Docking kinetics and equilibrium of a GAAA tetraloop-receptor motif probed by single-molecule FRET

Abstract

Docking kinetics and equilibrium of fluorescently labeled RNA molecules are studied with single-molecule FRET methods. Time-resolved FRET is used to monitor docking/undocking transitions for RNAs containing a single GAAA tetraloop-receptor tertiary interaction connected by a flexible single-stranded linker. The rate constants for docking and undocking are measured as a function of Mg²⁺, revealing a complex dependence on metal ion concentration. Despite the simplicity of this model system, conformational heterogeneity similar to that noted in more complex RNA systems is observed; relatively rapid docking/undocking transitions are detected for approximately two-thirds of the RNA molecules, with significant subpopulations exhibiting few or no transitions on the 10- to 30-s time scale for photobleaching. The rate constants are determined from analysis of probability densities, which allows a much wider range of time scales to be analyzed than standard histogram procedures. The data for the GAAA tetraloop receptor are compared with kinetic and equilibrium data for other RNA tertiary interactions.

The discovery that RNAs can catalyze biological reactions has led to intensive effort aimed at identifying additional biological functions for RNA. More than 20 years later, we now know that RNAs play critical functional roles in metabolism, replication, regulation, and development in cells. Extensive biochemical and biophysical studies have led to a better understanding of the molecular mechanisms by which RNAs achieve their biological function, highlighting the important roles of both structure and dynamics. In this regard, single-molecule methods have recently emerged as particularly powerful tools. The folding dynamics of various functional RNAs have been investigated by single-molecule FRET experiments, which probe dynamics under equilibrium conditions via observation of the stochastic fluorescence trajectories as a function of time (1–6). These techniques offer a unique glimpse into subpopulations of a system and, in some cases, have identified conformational heterogeneity or the presence of intermediates that would otherwise be undetectable by ensemble methods (3–5, 7, 8).

Unlike the highly cooperative, all-or-none, folding process observed for most protein domains, RNAs generally fold in a noncooperative manner, where the secondary structure forms independently of the tertiary structure. The thermodynamics for formation of helical secondary structure in RNA is well understood, but relatively little information exists on the thermodynamics for stabilization of RNA tertiary interactions. Biochemical and structural studies have identified a variety of motifs that form tertiary interactions and thereby help stabilize RNA folding. One of the best-characterized tertiary interactions is the GRNA tetraloop-receptor interaction. This tertiary motif was first identified from phylogenetic and biochemical studies and has been found in a variety of RNAs including group I and group II introns and RNase P (9–11). The high-resolution structure of the GAAA tetraloop docked to its canonical 11-nt receptor, in the P4–P6 domain of the Tetrahymena group I intron, has been determined by x-ray crystallographic studies, which revealed an intricate network of hydrogen-bonding and base-stacking interactions stabilizing this tertiary motif (12). Solution NMR and x-ray crystal structures of the GAAA tetraloop for the free and receptor-bound states indicate that the tetraloop structure changes little upon receptor docking (12, 13). However, there are substantial differences between the structures of the free and docked receptor (12, 14), suggesting that the free receptor undergoes a major conformational rearrangement to form a tertiary interaction with the tetraloop. Biochemical studies have also examined thermodynamics for docking of the GAAA tetraloop with its receptor (15, 16). However, these studies were performed on bimolecular complexes, whereas in natural RNAs GAAA tetraloop-receptor docking is intramolecular. In this work, we use single-molecule FRET to probe the kinetics and thermodynamics for docking/undocking of a GAAA tetraloop with its receptor in an intramolecular reaction.

Folding dynamics for various RNA systems have been studied by single-molecule FRET (1–6). However, we still have only a very limited understanding of the role of individual tertiary interactions on folding dynamics (17). Indeed, the folded structure of a globular RNA relies on a number of tertiary interactions, each of which may be formed and broken at significantly different rates. Conformational heterogeneity (3, 4, 8, 18) has also been observed in single-molecule studies of RNAs. These effects can make it difficult to isolate and extract the contributions of individual tertiary contacts to the folding dynamics of RNA.

