Past exam of the biological physics course of the University of Oxford / 2021 / 3 / b / Solution

For an inextensible worm-like chain, the DNA bending energy iswhere $\kappa$ is curvature and $\ell$ is polymer contour length. A circle of radius $R_i$ has $\kappa =1/R_i$ and length $2\pi R_i$. Its elastic energy is therefore $(k_{B}T{l}_p/2)(2\pi R_i)/R_i^2=\pi l_p{k}_{B}T/R_i$. Adding the contributions of $n(R_i)$ rings at each radius givesThis ignores short connecting segments, twist, interactions between neighboring DNA segments and end effects. In a cylindrical stack with axial spacing $s$, $n(R_i)\simeq h/s$ away from the ends.