Fluctuations of a long, semiflexible polymer in a narrow channel

We consider an inextensible, semiflexible polymer or worm-like chain, with persistence length $P$ and contour length $L$, fluctuating in a cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$, corresponding to a long, tightly confined polymer, the average length of the channel $<R_{\parallel }>$ occupied by the polymer and the mean square deviation from the average vary as $<R_{\parallel }>=[1-{\alpha }_{\circ }(D\mathrm{/}P{)}^{2\mathrm{/}3}]L$ and $<\Delta R_\parallel^{\thinspace 2}\thinspace>=\beta_\circ(D^2/P)L$,respectively,where$\alpha_\circ$and$\beta_\circ$ are dimensionless amplitudes. In earlier work we determined ${\alpha }_{\circ }$ and the analogous amplitude ${\alpha }_{\mathrm{\square }}$ for a channel with a rectangular cross section from simulations of very long chains. In this paper we estimate ${\beta }_{\circ }$ and ${\beta }_{\mathrm{\square }}$ from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of $R_{\parallel }$ or radial distribution function, which is asymptotically exact for large $L$ and has the skewed shape seen experimentally.