A Kinetic Zipper Model with Intrachain Interactions Applied to Nucleic Acid Hairpin Folding Kinetics

Abstract

Single-stranded DNA and RNA hairpin structures with 4–10 nucleotides (nt) in the loop and 5–8 basepairs (bp) in the stem fold on 10–100 μs timescale. In contrast, theoretical estimate of first contact time of two ends of an ideal semiflexible polymer of similar lengths (with persistence length ∼2-nt) is 10–100 ns. We propose that this three-orders-of-magnitude difference between these two timescales is a result of roughness in the folding free energy surface arising from intrachain interactions. We present a statistical mechanical model that explicitly includes all misfolded microstates with nonnative Watson-Crick (WC) and non-WC contacts. Rates of interconversion between different microstates are described in terms of two adjustable parameters: the strength of the non-WC interactions (ΔG_nWC) and the rate at which a basepair is formed adjacent to an existing basepair ($k_{\mathit{bp}}^{+}$). The model accurately reproduces the temperature and loop-length dependence of the measured relaxation rates in temperature-jump studies of a 7-bp stem, single-stranded DNA hairpin with 4–20-nt-long poly(dT) loops, with ΔG_nWC ≈ −2.4 kcal/mol and $k_{\mathit{bp}}^{+}$ ≥ (1 ns)⁻¹, in 100 mM NaCl. Thus, our model provides a microscopic interpretation of the slow hairpin folding times as well as an estimate of the strength of intrachain interactions.

Introduction

Nucleic acid hairpins have numerous biological functions. In single-stranded DNA (ssDNA), they play an important role in gene expression, DNA recombination, and DNA transposition (1, 2). Hairpin formation of triplet repeat sequences during DNA replication is thought to be critical in the expansion of such repeats, leading to several genetic disorders (3, 4, 5). In RNA, hairpins serve as important structural motifs in RNA-protein recognition and gene regulation (6, 7). They are the most common secondary structural element in RNA and serve as nucleation sites that appear early in the folding pathway of RNA molecules. Thus, a deep understanding of the stability and dynamics of these structures is of fundamental importance in biology and is central to understanding the overall RNA folding problem. Furthermore, the relatively rapid folding of nucleic acid hairpins makes them amenable to computer simulations for further insight into their folding mechanisms (8, 9, 10, 11, 12, 13, 14, 15, 16).

Hairpin folding in ssDNA or RNA is nucleated by the formation of looped configurations stabilized by one or two basepairs, followed by zipping of the stem. Several kinetics measurements of hairpin folding, starting from early relaxation measurements in response to an electric discharge temperature-jump (T-jump) (17, 18, 19), together with more recent laser T-jump studies (20, 21, 22, 23, 24, 25), as well as fluorescence fluctuation and single-molecule FRET studies (26, 27, 28, 29, 30), have revealed the following features: 1), hairpins with short (5, 6, 7, 8) basepairs (bp) in the stem, and 4–10 bases in the loop, form on timescales of tens of microseconds; 2), the folding of small hairpins is largely cooperative and, to a good approximation, described by single-exponential kinetics near the melting temperature T_m (20, 23, 25, 26, 31); and 3), the folding times scale with the length L of the loop as L² to L³ (23, 26, 30, 31, 32).

The question remains: What is the rate-determining step in hairpin folding? A comparison of hairpin folding times with the timescales for loop closure for an ideal semiflexible polymer of similar lengths and flexibility as the nucleic acid strands indicates that hairpin folding times are ∼1000-fold slower than the loop-closure times (32, 33). One scenario for the slow folding of hairpins is an additional entropic barrier arising from the penalty of ordering the backbone and the nucleotides for Watson-Crick (WC) pair formation in the nucleation step (7, 34). Another scenario is an apparent reduction in the configurational diffusion coefficient of the semiflexible polymer from intrachain interactions not included in the theoretical description of an ideal chain (20, 35).

Two pieces of experimental evidence point to the latter as the dominant contributor to the slow folding kinetics. One is the direct measurements of the collision of two ends of short single-stranded (ss) nucleic acid chains, which demonstrated that loop closure in these “real” chains occurs on the timescale of ∼400 ns for 4-nucleotide (nt) long poly(dT) strands and ∼8 μs for 4-nt poly(dA) strands (30, 36), and not tens of nanoseconds as estimated for an ideal chain of similar lengths (32). The second is the observation that nucleic acid hairpin folding/unfolding times scale linearly with solvent viscosity, thus suggesting that chain diffusion plays a role in the rate-determining step (37). Evidence of intrachain interactions also comes from force-extension measurements on ss-polynucleotides that show significant deviations from the behavior expected for an ideal semiflexible polymer, especially under conditions of low forces and high ionic strength (38, 39).

Theoretical and computational studies have postulated that the effective diffusion coefficient of a polymer may be reduced in comparison with the intrinsic diffusion coefficient by a factor exp(−(ε/k_BT)²), as a result of roughness (of magnitude ε) in the free energy surface of the polymer from intrachain interactions (40, 41, 42, 43). A reduced diffusion coefficient with significant temperature-dependence, consistent with theory, has been invoked in the description of end-to-end contact measurements in polypeptides (44, 45), and to partially explain the apparently anomalous behavior of the activation enthalpy for the closing step observed in hairpins with long poly(dA) loops (31, 32, 35).