The focus of this work is characterizing the docking of a single isolated RNA tertiary interaction. We designed a model RNA system containing one GAAA tetraloop and one tetraloop receptor domain joined by a flexible single-stranded linker that separates this tertiary interaction from the effects of other tertiary motifs. As a result, this study directly probes the kinetics and equilibrium associated with forming/breaking the GAAA tetraloop-receptor interaction. Intramolecular docking of the domains is detected by single-molecule FRET between attached donor and acceptor dyes. Single-molecule FRET experiments are especially well suited for these kinetic studies because they can measure rates of docking/undocking processes for a system at equilibrium. The kinetic data can then be used to investigate how folding dynamics affect the stability of the commonly occurring GNRA tetraloop-receptor tertiary motif.

Materials and Methods

RNA Preparation. Chemically synthesized RNA strands containing an aliphatic three-carbon amino modification at the 5′ phosphate were purchased from Dharmacon Research (Lafayette, CO), and the 5′-biotin DNA was from Integrated DNA Technologies (Coralville, IA). The RNAs were fluorescently labeled at the 5′ amines by reacting with Cy3 or Cy5 N-succinimidyl esters (Amersham Pharmacia Biosciences). Unreacted dyes were removed by microfiltration, followed by C-18 reverse-phase HPLC to separate the labeled and unlabeled RNAs. The RNA construct used for the experiments was composed of 5′-Cy3- and 5′-Cy5-labeled RNA strands and 5′-biotin DNA (Fig. 1). The assembled RNA formed one domain containing the GAAA tetraloop and another containing the tetraloop receptor, with the two connected by an A₇ linker. The 5′-biotin DNA strand hybridized to the Cy5-RNA strand, enabling surface immobilization. Dilute (≈25 pM) RNA samples were immobilized on a glass cover slide by biotin-streptavidin binding (1). All experiments were carried out in 50 mM Hepes (pH 7.5), 100 mM NaCl, and up to 0.10 mM EDTA, with an enzymatic oxygen scavenger system to minimize photobleaching (19).

Fig. 1.

Docking reaction of the model GAAA tetraloop-receptor system. The GAAA tetraloop is shown in bold, and the receptor is in boxed type. The curved arrows illustrate the freedom of motion between the tetraloop and receptor domains allowed by the A₇ single-stranded linker. In the docked configuration, the Cy3–Cy5 distance is ≈35 Å and increases to as much as 70 Å in the undocked state. The 5′-biotin strand allows surface immobilization.

Single-Molecule Fluorescence Spectroscopy. Single-molecule excitation and detection uses a water immersion confocal fluorescence microscope (Olympus IX-70) with a mode locked Nd:YAG laser (model 3800, Spectra-Physics) at 532 nm. Dichroic mirrors isolate fluorescence emission from the excitation wavelength (550 LP, Chroma Technology, Rockingham, VT) and split emission into donor and acceptor channels (645DCXR). Photon wavelength is further defined by band pass filters (HQ585/70M and HQ725/120M) and detected with avalanche photodiodes (SPCM-AQR-14, PerkinElmer). A 3D nano-piezo stage (Physik Instruments, Karlsruhe, Germany) raster-scans the microscope coverslip at 2 ms per pixel to collect 12.5 × 12.5 μm (256 × 256 pixel) surface images. A microfluidic flow system allows solutions to be changed without moving the microscope cover glass on the stage, thereby permitting the same RNAs to be probed for different Mg²⁺ conditions. Time-resolved FRET trajectories are acquired by locating individual RNA molecules in the focus of the laser via an intensity optimization algorithm, with resulting fluorescence detected by time-correlated single photon counting (SPC-134, Becker & Hickl, Berlin). FRET efficiency, E_FRET, is calculated as E_FRET = I_A/(I_D + I_A), where I_A and I_D are acceptor and donor fluorescence intensities, respectively, after explicit correction for (i) background, (ii) cross-talk between the donor and acceptor channels, and (iii) direct excitation of the acceptor. Because the same analysis procedure is applied to all molecules, including subtraction of direct excitation of the Cy5, molecules with no or inactive acceptors have negative E_FRET values.

