Elasticity of two-dimensional filaments with constant spontaneous curvature

We study the mechanical property of a two-dimensional filament with constant spontaneous curvature and under uniaxial applied force. We derive the equation that governs the stable shape of the filament and obtain analytical solutions for the equation. We find that for a long filament with positive initial azimuth angle (the azimuth angle is the angle between $x$ axis and the tangent of the filament) and under large stretching force, the azimuth angle is a two-valued function of the arclength, decreases first, and then increases with increasing arclength. Otherwise, the azimuth angle is a monotonic function of arclength. At finite temperature, we derive the differential equation that governs the partition function and find exact solution of the partition function for a filament free of force. We obtain closed-form expressions on the force-extension relation for a filament under low force and for a long filament under strong stretching force. Our results show that for a biopolymer with moderate length and not too small spontaneous curvature, the effect of the spontaneous curvature cannot be replaced by a simple renormalization of the persistence length in the wormlike chain model.

Figure 1

Schematic picture of a filament under different boundary conditions. (a) ${\varphi }_0$ is fixed by grafting a small portion of filament to a substrate. ${\varphi }_0=0.2$. (b) The same as (a) except for ${\varphi }_0=-0.2$. (c) One end of the filament is attached to a bead, which is confined by a pivot. ${\varphi }_0$ is not fixed so the bead can rotate freely around the pivot. $c_0=0.1$ and $L=10$ for all three cases.

Figure 2

Typical shapes of a two-dimensional filament under stretching. The parameters are $c_0=0.1$, $L=15$, and (a) (solid) ${\varphi }_0=0.2$, $F_x=0.0373$; (b) (dash) ${\varphi }_0=0.2$, $F_x=0.04$; (c) (solid) ${\varphi }_0=0.2$, $F_x=0.3$; (d) (dot) ${\varphi }_0=-0.2$, $F_x=0.3$. The unit of length and $F_x$ are the same as $1∕c_0$ and $c_0^2$, respectively.

Figure 3

Relative extension $X_L\equiv x\left(L\right)∕L$ (solid), $Y_L\equiv y\left(L\right)∕L$ (dash), and end-to-end distance (dot) $r_L\equiv \sqrt{X_L^2+Y_L^2}∕L$ vs reduced force for a filament under stretching. The parameters are $c_0=0.1$, ${\varphi }_0=0.2$, and $L=15$. The unit of length and $F_x$ are the same as $1∕c_0$ and $c_0^2$, respectively.

Figure 4

Reduced effective persistence length, $l_p^{\mathrm{eff}}∕l_p$, vs reduced total arclength $L∕l_p$ for (a) present model [Eq. (44)], (b) WLC model [Eq. (45)], with a replacement of $l_p$ by $l_p^{\mathrm{eff}1}$ [Eq. (48)]. $c_0{l}_p=0.3$ in figure.

Figure 5

$l_p^{\mathrm{eff}}∕l_p$ vs $L∕l_p$ for a filament. The parameters are (a) $c_0{l}_p=0.1$, (b) $c_0{l}_p=0.2$, (c) $c_0{l}_p=0.3$, (d) $c_0{l}_p=0.5$, (e) $c_0{l}_p=0.7$, and (f) $c_0{l}_p=3$.

Figure 6

Filled circles represent the relationship between $L_m$ and $1∕c_0$. The solid straight line and the dashed straight line have slopes $\pi$ and $2.33$, respectively.