Bistable collective behavior of polymers tethered in a nanopore

Polymer-coated pores play a crucial role in nucleo-cytoplasmic transport and in a number of biomimetic and nanotechnological applications. Here we present Monte Carlo and Density Functional Theory approaches to identify different collective phases of end-grafted polymers in a nanopore and to study their relative stability as a function of intermolecular interactions. Over a range of system parameters that is relevant for nuclear pore complexes, we observe two distinct phases: one with the bulk of the polymers condensed at the wall of the pore, and the other with the polymers condensed along its central axis. The relative stability of these two phases depends on the interpolymer interactions. The existence the two phases suggests a mechanism in which marginal changes in these interactions, possibly induced by nuclear transport receptors, cause the pore to transform between open and closed configurations, which will influence transport through the pore.

Figure 1

(a) Illustration of the polymer model where bead $n$ is constrained to lie on a circle such that the length of the bonds to beads $n-1$ and $n+1$ is unchanged. (b) Snapshot of a Monte Carlo simulation with 40 noninteracting polymers of contour length 100 nm, bead diameter $d=1$ nm, and bond length $b=1$ nm, end-grafted to the thick dots in a cylindrical pore of 25 nm radius.

Figure 2

Snapshots taken from converged Monte Carlo simulations as in Fig. 1, showing different results for polymers that are subject to the same excluded volume and attractive interactions, as defined by Eq. (1) with $\epsilon =0.1k_{B}T$ and $\lambda =1.0$ nm. (a) Condensation in the center. (b) and (c) Different numbers of clumps can be found when the polymers condense closer to the wall.

Figure 3

Mean bead density profiles for Monte Carlo (MC) simulations (a) and DFT calculations (b) for 40 polymers, composed of 100 beads each, tethered within a cylinder, and interacting only through hard sphere interactions. The plots are given for a range of radial ($r$) and axial ($z$) coordinates. (c) and (d) More detailed comparisons of the profiles as a function of, respectively, the radial direction in the plane of the tethering ring ($z=0$) and the axial direction at a radius $r=17.5$ nm. DFT results are given by the solid lines, and the statistical uncertainty in the MC results is indicated with error bars.

Figure 4

Converged polymer configurations and bead density profiles calculated for an attractive pair potential of depth $\epsilon =0.1k_{B}T$ and range $\lambda =1.0$ nm in a pore of 25 nm radius. The data in the left and right column correspond to initial conditions with the 40 polymers concentrated at the wall and at the center, respectively. (a–b) DFT results with parameters corresponding to the Monte Carlo simulations, as a function of radial ($r$) and axial ($z$) position. (c–d) Comparison of Monte Carlo (with error bars) and DFT (smooth curves) radial profiles for $z=0$, the plane of the tethering points. Axial profiles (e) for the wall phase at $r=22.5$ nm and (f) for the central phase at $r=2.5$ nm.

Figure 5

Metastability diagram of the polymer phases as a function of strength $\epsilon$ and range $\lambda$ of the attractive interactions. $\Delta F_{\mathrm{p}}$ indicates the free energy difference per polymer between the wall and central phases, for parameters where they can both exist as a stable and metastable state. The coexistence conditions lie along the boundary to the right of the “Center metastable” region. In the green (light gray) region, only one phase is found to be possible: the metastability of the other has been lost.

Figure 6

Free energies per polymer of the central phases (solid) and wall phases (dashed) for systems of 40 polymers with a range of bead number $N$, for interaction range $\lambda =0.5$ nm, plotted as a function of interaction strength $\epsilon$. The phase with the lower free energy is thermodynamically stable. The limited extent of the curve of higher free energy, in some cases, is an illustration of the limits of the metastability of that phase, equivalent to spinodal behavior. As $N$ increases, the central phase becomes stable over a larger range of $\epsilon$.