Here we present a statistical mechanical kinetic zipper model designed to examine the effect of introducing intrachain interactions before the nucleation event on the folding kinetics. This model explicitly includes all misfolded microstates with nonnative WC pairs, as well as all looped configurations held together by non-WC interactions and is an extension of the equilibrium zipper model that we used previously to describe the melting profiles of nucleic acid hairpins with loops of different sizes (23, 46).

Our kinetic model accurately reproduces both the temperature and loop-size dependence of the relaxation rates measured on a 7-bp stem ssDNA hairpin, with 4–20 nt long poly(dT) loops, in terms of two adjustable parameters: the strength of the non-WC interactions and the rate at which the stem is zipped, and correctly reproduces the magnitude and the ∼L² dependence of the folding times for these hairpins. Comparison of experiments and model calculations yields a characteristic value of ∼2.4 kcal/mol for the strength of the intrachain interactions.

Kinetic zipper model: description of microstates

The thermodynamics of hairpin melting and relaxation kinetics as a result of a rapid change in conditions to initiate folding or unfolding (e.g., a temperature-jump) is described in terms of a statistical mechanical model that includes all microstates consisting of looped configurations closed with contiguous WC pairs, whether native or nonnative, together with microstates with non-WC interactions in which any two nucleotides that cannot form a WC pair can nevertheless make a hydrophobic/stacked contact (Fig. 1 and see Fig. S1 in the Supporting Material). The strength of the non-WC interaction is defined in terms of a single parameter ΔG_nWC, which, for sake of simplicity, is assumed independent of the nucleotide type.

Figure 1.

Schematic representation of the ensemble of microstates including in the kinetic zipper model. The microstates include unfolded (red), misfolded with non-WC contacts (purple), partially folded with native WC pair(s) (green), microstates with mismatched stems and nonnative WC pair(s) (orange), and fully folded native conformation (blue). (Red arrows) Transitions represent closing/opening a loop; (blue arrows) transitions represent forming/breaking a WC pair adjacent to an existing contact.

Nucleation can occur for any configuration of the chain for which one or more basepairs form to stabilize the looped conformations, and the stem can grow or zip if the adjacent pair can form a WC pair. Configurations with nonnative WC pairs or non-WC contacts act as transient traps before the formation of a folding nucleus (with a native WC contact) that leads to complete zipping of the stem.

Thermodynamics of hairpin formation

To assign statistical weights to each of the microstates, we followed our previous formulation (46), motivated by the work of Wartell and Benight (47) and Paner et al. (48, 49), which included only microstates with native contacts, and added misfolded microstates as described above (Fig. 1). The statistical weight of each microstate with WC pairs in the stem (basepaired region) is written as z_WC = z_stemz_loop, where z_stem and z_loop are the contributions to the statistical weight from the stem and loop, respectively. We write z_stem as described in Eqs. S1–S3 in the Supporting Material. The loop contribution to the statistical weight of each of the microstates is given by

$$z_{loop}\left(N\right)=z_{wlc}\left(N\right){\sigma }_{loop}\left(N\right),$$ (1)

where z_wlc is the end-loop weighting function from a wormlike chain description of the probability of loop formation, and written as in Eq. S5 in the Supporting Material.

The σ_loop(N) in Eq. 1 is a phenomenological parameterization of the experimental observation on several ssDNA and RNA hairpins that showed a much steeper dependence of hairpin stability with changing loop length than expected from entropic considerations alone (23, 32, 46), which we interpreted as indicating that smaller loops are additionally stabilized as a result of stem-loop and/or intraloop interactions. We parameterize σ_loop(N) as

$${\sigma }_{\mathit{loop}}\left(N\right)={\langle \sigma \rangle }^{1/2}+\frac{C_{\mathit{loop}}}{N_{\text{b}}^{\gamma }},$$ (2)

where 〈σ 〉 is a cooperativity parameter, defined in Eq. S3 in the Supporting Material, C_loop parameterizes the additional loop contribution to hairpin stability, and γ parameterizes the dependence of this stabilizing term on loop size (32, 46).

For all microstates with non-WC contacts, the statistical weights are written as

$$z_{nWC}=s_{nWC}{z}_{loop}=\text{exp}\left(\frac{-\Delta G_{nWC}}{RT}\right)z_{loop},$$ (3)

with z_loop calculated as in Eq. 1.

Relaxation kinetics obtained from a master equation

The transitions between the various microstates in the ensemble are described in terms of a set of coupled differential equations,

$$\frac{d{p}_i}{dt}=\sum _{j\ne i}\left[k_{j\to i}{p}_j-k_{i\to j}{p}_i\right],$$ (4)

where p_i (p_j) is the population of the i^th (j^th) microstate and k_j→i and k_i→j are the rates for transitions from state j to state i and from state i to state j, respectively. The solution to Eq. 4 is obtained as described in Eqs. S7 and S8 in the Supporting Material.

To model relaxation kinetics in response to a T-jump from an initial temperature T_i to a final temperature T_f, the initial population of the microstates immediately after the T-jump is assumed to be identical to the equilibrium population at T_i, i.e., p_j(t = 0, T_f) = z_j(T_i)/Q(T_i), where z_j is the statistical weight of the j^th microstate and Q(T) is the partition function obtained by summing the statistical weights of all microstates (with native and nonnative contacts) in the model. The final populations, after the relaxation is complete, should be consistent with the equilibrium populations at the final temperature T_f, i.e., p_j(t = ∞, T_f) = z_j(T_f)/Q(T_f). This consistency check is used to verify the accuracy of the solution obtained for the coupled rate equations.

