Supplementary Figures

Supplementary Fig. 1. Comparison of background levels of single-molecule measurement. (a-b) Typical levels of background signals of buffer solution at the power of 1 mW for 532 nm and 0.3 mW for 633 nm lasers in alternating mode. The excitation lasers were focused at a point 20 μm from the surface of a coverslip in solution: (a) without liposome and (b) with 500 μM liposome in lipid concentration. No difference in the background signals was observed with and without liposome in solution.

dynamic light scattering (DLS) measurement was performed to confirm the size of the liposomes. The DLS measurement was performed using a Zetasizer Nano S instrument (Malvern) with a HeNe laser (633 nm). The scattering was detected at a back-scattering angle of 173° at 25℃. The sample was diluted to a final lipid concentration of 50 μM. The size distribution of the vesicles was analyzed based on the intensity of the vesicles. We observed a monodisperse distribution of vesicles. The average size and polydispersity index of liposomes were 96 nm and 0.073, respectively.

Supplementary Discussion
The probability of multiple binding of HJ to a liposome When a binding event is an independent process, the probability of binding event has Poisson distribution, x is the number of bound HJ and k is the ratio of HJ and neutravidin, [HJ]/[neutravidin]. Therefore, the probability of multiple binding of HJ to neutravidin is P(x≥2|x≥1) = (P(2) + P(3) + P(4)) / (P(1) + P(2) + P(3) + P(4)). Because we used HJ: NeutrAvidin=1:100, the probability of multiple binding is ~0.005 (0.5%), which is insignificant. It is to be noted that 0.1% Biotin in liposome gives approximately 50 biotin molecules on outer membrane of a 100-nm sized liposome: here, we didn't intend to put a single biotin to a liposome. We used 1:1 molar ratio of NeutrAvidin: liposome. Because only ~ 1% of NeutrAvidin have HJ, the ratio between HJ-bound NeutrAvidin:liposome is 1:100. Then, again the probability of multiple binding of HJ-bound NeutrAvidin to a liposome is less than 0.5% from Poisson distribution. Considering these two factors, the probability of multiple binding to a liposome is less than 1%. Indeed, when we analyzed time-traces longer than 30 ms with photobleaching steps (22 total), all trace showed single photobleaching step. Thus, the contribution of multiple binding in our measurement is negligible.

Negligible effect of the orientaional fluctuation of dyes
The orientational fluctuation of the probe dyes can induce the fast dynamics we observed. For SSB diffusion on ssDNA, however, the FRET time trace of (dT) 60 only in Fig

Single-molecule alternating-laser-excitation setup
The microscope setup for alternating-laser excitation (ALEX) has been extensively described elsewhere 1 .
In brief, two lasers, a 532-nm solid-state green laser (Cobolt Samba, Cobolt) and a 633-nm HeNe laser

Fluorescence correlation spectroscopy measurements
The fluorescence correlation spectroscopy (FCS) measurements were performed using the same instrumental setup as for ALEX, but a 532-nm continuous laser at a power of 15 μW was used instead of the alternating-laser conditions. A sample of 2.5 nM oligonucleotide labeled with Atto-647N dye was measured for 4 min at room temperature, and an auto-correlation curve was then obtained using a custom LabVIEW analysis program. For tethering to liposomes, 100 nM NeutrAvidin and 100 nM biotinyl liposomes were added, and the sample was incubated for 10 min prior to the FCS measurement.

Hidden Markov and dwell-time analyses
To identify hidden states and their transitions in the collected time traces of single diffusing molecules, hidden Markov modeling (HMM) was performed using the vbFRET program, which uses a maximum evidence algorithm 2 . We performed an evaluation of the HMM method we used for FRET trace analysis.
Among HMM methods, HaMMy and vbFRET are most widely used to analyze FRET traces. The vbFRET uses a maximum evidence (ME), which is more suitable than the maximum likelihood (ML) method used in HaMMy for the analysis of time traces with fast transitions. To verify the accuracy of the HMM methods, we performed HMM analysis using HaMMy and vbFRET on synthetic FRET traces. We generated synthetic FRET traces by simulation using the parameters summarized in Supplementary Table   1; these traces show two-state dynamics with a mean dwell time of 1.44 ms -9.0 ms between transitions and 0.5 ms temporal resolution. We built analysis code using MATLAB based on the Kevin Murphy Toolbox (http://www.cs.ubc.ca/~murphyk/Software/), which generates discrete FRET time traces that have random transitions between two states. We set two states: E = 0.22 and 0.65. Then, we added Gaussian noise to each state with a standard deviation (σ) 2,3 . We used the standard deviation σ = 0.1 for each state, which was determined from our experimental result ( Supplementary Fig. 5). Then, we used two HMM programs to find hidden states in the synthetic FRET traces that had appropriate emission noise. Supplementary Fig. 6 shows a comparison of the performance of the HMM programs that were used to find the hidden states. We found that vbFRET, which is based on ME, correctly found the transitions between the two states for all synthetic FRET traces having 1.44 ms dwell times with a 25 ms trace length. On the contrary, HaMMy, which is based on ML, correctly found two states for only 69% of all the synthetic FRET traces (Supplementary Fig. 6a). Thus, HaMMy failed to correctly find two state for the 31% of the synthetic FRET traces. Next, we obtained the transition rates from the dwell time distributions obtained from HMM analysis. The theoretical transition rate was 693 s -1 , which was calculated from the transition probabilities 4 . The vbFRET (ME) analysis found a value of 624 ± 7 s -1 , corresponding to 91 % accuracy. However, the HaMMy (ML) analysis significantly underestimated the transition rate at 493 ± 18 s -1 ; this value corresponds to a 71% accuracy of the theoretical transition rate, although we excluded the 31% of traces that failed to correctly show two states in the dwell time analysis ( Supplementary Fig. 6b). Then, we varied the dwell times from 1.44 ms to 6 ms while fixing the synthetic time-trace length to 25 ms ( Supplementary Fig. 6c). vbFRET estimated the transition rates better than HaMMy for fast transitions. When we increased the length of the synthetic time trace to 50 ms, vbFRET was slightly better than HaMMy in estimating the transition rates. Importantly, the accuracy of vbFRET at determining the transition rates was always better than 90%. Thus, vbFRET is appropriate for analyzing the fast dynamic time traces in this work. After HMM, the transition-density plot and dwell-time distributions for each transition were obtained.