Variational Bayes analysis of a photon-based hidden Markov model for single-molecule FRET trajectories

Abstract

Single-molecule fluorescence resonance energy transfer (smFRET) measurement is a powerful technique for investigating dynamics of biomolecules, for which various efforts have been made to overcome significant stochastic noise. Time stamp (TS) measurement has been employed experimentally to enrich information within the signals, while data analyses such as the hidden Markov model (HMM) have been successfully applied to recover the trajectories of molecular state transitions from time-binned photon counting signals or images. In this article, we introduce the HMM for TS-FRET signals, employing the variational Bayes (VB) inference to solve the model, and demonstrate the application of VB-HMM-TS-FRET to simulated TS-FRET data. The same analysis using VB-HMM is conducted for other models and the previously reported change point detection scheme. The performance is compared to other analysis methods or data types and we show that our VB-HMM-TS-FRET analysis can achieve the best performance and results in the highest time resolution. Finally, an smFRET experiment was conducted to observe spontaneous branch migration of Holliday-junction DNA. VB-HMM-TS-FRET was successfully applied to reconstruct the state transition trajectory with the number of states consistent with the nucleotide sequence. The results suggest that a single migration process frequently involves rearrangement of multiple basepairs.

Figure 1

Example of TS-FRET signals, illustrating the relationship between TS and SPC signals. (A) The TS-FRET signal records arrival times of every single photon on the donor (green) and the acceptor (red) detector channels. (Inset) Experimental observables t, Δt, and ρ. (B and C) The SPC signals are made by binning photons with a fixed time period. (D) The time bin-based FRET trajectory (blue) is calculated from the SPC signals. The true FRET trajectory (purple) changes stepwise, showing the typical feature of single-molecule signals. (Color online.)

Figure 2

Example results of analyses of an intensity signal. (A) The SPC-signal with 1-ms (dashed), 5-ms (black), and 10-ms (gray) time bins are converted from a simulated TS signal (red). (B) The state transition trajectories. The original simulated data (red), the results of VB-HMM-TS (blue), and CPD (green) assign a state to each photon. The VB-HMM-PC results with 1-ms (gray), 5-ms (dashed), and 10-ms (light-shaded) bins are time-bin-based. I_ratio = 0.5 and {λ₁, λ₂, λ₃} = {20, 10, 5} ms were applied for this simulation. Full-length data are shown in Fig. S1 in the Supporting Material.

Figure 3

Example results of analyses of a dual-channel FRET signal. (A) The SPC-signals with 5-ms time bins (donor, green; acceptor, red) are converted from the simulated TS signal (not shown). The FRET trajectory (blue) is calculated from the SPC trajectories. Light-colored lines are plots using 1-ms time bins. (B) The state transition trajectories. The original simulated data (red), the results of VB-HMM-TS-FRET (blue), and VB-HMM-TS (light blue) assign a state to each photon. The results of VB-HMM-PC-FRET (dashed gray) and VB-HMM-PC (light gray) only with 1-ms bin are shown. VB-HMM-TS/-PC analyses treat only the donor photons. ΔE = 0.1, I = 10,000, and λ = 100 ms were applied to all states. Full-length data are shown in Fig. S3.

Figure 4

Statistics of an intensity analysis on 1000 Monte Carlo-simulated time series data with I_ratio = 0.5. (A) The accuracy of the NoS estimation. VB-HMM-TS achieves almost 100% accuracy over this parameter range, while VB-HMM-PC with 1-ms bin is close. (B) The accuracy in reproducing the state transition trajectory. VB-HMM-TS and VB-HMM-PC (1 ms) are again superior to others. CPD, another TS-based analysis, is equivalent at large λ. (C) The accuracy of parameter estimation of the transition rates given by the state lifetime λ. VB-HMM-TS shows almost perfect results. Error bars designate the standard deviation. Further results are summarized in Fig. S5.

Figure 5

Evaluation of the FRET signal analysis on 1000 Monte Carlo-simulated time series data with ΔE = 0.1. (Solid lines) Results of the FRET analyses. (Dashed lines) Intensity analyses. (A) The accuracy of the NoS estimation. (B) The reproducibility of the state transition trajectory. (C) Estimation of the transition rates, given by the state lifetime λ. VB-HMM-TS-FRET again shows almost perfect results. Overall, the FRET analyses are superior to the intensity-based analyses. Error bars designate the standard deviation. Further results are summarized in Fig. S6.

Figure 6

(A) Schematic of branch migration of the Holliday junction. “D” (donor) and “A” (acceptor) represent fluorescent labels. (B) Time series of fluorescence intensities from the donor (green) and the acceptor (red), respectively, acquired by smFRET observation of spontaneous branch migration. (C) Compensated intensity I_C (yellow) and (D) FRET efficiency E_FRET (purple) calculated from fluorescence intensities. (E) Variational lower bounds given by VB-HMM-TS-FRET analysis. The maximum lower bounds were obtained with the NoS of three. I_C and E_FRET trajectories reconstructed from the VB-HMM-TS-FRET result are overlaid (blue) in panels C and D, respectively. (Color online.)

Figure 7

Histograms of the FRET efficiency created from nine sets of smFRET time series. (Green histogram) Constructed from the FRET trajectories with a 5-ms bin, calculated directly from the fluorescence signals, forms a single broad peak. Individual states are not distinguishable. (Blue histogram) Constructed from the VB-HMM-TS-FRET results, in which three distinct distributions are clearly shown. (Color online.)