Single-Molecule Raster-Scan Imaging and Time-Resolved Fluorescence. Two different types of single-molecule FRET experiments were performed. In the first experiment, RNAs were spatially identified from the 2D raster-scanned fluorescence images and a single average FRET value per molecule was calculated from total acceptor and donor photon counts. This process permits rapid screening of 50–100 single-molecules per image; however, quantitative rate information on docking/undocking dynamics is lost because multiple docking events can occur during the raster scan over a single RNA. To generate quantitative kinetic data, time-resolved fluorescence trajectories were obtained with the laser focused on a single RNA, with docking/undocking events directly monitored by discrete changes in the FRET values with ms time scale resolution. The results presented herein represent >2,600 individual RNAs for all of the raster-scanned images and a total of ≈250 molecules for all of the time-resolved trajectories.

Calculation of Rate Constants of Docking and Undocking. To determine docking and undocking rate constants at each Mg²⁺ concentration, docked and undocked FRET events are defined as every time the FRET trajectory crosses a threshold set halfway between the average high and low FRET. The event duration is determined as the time the FRET value remains above or below the threshold. Histograms were constructed from the durations of docked and undocked events at a given Mg²⁺ concentration, which can be least-squares fit to a single exponential function. Because of experimental limitations imposed by irreversible photobleaching of the donor or acceptor, there were relatively few events observed for increasingly long event durations, τ. This analysis process leads to values of 0 or 1 for individual bins in the histogram at the longer times, which resulted in a very limited dynamic range in the time axis, typically <10:1. This limitation can be improved somewhat by increasing the time bin width, but with a corresponding loss of temporal resolution in the data.

However, the experimental dynamic range can be increased dramatically, and without loss of temporal resolution, by conversion of the standard histograms to probability densities, P(τ), as a function of event duration (20). Specifically, in the regions of low sampling (i.e., with single events occurring in widely spaced bins of time duration), the best statistical estimate for the true probability density is proportional to the frequency of the rare events. For a random sampling of single-molecule docking/undocking processes, this frequency can be approximated as the number of such events observed in a given discrete time bin, scaled by the average time separation between a given bin and its two neighboring nonzero bins. Thus, the probability density can be simply estimated from

where H(τ_i) is the standard histogram value and τ_i represents an ordered list of nonzero time bins. This procedure has negligible impact (other than a constant scaling factor) on the short time data where there are multiple events in each time bin, yet maps continuously onto the best representation of probability densities at longer times where the event frequency is low. The result is a smoothly decreasing normalized probability density, P(τ)/P(0), with >100–to 1,000-fold enhancement in dynamic range. The improved dynamic range enables detection of complex kinetics occurring over a wide range of time scales that would be difficult to resolve with a standard histogram (20).

Results

The GAAA Tetraloop-Receptor Construct Docks Reversibly with Mg²⁺. Raster-scanned FRET images were measured as a function of Mg²⁺ concentration on the GAAA tetraloop-receptor construct, as shown in Fig. 2. Fig. 2 a and b shows raster-scanned images of the same RNA molecules at low (≈0 mM) and high (5 mM) Mg²⁺, indicating a clear shift from undocked to docked conformations with increasing Mg²⁺ concentration. At the lowest concentrations, the total Mg²⁺ could be increased by Mg²⁺ in the oxygen scavenger or stock RNA solutions, but this change was <0.05 mM. As a control for tetraloop-receptor docking specificity, similar experiments were performed on RNAs with the GAAA loop replaced by a UUCG loop, which cannot dock to the receptor. No change in FRET was observed in the raster-scanned images or the single-molecule fluorescence trajectories for this RNA at any Mg²⁺ concentration (data not shown). These results demonstrate specific, Mg²⁺-dependent docking in our model GAAA tetraloop-receptor system.

Fig. 2.

