Constrained thermal denaturation of DNA under fixed linking number

A DNA molecule with freely fluctuating ends undergoes a sharp thermal denaturation transition upon heating. However, in circular DNA chains and some experimental setups that manipulate single DNA molecules, the total number of turns (linking number) is constant at all times. The consequences of this additional topological invariant on the melting behaviour are nontrivial. Below, we investigate the melting characteristics of a homogeneous DNA where the linking number along the melting curve is preserved by supercoil formation in duplex portions. We obtain the mass fraction and the number of loops and supercoils below and above the melting temperature. We also argue that a macroscopic loop appears at T (c) and calculate its size as a function of temperature.