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Minimal Mechanisms of Microtubule Length Regulation in Living Cells

Anna C Nelson, Melissa M Rolls, Maria-Veronica Ciocanel, Scott A McKinley

TL;DR

It is found 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, so two minimally-complex length-limiting factors are proposed and investigated.

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 \textit{in vivo} experimental data on microtubule behavior in \textit{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.

Minimal Mechanisms of Microtubule Length Regulation in Living Cells

TL;DR

It is found 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, so two minimally-complex length-limiting factors are proposed and investigated.

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 \textit{in vivo} experimental data on microtubule behavior in \textit{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.
Paper Structure (14 sections, 1 theorem, 31 equations, 10 figures, 2 tables)

This paper contains 14 sections, 1 theorem, 31 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview of modeling decisions
  3. Overview of the paper
  4. Stochastic model of microtubule dynamics
  5. Stochastic model observables
  6. Reduced deterministic model of microtubule dynamics
  7. Statement of ODE model and justification
  8. Non-dimensionalization and basic mathematical properties
  9. Strategy for parameterization
  10. Results
  11. Comparison of the impact of rate-limited mechanisms
  12. The predicted response to tubulin manipulation
  13. Discussion
  14. Stochastic model implementation details

Key Result

Proposition 3.3

Suppose that $\eta^\pm \geq 0$, $\widetilde{T} > N \ell_0$, and all other nondimensional constants in Table tab:nondim are positive. Let $\mathcal{O} = (0,\tilde{T}/N) \times(0,1)\times(0,1)$. Then for any initial condition $(\ell,g^+,g^-)$ in $\mathcal{O}$ there exists a unique global solution to t

Figures (10)

  • Figure 1: Overview of MT dynamics in a stochastic model simulation with two growth regulation mechanisms: tubulin-dependent growth and length-dependent catastrophe. Each microtubule has a length $L$ and its plus and minus end are in either growth or shrinking states. In growth, the MTs utilize the available tubulin pool; while shrinking, they replenish tubulin into the unavailable pool.
  • Figure 2: Schematic outlining model parameters of the stochastic simulations, model observables that are used in parameterization of the stochastic model based on the deterministic model, and model outputs that provide predictions for experimental measurements.
  • Figure 3: Outputs of a single stochastic simulation for (a) no length-regulating mechanism and for (b) length-dependent catastrophe ($\gamma = 0.005$) and tubulin limited growth ($T_{\mathrm{tot}} = 1000\mu$m) for $N=20$ microtubules. The left panel shows positions of the plus end (blue) and minus end (yellow) of a sample MT throughout the entire simulation. Black dots indicate times when the MT reaches length zero. The right panel shows the mean and interquartile range of the MT lengths (orange) and each MT length (grey) throughout the simulation time. The horizonal line denotes the target MT length, $L_* = 35 \mu$m. Note the difference in the $y$-axis range for both panels in (a) and (b).
  • Figure 4: Cartoon of photonvertible tubulin experiment, inspired from Rolls2021. (Top) Tubulin in existing MTs in a window of interest is fluoresced at the photoconversion time. (Bottom) The intensity of the remaining fluoresced tubulin in the same window is tracked at subsequent times.
  • Figure 5: (Left) Various evaluations of the functions $H_0(\ell)$ and $H_1(\ell)$, see Equation \ref{['eq:defn-H']}. We use the baseline parameter set in both cases, with the modifications that $H_0$ is evaluated over a range of $\gamma$ (blue, red, and green), while the function $H_1$ is evaluated over a range of total tubulin (light gray to black as $T_{\mathrm{tot}}$ increases.) The steady state values (filled circles) occur at intersections of $H_0$ and $H_1$. (Right) The steady state $L_*$ is displayed as a function of $T_{\mathrm{tot}}$ for the fixed values of $\gamma$. The displayed steady state values (filled circles) correspond to those displayed on the left.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Remark 3.4
  • proof