Chain persistency in single-stranded DNA

We develop a theoretical approach to hairpin-loop formation of single-stranded (ss) DNA by treating the strand as a two-state system in which bases are either “stacked” or “unstacked.” The looping kinetics of ssDNA is shown to be intrinsically different from that of a wormlike chain; it is mainly controlled by stacking-breakage probability, not by the mean curvature of loops, and highly sensitive to the composition of the loop as seen in recent experiments. Our estimate of a stacking energy for poly$\left(dA\right)$, $-3.9\text{kcal}∕\mathrm{mol}$, is consistent with known results.

Figure 1

Schematic representation of (a) a wormlike chain (WLC) that tends to bend smoothly (it remains rodlike within ${\ell }_p$, the persistence length), and (b) a stacking chain that models ssDNA as schematically shown in (c). Unlike the WLC, the stacking chain allows abrupt bending or stacking breakage, as marked by an arrow in (b).

Figure 2

Conformations considered in text: (a) open poly$\left(dT\right)$ conformation, (b) misfolded poly$\left(dT\right)$ conformation, (c) closed poly$\left(dT\right)$ conformation, (d) open poly$\left(dA\right)$ conformation, (e) misfolded poly$\left(dA\right)$ conformation, and (f) closed poly-$A$ conformation. Using an analysis of the probability densities and binding energy scales for each of these plots, it can be shown that (b) and (c) should be included in the interpretation of poly$\left(dT\right)$ data, while (d) and (f) are crucial for the interpretation of poly$\left(dT\right)$ data in Ref. 7.

Figure 3

The closing time $\left({\tau }_c\right)$ of a hairpin loop (in $\mu \mathrm{s}$) as a function of temperature $T$: (a) poly$\left(dT\right)$ and (b) poly$\left(dA\right)$. Our results for the poly$\left(dT\right)$ and poly$\left(dA\right)$ are represented by the dotted and solid curves, respectively. Our scaling analysis predicts a straight line in this semilogrithmic plot. Our estimate of staking energy, deduced from the slope of the solid curve, is favorably comparable to previous estimates [see the relevant discussion below Eq. (10)]. Experimental data adopted from Ref. 7 are represented by various symbols: filled circles $\left(N=30\right)$, filled squares $\left(N=21\right)$, filled diamonds $\left(N=16\right)$, triangles $\left(N=12\right)$, crosses $\left(N=10\right)$, and plus signs $\left(N=8\right)$. For poly$\left(dT\right)$, the closing time has been multiplied by $10∕[N^{1.5}B\left(N\right)]$ [Eq. (8)], where the factor 10 was introduced for clarity of the plot. For poly$\left(dA\right)$, the closing time has been multiplied by $1∕N^{1.5}$ [Eq. (10)]. The large-$N$ data, represented by the filled symbols, approach the asymptotic straight line.