Force-dependent hopping rates of RNA hairpins can be estimated from accurate measurement of the folding landscapes

Abstract

The sequence-dependent folding landscapes of nucleic acid hairpins reflect much of the complexity of biomolecular folding. Folding trajectories, generated by using single-molecule force-clamp experiments by attaching semiflexible polymers to the ends of hairpins, have been used to infer their folding landscapes. Using simulations and theory, we study the effect of the dynamics of the attached handles on the handle-free RNA free-energy profile F^o_eq(z_m), where z_m is the molecular extension of the hairpin. Accurate measurements of F^o_eq(z_m) requires stiff polymers with small L/l_p, where L is the contour length of the handle, and l_p is the persistence length. Paradoxically, reliable estimates of the hopping rates can only be made by using flexible handles. Nevertheless, we show that the equilibrium free-energy profile F^o_eq(z_m) at an external tension f_m, the force (f) at which the folded and unfolded states are equally populated, in conjunction with Kramers' theory, can provide accurate estimates of the force-dependent hopping rates in the absence of handles at arbitrary values of f. Our theoretical framework shows that z_m is a good reaction coordinate for nucleic acid hairpins under tension.

A molecular understanding of how proteins and RNA fold is needed to describe the functions of enzymes (1) and ribozymes (2), interactions between biomolecules, and the origins of misfolding that is linked to a number of diseases (3). The energy-landscape perspective has provided a conceptual framework for describing the mechanisms by which unfolded molecules navigate the large conformational space in search of the native state (4–6). Recently, single-molecule techniques have been used to probe features of the energy landscape of proteins and RNA that are not easily accessible in ensemble experiments (7–18). It is possible to construct the shape of the energy landscape, including the energy scales of ruggedness (19, 20), by using dynamical trajectories that are generated by applying a constant force (f) to the ends of proteins and RNA (14, 15, 21, 22). If the observation time is long enough for the molecule to sample the accessible conformational space, then the time average of an observable X recorded for the αth molecule should equal the ensemble average , and the distribution P(X) should converge to the equilibrium distribution function P_eq(X). By using this strategy, laser optical tweezer (LOT) experiments have been used to obtain the sequence-dependent folding landscape of a number of RNA and DNA hairpins (8, 14, 15, 23), by using X = R_m, the end-to-end distance of the hairpin that is conjugate to f, as a natural reaction coordinate. In LOT experiments, the hairpin is held between two long handles [DNA (15) or DNA/RNA hybrids (8)], whose ends are attached to polystyrene beads (Fig. 1a). The equilibrium free-energy profile βF_eq(R_m) = −logP_eq(R_m) (β ≡ 1/k_BT, k_B is the Boltzmann constant, and T is the absolute temperature) may be useful in describing the dynamics of the molecule, provided R_m is an appropriate reaction coordinate.

Fig. 1.

A schematic diagram of the optical-tweezers setup used to measure the hairpin's folding landscape. (a) Two RNA/DNA hybrid linkers are attached to the 5′ and 3′ ends of the RNA hairpin, and a constant force is applied to one end through the bead. (b) Ensemble of sampled conformations of the H–RNA–H system during the hopping transitions obtained by using L = 25 nm and l_p = 70 nm. The illustration is created by using the simulated structures collected every 0.5 ms. An example of the time trace of each component of the system, at f = 15.4 pN, given L for both linkers is 25 nm. z_m (= z_5′ − z_3′) and z_sys(= z_p − z_o) measure the extension dynamics of the RNA hairpin and of the entire system, respectively. The time-averaged value z̄_r(t) = (1/τ)∫_dτ^tZ_sys(τ) for the time trace of z_sys is shown as the bold line. The histograms of the extension are shown on top of each column.

