Free-energy cost of localizing a single monomer of a confined polymer

We describe a simple Monte Carlo simulation method to calculate the free-energy cost of localizing a single monomer of a polymer confined to a cavity. The localization position is chosen to be on the inside surface of the confining cavity. The method is applied to a freely jointed hard-sphere polymer chain confined to cavities of spherical and cubic geometries. In the latter case, we consider localization at a corner and at the center of a face of the confining cube. We consider cases of end-monomer localization both with and without tethering of the other end monomer to a point on the surface. We also examine localization of monomers at arbitrary positions along the contour of the polymer. We characterize the dependence of the free energy on the cavity size and shape, the localization position, and the polymer length. The quantitative trends can be understood using standard scaling arguments and use of a simple theoretical model. The results are relevant to those theories of polymer translocation that focus on the importance of the free-energy barrier as the translocation process requires an initial localization of a monomer to the position of a nanopore.

Figure 1

Schematic illustration of the method used to calculate the chain-end localization free energy for volume subdivision fraction ${\alpha }_m=\frac{1}{2}$. (a) The volume subdivisions used to localize the chain end to a point (labeled as a black dot). Each successive subvolume is half of the previous subvolume in the sequence. (b) Illustration of method for calculating $\mathrm{\Delta }{F}_m$ for $m=2$. The center of the selected end monomer (colored green) is confined to lie within the volume $V/2$ (shaded red), while the centers of the remaining monomers (colored blue) can be anywhere in the full volume $V$. $\mathrm{\Delta }{F}_2$ is determined in a MC simulation by calculating the fraction of monomer center positions of the end monomer that lie within the volume $V/4$ (double-shaded red). Note that the localization point lies on the surface enclosing the volume accessible to the monomer centers. Thus, the subvolumes can be made arbitrarily small regardless of the finite size of the monomers.

Figure 2

Illustration of the convergence of the summation of Eq. (21) for the case of confinement in a cubic cavity of side length $D=30$ and a polymer of length $N=100$. Here, the volume subdivision ratio is ${\alpha }_m=0.16069$ for the $m\mathrm{th}$ subdivision. $\mathrm{\Delta }{F}_m$ represents the free-energy difference between localizing the end-monomer volumes $V_m$ and $V_{m-1}$, where these subvolumes are defined in Eq. (20). In addition, $ln\left({\alpha }_m^{-1}\right)$ is the free-energy difference (in units of $k_{\mathrm{B}}T$) for a single isolated particle for confinement in these two subvolumes. These two quantities appear on the right side of Eq. (21). $\mathrm{\Delta }{F}_{\mathrm{cum}}\left(m\right)$ is the cumulative sum of the difference in these two quantities. As the number of volume subdivisions $m$ increases, $\mathrm{\Delta }{F}_{\mathrm{cum}}\left(m\right)$ converges to $\mathrm{\Delta }{F}_{\mathrm{loc}}$.

Figure 3

(a) $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs confining $D/R_{\mathrm{g}}$ for chains of length $N=20$, 50, 100, and 200. Results are shown for both spherical cavities (solid circles) and cubic confinement cavities (open squares). For the cubic cavities, chain end localization occurs at the center of one side of the cube. Here, $D\equiv V^{1/3}$, where $V$ is the volume of the confining cavity accessible to the monomer centers, and $R_{\mathrm{g}}$ is the radius of gyration of an unconfined polymer. (b) As in panel (a), except results are shown for localization of a chain end to the corner of a cube (solid squares) and the center of one face of a cube (open squares).