Raster-scanned single-molecule fluorescence images demonstrating the effect of Mg²⁺ on docking equilibrium and the conformational heterogeneity of the GAAA tetraloop-receptor system. (a and b) Representative raster-scanned images of the same RNA molecules with no (a) and high (5.0 mM) (b) Mg²⁺ concentrations are shown. Each pixel depicts a false color representation of donor/acceptor emission with intensity proportional to number of donor (green) vs. acceptor (red) fluorescence photons. The size of each image is 12.5 × 12.5 μm; the intensity scale is 0–10 kHz for an incident power of 1.1 μW. (c and d) Corresponding histograms of the average E_FRET of each molecule are shown for no Mg²⁺ (1,841 molecules) (c) and 5.0 mM Mg²⁺ (823 molecules) (d) conditions. The histograms are overlaid with fits to a sum of two or three Gaussian distributions, respectively.

Single-Molecule Docking Kinetics and Equilibria. As described above, time-resolved FRET trajectories were measured for individual molecules and used to calculate the rate constants for both docking and undocking reactions. The FRET trajectories were typically measured until one or both dyes irreversibly photobleached. The single-step loss of fluorescence upon photobleaching ensures that only a single molecule is being observed, while also providing quantitative correction for background, cross-talk between channels, and direct excitation events. Fig. 3 shows representative single-molecule donor/acceptor (Upper) and corresponding FRET (Lower) trajectories for concentrations 0, 0.50, and 5.0 mM Mg²⁺. Under all Mg²⁺ concentrations, the FRET value exhibited two-level behavior, fluctuating between E_FRET = 0.22 ± 0.05 and E_FRET = 0.68 ± 0.08, where the error reflected 1σ distributions in time-resolved trajectories for many molecules. No intermediate FRET states were resolved, and transitions between the two states were rapid relative to the 5-ms binning times shown in Fig. 3. The high and low FRET states were assigned as docked and undocked, respectively.

Fig. 3.

Typical time-resolved fluorescence intensity and corresponding FRET efficiency, E_FRET, traces for the GAAA tetraloop-receptor system at ≈0 mM Mg²⁺ (a), 0.50 mM Mg²⁺ (b), and 5.0 mM Mg²⁺ (c). The donor and acceptor signals are plotted in green and red, respectively. At zero Mg²⁺ concentrations, decreased oxygen scavenger efficacy led to very occasional photophysical blinking events (e.g., a), which are readily identified by a threshold in total count rate and eliminated from the docking/undocking analysis. Two E_FRET states are resolved at all Mg²⁺ concentrations. The high E_FRET state (E_FRET = 0.68) is assigned as docked, and the low E_FRET state (E_FRET = 0.22) is assigned as undocked.

As evidenced in Fig. 3, the event durations for the docked and undocked states were significantly influenced by Mg²⁺ concentration. As described above, histograms of these event durations were converted into probability densities. Sample logarithmic normalized probability densities for docked and undocked events at 0, 0.50, and 5.0 mM Mg²⁺ are shown in Fig. 4, demonstrating the greatly enhanced dynamic range in the time axis (10³:1) facilitated by this analysis. Linear fits to these probability densities yield first-order rate constants for docking and undocking (k_dock and k_undock, respectively), as summarized in Table 1, where the errors were obtained from the least-squares-fitting results. Both the docking and undocking reactions exhibit a significant Mg²⁺ dependence, with k_dock increasing 12-fold and k_undock decreasing 3-fold over the range of Mg²⁺ sampled (0–10 mM).

Fig. 4.

Normalized probability density plots of the docked (Upper) and undocked (Lower) states for ≈0 mM (black), 0.50 mM (red), and 5.0 mM Mg²⁺ (green). The regions at high probability densities are quite well fit by single exponential functions (solid line) but deviate somewhat from pure exponential behavior at longer times, suggesting possible heterogeneity in the docking kinetics.

Table 1. Rate constants and associated equilibrium for docking and undocking at various [Mg²⁺].

