Oligomerization of the tetramerization domain of p53 probed by two- and three-color single-molecule FRET

Significance

Intrinsically disordered proteins often form pathological oligomers implicated in various diseases. In many cases, these oligomers cannot be separated and characterizations of their sizes and conformations are difficult. We develop a single-molecule fluorescence method that can probe individual oligomers without separation and determine the equilibrium constants and oligomerization kinetics. By combining two- and three-color single-molecule FRET spectroscopy with fluorescence lifetime analysis, it is possible to determine conformations and flexibility of individual oligomers unambiguously. We apply this method to the oligomerization of the tetramerization domain of p53 and compare conformations of monomer, dimer, and tetramer. This method will be useful in exploring other protein oligomerization systems involved in important biological and disease processes.

Abstract

We describe a method that combines two- and three-color single-molecule FRET spectroscopy with 2D FRET efficiency–lifetime analysis to probe the oligomerization process of intrinsically disordered proteins. This method is applied to the oligomerization of the tetramerization domain (TD) of the tumor suppressor protein p53. TD exists as a monomer at subnanomolar concentrations and forms a dimer and a tetramer at higher concentrations. Because the dissociation constants of the dimer and tetramer are very close, as we determine in this paper, it is not possible to characterize different oligomeric species by ensemble methods, especially the dimer that cannot be readily separated. However, by using single-molecule FRET spectroscopy that includes measurements of fluorescence lifetime and two- and three-color FRET efficiencies with corrections for submillisecond acceptor blinking, we show that it is possible to obtain structural information for individual oligomers at equilibrium and to determine the dimerization kinetics. From these analyses, we show that the monomer is intrinsically disordered and that the dimer conformation is very similar to that of the tetramer but the C terminus of the dimer is more flexible.

It is well known that intrinsically disordered proteins (IDPs) can fold into different structures when attaching to their binding targets. This structural flexibility and binding promiscuity are required for the formation of protein–protein interaction networks in various biological processes such as signal transduction and gene transcription (1–3). On the other hand, some IDPs self-assemble and form oligomers, many of which are implicated in the development of diseases such as Alzheimer’s disease (amyloid-β protein) (4, 5) and Parkinson’s disease (α-synuclein) (6). An ensemble of these oligomers with different sizes and conformations is not easy to separate, and therefore, their characterization is very difficult. However, single-molecule spectroscopy can be a very powerful tool because it can probe subpopulations in a mixture without the need for separation. Single-molecule spectroscopy has been successfully used for characterizing individual molecular states such as the folded and unfolded states of proteins (7–9), intermediate states (10, 11), and transition paths (12–17) and for identifying specific molecular species and complexes (18–22). In this paper, we describe the development of a single-molecule fluorescence method that probes individual oligomers in a mixture, characterizes their conformations, and determines oligomerization kinetics.

We have used two- and three-color Förster resonance energy transfer (FRET) spectroscopy. Compared with two-color FRET that monitors a single distance, three-color FRET can determine three distances and therefore has great potential to obtain 3D structural information for a molecule or a molecular complex. Multicolor FRET has been demonstrated for well-known and designed molecular systems (23–27) and used for the molecular identification (24, 28, 29) and investigations of conformational changes and dynamics of proteins and nucleic acids (30–35) and their interactions (36–38). In some cases, the experiment and analysis can be simplified by preventing the energy transfer between one dye pair (e.g., much larger separation than the Förster radius). However, multicolor FRET experiments are generally complex due to technical difficulties in site-specific labeling and poor photophysical properties of an additional fluorophore. This complication makes the accurate determination of FRET efficiencies challenging, and only a few studies have determined and used all FRET efficiency values (24–26, 36, 37). In this paper, we present various analysis methods to overcome these problems and determine FRET efficiencies accurately. In addition to the FRET efficiency, we also use fluorescence lifetimes to analyze the correlation between the FRET efficiency and the fluorescence lifetime in the two-color experiment. From the 2D FRET efficiency–lifetime analysis (39–44), it is possible to not only distinguish conformational states but also estimate the conformational flexibility of each state. For this analysis, accurate determination of the FRET efficiency and lifetime is very important. In addition to typical correction procedures for background, donor leak into the acceptor channel, and detection efficiencies and quantum yields of dyes (γ-factor), we present a correction method for fast acceptor blinking, which causes a 5–10% error in the determination of the FRET efficiency and donor lifetime. The acceptor blinking effect can be easily missed because it is faster than the bin time of 10–20 ms and does not clearly appear in FRET efficiency trajectories.

In this work we have applied this development in single-molecule FRET spectroscopy to the oligomerization of the tetramerization domain (TD) of the tumor suppressor protein p53. Atomic-resolution structures are available for the tetramer (45–47), but due to the very small dissociation constant, it is not possible to characterize the monomer and dimer conformations using ensemble measurements. [The solution NMR structure has been obtained for a destabilized mutant dimer that does not form a tetramer (48).] Although the dimerization and the tetramerization occur sequentially as the concentration is increased (49), the two dissociation constants are very similar for the TD construct used in this work (residues 319–360 of the full-length p53), which makes the dimer characterization difficult even by single-molecule methods. We show that the dissociation constants of the dimer and the tetramer and the dimerization kinetics can, however, be determined in single-molecule free-diffusion experiments. By immobilizing molecules, we could selectively detect dimers and determine the FRET efficiency and fluorescence lifetime more accurately from longer trajectories. Combination of the two- and three-color FRET experiments and the 2D FRET efficiency–lifetime analysis shows that the monomer is disordered and the dimer conformation is very similar to that of the tetramer but the C terminus of the chain is more flexible.

Results

Two- and Three-Color FRET Experiments and Fluorescence Lifetime Analysis.

In the two- and three-color FRET experiments, we measured both fluorescence intensities and lifetimes by using picosecond-pulsed laser excitation (Fig. 1A) and a confocal microscope. Compared with three-color FRET, two-color FRET experiments and the data analysis are relatively simple and straightforward because energy transfer occurs between only one pair of fluorophores. On the other hand, in the three-color experiment, there are three energy transfer efficiencies, which cannot be determined by a single laser excitation. The excitation of the donor (D) leads to the energy transfers to both acceptors (FRET efficiencies E₁ and E₂). In addition, the transferred energy to acceptor 1 (A1) can be further transferred to the second acceptor (A2) (Fig. 1B). To determine all three FRET efficiencies, an additional excitation of A1 is required. In this excitation, there is only a single energy transfer from A1 to A2. After determining this FRET efficiency (E₁₂), it is possible to obtain E₁ and E₂ as well (SI Materials, Methods, and Theory, FRET Efficiency Calculation and Standard Corrections in Two- and Three-Color Experiments). We used the alternating laser excitation scheme (24, 50, 51) with two picosecond-pulsed lasers (485 nm and 640 nm) for the excitation of Alexa Fluor 488 (Alexa 488, D) and Alexa Fluor 647 (Alexa 647, A1) with the alternating frequency of 40 MHz.

Fig. 1.

Three-color FRET. (A) A schematic representation of a photon sequence of three fluorophores detected after alternating laser excitation (40 MHz). For each photon, the absolute arrival time (t) and the delay time δt between the laser pulse and the photon arrival are recorded. All three types of photons [donor (D), green circles; acceptor 1 (A1), red circles; acceptor 2 (A2), purple circles] are detected by excitation of the donor (485 nm, blue dashed lines) whereas only A1 and A2 photons are detected by excitation of A1 (640 nm, red dashed lines). Photons from the two excitations (i and j are indexes of photons from D and A1 excitation) can be separated (second and third rows) using their delay times. (Fluorescence lifetimes are much shorter than the interval between alternating laser pulses, 25 ns.) (B) FRET efficiencies between dye pairs. E₁ and E₂ are the energy transfer efficiencies from D to A1 and A2, respectively, and E₁₂ is the FRET efficiency from A1 to A2. (C) Kinetic scheme for three-color FRET. After excitation of the donor (DA₁A₂ → D*A₁A₂), the donor decays to the ground state radiatively (rippled arrow) or nonradiatively (dashed arrow), or the energy is transferred to either A1 (DA₁*A₂) or A2 (DA₁A₂*), with the energy transfer rates of k_ET1 and k_ET2. Excited A1 decays to the ground state or the energy is further transferred to A2 with the rate of k_ET12. A1 excitation results in the energy transfer only from A1 to A2. k_D, k_A1, and k_A2 are the sums of the radiative and nonradiative relaxation rates of the three dyes, which are equal to the inverse of the excited-state lifetimes in the absence of the energy transfer, τ_D⁰, τ_A1⁰, and τ_A2⁰, respectively.

The pulsed excitation of fluorophores allows for recording delay times between the laser excitation pulse and the photon arrival in addition to the absolute photon arrival times (Fig. 1A). The absolute arrival times are used to construct fluorescence trajectories. The mean delay times determine fluorescence lifetimes (SI Materials, Methods, and Theory, Instrument Response Function and Determination and Correction of Fluorescence Lifetimes and Fig. S1). The mean FRET efficiency and donor (or acceptor) lifetime values are used to construct 2D FRET efficiency–lifetime distributions.

Fig. S1.

Extracting lifetime information. (A) IRFs in the two-color (Top, 485-nm excitation) and three-color (Middle, 485-nm excitation; Bottom, 640-nm excitation) experiments. Solid curves are the fits to the Gamma distribution (Eq. S16). For the two-color experiment, a = 6.76, and k_γ = 8.69 ns⁻¹ for the acceptor channel and a = 6.80 and k_γ = 8.02 ns⁻¹ for the donor channel. The parameters for the three-color experiment are a = 6.96 and k_γ = 8.74 ns⁻¹ for the donor, a = 6.21 and k_γ = 7.42 ns⁻¹ for A1, and a = 5.40 and k_γ = 7.64 ns⁻¹ for A2 in 485-nm excitation and a = 11.5 and k_γ = 14. 9 ns⁻¹ for A1 and a = 11.8 and k_γ = 16.0 ns⁻¹ for A2 in 640-nm excitation. (B) An example of the delay time distribution of the donor (Alexa 488) (gray, 17,571 photons) from a trajectory of a single molecule and the fit (red) to the convolution of a biexponential function with the IRF (Eq. S20) (green dashed line). (C) The distribution of δt_D0 (Eq. S20) obtained from the fit in B. (D) The distribution of the difference between the mean delay times calculated using Eq. S19 and the lifetimes obtained from the fit (Eq. S20) in B. Data in B–D were obtained from the two-color binding experiment of immobilized Alexa 488-labeled TD (TD-D) and 10 nM Alexa 647-labeled TD (TD-A1).

In the two-color immobilization experiment, a TD monomer labeled with a donor dye (Alexa 488, D) at its C terminus (TD-D) was immobilized on a glass surface and incubated with 10 nM TD labeled with an acceptor dye (Alexa 647, A1) at either the N (A1-TD) or the C terminus (TD-A1) (Fig. 2B). Molecules were excited by a 485-nm laser in the pulsed mode at 20 MHz. To determine the dimerization kinetics, free-diffusion experiments were carried out to collect fluorescence bursts after manually mixing 40 pM of TD-D with A1-TD or TD-A1 at various concentrations. In the three-color immobilization experiment, the donor- and acceptor-labeled TD monomer (D-TD-A1) was immobilized and incubated with 10 nM TD labeled with acceptor 2 (Alexa 750, A2) at the C terminus (TD-A2).

Fig. 2.

Immobilization of dye-labeled proteins in two- and three-color FRET binding experiments. (A) Front (Upper) and top (Lower) views of the solution structure of the tetramer (PDB ID: 1SAE) (46). Four monomer chains are indicated by numbers from 1 to 4. Dimers are formed by chains 1 and 2 and by chains 3 and 4. The tetramer is a dimer of these two dimers. Quoted numbers are the average Cα distances between C-terminal glycine residues of 77 NMR structures (46). Note that there would be small differences between these values and the true average distances of the entire ensemble. (B) In the two-color binding experiments, Alexa 488 (D) is attached to the C-terminal cysteine residue of the TD, which contains biotin and an unlabeled unnatural amino acid. This molecule (blue) is immobilized on a polyethylene glycol (PEG)-coated glass surface via a biotin–streptavidin linkage and incubated with TD (orange) labeled with Alexa 647 (A1) at the N (A1-TD) or C terminus (TD-A1). (C) In the three-color binding experiment, biotin-attached TD (blue) is labeled with Alexa 488 and Alexa 647 at the N and C termini, respectively, and immobilized. This molecule is incubated with Alexa 750 (A2)-labeled TD (TD-A2). (D) Amino acid sequences of three TD constructs. In UA-TD-Cys, a biotin molecule is attached to the lysine residue (blue) in the biotin-accepting sequence (AviTag), and an unnatural amino acid (green U, 4-acetylphenylalanine) and cysteine (red C) are incorporated between the spacer and the TD sequence and at the C terminus, respectively, for site-specific dye labeling. In Cys-TD and TD-Cys, dyes (A1 or A2) are attached to the cysteine residues.

For the accurate determination of the FRET efficiencies and lifetimes, we performed various corrections (52). The detailed correction procedures of the FRET efficiency and lifetime are described in SI Materials, Methods, and Theory, FRET Efficiency Calculation and Standard Corrections in Two- and Three-Color Experiments. The corrections for the background photons and donor leak are straightforward. We found that the difference of the detection efficiencies and quantum yields (γ-factor) and direct excitation of the acceptor by a donor excitation laser can be corrected together. In addition, although fast photoblinking on the timescale of milliseconds and shorter was not visually detectable in the trajectories with the bin time of 20 ms (Fig. 3A), the correlation analysis of photon trajectories clearly shows blinking of fluorophores (Fig. S2A). No correction was needed for donor blinking because it does not change FRET efficiencies and lifetimes. However, during acceptor blinking, only donor photons are detected and this decreases the mean FRET efficiency. In addition, the fluorescence lifetime of these photons is longer than that in the presence of the active acceptor, which increases the donor lifetime. The effect of acceptor blinking is larger in the three-color experiment, in which A1 is excited more by alternating excitation. For each individual trajectory, we made corrections for acceptor blinking, using the population of the acceptor in the bright state. The acceptor bright-state population was determined using the maximum-likelihood method, analyzing photon trajectories directly without binning (53–55). The detailed correction procedures of the FRET efficiency and lifetime are described in SI Materials, Methods, and Theory, FRET Efficiency and Lifetime Corrections for Acceptor Blinking. We also discuss the complex photophysics of Alexa 488 and Alexa 647 (SI Materials, Methods, and Theory, Photophysics of Alexa 488 and Alexa 647) that should be carefully scrutinized before the analysis.

Fig. 3.

Dimerization and tetramerization of TD probed by two-color FRET. Donor-labeled TD (TD-D) was immobilized and incubated with 10 nM acceptor-labeled TD (A1-TD, A–C, Left, columns 1 and 2; TD-A1, A–C, Right, columns 3 and 4). (A) Representative fluorescence trajectories (20-ms bin time) in the donor and acceptor channels. (B) FRET efficiency histograms (Left) and donor lifetime histograms (Right) were constructed from the mean FRET efficiency and mean donor delay time values obtained from the initial segment of the trajectory of each molecule. Segments containing more than 3,000 photons were included in the analysis. The FRET efficiency was corrected for background, donor leak into the acceptor channel, γ-factor, and acceptor blinking. The donor lifetime was corrected for background and acceptor blinking. The donor-only peak at E = 0 (green bars) has contributions from the monomers and a small fraction of the oligomers with inactive acceptors. The dimer and tetramer (with active acceptors) distributions appear at higher FRET efficiency and shorter donor lifetime (orange bars). B, Upper and Lower shows the distribution obtained before and after the addition of a large excess (2.5 μM) of unlabeled TD, respectively. (C) Two-dimensional FRET efficiency–donor lifetime plot. The distribution of the donor-only molecules is centered at E = 0 and the relative donor lifetime (τ_D/τ_D⁰) of 1 (green dots). The distributions of the molecules with both the donor and acceptor dyes are shifted upward from the diagonal line, suggesting the presence of conformations that interconvert rapidly (main text). The data inside the green rectangles were used to calculate the average FRET efficiency and lifetime values for the estimation of the variance of the FRET efficiency due to these conformational distributions (Eq. 2). Two-dimensional plots using the donor delay times obtained from acceptor photons and comparison with the results without acceptor blinking correction are presented in SI Materials, Methods, and Theory (Fig. S3).

