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 $<R_\parallel>=[1-α_\circ(D/P)^{2/3}]L$ and $<ΔR_\parallel^{\thinspace 2}\thinspace>=β_\circ(D^2/P)L$, respectively, where $α_\circ$ and $β_\circ$ are dimensionless amplitudes. In earlier work we determined $α_\circ$ and the analogous amplitude $α_\Box$ for a channel with a rectangular cross section from simulations of very long chains. In this paper we estimate $β_\circ$ and $β_\Box$ 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.

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