Statics and dynamics of the wormlike bundle model

Bundles of filamentous polymers are primary structural components of a broad range of cytoskeletal structures, and their mechanical properties play key roles in cellular functions ranging from locomotion to mechanotransduction and fertilization. We give a detailed derivation of a wormlike bundle model as a generic description for the statics and dynamics of polymer bundles consisting of semiflexible polymers interconnected by crosslinking agents. The elastic degrees of freedom include bending as well as twist deformations of the filaments and shear deformation of the crosslinks. We show that a competition between the elastic properties of the filaments and those of the crosslinks leads to renormalized effective bend and twist rigidities that become mode-number dependent. The strength and character of this dependence is found to vary with bundle architecture, such as the arrangement of filaments in the cross section and pretwist. We discuss two paradigmatic cases of bundle architecture, a uniform arrangement of filaments as found in $F$-actin bundles and a shell-like architecture as characteristic for microtubules. Each architecture is found to have its own universal ratio of maximal to minimal bending rigidity, independent of the specific type of crosslink-induced filament coupling; our predictions are in reasonable agreement with available experimental data for microtubules. Moreover, we analyze the predictions of the wormlike bundle model for experimental observables such as the tangent-tangent correlation function and dynamic response and correlation functions. Finally, we analyze the effect of pretwist (helicity) on the mechanical properties of bundles. We predict that microtubules with different number of protofilaments should have distinct variations in their effective bending rigidity.

Figure 1

(Color online) Wormlike bundle model. We consider bundles that consist of regular arrangements of filaments. These are assumed to be locked in place by crosslinking proteins. When the bundle bends and twists in space, the filaments start to slide along each other. This effect leads to shear deformation in the crosslinks.

Figure 2

(Color online) Illustration of the bundle geometries considered. The position of the $k\text{th}$ filament in the cross-sectional plane [$\left(y,z\right)$ plane] is given by ${\mathbf{R}}_k$. $F$ -actin bundle architecture (left): filaments are arranged on a square lattice. Microtubule architecture (right): filaments are arranged on the surface of a cylinder.

Figure 3

(Color online) Illustration for the definition of the generalized curvatures $\mathbf{\Omega }$. Under planar bending one can take ${\partial }_s{\mathbf{d}}_1=0$ and the third Frenet-Serret equations reduces to ${\partial }_s{\mathbf{d}}_3={\Omega }_1{\mathbf{d}}_2$. Thus, ${\Omega }_1\delta s$ is just the change in the tangent ${\mathbf{d}}_3$ when going from the arclength position $s$ to $s+\delta s$.

Figure 4

(Color online) Illustration of crosslink shear deformation for the case of a two-filament bundle. Bundle deflection through the angle $\theta$ leads to the arclength mismatch, $\Delta =b\theta$. The filaments have to stretch the relative amount $u_1-u_2=b\theta$, in order to keep the crosslink (dashed line) undeformed with zero shear energy.

Figure 5

(Color online) Illustration of crosslink shear deformation for a twisted microtubule. Indicated is a filament pair that winds around the microtubule cylinder (radius $R$) taking an angle $R{\Omega }_3$ with the cylinder axis $\mathbf{t}$. The resulting arclength mismatch is given by $\Delta =bR{\Omega }_3$.

Figure 6

(Color online) Effective bending rigidity, Eq. (10), as a function of mode-number, $q\lambda$, and for the set of bundle sizes $N=4,9,16,25$ (from bottom to top). In the fully coupled and the decoupled regimes (corresponding to small and large $q$) the bending rigidity is constant. At intermediate values of $q$ the bending rigidity scales as ${\kappa }_B\left(n\right)\sim k_{\times }{q}_n^{-2}$ (shear-dominated regime).

Figure 7

(Color online) Dependence of axial strain $u_k$ on $k$, the distance from the bundle central line. Decreasing the dimensionless shear-stiffness $\alpha =k_{\times }{L}^2/k_s{\delta }^2$ (from top to bottom) the strain is reduced but not in a linear fashion. The outer layers of the bundle remain stretched stronger than the inner ones.

Figure 8

(Color online) Continuum limit of effective bending rigidities as a function of $x={\left(q_{n}D\right)}^{2}E/12G$. Comparison of exact solution (solid line), Eq. (13), with approximation (dotted line), Eq. (12). Inset: relative error between both expressions. The approximation overestimates the bending stiffness by maximal $\sim 6\mathrm{\%}$.

Figure 9

(Color online) Comparison of the different persistence lengths as function of arclength $s$. The $l_p$ as defined via Eq. (16) increases with increasing distance and saturates at the value $l_p\left(s\to \infty \right)=n^2$ (here, $n=16$). On the other hand, $l_p^{\star }$ as taken from Eq. (17) decays to zero.

Figure 10

(Color online) Comparison of transverse response function ${\chi }_{\perp }\left(\omega \right)$ (top figure) with longitudinal response function ${\chi }_{\parallel }\left(\omega \right)$ (bottom figure) for a bundle of $N=20$ filaments (in units of $N{\kappa }_b/L^3$ and assuming $L{k}_{B}T/N{\kappa }_b=1$). Frequencies are plotted in units of $\zeta L^4/N{\kappa }_b$. The different curves correspond to different values of $\lambda /L$. The two asymptotic scaling regimes with ${\omega }^{-3/4}$ correspond to the decoupled (top curve) and fully coupled regime (bottom curve), respectively. The intermediate ${\omega }^{-1/2}$ is sharper in ${\chi }_{\perp }$ than in ${\chi }_{\parallel }$ even though the latter function shows overall a stronger variation with frequency.

Figure 11

(Color online) Cross-sectional warping $u_k$ ($k=-M+1,\dots ,M$, corresponding to one row of the rectangular array) for different nondimensional crosslink shear-stiffness $\alpha =k_{\times }{L}^2/k_s{\delta }^2$ (decreasing $\alpha$ from top to bottom). The classic solution for a beam (dashed line) is approached as $k_{\times }\to \infty$. In the opposite limit, $k_{\times }\to 0$, there is no warping and $u\equiv 0$, as filaments remain uncoupled in this limit.

Figure 12

(Color online) Effective bending stiffness (in units of $N{\kappa }_b$) as a function of shear-stiffness $k_{\times }$ (in units of ${\kappa }_b\delta /b^2{L}^2$) of a bundle of four filaments under a tip force $F$. The bending stiffness is calculated from the determined end-deflection $y\left(L\right)$ by ${\kappa }_{\text{eff}}=F{L}^3/12y\left(L\right)$. The full curves are for finite pretwist ${\omega }_{0}L/\pi =\left(0.25,0.5,1,2,3\right)$ (increasing ${\omega }_0$ from top to bottom), while the dashed curves represent the limiting cases of zero and infinite pretwist, respectively.

Figure 13

(Color online) Illustration of the two types of solution obtained for the stretching $u_{ij}$ of filament $\left(i,j\right)$. The symbolds (plus/minus) indicate the sign of u in the respective region. Left: square cross-section of $40\star 40$ filaments. Right: $30\star 40$ filaments.