_{d}) and the intrinsic FRET efficiency (E

_{max}) of an interacting donor-acceptor pair. Assuming bimolecular interaction, the predicted FRET efficiency is a function of donor concentration, acceptor concentration, K

_{d}, and E

_{max}. We estimate K

_{d}and E

_{max}by minimizing the sum of the squared error (SSE) between the predicted and measured FRET efficiency. This is accomplished by examining the topology of SSE values for a matrix of hypothetical K

_{d}and E

_{max}values. Applying an F-test, the 95% confidence contour of K

_{d}and E

_{max}is calculated. We test the method by expressing an inducible FRET fusion pair consisting of FKBP12-Cerulean and Frb-Venus in HeLa cells. As the K

_{d}for FKBP12-rapamycin and Frb has been analytically determined, the relative Kd (in fluorescence units) could be calibrated with a value based on protein concentration. The described methodology should be useful for comparing protein-protein interaction affinities in living cells.

## 1.

## Introduction

Förster resonance energy transfer (FRET) is a physical process in which a donor fluorophore molecule in the excited state transfers energy nonradiatively to an acceptor molecule (often also a fluorophore) in the ground state.^{1, 2} FRET occurs over distances of molecular scale
$\left(1\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}\right)$
and can thus be used to evaluate protein-protein interaction in living cells. The recent development of spectrally diverse fluorescent proteins (FP) that serve as genetically encoded donor and acceptor fluorophore tags has greatly facilitated FRET measurements in living cells.^{3, 4}

FRET is most appropriately quantified using the FRET efficiency, an instrument-independent index defined as the proportion of donor molecules that transfer excitation state energy to acceptor molecules.^{4} In living cells expressing proteins fused with FPs, FRET efficiency can be accurately determined using donor fluorescence lifetime or sensitized acceptor emission intensity.^{5} However, equating FRET efficiency with protein-protein interaction presents a challenge, as FRET efficiency is a function of multiple variables. The variables include the intracellular concentrations of donor and acceptor fluorophores (usually proteins fused to FPs), the affinity (i.e., the equilibrium dissociation constant
${K}_{d}$
) of the interacting proteins, and the intrinsic FRET efficiency
$\left({E}_{\mathrm{max}}\right)$
of the donor∕acceptor pair. Moreover, FRET efficiency is not linearly related to these variables with the exception of
${E}_{\mathrm{max}}$
. Therefore, protein pairs with disparate
${K}_{d}$
values may display indistinguishable measured FRET efficiencies, while protein pairs with identical
${K}_{d}$
values may display different measured FRET efficiencies.^{4} Hence, the measured FRET efficiency *per se* is not an appropriate estimator of the protein-protein binding affinity.

Estimation of the binding affinity is an important goal of protein-protein interaction assays. Erickson, Alseikhan, and Peterson, and Erickson
^{6, 7} were the first to demonstrate the feasibility of using quantitative “three-cube” FRET measurements^{8, 9} to estimate the relative
${K}_{d}$
of two proteins tagged with genetically encoded fluorophores expressed in living cells. However, the methodology required determination of several parameters measured from purified donor and acceptor fluorescent proteins, and detailed knowledge of the optical filter characteristics (e.g., the average molar extinction coefficient of donor and acceptor fluorophores over the bandwidth of the FRET cube excitation filter).^{6, 7} Moreover, the equivalence of the fluorescent protein spectra determined *in vitro* and *in vivo* (i.e., a cellular context) was assumed.

Here, we describe a method to estimate the relative
${K}_{d}$
of two FP-tagged interacting proteins, which avoids these requirements and is performed using standard fluorescence microscopy and quantitative imaging techniques. The method was validated using an inducible FRET pair consisting of the FPs Cerulean and Venus fused to FKBP12 (a rapamycin-binding protein) and the Frb domain of FRAP1∕mTOR, respectively. Upon addition of rapamycin, a complex with defined structure, stoichiometry, and affinity forms.^{10}

