Elasticity of a semiflexible filament with a discontinuous tension due to a cross-link or a molecular motor

We analyze the stretching elasticity of a wormlike chain with a tension discontinuity resulting from a Hookean spring connecting its backbone to a fixed point. The elasticity of isolated semiflexible filaments has been the subject in a significant body of literature, primarily because of its relevance to the mechanics of biological matter. In real systems, however, these filaments are usually part of supramolecular structures involving cross-linkers or molecular motors, which cause tension discontinuities. Our model is intended as a minimal structural element incorporating such a discontinuity. We obtain analytical results in the weakly bending limit of the filament, concerning its force-extension relation and the response of the two parts in which the filament is divided by the spring. For a small tension discontinuity, the linear response of the filament extension to this discontinuity strongly depends on the external tension. For large external tension $f$, the spring force contributes a subdominant correction $\sim 1/f^{3/2}$ to the well-known $\sim 1/\sqrt{f}$-dependence of the end-to-end extension.

Figure 1

Schematic presentation of a filament attached to a spring with clamped-free and hinged-hinged boundary conditions at the ends. One end of the spring is fixed to the substrate. The spring pulls one side of the filament and pushes the other side. Therefore, the two sides of the filament are at different tension.

Figure 2

Extension of the filament, $\langle R\rangle$, as a function of the external force $f_{\text{ext}}$, which can be compressive or extensile; full line, no motor attached; dashed line, $\beta L{f}_m=+50$; dotted line, $\beta L{f}_m=-50$. (Parameters: $L=1, l_p=10, L/L_m=2$; hinged-hinged filament.)

Figure 3

Extension of the two separate parts of the filament, $\langle R_1\rangle$ and $\langle R_2\rangle$, as a function of the external force, $f_{\text{ext}}$, for several values of the motor force; red lines, clamped-free filament; blue lines, hinged-hinged filament. (Full lines, no motor attached; dashed lines, $\beta L{f}_m=+50$; dotted lines, $\beta L{f}_m=-50$; parameters as described in the caption of Fig. 2.)

Figure 4

Extension of the filament, $\langle R\rangle$, as a function of the motor force $f_m$ for several values of the external force; red lines, clamped-free filament; blue lines, hinged-hinged filament. (Full lines, no motor attached; dashed lines, $\beta L{f}_{\text{ext}}=+50$; dotted lines, $\beta L{f}_{\text{ext}}=-10$; parameters as described in the caption of Fig. 2.)

Figure 5

Change in the differential stiffness relative to the case without a spring as a function of the external force; red lines, clamped-free filament; blue lines, hinged-hinged filamen. (Dashed lines, $\beta L{f}_m=+50$; dotted lines, $\beta L{f}_m=-50$; parameters as described in the caption of Fig. 2.)

Figure 6

Effective stiffness, $k\left(f_{\text{ext}}\right)$, of the filament with respect to the motor force as a function of external force $f_{\text{ext}}$. The range of the external force is chosen in a way that the filament is well approximated as weakly bending ($\frac{R}{L}>0.9$); parameters as described in the caption of Fig. 2.

Figure 7

Comparison of the complete solution for the end-to-end distance $\langle R\rangle /L$ and the one linearized around $f_m=0$; full lines, $\beta L{f}_{\text{ext}}=0$; dashed lines, $\beta L{f}_{\text{ext}}=50$; dashed-dotted lines, $\beta L{f}_{\text{ext}}=100$; dotted line, $\beta L{f}_{\text{ext}}=-10;$ parameters as described in the caption of Fig. 2.

Figure 8

Coefficient of the asymptotic expansion, $\alpha \left(f_{\text{ext}}\right)$, as function of external force $f_{\text{ext}}$; parameters as described in the caption of Fig. 2.