Shot-noise limited single-molecule FRET histograms: comparison between theory and experiments

Abstract

We describe a simple approach and present a straightforward numerical algorithm to compute the best fit shot-noise limited proximity ratio histogram (PRH) in single-molecule fluorescence resonant energy transfer diffusion experiments. The key ingredient is the use of the experimental burst size distribution, as obtained after burst search through the photon data streams. We show how the use of an alternated laser excitation scheme and a correspondingly optimized burst search algorithm eliminates several potential artifacts affecting the calculation of the best fit shot-noise limited PRH. This algorithm is tested extensively on simulations and simple experimental systems. We find that dsDNA data exhibit a wider PRH than expected from shot noise only and hypothetically account for it by assuming a small Gaussian distribution of distances with an average standard deviation of 1.6 A. Finally, we briefly mention the results of a future publication and illustrate them with a simple two-state model system (DNA hairpin), for which the kinetic transition rates between the open and closed conformations are extracted.

Figure 1

A schematic comparison of the all-photon-burst-search (APBS) with the dual-channel-burst-search (DCBS) for three possible scenarios. The green line represents the sum of donor photons and acceptor photons detected during the green laser periods, while the red line represents the red photons detected during the red laser periods. A. Schematic relationship between photon streams and binned time traces. B. Bleaching scenario. In the case shown here, the acceptor bleaches toward the end of the burst. The APBS algorithm considers all the detected photons as belonging to the same burst and, hence, results in a calculated PR value that is lower than the true value. On the contrary, the DCBS algorithm interrupts the burst where the acceptor is bleached, resulting in an accurate PR value. C. Blinking scenario. In the case considered here, the acceptor blinks during the burst. As in A, the APBS algorithm results in a PR value that is lower than the true value. The DCBS algorithm results in two successive, smaller bursts with an accurate PR value. D. Coincidence scenario. In the case shown here, the D–A species is mixed with a donor-only molecule toward the end of the burst. The APBS algorithm considers all photons as belonging to a unique burst, resulting in an inaccurate PR value. The DCBS algorithm rejects part of the donor photons, resulting in an improved PR value.

Figure 2

A. Distribution of values {i/N; 0 ≤ i ≤ N} for N = 1–10. B. Distribution of values {i/N; 0 ≤ i ≤ N; 1 ≤ N ≤ 500}. The zero-count is not represented (=500).

Figure 3

A. 40 base pair dsDNA molecules used in this work. The top strand is labeled at position 5 (from the 5′ end) with Cy3B (donor). Five different bottom strands with complementary sequence are labeled with ATTO-647N (acceptor) at position 13, 18, 23, 28, and 35, respectively, from the 3 ′ end. B. DNA hairpin used in this work (Figure 12). The 35 bp dsDNA stem is followed on the top strand by a 5 bp ssDNA stem, a 21 bp poly-T loop, and a complementary sequence to the previous 5 bp. The acceptor dye is attached to the 5′ end of the top strand, while the donor dye is attached on the bottom strand at a 10 bp distance from the acceptor dye in the closed conformation.

Figure 4

Simulated PRH (gray box histogram) and best-fit shot-noise limited theoretical PRH (black curve) for different E values. E is the FRET efficiency used in the simulation, and ε is the best-fit parameter. A–C. PRH comparison for E = ε = 0.2, 0.5, and 0.8. Bursts were defined using an “all-photon” burst search algorithm with parameters L = 50, M = 30, and T = 500 μs. D,E. PRH for the same data as B, but using different burst size (S) subpopulations (D, 50 < S < 100; E, 300 < S < 4000), illustrating the perfect agreement between experimental and theoretical PRH. Histogram bin size: 0.01. Parameters of the simulation: k_bl = k_ISC = 0; k_e^D = 6 × 10⁵ s⁻¹; k_r^D = 10⁹ s⁻¹; k_e^A = 6 × 10⁵ s⁻¹; k_r^A = 10⁹ s⁻¹; η_D =η_A = 1 at the center of the confocal spot.

Figure 5

Influence of background counts on the PRH. Simulated bursts corresponding to molecules with a FRET efficiency of 0.2 were contaminated with 6 kHz of background. A. Using eq 17, a reasonably good fit is obtained for a PR value ε = 0.16. The residual curve shows a slight discrepancy. B. Using the known background level in the donor channel and eq 36, a better fit is obtained, with a PR value ε = 0.195, almost indistinguishable from the exact FRET efficiency. The residual curve shows a much better agreement than in the previous case.