Fig. S2.

Blinking characterization. (A, Left and Center) Auto- and cross-correlation functions calculated for the trajectories with active donor and acceptor dyes (Left) and donor-only trajectories (Center) in the two-color immobilization experiment [Alexa 488-labeled TD (TD-D) + 10 nM Alexa 647-labeled TD (A1-TD)]. (A, Right) Autocorrelation function calculated from trajectories with only A1 in the three-color immobilization experiment (640 nm excitation of A1). The blue dashed curve is a fit to a single exponential function with the relaxation rate of 2.4 ms⁻¹. The positive sign of the donor–acceptor cross-correlation (Left, blue curve) results from donor blinking. All correlation curves show that photoblinking occurs on the timescale of several hundred microseconds. (B) Two-state (bright and dark states of the acceptor) maximum-likelihood analysis (details in SI Materials, Methods, and Theory). Shown are the distributions of the rate coefficient from the dark state to the bright state (k_b, Left) and the bright-state population (p_b, Right). k_b and p_b values were obtained from the analysis of the data in A. The average value of k_b is 2.5 ms⁻¹.

Characterization of TD Oligomerization by Two-Color FRET Experiment.

We first investigated the oligomerization of TD using two-color FRET, which is simpler to interpret. Because the atomic resolution structure of the tetramer conformation is known (Fig. 2A) (46), our main focus was to obtain the dimer conformation in isolation at a low concentration. For this purpose, we performed experiments with two binding constructs shown in Fig. 2B. A TD monomer labeled with a donor dye (Alexa 488, D) at its C terminus (TD-D) was immobilized on a glass surface and incubated with 10 nM TD labeled with an acceptor dye (Alexa 647, A1) at either the N (A1-TD) or the C terminus (TD-A1), so that we could obtain distance information between the N and C termini and between the C termini of the two monomer chains in the dimer state (Fig. 3 and Fig. S3).

Fig. S3.

Comparison of the distributions of the FRET efficiencies and lifetimes before and after the acceptor blinking correction in the two-color experiment (Fig. 3). (A) Histograms of FRET efficiency corrected for background, donor leak into the acceptor channel, and γ-factor. The donor lifetime was corrected for background. (B) Distributions after the correction for acceptor blinking in addition to the corrections in A. FRET efficiencies become higher and donor lifetimes become shorter. E′ and E are the FRET efficiencies before and after the acceptor blinking correction. (C and D) Two-dimensional FRET efficiency–lifetime plots before (C) and after (D) the acceptor blinking correction. The distribution located at the upper left corner (E = 0, τ/τ_D = 1, dark green dots) results from donor-only molecules and therefore is independent of this correction. Lifetimes (τ) were obtained from the mean delay times of donor photons (light green dots) or those of the acceptor photons (orange dots), using Eqs. S19, S21, and S22. The data inside green and red rectangles were used to calculate the average FRET efficiency and lifetime for the calculation of the variance of the underlying FRET efficiency distributions (Eqs. 2 and S26 and Table S1). After the acceptor blinking correction, the distributions by donor photons move toward the lower right direction along the diagonal line because the FRET efficiency increases while the donor delay time decreases after the correction (Eqs. S29 and S31). On the other hand, the distributions by acceptor photons move parallel to the FRET efficiency axis because the acceptor delay time is not affected by acceptor blinking.

Representative donor and acceptor fluorescence trajectories are shown in Fig. 3A. A majority of the trajectories exhibited a constant FRET efficiency without any transition, followed by either donor or acceptor photobleaching (Fig. S4A). This result indicates that the dimerization kinetics are much slower than the tens-of-seconds duration of the trajectories. Therefore, the FRET efficiency and donor lifetime distributions in Fig. 3B were constructed from the mean values of the FRET efficiency and donor delay time of the first segment (before photobleaching) of each trajectory (i.e., each molecule). There are three peaks both in the FRET efficiency distributions and in the donor lifetime distributions. The peak colored in green at E = 0 corresponds to the monomer and a small fraction of the oligomers with only an active donor dye (due to incomplete acceptor labeling or inactive acceptor), whereas the orange peaks (E > 0) correspond to oligomers containing both active donor and acceptor dyes. The peaks at E ∼ 0.5 of A1-TD and E ∼ 0.45 of TD-A1 (Fig. 3B, Upper) are expected to correspond to the dimer, assuming that the structure of the isolated dimer is similar to that in the tetramer (48). (The distance between the C-terminal residues of the labeled chains 1 and 2 is comparable to the Förster radius R₀ = 5.2 nm; Fig. 2A.) In addition to these peaks, there are unexpected peaks at E ∼ 0.7 (A1-TD) and E ∼ 0.8 (TD-A1). Interestingly, these high FRET efficiency peaks disappear after the addition of a large excess of unlabeled TD (Fig. 3B, Lower), which leads to the formation of the tetramer consisting of a dimer with a donor-labeled monomer and an acceptor-labeled monomer (chains 1 and 2) and an unlabeled dimer (chains 3 and 4). The high-E peak is not an artifact caused by immobilization of the proteins because similar distributions are observed in the free diffusion experiment (Dimerization Kinetics Measured by Free Diffusion Experiment). Anisotropy measurement also rules out the possibility of fluorophore sticking (Fig. S5).

Fig. S4.

Transition maps of the FRET efficiency and donor lifetime in the two- and three-color experiments. (A–C, Left and Right) The two-color data with 10 nM A1-TD and TD-A1 are shown, respectively. (A) Representative fluorescence trajectories (light green, donor, D; orange, acceptor, A1) exhibiting transitions. Quoted numbers in blue are the donor lifetimes (in nanoseconds) of individual segments. (B and C) Transition maps obtained from the FRET efficiencies (B) and donor lifetimes (C) before and after transitions. Blue and green ellipses indicate the FRET efficiency and lifetime transition pairs expected when transitions are reversible. (D) Transition maps constructed from the three-color experiment data with 10 nM Alexa 750-lableled TD (TD-A2). E₁₂ and E₂′ values were calculated from the segments containing only A1 and A2 with A2 blinking correction and from those containing D and A2 without A2 blinking correction, respectively. Similar to the data in B, transitions are not symmetric, indicating the transitions result from A2 photobleaching.

Fig. S5.

Anisotropy measurement of the donor (Alexa 488) and acceptor (Alexa 647). Alexa 488-labeled TD (TD-D) was immobilized and incubated with 10 nM Alexa 647-labeled TD (TD-A1). (A) FRET efficiency histograms were constructed from the efficiency values obtained from the initial segment of the trajectory of each molecule. The FRET efficiency was corrected for background, donor leak into the acceptor channel, and γ-factor. (B) The mean anisotropy values of the donor-only molecules (Left) and molecules with low E (Center) and high E (Right) values are 0.14, 0.17, and 0.21, respectively. With the corresponding donor lifetimes of 3.2 ns, 2.4 ns, and 1.3 ns, the reorientational correlation times of the donor calculated using Eq. S41 are 1.7 ns, 1.7 ns, and 1.5 ns, respectively. (C) The mean acceptor anisotropy values calculated for the molecules with low and high E are very small (0.01 and −0.01).

The presence of the two peaks at E = 0.45–0.5 and E = 0.7–0.8 may suggest that two stable conformations exist in the dimer state. However, it turns out that the high-E peak is not a second dimer state, but corresponds to the tetramer. To show this, we constructed a transition map from the trajectories exhibiting transitions between high and low FRET efficiencies. If there were two dimer states, these transitions would be reversible because there is only one tetramer conformation (46). However, the transitions are almost unidirectional toward the low-E state (Fig. 4A and Fig. S4B). In addition, multiple transitions are also observed (Fig. S4A, Lower Right trajectory). Therefore, it is more likely that these transitions result from an irreversible photobleaching of multiple acceptors in the tetramer. The acceptor labeling efficiency is not 100%, so the number of photobleaching steps in the tetramer varies from one to three, which is consistent with the trajectories shown in Fig. S4A. In the case of C-terminal–labeled TD (TD-A1), the FRET efficiency histogram is more widely distributed, and it is possible to assign the dimers and tetramers with combinations of the active and inactive acceptors (Fig. 4A). The peak at E = 0.8 results from the tetramers that have an active acceptor in chain 3 and the peak at E ∼ 0.4 consists of both the dimer and the tetramers without an active acceptor in chain 3. The transition map of the E distributions of A1-TD can be explained similarly (Fig. S4B). In the following section, we show that the dissociation constant of the tetramer is very low and the tetramer is formed even at 10 nM.

Fig. 4.

Photobleaching of the acceptors in the tetramer and characterization of the dimer conformation in the two-color experiment. Trajectories showing transitions in the FRET efficiency are shown in Fig. S4A. (A, Left) Transition map constructed from the FRET efficiencies before [E (initial)] and after [E (final)] transitions in the experiment with the immobilized donor-labeled TD incubated with 10 nM acceptor-labeled TD (TD-A1). (A, Right) The expected E values of the three clusters of the tetramer species with various active acceptor labels (enclosed in black rectangles). The donor is attached to chain 1. All of the tetramer species with an active acceptor in chain 3 are aggregated in the peak at E ∼ 0.8 regardless of the acceptor status of the other sites. (Additional transfer to the acceptor in chain 2 or 4 increases the FRET efficiency by only 0.03.) The FRET efficiency of the tetramer with two active acceptors in chains 2 and 4 is expected to be ∼0.5. The tetramers with a single active acceptor in chain 2 or 4 and the dimer are expected to appear at E ∼ 0.4. In A, Left, blue and green ellipses indicate the locations of the transitions between these clusters above (dashed) and below (solid) the diagonal. (B) FRET efficiency histograms of the dimers constructed from the trajectories with a single acceptor bleaching followed by donor bleaching. (B, Right) A small peak at E ∼ 0.8 for TD-A1 results from the tetramer with a single active acceptor in chain 3. (C) Two-dimensional FRET efficiency– donor lifetime plot of the data in B. FRET efficiencies and donor lifetimes were obtained as described in Fig. 3.

Because the NMR structure is known for the tetramer, our goal was to characterize the dimer at a low concentration where the tetramer is not formed. However, the coexistence of the dimer and tetramer due to the similar dimer and tetramer dissociation constants makes this difficult. Our problem, then, is to obtain dimer information in the presence of the tetramer. Because the dimer contains only one donor and one acceptor, we selectively analyzed trajectories that exhibit a single acceptor photobleaching followed by a donor photobleaching (Fig. 3A), which would be dimer trajectories in most cases. Although these trajectories are only a small fraction of the entire data (compare Figs. 3B and 4B), it is clear that there is a significant reduction of the high-E population (E ∼ 0.7 for A1-TD and E ∼ 0.8 for TD-A1), suggesting the majority of the data result from the dimers. (The remaining high-E population corresponds to the tetramer with only one active acceptor.) The low FRET efficiency of the dimer suggests that the dimer conformation is similar to that in the tetramer state.

A remaining question is why only the high-E tetramer peak disappears when an excess of the unlabeled TD is added (Fig. 3B). The simplest explanation is that the dissociation/association kinetics between the tetramer and the dimer are much faster than those between the dimer and monomer (49); once a dye-labeled dimer dissociates from a tetramer, it quickly binds an unlabeled dimer, which is in large excess, to form a new tetramer before it dissociates into monomers. (All acceptor-labeled TDs will be eventually replaced with unlabeled TD after a long time.) In this case, only one pair of the donor and acceptor locations is possible in the tetramer (i.e., chains 1 and 2 are labeled with a donor and an acceptor, respectively, and chains 3 and 4 are unlabeled; Fig. 2A). We also note that the mean FRET efficiency of this tetramer (Fig. 3B, Lower) is lower than that of the dimers in Fig. 4B both for A1-TD and TD-A1, reflecting a small conformational difference between the dimer and tetramer (Table S1 shows the comparison of the FRET efficiencies and corresponding distances). The FRET efficiency difference is slightly larger for TD-A1, suggesting that the conformation of the C terminus of the helix may be more disordered and flexible in the dimer state compared with that in the tetramer state, consistent with the NMR structural data of the mutant dimer (48) and the molecular dynamics simulation result (56) (Discussion for more explanation).

Table S1.

Mean FRET efficiency and average distance obtained from the FRET efficiency histograms and variance of the FRET efficiency distribution (σ_c²) from the 2D FRET efficiency–lifetime analysis of donor (Eq. 2) and acceptor (Eq. S26) photons before and after the acceptor blinking correction

Acceptor blinking correction			After	After	Before	After	Before	After
Color	Experiment	Oligomer	FRET efficiency	Distance, nm*	σc2 from donor delay times		σc2 from acceptor delay times
2	TD-D + 10 nM A1-TD	Dimer + tetramer†	—	—	0.07 (± 0.02)	0.06 (± 0.02)	0.07 (± 0.02)	0.06 (± 0.02)
		Dimer‡	0.525 (± 0.012)	dC1N2 = 5.1	0.08 (± 0.02)	0.07 (± 0.02)	0.07 (± 0.01)	0.06 (± 0.02)
	TD-D + 10 nM A1-TD + 2.5 μM unlabeled TD	Tetramer†	0.457 (± 0.004)	dC1N2 = 5.4	0.06 (± 0.02)	0.05 (± 0.02)	0.07 (± 0.02)	0.06 (± 0.02)
2	TD-D + 10 nM TD-A1	Dimer + tetramer†	—	—	0.06 (± 0.03)	0.06 (± 0.02)	0.07 (± 0.02)	0.06 (± 0.02)
		Dimer‡	0.415 (± 0.004)	dC1C2 = 5.5§	0.06 (± 0.03)	0.06 (± 0.02)	0.07 (± 0.02)	0.06 (± 0.02)
	TD-D + 10 nM TD-A1 + 2.5 μM unlabeled TD	Tetramer†	0.338 (± 0.002)	dC1C2 = 5.8§	0.03 (± 0.02)	0.02 (± 0.02)	0.06 (± 0.01)	0.05 (± 0.01)
2	D-TD-A1¶	Monomer	0.671 (± 0.006)	dN1C1 = 4.6	0.10 (± 0.03)	0.06 (± 0.03)	0.10 (± 0.04)	0.06 (± 0.04)
	D-TD-A1 + 2.5 μM unlabeled TD¶	Tetramer	0.817 (± 0.003)	dN1C1 = 4.1	—	—	—	—
3	D-TD-A1 + 10 nM TD-A2#	Dimer	E1 = 0.824 (± 0.004)	dN1C1 = 4.0	—	—	—	—
			E12 = 0.660 (± 0.009)	dC1C2 = 6.0§	0.06 (± 0.02)	0.03 (± 0.02)	0.06 (± 0.04)	0.03 (± 0.05)
	D-TD-A1 + 10 nM TD-A2 + 2.5 μM unlabeled TD#	Tetramer	E1 = 0.817 (± 0.006)	dN1C1 = 4.1	—	—	—	—
			E12 = 0.593 (± 0.009)	dC1C2 = 6.3§	0.05 (± 0.02)	0.02 (± 0.03)	0.08 (± 0.04)	0.06 (± 0.04)

Errors are SDs obtained from Gaussian fitting of FRET efficiency histograms or calculated by error propagation using the SD and covariance values of the FRET efficiencies and donor lifetimes in the 2D FRET efficiency–lifetime distributions.