The dynamics of the RNA extension in the presence of f (z_m = z_3′ − z_5′ ≈ R_m, provided transverse fluctuations are small) is indirectly obtained in an LOT experiment by monitoring the distance between the attached polystyrene beads (z_sys = z_p − z_o), one of which is optically trapped at the center of the laser focus (Fig. 1a). The goal of these experiments is to extract the folding land-scape [βF^o_eq(z_m)] and the dynamics of the hairpin in the absence of handles by using the f-dependent trajectories z_sys(t). To achieve these goals, the fluctuations in the handles should minimally perturb the dynamics of the hairpin to probe the true dynamics of a molecule of interest. However, depending on L and l_p (L is the contour length of the handle, and l_p is its persistence length), the intrinsic fluctuations of the handles cannot only distort the signal from the hairpin, but also directly affect its dynamics. The first is a problem that pertains to the measurement process, whereas the second is a problem of the coupling between the instruments and the dynamics of RNA.

Here, we use coarse-grained molecular simulations of a RNA hairpin and theory to show that, to obtain accurate βF_eq(R_m), the linkers used in the LOT have to be stiff, i.e., the value of L/l_p has to be small. To investigate the handle effects on the energy landscape and hopping kinetics, we simulated the hairpin dynamics under force-clamp conditions by explicitly modeling the linkers as polymers with varying L and l_p. Surprisingly, the force-dependent folding and unfolding rates that are directly measured by using the time traces, z_m(t), are closest to the ideal values (those that are obtained by directly applying f, without the handles, to the 3′ end with a fixed 5′ end) only when the handles are flexible. Most importantly, accurate estimates of the f-dependent hopping rates over a wide range of f values, in the absence of handles, can be made by using βF_eq(R_m), in the presence of handles obtained at f = f_m, the transition midpoint at which the native basin of attraction (NBA) and the unfolded basin of attraction (UBA) of the RNA are equally populated. The physics of a hairpin attached to handles is captured by using a generalized Rouse model (GRM), in which there is a favorable interaction between the two noncovalently linked ends. The GRM gives quantitative agreement with the simulation results. The key results announced here provide a framework for using the measured folding landscape of nucleic acid hairpins at f ≈ f_m to obtain f-dependent folding and unfolding times and the transition state movements as f is varied (24–29).

Results and Discussions

Modeling the LOT Experiments.

To extract the folding landscape from LOT experiments, the time scales associated with the dynamics of the beads, handles, and the hairpin have to be well separated (30–33). The bead fluctuations are described by the overdamped Langevin equation γdx_p/dt = −kx_p + F(t), where k is the spring constant associated with the restoring force, and the random white-noise force F(t) satisfies 〈F(t)〉 = 0, and 〈F(t)F(t′)〉 = 2γk_BTδ(t − t′). The bead relaxes to its equilibrium position on a time scale τ_r = γ/k. In terms of the trap stiffness, k_p, and the stiffness k_m associated with the Handle–RNA–Handle (H–RNA–H; see Fig. 1), k = k_p + k_m. With γ = 6πηa, a = 1μm, η ≈ 1cP, k_p ≈ 0.01 pN/nm (34), and k_m ≈ 0.1 pN/nm, we find τ_r ≲ 1 ms. In LOT experiments (30, 32, 33), separation in time scales is satisfied such that τ^o_U ≈ τ^o_F ≫ τ_r at f ≈ f_m, where τ^o_U and τ^o_F are the intrinsic values of the RNA (un)folding times in the absence of handles.

Because z_m is a natural reaction coordinate in force experiments, the dispersion of the bead position may affect the measurement of F_eq(z_m). At equilibrium, the fluctuations in the bead positions satisfy 〈δx_eq²〉 ∼ k_BT/(k_p + k_m) ∼ k_BT/k_m, and hence k_m should be large enough to minimize the dispersion of the bead position. The force fluctuation, 〈δf_eq²〉 ∼ k_BTk_p²/(k_p + k_m), is negligible in the LOT because k_p ≪ k_m, and as a result δf_eq/f_m ≈ 0, because δf_eq ≈ 0.1 pN, whereas f_m ≈ 15 pN. Thus, we model the LOT setup by assuming that the force and position fluctuations due to the bead are small and exclusively focus on the effect of handle dynamics on the folding landscape and hopping kinetics of RNA (Fig. 1 a and b).

Short, Stiff Handles Are Required for Accurate Free-Energy Profiles.

