Fluctuations of a Long, Semiflexible Polymer in a Narrow Channel

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(∥)> occupied by the polymer and the mean-square deviation from the average vary as <R(∥)>=[1-α(∘)(D/P)(2/3)]L and <ΔR(∥)(2)>=β(∘)(D(2)P)L , respectively, where α(∘) and β(∘) are dimensionless amplitudes. In earlier work we determined α(∘) and the analogous amplitude α(square) for a channel with a rectangular cross section from simulations of very long chains. In this paper, we estimate β(∘) and β(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(∥) or radial distribution function, which is asymptotically exact for large L and has the skewed shape seen experimentally.