Figure 4

(a) $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs polymer length $N$ for polymers confined to a cubic space. Results are shown for various ratios of $D/R_{\mathrm{g}}$ for the cases of chain-end localization to the center of one face of the cube (closed squares) and to one corner of the cube (open squares). The curves overlaid on the data are fits to $\mathrm{\Delta }{F}_{\mathrm{loc}}=c+\alpha lnN$. The fitting parameter values are as follows for localization to a corner: $\alpha =1.54\pm 0.01$ and $c=0.82\pm 0.06$ for $D/R_{\mathrm{g}}=15$; $\alpha =1.48\pm 0.01$ and $c=0.51\pm 0.05$ for $D/R_{\mathrm{g}}=7.5$; $\alpha =1.28\pm 0.02$ and $c=-1.04\pm 0.08$ for $D/R_{\mathrm{g}}=2.5$. For localization to the center of a face, the values are $\alpha =0.51\pm 0.01$ and $c=-0.09\pm 0.04$ for $D/R_{\mathrm{g}}=15$; $\alpha =0.47\pm 0.01$ and $c=-0.49\pm 0.04$ for $D/R_{\mathrm{g}}=7.5$; $\alpha =0.36\pm 0.01$ and $c=-0.90\pm 0.04$ for $D/R_{\mathrm{g}}=2.5$.

Figure 5

Predictions for $\mathrm{\Delta }{F}_{\mathrm{loc}}$ using the theoretical model of Appendix pp3. (a) $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs $D/R_{\mathrm{g}}$ for $N=20$, 50, 100, and 200. In each case, results are shown for end-monomer localization to the center of one face of the cubic box and to a point on the wall of a confining sphere. (b) As in panel (a), except results are shown for end-monomer localization to the center of one face of the cubic box and localization to one corner of the cube.

Figure 6

Predictions for $\mathrm{\Delta }{F}_{\mathrm{loc}}$ using the theoretical model of Appendix pp3. $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs $N$ for $D/R_{\mathrm{g}}=2.5$, 7, and 15. In each case, results are shown for end-monomer localization to the center of one face of the cubic box and localization to one corner of the cube.

Figure 7

(a) $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs scaled confining cavity size $D/R_{\mathrm{g}}$, where $\mathrm{\Delta }{F}_{\mathrm{loc}}$ is the free-energy cost of localizing a middle monomer to a point in the center of a face on a confining cubic surface. Results are shown for polymer chains of length $N=20$, 50, 100, and 200. The dashed line shows the theoretical prediction of Eq. (26) for the limiting case of $D/R_{\mathrm{g}}\to \infty$. The constant in Eq. (26) is chosen simply to shift the theoretical curve above the simulation data for clarity. (b) $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs monomer localization index $m_{\mathrm{loc}}$ for a $N=200$ polymer confined to a spherical cavity. Results for different spherical cavity sizes are shown.

Figure 8

$\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs polymer length $N$ for a polymer confined to a sphere in the case of localization of the middle monomer of the polymer. Results are shown for $D/R_{\mathrm{g}}=2.5$ and 10. For convenient comparison, results are also shown for end-monomer localization with $D/R_{\mathrm{g}}=10$. The curves overlaid on the data are fits to $\mathrm{\Delta }{F}_{\mathrm{loc}}=c+\alpha lnN$. The fitting parameter values are as follows for localization of the middle monomer: $\alpha =1.01\pm 0.02$ and $c=0.46\pm 0.07$ for $D/R_{\mathrm{g}}=10$ and $\alpha =0.928\pm 0.005$ and $c=0.28\pm 0.02$ for $D/R_{\mathrm{g}}=2.5$. For end-monomer localization with $D/R_{\mathrm{g}}=10, \alpha =0.54\pm 0.02$ and $c=0.0\pm 0.1$.

Figure 9

$\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs confining sphere diameter $D$ for a chain of length $N=200$. Here, $\mathrm{\Delta }{F}_{\mathrm{loc}}$ is the change in the conformational free energy upon localizing a monomer to a point on the inner surface of the sphere in the case where the other monomer is tethered to a different point on the sphere. Results are shown for four different values of the angular distance $\theta$ between the tethered point and the localization point for real (i.e., self-avoiding) polymers and ideal polymers (for which the excluded-volume interaction between nonbonded monomers is turned off). The inset shows the scaled free energy ${\left(\mathrm{\Delta }{F}_{\mathrm{loc}}\right)}^{1/(1-\nu )}$ vs scaled end-to-end distance $d_{\mathrm{ee}}/2R_{\mathrm{g}}$ where $d_{\mathrm{ee}}\equiv {(6/\pi )}^{1/3}Dsin(\theta /2)$. The results were calculated using the raw data with $\nu =\frac{1}{2}$ for ideal chains and $\nu =\frac{3}{5}$ for real chains. The dashed curve is a linear fit to the data in the domain $d_{\mathrm{ee}}/2R_{\mathrm{g}}>1.7$. The illustration in the inset shows the definition of the angle $\theta$.