The transitions between the microstates in this model are described in terms of four types of elementary rates: $k_{\mathit{loop}}^{+}$, $k_{\mathit{loop}}^{-}$, $k_{\mathit{bp}}^{+}$, and $k_{\mathit{bp}}^{-}$ (Fig. 1). Here, $k_{\mathit{loop}}^{+}$ is the rate constant for closing a loop of length L, starting from an unfolded, random coil configuration, to form a looped configuration with a single WC or non-WC contact to close the loop, $k_{\mathit{loop}}^{-}$ is the rate constant for the reverse step, from the looped configuration back to a random coil configuration, $k_{\mathit{bp}}^{+}$ is the rate constant for adding a WC contact adjacent to an existing WC or non-WC contact, and $k_{\mathit{bp}}^{-}$ is the corresponding reverse step. The rate constant $k_{\mathit{loop}}^{+}$ is assumed to be diffusion-limited and is calculated using loop-closure rates for wormlike chains (see Fig. S2), as derived by Toan et al. (50):

$$k_{loop}^{+}=\frac{24\sqrt{6}{D}_{M}ag\left(N\right)}{b^3\pi N_b^{1/2}}.$$ (5)

Here D_M is the monomer diffusion coefficient, a is the reaction distance (assumed to be 1 nm), g(N) is the loop-closure probability for wormlike chains, defined in Eq. S6 in the Supporting Material, b (= 2P) is the statistical segment length with P the persistence length, and N_b is the number of statistical segments in the loop. There are no free parameters in the calculation of $k_{\mathit{loop}}^{+}$ from Eq. 5, once we assign a value to the persistence length, calculate N_b = (N+1)h/b, where h is the internucleotide distance, and calculate D_M from the Stokes-Einstein relation D_M = k_BT/6πηr_M with a suitable value for the monomer radius r_M. We assigned P ≈ 1.4 nm, h ≈ 0.5 nm, and r_M ≈ 1 nm (46).

The reverse rate, $k_{\mathit{loop}}^{-}$, is calculated if the thermodynamics is known, from

$$k_{loop}^{-}=k_{loop}^{+}\text{exp}\left(\frac{\Delta G_{loop}}{RT}\right),$$ (6)

where ΔG_loop is the difference in free energy between the microstate with a single WC or non-WC contact to close the loop and the fully unfolded, random coil state and is calculated from

$$\Delta G_{loop}=-RT\text{ln}\left({\sigma }_{end}{s}_1{z}_{loop}\right)$$ (7a)

or

$$\Delta G_{loop}=-RT\text{ln}\left(s_{nWC}{z}_{loop}\right)$$ (7b)

Equation 7a is for microstates with WC contacts (with σ_end defined in Eq. S3), whereas Eq. 7b is for microstates with non-WC contacts. The only free parameter in Eq. 7b is ΔG_nWC (see Eq. 3). Thus, all the reverse rates for unfolding from any of the looped configurations are determined once ΔG_nWC is fixed.

The rate constant $k_{\mathit{bp}}^{+}$ is a parameter in the kinetic model and is assumed to be independent of temperature and sequence. All the sequence- and temperature-dependence appears in the reverse rate $k_{\mathit{bp}}^{-}$, which is determined from

$$k_{bp}^{-}=k_{bp}^{+}\text{exp}\left(\frac{\Delta G_{bp}}{RT}\right),$$ (8)

where ΔG_bp is the difference in free energy between two microstates that are linked together by the zipping/unzipping of a single WC pair adjacent to an existing contact.

To summarize, all the rate constants in the kinetic scheme can be calculated in terms of two free parameters, ΔG_nWC and $k_{\mathit{bp}}^{+}$, once the parameters that define the statistical weights of each of the microstates in the ensemble are determined from the equilibrium melting profiles.

Comparison of model calculations with experiments

In our equilibrium melting profile measurements and kinetics measurements, we monitor the absorbance at 266 nm, and interpret the change in absorbance as reflecting a change in the fraction of native WC contacts (θ_N) that we define as our order parameter. To compare the equilibrium thermodynamics and relaxation kinetics obtained from the model with experimental data, we calculate the order parameter from our model as

$${\theta }_N\left(t,T\right)=\frac{1}{N_s}\sum _{\left\{j\right\}}{n}_j{p}_j\left(t,T\right),$$ (9a)

$${\theta }_N^{eq}\left(T\right)=\frac{1}{N_s}\sum _{\left\{j\right\}}\frac{n_j{z}_j\left(T\right)}{Q\left(T\right)},$$ (9b)

where N_s is the total number of basepairs in the stem of a fully folded native hairpin, p_j(t,T) is the time evolution of the probability of the j^th microstate with n_j native WC contacts, z_j(T)/Q(T) is the corresponding equilibrium population at temperature T, ${\theta }_N^{eq}$(T) is the equilibrium value of the order parameter at temperature T, and the sum in Eq. 9 is over the subset {j} of microstates with only native WC contacts. To compare results from our calculations with experiments, we calculate an average relaxation time, as follows:

$$\langle \tau \rangle =\int {\theta }_N\left(t\right)dt.$$ (10)

Results

Determining the statistical weights of microstates from loop dependence of equilibrium melting profiles