For purposes of illustration, we used the self-organized polymer (SOP) model of the P5GA hairpin (35) and applied a force f = f_m ≈ 15.4 pN. The force is exerted on the end of the handle attached to the 3′ end of the RNA (P in Fig. 1a) while fixing the other end (O in Fig. 1a). Simulations of P5GA with handles of length L = 25 nm and persistence length l_p = 70 nm show that the extension of the entire system (z_sys = z_p − z₀) fluctuates between two limits centered around z_sys ≈ 50 nm and z_sys ≈ 56 nm (Fig. 1b). The time-dependent transitions in z_sys between 50 and 56 nm correspond to the hopping of the RNA between the NBA and UBA. Decomposition of z_sys as z_sys = z_H^5′ + z_m + z_H^3′, where z_H^5′ (= z_5′ − z_o) and z_H^3′ (= z_p − z_3′) are the extensions of the handles parallel to the force direction (Fig. 1a), shows that z_sys(t) reflects the transitions in z_m(t) (Fig. 1b). Because the simulation time is long enough for the hairpin to ergodically explore the conformations between the NBA and UBA, the histograms collected from the time traces amount to the equilibrium distributions P_eq(X), where X = z_sys, z_H^5′, z_m, or z_H^3′ [Fig. 1b; for P_eq(z_H^5′) and P_eq(z_H^3′), see the supporting information (SI) Fig. S1a]. To establish that the time traces are ergodic, we show that z̄_T(t) = (1/τ)∫_dτ^tZ_sys(τ) reaches the thermodynamic average (≈∫_−∞^∞z_sysP_eq(z_sys)dz_sys = 53.7 nm) after t ≳ 0.1 sec (the magenta line on z_sys(t) in Fig. 1b).

Fig. 1b shows that the positions of the handles along the f direction fluctuate, even in the presence of tension, which results in slight differences between P_eq(z_sys) and P_eq(z_m). Comparison between the free-energy profiles obtained from the z_sys(t) and z_m(t) can be used to investigate the effect of the characteristics of the handles on the free-energy landscape. To this end, we repeated the force-clamp simulations by varying the contour length (L = 5–100 nm) and persistence length (l_p = 0.6 and 70 nm) of the handles. Fig. 2 shows that the discrepancy between the measured free energy F_eq(z_sys) (dashed lines in blue) and the molecular free energy F_eq(z_m) (solid lines in red) increases for the more flexible and longer handles (see SI Text and Fig. S1 for further discussion of the dependence of the handle fluctuations on L and l_p). For small l_p and large L, the basins of attraction in F_eq(z_m) are not well resolved. The largest deviation between F_eq(z_sys) and F_eq(z_m) is found when l_p = 0.6 nm and L = 25 nm (L/l_p ≈ 40) (the graph enclosed by the orange box in Fig. 2a). In contrast, the best agreement between F_eq(z_sys) and F_eq(z_m) is found for l_p = 70 nm and L = 5 nm (the graph inside the magenta box in Fig. 2), which corresponds to L/l_p ≈ 0.07. In the LOT experiments, L/l_p ≈ 6–7 (8, 14, 15).

Fig. 2.

The free-energy profiles, F_eq(z_sys) (dashed line in blue) and F_eq(z_m) (solid line in red), calculated by using the histograms obtained from the time traces z_sys(t) and z_m(t) for varying L and l_p. F_eq(z_sys) and F_eq(z_m) for a given l_p and different L are plotted in the same graph to highlight the differences. The intrinsic free energy F^o_eq(z_m), the free-energy profile in the absence of handles, is shown in black. The condition that produces the least deviation (l_p = 70 nm, L = 5 nm) and the condition of maximal difference (l_p = 0.6 nm, L = 25 nm) between F_eq(z_m) and F_eq(z_sys) are enclosed in the magenta and orange boxes, respectively.

GRM Captures the Physics of H–RNA–H Under Tension.

