Helical polymer in cylindrical confining geometries

Using an algorithm for simulating equilibrium configurations, we study a fluctuating helical polymer either (i) contained in a cylindrical pore or (ii) wound around a cylindrical rod. We work in the regime where both the contour length and the persistence length of the helical polymer are much larger than the diameter of the cylinder. In case (i) we calculate the free energy of confinement and interpret it in terms of a wormlike chain in a pore with an effective diameter that depends on the parameters of the helix. In case (ii) we consider the possibility that one end of the helical polymer escapes from the rod and wanders away. The average numbers of turns at which the helix escapes or intersects the rod are measured in the simulations, as a function of the pitch $p_0$. The behavior for large and small $p_0$ is explained with simple scaling arguments.

Figure 1

Probability $P\left(n\right)$ that a helical polymer with radius $r_0=0.3$. pitch $p_0=0.3$, and persistence length $L_p=8000$ in a cylindrical pore does not intersect the pore wall in the first $n$ steps of the algorithm. The solid circles (●) correspond to a pore with a circular cross section with diameter $D=1$ and the triangles (▵) to a square cross section with edge length $D=1$. Fitting the data to Eq. (19) for large $n$ yields ${\lambda }_0=6.57\times {10}^{-6}$ and $6.01\times {10}^{-6}$, respectively. The solid line corresponds to the exact exponential decay $e^{-{\lambda }_{0}n}$.

Figure 2

The effective diameter $D_{\mathrm{eff}}$ as a function of the pitch $p_0$ for a helical polymer with radius $r_0=0.3$ and persistence length $L_p=8000$ in a cylindrical pore with a circular cross section with diameter $D=1$. The data interpolate between the limiting values $D-2r_0=0.4$ and $D=1$ for small and large $p_0$, respectively.

Figure 3

Probability $P\left(n\right)$ that a helical polymer with radius $r_0=0.3$ and persistence length $L_p=8000$, wound at the fixed end around a cylindrical rod of diameter $D=0.2$, does not intersect the rod in the first $n$ steps of the algorithm. The pitch of the helix is $p_0=10$, (●) 30 (▵), and 100 (엯).

Figure 4

Average numbers of turns $N_e^1$ (▵), $N_e^2$ (●) at which the helix escapes from the rod and the average number of turns $N_i$ (엯) at which the helix intersects the rod as a function of $p_0$. Here $N_e^1$ is based on all the configurations which escape, independent of whether they return to intersect the rod or not; $N_e^2$ is based on the configurations which escape and in $5\times {10}^6$ steps of the algorithm do not return to intersect the rod. The dashed lines on the left and right have slopes $-2∕3$ and $-1$, respectively, and the solid line shows the prediction (25).

Figure 5

Fractions $f_e^1$ (▵), $f_e^2$ (●), and $f_i$ (엯) of the 10 000 configurations which contribute to $N_e^1$, $N_e^2$, and $N_i$ in Fig. 4, as a function of $p_0$.