We analyzed the absorbance melting profiles of our hairpins with sequence 5′-CGGATAA(T_N)TTATCCG-3′, with N ranging from 4 to 20 (see Fig. S3 a). We chose this sequence because we have previously carried out a series of thermodynamics and kinetics measurements on these hairpins (32, 37, 46). We first determined the parameters C_loop and γ, which in our equilibrium statistical mechanical model describe the steep dependence of nucleic acid hairpin stability on the loop size (parameterized in Eq. 2), by calculating the thermal melting profiles from Eq. 9b and comparing the melting temperatures $T_m^{\mathit{calc}}$, defined as the temperature at which θ_N = 1/2, with the experimentally obtained melting temperatures $T_m^{\mathit{exp}}$ from the measured absorbance melting profiles (see Fig. S3 b). For equilibrium thermodynamics, we included only microstates with native and nonnative WC contacts and ignored the microstates with non-WC contacts, which are assumed to contribute to the unfolded ensemble. The parameters that best fit the data are obtained from a Monte Carlo search in parameter space, as described in Kuznetsov et al. (23), which yields C_loop = 250 ± 125 and γ = 6.5 ± 0.5, in 100 mM NaCl.

In an earlier study we found that salt contributed significantly to the dependence of hairpin stability on loop size, with γ = 2.5 ± 0.5 in 2.5 mM MgCl₂, for both poly(dT) loops in ssDNA hairpins and poly(rU) loops in RNA hairpins, in comparison with γ ≈ 7–8 in 100 mM NaCl for poly(dT) or poly(dA) loops in ssDNA (23, 32). Thus, measurements in 2.5 mM Mg²⁺ exhibited loop dependence that was closer to that expected from a wormlike chain description, for which γ ≈ 1. A further increase in the Mg²⁺ concentration from 2.5 mM to 33 mM did not affect γ (23).

One plausible explanation for the unusually large value of γ in 100 mM NaCl is that Na⁺ ions may specifically stabilize smaller loops in comparison with large loops, and that the strength of these stabilizing interactions is diminished when Na⁺ is replaced by Mg²⁺. An alternative explanation is that the negative charge on the phosphates is not completely neutralized in 100 mM NaCl, and that intrastrand charge repulsion contributes an unfavorable term to loop free energy, which increases as the loops get longer. Tan and Chen (51) have developed a statistical mechanical model that predicts ion-dependent loop stability contribution to nucleic acid hairpin thermodynamics, in which they indeed find a strong dependence of hairpin stability on loop size in 100 mM Na⁺ in comparison with 2.5 mM Mg²⁺, as a result of charge repulsion effects. Measurements in 1 M NaCl should help distinguish between these two scenarios, with γ expected to increase or remain unchanged at higher [Na⁺] if these specific ions help stabilize smaller loops, but expected to decrease if incomplete neutralization of charge destabilizes larger loops.

Simulation of relaxation kinetics

To obtain the best-fit values of the two additional parameters required in the description of the kinetic zipper model, ΔG_nWC and $k_{\mathit{bp}}^{+}$, we first simulated the relaxation kinetics for a reference hairpin, chosen as the one with the T₈ loop, using the statistical weights of each of the microstates as obtained from fitting the equilibrium melting profiles. This hairpin has 28 microstates with native WC contacts, 40 microstates with nonnative WC contacts, and 130 microstates with non-WC hydrophobic contacts, which includes all allowed conformations with a minimal loop size of three nucleotides. Thus the total number of microstates for this hairpin is Ω = 199, including the completely unfolded random-coil state. The inclusion of microstates with non-WC contacts to describe the kinetics disturbs the equilibrium thermodynamics parameters, which were obtained from a zipper model that did not explicitly include these microstates (47). To compensate, we renormalized the statistical weights of the microstates with the non-WC contacts (z_nWC) and that of the random-coil microstate z_rc as

$$z_{nWC}^{new}=\frac{z_{nWC}}{\Sigma z_{nWC}+z_{rc}},$$ (11a)

$$z_{rc}^{new}=\frac{z_{rc}}{\Sigma z_{nWC}+z_{rc}},$$ (11b)

where the sum in Eq. 11 is over all microstates with non-WC contacts. Thus, we recover the melting profiles and $T_m^{\mathit{calc}}$ as before, with the reassignment of the statistical weight of the ensemble of unfolded states as $z_{un}=\sum z_{nWC}^{new}+z_{rc}^{new}=1$.

From our kinetic zipper model we calculated θ_N(t) from Eq. 9a using the populations of the microstates as obtained from the solution to the master equation (see Eqs. S7 and S8 in the Supporting Material), in response to a T-jump from an initial temperature T_i to the final temperature T_f. The relaxation kinetics traces thus obtained exhibit a predominantly single-exponential phase on timescales longer than ∼10 ns (see Fig. S4 in the Supporting Material). At temperatures below T_m (≈51°C for the T₈ hairpin), a rapid phase appears on <10-ns timescale, whose amplitude increases as we lower the temperature. The time-resolution in our T-jump measurements is not sufficient to resolve this rapid phase, and we observe relaxation kinetics that are well described by a single exponential.

