Denaturation of circular DNA: Supercoil mechanism

The denaturation transition which takes place in circular DNA is analyzed by extending the Poland-Scheraga (PS) model to include the winding degrees of freedom. We consider the case of a homopolymer whereby the winding number of the double-stranded helix, released by a loop denaturation, is absorbed by supercoils. We find that as in the case of linear DNA, the order of the transition is determined by the loop exponent $c$. However the first-order transition displayed by the PS model for $c>2$ in linear DNA is replaced by a continuous transition with arbitrarily high order as $c$ approaches 2, while the second-order transition found in the linear case in the regime $1<c⩽2$ disappears. In addition, our analysis reveals that melting under fixed linking number is a condensation transition, where the condensate is a macroscopic loop which appears above the critical temperature.

Figure 1

A typical configuration of a portion of the circular DNA model used in this study.

Figure 2

A plot of $G\left(x\right)$, as defined in (20), as function of $x$. The solid line corresponds to $c=1.5$ while the dashed line to $c=2.5$. The horizontal lines corresponds to different temperatures. The arrows point to the solutions of Eq. (18) which is equivalent to the thermodynamic limit. While for $c=1.5$ there exists a solution for all temperatures, for $c=2.5$ there is a solution only up to some finite temperature, in this case ${\nu }_c^{1/2}{\omega }_c^{-1}\approx 0.25$. The parameters which were used in this plot are $E_b=-3$, $E_s=-2$, $s=5$, and $A=0.1$.

Figure 3

The loop size distribution $p\left(l\right)$ in the canonical (solid line) and the regularized grand-canonical (dashed line) ensembles. For small values of $l$ the critical phase can be identified [where $p\left(l\right)\sim l^{-c}$]. For $l\sim L$ the canonical curve shows the “bump” around $l\approx \xi$ (see text) while the grand-canonical curve behaves in a somewhat different manner. The parameters which were used for this plot are $c=3.5$, $L=400$, $M=200$, $E_b=-3$, $E_s=-2$, $s=5$, $A=0.1$, and $T=3$ ($T_c=1.167$).

Figure 4

The integration procedure used for the canonical partition function: (a) In the $z$ plane, the contour $C^{\left(z\right)}$ which encircles the origin can be replaced by $C_p^{\left(z\right)}$ around the pole at $z_0$ and $C_{bc}^{\left(z\right)}$ which wraps the branch cut (thick line) and closes at infinity; (b) in the $\mu$ plane, the contour $C^{\left(\mu \right)}$ can be deformed to pass through a saddle point ${\mu }_0$ when it exits. Otherwise the dominant contribution comes from the vicinity of the branch point (see text).