To establish the generality of the relationship between the free-energy profiles as measured by z_m and those measured by z_sys, we introduce an exactly solvable model that minimally represents the RNA and handles (Fig. 3a). We mimic the hairpin using a Gaussian chain with N₀ monomers and Kuhn length a. The endpoints of the RNA mimic are harmonically trapped in a potential with stiffness k as long as they are within a cutoff distance c = 4 nm. Two handles, each with N_h monomers and Kuhn length b, are attached to the ends of the RNA (see Methods). We fix one endpoint of the entire chain at the origin and apply a force f_m ≈ 15.4 pN to the other end. The free energies as a function of both the RNA's extension, R_m = ∣r_3′ − r_5′∣ (≈z_m at high f) and the system's extension R_sys = ∣r_p − r₀∣ (≈z_sys at high f) are exactly solvable in the continuum representation. We choose k such that f_m is near the midpoint of the transition, so that ∫₀^cd³rP_eq(r) ≈ ∫_c^∞d³rP_eq(r). We tune N₀ so that the barrier heights for the GRM and P5GA are similar at f = f_m. These requirements give N₀ = 20 and k ≈ 0.54 pN/nm. Although the stiffness in the handles of the simulated system (Fig. 1) cannot be accurately modeled by using a Gaussian chain, the primary effect of attaching the handles is to alter the fluctuations of the endpoints of the RNA. By equating the longitudinal fluctuations for the WLC, 〈δR_‖²〉_WLC ∼ Ll_p^−1/2(βf)^−3/2, with the fluctuations for the Gaussian handles, 〈δR_‖²〉_G ∼ Lb, we estimate that the effective persistence length of the handles scales as l_p^eff ∼ b⁻²f⁻³ (see SI Text for details). Thus, smaller spacing in the Gaussian handles in the GRM will mimic stiffer handles in the H–RNA–H system. The free energies computed for the GRM, shown in Fig. 3 b and c, are consistent with the results of the simulations. The free-energy profiles deviate significantly from F^o_eq(z_m) as N_h increases or stiffness decreases. The relevant variable that determines the accuracy of F_eq(z_sys) is N_hb² ∼ L/l_p^eff, with the free energies remaining unchanged if N_hb² is kept constant. The barrier height and well depths as a function of z_m are unchanged as a function of L and b. However, the apparent free energy of activation is decreased as measured by z_sys (seen in Fig. 2 as well). The GRM confirms that accurate measurement of the folding landscape by using z_sys requires stiff handles.

Fig. 3.

Free-energy profiles for the GRM. (a) A schematic diagram of the GRM, showing the number of monomers (N₀ and N_h) and Kuhn lengths (a and b) in each region of the chain, the harmonic interaction between the ends of the RNA mimic, and the external tension. (b) The free-energy profile for a fixed b = (a/3) and increasing N_h as a function of R_sys ≈ z_sys. The barrier heights decrease, and the well depths increase for increasing N_h. (c) The free-energy profile for fixed N_h = 20 and varying b. The barrier heights decrease, and the well depths increase for increasing b. In both (b) and (c), the profiles are shifted so that the positions of the local maxima and minima coincide with those of the intrinsic free energy (with N_h = 0).

Accurate Estimates of the Hopping Kinetics Requires Short and Flexible Handles.

Because LOT experiments can also be used to measure the force-dependent rates of hopping between the NBA and the UBA, it is important to assess the influence of the dynamics of the handles on the intrinsic hopping kinetics of the RNA hairpin. In other words, how should the structural characteristics of the linkers be chosen so that the measured hopping rates using the time traces z(t) and the intrinsic rates are as close as possible?

Folding and unfolding rates of P5GA and the free-energy profile without handles.

We first performed force-clamp simulations of P5GA in the absence of handles to obtain the intrinsic (or ideal) folding (τ^o_F) or unfolding (τ^o_U) times, that serve as a reference for the H–RNA–H system. To obtain the boundary conditions for calculating the mean refolding and unfolding times, we collected the histograms of the time traces and determined the positions of the minima of the NBA and UBA, z_F = 1.9 nm, and z_U = 7.4 nm (Fig. 4a). The analysis of the time traces provides the transition times in which z_m reaches z_m = z_U starting from z_m = z_F. The mean unfolding time τ_U is obtained by using either τ_U = 1/NΣ_iτ_U(i) or from the fits to the survival probability P_F(t) = e^−t/τ_U (Fig. S3). The mean folding time is similarly calculated, and the two methods give similar results. The values of τ^o_U and τ^o_F computed from the time trace of z_m(t) are 2.9 and 1.9 ms, respectively. At f_m = 15.4 pN and L = 0 nm, the equilibrium constant K_eq = τ^o_F/τ^o_U = 0.67, which shows that the bare molecular free energy is slightly tilted toward NBA at f = 15.4 pN.

