Fluctuations of a long, semiflexible polymer in a narrow channel

Abstract

We consider an inextensible, semiflexible polymer or wormlike chain, with persistence length $P$ and contour length $L$, fluctuating in a cylindrical channel of diameter $D$. In the regime $D⪡P⪡L$, corresponding to a long, tightly confined polymer, the average length of the channel $⟨{R}_{\ensuremath{\parallel}}⟩$ occupied by the polymer and the mean-square deviation from the average vary as $⟨{R}_{\ensuremath{\parallel}}⟩=[1\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\circ}}{(D/P)}^{2/3}]L$ and $⟨\ensuremath{\Delta}{R}_{\ensuremath{\parallel}}^{2}⟩={\ensuremath{\beta}}_{\ensuremath{\circ}}({D}^{2}/P)L$, respectively, where ${\ensuremath{\alpha}}_{\ensuremath{\circ}}$ and ${\ensuremath{\beta}}_{\ensuremath{\circ}}$ are dimensionless amplitudes. In earlier work we determined ${\ensuremath{\alpha}}_{\ensuremath{\circ}}$ and the analogous amplitude ${\ensuremath{\alpha}}_{◻}$ for a channel with a rectangular cross section from simulations of very long chains. In this paper, we estimate ${\ensuremath{\beta}}_{\ensuremath{\circ}}$ and ${\ensuremath{\beta}}_{◻}$ 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 ${R}_{\ensuremath{\parallel}}$ or radial distribution function, which is asymptotically exact for large $L$ and has the skewed shape seen experimentally.