^*The distance was calculated using E = 1/[1 + (r/R₀)⁶], with the corrected FRET efficiency and the Fӧrster radius (R₀) of 5.2 nm for the Alexa 488/Alexa 647 pair and 6.7 nm for the Alexa 647/Alexa 750 pair calculated from the measured absorption and fluorescence dye spectra. Subscripts indicate labeling positions. For example, d_C1N2 is the distance between the dyes at the C terminus of chain 1 and N terminus of chain 2. We used the quantum yields of 0.9 for Alexa 488 and 0.33 for Alexa 647, and molar extinction coefficients of 270,000 M⁻¹cm⁻¹ for Alexa 647 and 290,000 M⁻¹cm⁻¹ for Alexa 750. This calculation does not account for dye linker dynamics and implies κ² = 2/3. This results in the overestimation of the distance especially for a short distance like d_C1N1, in which case the short donor fluorescence lifetime is comparable to its reorientational correlation time (Fig. S5) (52, 80).

^†Obtained from the FRET efficiency histograms in Fig. 3B and the distributions inside the rectangles in Fig. 3C and Fig. S3C.

^‡Obtained from the FRET efficiency histograms in Fig. 4B and the distributions inside the rectangles in Fig. 4C.

^§The difference in the estimation of d_C1C2 between the experiments with the Alexa 488/Alexa 647 and Alexa 647/Alexa 750 pairs may result from the inaccuracy of the parameters such as the molar extinction coefficient and quantum yield, and the difference in κ² between the dye pairs due to the different fluorescence lifetimes.

^¶Obtained from the FRET efficiency histograms in Fig. S8A (Lower) and the distributions inside the rectangles in Fig. S8D (Lower Left).

^#E₁ was calculated from the histograms in Fig. 6E (Top Left) for the dimer and in Fig. 6D (Top Right) for the tetramer. E₁₂ was calculated from the lower-E distribution in Fig. 6D (Bottom Left) for the dimer and the histogram in Fig. 6D (Bottom Right) for the tetramer. σ_c² of the dimer was calculated from the distributions inside the rectangles in Fig. S8F. The data for the calculation of σ_c² of the tetramer is not shown.

This interpretation on the more flexible C terminus of the dimer is supported by the estimation of the conformational flexibility by 2D FRET efficiency–lifetime distributions in Figs. 3C and 4C. For a state with a single conformation, where the distance between the two dyes is fixed, the mean FRET efficiency and the lifetime are related as (57)

$${\tau }_D/{\tau }_D^0=1-E,$$ [1]

where τ_D and τ_D⁰ are donor lifetimes in the presence and absence of the acceptor. On the other hand, when there is a distribution of conformations that interconvert so rapidly that it does not appear in binned trajectories as those in Fig. 3A, the lifetime becomes longer and the peak shifts above the diagonal in the 2D plot (39–44). In this case, the donor lifetime is related to the mean FRET efficiency E as (43)

$${\tau }_D/{\tau }_D^0=1-E+\frac{{{\sigma }_c}^2}{1-E},$$ [2]

where σ_c² is the variance of the FRET efficiency distribution of the underlying rapidly interconverting conformational substates. (Note that this is not the variance of the FRET efficiency peak in the histograms plotted in Figs. 3 and 4.) For the normalized donor–acceptor distance distribution P(r), the mean FRET efficiency is $E={\displaystyle {\int }_0^{\infty }E\left(r\right)P\left(r\right)dr}$ and the variance is ${{\sigma }_c}^2={\displaystyle {\int }_0^{\infty }E{\left(r\right)}^{2}P\left(r\right)dr}-E^2,$ where E(r) = 1/[1 + (r/R₀)⁶] is the FRET efficiency when the donor–acceptor distance is r and R₀ is the Förster radius. The upward shift in the 2D plot indicates the conformational flexibility of a state. σ_c² ranges between 0 (fixed distance) and 0.25 (a system with two equally populated interconverting states with the FRET efficiency values of 0 and 1). For the Gaussian chain model (58) with the root mean-square end-to-end distance equal to the Förster radius, σ_c² = 0.11. For this analysis, it is important to determine the FRET efficiency and lifetime accurately, which requires the corrections mentioned above, including acceptor blinking. (SI Materials, Methods, and Theory, FRET Efficiency and Lifetime Corrections for Acceptor Blinking for the details of the correction procedure.) σ_c² can also be determined from the mean acceptor delay times (Eq. S26), which are not affected by acceptor blinking (44). The variance of the FRET efficiency distribution σ_c² in the dimer state (Fig. 4C) is larger than that of the tetramer state (Fig. 3C after the addition of unlabeled TD), especially in the experiment with C-terminal labeled TD (TD-A1). This result indicates that the C-terminal region of the isolated dimer is more disordered as mentioned above (Discussion and Table S1).

Dimerization Kinetics Measured by Free Diffusion Experiment.

In the immobilization experiment above, binding and dissociation are not detectable because the dimerization kinetics are very slow. Fersht and coworkers showed that the dimer dissociates on the timescale of tens of minutes, using fluorescence correlation spectroscopy (FCS) (49). Therefore, we performed free diffusion experiments after mixing solutions manually to determine the dimerization kinetics and the dissociation constants of the dimer and tetramer. Donor (Alexa 488)-labeled TD (TD-D) was incubated at a low concentration (40 pM) before the experiment to ensure the dissociation of molecules into monomers. Then the acceptor (Alexa 647)-labeled TD in a stock solution (100 nM, measured by absorbance of Alexa 647) was diluted into the donor solution (Fig. 5A). After dilution, acceptor-labeled TD dissociates and then associates with the donor-labeled monomer.

Fig. 5.

Oligomerization kinetics measured by two-color free diffusion experiment (CW excitation). (A) Acceptor (Alexa 647)-labeled TD (A1-TD or TD-A1, 100 nM) was diluted into a solution of donor (Alexa 488)-labeled TD (40 pM) manually. The final concentrations of acceptor-labeled TD were 2.5 nM, 5 nM, and 10 nM. After mixing, bursts of fluorescence emitted by molecules briefly residing in the confocal volume were measured. (B) Apparent FRET efficiency histograms constructed from photons (≥60) collected in 2-ms bins (Left, 5 nM A1-TD; Right, 5 nM TD-A1). Top Left and Top Right show the total FRET efficiency distributions, which are the sum of the distributions (Bottom five rows) collected in every 30-min interval after mixing. In the total distribution, there are three components. The donor-only component at E ∼ 0.1 was fitted to a log-normal function, and the components with higher FRET efficiencies were fitted to a double-Gaussian function. The fitted curves of the individual and the whole distributions are shown in black and red, respectively. All three components have contributions from the monomer, dimer, and tetramer (Eq. S52). (C) The fitted parameters in B were used to calculate the fraction of each component as a function of the time after mixing as described in SI Materials, Methods, and Theory. The fractions of component 1 (mid-FRET efficiency peak, p₁, open circles) and component 2 (high-FRET efficiency peak, p₂, solid squares) are plotted. The time-dependent changes of p₁ and p₂ at different acceptor-labeled TD concentrations were globally fitted as described in SI Materials, Methods, and Theory (solid curves) to obtain four parameters: the dissociation constants of the dimer (K_d^D) and tetramer (K_d^T), the acceptor-labeling efficiency (φ, fraction of the active acceptor), and the dimer dissociation rate (k_d). K_d^D = 2.2 (± 0.4) nM, K_d^T = 1.8 (± 0.2) nM, φ = 0.31 (± 0.01), and k_d = 1.6 (± 0.2) h⁻¹ for A1-TD and K_d^D = 3.1 (± 0.5) nM, K_d^T = 1.0 (± 0.2) nM, φ = 0.44 (± 0.03), and k_d = 2.0 (± 0.2) h⁻¹ for TD-A1.

The time-dependent FRET efficiency histograms are shown in Fig. 5B. They consist of a donor-only peak and a distribution corresponding to the donor- and acceptor-labeled species (dimers and tetramers). The population of the donor- and acceptor-labeled species grows with time (also see Fig. S6 for the time-dependent histograms at three different final concentrations of acceptor-labeled TD). The distributions were fitted to the sum of a log-normal function (for the donor-only peak) and a two-component Gaussian function (details of the fitting procedure in Materials and Methods and SI Materials, Methods, and Theory, Measurement of Dimerization Kinetics and Dissociation Constants of Dimer and Tetramer Using Two-Color FRET). The time-dependent relative fractions of the two Gaussian components were calculated and used in model fitting (Fig. 5C).

Fig. S6.

Apparent FRET efficiency histograms constructed from photons (≥60) collected in 2-ms bins (Left, A1-TD; Right, TD-A1) by CW excitation at various final concentrations of acceptor-labeled TD. (See the main text and Fig. 5 for the details of the experiments.) Top row shows the total FRET efficiency distributions, which are the sum of the distributions (Bottom five rows) collected in every 30-min interval after mixing. The total distributions were fitted to a log-normal function and a double-Gaussian function.

To fit the oligomerization kinetics data, we related the fractions of these two components to the concentrations of the monomer, dimer, and tetramer (details in Materials and Methods and SI Materials, Methods, and Theory, Measurement of Dimerization Kinetics and Dissociation Constants of Dimer and Tetramer Using Two-Color FRET). Due to the incomplete acceptor labeling and the presence of inactive (or photobleached) acceptors, each component has contributions from different oligomeric species with a different number of active acceptors as seen in the immobilization experiment (Fig. 4A). Therefore, we incorporated the acceptor labeling efficiency (i.e., the fraction of the active acceptor) into the model as an additional fitting parameter. The concentrations of the oligomer species were found by solving differential kinetic equations numerically (Eqs. S53 and S54). By fitting the experimentally determined fractions of the two components in the histograms to those calculated using the model (solid curves in Fig. 5C), we determined the dissociation constants and the dissociation rate of the dimer. The dissociation constants of the dimer and tetramer are 2.2 (± 0.4) nM and 1.8 (± 0.2) nM for A1-TD and 3.1 (± 0.5) nM and 1.0 (± 0.2) nM for TD-A1, respectively, and the dimer dissociation rate is 1.6 (± 0.2) h⁻¹ for A1-TD and 2.0 (± 0.2) h⁻¹ for TD-A1. These dissociation constants are similar to those obtained from the FCS measurement (Materials and Methods and SI Materials, Methods, and Theory, Determination of the Dimer and Tetramer Dissociation Constants Using FCS Measurement and Fig. S7).

Fig. S7.

Dissociation of TD probed by FCS. (A) TD monomer labeled with Alexa 488 (TD-D) was incubated with unlabeled TD at various concentrations (C₀ is the total concentration of TD in terms of monomer) in the presence of 2 mM DTT for several hours and the half-time (τ_1/2) was measured (Eq. S49). The data were fitted to the dissociation curve calculated using the ratio of the monomer, dimer, and tetramer determined by the dissociation constants (SI Materials, Methods, and Theory). K_d^D = 2.4 (± 0.3) nM, K_d^T = 1.3 (± 0.2) nM, and the diffusion times of the monomer, dimer, and tetramer are 0.105 ms, 0.117 ms, and 0.135 ms, respectively. (B and C) The fractions of labeled oligomers (B) (containing one donor-labeled monomer) and unlabeled oligomers (C) calculated using the fitted parameters. L, donor-labeled TD; U, unlabeled TD.

Dimer Conformation Probed by Three-Color FRET Experiment.

In the two-color experiments described above, only intermolecular FRET can be monitored for binding. Therefore, there is no direct information on the conformation of the monomer chain in different oligomeric states. However, the conformational difference of the monomer chain can be probed by three-color FRET. In a three-color FRET experiment, an immobilized TD monomer (D-TD-A1) labeled with the donor (D, Alexa 488) and acceptor 1 (A1, Alexa 647) at the N and C termini is incubated with TD molecules labeled with acceptor 2 (A2, Alexa 750) at the C terminus in solution (TD-A2, Fig. 2C). In this way, the intramolecular FRET between D and A1 on the same monomer chain is observed simultaneously with the intermolecular FRET from D to A2 or A1 to A2. Fig. 6A shows representative three-color trajectories. After determining E₁₂ (between A1 and A2) from the trajectory with A1 excitation (Fig. 6A, Lower, 640 nm), E₁ (between D and A1) and E₂ (between D and A2) can be calculated using the trajectory with D excitation (Fig. 6A, Upper, 485 nm) (SI Materials, Methods, and Theory, Calculation of FRET efficiencies in three-color FRET).

Fig. 6.

Dimerization and tetramerization of TD probed by three-color FRET. Data were collected for immobilized Alexa 488- and Alexa 647-labeled TD (D-TD-A1) incubated with 10 nM Alexa 750-labeled TD (TD-A2). (A) Representative fluorescence trajectories (20-ms bin time) in the donor (D, Alexa 488), acceptor 1 (A1, Alexa 647), and acceptor 2 (A2, Alexa 750) channels obtained by D excitation (485 nm, Upper panels) and A1 excitation (640 nm, Lower panels) (also see Fig. S9). Red and purple arrows indicate photobleaching of A1 and A2, respectively. (B–D) Mean FRET efficiencies and lifetimes were calculated from the initial segment of the trajectories before (Left) and after (Right) the addition of 2.5 μM unlabeled TD. (B) E₁, E₂′, and E₁₂ were calculated from the two-color segments that miss A2, A1, and D and contain more than 1,500, 1,000, and 1,000 photons, respectively. (C) Two-dimensional E₁–donor lifetime plots of the data in B. (D) E₁ and E₁₂ were obtained from the segments (>1,000 photons) in which all three dyes are active. (E) Distributions of E₁ and E₂′ constructed for the low (Left)- and high (Right)-E₁₂ species. FRET efficiencies were calculated from the segments with three active dyes (E₁, Top) or from the two-color segments immediately following photobleaching of A2 (E₁, Middle) or A1 (E₂′, Bottom). All FRET efficiency and lifetime values were corrected as described in Fig. 3 except E₂′, which was not corrected for acceptor blinking.

However, as in the two-color experiment, many trajectories miss one or two dyes due to incomplete labeling. When one dye is missing, the experiment becomes a complex two-color one, because of various combinations of two dyes, as shown in Fig. 6B. According to the labeling positions of the three fluorophores in Fig. 2, E₂ and E₁₂ correspond to the FRET efficiencies in the two-color experiments with A1-TD and TD-A1, respectively. Importantly, the consistency of the analysis is shown by the fact that both E₁₂ and E₂ distributions reproduce those of the two-color experiments. There are two peaks in the E₁₂ histogram at 10 nM of TD-A2 and the high-E peak disappears upon the addition of 2.5 μM unlabeled TD. The transition map of E₁₂ (Fig. S4D) is very similar to that of the two-color experiment with 10 nM TD-A1 (Fig. 4A and Fig. S4B). The high-E₂ component (E₂ ∼ 0.4) also disappears after 2.5 μM of unlabeled TD is added (Fig. 6B) as in the case of the two-color experiment with A1-TD. In this comparison, instead of the true FRET efficiency E₂, we use E₂′ without acceptor blinking correction because it is difficult to extract the acceptor bright-state population when E₂ is very low and similar to the value of the A2 dark state.

The intramolecular energy transfer efficiency E₁ provides additional important information. At 10 nM TD-A2, there are two peaks at E ∼ 0.67 and 0.82 and the lower-E peak disappears after 2.5 μM unlabeled TD is added (Fig. 6B). This result indicates that E ∼ 0.82 is the FRET efficiency in the dimer and tetramer states as expected from the short distance between N and C termini in the tetramer structure (Fig. 2A) and E ∼ 0.67 is the FRET efficiency of the monomer. This interpretation is supported by the two-color experiment of D-TD-A1 without TD-A2 (Fig. S8A). A single peak at E ∼ 0.67 is observed when no unlabeled protein is added in the solution and this peak is shifted to E ∼ 0.82 by the addition of 2.5 μM unlabeled TD. The low FRET efficiency of 0.67 suggests that the monomer is unfolded (intrinsically disordered; Discussion). Indeed, a positive shift from the diagonal is observed in the 2D E₁-donor lifetime plot (σ_c² = 0.06, Fig. S8D), indicating conformational flexibility of the unfolded TD molecule.

Fig. S8.