[Mg2+], mM	Kdock, s–1	Kundock, s–1	Kdock (Kdock/Kundock)
≈ 0.0	5.1 ± 0.3	10.3 ± 0.4	0.49 ± 0.04
0.35	10.5 ± 0.2	7.7 ± 0.2	1.36 ± 0.05
0.5	17.7 ± 0.5	6.8 ± 0.2	2.6 ± 0.1
1.0	30.1 ± 1.3	7.2 ± 0.3	4.2 ± 0.2
2.0	38.6 ± 1.3	5.5 ± 0.1	7.0 ± 0.3
5.0	51.2 ± 1.1	4.2 ± 0.2	12.3 ± 0.6
10.0	63.1 ± 1.9	3.3 ± 0.1	19.1 ± 0.9

The equilibrium constant for docking, K_dock, was also determined as k_dock/k_undock at each Mg²⁺ concentration, as summarized in Table 1. K_dock increases 40-fold over the measured range of Mg²⁺ concentrations. Notably, a significant population of the docked state was observed without Mg²⁺ (K_dock = 0.49), indicating the interaction forms at physiological monovalent ion concentrations.

To test the effect of the linker on GAAA tetraloop-receptor docking, the A₇ sequence was replaced by U₇, and limited single-molecule studies were performed (see Fig. 6, which is published as supporting information on the PNAS web site). The U₇ linker shows only a small change in docking, with K_dock = 1.4 ± 0.1 for A₇ compared with 2.8 ± 0.1 for U₇ at 0.35 mM Mg²⁺. The corresponding docking/undocking rate constants show this modest difference arises primarily from a larger k_dock for the U₇ (k_dock = 19.5 ± 0.6 s^-1, k_undock = 6.9 ± 0.2 s^-1) versus the A₇ (k_dock = 10.5 ± 0.2 s^-1, k_undock = 7.7 ± 0.2 s^-1). This effect may result from reduced base stacking in the oligo U versus the oligo A sequence. In any case, these data demonstrate that the sequence of the linker does not significantly influence docking dynamics.

Single-Molecule Experiments Reveal Heterogeneity in Tetraloop-Receptor Docking. Analysis of FRET values and distributions from the raster-scanned FRET images indicates that the RNA molecules exist in multiple subpopulations (Fig. 2). First, there is a small subset of molecules (10–15%) lacking Cy5 acceptor emission. These donor-only molecules are unambiguously identified by their negative E_FRET values and are omitted from the analysis discussed below (see Materials and Methods).

For molecules with both active donor and acceptor emission, the histograms of E_FRET (Fig. 2 c and d) show distinct subpopulations. These histograms represent the FRET of individual molecules averaged over a much longer time scale than docking/undocking events. Thus actively docking/undocking molecules will result in a single peak with an E_FRET corresponding to the population-weighted average of the docked and undocked states. As a consequence, there is no obvious high E_FRET peak in the 0 mM Mg²⁺ histogram (Fig. 2c) despite the significant dwell time of molecules in the docked state (Fig. 3a). In contrast, at 5 mM Mg²⁺ there are two major peaks in the histogram with positive E_FRET (Fig. 2d). The largest peak (68% of the molecules) represents molecules that are interconverting between the docked and undocked states on the time scale of the raster-scanned images, with the resulting E_FRET = 0.67 ± 0.004 that is a population-weighted average of the E_FRET for the docked and the undocked states. The increased E_FRET for these molecules demonstrates that the relative dwell time in the docked state has increased with Mg²⁺. The other peak in Fig. 2d has an E_FRET of 0.27 ± 0.02 and represents 32% of the total molecules. These molecules do not dock on the time scale of the raster-scanned images. In the 0 mM Mg²⁺ histogram (Fig. 2c), these molecules cannot be distinguished from the actively docking/undocking molecules because under those conditions they have similar average E_FRET. Fluorescence experiments on freely diffusing molecules (21) also showed a similar distribution of subpopulations at high Mg²⁺ (see Fig. 7, which is published as supporting information on the PNAS web site). These additional experiments demonstrate that the heterogeneity in this RNA does not arise from surface effects. Additionally, the RNA with the U₇ linker described above has a similar distribution of subpopulations in raster-scanned images, demonstrating that this heterogeneity is not a property of a specific RNA construct (data not shown).

