Bistability induced by a spontaneous twisting rate for a two-dimensional intrinsically curved filament

We find that a moderate intrinsic twisting rate (ITR) can induce a bistable state for a force-free two-dimensional intrinsically curved filament. There are two different configurations of equal energy in a bistable state so that the filament is clearly different from its three-dimensional counterpart. The smaller the ITR or the larger the intrinsic curvature (IC), the clearer the distinction between two isoenergetic configurations and the longer the filament. In bistable states, the relationship between length and ITR is approximately a hyperbola and relationship between IC and critical ITR is approximately linear. Thermal fluctuation can result in a shift between two isoenergetic configurations, but large bending and twisting rigidities can prevent the shift and maintain the filament in one of these two configurations. Moreover, a filament can have a metastable state and at a finite temperature such a filament has the similar property as that of a filament with bistable state.

Figure 1

$L$ vs ${\psi }_0$ obtained from Eq. (24) when $B_1<0$. $B_0=0.067$, ${\zeta }_3=0.05$ (red dashed), 0.1 (solid black) and 0.2 (green dotted); $B_0=0.1$, ${\zeta }_3=0.05$ (blue dash-dotted), 0.1 (cyan dash-dot-dotted), and 0.2 (magenta shot-dashed). All ${\psi }_0\mathrm{s}$ begin from $B_1\approx 0$. The unit of length is nm so that the units of both $c_0$ and ${\zeta }_3$ are ${\mathrm{nm}}^{-1}$, and the unit of $B_0$ is ${\mathrm{nm}}^{-2}$.

Figure 2

$L$ vs ${\psi }_0$ obtained from Eq. (26) when $B_0=0.000267$ but with different ${\zeta }_3$ and $n$ (=0 in left and 1 in right). The same color for the data with the same parameters. ${\zeta }_3=0.013$ (black), 0.02 (red), and 0.055 (green). Three horizontal dashed straight lines indicate the corresponding ${\psi }_0$ for BSs when $c_0=0.02$. The corresponding GSCs are displayed in Fig. 3. The units are the same as that in Fig. 1.

Figure 3

GSCs in BS. The same color for the data with the same parameters. For black lines, $B_0=0.000267$, $c_0=0.02$, ${\zeta }_3=0.013$, $L=300$, $L_h=263.4$, and (1) $r_L=21.7$ (solid, $n=0$); (2) $r_L=225.4$ (dashed, $n=1$). For red lines, $B_0=0.000267$, $c_0=0.02$, ${\zeta }_3=0.02$, $L=163$, $L_h=222.1$, and (3) $r_L=112.0$ (dotted, $n=0$); (4) $r_L=151.8$ (dash-dotted, $n=1$). For green lines, $B_0=0.000267$, $c_0=0.02$, ${\zeta }_3=0.055$, $L=58$, $L_h=107.4$, and (5) $r_L=56.0$ (dash-dot-dotted, $n=0$); (6) $r_L=57.6$ (short-dashed, $n=1$). For purple lines, $B_0=0.08$, $c_0=0.2$, ${\zeta }_3=0.19$, $L=28$, $L_h=22.8,$ and (7) $r_L=3.56$ (short-dashed, $n=0$); (8) $r_L=21.13$ (short-dotted, $n=1$). The units are the same as that in Fig. 1.

Figure 4

$\psi$ vs $s$ in BS when $B_0=0.000267$. The same color for the data with the same parameters. $L=300$ and ${\zeta }_3=0.013$ for solid black ($n=0$) and dashed-black ($n=1$) lines; $L=163$ and ${\zeta }_3=0.02$ for red-dotted ($n=0$) and red dash-dotted ($n=1$) lines; $L=58$ and ${\zeta }_3=0.055$ for green dash-dot-dotted ($n=0$) and green short-dashed ($n=1$) lines. The units are the same as that in Fig. 1.

Figure 5

$L$ vs ${\zeta }_3$ in BS when $b_3/b_1=1.5$. $c_0=0.02$ (solid black square), 0.05 (empty red square), 0.1 (solid green circle), and 0.2 (empty blue circle). The solid black lines are given by $L=3.3/{\zeta }_3$. The inset presents the data with larger range in $L$. The units are the same as that in Fig. 1.

Figure 6

$L$ vs ${\zeta }_3$ in BS when $b_3/b_1=1$ and 1/2 (inset). $c_0=0.02$ (solid black square), 0.05 (empty red square), 0.1 (solid green circle), and 0.2 (empty blue circle). The solid black lines are given by $L=3.3/{\zeta }_3$. The units are the same as that in Fig. 1.

Figure 7

${\zeta }_c$ vs $c_0$ when $b_3/b_1=1.5$ (square), 1 (circle), and 1/2 (triangle). The solid straight line is given by ${\zeta }_c=0.52c_0$; the dashed straight line is given by ${\zeta }_c=0.64c_0$; the dotted straight line is given by ${\zeta }_c=0.96c_0$. The units are the same as that in Fig. 1.

Figure 8

$〈r_N〉$ and $〈E_T〉/N$ vs sample counts when $c_0=0.2$, ${\zeta }_3=0.19$ and $N=28$. Black square for $〈r_N〉$ when $k_1=12$ and $k_3=6$; red circle for $〈r_N〉$ when $k_1=100$ and $k_3=50$; green triangle for $〈r_N〉$ when $k_1=200$ and $k_3=100$. Blue diamond for $〈E_T〉/N$ when $k_1=200$ and $k_3=100$. The units are the same as those described in the caption of Fig. 1.

Figure 9

$〈r_N〉$ and $〈E_T〉/N$ vs sample counts when $c_0=0.02$, ${\zeta }_3=0.013$ and $N=300$. Green triangle for $〈r_N〉$ when $k_1=1000$ and $k_3=1500$; black square for $〈r_N〉$ when $k_1=10$ and $k_3=15$; blue open diamond for $〈E_T〉/N$ when $k_1=1000$ and $k_3=1500$; red circle for $〈E_T〉/N$ when $k_1=10$ and $k_3=15$. The units are the same as those described in the caption of Fig. 1.

Figure 10

$〈r_N〉$ (solid) and $〈E_T〉/N$ (empty) vs sample counts in BLS when $k_1=200$, $k_3=100,$ and $c_0=0.2$. Black square for $N=23$ and ${\zeta }_3=0.19$; red circle for $N=33$ and ${\zeta }_3=0.19$; green triangle for $N=28$ and ${\zeta }_3=0.15$; blue diamond for $N=28$ and ${\zeta }_3=0.22$. The units are the same as those described in the caption of Fig. 1.