Cooperative motor action to regulate microtubule length dynamics

Motivated by the recent experimental observations on motor induced cooperative mechanism controlling the length dynamics of microtubules (MTs), we examine how plus-end-targeted proteins of the kinesin family regulate MT polymerization and depolymerization routines. Here, we study a stochastic mathematical model capturing the unusual form of collective motor interaction on MT dynamics originating due to the molecular traffic near the MT tip. We provide an extensive analysis of the joint effect of motor impelled MT polymerization and complete depolymerization. The effect of the cooperative action is included by modifying the intrinsic depolymerization rate. We analyze the model within the framework of continuum mean-field theory and the resultant steady-state analytic solution is expressed in terms of Lambert $W$ functions. Four distinct steady-state phases including a shock phase have been reported. The significant features of the shock including its position and height have been analyzed. Theoretical outcomes are supported by extensive Monte Carlo simulations. To explore the system alterations between the regime of growth and shrinkage phase, we consider kymographs of the microtubule along with the length distributions. Finally, we investigated the dependence of MT length kinetics both on modifying factor of depolymerization rate and motor concentration. The overall extensive study reveals that the flux of molecular traffic at the microtubule plus end initiates a cooperative mechanism, resulting in significant change in MT growth and shrinkage regime as also observed experimentally.

Figure 1

Schematic representation of the model. Allowed transitions are shown by arrows. Crossed arrows indicate impermissible transitions. (a) MT growth dynamics, (b) knockdown mechanism resulting in accelerated detachment of plus-end bound motor by incoming motors, (c) paused molecule before knockdown.

Figure 2

Knockdown mechanism explaining the accelerated depolymerization dynamics at the occupied MT tip taking into account the configuration of (a) adjacent neighboring site, (b) two nearest neighboring sites.

Figure 3

The real branches $W_0$ and $W_{-1}$ of Lambert $W$ function.

Figure 4

Phase diagrams for varied $c$ as (a) 20, (b) 100, (c) 500, (d) 2000 with $N=1000$, ${\omega }_D=0.2$, ${\mathrm{\Omega }}_a=5.3\times {10}^{-4}$, and ${\mathrm{\Omega }}_d=7.6\times {10}^{-4}$. Solid lines denote phase transition for $\delta =1.1$, while dotted lines are those obtained for $\delta =1.5$. Color code running from orange to red represents the transition from shrinkage to growth phase as calculated from Eq. (27) for $\delta =1.1$.

Figure 5

Phase transitions from (a) low density (LD) to high density (HD) phase for $\gamma =0.18$ with varied $\alpha$ as 0.15, 0.33, and 0.55, (b) S to LH phase for $\alpha =0.33$ with varied $\gamma =0.18$, 0.26, and 0.40. These values are for curves from bottom to top in (a) with $N=1000$, ${\omega }_D=0.20$, $c=100$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d$ = $7.6\times {10}^{-4}$, and $\delta =1.1$. In (b) other parameters are same as for (a) but dashed red and pink color curves are for $\gamma =0.18$ while solid yellow and blue color are for $\gamma =0.26$ and 0.40, respectively. Shock profile in both (a) and (b) shown in black color is obtained by solving Eq. (7) numerically and all other profiles are obtained using a suitable branch of Lambert $W$ function as discussed in Sec. 3.

Figure 6

Density profiles of (a) LD ($\rho <0.5$), (b) HD ($\rho >0.5$), (c) LH, (d) S phases with (a) $\alpha =0.22,\gamma =0.22$, (b) $\alpha =0.8,\gamma =0.23$, (c) $\alpha =0.35,\gamma =0.35$, (d) $\alpha =0.32,\gamma =0.21$; $N=1000$, ${\omega }_D=0.20$, $c=100$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d=7.6\times {10}^{-4}$, and $\delta =1.1$. Solid and dotted lines denote profiles obtained from Lambert $W$ function while markers (squares) represent the Monte Carlo simulation results. HD and LD profiles are captured by Lambert $W$ branch given in Eqs. (23) and (24) along suitable boundary conditions as discussed in Sec. 3, respectively. In (d), solid curve is obtained by solving Eq. (7) numerically.

Figure 7

(a) Shock position, (b) shock height for $\gamma =0.03$ with $N=1000$, ${\omega }_D=0.20$, $c=100$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d=7.6\times {10}^{-4}$, and $\delta =1.1$. The shaded portion shows magnified region of shock phase.

Figure 8

Effect of motor concentration on kymographs and corresponding length distribution exhibiting the MT dynamics with time for (a) $c=20$, (b) $c=50$, (c) $c=180$ with $N=100$, $\alpha =0.15$, $\gamma =0.16$, ${\omega }_D=0.20$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d$ = $7.6\times {10}^{-4}$, and $\delta =1.1$ obtained from Monte Carlo simulations. Empty sites are shown in cyan, occupied sites in red. $P\left(N\right)$ denotes probability density function for length $N$. Note that kymographs are plotted by left aligning the lattice.

Figure 9

Effect of the motor concentration on the steady-state length distribution for various values of (a) $c=1000$, (b) $c=3000$, (c) $c=5000$ with $\alpha =0.8$, $\gamma =0.15$, ${\omega }_D=0.20$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d$ = $7.6\times {10}^{-4}$, and $\delta =1.1$ obtained from Monte Carlo simulations.

Figure 10

Drift velocity of the MT tip $\frac{\partial N}{\partial t}$ as a function of the polymerization and depolymerization rates for (a) $c=100$, (b) $c=5000$ with $N=1000$, $\alpha =0.8$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d$ = $7.6\times {10}^{-4}$, and $\delta =1.1$ as obtained from theoretical analysis. Color code ranging from orange to red indicates the magnitude of the drift velocity. Dashed white line indicates where the zero MT velocity as retrieved from the analytical calculations, using Eq. (27).

Figure 11

Effect of modifying factor $\delta$ on the system dynamics in (a) $(\gamma ,{\omega }_D)$ parameter space for $\alpha =0.8$, (b) $(\alpha ,c)$ parameter space for $\gamma =0.22$ as obtained from theoretical analysis with $N=1000$, ${\omega }_D=0.20$, ${\mathrm{\Omega }}_a$ = $5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d$ = $7.6\times {10}^{-4}$, and $\delta =1.1$. (c) Same parameters as in (b). Color code ranging from orange to red indicates the magnitude of the drift velocity and current in (a), (b), and (c), respectively, for $\delta =1.1$.

Figure 12

Phase diagrams for $c=100$, $N=1000$, ${\omega }_D=0.2$, ${\mathrm{\Omega }}_a=5.3\times {10}^{-4}$, ${\mathrm{\Omega }}_d=7.6\times {10}^{-4}$, ${\gamma }^{*}=0.2$, and ${\omega }_{D^{*}}=0.1$. Solid lines denote phase transition for $\delta =1.1$, while dotted lines are those obtained for $\delta =1.5$. Color code running from orange to red represents the transition from shrinkage to growth phase as calculated from Eq. (A3) for $\delta =1.1$.