Polymerisation force of a rigid filament bundle: diffusive interaction leads to sublinear force-number scaling

Abstract

Polymerising filaments generate force against an obstacle, as in, e.g., microtubule-kinetochore interactions in the eukaryotic cell. Earlier studies of this problem have not included explicit three-dimensional monomer diffusion, and consequently, missed out on two important aspects: (i) the barrier, even when it is far from the polymers, affects free diffusion of monomers and reduces their adsorption at the tips, while (ii) parallel filaments could interact through the monomer density field ("diffusive coupling"), leading to negative interference between them. In our study, both these effects are included and their consequences investigated in detail. A mathematical treatment based on a set of continuum Fokker-Planck equations for combined filament-wall dynamics suggests that the barrier-induced monomer depletion reduces the growth velocity and also the stall force, while the total force produced by many filaments remains additive. However, Brownian dynamics simulations show that the linear force-number scaling holds only when the filaments are far apart; when they are arranged close together, forming a bundle, sublinear scaling of force with number appears, which could be attributed to diffusive interaction between the growing polymer tips.

Figure 1

Schematic diagram for a bundle of inflexible filaments pushing against a movable rigid barrier acted upon by a constant force f. The rigid barrier also undergoes thermal motion characterised by diffusion coefficient D_w.

Figure 2

Schematic diagram of the cubical box, containing a bundle of filaments growing by a diffusion-limited reaction used in Brownian dynamics simulations. One face of the cubical box facing the filament tip is movable (the barrier); it is acted on by a constant force f in the backward direction and also undergoes random motion characterised by diffusion coefficient D_w.

Figure 3

A schematic diagram of a multi-stranded filament with microtubule-like geometry.

Figure 4

The force-velocity curve for a single filament versus two filaments, obtained from Brownian dynamics simulations, for (a) a = 10 nm and (b) a = 2 nm. For (a), the off-rate of monomers is k_off = 2 s⁻¹ while for (b), k_off = 0.2 s⁻¹. Analytical results are shown for best fit value of λ; 0.275 nm⁻¹ in (a) and 0.4 nm⁻¹ in (b). In the insets, fits for (scaled) separation-dependent on-rate α (d) = k_on (d)/k_on (∞), using the same λ are shown. The other parameter values are given in Table 1. Here, near means zero base separation between polymers, wheras far refers to a base separation 10a. The error bars are typically smaller than the size of the symbols.

Figure 5

The gap distribution for a = 10 nm, for two forces, far from and near to stall, with (a) N = 1 and (b) N = 2. Fits of the analytical results, Eq. 28 for N = 1 and Eq. S43(Supplementary Information) for N = 2 are also shown for the best fit value λ = 0.275 nm⁻¹, with k_on (∞) = 7.5 s⁻¹. For both (a) and (b), k_off = 2 s⁻¹. The other parameter values are listed in Table 1.

Figure 6

The gap distribution for a = 2 nm, for two forces, far and near to stall. (a) is shown for N = 1 and (b) is shown for N = 2. A fit of the analytical results (Eq. 28 for N = 1 and Eq. S43(Supplementary Information) for N = 2) also shown for the best fitting value of λ = 0.4 nm⁻¹, with k_on (∞) = 0.73 s⁻¹. For both (a) and (b), k_off = 0.2 s⁻¹. The other parameter values are listed in Table 1.

Figure 7

Force-velocity relation for a single microtubule and two microtubules, both near (zero base separation) and far (base separation of 150 nm). The values of the parameters used in the simulation are listed in Table 1. The inset zooms the force range where the velocity vanishes. Note that while the single filament curve crosses the x-axis after touching zero at stall, the two-filament velocity remains close to zero after stall. In most cases, the error bars are smaller than the size of the symbols. For two filaments, ‘near’ means zero base separation, while ‘far’ refers to a base separation 150 nm.

Figure 8

The time evolution of the average positions of the wall and one of the protofilaments is shown for (a) one microtubule, (b) two microtubules (near) and (c) two microtubules (far). In the inset of (a), the average position of wall alone shown. The parameters common for all the three cases are listed in Table 1. For two filaments, ‘near’ means zero base separation, while ‘far’ refers to a base separation 150 nm.