Comparison of the 1D FRET efficiency and 2D lifetime–FRET efficiency distributions before and after acceptor blinking correction in the three-color experiment. (A–F) Data were collected for immobilized Alexa 488 and Alexa 647-labeled TD (D-TD-A1) with (B, C, E, and F) and without (A and D) incubation of 10 nM Alexa 750-labeled TD (TD-A2). FRET efficiency and lifetime data without (E′) and with (E) the acceptor blinking correction are compared in A–F, Upper and Lower panels. (A) E₁ histograms of D-TD-A1 before (i.e., monomer, Left) and after (i.e., tetramer, Right) the addition of 2.5 μM unlabeled TD. (B) FRET efficiency E₁ (orange) and E₁₂ (purple) histograms of D-TD-A1 incubated with 10 nM TD-A2 before and after the subsequent addition of 2.5 μM unlabeled TD. FRET efficiencies were calculated from the initial segment of a trajectory of each molecule with only two active dyes: D and A1 (orange) or A1 and A2 (purple). (C) E₁ (orange) and E₁₂ (purple) histograms of the same experiment in B for the initial segments in which all three dyes are active. (D–F) Two-dimensional FRET efficiency–lifetime plots. Two-dimensional plots in D, E, and F correspond to E₁ histograms in A and E₁ and E₁₂ histograms in B, respectively. Segments containing more than 1,000 photons were included in the analysis except for the E₁ data in B and E, in which the threshold level is 1,500 photons. In 2D plots, lifetimes (τ) were obtained from the mean delay times of D (light green dots), A1 (orange dots), and A2 (purple dots). The data inside rectangles were used to calculate the average FRET efficiency and lifetime for the calculation of the variance of the underlying FRET efficiency distributions (Eqs. 2 and S26 and Table S1). The changes of the distributions after the acceptor blinking correction are similar to those described in the two-color experiment shown in Fig. S3.

Fig. 6D shows the E₁ and E₁₂ distributions when all three dyes are active. E₁ is always high regardless of the addition of unlabeled TD because the detection of the three fluorophores means that the dimer (or tetramer) is formed. The two peaks in the E₁₂ histogram are very similar to the results in Fig. 6B, in which Alexa 488 is absent, indicating no influence of Alexa 488 on the determination of E₁₂. The unique feature of three-color FRET is its capability of determining all three FRET efficiencies simultaneously. It should be possible to find the correlation between the E₁₂ values in the two peaks and the other two FRET efficiencies. For example, even though E₁ shows a single peak (Fig. 6D), there may be a difference between the high- and low-E₁₂ species (i.e., tetramer and dimer). However, for the TD constructs in this work, E₂ is too low (before γ correction, E₂ ∼ 0.1) to be determined accurately in the three-color experiment. (Although E₂ is not accurate, its value is so low that E₁ is reasonably accurate as shown in Fig. 6E, Top.) Instead, we avoided this problem by analyzing the segments that immediately follow photobleaching of A1 or A2. For example, in the two trajectories in Fig. 6A, Left, A2 photobleaches earlier than A1. In the two trajectories in Fig. 6A, Right, A1 photobleaches earlier than A2. Because the dissociation/association kinetics are very slow, the oligomeric state would be the same before and after photobleaching of these dyes. Therefore, it is possible to determine E₁ and E₂ more accurately using two-color segments after photobleaching of A2 and A1, respectively, and correlate these values with E₁₂ in the preceding three-color segment. Fig. 6E shows this analysis. E₂ is low when E₁₂ is low (Fig. 6E, Bottom Left) and E₂ is high when E₁₂ is high (Fig. 6E, Bottom Right). (The number of trajectories for high E₁₂ is very small because photobleaching of A1 earlier than A2 especially with high E₁₂ is a very rare event.) The difference of E₁ (Fig. 6E, Middle) is very small between the high- and low-E₁₂ states, indicating that the average distances between N and C termini of a monomer chain in the dimer and tetramer states are similar (Fig. 7).

Fig. 7.

Average distance between dye labels and the flexibility of the N and C termini in the dimer and tetramer states. The size of the dashed arrows indicates the relative flexibility of the donor (green)- and acceptor (red)-labeled termini. In the dimer state, both N and C termini are largely flexible and the variance of the FRET efficiency distribution (σ_c²) is relatively large (Figs. 3C and 4C and Table S1). In the tetramer state, the flexibility of the C terminus is smaller (indicated by smaller arrows) because the lower part of the labeled dimer is blocked by the unlabeled dimer. Therefore, σ_c² of the FRET efficiency between C termini (TD-A1, Center) is small. On the other hand, σ_c² of the tetramer with A1-TD (Left) is still large because the flexibility of the acceptor-labeled N terminus will not be affected by the tetramer formation. The corrected FRET efficiencies, average distances (calculated using κ² = 2/3), and σ_c² values obtained from the two- and three-color experiments are listed in Table S1.

SI Materials, Methods, and Theory

Protein Expression, Purification, and Dye Labeling.

An expression plasmid 6His-GB1-Avi-UA-TD-Cys encoding the sequences MGSSHHHHHHSSGMQYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTE [6His-56–residue long Ig-binding domain B1 of streptococcal protein G (GB1)], SSGLVPRGSGHM (thrombin cleavage site flanked by spacer residues), GMSGLNDIFEAQKIEWHE (biotin acceptor peptide termed Avi; Avidity LLC), SSGLVAGGGGSGGGGSGGGGS (long spacer), and UKKKPLDGEYFTLQIRGRERFEMFRELNEALELKDAQAGKEPGC (UA-TD-Cys) was engineered (DNA2.0 Inc.). UA-TD-Cys denotes incorporation of an unnatural amino acid (UA, 4-acetylphenylalanine, SC-35005; SynChem Inc.) and a cysteine residue at the N and C termini, respectively, in the 42-aa TD of the tumor suppressor protein p53 (46). The synthetic gene was cloned in PJ414 vector (DNA2.0 Inc.) flanked by Nde1 and BamH1 restriction sites at the 5′ and 3′ ends of the DNA insert, respectively. The resulting construct was verified by DNA sequencing and electrospray ionization mass spectrometry and upon its expression and isolation. TDs as binding partners contain a cysteine residue either at the N (Cys-TD) or the C terminus (TD-Cys). Chemically synthesized Cys-TD was purchased (California Peptide Research Inc.) and used without further purification. TD-Cys was expressed and purified from a second recombinant plasmid encoding just the 6His-GB1 and TD-Cys separated by a thrombin site (6His-GB1-Thrombin-TD-Cys; DNA2.0 Inc.), similar in strategy to the above.

The expression construct 6His-GB1-Avi-UA-TD-Cys, a plasmid with an isopropylthiogalactoside (IPTG)-inducible birA gene to overexpress the biotin ligase (Avidity LLC), and the pEVOL plasmid (71) for the incorporation of the UA were cotransformed into Escherichia coli BL-21 (DE3; 200131, Agilent). Cells were grown in Luria–Bertani medium, and expression was induced at an absorbance of 0.7 monitored at 600 nm with a final concentration of 1 mM IPTG for a period of 3–4 h. A final concentration of 50 µM d-biotin (B4501; Sigma-Aldrich) was added to the medium ∼30 min before induction. Typically, cells harvested from a 500-mL culture were lysed by uniform suspension in 20 mL bacterial protein extraction reagent (B-PER; Pierce) and sonication. The lysate was centrifuged at 20,000 × g for 30 min at 4 °C. The supernatant was subjected to Ni-NTA affinity chromatography in 1× PBS (1.7 mM KH₂PO₄, 5 mM Na₂HPO₄, 150 mM NaCl, pH 7.4). The bound protein was eluted in the same buffer containing 0.22 M imidazole, concentrated using Amicon Ultra centrifugal filters (EMD Millipore) to ∼2 mL, and loaded onto a Superdex 75 column (S75, 1.6 cm × 60 cm; GE HealthCare Life Sciences) equilibrated in 25 mM sodium acetate, pH 4, 150 mM NaCl, and 2 M guanidine hydrochloride (GdmCl) at a flow rate of 1.5 mL/min at room temperature. A fraction of the full-length 6His-GB1-Avi-UA-TD-Cys was labeled with fivefold molar excess of Alexa Fluor 488 hydroxylamine (A30629; Thermo Fisher Scientific) for ∼16 h. The reaction mixture was fractionated on an S75 column (1 × 30 cm) equilibrated in 25 mM Tris⋅HCl, pH 7.5, and 2 M GdmCl to remove the excess dye. Peak fractions of 6His-GB1-Avi-UA-TD-Cys were combined, concentrated, treated with a final concentration of 1.5 mM TCEP [Tris(2-carboxy-ethyl)phosphine hydrochloride; Sigma-Aldrich] for 2 h, and then reacted with fivefold molar excess of Alexa Fluor 647 maleimide (Alexa 647; A20347, Thermo Fisher Scientific) and subjected again to column fractionation on S75 in 25 mM Tris⋅HCl, pH 7.5, 100 mM NaCl, and 2 mM CaCl₂. Peak fractions containing the dye pair were concentrated, subjected to thrombin cleavage, and verified by SDS/PAGE. The cleaved, fully biotinylated Avi-UA-TD-Cys was separated away from 6His-GB1 by first collecting the unbound fraction after Ni-NTA affinity chromatography and then passing this fraction through a streptavidin Mutein column (03708152001; Roche Diagnostics) followed by elution of the bound fraction in 1× PBS containing 2 mM d-biotin. For the two-color experiments, the C-terminal cysteine residue of Avi-UA-TD-Cys was labeled with Alexa Fluor 488 (Alexa 488; A10254, Thermo Fisher Scientific).

TD-Cys was recovered from the unbound fraction following cleavage of purified 6His-GB1-thrombin-TD-Cys with thrombin and Ni-NTA affinity chromatography. Cys-TD and TD-Cys were labeled following incubation of the peptide in 6 M GdmCl, 50 mM Tris⋅HCl, pH 8, and 1.5 mM TCEP and purified on a Superdex peptide column (1 × 30 cm) in 0.5× PBS. The N- or C-terminal cysteine residue of Cys-TD or TD-Cys was labeled with Alexa Fluor 647 for the two-color experiment. For the three-color experiment, TD-Cys was labeled with Alexa Fluor 750 (Alexa 750; A30459, Thermo Fisher Scientific).

Single-Molecule Spectroscopy.

Single-molecule FRET experiments were performed using a confocal microscope system (MicroTime200; Picoquant) with a 75-μm diameter pinhole, a dichroic beamsplitter (ZT405/488/635rpc; Chroma Technology), and an oil-immersion objective (UPLSAPO, NA 1.4, × 100; Olympus). In the two-color experiment, Alexa 488 was excited by a 485-nm diode laser (LDH-D-C-485; PicoQuant) in the pulsed mode at 20 MHz. Molecules were illuminated at 0.2 μW in the immobilization experiment and 30 μW in the free diffusion experiment. Alexa 488 and Alexa 647 fluorescence was split into two channels using a beamsplitter (585DCXR; Chroma Technology) and focused through optical filters (ET525/50m for Alexa 488 and E600LP for Alexa 647; Chroma Technology) onto photon-counting avalanche photodiodes (SPCM-AQR-16; PerkinElmer Optoelectronics). In the three-color experiment, Alexa 488 and Alexa 647 were excited alternately by 485-nm (0.2–0.3 μW) and 640-nm (0.1 μW) (LDH-D-C-640; PicoQuant) lasers at 40 MHz (20 MHz each with a 25-ns delay). Fluorescence was split into three channels using two beamsplitters (720LPXR and 585DCXR; Chroma Technology) through optical filters: ET525/50m for Alexa 488, ET650/100m and ZET635NF for Alexa 647, and ET730LP for Alexa 750 (Chroma Technology).

In the immobilization experiment, biotinylated TD molecules were immobilized on a biotin-embedded, PEG-coated glass coverslip (Bio_01; Microsurfaces Inc.) via a biotin (surface)–streptavidin–biotin (protein) linkage and incubated with labeled or unlabeled TD molecules for binding. All experiments were performed in 50 mM phosphate buffer (pH 7.5). To reduce photobleaching of dyes, an oxygen scavenging system, 50 nM protocatechuate 3,4-dioxygenase (PCD) (P8279-25UN; Sigma) and 2.5 mM 3,4-dihydroxybenzoic acid (PCA) (37580–25G-F; Sigma) (72), was used. The removal of oxygen can increase triplet blinking. Therefore, a triplet quencher, 1 mM trolox, and 1 mM of l-ascorbic acid and methyl viologen (73) were also added in the solution.

For the anisotropy measurement in the two-color experiment, Alexa 488-labeled TD (TD-D) was immobilized and incubated with 10 nM Alexa 647-labeled TD (TD-A1). Photons of parallel and perpendicular polarizations were separated using a polarization cube, and donor and acceptor photons of each polarization were subsequently separated by dichroic beamsplitters (585DCXR; Chroma Technology).

In the free diffusion experiment to measure the dissociation kinetics, Avi-UA-TD-Cys labeled with Alexa 488 at its C terminus (TD-D) was preincubated at 40 pM for several hours to ensure the dissociation of TD into monomers. Then, a 100-nM solution of Alexa 647-labeled TD at the N or C terminus (A1-TD or TD-A1) was added to the final concentrations of 2.5 nM, 5 nM, and 10 nM. [The concentration was measured by Alexa 647 absorption. The actual TD concentrations are higher due to incomplete labeling. The labeling efficiency was determined by fitting (SI Materials, Methods, and Theory, Measurement of Dimerization Kinetics and Dissociation Constants of Dimer and Tetramer Using Two-Color FRET).] Cysteamine (10 mM) and β-mercaptoethanol (100 mM) (74) were used to reduce dye photobleaching and blinking. To prevent protein sticking to a glass surface, 0.01% Tween 20 was used. Fluorescence bursts were measured 10 μm above the glass surface.

The dissociation constants of TD were also measured by FCS. Alexa 488-labeled Avi-UA-TD-Cys (40 pM) was mixed with unlabeled Cys-TD at various concentrations (0–2.4 μM) and incubated in the presence of 2 mM DTT for several hours to overnight for a complete exchange of the labeled and unlabeled proteins. The FCS experiment was performed 10 μm above the glass surface at the illumination power of 12 μW.

All experiments were performed at room temperature (22 °C).

Additional details for the optical setup and single-molecule experiments have been described elsewhere (75, 76).

FRET Efficiency Calculation and Standard Corrections in Two- and Three-Color Experiments.

For the data from the immobilization experiment, trajectories were binned (20-ms bin time) and separated into segments by identifying transition points using fluorescence intensity and apparent FRET efficiency of each bin. Then, the mean FRET efficiency of each segment was calculated as described below. Accurate determination of the FRET efficiency requires several corrections. In this section we describe standard corrections that are typically used in FRET experiments for background photons, donor leak into the acceptor channel (cross-talk), direct excitation of the acceptor, and the difference in the detection efficiencies and the fluorescence quantum yields of the donor and the acceptor (γ-factor) (52).

Corrections for background and donor leak.

The correction for background photons is straightforward. In a trajectory, the mean photon count rates of each segment are subtracted by the background photon count rates of the corresponding detection channels. The background photon count rates can be obtained from the segment after photobleaching of all dyes.

