Length Regulation of Active Biopolymers by Molecular Motors

For biopolymers like cytoskeletal actin filaments and microtubules, assembly and disassembly are inherently dissipative processes. Molecular motors can affect the rates of subunit removal at filament ends. We introduce a driven lattice-gas model to study the effects of motor-induced depolymerization on the length of active biopolymers and find that increasing motor activity sharpens unimodal steady-state length distributions. Furthermore, for sufficiently fast moving motors, the relative width of the length distribution is determined only by the attachment rate of motors. Our results show how established molecular processes can be used to robustly regulate the size of cytoskeletal structures like mitotic spindles.

Figure 1

Motor-induced shortening of an active biopolymer. (a) Illustration of the model: Empty sites are added at the plus end at rate $\alpha$ and occupied sites are removed from the lattice at the minus end at rate $\beta$. Particles bind to empty sites at rate $\omega$ and hop at rate $\gamma$ to empty sites towards the minus end. In the following, all rates are scaled by $\alpha$. (b) Examples of the steady-state dynamics with $\beta =10$, $\omega =0.01$, and $\gamma =0$ (left) and $\gamma =2$ (right). Empty sites are shown in blue, occupied sites in red.

Figure 2

Effects of the motor hopping rate $\gamma$ on the length distribution. (a) Steady-state length distributions for various values of $\gamma$, $\beta =10$, and $\omega =0.01$. (b) Average occupation probability of lattice sites. Symbols as in (a). Red dashed lines represent the solution of Eq. (1), while full black lines are given by Eq. (2). Gray lines indicate the slope $\omega /(1+\gamma )$ at $i=1$; see text. Values are shown for every tenth data point.

Figure 3

Relative width $Q$ of the length distribution as a function of the motor hopping rate $\gamma$ and the removal rate $\beta$ for $\omega =0.01$. White lines connect loci of equal average length.

Figure 4

Mean length and variance of length distribution as a function of the removal rate $\beta$ for different values of the hopping rate $\gamma$ (a),(c) and of the motor attachment rate $\omega$ (b),(d). Symbols represent simulation results, full lines are given by Eqs. (8, 9) for $⟨L⟩$ and $\sigma$, respectively. Dashed lines are obtained from solutions of the jump process defined by Eqs. (4, 5, 6). In (a),(c) $\omega =0.01$ and $\gamma =0$ ($\times$), 0.5 (▵), 1 (□), 2 (⋄), ${10}^5$ (○); in (b),(d) $\gamma =2$ and $\omega =0.005$ (▿), 0.01 (○), 0.05 (⋄), 0.1 (□).