## 2.

## Methods

In plasmid constructs used in this study, transcription was driven by the human cytomegalovirus promoter and the start codon was under the optimal context for translational initiation.^{11} The constructs encoding monomeric (A206K mutation) Cerulean and Venus, variants of cyan fluorescent protein (CFP) and yellow fluorescent protein (YFP); respectively, have been previously described.^{12, 13, 14, 15} The open reading frames (ORFs) encoding the rapamycin-binding protein FKBP12 (GenBank accession number U69485) and the Frb domain of FRAP1∕mTOR (GenBank accession number NM̱019906; residues 2018-2114) were amplified using the polymerase chain reaction from rat whole brain cDNA (Clontech, Mountain View, California), and cloned into mammalian expression vectors encoding Cerulean and Venus, respectively. The linker sequence, KLRILQSTVPRARDPPVAT, in FKBP12-Cerulean and Frb-Venus fusion constructs was identical. The W2101F mutation was introduced into the Frb domain using the QuikChange® method (Stratagene, La Jolla, California).
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
was termed
${\mathrm{Frb}}_{\mathrm{KTF}}$
in the original description of the mutation.^{16}

HeLa cells (ATCC, Manassas, Virginia) suspended in minimal essential medium containing 10% bovine calf serum were plated onto glass-bottom tissue culture dishes (MatTek, Ashland, Massachusetts) and stored in a humidified atmosphere containing 5% $\mathrm{C}{\mathrm{O}}_{2}$ in air at $37\phantom{\rule{0.2em}{0ex}}\xb0\mathrm{C}$ . HeLa cells were transfected with plasmids using Lipofectamine 2000 (Invitrogen, Carlsbad, California) for approximately $24\phantom{\rule{0.3em}{0ex}}\mathrm{h}$ . Prior to image acquisition, culture medium was replaced with phosphate buffered saline (PBS) containing $1\phantom{\rule{0.3em}{0ex}}\mathrm{mM}$ $\mathrm{Ca}{\mathrm{Cl}}_{2}$ and $\mathrm{Mg}{\mathrm{Cl}}_{2}$ . To induce heterodimerization, rapamycin (Biomol, Plymouth Meeting, Pennsylvania) was added to the PBS ( $1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}$ final concentration) at least $40\phantom{\rule{0.3em}{0ex}}\mathrm{min}$ prior to imaging.

The imaging procedure has been described.^{17} Briefly, an Olympus IX-71 inverted microscope equipped with a
$75\text{\u2010}\mathrm{W}$
xenon arc lamp (Cairn Research, Kent, United Kingdom), a UNIBLITZ mechanical shutter (Vincent Associates, Rochester, New York), a
$60\times $
oil immersion objective (NA 1.4), a CFP filter set (
${I}_{DD}$
cube, the donor channel; excitation:
$436\pm 10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, emitter:
$480\pm 20\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, dichroic: 455LP), a YFP filter set (
${I}_{AA}$
cube, the acceptor channel; excitation:
$500\pm 10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, emitter:
$540\pm 15\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, dichroic: 520LP), and a FRET filter set (
${I}_{DA}$
cube, the FRET channel; excitation:
$436\pm 10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, emitter:
$540\pm 15\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$
, dichroic: 455LP). Using a
$12\text{\u2010}\text{bit}$
cooled charge-coupled device (CCD) camera (Retiga Exi, Qimaging, Burnaby, British Columbia, Canada) and a custom IGOR Pro (version 5, WaveMetrics, Lake Oswego, Oregon) based program, three fluorescence intensity images, i.e.,
${I}_{DD}$
,
${I}_{DA}$
, and
${I}_{AA}$
images of HeLa cells coexpressing Venus (or Venus-tagged protein) and Cerulean (or Cerulean-tagged protein) were acquired using the
${I}_{DD}$
,
${I}_{DA}$
, and
${I}_{AA}$
filter cubes. Background fluorescence measured from untransfected cells was subtracted from all images. Images were acquired at room temperature
$(22\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}23\phantom{\rule{0.2em}{0ex}}\xb0\mathrm{C})$
with exposure times of
$0.1\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}1.5\phantom{\rule{0.3em}{0ex}}\mathrm{s}$
, analog gain was set to 1, and binning set to
$2\times 2$
. Fluorescence intensities were normalized to an exposure duration of
$1\phantom{\rule{0.3em}{0ex}}\mathrm{s}$
. Due to the overlap of the excitation and emission spectra of Cerulean and Venus, the fluorescence intensity of the
${I}_{DD}$
,
${I}_{DA}$
, and
${I}_{AA}$
images had three potential sources: the donor fluorescence
$\left({I}_{dd}\right)$
, the sensitized acceptor emission
$\left({F}_{c}\right)$
due to FRET, and the direct acceptor emission
$\left({I}_{aa}\right)$
. To isolate these components in each pixel of intensity images, standard procedures and formulae have been employed.^{9, 18, 19}
${I}_{DD}$
,
${I}_{DA}$
, and
${I}_{AA}$
images of HeLa cells expressing only Venus were acquired, and two cross talk parameters were determined:
$a={I}_{DA}\u2215{I}_{AA}$
and
$b={I}_{DD}\u2215{I}_{AA}$
. In these experiments,
$a=0.0335\pm 0.0003$
$(\mathrm{n}=11)$
and
$b=0.0004\pm 0.0001$
$(\mathrm{n}=11)$
.
${I}_{DD}$
,
${I}_{DA}$
, and
${I}_{AA}$
images of HeLa cells expressing only Cerulean were acquired, and two cross talk parameters were determined:
$d={I}_{DA}\u2215{I}_{DD}$
and
$c={I}_{AA}\u2215{I}_{DD}$
. In these experiments,
$c=0.0013\pm 0.0001$
$(\mathrm{n}=12)$
and
$d=0.3769\pm 0.0005$
$(\mathrm{n}=12)$
. The pixel intensity of
${I}_{DD}$
,
${I}_{DA}$
, and
${I}_{AA}$
images in cells coexpressing Venus (or Venus-tagged protein) and Cerulean (or Cerulean-tagged protein) were characterized with the following equations:

^{9}and Eqs. 2, 3 are essentially the same as described by Gordon

^{9}and Tron

^{19}From these equations, ${I}_{aa}$ , ${I}_{dd}$ , and ${F}_{c}$ were isolated as follows:

Previously we have demonstrated that expressing two donor-acceptor fusion FPs with differing FRET efficiencies allows us to determine: 1. the ratio of sensitized acceptor emission to donor fluorescence quenching (
$G$
factor), and 2. the ratio of donor∕acceptor fluorescence intensity for equimolar concentrations in the absence of FRET (
$k$
factor).^{17}
$G$
and
$k$
factors are constant for given donor∕acceptor pairs and a particular imaging setup. Following determination of the
$G$
and
$k$
factors for the FPs Cerulean and Venus, the FRET efficiency
$\left(E\right)$
and the concentration ratio
$D\u2215A$
of donor (Cerulean or Cerulean-tagged protein) and acceptor (Venus or Venus-tagged protein) fluorophore can be determined:^{17, 18}

## 3.

## Theory

Assuming a bimolecular interaction between donor and acceptor molecules, a formalism derived from receptor-ligand binding studies can be adopted, in which we equate the donor and acceptor molecules to receptor and ligand, respectively. In this study, the acceptor was not expressed in great excess relative to the donor, thus we did not assume that “free” acceptor concentration was approximated by total acceptor concentration (i.e.,
$A$
). The situation is equivalent to binding experiments with ligand depletion:^{20}

## Eq. 11

$${D}_{\text{free}}+{A}_{\text{free}}\underset{{k}_{\text{off}}}{\overset{{k}_{\text{on}}}{\rightleftharpoons}}DA,$$^{20}The predicted FRET efficiency $\left({E}_{\mathrm{pred}}\right)$ in a two-molecule system, as described here, is:where $DA\u2215D$ is the fractional occupancy of donor with acceptor molecules, and ${E}_{\mathrm{max}}$ is the intrinsic FRET efficiency of an interacting donor-acceptor pair.

^{21}Thus, combining Eqs. 15, 16, ${E}_{\mathrm{pred}}$ for any given $D$ and $A$ is:

## Eq. 17

$${E}_{\mathrm{pred}}={E}_{\mathrm{max}}\times 0.5\times \{(A+D+{K}_{d})-{[{(A+D+{K}_{d})}^{2}-4A\times D]}^{1\u22152}\}\u2215D.$$We can determine
$D$
,
$A$
, and
$E$
for cells expressing varying concentrations and ratios of interacting proteins fused to Cerulean and Venus [see Eqs. 7, 9, 10]. Treating
$D$
and
$A$
as independent variables, and
$E$
as the dependent variable, the free parameters
${K}_{d}$
and
${E}_{\mathrm{max}}$
are adjusted to minimize the difference between the experimental and predicted results. Based on maximum likelihood analysis, the best estimates of
${K}_{d}$
and
${E}_{\mathrm{max}}$
result from using the sum of the squared residual error (SSE) as the figure-of-merit function:^{22, 23}

^{17}thus $D$ was also treated as an independent variable. Another assumption was a Gaussian distribution of $E$ residuals—a point we examine later in the manuscript.