The molecules in the 68% subpopulation change E_FRET at different Mg²⁺ concentrations (Fig. 2 a and b), clearly indicating they are actively docking/undocking. This subpopulation also shows many docking/undocking transitions in time-resolved fluorescence trajectories at all Mg²⁺ concentrations examined (Fig. 3) and is used to determine the rate constants reported in Table 1.

The molecules in the 32% subpopulation appear to be always undocked at any Mg²⁺ concentration. Examples are seen in Fig. 2 as molecules that remain primarily yellow-green in both Fig. 2 a and b. This subpopulation shows no docking/undocking transitions over the time scale of the time-resolved FRET measurements (≈10–30 s, limited by photobleaching). A simple interpretation is that these molecules have inactive conformations of the tetraloop and/or receptor that prevent the docking interaction.

There is also a small, but reproducible, subpopulation (1–5%) with FRET values higher than the docked state (E_FRET ≈ 0.95) for both Mg²⁺ conditions in Fig. 2 a and b. The specific identity of this subpopulation is not clear, although single-step photobleaching confirms that such high FRET activity cannot be ascribed to aggregates of RNA molecules.

The normalized probability density plots in Fig. 4 also show evidence of heterogeneity within the actively docking/undocking subpopulation. For single exponential kinetics, a plot of log(P(τ)/P(0)) vs. τ would yield a straight line. However, the plots in Fig. 4 deviate from linearity at longer event durations (τ_docked or τ_undocked > 0.3 s, Fig. 4), which indicates there are more long τ events than predicted by a single exponential rate. This possible kinetic heterogeneity would not be detectable from fits of standard histograms to a single exponential, because of the more limited dynamic range of such an analysis. These results demonstrate the power of analyzing probability densities vs. standard histograms in extracting kinetic data for single-molecule systems.

Discussion

Global folding of RNA is stabilized by long-range tertiary structure motifs such as the GAAA tetraloop-receptor (10), the A-minor motif (22), the ribose zipper (12, 23), or kissing hairpin loops (24, 25). The linker regions between these tertiary motifs can also help stabilize the folded structure, either by introducing flexibility into the RNA chain or positioning the tertiary domains in an optimal orientation. Examples of stabilizing linkers include the J5/5a region of the P4–P6 domain of the group I intron (26), the K-turn motif (27), and various three- and four-way junctions (28–31).

In practice, it is usually difficult to quantify the contributions of individual structural elements to the overall RNA folding. For example, catalytic activity is retained in the hairpin ribozyme when a two-way junction containing either a nick or flexible linker replaces the wild-type four-way junction. However, ribozyme folding differs significantly from the wild type, illustrating that this junction actively positions the helices to form the tertiary contact (5, 28, 29, 32). Another example of interplay between a junction and a tertiary interaction comes from recent studies on the hammerhead ribozyme (33, 34) that show greatly enhanced activity with inclusion of a naturally occurring loop-loop tertiary contact between stems I and II. A truncated form of the hammerhead, lacking the loop-loop tertiary interaction, was extensively studied before it was realized that this interaction is critical for optimal folding and activity.

Separating the energetic and kinetic contribution of individual junctions, linkers, or tertiary interactions is even more complicated in larger RNAs such as the group I intron or RNase P, which include large numbers of these interactions. The present work addresses this issue by probing an isolated tertiary interaction with the two interacting domains connected by a flexible A₇ linker. In contrast to the four-way junction in the hairpin ribozyme (5), this A₇ linker is designed to reduce the effective volume sampled but not orient the domains. Thus the tertiary interaction becomes the primary determinant of docking efficiency. Single-molecule FRET is an ideal assay to directly determine both the equilibrium and the kinetics associated with a single GAAA tetraloop-receptor interaction.

Similar to many RNA systems, the GAAA tetraloop-receptor interaction requires divalent metal ions for optimal folding and function. The K_dock increases ≈40-fold between 0 and 10 mM Mg²⁺, which is similar to the change observed in the two-way junction hairpin ribozyme (6). A significant docked population is also observed at ≈0 mM Mg²⁺ (K_dock = 0.49), indicating that the GAAA tetraloop-receptor interaction forms in the absence of divalent ions at physiological monovalent ion concentrations. Larger RNAs, such as RNase Ps or group I introns, typically do not achieve native structure without Mg²⁺, which means that the requirement of Mg²⁺ is not as stringent for docking of the GAAA tetraloop receptor as it is for folding of most other RNAs.

