Elasticity of connected semiflexible quadrilaterals

A tile model of circuit topology for self-entangled biopolymers

Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and connected sums of these knots. This includes all proteins knots that have thus far been identified.

Elasticity of Semiflexible ZigZag Nanosprings with a Point Magnetic Moment

Kinks can appear along the contour of semiflexible polymers (biopolymers or synthetic ones), and they affect their elasticity and function. A regular sequence of alternating kink defects can form a semiflexible nanospring. In this article, we theoretically analyze the elastic behavior of such a nanospring with a point magnetic dipole attached to one end while the other end is assumed to be grafted to a rigid substrate. The rod-like segments of the nanospring are treated as weakly bending wormlike chains, and the propagator (Green's function) method is used in order to calculate the conformational and elastic properties of this system. We analytically calculate the distribution of orientational and positional fluctuations of the free end, the force-extension relation, as well as the compressional force that such a spring can exert on a planar wall. Our results show how the magnetic interaction affects the elasticity of the semiflexible nanospring. This sensitivity, which is based on the interplay of positional and orientational degrees of freedom, may prove useful in magnetometry or other applications.

Grafted Semiflexible Nunchucks with a Magnetic Bead Attached to the Free End

Semiflexible nunchucks are block copolymers, which consist of two long blocks of high bending stiffness jointed together by a short block of low bending stiffness. Semiflexible nunchucks that consist of two DNA nanorods jointed by a short segment of double-stranded (ds) DNA and confined in two dimensions have been used in recent experiments by Fygenson and coworkers as a tool to magnify the bending fluctuations of the linking dsDNA, which in turn are used to deduce the persistence length of dsDNA. In a recent theoretical analysis, we showed that in a semiflexible nunchuck with one end grafted, the fluctuations of the position of the free end that is transverse to the grafting direction exhibit a pronounced bimodality, provided that the bending stiffness of the hinge is not very large. In this article, we theoretically analyse a grafted semiflexible nunchuck with a magnetic bead attached to its free end. We show that a transverse magnetic field induces an asymmetry in the bimodal distribution of the transverse fluctuations of the free end. This asymmetry is very sensitive to interactions with a magnetic field and, in principle, could be used in magnetometry (the measurement of a magnetic field or the magnetic moment of the bead). We also investigate how the response of the bimodal distribution of the transverse fluctuations of the free end to a magnetic field depends on the bending stiffness of the nunchuck hinge. In addition, we analyse the closely related systems of a single filament and two filaments jointed at a kink point with one end grafted and the other end attached to a magnetic bead.

Orientational Fluctuations and Bimodality in Semiflexible Nunchucks

Semiflexible nunchucks are block copolymers consisting of two long blocks with high bending rigidity jointed by a short block of lower bending stiffness. Recently, the DNA nanotube nunchuck was introduced as a simple nanoinstrument that mechanically magnifies the bending angle of short double-stranded (ds) DNA and allows its measurement in a straightforward way [Fygenson et al., Nano Lett. 2020, 20, 2, 1388-1395]. It comprises two long DNA nanotubes linked by a dsDNA segment, which acts as a hinge. The semiflexible nunchuck geometry also appears in dsDNA with a hinge defect (e.g., a quenched denaturation bubble or a nick), and in end-linked stiff filaments. In this article, we theoretically investigate various aspects of the conformations and the tensile elasticity of semiflexible nunchucks. We analytically calculate the distribution of bending fluctuations of a wormlike chain (WLC) consisting of three blocks with different bending stiffness. For a system of two weakly bending WLCs end-jointed by a rigid kink, with one end grafted, we calculate the distribution of positional fluctuations of the free end. For a system of two weakly bending WLCs end-jointed by a hinge modeled as harmonic bending spring, with one end grafted, we calculate the positional fluctuations of the free end. We show that, under certain conditions, there is a pronounced bimodality in the transverse fluctuations of the free end. For a semiflexible nunchuck under tension, under certain conditions, there is bimodality in the extension as a function of the hinge position. We also show how steric repulsion affects the bending fluctuations of a rigid-rod nunchuck.