Fig. 4.

Hopping rates from free energy profile. (a) The free-energy profile for P5GA with L = 0 nm. (b) The transition times at f = f_m, obtained by using z_m(t) (filled symbols) and z_sys(t) (open symbols). The ratio τ_U(F)^m/τ_U(F)^sys ≈ 1, which shows that z_sys(t) mirrors the hopping of P5GA. (c) Folding rate k_F^m(L)/k_F^m (0) as a function of L for varying b, by using the GRM. The plots show b/a = 1, 1/2, 1/5, and 1/10. The simulation results for P5GA are also shown as symbols to emphasize that the GRM accounts for the hopping kinetics in the H–RNA–H system accurately.

Hopping times depend on the handle characteristics.

The values of the folding (τ_F^m) and unfolding(τ_U^m) times were also calculated for the P5GA hairpin with attached handles (Fig. 1). As the length of the handles increase both τ_U^m and τ_F^m increase gradually, and the equilibrium distribution shifts toward the UBA, i.e., K_eq = τ_F^m/τ_U^m increases (Fig. 4b). Strikingly, the use of flexible handles results in minimal deviations of τ_U^m and τ_F^m from their intrinsic values (Fig. 4b). Attachment of handles (stiff or flexible) to the 5′ and 3′ ends restricts their movement, which results in a decrease in the number of paths to the NBA and UBA. Thus, both τ_U^m and τ_F^m increase (Fig. 4b). As the stiffness of the handle increases the extent of pinning increases. These arguments show that flexible and short handles, that have the least restriction on the fluctuations of the 5′ and 3′ ends of the hairpin, cause minimal perturbation to the intrinsic RNA dynamics and, hence, the hopping rates. Because the experimentally accessible quantity is the extension of the H–RNA–H, it is important to show that the transition times can be reliably obtained by using z_sys(t). Although z_sys(t) differs from z_m(t) in amplitude, the “phase” between the two quantities track each other reliably throughout the simulation, even when the handles are long and flexible (see Fig. S2). We calculated τ_U^sys and τ_F^sys by analyzing the trajectories z_sys(t) using the same procedure used to compute their intrinsic values. Comparison of τ_U^sys (τ_F^sys) and τ_U^m (τ_F^m) for both stiff and flexible handles shows excellent agreement at all L values (Fig. 4b). Thus, it is possible to infer the RNA dynamics z_m(t) by measuring z_sys(t).

Theoretical predictions using the GRM are consistent with the simulations.

The simulation results can be fully understood by using the GRM (Fig. 3a), for which we can exactly solve the overdamped Langevin equation using the discrete representation of the Gaussian chain (see Methods). By assuming that transverse fluctuations are small (which is reasonable under the relatively high tension of f = 15.4 pN), we use the Wilemski and Fixman (WF) theory (36) to determine an approximate time of contact formation (τ_F^m = (k_F^m)⁻¹) as a function of b (i.e., increasing handle stiffness) and N_h. The refolding rate of the RNA hairpin under tension is analogous to k_F^m. A plot of k_F^m(L)/k_F^m(0) versus L (Fig. 4c) illustrates that smaller deviations from the handle-free values occur when l_p is small. Moreover, Fig. 4c shows that the refolding rate decreases for increasing N_h regardless of the stiffness of the chain. The saturating value of k_F^m as N_h → ∞ depends on b, with stiffer handles having a much larger effect on the folding rate. Although the handles used in LOT experiments are significantly longer than the handle lengths considered here, the saturation of the folding rate suggests that L ≈ 100 nm is sufficiently long for finite-size effects to be negligible. We also find the dependence of k_F on L agrees well with the behavior observed in the simulation of P5GA. The ratio k_F^m(L)/k_F^m(0) for b = a agrees well with the trends of the flexible linker (l_p = 0.6 nm) for all of the simulated lengths, with both saturating at k_F(L) ≈ 0.35k_F (0) for large L. The trends for stiffer chains (smaller b) in the GRM qualitatively agree with the P5GA simulation with stiff handles (l_p = 70 nm), with remarkably good agreement for 0.1 ≤ b/a ≤ 0.2 over the entire range of L. The GRM, which captures the physics of both the equilibrium and kinetic properties of the more complicated H–RNA–H, provides a theoretical basis for extracting kinetic information from experimentally (or computationally) determined folding landscapes.