After the background noise correction, the photon count rates in a segment are corrected for donor leak into the acceptor channel. The donor leak is defined as the fraction of photons emitted by the donor and detected in the acceptor channel. This value is determined using donor-only segments (no active acceptor). For the two-color experiment, the donor leak is the same as the apparent FRET efficiency of this segment,

$$l=n_A^0/(n_A^0+n_D^0),$$ [S1]

where n_A⁰ and n_D⁰ are the background-corrected photon count rates in the acceptor and donor channels of a donor-only segment. The background-corrected count rates in the acceptor and donor channels of a segment with active acceptor fluorescence (n_A and n_D) are related to those in the absence of donor leak (n_A^c and n_D^c) as n_A = n_A^c + ln_D^c and n_D = (1 – l)n_D^c, respectively. Therefore, the leak-corrected count rates n_A^c and n_D^c can be calculated as

$$n_D^c=n_D/\left(1-l\right),\hspace{1em}{n}_A^c=n_A-n_{D}l/\left(1-l\right).$$ [S2]

In the three-color experiment, the leak of the donor photons into the two acceptor channels and the leak of A1 into the A2 channel are

$$\begin{array}{c}{l}_1=n_{A1}^0/(n_{A2}^0+n_{A1}^0+n_D^0)\\ l_2=n_{A2}^0/(n_{A2}^0+n_{A1}^0+n_D^0)\\ l_{12}=n_{A2}^{Aex0}/(n_{A2}^{Aex0}+n_{A1}^{Aex0}).\end{array}$$ [S3]

Here, n_A2⁰, n_A1⁰, and n_D⁰ are the background-corrected photon count rates in the A2, A1, and donor channels of donor-only (for l₁ and l₂) segments obtained from D excitation (485 nm). $n_{A2}^{Aex0}$ and $n_{A1}^{Aex0}$ are the background-corrected count rates in the A2 and A1 channels of A1-only (for l₁₂) segments obtained from A1 excitation (640 nm). The photon count rates by A1 excitation can be corrected as in Eq. S2, and the photon count rates by D excitation can be corrected as

$$\begin{array}{l}{n}_D^c=n_D/(1-l_1-l_2)\\ n_{A1}^c=(n_{A1}-n_D^c{l}_1)/(1-l_{12})\\ n_{A2}^c=n_{A2}-n_{A1}^c{l}_{12}-n_D^c{l}_2.\end{array}$$ [S4]

The leak values were averaged over the available trajectories. In the two-color experiment, l = 0.07, and in the three-color experiment, l₁ = 0.05, l₂ = 0.002, and l₁₂ = 0.16.

γ correction in two-color FRET.

γ is the ratio of the detection efficiencies (η) and quantum yields (ϕ) of the acceptor and donor, γ = (η_Aϕ_A)/(η_Dϕ_D). In the two-color experiment, this factor can be obtained experimentally by comparing the photon count rates of the segments before and after acceptor photobleaching, using the fact that n_A + γn_D does not depend on the energy transfer efficiency. Then,

$$\gamma =n_A/(n_D^0-n_D),$$ [S5]

where n_A and n_D are the background– and donor-leak–corrected count rates of the acceptor and donor before acceptor photobleaching and n_D⁰ is that of the donor after acceptor photobleaching. The γ-corrected FRET efficiency is calculated as

$$E=n_A/(n_A+\gamma n_D).$$ [S6]

Correction for direct acceptor excitation.

In the acceptor channel, there is a fraction of photons from direct excitation of the acceptor by a donor excitation laser instead of the energy transfer from the donor, which may affect the FRET efficiency as well as the γ-factor. The true γ-factor in the absence of the direct acceptor excitation, denoted as γ^t, can be found by the same way as in Eq. S5, however, with the acceptor count rate n_A replaced by that without photons emitted by a directly excited acceptor. We find that γ^t is related to γ as

$${\gamma }^t=\frac{n_A-n_A^{dir}}{n_D^0-n_D}=\gamma (1-\frac{n_A^{dir}}{n_A}),$$ [S7]

where $n_A^{dir}$ is the count rate of photons emitted after direct acceptor excitation. The corrected FRET efficiency is calculated as

$$E=\frac{n_A-n_A^{dir}}{n_A-n_A^{dir}+{\gamma }^t{n}_D}=\frac{n_A({\gamma }^t/\gamma )}{n_A({\gamma }^t/\gamma )+{\gamma }^t{n}_D}=\frac{n_A}{n_A+\gamma n_D},$$ [S8]

where we use Eq. S7 in the second equality. Interestingly, the FRET efficiency calculated using the true γ-factor is identical to the FRET efficiency calculated using the γ-factor (Eq. S6) including photons by direct acceptor excitation, which means the direct acceptor excitation may not need to be corrected. However, γ actually depends on n_A and therefore on the FRET efficiency, as follows from Eq. S7 assuming that γ^t is invariant. This fact can make the accurate determination of the FRET efficiency complicated. Experimentally, we determined and used different γ values for different states (e.g., γ is different for the states at E₁₂ ∼ 0.6 and 0.9 in Fig. 6B). γ can be state dependent because of the different local environment that affects the fluorescence quantum yields of dyes (76, 77). In each state, the width of the FRET efficiency distribution is narrow enough to use the same γ value. By using different γ values for different states, therefore, the direct acceptor excitation effect is automatically accounted for and we can use the γ-corrected FRET efficiency as the true FRET efficiency (Eq. S8).

The fraction of photons by direct acceptor excitation (f^dir) can be experimentally measured by the ratio of the photon count rates before (n_A and n_D) and after donor bleaching but before acceptor bleaching ($n_A^{dir}$) as

$$f^{dir}=n_A^{dir}/(n_A/\gamma +n_D)=\gamma E{f}_A^{dir}.$$ [S9]

In the second equality of Eq. S9, f^dir is related to the ratio of the acceptor count rates by direct acceptor excitation and donor excitation, $f_A^{dir}=n_A^{dir}/n_A.$ This ratio, which depends on the FRET efficiency value, is used in the lifetime correction below in Eq. S25. The ratio f^dir is independent of the FRET efficiency. For the excitation of Alexa 488 in both two- and three-color experiments, direct acceptor excitation is only 2% and γ is very similar over the states. Therefore, we used the same γ value of 1.0 for the two-color experiment and 0.77 for the three-color experiment. The direct excitation of A2 by the donor excitation laser is negligible. On the other hand, in the excitation of Alexa 647 in the three-color experiment, the direct excitation of Alexa 750 is much higher, 6%. Therefore, we used different γ values of 0.46 and 0.40 for the distributions at E₁₂ ∼ 0.6 and 0.9 (Fig. 6B), respectively.

Calculation of FRET efficiencies in three-color FRET.

Photon count rates (corrected for background and donor leak) in three-color FRET after donor excitation are (scheme in Fig. 1C)

$$\begin{array}{c}{n}_D={\eta }_D{\varphi }_D{k}_D^{ex}\frac{k_D}{k_D+k_{ET1}+k_{ET2}}\\ n_{A1}={\eta }_{A1}{\varphi }_{A1}{k}_D^{ex}\frac{k_{ET1}}{k_D+k_{ET1}+k_{ET2}}\frac{k_{A1}}{k_{A1}+k_{ET12}}\\ n_{A2}={\eta }_{A2}{\varphi }_{A2}{k}_D^{ex}\left(\frac{k_{ET2}}{k_D+k_{ET1}+k_{ET2}}+\frac{k_{ET1}}{k_D+k_{ET1}+k_{ET2}}\frac{k_{ET12}}{k_{A1}+k_{ET12}}\right),\end{array}$$ [S10]

where $k_D^{ex}$ is the donor excitation rate constant and η_I and ϕ_I are the detection efficiency and quantum yield of fluorophore I (= D, A1, and A2). The first term in the parentheses for n_A2 corresponds to the transfer from the donor to A2, whereas the second term corresponds to the transfer from the donor to A1 and then to A2.

The photon count rates are related to the FRET efficiencies defined as

$$\begin{array}{c}{E}_1=\frac{k_{ET1}}{k_D+k_{ET1}}\\ E_2=\frac{k_{ET2}}{k_D+k_{ET2}}\\ E_{12}=\frac{k_{ET12}}{k_{A1}+k_{ET12}}.\end{array}$$ [S11]

Then it follows from Eq. S10 that

$$\begin{array}{c}\frac{n_{A1}}{{\gamma }_1{n}_D}=\frac{(1-E_{12})E_1}{1-E_1}\\ \frac{n_{A2}}{{\gamma }_2{n}_D}=\frac{E_{12}{E}_1}{1-E_1}+\frac{E_2}{1-E_2},\end{array}$$ [S12]

where γ-factors are defined as γ₁ = (η_A1ϕ_A1)/(η_Dϕ_D) and γ₂ = (η_A2ϕ_A2)/(η_Dϕ_D).

Using Eq. S12, one can express FRET efficiencies E₁ and E₂ in terms of the measured photon count rates,

$$\begin{array}{c}{E}_1=\frac{n_{A1}}{n_{A1}+{\gamma }_1{n}_D(1-E_{12})}\\ E_2=\frac{n_{A2}-\alpha n_{A1}}{n_{A2}-\alpha n_{A1}+{\gamma }_2{n}_D}\\ \alpha =\frac{{\gamma }_{12}{E}_{12}}{1-E_{12}},\end{array}$$ [S13]

where γ₁₂ = (η_A2ϕ_A2)/(η_A1ϕ_A1). Note that γ₁₂ = γ₂/γ₁.

The FRET efficiency E₁₂ from A1 to A2 in the above equation is determined from the photon count rates by A1 excitation. In this case, the photon count rates of A1 ($n_{A1}^{Aex}$) and A2 ($n_{A2}^{Aex}$) are

$$\begin{array}{c}{n}_{A1}^{Aex}={\eta }_{A1}{\varphi }_{A1}{k}_{A1}^{ex}\frac{k_{A1}}{k_{A1}+k_{ET12}}\\ n_{A2}^{Aex}={\eta }_{A2}{\varphi }_{A2}{k}_{A1}^{ex}\frac{k_{ET12}}{k_{A1}+k_{ET12}}.\end{array}$$ [S14]

Here, the superscript in $n_{A1}^{Aex}$ and $n_{A2}^{Aex}$ indicates that the photons are emitted after excitation of A1. Therefore, FRET efficiency E₁₂ and factor α in Eq. S13 in terms of the measured photon count rates are

$$\begin{array}{c}{E}_{12}=\frac{n_{A2}^{Aex}}{n_{A2}^{Aex}+{\gamma }_{12}{n}_{A1}^{Aex}}\\ \alpha =\frac{n_{A2}^{Aex}}{n_{A1}^{Aex}}.\end{array}$$ [S15]

The expressions of the FRET efficiencies in Eqs. S13 and S15 are equivalent to those in equations 19–21 in ref. 25 and equations S41–S43 in ref. 24.

Instrument Response Function and Determination and Correction of Fluorescence Lifetimes.

Instrument response functions (IRFs) of detection channels by 485-nm and 640-nm excitation were measured using the reflected light from a glass surface and fitted to the Gamma distribution (44),

$$IRF\left(t\right)=\frac{k_{\gamma }}{\mathrm{\Gamma }\left(a\right)}{\left(k_{\gamma }t\right)}^{a-1}{e}^{-k_{\gamma }t},$$ [S16]

where Γ(a) is the Gamma function and a and k_γ are positive parameters. These parameters are found by fitting the experimental IRF to C × IRF(t − t₀), with two additional parameters, the amplitude C and time t₀ (Fig. S1A). The mean delay time of the IRF is ${\tau }_{IRF}^0={\displaystyle {\int }_0^{\infty }tIRF\left(t\right)dt}=a/k_{\gamma }.$

The convolution of the Gamma distribution and an exponential function can be found analytically (44),

$${\displaystyle {\int }_0^{t}IRF(t-t\prime )e^{-kt\prime }dt\prime }={\left(k_{\gamma }t\right)}^a{\gamma }^{\ast }\left(a,(k_{\gamma }-k)t\right)e^{-kt},$$ [S17]

where γ*(a, z) = z^−aP(a, z) and P(a, z) is the incomplete gamma function defined as $P(a,z)={\displaystyle {\int }_0^z{x}^{a-1}{e}^{-x}dx}/\mathrm{\Gamma }\left(a\right).$ This function has the following series expansion (78):

$${\gamma }^{\ast }(a,z)=\exp (-z){\displaystyle \sum _{n=0}^{\infty }\frac{z^n}{\mathrm{\Gamma }(a+n+1)}}=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\displaystyle \sum _{n=0}^{\infty }\frac{{(-z)}^n}{(a+n)n!}}.$$ [S18]

In this work, the fluorescence lifetime was determined from the mean delay times of photons. The mean donor delay time (the donor lifetime) was calculated as

$${\tau }_D=\langle \delta t_D\rangle -\delta t_{D0}-{\tau }_{IR{F}_D}^0,$$ [S19]

where $\langle \delta t_D\rangle$ is the average time delay of the donor photons from the laser trigger signal, ${\tau }_{IR{F}_D}^0$ is the mean delay time due to the donor IRF determined from the IRF fitting, and δt_D0 is the origin of the delay times. To determine δt_D0, the donor delay time distribution in the absence of the active acceptor (obtained from the donor-only trajectories) was fitted to the convolution of the IRF and a biexponential function as (Fig. S1B)

$$P_D\left(\delta t\right)={\displaystyle {\int }_0^{\delta t}IR{F}_D(\delta t-\delta t_{D0}-t)\left[A_1{\tau }_1^{-1}\exp (-t/{\tau }_1)+A_2{\tau }_2^{-1}\exp (-t/{\tau }_2)\right]dt}+B_D,$$ [S20]

where constant B_D accounts for the contribution from background photons. Note that the biexponential function is used as a fitting function and the two lifetime components are not related to any fluorescence or molecular species. This fitting allows one to determine the origin of the delay times, δt_D0, as well as the donor lifetime τ_D = (A₁τ₁ + A₂τ₂)/(A₁ + A₂). We found that the donor lifetime in the absence of the acceptor τ_D⁰ is 3.1 ns (Alexa 488) in the two-color experiment and 3.2 ns (Alexa 488) and 1.3 ns (Alexa 647) in the three-color experiment. Fig. S1D shows that the donor lifetime distributions obtained by Eq. S19 and by the more sophisticated procedure in Eq. S20 are very similar.

The mean delay time of the acceptor photons was obtained similarly,

$${\tau }_A=\langle \delta t_A\rangle -\delta t_{A0}-{\tau }_{IR{F}_A}^0.$$ [S21]

The origin of the delay time δt_A0 in the acceptor channel was obtained from fitting the delay time distributions of the donor photons leaking into the acceptor channel in the analysis of donor-only segments. The donor lifetime determined from the mean acceptor delay time τ_D^A is given by (44)

$${{\tau }_D}^A={\tau }_A-{\tau }_A^0,$$ [S22]

where ${\tau }_A^0$ is the acceptor lifetime by direct acceptor excitation. ${\tau }_A^0$ for Alexa 647 was obtained by 640-nm excitation and ${\tau }_A^0$ for Alexa 750 was determined by fitting the delay time distribution of the background photons from the segment after photonbleaching of all dyes in the three-color experiment (Eq. S20). A significant fraction of the background photons are A2 photons from the A2-labeled proteins in solution (10 nM) by 640-nm excitation. ${\tau }_A^0$ values for Alexa 647 and Alexa 750 are 1.3 ns and 0.6 ns, respectively.

Similar to the calculation of FRET efficiencies, accurate determination of lifetimes requires several corrections (44). First, we correct the lifetimes for background noise. The mean delay times (τ_A and τ_D) that include the contribution from background photons are related to the mean delay times (τ_A^c and τ_D^c) without background photons as (n_I^c + b_I)τ_I = n_I^cτ_I^c + b_Iτ_I^b, where n_I^c is the background-subtracted count rate, b_I is the background photon count rate, τ_I^b is the mean delay time of the background photons, and the subscript I = D, A indicates the photon color. Therefore, the contribution of background photons can be corrected as

$${\tau }_I^c={\tau }_I-\left({\tau }_I^b-{\tau }_I\right)b_I/n_I^c,\hspace{1em}I=D,A.$$ [S23]

The acceptor mean delay time needs to be further corrected for the leak of donor photons. The mean acceptor delay times in the presence (τ_A) and absence (τ_A^c) of donor leak are related by (n_A^c + ln_D^c)τ_A = n_A^cτ_A^c + ln_D^cτ_D, where n_A^c and n_D^c are the photon count rates corrected for donor leak (Eq. S2) in addition to the background photon correction performed earlier. Therefore, the donor-leak–corrected delay time is

$${\tau }_A^c={\tau }_A+\left({\tau }_A-{\tau }_D\right)\mathit{l}{\mathit{n}}_{\mathit{D}}^{\mathit{c}}/n_A^c.$$ [S24]

Finally, the acceptor mean delay time needs to be corrected for direct acceptor excitation. The mean acceptor delay times in the presence (τ_A) and absence (τ_A^c) of direct acceptor excitation are related by n_Aτ_A = (n_A – n_A^dir)τ_A^c + n_A^dirτ_A⁰, where τ_A⁰ is the acceptor excited-state lifetime, n_A^dir is the count rate of the photons emitted after the direct acceptor excitation, and n_A is the acceptor count rate of photons emitted after both the energy transfer and direct excitation. Therefore, the mean delay time corrected for the direct acceptor excitation is

$${\tau }_A^c={\tau }_A+\left({\tau }_A-{\tau }_A^0\right)f_A^{dir}/\left(1-f_A^{dir}\right).$$ [S25]

Here, $f_A^{dir}$ = n_A^dir/n_A = f^dir/γE is found using Eq. S9.

