Experimental determination of upper bound for transition path times in protein folding from single-molecule photon-by-photon trajectories

Abstract

Transition paths are a uniquely single-molecule property not yet observed for any molecular process in solution. The duration of transition paths is the tiny fraction of the time in an equilibrium single-molecule trajectory when the process actually happens. Here, we report the determination of an upper bound for the transition path time for protein folding from photon-by-photon trajectories. FRET trajectories were measured on single molecules of the dye-labeled, 56-residue 2-state protein GB1, immobilized on a glass surface via a biotin-streptavidin-biotin linkage. Characterization of individual emitted photons by their wavelength, polarization, and absolute and relative time of arrival after picosecond excitation allowed the determination of distributions of FRET efficiencies, donor and acceptor lifetimes, steady state polarizations, and waiting times in the folded and unfolded states. Comparison with the results for freely diffusing molecules showed that immobilization has no detectable effect on the structure or dynamics of the unfolded protein and only a small effect on the folding/unfolding kinetics. Analysis of the photon-by-photon trajectories yields a transition path time <200 micros, >10,000 times shorter than the mean waiting time in the unfolded state (the inverse of the folding rate coefficient). Szabo's theory for diffusive transition paths shows that this upper bound for the transition path time is consistent with previous estimates of the Kramers preexponential factor for the rate coefficient, and predicts that the transition path time is remarkably insensitive to the folding rate, with only a 2-fold difference for rate coefficients that differ by 10(5)-fold.

Fig. 1.

Schematics of a transition path (TP) (A) and an equilibrium single-molecule FRET efficiency (E) trajectory (B). The transition path time is the duration of a successful folding trajectory that passes x₀ and reaches x₁ for the first time, without visiting values of x < x₀ (8). Compared with the waiting times in each state, the transition path (TP) appears as an instantaneous jump in the FRET efficiency.

Fig. 2.

Immobilization of dye labeled proteins via biotin (protein)-streptavidin-biotin (surface) linkage to PEG-coated glass coverslip. Donor (Alexa Fluor 488) and acceptor (Alexa Fluor 594) dyes are labeled at the cysteine residues at positions 10 and 57 (Avi-GB1^K10C/C57). Spacer and AviTag sequences with a biotin molecule were added at the N terminus of the protein.

Fig. 3.

Representative FRET trajectories. FRET trajectories in Right were calculated from the donor and acceptor trajectories in Left. The photons were collected into 10-ms bins at 3 M urea and 20-ms bins at other urea concentrations.

Fig. 4.

FRET efficiency distributions. FRET efficiency distributions of immobilized proteins were obtained from the efficiencies of the initial segments of the trajectories. The ranges for folded (red), unfolded (yellow), and donor-only (green) states are 0.85 ≤ E, 0.4 ≤ E < 0.85, and E < 0.4, respectively. The equilibrium constant is equal to the ratio of the fractions belonging to the folded and unfolded states. (K = f_F/f_U) The Gaussian distribution in the unfolded peak (red dashed lines) was estimated from the distribution of the acceptor lifetime (τ_A) in Fig. 5B and does not include the small (<10%) contribution to the width from the shot noise (see Fig. 5A). Orange dashed lines are the mean FRET efficiencies of the unfolded state measured by free diffusion experiment (Inset). Yellow dashed line in the distribution from free diffusion experiment is the upper bound of the width because of shot noise (σ = [E(1 − E)/(n_A + n_D)]^1/2, n_A + n_D = 30).

Fig. 5.

