Polymer-Based Accurate Positioning: An Exact Worm-like-Chain Study

Abstract

Precise positioning of molecular objects from one location to another is important for nanomanipulation and is also involved in molecular motors. Here, we study single-polymer-based positioning on the basis of the exact solution to the realistic three-dimensional worm-like-chain (WLC) model. The results suggest the possibility of a surprisingly accurate flyfishing-like positioning in which tilting one end of a flexible short polymer enables positioning of the other diffusing end to a distant location within an error of ∼1 nm. This offers a new mechanism for designing molecular positioning devices. The flyfishing effect (and reverse process) likely plays a role in biological molecular motors and may be used to improve speed of artificial counterparts. To facilitate these applications, a new force-extension formula is obtained from the exact WLC solution. This formula has an improved accuracy over the widely used Marko-Siggia formula for stretched polymers and is valid for compressed polymers too. The new formula is useful in analysis of single-molecule stretching experiments and in estimating intramolecular forces of molecular motors, especially those involving both stretched and compressed polymer components.

Figure 1

Mechanics of a worm-like-chain polymer with its one end located at the origin (x = 0, y = 0, z = 0) and the other end diffusing to reach a location on the z axis. The probability distribution of the free end along the z axis, Q(z) = Q(0, 0, z), and the free energy F(z) = F(0, 0, z), the intrachain force f(z) = F′(z) derived are shown. The results are for two scenarios: the end at the origin either has free orientation or is aligned at a fixed angle θ with respect to the z direction. The polymer has a contour length l_c = 10 nm and a persistence length l_p = 4 nm.

Figure 2

Most probable end location and the fwhm probability vs the angle of alignment at the other end for the polymer of Figure 1. The results (symbols) are extracted from the Q(z) = Q(0, 0, z) distributions in Figure 1.

Figure 3

Dependence of the end probability, free energy and intrachain force on persistence length for WLC polymers as for Figure 1 for the alignment angle θ = 0. The polymers have the same contour length l_c = 10 nm but different persistence length (l_p). The exact solutions are shown by symbols with the lines to guide eyes.

Figure 4

Most probable end location and the fwhm probability vs persistence length for WLC polymers. The results for the scenarios of the end at the origin having free orientation or being aligned parallel to the z axis (θ = 0) are shown. The polymers have the same contour length (l_c = 10 nm) but different persistence length (l_p). The results (symbols) are extracted from the Q(z) distributions as in Figure 1. In the left panel, a fitting is also shown for the results for one scenario (the nonlinear part for l_p > 2.5 nm is fitted by a polynomial function for spline interpolation, and the part for lower l_p is fitted by a linear line).

Figure 5

New approximate formula for the force–extension relation of WLC polymers. The force–extension relation from the exact solutions in three dimensions for a l_c = 10 nm polymer in comparison with the predictions by the new formula (eq 1) and the Marko–Siggia formula are shown. Both ends of the polymer have free orientation.

Figure 6

Probability distribution Q(x, y, z) at different y planes for the diffusing end of a WLC polymer with its other end fixed at the origin of the x, y, z axes and aligned in parallel to the z axis. The polymer is characterized by different values of β = l_c/l_p. The x, y, z values shown in the figure are in unit of l_c.