The remaining task was to determine a method for adjusting the parameters
${K}_{d}$
and
${E}_{\mathrm{max}}$
to find the minimum SSE
$\left({\mathrm{SSE}}_{\mathrm{min}}\right)$
. Although various iterative algorithms exist for curve-fitting multivariate functions, we chose the simple, but computationally intense, approach of computing the SSE for each dataset from a large matrix of hypothetical
${K}_{d}$
and
${E}_{\mathrm{max}}$
values. The approach is feasible, because meaningful values of
${K}_{d}$
and
${E}_{\mathrm{max}}$
are constrained. For example,
${E}_{\mathrm{max}}$
must lie in the range 0 to 1 to be interpretable. Moreover, given the crystal structure of fluorescent proteins, we can restrict this range even more, as the beta barrel encapsulating the fluorophore residues prevents close apposition of the donor and acceptor fluorophores. Similarly,
${K}_{d}$
cannot be negative and the practical upper limit (that can be determined) is influenced by other factors (e.g., collisional FRET, see next). Computing the
${\mathrm{SSE}}_{\mathrm{min}}$
in this manner provides several advantages. First, since the error for a large parameter space is computed, the method is not susceptible to finding local minimums or dependent on initial parameter values. Second, the topology of the error space provides a visual clue as to parameter correlation. Third, the method facilitates generation of a “confidence interval contour” based on a critical value of SSE determined from the
$F$
distribution.^{22} Projection of this elliptical contour provides estimates of parameter confidence intervals.

## 4.

## Results

We tested our method of
${K}_{d}$
estimation using a well-characterized inducible protein-protein heterodimerization system.^{10, 16, 24} As shown schematically in Fig. 1a
, the protein FKBP12 has a high affinity
$({K}_{d}=0.2\phantom{\rule{0.3em}{0ex}}\mathrm{nM})$
for the cell permeable immunosuppressant, rapamycin. Subsequently, the FKBP12-rapamycin complex can bind the Frb domain of the FRAP1∕mTOR protein with high affinity
$({K}_{d}=12\phantom{\rule{0.3em}{0ex}}\mathrm{nM})$
, thus bringing the C-termini of FKBP12 and the Frb domains into close apposition
$(\sim 15\phantom{\rule{0.3em}{0ex}}\mathrm{\AA})$
. When expressed in HeLa cells, fluorescence of the fusion proteins FKBP12-Cerulean and Frb-Venus appeared distributed throughout the cytosolic and nuclear compartments (data not shown). As shown in Figs. 1b and 1c (left panels),
$E$
was negligible (
$0.008\pm 0.002$
,
$\mathrm{n}=120$
) in HeLa cells coexpressing FKBP12-cerulean and Frb-Venus, consistent with the idea that FKBP12 does not bind the Frb domain in the absence of rapamycin.^{10} In a few cells expressing a high concentration of Frb-Venus (
$>5000$
arbitrary fluorescence units, AFU), however,
$E$
was about 0.05. This was likely due to “collisional” FRET, as it also occurred when similar concentrations of Cerulean and Venus were expressed (data not shown). To avoid this issue, cells with Frb-Venus concentrations
$>5000$
AFUs were excluded from further analysis. In addition, it has been reported that three-cube FRET measurements were prone to error when
$D\u2215A$
fell outside the range 0.1 to 10; therefore only cells with
$D\u2215A$
between 0.2 to 5 were chosen for further analysis.^{25}

Application of rapamycin to cells coexpressing FKBP12 and Frb induces formation of the FKBP12-rapamycin-Frb complex.^{10, 16} Consistent with this idea,
$40\phantom{\rule{0.3em}{0ex}}\mathrm{min}$
following application of a saturating concentration of rapamycin
$\left(1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}\right)$
, we detected robust FRET [
$E=0.153\pm 0.004$
,
$\mathrm{n}=248$
, Fig. 1b and 1c, right panels] in HeLa cells coexpressing FKBP12-Cerulean and Frb-Venus. Under these conditions, all molecules of FKBP12-Cerulean should be bound to rapamycin. Thus, the trimolecular system reduces to a bimolecular system with the donor∕receptor being the rapamycin-FKBP12-Cerulean complex, and the acceptor∕ligand being Frb-Venus, thereby satisfying the conditions of Eqs. 15, 17 that assume a bimolecular interaction.