To compare the results of our model calculation with experimentally obtained relaxation times, we calculated an average relaxation time as given by Eq. 10. The parameters ΔG_nWC and $k_{\mathit{bp}}^{+}$ were varied to find the best agreement with the relaxation time measured for our reference hairpin near its T_m. We obtain a continuum of parameters that yield relaxation time of ∼18 μs (Fig. 2), close to the measured value for T-jump from 42°C to 51°C on the T₈ hairpin. The calculated relaxation rates are insensitive to $k_{\mathit{bp}}^{+}$ when it exceeds ∼(1 ns)⁻¹. Thus, our analysis yields values of $k_{\mathit{bp}}^{+}$ in the range from (1 ns)⁻¹ to (22 ns)⁻¹, which are within the range of previous estimates of this zipping rate of (125 ns)⁻¹ estimated by Pörschke (52), (300 ns)⁻¹ estimated by Cocco et al. (53), and (12 ns)⁻¹ and (780 ns)⁻¹ for A·U and C·G basepairs, respectively, estimated by Zhang and Chen (54).

Figure 2.

Parameter space that describes experimental relaxation rates. The free energy of non-WC interactions ΔG_nWC is plotted versus the corresponding basepair closing rate $k_{\mathit{bp}}^{+}$ (solid circles), and represents the set of parameters that yield relaxation time of 18 μs for the reference hairpin 5′-CGGATAA(T₈)TTATCCG-3′ hairpin for a T-jump from 42°C to 51°C. (Inset) Comparison of nearest-neighbor (nn) stacking free energies (ΔG_ST) for the 10 different dinucleotide steps of duplex DNA. (Vertical shaded bars) The nn stacking free-energy parameters, extracted from thermal denaturation experiments on oligonucleotide duplexes. Length of each vertical shaded bar indicates the range of the stacking parameters from seven independent research groups, obtained under different salt conditions and for varying lengths of duplex DNA, and unified by SantaLucia (55). (Solid circle) Stacking free energies obtained from electrophoretic mobility measurements on DNA fragments containing a nick in the sugar-phosphate backbone, between all possible combinations of dinucleotide steps, obtained from Protozanova et al. (56).

The set of parameters in Fig. 2 fall into two broad categories. In one limit, defined by $k_{\mathit{bp}}^{+}$ ≈ 4.7 × 10⁷ s⁻¹ ≈ (22 ns)⁻¹, the calculated relaxation rates are not sensitive to the value of ΔG_nWC from 0 to −1 kcal/mol. In this limit, the rate-determining step for forming hairpins is the addition of the second basepair to the equilibrium population of the looped conformations with native WC contacts, and hence is not sensitive to the search time to find the correct nucleating conformations. The closing rate in this limit is given by k_c ≈ K_nuc $k_{\mathit{bp}}^{+}$, where K_nuc is the equilibrium population of the nucleus that leads to zipping. In the other limit, ΔG_nWC ≈ −2.4 kcal/mol and $k_{\mathit{bp}}^{+}$ ≥ 10⁹ s⁻¹ ≈ (1 ns)⁻¹. For this set of parameters, the closing rate k_c ≈ $k_{\mathit{nuc}}^{+}$, where $k_{\mathit{nuc}}^{+}$ is the overall rate to form the ensemble of correct nucleating conformations, which in this limit is the rate-determining step, followed by the rapid zipping of the stem. This nucleation rate is significantly slower than the loop-closure rates obtained from Eq. 5 for a wormlike chain (see Fig. S2) and illustrates the effect of the intrachain interactions, described by the parameter ΔG_nWC, in decreasing the effective diffusion coefficient for sampling configurational space, as suggested by earlier theoretical studies (40, 41, 42, 43). The strength of the intrachain interactions is in good agreement with previous estimates of internucleotide stacking energies (55, 56), which is consistent with our interpretation that the intrachain interactions are dominated by misstacked bases (Fig. 2, inset).

Loop dependence of opening and closing times

To determine which set of parameters best captures the loop-length dependence of the measured relaxation traces, we compared the experimental relaxation times and the corresponding opening/closing times for hairpins with identical stems and varying loop lengths, from T₄ to T₂₀, with the values calculated from our model for different sets of parameters. This comparison is best carried out at a fixed temperature, as shown in Fig. 3 at 37°C and at 51°C. The temperature range over which there is significant amplitude in the relaxation kinetics measured in our T-jump spectrometer is limited to within ∼±10°C of the T_m for each hairpin. Therefore, a comparison of relaxation rates at a fixed temperature for hairpins with a broad range of T_m values, as shown in Fig. S3, necessitated an extrapolation of the measured rates for T₄ and T₂₀ hairpins at 37°C, and for T₁₆ and T₂₀ hairpins at 51°C. This extrapolation was carried out by first obtaining the opening and closing times over the temperature range where kinetics were observed, from the measured relaxation times and the equilibrium constants, assuming a two-state system, and then describing the temperature dependence of the opening/closing times in terms of an Arrhenius expression, which allowed us to estimate the opening, closing, and relaxation times for temperatures outside the measured range.

Figure 3.

Loop dependence of relaxation, opening and closing times. (a and d) The relaxation times, (b and e) the closing times, and (c and f) the opening times are plotted versus the length of the loop (L = N+1) for the hairpin 5′-CGGATAA(T_N)TTATCCG-3′, at 37°C (a–c) and 51°C (d–f). Symbols in each panel represent (●, black): experimental values obtained as described in the text; (Δ, blue): values obtained from the kinetic zipper model with ΔG_nWC = − 2.4 kcal/mol and $k_{\mathit{bp}}^{+}$ = 1 × 10⁹ s⁻¹; (∇, red): values obtained from the model with ΔG_nWC = 0 kcal/mol and $k_{\mathit{bp}}^{+}$ = 4.7 × 10⁷ s⁻¹. (Continuous lines in b, c, e, and f) Linear fits through the experimental points, and through the values obtained from the model with the two sets of parameters; the respective slopes are: 2.6 (black), 3.8 (blue), and 6.7 (red) in panel b; −2.3 (black), −1.5 (blue), and 1.4 (red) in panel c; 2.2 (black), 2.3 (blue), and 5.3 (red) in panel e; and −4.2 (black), −3.4 (blue), and −0.4 (red) in panel f. (Continuous lines in a and d) Relaxation times calculated from the linear fits to the closing and opening times shown in panels b, c, e, and f.