The single-molecule FRET experiments also clearly demonstrate that Mg²⁺ affects both the docking and undocking rate constants. The observed docking rate constant (k_dock) increases 12-fold from 0 to 10 mM Mg²⁺. Additionally, k_undock decreases 3-fold over the same range of Mg²⁺ concentrations, suggesting a complex metal ion dependence. This Mg²⁺ dependence was fit to a model proposed for Mg²⁺-dependent folding of the S15 binding domain RNA (2), with the results shown in Fig. 5. The fit yields an apparent Mg²⁺ dissociation equilibrium constant, K_Mg, of 1.84 ± 0.46 mM and a Hill coefficient of n = 1.06 ± 0.19. Thus there is no evidence of cooperative Mg²⁺ binding associated with the docking event.

Fig. 5.

Mg²⁺ dependence of k_dock (○) and k_undock (▵) of the GAAA tetraloop-receptor interaction. The Mg²⁺ dependence of the docking rate constants is fit to a Hill-type equation of the form derived by Kim et al. (2). In this model, k_dock = {k₁(K_Mg)ⁿ + k₂[Mg²⁺]ⁿ}/{(K_Mg)ⁿ + [Mg²⁺]ⁿ}, where k₁ and k₂ are the rate constants for docking in the absence and presence of Mg²⁺, respectively, n is the Hill coefficient, and K_Mg is the apparent dissociation equilibrium constant for Mg²⁺ binding to undocked RNA. Similarly, k_undock = {k_-1(K_Mg′)ⁿ + k_-2[Mg²⁺]ⁿ}, where k_-1 and k_-2 are the undocking rate constants in the absence and presence of Mg²⁺, respectively, and K_Mg′ is the apparent dissociation equilibrium constant for Mg²⁺ binding to docked RNA. A combined fit to the measured rate constants for k_dock (solid line) and k_undock (dashed line) is shown, with k₁ = 4.1 ± 2.1 s^-1, k₂ = 71 ± 7 s^-1, K_Mg = 1.84 ± 0.46 mM, k_-1 = 10.0 ± 2.1 s^-1, k_-2 = 3.2 ± 2.2 s^-1, K_Mg′ = 0.9 ± 1.2 mM, and a Hill coefficient of n = 1.06 ± 0.19.

As previously observed in the hairpin ribozyme (6), k_dock is more strongly affected than k_undock by Mg²⁺. In general, binding of Mg²⁺ can affect the docking rate either by increasing the frequency of encounters between the tertiary domains, increasing the fraction of encounters that result in docking, or both. Previous NMR and x-ray crystallographic studies showed that the GAAA tetraloop receptor has different conformations in the free and the tetraloop-bound states (12, 14), indicating the receptor undergoes a significant conformational change upon docking. One model is that binding of Mg²⁺ populates receptor conformations closer to that of the tetraloop-bound state, thereby increasing the fraction of encounters that result in docking. Another possibility is that nonspecific associations of Mg²⁺ ions promote docking by neutralizing charge–charge repulsion between the two helical domains. This is the model proposed by Zhuang and coworkers (6) for the two-way hairpin ribozyme, where diffusive binding of Mg²⁺ facilitates collapse of the undocked state into a compact intermediate closer to the docked state (35). This intermediate leads to a higher effective concentration of the two domains and therefore a higher frequency of encounter. Similar to our GAAA tetraloop-receptor system, this two-way hairpin consists of two interacting helical domains connected by a flexible linker (sequence AC₅). Thus the GAAA tetraloop-receptor system may use a similar docking mechanism, which could explain the Mg²⁺ dependence of k_dock in the tetraloop-receptor system.