To estimate the effective binding affinity $\left({K}_{d}\right)$ and the intrinsic FRET efficiency $\left({E}_{\mathrm{max}}\right)$ of the rapamycin-FKBP12-Cerulean and Frb-Venus complex, HeLa cells were transfected with several different cDNA molar ratios, and an effort made to image a wide range of fluorescence intensities. $E$ and $D\u2215A$ for FKBP12-Cerulean and Frb-Venus were determined using the three-filter cube approach following application of rapamycin $\left(1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}\right)$ . In the absence of rapamycin, $E$ varied between about $-3$ to 3% in cells with Frb-Venus intensity less than 5000 AFU [Fig. 1c, left). Therefore, to increase signal-to-noise ratio, seven cells with $E$ less than 3% in the presence of rapamycin were excluded from further analysis.

We then calculated the SSE
$(\mathrm{n}=241)$
between the measured and predicted
$E$
using Eqs. 7, 17, 18, custom Igor Pro-based computer program. A
$448\times 448$
array with row and column indices represented by hypothetical
${E}_{\mathrm{max}}$
(0.1 to 0.4) and
${K}_{d}$
(0 to 2000 AFU) values, respectively, was filled with the computed SSE values for the dataset. In other words, each “pixel” in the array contained an “intensity” (i.e.,
$z$
value) comprised of the SSE for the dataset computed from the corresponding
${K}_{d}$
(ordinate) and
${E}_{\mathrm{max}}$
(abscissa). These data could be represented as an image with the SSE “intensity” coded by a color scale, as shown in Fig. 2a
(color online only), allowing visualization of the error space topology. The band of similar colors extending diagonally across the plot indicates a “trough” in the SSE and thus parameter correlation. Expanding the color scale over a narrower range of SSE values, both the gradient [Fig. 2b] and convergence to the
${\mathrm{SSE}}_{\mathrm{min}}$
(0.125, marked with a cross symbol) at
${K}_{d}=148$
AFU and
${E}_{\mathrm{max}}=0.244$
become more apparent. A plot of the measured versus predicted
$E$
, and the corresponding residuals, is shown in Fig. 2d. A histogram of the residuals is plotted in Fig. 2e, and the resulting distribution fit with a Gaussian equation (solid line,
$p=0.12$
, chi-square test). This result supports an assumption of maximum likelihood analysis, on which the
${\mathrm{SSE}}_{\mathrm{min}}$
as best estimator of
${K}_{d}$
and
${E}_{\mathrm{max}}$
was based.^{22}

To generate estimated confidence limits for ${K}_{d}$ and ${E}_{\mathrm{max}}$ , we defined a region within the SSE space that contained all combinations of ${K}_{d}$ and ${E}_{\mathrm{max}}$ that generated SSEs not significantly different, at an arbitrarily defined level (e.g., $P=0.05$ ), from the best-fit values. The critical value of the SSE for $P=0.05$ is described by:

where $n$ is the number of data points and $F$ is the critical value of the $F$ distribution for a $P$ value of 0.05, with the number of parameters $(p=2)$ as degrees of freedom in the numerator, and $n\text{\u2010}p$ degrees of freedom in the denominator.^{22}The critical $F$ value can be determined in several ways, e.g., statistical tables, numerically solving the $F$ distribution cumulative probability function, or built-in computer functions. We used the Excel (Microsoft Corporation, Redmond, Washington) function $\mathrm{FINV}(0.05,p,n\text{\u2010}p)$ to determine the critical $F$ value (version 6.0 of Igor Pro now includes the equivalent function, StatsInvFCDF). For the data illustrated in Fig. 2a, ${\mathrm{SSE}}_{0.05}$ was about 0.128, resulting in the confidence ellipse or contour (in red) illustrated in Fig. 2c. The confidence interval for the individual parameters was calculated by projection of the confidence contour extremes onto the respective axes [Fig. 2c, black lines]. Hence, the 95% confidence intervals of ${K}_{d}$ and ${E}_{\mathrm{max}}$ were 83 to 230 AFU and 0.230 to 0.258, respectively. Note that the confidence intervals need not be symmetric around the best-fit values.

The data can be plotted as $E$ versus free acceptor concentration $\left({A}_{\text{free}}\right)$ , thereby emulating a typical saturation binding isotherm [Fig. 2f]. As mentioned earlier, $E$ is proportional to fractional donor occupancy. ${A}_{\text{free}}$ was calculated using the formula: ${A}_{\text{free}}=A-DA$ . The data points were compared with a theoretical line generated from the previously determined ${E}_{\mathrm{max}}$ and ${K}_{d}$ parameters according to: $E={E}_{\mathrm{max}}\times {A}_{\text{free}}\u2215({K}_{d}+{A}_{\text{free}})$ . The data points span much of the binding curve around the ${K}_{d}$ value—a prerequisite for accurate parameter estimation. It should be emphasized that the ${K}_{d}$ value cannot be derived from $E$ versus ${A}_{\text{free}}$ relationship, as this represents a circular dependency. The main purpose of Fig. 2f is to provide a visual data representation analogous to radioligand binding studies.