The dependence of relaxation times, and the corresponding opening and closing times, on the length of the loop, is shown in Fig. 3 together with the loop dependence calculated from the kinetic zipper model for the two limiting sets of parameters. The loop dependence is best reproduced when we pick ΔG_nWC ≈ −2.4 kcal/mol and $k_{\mathit{bp}}^{+}$ ≥ 10⁹ s⁻¹ from the set of allowed parameters shown in Fig. 2. These values of the parameters fall in the limit for which the rate-determining step in hairpin closing is the configurational diffusion to find the correct nucleating loop, which yields a loop dependence for the closing time of τ_c ∼ L^2.3 at 51°C, in very good agreement with our experimentally measured loop dependence of τ_c ∼ L^{2.2 ± 0.6}. These parameters also reproduce nicely the loop dependence of the opening times and yield τ_o ∼ L^−3.4 at 51°C, in comparison with L^{−4.2 ± 1.2} obtained from experiments.

In contrast, the parameters in the limit of no intrachain interactions, specified by ΔG_nWC = 0, with a corresponding value of $k_{\mathit{bp}}^{+}$ = 4.7 × 10⁷ s⁻¹, yield a much stronger loop dependence for the closing times as τ_c ∼ L^5.3 at 51°C (Fig. 3). As expected in this limit, the loop dependence appears in the equilibrium constant K_nuc, which in turn depends on the free energy of the loops, ΔG_loop, and which scales with the length of the loop with an exponent α = γ + 1.5 ≈ 8 (23, 46). The strong loop dependence for the closing rates in this limit is demonstrated by a direct calculation of k_c from k_c ≈ K_nuc $k_{\mathit{bp}}^{+}$, where K_nuc is estimated from our equilibrium model as the sum of the statistical weights of all conformations with a single, native WC contact. The estimate of the closing rate using this approximation, with $k_{\mathit{bp}}^{+}$ ≈ 4.7 × 10⁷ s⁻¹, yields k_c ≈ (0.51 μs)⁻¹, (103 μs)⁻¹, and (1.6 ms)⁻¹ for hairpins with 4, 12, or 20 nucleotides in the loop, respectively, at 51°C, in reasonable agreement with the closing rates obtained from the full kinetic model, with ΔG_nWC = 0, which gave k_c ≈ (2.2 μs)⁻¹, (254 μs)⁻¹, and (6.3 ms)⁻¹, and which deviate significantly from the experimental values, as shown in Fig. 3.

Temperature dependence of opening and closing times

We also computed the temperature dependence of the relaxation times for our reference hairpin with T₈ loop, using the same two limiting sets of parameters that were used to calculate the loop dependence in Fig. 3. Again, only one set of parameters, ΔG_nWC ≈ −2.4 kcal/mol and $k_{\mathit{bp}}^{+}$ > 10⁹ s⁻¹, reproduces well the temperature dependence of the measured relaxation times (Fig. 4).

Figure 4.

Temperature dependence of the relaxation times. (●) Relaxation times for the hairpin 5′-CGGATAA(T₈)TTATCCG-3′, obtained from T-jump measurements, are plotted versus inverse temperature. The continuous (black) line is drawn through the experimental points to guide the eye. The dashed (blue) line represents the relaxation times obtained from the kinetic zipper model with ΔG_nWC = − 2.4 kcal/mol and $k_{\mathit{bp}}^{+}$ = 1 × 10⁹ s⁻¹. The dashed-dot-dot (red) line represents relaxation times obtained from the model with ΔG_nWC = 0 kcal/mol and $k_{\mathit{bp}}^{+}$ = 4.7 × 10⁷ s⁻¹.

Discussion

This article focuses on a fundamental question regarding folding of ssDNA and RNA hairpins: Why do hairpins fold on timescales of microseconds, and what is the rate-determining step in these folding dynamics?

To explain the relatively slow folding times of 10–100 μs for hairpins with small loops (4–10 nucleotides), in comparison with much faster loop closure times expected for an ideal semiflexible polymer of similar length, we had proposed that roughness in the free energy landscape (as a result of nonnative intrachain interactions) reduces the effective configurational diffusion coefficient for chain dynamics, thus slowing down the critical nucleation step before zipping of the stem (20, 35). We estimated this roughness to be 1–2 kcal/mol, with the larger values for hairpins with long poly(dA) loops (32, 35).

The kinetic zipper model presented here explicitly accounts for this roughness by including all possible microstates with native and nonnative WC pairs, without internal bulges, as well as looped configurations with non-WC contacts, as illustrated in Fig. 1. The motivation to develop this model came from two key computational studies of the dynamics of hairpin formation. The first study, developed by Zhang and Chen (54), presented a detailed folding kinetics analysis of a small (9-bp stem, 3-nt loop) RNA hairpin, using a kinetic model that enumerated all conformations of the RNA chain with native and nonnative WC basepairs, with the free energy of each conformation obtained by calculating loop entropies from lattice model enumerations (57), and enthalpies and entropies of basepair stacks from the Turner rules (58). This computational study first demonstrated in detail a rugged energy landscape for RNA folding, with folding pathways that lead to dead-ends or traps, especially at temperatures below what they defined as the glass transition temperature T_g < T_m.