Figure 10

As in Fig. 9, except $\mathrm{\Delta }{F}_{\mathrm{loc}}$ vs polymer length $N$ for confinement in a sphere of diameter $D=50$. The inset shows ${\left(\mathrm{\Delta }{F}_{\mathrm{loc}}\right)}^{1-\nu }$ vs $d_{\mathrm{ee}}/2R_{\mathrm{g}}$, calculated as for the data in the inset of Fig. 9.

Figure 11

Free-energy differences vs polymer length calculated from PERM simulations. These differences are defined $\mathrm{\Delta }{F}_{\mathrm{ab}}\equiv F_{\mathrm{b}}-F_{\mathrm{a}}, \mathrm{\Delta }{F}_{\mathrm{ac}}\equiv F_{\mathrm{c}}-F_{\mathrm{a}}$, and $\mathrm{\Delta }{F}_{\mathrm{bc}}\equiv F_{\mathrm{c}}-F_{\mathrm{b}}$, where (i) $F_{\mathrm{a}}$ is the free energy of a free polymer; (ii) $F_{\mathrm{b}}$ is that for a polymer tethered to a hard flat wall, and (iii) $F_{\mathrm{c}}$ is that for a polymer tethered to a corner of an infinitely large confining cube. The dashed lines overlaid on the curves are fits to the function $\mathrm{\Delta }{F}_{\lambda \mu }=C_{\lambda \mu }+{\alpha }_{\lambda \mu }lnN$, where $C_{\mathrm{ac}}=1.476\pm 0.004, {\alpha }_{\mathrm{ac}}=1.451\pm 0.001$; $C_{\mathrm{bc}}=1.031\pm 0.002, {\alpha }_{\mathrm{bc}}=1.006\pm 0.001$; and $C_{\mathrm{ab}}=0.444\pm 0.002, {\alpha }_{\mathrm{ab}}=0.446\pm 0.002$. The inset shows the free energies $F_{\mathrm{a}}, F_{\mathrm{b}}$, and $F_{\mathrm{c}}$ vs $N$.

Figure 12

Conformational free energy of a polymer near a flat wall $F_{\mathrm{wall}}$ vs scaled distance of an end monomer away from the wall, $z/R_{\mathrm{g}}$. Results are shown for different polymer lengths. The inset shows the difference $\mathrm{\Delta }{F}_{\mathrm{ab}}\equiv F_{\mathrm{wall}}\left(0\right)-F_{\mathrm{wall}}\left(\infty \right)$ vs $N$. The solid line shows the fit to $\mathrm{\Delta }{F}_{\mathrm{ab}}=C_{\mathrm{ab}}+{\alpha }_{\mathrm{ab}}lnN$, which yields ${\alpha }_{\mathrm{ab}}=0.463\pm 0.003$ and $C_{\mathrm{ab}}=0.40\pm 0.01$.

Figure 13

Illustration of the various quantities used for the theoretical model. (a) Region A is cubic subspace of the confining cube. Region C is composed of eight small cubes near the corners of the confining cube, and region B is the remaining subspace located near the walls of the cube. When a polymer is located in region A, it does not interact with the walls of the box. However, when it is located in regions B or C, it does interact, leading to an increase in the conformational entropy, as explained in the text. (b) As in panel (a), except for a confining sphere. There is only a single region (B) where the polymer interacts with the confining wall. Note that $D^{\prime }={(4\pi /3)}^{1/3}D$, as explained in the text.