Figure 6

Effects of bleaching, blinking, and coincident detection on the PRH. Two different burst search algorithms were used to analyze simulated data: the all-photon-burst-search (APBS) algorithm (left column) and the dual-channel-burst-search (DCBS) algorithm (middle and right columns). A. in the case of bleaching, the APBS algorithm (A1) reveals a trail of bursts connecting a donor-only population (PR = 0, S = 1) and the population of interest (PR = S = 0.5). The DCBS algorithm (A2) eliminates those bursts, resulting in a good fit of the remaining PRH (A3). B. In the case of blinking, the APBS algorithm (B1) reveals a trail of bursts directed toward the location of donor-only bursts and the population of interest. As in A, the DCBS algorithm (B2) eliminates those bursts, resulting in a good fit of the remaining PRH (B3). C. In the case of coincident detection, the APBS algorithm (C1) reveals a trail of bursts directed toward the location of donor-only bursts or the location of acceptor-only bursts (S = 0) and the population of interest. Again, the DCBS algorithm (C2) eliminates most of these bursts, resulting in a relatively good fit of the remaining PRH (C3), although an obvious discrepancy exists (the actual PRH is wider than the fitted shot-noise limited PRH).

Figure 7

Effect of detection volume mismatch on the PRH. A,B. In the presence of a simulated 10% offset of the acceptor (Gaussian) detection volume with respect to the donor (Gaussian) detection volume, the PRH (B) resulting from the APBS algorithm (A) is reasonably well fitted by a shot-noise limited PRH, although the measured PRH is clearly wider than the fitted one. C,D. For a 20% offset, the PRH (D) resulting from the APBS algorithm (C) is not well fitted by a shot-noise limited PRH. E. In actual experiments, the measured offset of the donor and acceptor detection volume is smaller than 3%, as shown by the XY and YZ cross sections. The size of each pixel is 33.33 nm and the size of the scan is 5 μm on 7 μm.

Figure 8

Artificial FRET data. Singly labeled DNA molecules were detected using a standard SMS confocal microscope in which the emitted signal was split equally by a nonpolarizing beam-splitter cube and sent to two SPAD’s. Different neutral density filters were used in front of the detectors to modulate their detected signal ratio. A. Experimental PRH (gray box histogram) and the shot-noise limited fit (black curve) for 40% attenuation in front of the acceptor SPAD. B. Same for no attenuation. C. Same with 40% attenuation in front of the donor SPAD. D. Same with 15% attenuation in front of the donor SPAD.

Figure 9

Comparison of APBS and DCBS algorithms for dsDNA data. A 50 pM dsDNA composed of Cy3B-labeled ssDNA molecules hybridized to complementary ATTO-647N-labeled ssDNA molecules was observed using a μs-ALEX setup. A. Photon stream analyzed with an APBS algorithm (L = 50, M = 30, T = 500 μs), exhibiting a small donor-only (PR ≈ 0, S ≈ 1) and acceptor-only (S ≈ 0) contamination. The corresponding PRH (upper graph) exhibits a small peak for PR ≈ 0 and small tails in both directions. B. The analysis using a DCBS algorithm (L = 25, M = 15, T = 500 μs) suppresses this contamination and the corresponding artifacts on the PRH.

Figure 10

Experimental PRH of doubly labeled dsDNA samples. Five different samples containing the same Cy3-labeled ssDNA top strand and a complementary ssDNA bottom strand labeled with ATTO-647N at different distance from the top strand label (measured in base pairs) were used. A–E. (left and center column) 8, 13, 18, 23, and 30 bp, respectively. Experimental PRH (gray bar histogram) obtained with a DCBS algorithm (L = 25, M = 15, T = 500 μs) were fitted with eq 17 (black curve). Histograms were binned with either a fixed number of bins M = 100 (left column) or the optimal number of bins (center columns) calculated as described in Appendix B. A–E. (right column) Comparison of the experimental PRH and the shot-noise limited prediction in the presence of a Gaussian distribution of distance of width σ. Fitted values are σ = 1.5, 1.7, 1.8, 1.6, and 1.5 Å, respectively.

Figure 11

Relation between energy landscape and PRH in a simple two-state system. Left: System studied, with characteristic time scales of opening and closing. Center: Schematic energy landscape; simulated shot-noise limited PRH. A. For a rigid dsDNA molecule, the unique energy minimum results in a distribution of distances, which, after convolution with the shot noise effect, yields a single-peak PRH. B. For hairpins characterized with opening and closing time scales much longer than the typical diffusion time (~1 ms), each molecule is virtually frozen in one of the two energy minima characteristic of the system. After convolution with the shot noise effect, the PRH exhibits two separate peaks. C. For hairpins with a low energy barrier between the two minima, rapid transitions between the two states are expected, and each molecule samples the two typical conformations several times during each burst, resulting in a broad peak located between the expected position of each characteristics conformation. D. For intermediate time constants comparable with the characteristic diffusion time of the molecule, the two separate peak locations are still visible, but one also expects that intermediate values will be observed. A combined analysis of the influence of shot noise and of a simple two-state kinetic scheme on the PRH allows extraction of the kinetic parameters of the system.

Figure 12

DNA hairpin and two-state model. Experimental PRH (gray columns) and two-state model, shot-noise limited PRH (black curve) calculated with 100 bins. The best fit parameters are as follows: closing rate k_C = 1425 s⁻¹, opening rate k_O = 835 s⁻¹, average closed PR ε_C = 0.81, average open PR ε_O = 0.36.