Similar to Eq. 2, one can find the relationship between the donor lifetime determined from the mean acceptor delay time, τ_D^A, and the mean FRET efficiency (44)

$${{\tau }_D}^A/{\tau }_D^0=1-E-\frac{{{\sigma }_c}^2}{E},$$ [S26]

where σ_c² is the variance of the FRET efficiency due to the interdye distance distribution in a state. The two donor lifetimes determined from the donor and acceptor delay times in Eqs. S19 and S22 are equal when there is no conformational distribution in a state (i.e., σ_c² = 0). When there are dynamics between different conformations with different FRET efficiencies in a state, τ_D becomes larger whereas τ_D^A becomes smaller than τ_D⁰ (1 – E) (Eqs. 2 and S26).

FRET Efficiency and Lifetime Corrections for Acceptor Blinking.

Donor blinking affects the FRET efficiency and fluorescence lifetimes very little because no photon is detected except a very small fraction of background photons. In the presence of acceptor blinking, however, the FRET efficiency becomes lower and the donor lifetime becomes longer because only donor photons with long lifetimes are detected when the acceptor is in the dark state. When the acceptor blinking time is longer than the bin time, the acceptor blinking segment can be easily separated out. In this work, however, the timescale of acceptor blinking is much shorter than the bin time of 20 ms as shown in the correlation analysis (Fig. S2A). In this section, we describe how to correct the acceptor blinking effect after determining the acceptor bright-state population.

Determination of the acceptor bright-state population.

To determine the population of the acceptor bright state, we used a maximum-likelihood method analyzing photon trajectories directly without binning. The likelihood function for a photon trajectory with records of photon colors and arrival times is (53)

$$L={\mathbf{1}}^{\text{T}}{\displaystyle \prod _{i=2}^N\left[\mathbf{F}\left(C_i\right)\exp (\mathbf{K}(t_i-t_{i-1}))\right]\hspace{0.17em}\mathbf{F}\left(C_1\right)}{\mathbf{p}}_{eq},$$ [S27]

where N is the number of photons in a trajectory, C_i is the color of the ith photon (donor or acceptor), and t_i − t_i−1 is a time interval between the (i − 1)th and ith photons. K is the rate matrix; the photon color matrix F depends on the color C of a photon as F(acceptor) = E and F(donor) = I – E, where E is a diagonal matrix with the apparent FRET efficiencies of the individual states on the diagonal; I is the unity matrix; 1^T is the unit row vector (T means transpose); and p_eq is the vector of equilibrium populations. The parameters for each segment were determined by maximizing the likelihood function calculated by the diagonalization of K in Eq. S28 as described in ref. 53.

For the two-state model consisting of a bright state and a dark state, the matrix of FRET efficiencies, the rate matrix, and the vector of the equilibrium populations are given by

$$\mathit{E}=\left(\begin{array}{cc}{E_b}^{app}& 0\\ 0& {E_d}^{app}\end{array}\right),\hspace{1em}\mathbf{K}=\left(\begin{array}{cc}-k_d& k_b\\ k_d& -k_b\end{array}\right),\hspace{1em}{\mathbf{p}}_{eq}=\left(\begin{array}{c}{p}_b\\ 1-p_b\end{array}\right),$$ [S28]

where k_b (k_d) is the rate coefficient for the transition from the dark (bright) state to the bright (dark) state of the acceptor and p_b = k_b/(k_b + k_d) is the equilibrium population of the acceptor bright state. E_b^app and E_d^app are the apparent FRET efficiencies of the acceptor bright and dark states, respectively. The apparent FRET efficiency is the ratio of the acceptor count rate to the total count rate (including background photons). E_b^app is a fitting parameter in the maximum-likelihood method. In the acceptor dark state, acceptor photons are detected because of donor leak and background noise; therefore, the (uncorrected) count rate in the acceptor channel is (n – b_A – b_D)l + b_A, where n is the (uncorrected) total photon count rate, b_A and b_D are the acceptor and donor background count rates, and l is the donor leak. Then, E_d^app can be calculated for each trajectory as E_d^app = [(n – b_A – b_D)l + b_A]/n and used in Eq. S28.

Fig. S2B shows the result of this analysis on the data obtained from the two-color experiment. The blinking timescale obtained from the maximum-likelihood method (∼400 μs) agrees with the timescale of the decays of the correlation functions in Fig. S2A. The bright-state population p_b is ∼0.95. Once p_b is determined for each segment, this value can be used to correct the FRET efficiency and mean donor delay time (lifetime). Acceptor delay time is not affected by acceptor blinking.

Acceptor blinking correction in two-color experiments.

In the two-color experiment, acceptor photons are emitted only when the acceptor is in the bright state. (We assume that there is no energy transfer from the donor to the acceptor dark state.) Therefore, the acceptor photon count rate of the molecule switching between the bright and dark states is n_A = p_bn_A^b, where p_b is the population of the bright state and n_A^b is the acceptor photon count rate when the acceptor is in the bright state (here and below, superscript “b” stands for “bright”). Because donor photons are emitted from both the bright and dark states of the acceptor, the donor count rate is n_D = p_bn_D^b + (1 – p_b)n_D^d, where n_D^b and n_D^d are the donor photon count rates when the acceptor is in the bright and dark states, respectively. All photon count rates in this section are considered without background noise and donor leak (i.e., corrected). Note that because n_A + γn_D does not change during blinking, the count rate in the dark state is related to those in the bright state as n_A^b + γn_D^b = γn_D^d. The FRET efficiency of the blinking molecule is E = n_A/(n_A + γn_D) (Eq. S6) and the FRET efficiency in the bright state (i.e., corrected for acceptor blinking) is E^c = n_A^b/(n_A^b + γn_D^b). Using the above expressions for the count rates, we find that E = p_bE^c. Therefore, the FRET efficiency is corrected for acceptor blinking by

$$E^c=E/p_b.$$ [S29]

Now consider acceptor blinking correction for the mean delay times of the donor (donor lifetimes). The donor lifetime in the presence of acceptor blinking (τ_D) is the fluorescence-weighted average (40, 57) of the donor lifetimes in the acceptor bright state (τ_D^c) and dark state (τ_D⁰, donor lifetime in the absence of the acceptor),

$$n_D{\tau }_D=p_b{n}_D^b{\tau }_D^c+\left(1-p_b\right)n_D^d{\tau }_D^0,$$ [S30]

where the donor count rates (fluorescence intensities) are as explained above. Using n_D = p_bn_D^b + (1 – p_b)n_D^d and rewriting n_D^d/n_D^b = (n_A^b + γn_D^b)/γn_D^b = 1 + n_A^b/γn_D^b in terms of the FRET efficiency in the bright state, E^c = n_A^b/(n_A^b + γn_D^b), as n_D^d/n_D^b = 1/(1 – E^c), we get the following correction of the donor lifetime for acceptor blinking:

$${{\tau }_D}^c={\tau }_D-\frac{1-p_b}{p_b(1-E^c)}({\tau }_D^0-{\tau }_D).$$ [S31]

The acceptor blinking corrections in Eqs. S29 and S31 involve FRET efficiency E = p_bE^c and donor lifetime τ_D that have been corrected for background photons and donor leak.

Acceptor blinking correction in three-color FRET.

In the three-color experiment, E₁₂ can be corrected using the same correction as in Eq. S29,

$$E_{12}^c=E_{12}/p_2,$$ [S32]

where p₂ is the population of A2 in the bright state and E₁₂ is the measured (γ-corrected) FRET efficiency calculated using Eq. S15 without considering acceptor blinking.

To find FRET efficiencies E₁^c and E₂^c of A1 and A2 in the bright state using the observed (background and leak-corrected) photon count rates in the presence of acceptor blinking, we have to modify the procedure in Eq. S13. In the case of two blinking acceptors, there are four different states. The observed photon count rates are averaged over the bright and dark states of both acceptors. Assuming that blinking of the two acceptors is independent, photon count rates n_I (I = D, A1, A2) can be found as

$$n_I=p_1{p}_2{n}_I^{bb}+p_1(1-p_2)n_I^{bd}+(1-p_1)p_2{n}_I^{db}+(1-p_1)(1-p_2)n_I^{dd},$$ [S33]

where p₁ and p₂ are the populations of A1 and A2 in the bright state, and $n_I^{ij}$ is the photon count rate of color I (= D, A1, A2) in the bright (b) and dark (d) states of A1 (i) and A2 (j). When both acceptors are in the bright state, the photon count rates are given by Eq. S10. When A1 is in the dark state, we set k_ET1 → 0 in Eq. S10. When A2 is in the dark state, the photon count rates are given by Eq. S10 with k_ET2 → 0 and k_ET12 → 0. These photon count rates can be written explicitly in terms of the FRET efficiencies in the acceptor bright states (i.e., corrected for blinking and denoted by the superscript c) E₁^c, E₂^c, and E₁₂^c defined in Eq. S11:

• i)both A1 and A2 are in the dark state

$$n_D^{dd}=n,\hspace{1em}{n}_{A1}^{dd}=n_{A2}^{dd}=0,$$ [S34a]

• ii)only A1 is in the bright state

$$n_D^{bd}=n(1-E_1^c),\hspace{1em}{n}_{A1}^{bd}={\gamma }_{1}n{E}_1^c,\hspace{1em}{n}_{A2}^{bd}=0,$$ [S34b]

• iii)only A2 is in the bright state

$$n_D^{db}=n(1-E_2^c),\hspace{1em}{n}_{A1}^{db}=0,\hspace{1em}{n}_{A2}^{db}={\gamma }_{2}n{E}_2^c,$$ [S34c]

• iv)both A1 and A2 are in the bright state

$$\begin{array}{c}{n}_D^{bb}=n\frac{(1-E_1^c)(1-E_2^c)}{1-E_1^c{E}_2^c}\\ n_{A1}^{bb}={\gamma }_1{n}_D^{bb}(1-E_{12}^c)\frac{E_1^c}{1-E_1^c}\\ n_{A2}^{bb}={\gamma }_2{n}_D^{bb}\left(E_{12}^c\frac{E_1^c}{1-E_1^c}+\frac{E_2^c}{1-E_2^c}\right).\end{array}$$ [S34d]

Here, $n={\eta }_D{\phi }_D{k}_D^{ex}$ is the donor photon count rate in the absence of both acceptors.

Using these count rates and Eq. S33, we obtain the average photon count rates in the presence of blinking of both acceptors,

$$\begin{array}{c}{n}_{A1}={\gamma }_{1}n{p}_1\left[p_2(1-E_{12}^c)\frac{E_1^c(1-E_2^c)}{1-E_1^c{E}_2^c}+(1-p_2)E_1^c\right]\\ n_{A2}={\gamma }_{2}n{p}_2\left[p_1\frac{E_1^c(1-E_1^c)+E_{12}^c{E}_1^c(1-E_2^c)}{1-E_1^c{E}_2^c}+(1-p_1)E_2^c\right],\end{array}$$ [S35]

and a similar expression for n_D. It can be verified using Eq. S33 that n_D + n_A1/γ₁ + n_A2/γ₂ = n.

In the next step, we find E₂^c from the first equation in Eq. S35 and substitute it into the second equation. As a result, we have the following equation for E₁^c (which is actually a quartic equation with respect to E₁^c),

$$\begin{array}{l}{a}_3{E}_1^c{f}^2\left(E_1^c\right)+a_2{E}_1^{c}f\left(E_1^c\right)-f\left(E_1^c\right)-a_1{E}_1^c+a_0=0\\ f\left(x\right)=\frac{b_0-b_1/x}{c_0+c_{1}x},\end{array}$$ [S36]

where the coefficients are expressed in terms of the measured quantities, a₀ = n_A2/(nγ₂p₂), a₁ = p₁E₁₂, a₂ = p₁(1 + E₁₂^c) – a₀, a₃ = 1 – p₁, b₀ = 1 – p₂E₁₂^c, b₁ = n_A1/(nγ₁p₁), c₀ = p₂(1 – E₁₂^c) – b₁, c₁ = 1 – p₂, and n = n_D + n_A1/γ₁ + n_A2/γ₂. Using the solution of Eq. S36, E₁^c, the second unknown FRET efficiency E₂^c is found from

$$E_2^c=f\left(E_1^c\right)=\frac{b_0-b_1/E_1^c}{c_0+c_1{E}_1^c}.$$ [S37]

To summarize, the FRET efficiencies corrected for blinking of two acceptors, E₁^c and E₂^c, are obtained by Eqs. S36 and S37, using experimentally determined photon count rates in the coefficients a₀ and b₁, and E₁₂^c is determined in Eq. S32.

One can also express a₀ and b₁ in terms of the measured (γ-corrected) FRET efficiencies calculated in Eqs. S13 and S15 without the acceptor blinking correction (Eq. S35 with p₁ = p₂ = 1):

$$\begin{array}{c}{a}_0=\frac{1}{p_2}\frac{E_2(1-E_1)+E_{12}{E}_1(1-E_2)}{1-E_1{E}_2}\\ b_1=\frac{1}{p_1}\frac{(1-E_{12})E_1(1-E_2)}{1-E_1{E}_2}.\end{array}$$ [S38]

In these equations, there are five experimentally determined parameters, E₁, E₂, E₁₂, p₁, and p₂. The bright-state populations of A1 and A2, p₁ and p₂, are obtained from the maximum-likelihood analysis of the photon trajectories by Alexa 488 excitation and Alexa 647 excitation, respectively, in three color segments. In the determination of p₁, photons detected in both A1 and A2 channels were not distinguished but treated as a single kind to calculate the likelihood function in Eq. S27.

Summary of acceptor blinking correction.

The corrected mean donor (τ_D) delay times are given in Eq. S31 (two color) and the corrected FRET efficiencies are given in Eq. S29 (two color) and Eqs. S32, S36, and S37 (three color).

Correlation Analysis.

Autocorrelation functions of the donor and acceptor trajectories obtained from the immobilization experiment and donor–acceptor cross-correlation function were calculated as

$$C_{\alpha \beta }\left(\tau \right)=\overline{\frac{\langle N_{\alpha }(t+\tau )N_{\beta }\left(t\right)\rangle }{\langle N_{\alpha }\rangle \langle N_{\beta }\rangle }-1},\hspace{1em}\alpha ,\beta =A,D.$$ [S39]

N_D(t) and N_A(t) are the number of donor and acceptor photons in a bin at time t,〈…〉 denotes an average in a given segment in a trajectory, and the upper bar indicates the average over segments that are longer than 1 s. The results are shown in Fig. S2A.

Measurement of Fluorescence Anisotropy.

Fluorescence anisotropy (r_a) was calculated for each segment as

$$r_a=\frac{G{n}_{\left|\right|}-n_{\perp }}{G{n}_{\left|\right|}(1-3k_2)+n_{\perp }(2-3k_1)}.$$ [S40]

Here, n_|| and n_⊥ are the photon count rates in the parallel and perpendicular polarization channels for a given segment, k₁ (= 0.17) and k₂ (= 0.20) are objective calibration parameters, and G [= 0.84 (Alexa 488), 0.68 (Alexa 647)] is the correction factor for the detection efficiency of the two polarization channels (76). The anisotropy of a freely rotating donor dye is given by

$$r_a=\frac{3{\cos }^2\theta -1}{5(1+{\tau }_D/{\tau }_c)},$$ [S41]

where θ is the angle between absorption and emission dipoles, assumed to be zero, τ_D is the donor lifetime, and τ_c is the reorientational correlation time for the transition dipole moments of the dyes. Therefore, the anisotropy values depend on the FRET efficiency with different τ_D even for the same τ_c. The anisotropy values of different states and corresponding donor reorientational times are compared in Fig. S5.

