Orientational Fluctuations and Bimodality in Semiflexible Nunchucks

Abstract

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.

Keywords: bimodality; conformations; hinged polymers; tensile elasticity; wormlike chain.

Figure 1

A typical configuration of a rather stiff grafted semiflexible filament. The persistence length of the filament is $l_p$ and it has contour length of $L<l_p$. The filament is grafted in a substrate with grafting angle, $\omega$.

Figure 2

The probability of $\theta \left(L\right)$ (assuming $\omega =0$) from the Gaussian solution (solid lines) and from the quantum analogy (dashed lines) for (a) $L/l_p=3$, (b) $L/l_p=1$, and (c) $L/l_p=0.1$.

Figure 3

Upper panel: A configuration of two jointed weakly bending semiflexible filaments. The stiff joint (kink point) has a kink angle $\gamma$. The first filament has length $L_1$ and persistence length $l_{p1}$. The second filament has length $L_2$ and persistence length $l_{p2}$. The first filament is grafted on a substrate with a grafting angle $\omega$. Lower panel: A configuration of two jointed semiflexible filaments with a hinged point. It differs from the system in the upper panel in that the kink angle $\gamma$ fluctuates about an average value ${\gamma }_0$. The hinge point has a rotational stiffness $K_h$. For $K_h\to \infty$, the hinge point becomes a kink point and we recover the upper panel case.

Figure 4

The probability density of the transverse component of the free end of a grafted system of two hinged WLCs. The curve is obtained from Equation (27). The varying parameter is the rotational stiffness of the hinge point (in units of the thermal energy, $k_{B}T$): $K_h=0.01+0.5\times i$ where $i=0,1,2,3,4,5,6$. The lower the hinge stiffness, the sharper the bimodality of the distribution. The fixed parameters are: $L_1=L_2=3\mathsf{\mu }$m, $l_{p1}=l_{p2}=27\mathsf{\mu }$m, $\omega =0$.

Figure 5

The probability density of the transverse component of the free end of a grafted system of two hinged WLCs. The curve is obtained from Equation (27). The varying parameter is $l_p$; $l_p\equiv l_{p1}=l_{p2}=4.5\times i\mathsf{\mu }$m where $i=1,2,3,4,5,6$. The lower the value of $l_p$, the smoother the bimodality of the curves. The fixed parameters are: $L_1=L_2=3\mathsf{\mu }$m, $\frac{K_h}{2}=0.005$ and $\omega =0$.

Figure 6

The probability density of the bending angle of a WLC hinge with ratio of contour length to persistent length $s/l_p=1.2$. The dashed line is the usual Gaussian, whereas the solid line is obtained by imposing absorbing boundary conditions to Equation (3).

Figure 7

The average projection of the end-to-end distance n the direction of the force as a function of the hinge position for (a) $f=k_{B}T/L$ and (b) $f=20k_{B}T/L$

Figure 8

The mean extension of a semiflexible nunchuck in $2d$ as a function of the hinge position b for $l_p=L$ and $f=150k_{B}T/L$. The solid line corresponds to the weakly bending approximation of the hinged WLC (Equation (38)). The dotted line corresponds to a nunchuck whose small arm is a rigid rod and the long arm a weakly bending WLC (Equation (43)). The dashed line gives the extension of the corresponding intact WLC (without any hinge).