Single molecule fluorescence resonance energy transfer (SM FRET) is a powerful tool to study the structural dynamics of many biological molecules (1–4). This method is capable of yielding subpopulation dynamics with high signal/noise and requires very small numbers of molecules compared to bulk methods. There have been many measurements of single molecule structural dynamics ~1 ms or longer using SM FRET (5–8), but measurements with higher time resolution have been elusive due to the rate of photon emission from the commonly used fluorophores (typically 5 kHz). Fluorescence correlation spectroscopy (FCS), which is sensitive to transient fluctuations in fluorescence intensity (9–14), can extend the time resolution down to the average arrival time of individual photons (5,15). Structural dynamics of only a few biological systems have been probed so far by SM FRET and FCS, but these experiments have only measured the residence times of interconverting states, not the transition time (5,7,16).

In this article we introduce a new method to measure the folding transition time of a single RNA molecule using FCS and SM FRET. By filtering and correlating only the relevant photons surrounding the folding transitions, fast dynamics on the timescale of the average photon arrival interval can be measured by observing ~1000 transitions under typical SM FRET experimental conditions.

The experimental system for this study comprises a fragment of the group I self-splicing intron from Tetrahymena (17,18). The fragment, the P4–P6 domain (19–23), has been shown to fold in isolation upon the addition of divalent ions such as magnesium (21,24,25). For this study we have labeled the ends of the P4 and P6 helices with the fluorophores Cy3 and Cy5 (Fig. 1 a). Cy3 acts as a donor for Cy5 such that the observed intensities from each dye can be compared as

(1)

FIGURE 1 (a) Structure of the P4–P6 molecule labeled with Cy3 (light gray) and Cy5 (dark gray). At the conditions of these experiments the molecule spends about half the time in the folded and unfolded states corresponding to high FRET and low FRET, respectively. (more ...)

Fig. 1 b shows the time trace of a single molecule observed with SM FRET. The photon arrival times were binned into 10-ms windows for this figure. The ionic conditions were chosen such that each molecule spends about half the time in low FRET (unfolded) state and half the time in the high FRET (folded) state to maximize the number of transitions observed. Fig. 1 d shows a histogram of FRET values for 15 representative molecules among 1133 studied. The average time between transitions is 3 s, and on the timescale of bin width, the transition time between high and low FRET states is instantaneous. The vast majority of the photons collected are from molecules in either of the two states, folded or unfolded. Thus, their inclusion in the correlation would merely mask the small number of photons emitted during a folding transition.

The major result of this article is that we can measure the timescale of intermediate changes of RNA folding/unfolding that is as short as the average photon arrival interval from FRET. This measurement is done by time-aperturing the photon stream from each fluorophore. Each single molecule trace was parsed to only include photons within 10 ms of a transition. The transitions were found automatically by searching for spikes in the derivative of the FRET trace. However, each identified transition was also reviewed and approved by hand. For a transition that lasts 200 μs, the probability of more than one pair of photons being emitted during that period is only 26%.

Because the donor fluorescence is anticorrelated with the acceptor fluorescence, the autocorrelation of either channel should yield the same information as the cross correlation between the two channels. However, the cross correlation is much less sensitive to channel-specific dynamics such as dye isomerism (29) or autofluorescence of optical filters. The cross correlation for photons detected in the acceptor channel to photons detected in the donor channel is given by

(2)

An equivalent and computationally inexpensive method for calculating the cross correlation between individual photons is to make a histogram of times between photons from the acceptor channel and the donor channel. Fig. 1 e shows the averaged cross correlation from 1133 folding transitions. We model the change in FRET through the folding transition as shown in Fig. 2 a. This or an analogous assumption about the nature of the transition is required to build an analytical framework with which one uses cross correlations to extract transition times. The model assumes that the transition from low to high FRET state (i.e., the folding transition) takes place linearly during time t (Fig. 2 a). Therefore the cross correlation is given by

(3)

(4)

FIGURE 2 A model of SM FRET in RNA folding and photon correlations from simulated folding events. (a) A model of SM FRET signal changes during RNA folding. The FRET pair is assumed to be attached to an RNA molecule such that FRET efficiency increases as it folds; (more ...)