Free-Energy Landscapes and Hopping Rates.

Stiff handles are needed to obtain F_eq(z_sys) (15) that resembles F^o_eq(z_m), whereas the flexible handles produce hopping rates that are close to their handle-free values. These two findings appear to demand two mutually exclusive requirements in the choice of the handles in LOT experiments. However, if z_m is a good reaction coordinate, then it should be possible to extract the hopping rates by using accurately measured F_eq(z_sys) [≈F_eq(z_m) ≈ F^o_eq(z_m)] at f ≈ f_m, by using handles with small L/l_p. The (un)folding times can be calculated by using the mean first passage time (Kramers' rate expression) with appropriate boundary conditions (37),

where z_min, z_max, z_U, and z_F are defined in Fig. 4a. The effective diffusion coefficient D_F(D_U) is obtained by equating τ_F^KR (τ_U^KR) in Eq. 1, with F_eq(z_m) = F^o_eq(z_m), to the simulated τ^o_F (τ^o_U). We calculated the f-dependent τ_U^m(f) and τ_F^m(f) by evaluating Eq. 1 using F^o_eq(z_m∣f) = F^o_eq(z_m∣f_m) − (f − f_m)·z_m. The calculated and simulated results for P5GA are in good agreement (Fig. 5 a and b). At the higher force (f = 16.8 pN), the statistics of hopping transitions within our simulation time is not sufficient to establish ergodicity. As a result, the simulation results are not as accurate at high forces (see Fig. S4). To further show that the use of F^o_eq(z_m∣f) in Eq. 1 gives accurate hopping rates, we calculated τ^o_U(f) for the GRM and compared the results with direct simulations of the handle-free GRM, which allows the study of a wider range of forces (see Methods). The results in Fig. 5c show that F^o_eq(z_m∣f) indeed gives very accurate values for the transition times from the UBA and NBA over a wide force range.

Fig. 5.

Force-dependent hopping rates from free energy profile at f = f_m. (a) Comparison of the measured free-energy profiles (symbols) with the shifted free-energy profiles βF^o_eq(z_m∣f_m) − β(f − f_m)·z_m. (b) Folding and unfolding times as a function of force f = 14 pN < f_m, f = 15.4 pN ≈ f_m, and f = 16.8 pN > f_m. τ_F(U)^m is obtained from the time trace in figure 2B in ref 35 at each force, whereas τ_F(U)^m is computed by using the tilted profile βF^o_eq(z_m∣f) = βF^o_eq(z_m∣f_m) − β(f − f_m)·z_m in equation (1). (c) Folding and unfolding times using the GRM. Symbols are a direct simulation of the GRM (error bars are standard deviation of the mean). The solid lines are obtained by using the Kramers theory (Eq. 1). We choose D_U ≈ 3D₀, so that that the simulated and Kramers times agree at f = f_m. The position of each basin of attraction as a function of force for the GRM is given by z_U ≈ N₀a² βf/3 and z_F ≈ N₀a² βf/(3 + 2N₀a²βk).

