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Department of Physics and Astronomy and Department of Biochemistry, University of Missouri, Columbia, Missouri
Correspondence: Address reprint requests to Shi-Jie Chen, E-mail: chenshi{at}missouri.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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–18). RNA hairpin folding kinetics is found to give a wide range in magnitude and sign of the folding activation barriers for different sequences and for different temperatures (12–18). Furthermore, from the general theory developed in the previous article, we find that RNA hairpins, even though simple in structure, can be very complex in the folding kinetics. For example, depending on the nucleotide sequence, the folding can be rate-limited by the formation of the loop; or by the slow formation of the base stacks; or by the slow disruption of a misfolded non-native base stack. Moreover, the hairpin structure can form cooperatively through a two-state transition, or noncooperatively through multiple intermediate states. And a decrease in temperature can accelerate or decelerate the folding process. Different sequences of the hairpins can have a wide range of very different folding kinetics behaviors. Most of the previous studies are focused on isolated sequences and the effect of loop closure on the folding kinetics. In this study, we go beyond the isolated sequences by exploring systematically the sequence and structural dependence of the folding kinetics by investigating how the loop length, loop sequence, stem length, and stem sequence affect the hairpin folding kinetics. In addition, we investigate the effect of the kinetic intermediates, especially the misfolded intermediates, on the folding kinetics. We found that certain misfolded intermediates may assist the folding process by lowering the entropic barrier of folding.
Since this study is based on the general RNA hairpin folding theory developed in the previous article, we first briefly summarize major conclusions from the general theory.
We describe the chain conformations according to base stacks. Different conformations are kinetically connected through a kinetic move set defined as the formation and disruption of a base stack or a stacked basepair. The rate of a kinetic move is given by 
and 
for the formation and breaking of a base stack (or a basepair), respectively. Here 
S and H are the corresponding entropy and enthalpy changes. As a result, the rate-limiting steps of folding correspond to the formation of the native base stacks with the largest entropy decrease S and the disruption of the non-native base stacks with the largest enthalpy cost H.
RNA hairpin folding can involve the following four types of rate-limiting steps:
where Sstack and Sloop are the corresponding entropy losses. Sstack* and thus has a slow rate:
 | (1) |
 | (2) |
where Hnn is the enthalpy cost for the disruption of the non-native base stack. kdetrap is slow for large Hnn or low temperature T.
 | (3) |
The folding kinetics is a result of the intercluster transitions. In a cluster, there are two types of conformations: pathway conformations and nonpathway conformations. Conformations that directly participate in the intercluster transitions are called pathway conformations. All other conformations are nonpathway conformations. Therefore, the intercluster transitions (between cluster U and N) are realized by the kinetic moves between the pathway conformations Ui in cluster U and the pathway conformations Ni in cluster N, and the resultant rate constant is given by
 | (4) |
Ni) is determined by
 | (5) |
are the dominant pathways for U N. Higher stability (larger [Ui] in Eq. 4) of the pathway conformations (versus nonpathway conformations) and higher stability of the fast-rate pathway conformations (larger 
in Eq. 4) result in a faster kinetics.
Depending on the nucleotide sequence, the Arrhenius plot of the rate-temperature dependence can show non-Arrhenius behavior: there exists a rollover temperature Tr such that the folding activation barrier changes from positive for T 
Tr to negative for T > Tr, and the folding kinetics changes from noncooperative (multi-state) to cooperative. Summarized in Table 1 are the four folding kinetic scenarios in different temperature regimes.
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S and H (see Fig. 1). We find that as loop size is increased, the folding rate decreases, but the unfolding rate nearly does not change (data not shown). Moreover, we find that the folding rate kf scales with the loop size n as kf 
n–1.8 at T = 30°C (Fig. 2). These findings agree with the experimental measurements for hairpin-folding kinetics (18).
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) and use the two-cluster model with the native cluster N and unfolded cluster U = C + I. We first consider the unfolding transition N U for the breaking of the rate-limiting stack s*. The rate kNU is given by the sum over all the pathway conformations 
Because both [Ni] (= the fractional population of Ni) and 
are independent of the loop length n, the unfolding rate is independent of the loop size.
The folding transition U N corresponds to the formation of s*. The rate kUN is given by 
Except the direct folding pathway U1 N1, which has an extremely small rate kdirect (see Eq. 2), the other 19 pathways have the rate 
for the formation of s*. The fractional population [Ui] (i > 1) depends on the loop size through 
So 
where Sloop is the entropy of the native hairpin loop. From the experimental measurements (19) and the theoretical modeling (20), the loop entropy is Sloop kB ln n–1.8. So we have kf n–1.8 (see Fig. 2). This scaling law for the folding rate, which is obtained from the kinetic cluster analysis, agrees nearly exactly with the experimental data (18).