The second study that motivated this work, in particular the notion to explicitly include microstates with non-WC contacts that were not included in the model of Zhang and Chen (54), was the molecular dynamics simulations of Sorin et al. (8, 59), on an all-atom model of another small (4-bp stem, 4-nt loop) RNA hairpin. These simulations revealed at least two dominant mechanisms by which this hairpin folded from a fully extended, denatured state: the first involved loop formation followed by zipping, and the other involved nonspecific collapse, similar to the hydrophobic collapse in proteins (60, 61). The individual conformations observed in the collapsed state showed an ensemble of misfolded traps with nonnative WC basepairing interactions and non-WC hydrogen bonding and base-stacking interactions. The collapse rate of ∼(8 μs)⁻¹ at ∼300 K obtained from the simulations was very close to experimental observations of closing rates for similar size hairpins, suggesting that the initial collapse and reorganization of the intrastrand contacts is the rate-determining step in hairpin formation.

In this article, we obtain the time-dependent evolution of all the microstates from the solution to a master equation that describes the rates of interconversion of all microstates included in the kinetic model. The microscopic rates required in our model are: 1), loop-closure rates, which are calculated from theoretical estimates for wormlike chains (50); 2), rate of closing a WC pair adjacent to an already existing WC or non-WC contact, which is a free parameter in the model and is assumed to be sequence- and temperature-independent; and 3), microscopic rates for the reverse steps, which are determined from the forward rates and the statistical weights of each of the microstates. These statistical weights are fixed from known thermodynamic parameters for duplex stem stability together with loop stability parameters that accurately reproduce the hairpin melting temperatures, and that are not varied in the calculations involving the master equation. The only other free parameter in the kinetic model is the strength of the non-WC interactions that is used to calculate the statistical weights of microstates with non-WC contacts. The experimental observable in a T-jump relaxation measurement, which is the change in absorbance as a function of time, is simulated by calculating the time-dependence of an order parameter, defined as the fraction of intact native WC pairs. The average relaxation times obtained from the simulated kinetic traces are compared with the experimental results, and the two free parameters in the kinetic model are adjusted to reproduce the observed relaxation times.

We find a range of parameters that can reproduce the relaxation time for our reference (7-bp stem, T₈ loop) ssDNA hairpin at a single temperature near its T_m. This range falls into two sets. The first set corresponds to the scenario in which the rate-determining step is not the formation of a looped configuration with a native WC contact, but the addition of the next WC pair to zip up the stem, with ΔG_nWC = 0 and $k_{\mathit{bp}}^{+}$ ≈ 4.7 × 10⁷ s⁻¹. The second set corresponds to the scenario in which the rate-determining step is the formation of a native looped configuration, which is slowed down as a result of nonnative intrachain interactions characterized by ΔG_nWC ≈ −2.4 kcal/mol, followed by rapid zipping of the stem.

Only the second set of parameters accurately reproduces both the temperature dependence of the observed relaxation times for the reference hairpin as well as the loop dependence for hairpins with varying loop lengths. Our estimate of the intrachain interaction ΔG_nWC is in reasonable agreement with previous estimates of stacking interactions between nearest-neighbor WC pairs (55, 56), as is to be expected if the primary contribution to the stickiness of the chain is from misstacked bases.

Cocco et al. (53) also applied a kinetic zipper model (without any misfolded microstates) to simulate folding and unfolding rates observed in force-induced unfolding measurements of RNA hairpins. In their model, the timescale for each elementary step is set by a microscopic rate coefficient (r), which corresponds to the sequence-independent rate of closing each successive basepair in the limit of zero applied force, with the opening rate determined from the statistical weights assigned to each of the microstates. They obtained a value of r ≈ 3.6 × 10⁶ s⁻¹ ≈ (280 ns)⁻¹ at 25°C to reproduce the opening and closing times of ∼1 s measured by Liphardt et al. (62) in their force-induced unfolding experiments on a 22-bp stem, 4-nt loop RNA hairpin, under conditions for which the opening and closing times were similar (∼14 pN of applied force in 10 mM Mg²⁺).

By extrapolating their results to zero applied force, Cocco et al. (53) estimated closing times of ∼3 μs for a hairpin with 10 bases in the stem. In their model, the observed hairpin closing times are not from the slow formation of the looped conformations but from the slow, successive closing of basepairs along the stem, one pair at a time. The implication is that the closing times should scale linearly with the length of the stem. These predictions have yet to be tested in a systematic way for ssDNA and RNA hairpins, although measurements on two ssDNA hairpins (one 5-bp stem with a T₁₂ loop (26), and another 2-bp stem with a T₉ loop (30)) show similar closing times of ∼25 μs at 25°C. In another series of micromanipulation measurements of the folding/unfolding of ssDNA hairpins, Woodside et al. (63) investigated in some detail both the stem- and loop-length dependence of the folding and unfolding times of these hairpins as a function of applied force. However, extrapolation of these measurements to zero force conditions may be problematic, as discussed later.

