Diffusible Cross-linkers Cause Superexponential Friction Forces

The friction between cytoskeletal filaments is of central importance for the formation of cellular structures such as the mitotic spindle and the cytokinetic ring. This friction is caused by passive cross-linkers, yet the underlying mechanism and the dependence on cross-linker density are poorly understood. Here, we use theory and computer simulations to study the friction between two filaments that are cross-linked by passive proteins, which can hop between discrete binding sites while physically excluding each other. The simulations reveal that filaments move via rare discrete jumps, which are associated with free-energy barrier crossings. We identify the reaction coordinate that governs the relative microtubule movement and derive an exact analytical expression for the free-energy barrier and the friction coefficient. Our analysis not only elucidates the molecular mechanism underlying cross-linker-induced filament friction, but also predicts that the friction coefficient scales superexponentially with the density of cross-linkers.

Figure 1

Model of the microtubule overlap. The filaments are represented as one-dimensional lattices with spacing $\delta$ and are connected by cross-linking proteins, described as springs. The bottom filament is fixed, while the top filament can move in the longitudinal direction due to Brownian motion with diffusion constant $D_m$ or the pulling of stretched cross-linkers. Cross-linkers can make a diffusive step, causing them to switch from right-pulling linkers labeled $R$ to left-pulling linkers denoted $L$, or vice versa. Two of such possible transitions, together with their rates $h$, are indicated by orange arrows. The fixed microtubule is infinitely long and the mobile one has $\ell$ lattice sites. There are $N$ cross-linkers connecting the two filaments, which stay bound indefinitely.

Figure 2

A typical time trace of the mobile microtubule position shows that it moves with sudden jumps. Horizontal lines denote positions where the microtubules are in register. The jumps occur at a fixed rate in both directions, which can be estimated from the mean waiting time in simulations, $r=1/2\overline{\tau }$. (inset, top-left) A typical transition, where the microtubules begin and end in register. Cross-linkers are stretched in intermediate states, which energetically suppresses transitions. (inset, bottom-right) The observed rate of microtubule jumps appears to decrease exponentially with the number of cross-linkers $N$. Dots show simulation estimates of the rate, whereas the line shows a least square exponential fit. In the examples, $\ell =40$, and $N=12$ for the time trace.

Figure 3

Helmholtz free-energy as a function of the position $x$ and the number of right-pulling cross-linkers $N_R$. Free-energy minima exist at the bottom-left and top-right corners, where the two filaments are in register and the cross-linkers are fully relaxed. These minima are separated by a free-energy barrier, where the cross-linkers are stretched (see Fig. 2). A transition over the barrier is observed as a jump of the microtubule, and two transition paths are shown for illustration. $N_R$ is a discrete parameter, and a weak sinusoidal $y$ offset was added to the paths to visualize their course. In this example, we use $N=12$ and $\ell =40$. (inset) The free-energy profile as a function of the reaction coordinate $\alpha$. The height of the barrier is the difference between the Helmholtz free-energies at $\alpha =0$ and $\alpha =1/2$. Discontinuities occur due to the discrete nature of $N_R$ in the definition of $\alpha$.

Figure 4

The free-energy barrier height $\mathrm{\Delta }{\mathcal{F}}^{‡}$ increases exponentially with the number of cross-linkers $N$. The exact values [given by Eqs. (2), (4)], plotted as points, are approximated well by the continuous exponential curves as given by Eq. (7). Notice that the number of cross-linkers cannot exceed the number of sites on the microtubule, $N\le \ell$. Furthermore, we plot the approximated barrier height for an infinitely long mobile microtubule, which demonstrates that the barrier height increases linearly with $N$ when the cross-linker density $N/\ell$ is low. Using Eqs. (1) and (6), we predict that the friction coefficient $\zeta$ increases exponentially with $N$ at low cross-linker densities and superexponentially with $N$ at high densities.