The assumption of transitions taking place in the middle of the window simplifies solution of the equation. The resulting equations are also valid for transitions taking place elsewhere in the window because the sum of “all” possible products between I(x,t) and I(x,t + τ) does not change by shifting the location of transition. Inserting these values for I_a(x,t) and I_d(x,t) into Eq. 2 gives

(5)

(6)

We need to get an averaged correlation from multiple FRET traces, as the cross correlation from a single FRET trace is too noisy to fit to Eq. 5 or Eq. 6. To get an analytical solution of an averaged correlation, a single folding intermediate is assumed. According to this assumption, individual folding transition times should be exponentially distributed, which gives the averaged cross correlation of

(7)

(8)

Intensity variations caused by environmental heterogeneity do not alter Eq. 8, provided that FRET efficiencies remain unchanged. This is because all the denominators and numerators in Eq. 8 are composed of the same order of intensity terms. By assuming FRET transition is from ~0 to ~1 and the total fluorescence count of donor and acceptor stays constant before and after the transition, Eq. 8 further simplifies to Eq. 9 where A and B are the FRET efficiencies before and after the transition, respectively.

(9)

Note that the intercept of Eq. 9 in τ < t T depends on A and B, which are experimentally determined, T, which is known exactly, and the average transition time t. Thus, only the intercept is accurately required to analytically determine the folding time. In case of multiple intermediate steps during the folding, transition time t in the intercept of Eq. 9 (τ < t T) becomes the sum of all intermediate state lifetimes during the folding process, because the terms in convolution (Eq. 7) will simply become multiple integrals over independent variables. Therefore, the folding process time determined from the intercept of Eq. 9 (τ < t T) is the total folding process time regardless of the number of intermediate states.

In practice, single photon traces always include background photons, and the background cannot be corrected since the origin of an individual photon is unknown. Indeed, conventional intensity analysis cannot be used if it requires direct background correction. When Eq. 9 is used with FRET values from traces with background, A and B should be replaced with A′ − n/I_tot and B′ − n/I_tot, respectively, where A′ and B′ are FRET values with background, n is the number of background photons in one channel, and I_tot is the total number of photons. In practice, n/I_tot can be determined by using Eq. 9 in t ≤ τ T, and fitting the correlation curve to a line within a proper time range. With the background corrected A and B, we determine the intercept of the correlation and use Eq. 9 (τ < t T) to measure folding transition time t. Although the individual intercepts depend on different background levels, the difference between them does not. Therefore, this method measures the transition time as a difference between the instantaneous correlations of two different time regions (t ≤ τ T and τ < t T). Note that, since FRET efficiencies are used in the analysis, slight variations in individual dye intensities do not alter the results significantly. To determine the intercept of the correlation, we fit the correlation to a third-order polynomial in the range of τ < t. We fix the first order coefficient from the background-corrected A and B to lower the fitting error.

To determine the precision and accuracy of the method, simulations were performed to randomly produce photons during a FRET transition of predetermined transition time, t_TRUE. The validity of simulation is confirmed as in Fig. 2. Single molecule cross correlations were calculated from the individual simulated photon traces of exponentially distributed folding times with the decay time of t_TRUE. The average of all single molecule cross correlations was then fit with Eq. 9 (τ < t T) to find the transition time, t_FIT, which was compared to t_TRUE to determine the error of the fit (Fig. 3). Average photon emission rate of donor, the window size, T, and t_TRUE were varied independently to determine the significance of each of these parameters (Fig. 3). Several important points regarding the confidence level of the measured transition time are confirmed or revealed by these simulations.

FIGURE 3 Simulated cross correlations based on the model in Fig. 2, fitting errors, and calibration curves. (a) Averaged cross correlations with single exponentially distributed transition times with tTRUE = 200, 400, and 600 μs. Three-thousand (more ...)

