Elucidating the mechanism of folding of polynucleotides depends on accurate estimates of free energy surfaces and a quantitative description of the kinetics of structure formation. Here, the kinetics of hairpin formation in single-stranded DNA are measured after a laser temperature jump. The kinetics are modeled as configurational diffusion on a free energy surface obtained from a statistical mechanical description of equilibrium melting profiles. The effective diffusion coefficient is found to be strongly temperaturedependent in the nucleation step as a result of formation of misfolded loops that do not lead to subsequent zipping. This simple system exhibits many of the features predicted from theoretical studies of protein folding, including a funnel-like energy surface with many folding pathways, trapping in misfolded conformations, and non-Arrhenius folding rates.H airpin loops are ubiquitous in single-stranded DNA and RNA. Knowing the time scales and mechanism of formation of these loops is an essential first step toward understanding the folding problem. Although the stability of hairpin loops and the kinetics of hairpin formation have been a subject of intense investigation for over 30 years (1-5), our understanding of the kinetics is limited. In particular, there is no simple physical model that describes in a consistent way both the thermodynamics and kinetics of hairpin formation. With the exception of some early work on the helix-to-coil transition, in which the kinetics were described in terms of a statistical mechanical kinetic ''zipper'' model (6, 7), the kinetics of hairpin-to-coil transition have been described more recently in terms of a two-state system with Arrhenius temperature dependence for the rates of hairpin formation and unwinding (5, 8). The equilibrium dynamics of hairpins, obtained from fluctuation correlation spectroscopy measurements of hairpins labeled with fluorescent donor and acceptor pairs (8, 9), have revealed a number of kinetic features that are not easily explained within the framework of a simple two-state analysis. First, the data of Libchaber and coworkers show that the rate coefficient corresponding to the closing of hairpins has a non-Arrhenius temperature dependence (8). Second, they report a puzzling result in which the apparent activation energy for forming hairpins with poly(dA) loops increases as the loop size increases. Third, Klenerman and coworkers report stretched exponential kinetics at temperatures well below the melting temperature (9). These observations suggest a failure of the simplest two-state analysis and require a modification of even the more rigorous kinetic ''zipper'' model in the form in which it was applied to helix-coil kinetics (6).Here we present a model for the dynamics of hairpins that is consistent with many of the apparently anomalous kinetic observations. The dynamics are described as configurational diffusion along a free energy profile that we calculate from a statistical mechanical ''zipper'' model that describes the equilibrium melt...

A statistical mechanical "zipper" model is applied to describe the equilibrium melting of short DNA hairpins with poly(dT) loops ranging from 4 to 12 bases in the loop. The free energy of loop formation is expressed in terms of the persistence length of the chain. This method provides a new measurement of the persistence length of single-stranded DNA, which is found to be approximately 1.4 nm for poly(dT) strands in 100 mM NaCl. The free energy of the hairpin relative to the random coil state is found to scale with the loop size with an apparent exponent of > or = 7, much larger than the exponent of approximately 1.5-1.8 expected from considerations of loop entropy alone. This result indicates a strong dependence of the excess stability of the hairpins, from stacking interactions of the bases within the loop, on the size of the loop. We interpret this excess stability as arising from favorable hydrophobic interactions among the bases in tight loops and which diminish as the loops get larger. Free energy profiles along a generalized reaction coordinate are calculated from the equilibrium zipper model. The transition state for hairpin formation is identified as an ensemble of looped conformations with one basepair closing the loop, and with a lower enthalpy than the random coil state. The equilibrium model predicts apparent activation energy of approximately -11 kcal/mol for the hairpin closing step, in remarkable agreement with the value obtained from kinetics measurements.

The kinetics of unwinding of DNA hairpins with varying loop sizes L was monitored using time-resolved absorbance measurements after a laser temperature jump. The characteristic time for forming hairpins is found to scale with the loop size as L 2.0(0.2 , for loops consisting of both poly(dT) and poly(dA) strands, in close agreement with the scaling of loop-closure probability expected for semiflexible polymers. In contrast, equilibrium measurements show that the hairpins with smaller loops are stabilized by a factor that is much larger than can be accounted for simply by the entropic cost of bringing two ends of the polymer together. This excess stability of smaller loops partitions into the opening times, which are found to decrease as L -2.0(0.3 . The temperature dependence of the observed relaxation times, together with the equilibrium measurements, yields negative activation energy (≈ -11 ( 2.3 kcal/mol) for the closing step at temperatures near the melting temperature of the hairpins. In contrast, temperature dependence of the relaxation times, obtained primarily at temperatures below the melting temperature from fluctuation correlation spectroscopy measurements on similar hairpins, yield activation energies for the closing step that are positive (

Hairpin loops are critical to the formation of nucleic acid secondary structure, and to their function. Previous studies revealed a steep dependence of single-stranded DNA (ssDNA) hairpin stability with length of the loop (L) as ∼L8.5 ± 0.5, in 100 mM NaCl, which was attributed to intraloop stacking interactions. In this article, the loop-size dependence of RNA hairpin stabilities and their folding/unfolding kinetics were monitored with laser temperature-jump spectroscopy. Our results suggest that similar mechanisms stabilize small ssDNA and RNA loops, and show that salt contributes significantly to the dependence of hairpin stability on loop size. In 2.5 mM MgCl2, the stabilities of both ssDNA and RNA hairpins scale as ∼L4 ± 0.5, indicating that the intraloop interactions are weaker in the presence of Mg2+. Interestingly, the folding times for ssDNA hairpins (in 100 mM NaCl) and RNA hairpins (in 2.5 mM MgCl2) are similar despite differences in the salt conditions and the stem sequence, and increase similarly with loop size, ∼L2.2 ± 0.5 and ∼L2.6 ± 0.5, respectively. These results suggest that hairpins with small loops may be specifically stabilized by interactions of the Na+ ions with the loops. The results also reinforce the idea that folding times are dominated by an entropic search for the correct nucleating conformation.