The results of our simulations reveal essentially single exponential kinetics after ∼10 ns (see Fig. S4), with a rapid phase appearing at shorter times, at temperatures lower than the T_m of the hairpin, corresponding to the rapid zipping and unzipping of the stem. Pörschke (52) made the first experimental observation of this zipping/unzipping phase, more than three decades ago, in T-jump measurements on short (14–18 bp) RNA duplexes, carried out at temperatures well below the T_m, and found zipping/unzipping occurring on timescales of 100–300 ns.

Single-exponential kinetics in the vicinity of T_m are observed in several T-jump measurements on ssDNA and RNA hairpins (21, 23, 24, 25, 37), although deviations from single-exponential behavior that indicate intermediate states in the folding/unfolding pathway have been reported by several groups. These include fluctuation correlation spectroscopy (FCS) measurements on freely diffusing ssDNA hairpins labeled with fluorescent dyes (27, 29, 30, 64), as well as T-jump measurements carried out at temperatures far from T_m (21, 22, 24, 25, 65). These measurements have raised the question as to whether a two-state description of the folding/unfolding of nucleic acid hairpins (in which the free energy landscape has primarily two distinct valleys, corresponding to the ensemble of folded and unfolded conformations, separated by a large free energy barrier) is adequate under all conditions.

In the T-jump measurements, these additional phases are still on the submillisecond timescales and reveal rapid dynamics in the folded or unfolded ensemble, such as fraying of the stem at temperatures well below T_m (25, 65), similar to the zipping/unzipping kinetics observed by Pörschke (52), and additional phases at temperatures well above T_m that are attributed to dynamics in the unfolded ensemble, e.g., from transiently populated misfolded loops and/or collapsed states (22, 24, 25, 65).

At the other end of the temporal scale, Jung and Van Orden (29) reported much slower (greater than a few milliseconds) phase in the folding kinetics of hairpins with large (T₂₁) loops, based on discrepancies between equilibrium constants determined from the amplitudes of their FCS correlation curves and equilibrium constants obtained directly from thermodynamic melting measurements. They postulated a long-lived compact intermediate state that lingers for hundreds of microseconds before folding to the thermodynamically stable native state on a much longer timescale than the FCS measurement time. The fact that T-jump measurements do not show any evidence of an additional, slow (millisecond) phase, even for hairpins with large loops, may well indicate that the two techniques, T-jump and FCS, are exploring different regions of the free energy landscape.

It is also instructive to compare the nucleic acid folding kinetics observed in T-jump and FCS measurements with those measured under conditions of force-induced unfolding (62, 63, 66). Very slow folding/unfolding kinetics, consistent with a two-state description but occurring on timescales of milliseconds-to-seconds, are observed when the stems of the hairpins are forced apart under applied tension. These slow kinetics are the consequence of the steep dependence of the folding/unfolding times on force (F) of the form exp(FΔx^‡/k_BT). Recent micromanipulation studies carried out on a series of ssDNA hairpins with varying loop lengths, stem lengths, and composition, demonstrated two-state behavior over the entire range of forces measured that were nicely reproduced by a statistical mechanical model that incorporated the effect of applied force on the free energy profiles (63).

It should be noted that extrapolation of measured rates in micromanipulation experiments to zero force can lead to large errors, in part because of the assumption inherent in the extrapolation that the position of the free energy barrier (Δx^‡) is force-independent. For instance, the folding times obtained from force-induced measurements on a T₄ loop ssDNA hairpin (63), when extrapolated to zero force, are found to decrease from 11 μs to 2 μs, 0.36 μs, and 0.0036 μs as the length of the stem is increased from 6-bp to 8-bp, 10-bp, and 15-bp, respectively—illustrating that a direct comparison of measurements under applied force with those obtained under spontaneous folding conditions may not be meaningful.

Finally, a question remains as to whether RNA hairpins are more likely to exhibit deviations from single-exponential behavior in comparison with ssDNA hairpins. T-jump measurements that have revealed multiple phases in the folding kinetics of RNA hairpins were performed on hairpins with UNCG tetraloop (21, 24, 25, 65), which belongs to the phylogenetically conserved and stable tetraloop family that also includes GNRA, and CUUG tetraloops (7).

Notably, thermodynamic studies reveal a lower extent of cooperativity in the intraloop and loop-stem interactions that stabilize these tetraloops (closed by a CG pair) in RNA hairpins, in comparison with similarly highly stable GNA or GNAB (with B = C, G, or T) loops, also closed by a CG pair, in ssDNA hairpins (67, 68). The lack of strong cooperativity in the thermodynamics of RNA loops was attributed to the promiscuity of hydrogen-bonding interactions possible in RNA because of the presence of multiple 2′OH groups (68), and which could contribute to metastable misformed loops in the folding kinetics.

In our T-jump measurements, carried out on ssDNA hairpins with poly(dT) loops and RNA hairpins with poly(rU) loops, we observe essentially single-exponential kinetics for both kinds of nucleic acid hairpins (23, 37). Whether this similar folding behavior that we observe for both ssDNA and RNA hairpins is a consequence of the simple loop composition of all the hairpins we have studied, or the lack of sensitivity in our T-jump spectrometer to detect additional kinetics phases at temperatures well above or below T_m, remains unclear.

Additional measurements of nucleic acid hairpin folding kinetics, starting from different initial conditions, e.g., by using stopped-flow or microfluidic mixing techniques, are needed to help unveil further the underlying ruggedness of the free energy landscape of these simple hairpin structures, and to reveal the extent of cooperativity or lack of it in their folding kinetics.

Editor: Kathleen B. Hall.