The ability to distinguish different affinities was tested by introducing a point mutation, W2101F, in the Frb domain. Previous studies indicated that the affinity between the rapamcyin-FKBP12 complex and
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
was reduced.^{16} The experimental procedure used before was repeated using
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
-Venus in place of the wild-type construct. As shown in Fig. 3a
(upper ellipse), the effective
${K}_{d}$
and
${E}_{\mathrm{max}}$
were estimated to be 492 AFU (95% confidence interval 292 to 775 AFU) and 0.255 (95% confidence interval 0.229 to 0.287), respectively. As before, we plotted the
$E$
measured from HeLa cells
$(\mathrm{n}=261)$
coexpressing FKBP12-Cerulean and
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
-Venus (in the presence of
$1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}$
rapamycin) against the
${E}_{\mathrm{pred}}$
[Fig. 3b]. The residuals were also calculated [Fig. 3b] and the distribution found to be adequately described by a Gaussian equation (data not shown). Expressing the data as a standard binding isotherm [i.e.,
$E$
versus free acceptor concentration, Fig. 3c] revealed a right-shifted sigmoid curve with data points spanning the estimated
${K}_{d}$
value. Based on these data, we conclude that the affinity between rapamycin-FKBP12-Cerulean complex and
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
-Venus was significantly lower when compared with wild-type Frb-Venus. This conclusion is visually evident when the 95% confidence ellipse from Fig. 2c is superimposed on Fig. 3a (lower ellipse). Our conclusion is consistent with the results from a rapamycin-induced transcriptional switch assay performed in yeast cells using
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
.^{16} In contrast to the
${K}_{d}$
, the estimated
${E}_{\mathrm{max}}$
for
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
was not significantly different from the wild-type Frb. Although structural changes resulting from point mutations are difficult to predict, the result is consistent with the position of the W2101F mutation within the context of the wild-type Frb crystal structure. The mutated residue lies within the Frb-rapamycin binding pocket and thus, barring dramatic long-range alterations in conformation, would not be expected to alter the position of the Frb C-terminus where the fluorescent protein is attached, and hence the intrinsic FRET efficiency (i.e.,
${E}_{\mathrm{max}}$
).

An advantage of the FKBP12-rapamycin-Frb system, aside from being inducible, is that the absolute affinity
$({K}_{d}=12\phantom{\rule{0.3em}{0ex}}\mathrm{nM})$
has been well characterized.^{10} Assuming that the *in-vivo* and *in-vitro* binding affinities are comparable and the fluorescent protein tagging has a negligible effect on binding affinity, we can calibrate our system with the literature value for the wild-type Frb and estimate that the affinity between FKBP12-rapamycin and
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
in living cells to be about
$40\phantom{\rule{0.3em}{0ex}}\mathrm{nM}$
(95% confidence interval
$24\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}63\phantom{\rule{0.3em}{0ex}}\mathrm{nM}$
).

## 5.

## Discussion

We have described a simple optical method for estimating the effective affinity between two protein domains in living mammalian cells. This method was extended from a previous study in which
$E$
and
$D\u2215A$
was determined in living cells using three-cube wide-field fluorescence microscopy.^{17} Assuming a simple bimolecular interaction,
$E$
is a function of donor and acceptor concentrations (total, not free) and the parameters
${K}_{d}$
and
${E}_{\mathrm{max}}$
. We estimated
${K}_{d}$
and
${E}_{\mathrm{max}}$
by minimizing the SSE between the predicted and the measured
$E$
. Minimization was accomplished by generating SSE values for a matrix of hypothetical
${K}_{d}$
and
${E}_{\mathrm{max}}$
values. From the topology, a 95% confidence contour for
${K}_{d}$
and
${E}_{\mathrm{max}}$
was generated using a critical
$F$
value. Projection of the contour allowed assignment of estimated confidence intervals to each parameter. The method was tested by expressing an inducible FRET fusion pair consisting of FKBP12-Cerulean and Frb-Venus (wild-type or mutant) in HeLa cells. We demonstrated that when coexpressed in HeLa cells and in the presence of the heterodimerization-inducer rapamycin at the saturating concentration, FKBP12-Cerulean has a higher affinity for Frb-Venus than to the mutant
${\mathrm{Frb}}_{\mathrm{W}2101\mathrm{F}}$
-Venus. This result was consistent with a report using a transcriptional switch assay based on the yeast two-hybrid system.^{16} Therefore, the described methodology is useful for comparing protein-protein interaction affinities based on FRET determinations in living cells.