Because the acceptor dye was not excited directly by a laser but the excitation energy was transferred from the donor, the value calculated using Eq. S40 is not the acceptor anisotropy. However, the average values are close to 0, indicating that orientational averaging of the acceptor is sufficiently fast.

Oligomerization Equilibrium Between Unlabeled Molecules and Between Labeled and Unlabeled Molecules.

In this section, we discuss the relative oligomer populations and the kinetics of oligomerization of indistinguishable (e.g., unlabeled) molecules and those of distinguishable (e.g., labeled and unlabeled) molecules.

First, consider dimerization of indistinguishable monomers ($\text{M}+\text{M}\leftrightarrow \text{D}$) with the concentrations of the monomer and dimer, M(t) and D(t). The rate equations are

$$\frac{dM}{dt}=-2\left(\frac{k_a{M}^2}{2}-k_{d}D\right)$$ [S42a]

$$\frac{dD}{dt}=\frac{k_a{M}^2}{2}-k_{d}D,$$ [S42b]

where k_a and k_d are the association and dissociation rate coefficients. In Eq. S42a, the stoichiometric coefficient is 2 (i.e., two molecules appear and disappear during a reaction event), whereas in Eq. S42b, the coefficient is 1 because only one molecule appears and disappears. The association rate is k_aM²/2 because the number of different pairs (and therefore, reaction events) is proportional to M²/2. The situation is the same for the equilibrium between the dimer and tetramer. The dissociation constants of the dimer (K_d^D) and the tetramer (K_d^T) are defined as

$$K_d^D=\frac{{\left(M^{eq}\right)}^2}{D^{eq}},\hspace{1em}{K}_d^T=\frac{{\left(D^{eq}\right)}^2}{T^{eq}},$$ [S43]

where M^eq, D^eq, and T^eq are the equilibrium concentrations of the monomer, dimer, and tetramer. The rate coefficients of the dimerization are related by $K_d^D=2k_d/k_a.$

Now consider dimerization of a labeled and an unlabeled monomer, ${\text{M}}_{\text{L}}+\text{M}\leftrightarrow {\text{D}}_{\text{L}}.$ We assume that the concentration of the labeled monomer is low enough that it can bind only to the unlabeled monomer. The rate equations are

$$\frac{d{M}_L}{dt}=-\left(k_a{M}_{L}M-k_d{D}_L\right),$$ [S44a]

$$\frac{d{D}_L}{dt}=k_a{M}_{L}M-k_d{D}_L,$$ [S44b]

where M_L is the concentration of the labeled monomer and D_L is the concentrations of the dimer consisting of a labeled and an unlabeled monomer. Here the stoichiometric coefficient is 1 for both reactants. Unlike the case of indistinguishable molecules, the association rate is proportional to M_LM (compare with Eq. S42). In these reactions, the dissociation constants are

$$K_d^{DL}=\frac{{M_L}^{eq}{M}^{eq}}{{D_L}^{eq}}=\frac{k_d}{k_a}=\frac{K_d^D}{2},\hspace{1em}{K}_d^{TL}=\frac{{D_L}^{eq}{D}^{eq}}{{T_L}^{eq}}=\frac{K_d^T}{2},$$ [S45]

where M_L^eq, D_L^eq, and T_L^eq are the equilibrium concentrations of the monomer, dimer, and tetramer bearing a single labeled molecule. Note that the dissociation constants of the labeled molecules (K_d^DL and K_d^TL) are smaller than those of unlabeled molecules by a factor of 2. Therefore, we find that the equilibrium population ratios of the labeled monomer, dimer, and tetramer are different from those of unlabeled species as

$$M_L^{eq}:D_L^{eq}:T_L^{eq}=M^{eq}:2D^{eq}:4T^{eq}.$$ [S46]

This difference should be carefully considered in the determination of the dissociation constants and rate coefficients from the experiments using labeled molecules as described below.

Determination of the Dimer and Tetramer Dissociation Constants Using FCS Measurement.

Fersht and coworkers have determined the dissociation constants of the dimer and tetramer of TD (31-residue peptide and full-length protein) by measuring translational diffusion times as a function of the TD concentration using FCS (49). Following their experimental method, we mixed Alexa 488-labeled TD at its C terminus (Avi-UA-TD-Cys) (40 pM) with unlabeled TD (Cys-TD) at various concentrations.

The correlation function of a mixture of donor-labeled monomer, dimer, and tetramer is given by

$$C\left(\tau \right)=\frac{1}{N}\hspace{0.17em}{\displaystyle \sum _J{f}_J{\left(1+\tau /{\tau }_J\right)}^{-1}{\left(1+{\alpha }^{-2}\tau /{\tau }_J\right)}^{-1/2}},\hspace{1em}J=M,D,T,$$ [S47]

where α is a constant determined by the ratio of the depth of focus and the beam waist at the focus, and N is the average number of molecules in the focal volume assuming the same brightness of the species. f_J is the relative fraction of the labeled oligomeric species (f_M + f_D + f_T = 1), which is related to the relative fraction of the unlabeled species according to Eq. S46.

The equilibrium concentration of the unlabeled monomer M^eq can be obtained by solving the following polynomial equation,

$$4{\left(M^{eq}\right)}^4+2K_d^T{K}_d^D{\left(M^{eq}\right)}^2+K_d^T{\left(K_d^D\right)}^2(M^{eq}-C_0)=0,$$ [S48]

where C₀ is the total concentration of the unlabeled protein in terms of monomer. This equation follows from the conservation condition (C₀ = M^eq + 2D^eq + 4T^eq) and Eq. S43. The concentrations of the dimer D^eq and tetramer T^eq are determined by Eq. S43. Then, using Eq. S46, the fraction (f_J) of the labeled oligomeric species can be calculated as a function of the total protein concentration C₀ for given dissociation constants.

One way to obtain the dissociation constants, K_d^D and K_d^T, is to fit all FCS data globally to the calculated FCS curves at various C₀, using Eq. S47. However, there are too many parameters including the amplitude of each FCS curve, and we found that the uncertainties of parameters are large. Instead, we took a simpler approach. We first fit individual correlation curves to find half-times, τ_1/2 (i.e., the time when the value of the correlation function is one-half of the amplitude). As a fitting function, we used a function for a 2D diffusion model with a single component. Because the depth of focus is much larger than the beam waist at the focus, this would be a good approximation:

$$C\left(\tau \right)=\frac{1}{N}{\left(1+\tau /{\tau }_{1/2}\right)}^{-1}.$$ [S49]

Then, a dissociation curve τ_1/2(C₀) was constructed as a function of C₀ (Fig. S7). This curve was fitted to the half-times calculated from the three-component correlation function in Eq. S47 with α² → ∞ as

$${\displaystyle \sum _J{f}_J{\left(1+{\tau }_{1/2}/{\tau }_J\right)}^{-1}}=\frac{1}{2},\hspace{1em}J=M,D,T.$$ [S50]

In the analysis shown in Fig. S7, there are four fitting parameters: $K_d^D,$ $K_d^T,$ τ_M (= 0.105 ms), and τ_T (= 0.135 ms). The dimer diffusion time, τ_D, was fixed to a value obtained using the relationship between the diffusion coefficient and the molecular weights (τ_J ∝ m_J^1/3) as ${\tau }_D={\tau }_M+({\tau }_T-{\tau }_M)({m_D}^{1/3}-{m_M}^{1/3})/({m_T}^{1/3}-{m_M}^{1/3})$ because the dimer and tetramer dissociation constants are very similar and fitting was not stable. The fitted dissociation constants are similar to those determined from the kinetic mixing experiment of the donor-labeled TD and the acceptor-labeled TD as described in the next section.

Measurement of Dimerization Kinetics and Dissociation Constants of Dimer and Tetramer Using Two-Color FRET.

The dissociation constants and dimer dissociation kinetics can be determined by free diffusion mixing experiments of the donor-labeled and acceptor-labeled proteins. As explained in the main text, Alexa 647-labeled TD (A1-TD or TD-A1) was diluted into a solution of Alexa 488-labled TD (Avi-UA-TD-Cys, 40 pM) from 100 nM to the final concentrations of 2.5 nM, 5 nM, and 10 nM, and fluorescence burst data were collected.

Fig. S6 shows the FRET efficiency histograms constructed from the bursts collected in 30-min intervals. Each histogram consists of a donor-only peak (component 0) and two overlapping components 1 and 2 at higher FRET efficiencies. Component 0 results from the donor-labeled monomer and dimer and tetramer with inactive acceptors. Component 1 corresponds to the dimer with one donor and one acceptor and the tetramers with an inactive (or missing) acceptor in chain 3 (Figs. 2A and 4A for the molecular labels). Component 2 consists of all tetramer species containing an active acceptor in chain 3. The equilibrium and kinetics parameters are obtained by fitting the time-dependent changes of the relative populations of these three components (Eq. S51) to the calculated values (Eq. S52).

First, the relative population of each component can be determined from the histogram data. The shape of each component in the histograms does not change with time, whereas the amplitude is time dependent. To accurately determine the shape of each component, histograms were summed (Fig. S6, Top) and fitted to the weighted sum of the log-normal distribution (component 0) and two Gaussians (components 1 and 2). Using these parameters for the shape of the distributions, it is possible to find the amplitudes of the components by fitting the histograms at each time interval. However, the number of events (bursts) of components 1 and 2 is small at early times (Fig. S6), and the amplitude obtained in this way may not be accurate. Instead, we determined the relative population (p_J) of component J in every 10-min interval as

$$p_J=\frac{1}{N_b}{\displaystyle \sum _{i=1}^{N_b}\left[P_J\left(E_i\right)/{\displaystyle \sum _{k=0}^2{P}_k\left(E_i\right)}\right]},\hspace{1em}J=0,1,2,$$ [S51]

where P_J(E_i) is the normalized probability density function of the Jth component (either log-normal or Gaussian) for the FRET efficiency (E_i) of the ith burst (2-ms duration) in a given 10-min interval. The quantity in the brackets is the probability for a burst to belong to the Jth component. N_b is the number of bursts containing ≥60 photons in that interval. p₀ + p₁ + p₂ = 1. p₁ and p₂ are plotted in Fig. 5C.

Next, we calculate the theoretical values of the relative populations. As explained above, the three components consist of monomer, dimer, and tetramer with one donor-labeled TD and various combinations of acceptor-labeled and unlabeled TDs because of incomplete acceptor labeling. With the labeling efficiency of the acceptor φ (more precisely, fraction of TD molecules with a fluorescing acceptor), the relative populations of the three components are obtained by summing the populations of all species according to the diagram in Fig. 4A (three combinations for p₀ and four combinations for both p₁ and p₂). After some algebra, we find

$$\begin{array}{c}{p}_0={C_{L0}}^{-1}\left[M_L+(1-\phi )D_L+{(1-\phi )}^3{T}_L\right]\\ p_1={C_{L0}}^{-1}\left[\phi D_L+\phi (1-\phi )(2-\phi )T_L\right]\\ p_2={C_{L0}}^{-1}\phi T_L,\end{array}$$ [S52]

where C_L0 (= M_L + D_L + T_L) is the total concentration of donor-labeled molecules and a normalization factor to make p₀ + p₁ + p₂ = 1. M_L, D_L, and T_L are the concentrations of the monomer, dimer, and tetramer with one donor dye, respectively (subscript L stands for the donor label). For example, the second and third terms in p₀ are the fractions of the dimer and tetramer with no active acceptor.

To calculate the relative populations in Eq. S52, we need to find the concentration ratio of the donor-labeled oligomers at time t. Because there is no preference of binding of the donor-labeled TD to the acceptor-labeled TD and unlabeled TD, there is no need to distinguish these two species. In this section, M, D, and T are the sum of the concentrations of acceptor-labeled TD and unlabeled TD. Once the concentrations of the oligomers are determined, the oligomers with acceptor-labeled and unlabeled TD are partitioned as described in Eq. S52.

Before mixing (dilution), the species are equilibrated. The total concentration of TD in a stock solution is C₀ = M₀^eq + 2D₀^eq + 4T₀^eq and each concentration can be obtained by solving Eqs. S43 and S48. After mixing, the equilibrium condition is changed. To find new concentrations, we use two simplifications. First, because the number of donor-labeled molecules is much smaller compared with the other (acceptor-labeled or unlabeled) molecules, the concentrations of the species without donor labels are not affected by the donor-labeled species. Therefore, we first find the time-dependent concentrations, M, D, and T, and then find the concentrations of the donor-labeled species, M_L, D_L, and T_L. Second, we assume that the equilibrium between the dimer and tetramer is instantaneous because it is much faster than the equilibration time between the monomer and dimer as indicated by the similar ratio of components 1 and 2 over time (Fig. 5C and Fig. S6).

The concentration of the dimer and tetramer after mixing can be obtained as a function of the monomer concentration at time t using Eq. S43 and βC₀ = M + 2D + 4T, where β is a dilution factor. Then, the monomer concentration M(t) is found by solving a differential equation, Eq. S42a, with the initial condition of M(0) = βM₀^eq:

$$\begin{array}{c}\frac{dM\left(t\right)}{dt}=-k_a{M}^2\left(t\right)+2k_{d}D\left(t\right)\\ D\left(t\right)=\frac{K_d^T}{4}\left(-1+\sqrt{1+4(\beta C_0-M\left(t\right))/K_d^T}\right)\\ T\left(t\right)=D^2\left(t\right)/K_d^T.\end{array}$$ [S53]

Note that the association rate constant is related to the dimer dissociation equilibrium and rate constants as k_a = 2k_d /K_d^D.

Now we consider the concentration of the donor-labeled oligomers. The oligomerization equilibrium between the donor-labeled TD and either acceptor-labeled TD or unlabeled TD is described by Eq. S45. Then, the concentrations of the donor-labeled TD oligomers can be obtained by solving another differential equation using Eq. S44a with the initial condition of M_L(0) = C_L0, the total donor-labeled TD concentration, and using the dissociation constants given by Eq. S45:

$$\begin{array}{c}\frac{d{M}_L(t)}{dt}=-k_a{M}_L(t)M(t)+k_d{D}_L(t)\\ D_L(t)=\left(C_{L0}-M_L(t)\right){\left(1+\frac{2M^2(t)}{K_d^D{K}_d^T}\right)}^{-1}\\ T_L(t)=2D_L(t)D(t)/K_d^T.\end{array}$$ [S54]

For the dimer concentration in Eq. S54, we used the conservation relation $C_{L0}=M_L+D_L+T_L=M_L+D_L(1+2M^2/K_d^D{K}_d^T).$ Because an analytic solution of Eq. S53 does not exist, Eqs. S53 and S54 were solved numerically (ode45 in MATLAB). With these M_L, D_L, and T_L values, the relative populations of the three components in FRET efficiency histograms can be calculated as a function of time after mixing, using Eq. S52. The experimental values p₁ and p₂ (Eq. S51) were fitted globally to these calculated values to determine k_d, $K_d^D,$ $K_d^T,$ and φ.

Photophysics of Alexa 488 and Alexa 647.

In the immobilization experiment, the analysis and interpretation of results are often complicated by the interference of dye photophysics in addition to photoblinking or photobleaching. In this work, we noticed two photophysical phenomena that result in apparent transitions in the FRET efficiency. First, we observed a red shift of Alexa 488 fluorescence of ∼25 nm (26, 76, 77) that changes both FRET efficiency and donor lifetime. Fig. S9 shows representative fluorescence trajectories exhibiting the spectral shift of Alexa 488 fluorescence. At the time when the spectral shift occurs, indicated by a black arrow, the FRET efficiency increases because the red shift of the donor fluorescence increases the Förster radius due to greater spectral overlap. It is expected that the donor lifetime decreases when the FRET efficiency increases. However, the donor lifetime appears unchanged or slightly increased (the distributions are on the diagonal of the transition map in Fig. S9C). This happens because the spectral shift also increases the donor lifetime in the absence of acceptor (76, 79) as indicated by the longer donor-only lifetime (∼4 ns) after acceptor photobleaching compared with that without a spectral shift (∼3.1 ns) in Fig. 3B and Fig. S3. This spectral shift occurs regardless of the presence of the acceptor dyes. Because the spectral shift is not reversible on the timescale of the measurement, the segments after the spectral shift were excluded from the analysis.