Photophysics of dyes. (A) Distribution of FRET efficiency differences (ΔE_U) between 2 unfolded states separated by a folded state in the same trajectory. Blue dashed line is the distribution calculated from a Gaussian fit to the unfolded peak in Fig. 4 at 6 M urea (see Fig. S4 for data at other urea concentrations), and the red dashed line is the maximum width because of shot noise for n_A + n_D > 1000 photons. The narrower width of the red distribution compared with the blue distributions suggests some heterogeneity in either the surface or the streptavidin. (B) Lifetime distribution of the acceptor in the folded and unfolded states. For trajectories showing multiple transitions, photons of all folded (unfolded) states in the same trajectory were added up to calculate the lifetimes. Trajectories containing >6,000 photons (donor + acceptor) were considered. Vertical dashed line indicates the lifetime in the unfolded state obtained from the free diffusion experiment. (C) Trajectories in Upper demonstrate an abrupt decrease and increase in the signals in donor and acceptor channels, respectively, for Alexa Fluor 488 labeled on Avi-GB1^C57 because of the formation of a red-shifted donor emission spectrum. We call the form of the dye with the red shifted spectrum Alexa Fluor 488^R. The lifetime and the fraction of donor photons detected in the acceptor channel (the donor “leak”) were calculated for the individual segments of the trajectories consisting of >3,000 photons. Dashed lines in the histogram plots are the lifetime and the leak obtained from the free diffusion experiment. Low- and high-leak states corresponding to Alexa Fluor 488 and Alexa Fluor 488^R emission, respectively, are indicated with a red rectangular box and a green ellipse. (D) Ensemble fluorescence (solid) and absorption (dashed) spectra of dyes labeled on Avi-GB1^C57 and transmission windows of the filter set (shaded area). λ_Ex = 470 nm is indicated with an arrow. (E) Complex trajectory resulting from change of donor emission spectrum. Each spectrum and fluorescence decay were obtained from different segments in a single-trajectory: folded state (1), unfolded state and Alexa Fluor 488 (2), unfolded state and Alexa Fluor 488^R (3), and acceptor bleached state and Alexa Fluor 488^R (4). The vertical dashed lines indicate the peak positions of the spectra in F–H. (F–H) Donor and acceptor fluorescence spectra averaged over segments distinguished by the fraction of donor photons detected in the acceptor channel. (F) Alexa Fluor 488 labeled on Avi-GB1^C57 (0 M urea). (G) Alexa Fluor 488 in the donor-only (inactive acceptor) state of Avi-GB1^K10C/C57 (5 M urea). (H) Alexa Fluor 488 and Alexa Fluor 594 labeled on Avi-GB1^K10C/C57 for the unfolded state. Spectra are scaled to the height of the acceptor spectra for an emphasis on the insensitivity of the acceptor spectrum to the protein and the donor. (I) Blinking kinetics of dyes. (J) Anisotropy of Alexa Fluor 488 labeled on Avi-GB1^K10C/C57 either in donor-only (inactive acceptor) or in the unfolded states.

Fig. 6.

Transition map at 6M urea. Transitions are categorized into 3 types according to the FRET efficiencies before and after the transition: between folded and unfolded states (red), within folded or unfolded states (blue), and acceptor bleaching or blinking (green). The ranges of FRET efficiencies for folded (F) and unfolded (U) states are determined from the E histogram in Fig. 4. Ellipses indicate interpretable transitions in terms of Alexa Fluor 488 (red) and red-shifted (light blue) Alexa Fluor 488^R fluorescence. The widths of the ellipses are determined by the width of the distributions in Fig. 4. (± 2.6 σ obtained from the Gaussian fits).

Fig. 7.

Donor and acceptor intensity cross correlation averaged over unfolded segments longer than 5 seconds (continuous curves). Dashed curves are simulated decays of cross correlations with correlation times of 1 μs (light blue), 5 μs (green) and 20 μs (red) and the correlation amplitude at τ → 0 (C_DA (0) = −0.53) calculated from the E distribution of a Gaussian chain (black).

Fig. 8.

Folding and unfolding rates obtained from the exponential fits of the waiting time histograms. Errors were calculated from the fit with 95% confidence level. Rates corrected for the finite length of trajectories are indicated in red.

Fig. 9.

Comparison of single-molecule folding kinetics (blue squares uncorrected with 2σ fitting error, blue crosses corrected) with ensemble kinetics (red circles with 2σ calculated from average of 4–7 experiments) from stopped flow measurement with dye labeled proteins (λ_Ex = 488 nm). (Inset) Fluorescence decay from the stopped flow measurement exhibits a single-exponential decay (red).

Fig. 10.

Estimation of the window time within which transition occurs. FRET trajectories with 2-ms binning show instantaneous transitions (first row). Red and green circles are time tagged acceptor and donor photons (second row). The transition interval found by Eq. S17 (of SI Text) is indicated with blue vertical dashed line in the photon strings. Likelihood values are shown in cyan, which are normalized to the maximum. Interval in which transition occurs with 95% confidence level estimated by Eq. 4 are shown as dashed orange lines. Panels in the third row show FRET trajectories near the transition, which are binned by the width of the transition window (orange numbers in row 2 and orange vertical dashed lines). Horizontal dashed lines in the binned trajectories show the range of ± 2σ, where σ is the mean standard deviation in the folded (red) and unfolded (green) parts of the trajectory (σ = [E(1 − E)/(n_A + n_D)]^1/2).

Fig. 11.

Time window (τ_window) within which unfolded → folded (squares) or folded → unfolded (circles) transitions occur as a function of the inverse of the photon counting rate on the folded side of the transition point within the time window (ν_F). τ_window is the time interval defined by the dashed orange lines in Fig. 10 and Fig. S10) estimated using Eq. 4 with a 95% confidence level in the low (green, light green) and high (red, orange) excitation intensity. For comparison, the same method was applied to the photon string presented in figure 2c of ref. after digitizing the data, which results in τ_window = 2.2 ms (purple x).