Conclusions

The self-assembly of RNA and proteins may be viewed as a diffusive process in a multidimensional folding landscape. To translate this physical picture into a predictive tool, it is important to discern a suitable low-dimensional representation of the complex energy landscape, from which the folding kinetics can be extracted. Our results show that, in the context of nucleic acid hairpins, precise measurement of the sequence-dependent folding landscape of RNA is sufficient to obtain good estimates of the f-dependent hopping rates in the absence of handles. It suffices to measure F_eq(z_sys) ≈ F_eq(z_m) ≈ F^o_eq(z_m) at f = f_m by using stiff handles, whereas F_eq(z_m∣f) at other values for f can be obtained by tilting F_eq(z_m∣f_m). The accurate computation of the hopping rates by using F_eq(z_m) show that z_m is an excellent reaction coordinate for nucleic acid hairpins under tension. Further theoretical and experimental work is needed to test whether the proposed framework can be used to predict the force-dependent hopping rates for other RNA molecules that fold and unfold through populated intermediates.

Methods

RNA Hairpin.

The Hamiltonian for the RNA hairpin with N nucleotides, which is modeled by using the SOP model (35), is

where r_i,j = ∣r_i − r_j∣ and r_i,j^o is the distance between monomers i and j in the native structure. The first term enforces backbone chain connectivity by using the finite extensible nonlinear elastic (FENE) potential, with k ≈ 1.4 × 10⁴ pN·nm⁻¹ and R₀ = 0.2 nm. The Lennard–Jones interaction (second term in Eq. 2) describes interactions only between native contacts (defined as r_i,j^o ≤ 1.4 nm for ∣i − j∣ > 2), with Δ_i,j = 1 if monomers i and j are within 1.4 nm in the native state, and Δ_i,j = 0 otherwise. Nonnative interactions are treated as purely repulsive (the third term in Eq. 2) with σ = 0.7 nm. We take ε_h = 4.9 pN·nm and ε_l = 7.0 pN·nm for the strength of interactions. In the fourth term, the repulsion between the ith and (i + 2)th interaction sites along the backbone has σ* = 0.35 nm to prevent disruption of the native helical structure.

Handle Polymers.

The handles are modeled by using the Hamiltonian

The neighboring interaction sites, with an equilibrium distance r₀ = 0.5 nm, are harmonically constrained with a spring constant k_S ≈ 1.4 × 10⁴ pN·nm⁻¹. In the second term of Eq. 3, the strength of the bending potential, k_A, determines the handle flexibility. We choose two values, k_A = 7.0 pN·nm and k_A = 561 pN·nm to model flexible and semiflexible handles respectively and assign k_A = 35 pN·nm to the junction connecting two ends of the RNA and the handles. We determine the corresponding persistence length for the two k_A values as l_p = 0.6 and 70 nm (see SI Text). The contour length of each handle is varied from n = 5–200.

GRM.

The Hamiltonian for the GRM (Fig. 3a) is

where

The first two terms in Eq. 4 are the discrete connectivity potentials for the two handles, each with N_h bonds (N_h + 1 monomers), and with Kuhn length b. The mechanical force f in the third term is applied along the z direction, with ∣f∣ = f_m = 15.4 pN. We also fix the 5′ end of the system with a harmonic bond of strength k₀ = 2.5 × 10⁴ pN·nm⁻¹ in the fourth term of Eq. 4. The fifth term mimics the RNA hairpin with N₀ bonds and spacing a = 0.5 nm. Interactions between the two ends of the RNA hairpin are modeled as harmonic bond with strength k ≈ 0.54 pN·nm⁻¹ that is cut off at c = 4 nm (Eq. 5). When ∣R_m∣ > 4 nm, the bond is broken, mimicking the unfolded state.

The free energies as a function of both R_m ≈ z_m and R_sys ≈ z_sys are most easily determined in the continuum limit of the Hamiltonian in Eq. 4, with Σ_i=1^N → ∫₀ⁿds. Because of the relatively large value of the external tension (f_m ≫ k_BT/l_p), we can neglect transverse fluctuations without significantly altering the equilibrium or kinetic properties of the GRM. The refolding time, τ_F^m of the RNA mimic (Fig. 3a), which is the WF closure time (36), can be determined by numerically diagonalizing the Rouse-like matrix with elements (38) $\frac{1}{2}{\delta }^{2}H/\delta {\text{r}}_{\text{i}}\delta {\text{r}}_{\text{j}}$.