Molecular Elasticity and the Geometric Phase

We present a method for solving the wormlike-chain (WLC) model for twisting semiflexible polymers to any desired accuracy. We show that the WLC free energy is a periodic function of the applied twist with period $4\pi$. We develop an analogy between WLC elasticity and the geometric phase of a spin-$\frac{1}{2}$ system. These analogies are used to predict elastic properties of twist-storing polymers. We graphically display the elastic response of a single molecule to an applied torque. This study is relevant to mechanical properties of biopolymers such as DNA.

Figure 1

The torque-twist relation for $\beta =L/L_{\mathrm{B}\mathrm{P}}=1$ and $\alpha =L/L_{\mathrm{T}\mathrm{P}}=0,1,7$.