Fig. S9.

Photophysics of Alexa 488 and Alexa 647. (A–C) A red shift of Alexa 488 fluorescence of ∼25 nm (26, 76, 77) changes the FRET efficiency and donor lifetime. (A) Representative fluorescence trajectories exhibiting the spectral shift of Alexa 488 fluorescence. The point of spectral shift is indicated by a black arrow. Quoted numbers in blue are the donor lifetimes (in nanoseconds) before and after spectral shift and after photobleaching of the acceptor. (B) Red shift of the donor fluorescence increases Förster radius due to the greater spectral overlap, which results in the increase of the FRET efficiency as indicated by the distribution located above the diagonal line. (C) The donor lifetime appears unchanged (distributions are on the diagonal) because the spectral shift also increases the donor lifetime (76, 79). The donor-only lifetime (∼4 ns) after acceptor photobleaching is longer than that without a spectral shift (∼3.1 ns) in Fig. 3B and Fig. S3. (D) There are trajectories of immobilized Alexa 488 and Alexa 647-labeled TD (D-TD-A1) exhibiting a decrease in the FRET efficiency (black arrows). The direct excitation (640 nm) of Alexa 647 shows that the acceptor fluorescence intensity also decreases at the same time. These transitions (more than 20% change in Alexa 647 fluorescence intensity by A1 excitation) are shown in the transition map on (D, Right). Unlike the transitions observed in the two-color experiments by photobleaching of the acceptors in the tetramer (Fig. 4 and Fig. S4), these transitions are not aligned with the FRET efficiency distributions but more scattered, suggesting these transitions result from a different process. (E) Similar decreases in E₁ and A1 fluorescence intensity (black arrows) are observed in the three-color trajectories. In the Right trajectory, there is no change in E₁₂ (same ratio of A1 and A2 intensity by A1 excitation) before and after E₁ change, indicating that this process may result from the decrease in the extinction coefficient of Alexa 647 without changing the fluorescence spectrum, which will decrease the Förster radius for the Alexa 488/Alexa 647 pair but not for the Alexa 647/Alexa 750 pair. Red and purple arrows indicate photobleaching of Alexa 647 and Alexa 750, respectively.

We also detected possible changes in the molar extinction coefficient of Alexa 647. Fig. S9D shows the immobilization trajectories of Alexa 488- and Alexa 647-labeled TD (D-TD-A1) exhibiting a decrease in the FRET efficiency (black arrows). At the same time, Alexa 647 fluorescence by direct excitation (640 nm) also decreases. The asymmetric transition map (Fig. S9D) indicated that this is not a reversible process on the timescale of the measurement. The same kind of transition was also observed in the three-color experiment (Fig. S9E) as well as in the trajectories of Alexa 647 alone. In this work, the segments after the change in Alexa 647 absorption were not included in the analysis.

Discussion

We have described two- and three-color single-molecule FRET experiments and a fluorescence lifetime analysis and have shown how to use these methods to characterize a specific oligomeric state in an equilibrium mixture that is not separable by ensemble methods and is difficult even with single-molecule methods. We performed two-color experiments with two different TD constructs having different labeling positions, in which the information of the distances between the C terminus of one chain and the N (A1-TD) or C terminus (TD-A1) of the other chain of the dimer can be obtained. In principle, all this information could be obtained in a single three-color experiment, and we confirmed that the results of the three-color experiment reproduced the observations in the two-color experiments. However, due to various problems, such as incomplete dye labeling, and low quantum yield and rapid photobleaching of the third fluorophore (Alexa 750), the results of the three-color experiment are complex and not as quantitative as those of the two-color experiments. Consequently, we focused on extracting information that can be uniquely obtained by a three-color experiment and demonstrated how to combine this with more quantitative two-color results to derive structural and dynamical information for the various oligomer conformations. We showed that corrections for acceptor blinking are essential for accurate determination of the FRET efficiency and donor lifetime, especially for the estimation of the conformational flexibility in the 2D E–lifetime analysis.

The protein oligomerization system studied here is the tetramerization domain of the tumor suppressor protein p53. Because an atomic-resolution structure is known for the tetramer (45–47), we focused on the characterization of the monomer and dimer. Our original strategy was to characterize the monomer and dimer at a low concentration where the tetramer does not exist. Therefore, it was initially puzzling to observe broad distributions and multiple peaks in the FRET efficiency histograms (Fig. 3) and irreversible transitions between these states (Fig. 4). For the 42-residue peptide (residues 319–360 of full-length p53) used in this work, it turned out that the tetramer dissociation constant is very low, similar to the dimer dissociation constant (Fig. 5 and Figs. S6 and S7), unlike a shorter 31-residue peptide (residues 325–355) and the full-length protein (49). Therefore, the tetramer with multiple acceptors coexists with the dimer at low nanomolar concentrations, which explains the observed FRET efficiency distributions and the asymmetric transition maps. Although these features make the analysis more complex, it actually demonstrated the capability of combining two- and three-color single-molecule FRET spectroscopic methods to characterize individual oligomeric species in a mixture. We also showed that the slow dimerization kinetics can be measured by the free diffusion experiment after manual mixing (Fig. 5 and Fig. S6).

The monomer can be characterized relatively easily by simply lowering the concentration until the dimer dissociates (<100 pM). In the two-color measurement of TD labeled with Alexa 488 and Alexa 647 at the N and C termini (D-TD-A1), the FRET efficiency for the monomer is 0.67 (Fig. S8A and Table S1). In addition, the 2D FRET efficiency–lifetime analysis shows a positive shift of the distribution from the diagonal. This shift results from a broad FRET efficiency distribution (σ_c² = 0.06, Fig. S8D) reflecting the presence of rapidly interconverting conformations with different donor–acceptor distances (i.e., conformational flexibility). The low E value (0.67) compared with those of the dimer and tetramer states (0.82) and the flexibility of the monomer chain suggest that the monomer is unfolded, consistent with the result of the monomeric mutant (L344P) that does not form oligomers (59). Single-molecule data of unfolded polypeptide chains have been well described by polymer models (9) such as the Gaussian chain model (7, 41, 42, 60–62), in which the end-to-end distance (r) distribution is given by P(r) = 4πr²[3/(2π〈r²〉)]^3/2exp[–3r²/(2〈r²〉)]. The great advantage of the Gaussian chain model is that there is only one free parameter, 〈r²〉, which can be determined from the experimentally obtained mean FRET efficiency as $E={\displaystyle {\int }_0^{\infty }E\left(r\right)P\left(r\right)dr}$ (Results). With the experimentally determined 〈r²〉^1/2 = 4.8 nm, the variance of the Gaussian chain is σ_c² = 0.10, which is larger than the measured value of 0.06 (± 0.03). In other words, the chain dynamics are more restricted than expected from the Gaussian chain model. It has been known that intrinsically disordered proteins behave like a random polymer even though they are more collapsed compared with the proteins unfolded by chemical denaturant (9). Moreover, the end-to-end distance distribution does not depend much on the specific polymer model [σ_c² = 0.09 for the self-avoiding walk model (9)]. The narrower end-to-end distance distribution may result from the unusually high fraction of the charged residues (43%) in the sequence of the TD that might cause less flexibility of the unfolded chain. However, the net charge is 0 at neutral pH and the residues with opposite charges are well mixed. In this case, the chain dynamics are not affected by charges but resemble those of random polymers (63). There is a possibility that some residual secondary structure is formed in the monomer state, but it is difficult to prove experimentally, because the TD will form tetramers at the concentrations necessary for high-resolution structural techniques.

Two- and three-color binding experiments showed that the structure of the monomer chain is similar in the dimer and tetramer states, but it is more flexible in the dimer state. Although the solution is a mixture of the monomer, dimer, and tetramer at the concentration of 10 nM, in the two-color binding experiment, the selective detection of the dimer was made possible by analyzing the trajectories exhibiting single-donor and single-acceptor photobleaching (with a small fraction of tetramers). In both cases of A1-TD and TD-A1, the FRET efficiencies in the dimer state (Fig. 4B) are higher than those in the tetramer state (Fig. 3B after adding excess unlabeled TD), suggesting more flexibility in the dimer state, which may bring the two termini close to each other (Fig. 7). The flexibility of the chains monitored by the 2D FRET efficiency–lifetime analysis also supports this interpretation. The 2D distributions of the dimer and tetramer in Figs. 4C and 3C show that the width of the distance distribution between the N terminus of one monomer chain and the C terminus of the other chain in the dimer state is similar to that in the tetramer state [σ_c² = 0.07 (± 0.02) and 0.05 (± 0.02) before and after the addition of the unlabeled TD in the A1-TD binding experiment (Table S1)]. On the other hand, the distance distribution between the two C termini in the tetramer [σ_c² = 0.02 (± 0.02)] is much narrower than that in the dimer [σ_c² = 0.06 (± 0.02)]. These two distinct results suggest that the C terminus of the chain is more flexible in the dimer state than in the tetramer state whereas the flexibility of the N terminus is similar in both states, consistent with the previous NMR and simulation results (48, 56).

This difference can be explained more clearly by the dimer and tetramer structures (46) in Fig. 7. In the tetramer state, the α-helices form the core in the middle, sandwiched by the β-strands outside. In fact, seven and five residues of the N and C termini (eight and six including the residues where dyes are attached), respectively, are unstructured, and this accounts for the relatively large variance of E even in the tetramer state. Especially, the acceptor at the N terminus (Fig. 7, Left) can access the region close to the donor side. This flexibility of the N terminus will be similar in the dimer and the tetramer states, which results in the similar variance of the FRET efficiency in the A1-TD experiment. On the other hand, the C terminus is much more restricted in the tetramer state compared with the dimer state (Fig. 7, Center). Therefore, the variance of the FRET efficiency distribution in the TD-A1 experiment is much smaller for the tetramer than for the dimer. Several groups have recently developed tools to calculate an accurate distribution of the distance between the donor and acceptor by appropriate modeling of fluorophores and linkers for macromolecules with well-defined structures (64–69). This calculation may allow for a more quantitative comparison. However, as mentioned above, there is a fair amount of disorder in both N- and C-terminal regions of the monomer unit in both dimer and tetramer states. Therefore, more quantitative analysis and comparison would be possible in conjunction with molecular dynamics simulations.

The above structural analysis is based on the assumption that the conformation of the monomer chain is similar in the dimer and tetramer states, which is almost certain but cannot be proved by the two-color experiment alone. The three-color experiment provides unambiguous and more complete information for the dimer conformation. We showed that the FRET efficiency distributions of E₂ and E₁₂ are consistent with those of the two-color experiment with A1-TD and TD-A1, respectively. The FRET efficiency E₁ between the N and C termini of the same chain is high (∼0.82) in both the dimer and tetramer states, which clearly indicates that the structure (β-strand, turn, and α-helix) found in the tetramer is also present in the dimer state.

In principle, three-color FRET experiments alone can extract 3D information of molecular structure and kinetics by measuring three distances. In practice, however, various complications caused by introducing an additional dye make it difficult to obtain information with the same accuracy as in the two-color experiment. However, as we have demonstrated in this work, by combining two- and three-color experiments it is possible to obtain 3D quantitative information with high accuracy. The capability of selectively detecting specific oligomeric species will be very useful in exploring other protein oligomerization systems involved in important biological and disease processes.

Materials and Methods

Materials.

In the binding experiment, three different TD constructs were used (Fig. 2D). TD with a biotin tag and an unnatural amino acid, 4-acetylphenylalanine (70, 71), at the N terminus (Avi-UA-TD-Cys) was labeled with Alexa 488 at the C terminus (TD-D) for the two-color experiment. The same protein was site-specifically labeled with Alexa 488 and Alexa 647 at the N and C termini, respectively (D-TD-A1) for the three-color experiment. TDs with an additional cysteine residue at the N (Cys-TD) or C terminus (TD-Cys) were labeled with Alexa 647 (A1-TD or TD-A1) and Alexa 750 (TD-A2) for the two- and three-color experiments, respectively. Details of the expression, purification, and dye labeling of proteins are described in SI Materials, Methods, and Theory.

Single-Molecule Spectroscopy.

To determine the FRET efficiency and donor lifetime, TD-D (D-TD-A1) was immobilized and incubated with A1-TD or TD-A1 (TD-A2) in the two-color (three-color) experiments. For the determination of the dissociation constants and the dimerization kinetics, TD-D was manually mixed with A1-TD or TD-A1 and fluorescence bursts were collected in the free diffusion experiment. The dissociation constants were also determined using equilibrium FCS measurement of the TD-D mixed with unlabeled TD (Cys-TD).

Among various corrections, the most complex step in the FRET efficiency determination is the acceptor blinking correction. In this step, first, the population of the acceptor bright state (p_b) was obtained using the maximum-likelihood method. With this population, the FRET efficiency corrected for blinking (E^c) in the two-color experiment can be calculated relatively easily as E^c = E/p_b (Eq. S29), where E is the FRET efficiency before the blinking correction. In three-color FRET, there are four different combinations of the bright and dark states of the two acceptors. The photon count rates of these four cases can be explicitly expressed in terms of three FRET efficiencies in the absence of acceptor blinking (Eq. S34). The actual photon count rates in the presence of acceptor blinking are linear combinations of these four cases with relative weights determined by the bright-state populations of the two acceptors (Eqs. S33 and S35). Then, the FRET efficiencies corrected for acceptor blinking can be found by solving Eqs. S36 and S37.

The donor fluorescence lifetime was determined using the mean delay time (Eq. S19), which was subsequently corrected for background and acceptor blinking. The origin of the delay time was determined by fitting the delay time distribution (Eq. S20) obtained from donor-only trajectories without the active acceptor.

Further details of single-molecule experiments, theories of two- and three-color FRET, lifetime determination, corrections of background, donor leak, γ-factor, direct acceptor excitation, and acceptor blinking (maximum-likelihood method) are described in SI Materials, Methods, and Theory.

Oligomerization Equilibrium and Kinetics.

In this work, the oligomerization kinetics of TD were measured by mixing donor-labeled TD with an excess of acceptor-labeled TD in the free-diffusion experiment. Using the acceptor-labeling efficiency (the fraction of the active acceptor), the relative amplitudes of the components observed in FRET efficiency histograms (Eq. S51) were related to the concentrations of the donor-labeled species with an arbitrary number of acceptor labels (Eq. S52). There is only one donor-labeled TD monomer in each oligomeric species because the concentration of the donor-labeled TD is much smaller than those of acceptor-labeled or unlabeled TD.

The concentrations of the donor-labeled species were found by numerically solving the kinetic equations that couple the concentrations of monomers, dimers, and tetramers with and without donor labels (Eqs. S53 and S54). To simplify the kinetic equations, we assumed that the dimer–tetramer equilibration is much faster than the dimer–monomer equilibration (49). The similar growth rates of the two components (E > 0) at a given acceptor-labeled TD concentration support this assumption. There are four fitting parameters in the kinetic equations: the equilibrium dissociation constants of the dimer and tetramer, the rate constant of dimer dissociation, and the acceptor labeling efficiency. Further details of the determination of the dissociation constants and dimerization kinetics are described in SI Materials, Methods, and Theory.

It should be noted that both the equilibrium constant and the kinetic equation for a reaction between unlabeled (indistinguishable) molecules differ from those between the labeled and unlabeled (distinguishable) molecules by a statistical factor. For example, the dissociation constant of the dimer of donor-labeled and unlabeled TD monomers is one-half of the dissociation constant of the dimer of unlabeled TD monomers. As a result, the relative populations of the donor-labeled oligomers are different from those without a donor label. This fact should be considered carefully in the analysis (SI Materials, Methods, and Theory, Oligomerization Equilibrium Between Unlabeled Molecules and Between Labeled and Unlabeled Molecules).