To explain the findings from the simulations, we must discuss the noise involved in the correlation. Assuming the only source of noise is Poissonian photon emission, the propagated noise to cross correlation is approximated to

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FIGURE 4 Noise in the cross correlation. (a) Reciprocal probability of detecting multiple photons within τ ~ τ + Δτ as in Eq. 10, i.e., the Poissonian noise in the correlation. (b) Noise of experimental data fitted (more ...)

We started the analysis on our experimental data with 200 ms aperture width (i.e., T = 200 ms) to find that the aperture size is too wide to resolve the timescale of the transitions under our experimental conditions. The noise of ΔCC is approximately proportional to 1/√T (from Eq. 10) whereas ΔCC is approximately proportional to 1/T, yielding 1/√T dependence in confidence level of the fitting. Thus, by narrowing the correlation window size T, we can get better confidence level by the factor of square root of the window size ratio. However, too narrow T gives too high noise level in the long-time regime, which makes it difficult to estimate the background from Eq. 9 (t ≤ τ T). From the simulation, with a 5-kHz photon emission rate, 10 ms is found to be the narrowest usable window size among the time windows examined. A comparison of simulations with a window of 5, 10, and 25 ms revealed that significantly less transitions are required for the same accuracy in fitting data from the 10-ms window compared to the 5- and 25-ms windows. For example, to measure a 200-μs transition time with a 10-ms window requires only 1000 transitions, whereas 1500 transitions was not sufficient for the 5- or 25-ms window.

The other practical point confirmed from the simulation is that too small τ region of data is detrimental to the confidence of results because of the high noise level. The error increases approximately exponentially as τ decreases (Fig. 4 a). Because ΔCC between t = 0 and t ≠ 0 is well approximated to a third-order polynomial and ΔCC(0) with t ≤ 800 μs is at most comparable to the average noise level of the correlation, there should be a point in τ in Fig. 4 where the noise starts dominating ΔCC as τ gets smaller. It will, thus, diminish the accuracy of the results to include too early time points. The earliest τ to be included in the fitting should be optimized considering the accuracy of the method and the shortest measurable t. To achieve the highest confidence in the result, the earliest τ of the fitting range was determined by comparing t_TRUE to t_FIT from simulations. We determined 80 μs was the earliest fitting point to yield t_TRUE from t_FIT with the best confidence level for t_TRUE = 200–800 μs.

Because Eq. 9 is only correct for τ < t, the following procedure was used to determine the upper limit of fitting. Multiple test fittings were done per averaged correlation by setting each time point in the entire data as the upper limit of fitting (the lower limit of fitting is always 80 μs). Five consecutive time points with the minimum discrepancy between the upper limit (i.e., the time point) and the measured transition time were identified, and the median time was chosen as the upper limit. By using these methods to fit the experimental cross correlation (Fig. 5) to a third-order polynomial, the folding transition time was found to be 440 μs in the fitting range of 80–594 μs. For comparison, the unfolding time is found to be too short to measure with this method (Fig. 5 b). Because the experimental conditions and analysis were the same for both folding and unfolding transitions, the significant difference in measured rates is evidence of the validity of this method.

FIGURE 5 Averaged cross correlations of photons from folding and unfolding events. (a) An average cross correlation from 1133 folding events with the 10-ms data window around the folding. Average photon emission rate from the dye is ~5 kHz. FRET efficiencies (more ...)