## 5.1.

### Comparison with Other Methods

The development of our
${K}_{d}$
estimation method was inspired by the seminal work of Erickson, Alseikhan, and Peterson, and Erickson, ^{6, 7} who first demonstrated the feasibility of estimating protein-protein interaction affinity in living cells using three-cube FRET measurements. Although similar in principle, there are several factors that differentiate the two techniques. First, the method described here relies solely on conventional fluorescent microscopy and live cell measurements. The FRET efficiency, relative donor concentration, and relative acceptor concentration were determined using methods developed in a previous study.^{17} The key requirement for calibrating the measurements (i.e., the
$G$
and
$k$
factors) was development of donor-acceptor pairs (i.e., FP fusion constructs) with widely differing
$E$
and known stoichiometries (e.g., 1:1). In contrast, Erickson, Alseikhan, and Peterson, and Erickson
^{6, 7} used spectroscopic data from purified FPs and optical filters to arrive at
$D$
and
$A$
. Thus, our method requires fewer assumptions (e.g., equivalence of *in-vitro* to *in-vivo* derived constants) and is generally more accessible. Second, the method described here is based on traditional donor-based FRET efficiency measurements, whereas FRET ratio, an acceptor-based FRET index, was used in Refs. 6, 7. Although both indices provide similar information, FRET efficiency is instrument-independent and thus comparable among different laboratories and experimental settings. Moreover, intrinsic FRET efficiency
$\left({E}_{\mathrm{max}}\right)$
provides the basis for limited inference about the distance between fluorophores. Third, although both methods are based on cellular fluorescence intensity measurements, our data were acquired with a cooled CCD camera (data for single cells represent the mean of many pixel values), whereas Erickson, Alseikhan, and Peterson, and Erickson
^{6, 7} used a photomultiplier tube to measure total cellular fluorescence intensity, and thus assume a similar cell size. Fourth, we minimized SSE by examining a subset of error space topology, while Erickson, Alseikhan, and Peterson, and Erickson used an iterative routine. Computing the error space topology allows estimation of a confidence contour helpful in interpreting the fitted parameters, and avoids finding the local (instead of global) minimum, which is a potential issue inherent to most iterative algorithms. The contour includes the parameter covariance in the evaluation of the confidence intervals, thus allowing more rigorous statistical comparison of the estimated parameters.^{26} The color-coded error topology graphs [e.g., Fig. 2a (color online only)] provide visual indication of the correlation between
${K}_{d}$
and
${E}_{\mathrm{max}}$
. A paradoxical consequence of parameter correlation is that nonoverlapping confidence contours can result in overlapping parameter confidence intervals. In this case,
${K}_{d}$
and∕or
${E}_{\mathrm{max}}$
significantly differ, but one cannot assign the difference to an individual parameter.

FRET efficiency can be determined using fluorescence lifetimes rather than intensities. In principle, the intrinsic FRET efficiency and fractional donor occupancy (by acceptor) are directly determined from the parameters (i.e., time constants and relative amplitudes) of a multiexponential fit to the donor fluorescence decay.^{27, 28} For a bimolecular reaction, the relationship between fractional donor occupancy and free acceptor concentration would be described by a Langmuir binding isotherm, thus allowing
${K}_{d}$
determination using conventional analyses. However, lifetime imaging requires very expensive equipment unavailable to most laboratories, and collecting sufficient photon counts for reliable analyses can require
$10\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}15\phantom{\rule{0.3em}{0ex}}\mathrm{min}$
per cell. Moreover, FRET efficiency and fractional donor occupancy determination based on FP lifetime analysis has proven more complex than anticipated. For example, linked constructs of Cerulean and Venus exhibited multiple time constants of donor fluorescence decay when only one component was predicted.^{12} The reason for this discrepancy is unclear, but the phenomenon demonstrates unexpected complexity in what superficially appears to be a simple process. Whether lifetime imaging or intensity-based methods are used to estimated
${K}_{d}$
(based on FRET efficiency) will be dictated by the resources and expertise available, and the conditions and goals of the experiment. Under some circumstances, lifetime imaging is a more reliable method to quantify FRET efficiency when compared with intensity-based measurments.^{27} Conversely, for FPs imaged in living cells, intensity- and lifetime-based methods produced comparable results.^{5}

