Macroscopic loop formation in circular DNA denaturation

The statistical mechanics of DNA denaturation under fixed linking number is qualitatively different from that of unconstrained DNA. Quantitatively different melting scenarios are reached from two alternative assumptions, namely, that the denatured loops are formed at the expense of (i) overtwist or (ii) supercoils. Recent work has shown that the supercoiling mechanism results in a picture similar to Bose-Einstein condensation where a macroscopic loop appears at $T_c$ and grows steadily with temperature, while the nature of the denatured phase for the overtwisting case has not been studied. By extending an earlier result, we show here that a macroscopic loop appears in the overtwisting scenario as well. We calculate its size as a function of temperature and show that the fraction of the total sum of microscopic loops decreases above $T_c$, with a cusp at the critical point.

Figure 1

Fraction of the denatured DNA (solid line) and the contribution from the microscopic loops (dashed line) for the supercoiling (left) and overtwisting (right) scenarios, as a function of temperature. The parameters $s=5$, ${\epsilon }_b=3$, $c=3.5$, and $A=0.1$ are the same for both figures, while the stiffness parameters for overtwisting ($\kappa =1.0$) and supercoiling (see Ref. [19]) are comparable.

Figure 2

Weight of the macroloop among all the denatured bases shown as a function of temperature for various values of $c$. The top curve for $c=3.5$ is indistinguishable from those obtained for higher values of $c$.

Figure 3

Loop fraction $m$ of the DNA as a function of the reduced temperature $t=(T-T_c)/T_c$ and the overtwist penalty $\kappa$. The surface labeled “microloop fraction” (red online) shows the contribution of the microscopic loops to the total denatured DNA. The size of the macroscopic loop is the vertical distance between the top (gray) and the bottom (red) surfaces for $t>0$. The back panel of the bounding box corresponds to the PS model. Note that the reduced temperature $t$ does not correspond to a single temperature value since $T_c$ is a function of $\kappa$.