Interestingly, kinetic studies of the two-way hairpin ribozyme yielded k_dock of 0.018 s^-1 and k_undock of 0.01 s^-1 (there were also subpopulations with k_undock of 0.1, 0.8, and 6 s^-1) at 12 mM Mg²⁺, 25°C (6). Despite its structural similarities to the two-way hairpin, the k_dock for the tetraloop-receptor system at 10 mM Mg²⁺ is >3,000-fold larger than the dominant rate constant for the hairpin at 12 mM Mg²⁺. The difference in docking rates indicates that either the ability to collapse into the proposed compact intermediate is strongly influenced by factors other than the linker length, or this collapse is not the rate-limiting step in docking of the two-way hairpin at all Mg²⁺ concentrations. The docking rate of the hairpin ribozyme is substantially increased by a four-way junction, which increases the effective concentration of the loops and helps orient them for docking (5). However, even this four-way hairpin has a smaller k_dock than observed for the tetraloop receptor at similar Mg²⁺ concentrations. Therefore, the differences in docking rates likely result from a higher fraction of encounters in the GAAA tetraloop receptor leading to a successful docking event. The docking and undocking rates for this tetraloop receptor are also significantly faster than those measured for P1 helix docking to the Tetrahymena group I intron (k_dock = 1.6 s^-1, k_undock = 0.224 s^-1 at Mg²⁺ = 10 mM) (4). Thus the docking rate constants for these natural RNAs appear to be much smaller than their encounter-controlled rates.

Our results for the GAAA tetraloop-receptor system can also be compared with the same tertiary interaction in larger RNAs, such as the P4–P6 domain of the group I intron. Mutations in the GAAA tetraloop have been shown to destabilize the P4–P6 folding equilibrium by 2–4 kcal/mol, but had a much smaller effect on the folding rate, corresponding to an increase of only 0.2–0.4 kcal/mol in the activation energy (36). These data were interpreted that formation of the GAAA tetraloop-receptor contact is not the rate-limiting step in P4–P6 folding. The results herein demonstrate that if a local conformational change in the tetraloop or receptor is required for docking, this process must be at least as fast as the 63-s^-1 docking rate constant measured at 10 mM Mg²⁺. Because the folding rate constant observed (36) for the P4–P6 domain is 3-fold slower (21 s^-1) at 10 mM Mg²⁺, our results are also consistent with a model that local conformational rearrangement in the GAAA tetraloop or the receptor is not the rate-limiting step in folding of the P4–P6 domain.

Conclusions

The docking/undocking equilibrium and kinetics have been studied for an isolated GAAA tetraloop-receptor tertiary interaction by single-molecule FRET techniques. A short single-stranded (A₇) linker is used to connect these two commonly occurring RNA domains, and this region is designed to restrict the effective concentration of the domains but not orient them for docking. This approach demonstrates that the energetics and dynamics of a RNA tertiary interaction can be studied successfully when separated from the contributions of linker regions.

Heterogeneity is observed in the docking/undocking behavior for this simple GAAA tetraloop-receptor system, including a permanently undocked subpopulation likely resulting from inactive conformers and deviation from simple first-order kinetics for long duration events in the actively docking/undocking population. Observation of conformational heterogeneity in this simple model system with a single, isolated tertiary interaction emphasizes the complexity of RNA folding. Single-molecule methods, however, make it possible to identify the actively docking molecules and accurately characterize their equilibrium and kinetic behavior in isolation.

The single-molecule FRET experiments on the GAAA tetraloop-receptor interaction allow direct observation of the effect of metal cations on docking/undocking. A complex metal ion dependence is observed, as evidenced by the significant effect of Mg²⁺ concentration on both k_dock and k_undock. No evidence for cooperativity in Mg²⁺ binding is found, with analysis of k_dock yielding a Hill coefficient of 1.06 ± 0.19. The observed rate constants for docking/undocking differ significantly from previously measured docking transitions in similarly sized RNA molecules, illustrating that a variety of events can be rate limiting in RNA tertiary contact formation. Also, there is a significant population of docked molecules in the absence of Mg²⁺, which indicates the GAAA tetraloop-receptor interaction has been explicitly shown to form without divalent metal ions.