In the present model, the unfolding is rate-limited by the disruption of the rate-limiting stack s*. Since the enthalpy cost H* for breaking s* is assumed to be n-independent (under 1 M NaCl condition), the unfolding rate 
would be nearly independent of the loop size n. However, for small loops under lower ionic concentrations, the loop can be stabilized by excess loop-stem interaction (21–23). Considering the n-dependence of such excess stabilization Hexcess, the unfolding rate 
can be n-dependent. Specifically, the loop would unfold faster for larger n. In fact, the n-dependence of the unfolding rate has been estimated from experiments as ku n2.3 for DNA hairpins under 0.1 M NaCl (18). However, ku for RNA hairpin folding (in 1 M NaCl) may scale differently.
Stem-length dependence
In this section, we investigate the stem-length dependence of folding rate. By adding AU or UA basepairs to the helical stem of the sequence with n = 5 in the previous section, we generate a series of sequences with the same loop size but different stem length: (AU)mCGC5CG(AU)m (m = 2, 3, ...). As shown in Fig. 3, we find the stem-length dependence of the folding and unfolding rate, as discussed below.
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50°C is the melting temperature (computed from the statistical mechanical model (20)) and Tr 10°C is the rollover temperature. The fast-folding pathway conformations in cluster U contain helical stems (see U19 and U20 in Fig. 1 b), and the longer helix stem enhances the stability of these fast-folding conformations. Therefore, a longer stem leads to faster folding. However, if the stem is too long, the nonpathway conformations (non-native states in Inn) can be very stable and can dominate the population. This would effectively destabilize the pathway conformation and cause a slow folding.
Noncooperative folding (T < Tr; scenarios 3 and 4 in Table 1)
In this case, detrapping is rate-limiting. As the chain is elongated, the number of non-native conformations quickly increases. This greatly enhances the probability for the chain to fold to the misfolded states, causing a slower folding.
Cooperative unfolding (T > Tm; scenario 2 in Table 1)
At the unfolding temperature T > Tm, the dominant kinetic process is the unfolding. The rate is determined by the (unfolding) rate of the disruption of the rate-limiting stack (N U),
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and H* is the enthalpy of the rate-limiting stack (U,C,G,A). Since the stem length only weakly affects the fractional population [Ni] of N, the unfolding rate ku is independent of the stem length.
Loop-sequence dependence
The loop sequence can affect the folding kinetics through two effects: (1), the sequence-dependent, single-stranded stacking in the loop region; and (2), the possible formation of non-native basepairs between the loop and the stem. Here we explore the loop-sequence dependence due to the formation of the non-native basepairs. We make a loop mutation C12 G for sequence (AU)2CGAUAC5UAUCG(AU)2 (see Fig. 4). The mutation does not alter the native structure (shown in Fig. 4 a) and the unfolding rate, but it notably changes the folding rate and its temperature-dependence (see Fig. 5 a): (1), the wild-type sequence folds much faster than the mutant sequence; and (2), they show opposite temperature-dependence: as the temperature is increased, the wild-type folds more slowly and the mutant sequence folds more quickly.
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; and 
(see Fig. 4). The formation of 
and 
have rate constants of 
and 
respectively. According to the two rate-limiting stacks, we classify the conformational ensemble into four clusters:
 | (6) |
through U I1 or through U I2) is extremely slow and is the bottleneck for the overall folding process. The rate is slow because in cluster U, the most populated state (= the fully unfold state) is slow-folding (through direct folding), with the extremely small rate kdirect (see Eq. 2), while the fast-folding conformations (i.e., stacked conformations) occupy <1% of total population in U. 
for or 
for 
Both and are much faster than the rate for the formation of the first stack. Therefore, the overall folding is rate-limited by the formation of the first stacks and the resultant folding kinetics is single-exponential with a rate of 
Equation 4 gives 
and 
so kf = 4.47 x 103 s–1, which is very close to the result from the rigorous eigenvalue 4.03 x 103 s–1. In Fig. 4 b, we show the dominant pathways predicted from the kinetic partitioning factor 
(see Eq. 5) in the kinetic cluster analysis. As temperature is increased, the slow-folding (fully unfolded) state in U is stabilized, causing a decrease in the folding rate.
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Is this misfolded state a kinetic trap that prevents the pre-equilibration process? No. In fact, it is the result of the pre-equilibration of cluster U. The emergence of the transient intermediate is due to its low free energy relative to all the other states in cluster U. Because its free energy is high relative to the states in N, the intermediate exists only transiently and would disappear when the chain folds into cluster N and the system relaxes to the final equilibrium state.
