  Citation: Razbin, M.; Benetatos, P. Grafted Semiflexible Nunchucks with a Magnetic Bead Attached to the Free End. Polymers 2022, 14, 695. https:// doi.org/10.3390/polym14040695 Academic Editor: Arash Nikoubashman Received: 13 January 2022 Accepted: 3 February 2022 Published: 11 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). polymers Article Grafted Semiflexible Nunchucks with a Magnetic Bead Attached to the Free End Mohammadhosein Razbin 1, * and Panayotis Benetatos 2, * 1 Department of Energy Engineering and Physics, Amirkabir University of Technology, Tehran 14588, Iran 2 Department of Physics, Kyungpook National University, 80 Daehakro, Bukgu, Daegu 41566, Korea * Correspondence: m.razbin@aut.ac.ir (M.R.); pben@knu.ac.kr (P.B.) Abstract: 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. Keywords: semiflexible polymers; hinged polymers; bimodality; magnetometry; signal enhancement 1. Introduction Semiflexible nunchucks are block copolymers consisting of two long blocks of high bending stiffness linked together by a short block of lower bending stiffness. Such nanos- tructures have been manufactured using DNA nanotubes linked by a segment of ds- DNA [1,2]. The DNA nanotubes are fluorescently labeled and confined in two dimensions between two glass plates. Their fluctuations are directly visualised and act as a magnifica- tion of the bending fluctuations of the short dsDNA segment. The spread of the distribution of the bending angle is used to deduce the persistence length of the dsDNA. The conformational fluctuations of a semiflexible nunchuck in two dimensions are amenable to analytical treatment [3]. Assuming one end to be grafted, we can calculate (up to a numerical integration) the probability distribution of the position of the free end. We assume that the contour length of the linking block is negligible compared to the contour length of the two arms, and we treat it as a harmonic orientational spring that is characterized by a bending stiffness. The probability distribution of the transverse position of the free end (after integrating out the longitudinal position) assumes a unimodal or bimodal form, depending on the bending stiffness of the hinge between the two arms. For large values of bending stiffness, the distribution is unimodal with one peak and as the bending stiffness decreases, the distribution flattens and eventually develops a pronounced bimodality with two peaks. In order to understand the origin of this emergent bimodality, Polymers 2022, 14, 695. https://doi.org/10.3390/polym14040695 https://www.mdpi.com/journal/polymers