Elasticity of semiflexible polymers in two dimensions

We study theoretically the entropic elasticity of a semiflexible polymer, such as DNA, confined to two dimensions. Using the worm-like-chain model we obtain an exact analytical expression for the partition function of the polymer pulled at one end with a constant force. The force-extension relation for the polymer is computed in the long chain limit in terms of Mathieu characteristic functions. We also present applications to the interaction between a semiflexible polymer and a nematic field, and derive the nematic order parameter and average extension of the polymer in a strong field.

Figure 1

(Color online) Eigenfunctions $c{e}_0$, $s{e}_2$, and $c{e}_2$ plotted for $4f\kappa =1$ and for $\theta$ between $-\pi$ and $\pi$.

Figure 2

Characteristic functions $a_0\left(q\right)$, $b_2\left(q\right)$, $a_2\left(q\right)$, $b_4\left(q\right)$, and $a_4\left(q\right)$ plotted against $q$.

Figure 3

Force extension curves for semiflexible polymers. The full line is the relative extension $⟨X⟩∕L$ plotted against the dimensionless force, $q=4F\kappa ∕k_{B}T$ using the exact formula, Eq. (19). The dashed line is the interpolation formula, Eq. (33). For comparison, the force-extension curve in three dimensions is plotted as a dot-dashed line, using the approximate interpolation formula described in Ref. 2.