oligomer (39). Furthermore, FRET experiments are ill suited to discern the exact size of oligomers when n > 6, as such oligomer sizes are of the order of 100 Å. This is twice the Förster radius and is thus a distance where fluorophores cannot engage in FRET. Finally, the measured FRET for a heterogeneous population of oligomers is the average of the FRET efficiencies for the different types of oligomers.

Recent years have witnessed the development and implementation of fluctuation-based methods such as PIE-FCCS, N&B, and FIF (49) for measurements of membrane protein oligomer sizes. These methods detect co-diffusion of fluorescently labeled receptors and report on their average oligomer size. Such methods have been used to measure EphA2 oligomer sizes and have revealed that EphA2 forms heterogeneous populations of oligomer sizes, with the most probable size between 3 and 4 (35,40,41). Thus, none of the simple models in Eq. 22 can faithfully describe EphA2 lateral interactions. It may be tempting to develop more complex models that assume the existence of intermediates (such as a dimer of dimers model) and to use them to fit the FRET data. However, each intermediate adds two additional unknown parameters to the association model, and a fit may appear better only because of the larger number of fitting parameters (40). Furthermore, different combinations of fitting parameters may yield fits of comparable quality.

It is now well established that although EphA2 monomers are inactive, the EphA2 molecules in the oligomers activate each other by attaching multiple phosphate groups to the intracellular tyrosines (35,50,51,52,53). Other RTKs, as well as other receptors, are also known to oligomerize (37,38). In all these cases, we are particularly interested in the propensity for self-association, irrespective of the oligomer size, although the oligomer size may fine-tune the signaling properties. Here we show that we can extract an effective equilibrium constant from an EphA2 FRET dataset, no matter what model of association is assumed. Thus, the effective equilibrium constant can be determined even if the oligomer size is poorly defined. By design, this equilibrium constant has a very well-defined physical meaning and biological significance. The effective dissociation constant is the EphA2 concentration for which half of the EphA2 molecules are monomeric and thus inactive, and the other half are associated into oligomers and are therefore active. Furthermore, we can say that, for EphA2 concentrations below the effective dissociation constants, EphA2 will be predominantly monomeric and inactive. On the other hand, EphA2 will be predominantly associated into active dimers or oligomers once its concentration exceeds the effective dissociation constant.

An ANOVA analysis in Fig. 1 D shows that we obtained the same value for the EphA2 effective dissociation constant, no matter what EphA2 association model is used. We cannot expect that this will be the case for any membrane protein, as the oligomerization curves obtained from the experimental FRET measurements depend on the contribution for proximity FRET. The proximity FRET contribution, in turn, depends on the oligomer size and this contribution is the highest for dimers and decreases as the oligomer size decreases (39,47). In cases when the relative contribution of proximity FRET is significant, the calculated effective dissociation constants for different n values can be expected to be different from each other. In such cases, a few recommendation can be made. If a well-defined oligomer size can be determined in the fluorescence fluctuation experiments, the effective dissociation constant obtained for this model should be considered the most likely one. If, however, different association models are all consistent with the fluorescence fluctuation data, then the experiment will produce a range of dissociation constants that have to be considered equally likely.

The concept of the effective dissociation constant will bring clarity to studies of the effect of pathogenic RTK mutations that cause dysregulated signaling while altering the size of the RTK oligomer (54). Although fluorescence fluctuation techniques can detect the changes in the oligomer size due to the mutations, fitting wild-type and mutant data with different models of association will yield dissociation constants of different units which cannot be compared to each other using statistical tools. The effective dissociation constants for wild-type and mutant receptors, however, can be directly compared using t-tests.

In conclusion, this work demonstrates the utility of FRET experiments in studies of receptor interactions even when the receptors form oligomers, not just dimers. It is highly recommended that FRET is used in conjunction with fluorescence fluctuation techniques, such as N&B (40,55,56), FIF (15,16,35), and PIE-FCCS (19,57,58), which directly report on the average oligomer size.

Author contributions

D.M.M. developed the theory, analyzed the data, and wrote the first draft. D.W. performed the research and edited the paper. T.V.P. and K.H. secured funding and edited the paper.

Editor: Manuel Jose Estevez Prieto.