Stem-sequence dependence
In this section, we study three sequences that have the same loop size and the same stem length, but different stem sequences: sequences 1, 2, and 3, which are shown in Fig. 7, b and c, and Fig. 4 a (wild-type), respectively. The three stem sequences differ by the different positions of two consecutive GC basepairs that form a stable (G, C, G, C) base stack as a clamp in the helix. Specifically, sequences 1, 2, and 3 have the GC clamp near the hairpin loop, at the tail of the stem, and in the middle of the stem, respectively. Sequences 1 and 2 contain one rate-limiting stack, and sequence 3 contains two rate-limiting stacks; see Fig. 7, b and c, and Fig. 4 a, respectively. Plotted in Fig. 7 a are the temperature-dependence of the rates. From the figure, we make the following two observations:
for sequence 2 and 
for sequence 1, where S* is the entropy change for the formation of the rate-limiting stack and Sloop is the entropy change due to the change of the loop size from length 7 to 5 in Fig. 7 b. Sloop is negative. So kseq1 > kseq2, i.e., sequence 1 folds faster than sequence 2.
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)). We focus on the kinetics at T = 40°C < Tm.
To understand the microscopic folding pathways, we use the kinetic-cluster analysis. For this sequence, there are three slow-forming native base stacks (with large S),
 | (7) |
H),
 | (8) |
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formed and Iij = the states with both 
and 
formed. The eigenvalues of the 12-state kinetic cluster system are (0, 1.13, 2.38, 3.93, 5.93, 10.4, ...) x 104 s–1. The eigenvalue spectrum of the 12-state system agrees well with that of the original 1021-state system: (0, 1.09, 2.31, 3.80, 5.72, 10.2, ...) x 104 s–1. This validates our kinetic cluster analysis based on the 12-cluster system.
As we discussed for the folding with two (multiple) rate-limiting stacks, the formation of the first rate-limiting native stack is the bottleneck for the overall folding. From the kinetic connectivity diagram in Fig. 8 a, there exist two types of pathways for the formation of the first rate-limiting native stack (s1, s2, or s3):
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 | (9) |
can be directly computed from Eq. 4. For 
considering the rebound from the two intermediate states I'i and I1i', we have (24)
 | (10) |
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22.8% population in cluster U folds through the on-pathway route U I1 and 68% folds through the off-pathway route U I1'. Therefore, the folding is dominated by the off-pathway process.
To further characterize the populational statistics, we plot in Fig. 8 b the net populational fluxes along pathways U I1', I11' I1, I1 I12, and I12 N. The populational flux PIJ is the (accumulated) probability for the molecule to fold through I J during time period 0 t. The populational flux from cluster I to cluster J is defined as (24):
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and that 
and 
quickly rise in the folding process, which confirms that the dominant pathway is the off-pathway route through the formation and disruption of the non-native base stack 
(U I1' I11' I1 I12 N). How does the formation of the non-native stack in I1' facilitate the folding process?
From the unfolded state U, the formation of the non-native base stack 
is much faster than the direct formation of the native base stack 
In the unfolded cluster U, except the fully unfolded state, which has negligible direct folding rate, the most stable pathway conformation is state 77 (see Fig. 8 c), which occupies 1.32% of the total population of U.
The dominant pathway for the formation of the native 
is through 77 582. This pathway involves the closure of an internal loop, and thus has a slow rate of due to the entropic loss (Sintloop) for the formation of the internal loop closed by basepairs (4,20) and (7,15) in state 582 (see Fig. 8 c): 
= 4.16 x 102 s–1. Here 
is the entropy parameter for the formation of stack 
On the other hand, the dominant pathway for the formation of the non-native is through 77 324. Since this pathway does not involve the closing of additional loops, it has a much faster rate 
= 6.92 x 105 s–1.
So most of the population in U would quickly fold along the off-pathway route 77 324 to form the non-native rate-limiting stack 
Once the non-native base stack is formed in state 324 in cluster I1', the pathway conformations in I1' can be quickly stabilized through the elongation of the helix stem (e.g., 324 995 in Fig. 8 c). These stabilized (non-native) pathway conformations would cause fast transitions from I1'. In addition, the stable non-native structures in I1' can serve as scaffolds to lower the entropic barrier for the further formation of the native rate-limiting stack This would accelerate the folding process. For example, transition 995 1017 is accompanied by an entropic change Sintloop < 0 for the decrease in the internal loop size. As a result, 
is much faster than both the direct on-pathway folding rate k77324 = 4.16 x 102 s–1 and the off-pathway rate k77324 = 6.92 x 105 s–1.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Hstack, Sstack) parameters for different base stacks. As a result, for most DNA sequences, hairpins fold through the formation of the stable loop (scenario 1) instead of the slow-folding native base stack (scenario 2).
