Minimal Mechanisms of Microtubule Length Regulation in Living Cells

Abstract

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Appendix A: Stochastic Model Implementation Details

To implement the stochastic model of MT polymerization and depolymerization in Sect. 2, we implement an Euler-based stochastic scheme that is time- rather than event-driven, and thus we update events at fixed time steps $\Delta t$. Similar to the leap condition in Gillespie’s tau-leaping scheme (Gillespie 2001; Gillespie et al. 2007), we choose a time interval $\Delta t$ that is small enough so that reaction propensities are relatively fixed in time interval $[t_i, t_i+\Delta t = t_{i+1}]$, and large enough so that many reactions are still happening in each step. Since we are interested in the emergent behavior of dynamic MT instability, our approach differs from a strict implementation of tau-leaping in that we avoid resolving tubulin unit scale reactions and instead focus on MT end state switching reactions. Tau-leaping methods face significant challenges when one or more of the state variables are near boundary values. In our case, the pool of available tubulin is often near zero. This also motivates our choice of time scale, and our decision to treat tubulin resources as a continuum, rather that in discrete units. To determine how small time step $\Delta t$ should be, we performed an experiment shown in Fig. 9 to see the effect of time step size on average length. We see MT behavior is qualitatively the same for time step sizes less than or equal to 1 s, so we utilize $\Delta t = 1 \text { s}$.

Fig. 9

Average MT length over time for different stochastic time steps, $\Delta t$, for $\gamma = 0$, $L_* = 35\,\upmu$m, and $N=20$ MTs for one realization

To determine how long each end spends in a certain state during time step $\Delta t$, we first update the length-dependent catastrophe rate based on the length of the microtubules at the previous time step, $t_i$, denoted as $L_i = L(t_i)$. Therefore, the catastrophe rate at $t_{i+1} = t_i + \Delta t$ will take the following form

$$\begin{aligned} \lambda ^{\pm }_{g\rightarrow s}(L_i) = \max \left( \lambda _{\min }, \lambda _{\textrm{g} \rightarrow \textrm{s}}^\pm + \gamma L_0 \phi \left( \frac{L_i}{L_0}\right) \right) , \end{aligned}$$

(A1)

where $\phi (x) = x - 1$. Again, for small L and large $\gamma$, it is possible that $\lambda _{min} > \lambda _{\textrm{g} \rightarrow \textrm{s}}^\pm + \gamma L_0(\frac{L(t)}{L_0} - 1)$, so then the catastrophe rate will be $\lambda _{min}$ for short MTs. The updated catastrophe rates are then used to draw times spent in growth phase.

Fig. 10

Sketch of admissible growth and shrinking times ($t_g$ and $t_s$, respectively) within a single time step $\Delta t$ of the stochastic simulation. Green color denotes growth and red denotes shrinking. In a single $\Delta t$, we only permit one shrinking phase to occur (Color figure online)

At each time step from $t_i$ to $t_i + \Delta t$, each microtubule end is in either a shrinking or growth state. For each end of the microtubule, the time spent in growth and shrinking, denoted as $t_g$ and $t_s$, respectively, are given by random variables drawn from the appropriate exponential distribution, where

$$\begin{aligned} t_g^{i+1} \sim \text {Exp}(\lambda _{g\rightarrow s}(L_i)), \qquad t_s^{i+1} \sim \text {Exp}(\lambda _{s\rightarrow g}). \end{aligned}$$

(A2)

We then determine if the microtubule exits the time step in either growth or shrinking based on the idea that only one shrinking phase can occur in a time step $\Delta t$ (see Fig. 10). For example, if a microtubule end entered in growth and $t_g > \Delta t$, then the microtubule would also exit the time step in growth (see Fig. 10d). Similarily, if a microtubule end entered in growth and $t_g + t_s < \Delta t$, then the microtubule also exits in growth, shown in Fig. 10e.

Once the time spent in each state has been determined, we then perform the shrinking events, which involve updating the microtubule lengths and adding the lost tubulin to unavailable tubulin bin. If the shrinking events resulted in a completely-catastrophed microtubule (that is, a microtubule with length less than or equal to zero), we reseed the microtubule at its previous position with length zero and both ends in a growth state. This reseeding procedure ensures that the number of microtubules remains the same throughout the simulation.

After performing shrinking events, we update the available tubulin pool from the unavailable tubulin pool at rate $\tau _{tub}$ and then implement growth events, which depend on the available tubulin pool. To allow for a smooth transition in growth speed as the available tubulin pool becomes very small, we model the dependence of the microtubule growth on the available pool using Michaelis–Menten kinetics. For example, suppose the desired amount of tubulin for growth for a single MT is $v_{g} t_g$, where $v_{g}$ is the growth velocity and $t_g$ is the time spent in growth for that MT in $\Delta t$. Since growth is dependent on tubulin availability, we update the desired growth length for the j-th MT to be

$$\begin{aligned} {\tilde{x}}_j = \frac{F}{F_{1/2} + F}v_g t_g, \end{aligned}$$

(A3)

where F is the available tubulin amount and $F_{1/2}$ is the same Michaelis–Menten constant from Eq. (18).

We implement the growth events for all MTs in $\Delta t$ and decrease F based on the total desired polymerization length, $\sum _{i=1}^N {\tilde{x}}_i$. If tubulin resources are limited, it is possible that the available tubulin amount is less than the total desired polymerization length, so that $F < \sum _{i=1}^N {\tilde{x}}_i$. To account for this, we divide the available tubulin amount proportionally to each MT based on the desired polymerization length for that time step. Therefore, the final length grown for the j-th MT, $x_j$, is

$$\begin{aligned} x_j = \frac{{\tilde{x}}_j}{\sum _{i=1}^N {\tilde{x}}_i} F . \end{aligned}$$

(A4)

We then deplete the tubulin used to grow MTs from the available tubulin pool, F, so that

$$\begin{aligned} F(t_{i+1}) = F(t_i) - \sum _{i = 1}^N x_i . \end{aligned}$$

(A5)