Although the sample includes ~20% of contaminants that could have somewhat different properties than the bulk (see Materials and Methods), the measurement is valid because the signal of interest dominates the contamination; i.e., the number of contaminating molecules (~200) is far less than the required number of molecules (~1000) to yield any significant difference in the correlation. Nevertheless, there appears to be a systematic bias in t_FIT particularly for fairly small numbers of transitions (<2000). The discrepancy between t_FIT and t_TRUE determined from simulations of 1000 transitions (Fig. 3 d) was used to correct the fitted t of the data in Fig. 5. Accordingly, the folding time of 440 μs determined from the fitting as described above (Fig. 5 a) is adjusted to 310 μs. The average ΔCC for t = 310 μs in the range of 80–310 μs is 0.0221 from Eq. 9. Using the approximated error in Fig. 4 b, signal/noise in ΔCC (ΔCC divided by Eq. 11) in the fitting range of 80–310 μs with T = 10 ms, p = 6.6 kHz, N = 1133, and n = 59 (as in our data, τ_i+1 = τ_i × 10^0.01) is 1.767 (Eq. 11). Because ΔCC is always positive, this value corresponds to 94% confidence level in the fitting.

Finally, the assumption of constant total counts before and after the transition introduces an error because we have ~23% more photons after the transition. By using Eq. 8, for the case of a 0.403 FRET change, a 23% increase in the total number of photons after the transition was found to introduce 24% overestimation in the transition time from the intercept. The error in calculating the first order coefficient in Eq. 9 with this assumption was neglected because it is too small (<1%) to significantly affect our accuracy level. Therefore, to compensate the overestimation from this nonconstant total counts, the folding time is corrected to 240 μs.

We have presented a method to measure an RNA folding transition time on the single molecule level. We have tested the method on the P4–P6 domain of the Tetrahymena group I intron ribozyme. Previous measurements of immobilized molecules using integrative photon counting have had the time resolution only to measure the dwell time in steady states (5,30), but not the transition times between states that appear instantaneous on the millisecond timescale. In this study we have used FCS only on photons detected within 10 ms of a transition between FRET states and have fitted the correlation to the proposed model to yield the average folding transition time of 240 μs.

Although this folding time is significantly faster than any process previously measured in P4–P6 and is one of the fastest measurements made on the single molecule level, it is nevertheless a very noisy measurement and the error in 200–400 μs transition time is ~40–50%. This is due to the inherently long dwell time (~3 s) in the steady states and the experimentally low photon counting rate set by the power of the excitation laser so that at least one transition was detected before photobleaching. This method would be more effective in a system that had an inherently shorter dwell, but dwell times need to be at least 50 ms so that each transition can be isolated in a 10-ms window. Raising the photon counting rate may raise this limit by allowing for shorter windows, but only if the photobleaching rate is not so high that it prevents collection of adequate amounts of data.

This new method measures the transition time between stable folding states on the single molecule level for times as short as the average photon arrival interval. However, due to the photon emission rate of our fluorophores, in this work we could only resolve the total transition time and not any intermediate steps. Nevertheless, this method provides information about the folding transition that cannot be obtained by more traditional stopped-flow methods that typically operate on the millisecond timescale. It should be interesting to compare this method with folding rates measured by faster mixing methods (31,32) because folding prompted by a large change in Mg²⁺ ion concentration may not duplicate folding in equilibrium.

The [Mg²⁺] in our experiments was chosen to ensure the maximum number of folding/unfolding transitions. The window of such [Mg²⁺] is too narrow to make it practical to use [Mg²⁺] as an experimental control. The strength of this particular system is that it illustrates the core idea of the article: it is possible to squeeze information out of single photons by time-aperturing photon stream.

The microsecond transition may represent one of the following: i), the lifetime of an intermediate mid-FRET state after leaving the low FRET state (33) or, ii), the kinetic time it takes to transition from the low FRET state to a compact, high FRET state, which may or may not be the fully folded state because the FRET probes cannot distinguish between the fully folded state and a highly compact partially unfolded state. It will be informative to use varied conditions and mutants to probe the nature and properties of this now-accessible transition. Future development of brighter fluorescence probes may further increase the information content of this approach, revealing details of the folding transition.

This work was supported by National Science Foundation (PHY-0420752 to Steven Chu) and National Institutes of Health (P01 GM66275-01A1RV to Steven Chu and Dan Herschlag). The research of Lisa Lapidus, PhD. was supported in part by a Career Award at the Scientific Interface from the Burroughs Wellcome Fund.