## 5.2.

### Factors Affecting Föster Resonance Eenergy Transfer Based Affinity Measurements

There are several factors that require consideration when using FRET to estimate affinity in living cells. First, signal-to-noise ratio is maximized when the proteins under study have both a high affinity and intrinsic FRET efficiency. Estimation of low affinity interactions is compromised by “collisional” FRET. For example, if the relative
${K}_{d}$
between two proteins is larger than
$\sim 5000$
AFU, then donor occupancy cannot span a large range without incurring nonspecific FRET. Consequently,
${K}_{d}$
estimation based on these data will be prone to error. Based on the reported *in-vitro* determined
${K}_{d}$
of FKBP12-rapamycin and Frb, we estimate that a
${K}_{d}$
greater than
$0.4\phantom{\rule{0.3em}{0ex}}\mu \mathrm{M}$
will be difficult to assess unless collisional FRET efficiency can be accounted for. Maximizing intrinsic FRET efficiency helps to minimize this problem by increasing the signal above both the inherent noise floor and any collisional FRET that might occur. In this regard, two novel variants of GFP, CyPet and yPet, have been engineered to maximize FRET signals.^{29} It is unclear at this time, however, whether CyPet and yPet will be superior to Cerulean and Venus for these types of studies.

Second, we assume that the mean fluorescence intensity (corrected for background and FRET) is proportional to fluorophore concentration. Ideally, each pixel would represent a constant cellular volume. With wide-field microscopy, this is unlikely, as one observes variation in fluorescence intensity within cells, even with a high NA objective (e.g., brighter over the nucleus, which is the thickest portion of the cell). Although determinations of $E$ and $D\u2215A$ are not affected by this requirement, determinations of $D$ and $A$ are. For cells with similar morphology (such as HeLa cells), however, the mean cellular fluorescence intensity should be proportional to cellular fluorophore concentration for small soluble proteins that distribute evenly throughout the nucleus and cytoplasm. Estimation of the ${K}_{d}$ between two plasma membrane proteins or between a plasma membrane and a cytosolic protein, however, will likely present greater challenges, as the heterogeneity in concentrations and FRET efficiencies within subcellular compartments complicates interpretation. Techniques that limit the excitation to (multiphoton or total internal reflection fluorescence microscopy) or collection of light from (confocal microscopy) a smaller cellular volume should be superior under these circumstances.

Third, the ${K}_{d}$ estimation based on FRET measurement assumes that negligible fractions of donor or∕and acceptor molecules bind to endogenous proteins. If this assumption does not hold, the ${K}_{d}$ value will be overestimated. However, the presence of potentially interacting endogenous proteins should not affect the assessment of the affinity change arising from a point mutation. It should be noted that ${K}_{d}$ values reported here (in AFUs) are instrument-dependent, whereas the ranking of ${K}_{d}$ values is not.

Fourth, we emphasize the need to image cells varying in both fluorescent protein concentration and
$D\u2215A$
ratio. The former is usually accomplished by exploiting the normally occurring variations in protein expression, i.e., imaging the entire range of dim to bright cells. The latter, however, requires transfecting several different ratios (empirically determined) of donor∕acceptor cDNA. Together, the two conditions are usually sufficient to generate a wide range of donor fractional occupancy—a requirement for accurate parameter estimation. An indicator of inadequate coverage of the donor fractional occupancy range is failure of the confidence ellipse to close at one end.^{22}

Finally, the current method is predicated on the assumption of a simple bimolecular interaction. It does not account for multiple binding affinities or energy transfer to multiple acceptors. Although a model accounting for additional complexity could be generated, the expansion of free parameters (and correlation between parameters) would increase parameter uncertainty and correlation—possibly beyond the point where meaningful parameter estimates are generated.

## 5.3.

### Utility

The use of FRET-based methods to indicate protein-protein interaction is becoming increasingly popular. However, FRET indices are primarily an indicator of proximity (if one ignores orientation factors), and any conclusions regarding protein-protein interaction require an interpretation within this context. Thus, the ability to estimate the affinity between two proteins in living cells using FRET measurements provides an additional means to evaluate FRET experiments and make judgments as to whether biologically relevant protein-protein interaction can be reasonably assumed from the measurements.

## Acknowledgment

We thank Steven S. Vogel for many helpful discussions. This work was supported by the intramural program of the NIH, National Institute on Alcohol Abuse and Alcoholism, Bethesda, Maryland 20892.