Free energy and extension of a semiflexible polymer in cylindrical confining geometries

We consider a long, semiflexible polymer with persistence length $P$ and contour length $L$ fluctuating in a narrow cylindrical channel of diameter $D$. In the regime $D⪡P⪡L$ the free energy of confinement $\Delta F$ and the length of the channel $R_{\Vert }$ occupied by the polymer are given by Odijk’s relations $\Delta F∕R_{\Vert }=A_{\circ }{k}_{B}T{P}^{-1∕3}{D}^{-2∕3}$ and $R_{\Vert }=L[1-{\alpha }_{\circ }{(D∕P)}^{2∕3}]$, where $A_{\circ }$ and ${\alpha }_{\circ }$ are dimensionless amplitudes. Using a simulation algorithm inspired by the pruned enriched Rosenbluth method, which yields results for very long polymers, we determine $A_{\circ }$ and ${\alpha }_{\circ }$ and the analogous amplitudes for a channel with a rectangular cross section. For a semiflexible polymer confined to the surface of a cylinder, the corresponding amplitudes are derived with an exact analytic approach. The results are relevant for interpreting experiments on biopolymers in microchannels or microfluidic devices.

Figure 1

The curve may be interpreted as a tightly confined semiflexible polymer in a channel with a circular cross section or as the trajectory $\vec{r}\left(t\right)$, plotted as a function of $t$, of a randomly accelerated particle moving in two dimensions, which has not yet left a circular domain.

Figure 2

$\ln Q\left(t^{\prime }\right)$ vs $t^{\prime }$ for a polymer on a two-dimensional strip (upper curve) and for a polymer in a three-dimensional channel with a circular cross section (lower curve).

Figure 3

${⟨v_x^{\prime 2}⟩}_{1∕2,1}^{\mid }$ vs $t^{\prime }$ for a polymer on a two-dimensional strip (lower curve) and ${⟨{\vec{v}}^{\prime 2}⟩}_{1∕2,1}^{\circ }$ vs $t^{\prime }$ for a polymer in a three-dimensional channel with a circular cross section (upper curve).

Figure 4

Distribution $P\left(⟨v^{\prime 2}⟩\right)$ for a polymer on a two-dimensional strip, as defined below Eq. (31). The curves correspond, from bottom to top, to $t^{\prime }=100$, 225, 400, 625, and 900.

Figure 5

Double-logarithmic plot of the half width or standard deviation $w$ of the distribution $P\left(⟨v^{\prime 2}⟩\right)$ (see Fig. 4) as a function of $t^{\prime }$. The straight line corresponds to $w=k{t}^{\prime -1∕2}$, where $k$ is a constant.