Furthermore, the cluster model can explain the ion concentration-dependence of the folding and unfolding rates. Following Santalucia (25), we note that the enthalpy Hstack for a base stack is nearly independent of [Na+], while the entropy is Sstack for a base-stack decrease for higher [Na+] (25).
If the hairpin folding is rate-limited by the formation of a slow-forming base stack, the folding rate 
would increase as [Na+] is increased, while the unfolding rate 
does not change with the ion concentration. These ion-dependences of kf and ku agree with the experimental results for RNA duplex association and dissociation kinetics (26).
If the hairpin folding is rate-limited by the loop formation (see Fig. 10 a), as the ion concentration is increased, the folding rate 
would increase due to the decrease in the entropy. The unfolding rate is given by 
where [c] is the fractional population of state c in Fig. 10 a. Higher ion-concentration stabilizes structures with longer helix stems, e.g., state d (rather than state c) in Fig. 10 a, causing a smaller [c] for state c, which has only one stack. As a result, ku decreases as [Na+] is increased. Moreover, the temperature-dependence of ku is dominated by the 
factor, so the apparent activation barrier of the unfolding does not change with the ion concentration (Hstack is assumed to be [Na+]-independent). This is in agreement with the experimental finding (12).
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These stem/loop length and sequence-dependence of the folding kinetics may be a paradigm for more complete and complex analysis of RNA folding kinetics. Moreover, the general length and sequence dependence can provide useful guidance for molecular design for folding rate, pathways, and cooperativity.
In this study, the effect of the specific loops such as the GNRA and UUCG tetraloops are not considered. These tetraloops can have excess stability due to the intraloop base stacking and hydrogen bonding (27–30). As shown below, it is possible to obtain a rough estimate for the kinetic effects by treating the tetraloop as a stable state (state b in Fig. 10 a) on the free energy landscape. To simplify the analysis, we use a rather crude energy landscape to represent the actual free energy landscape. Considering the rebound effect from the intermediate state b, we can estimate the forward folding rate kf (24):
 | (11) |
With the loop entropy Sloop and enthalpy Hloop for the tetraloop and the stacking entropy Sstack for the (a, c, g, u) stack (see the shaded stack in Fig. 10 b), our rate constant model gives 
The excess tetraloop stabilization parameter can be determined as Sexcess = Sloop – 
and Hexcess = Hloop, where is the entropy of the loop without the tetraloop stabilization.
To directly connect the theory to the experiment, we consider the YNMG RNA hairpins whose folding and unfolding rates have been measured by Proctor et al. (4). We specifically compare the folding rates for the following two sequences: ggacUUCGgucc (with tetraloop stabilization) and ggacUUUUgucc (without tetraloop stabilization). To extract the Sloop and Hloop for the experiment, we subtract the stem parameters from the experimentally measured hairpin parameters (4). Here the stem parameters are calculated from the Turner rule (19) with the salt corrections (with experimental condition of 10 mM Na+) (25).
For the UUCG tetraloop, we found that Sexcess = 25 eu and Hexcess = 12 kcal/mol. Proctor et al. (4) measured that 
= 6.1 x 104 s–1 at T = 65°C. Our theory (with Eq. 11) gives 
= 8.91 x 104 s–1, which is close to the experimental result. The unfolding rate can be estimated from the hairpin stability G(exp) = – 0.79 kcal/mol as 
which gives 
= 1.6 x 104 s–1 and ku(model) = 2.3 x 104 s–1, respectively.
For the UUUU loop, there is no unusual tetraloop stabilization interaction. By assuming Hexcess and Sexcess to be zero in the above equations (i.e., Hloop = 0 and Sloop = ), we found that 
at T = 65°C, which is close to the experimental result 
The experimental and theoretical unfolding rates are 
and 
respectively.
Consistent with the experimental finding, the theory predicts the acceleration in the folding process and the deceleration in the unfolding process due to the tetraloop stabilization. Physically, folding is accelerated because the excess intraloop stacking and basepairing can stabilize the transition state for the folding (see 
in Fig. 10 a) to lower the free energy barrier of folding. The unfolding is decelerated because the intraloop stacking and basepairing in the folded state can cause a higher (enthalpic) barrier for the disruption of the tetraloop.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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This research was supported by the National Institutes of Health (NIH/NIGMS) through grant GM No. 063732 (to S.-J. C).
Submitted on March 14, 2005; accepted for publication September 30, 2005.
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Ni, and the rate constants.
