Topology promotes length exploration in microtubule dynamic instability

TL;DR

The paper addresses how microtubules achieve broad length exploration during dynamic instability by proposing a topological two-component cap model with edge currents that support growth, a stutter phase, and catastrophe. Using a Kagome-lattice state space and two tunable parameters, the model reproduces the observed peaked catastrophe-length distribution and its concentration dependence, and provides an analytical condition for when the peak arises via a cap-length–dependent dissociation rate. An explicit 1D single-component cap fails to capture the key features, establishing the necessity of the two-component topological framework. The work suggests a general topology-driven mechanism for cellular search processes and makes testable predictions about cap length, catastrophe length, and the impact of tubulin mutants on catastrophe statistics.

Abstract

Microtubules stochastically switch between growth and shrinkage during catastrophe events across a very large range of filament lengths, with the length distribution at catastrophe peaking at a finite filament length, which can aid the search for chromosomes during mitosis. To model these distinct features, we introduce a topological model of a two-component microtubule cap, where protected edge states give rise to different phases of microtubule dynamics - growth, shrinkage, and a recently observed "stutter" phase. With only two free parameters, our model quantitatively reproduces the peaked catastrophe length distribution and its dependence on tubulin concentration from experimental data. The model further provides an analytical condition for when the catastrophe length distribution is peaked. Our work shows how microtubules may utilize topological edge states to promote length exploration, elucidating a novel mechanism for search and target reaching in cellular biology.

Figures (4)

• Figure 1: Topological model for microtubule dynamics. (a) Microtubules grow in the presence of a stabilizing cap, made from GTP-tubulin (purple) and GDP-P$_\text{i}$-tubulin dimers (blue). Cap loss leads to catastrophe events, followed by rapid microtubule shrinkage. (b) The cap is modified by cyclic external transitions (solid arrows): GTP-tubulin addition (top), GTP cleavage (left), and P$_\text{i}$ release (right). Brackets $(x,y)$ record the number of GTP-tubulin and GDP-P$_\text{i}$-tubulin dimers, which change with transitions shown by black arrows. Green arrows denote reactions that change the number of GDP-tubulin (recorded separately). (c) For each $(x,y)$, there are three internal states (A, B, C) that transition through dashed arrows; each internal state primes the cap for a different external reaction. (d) Repeating the reaction cycles along the $x$ and $y$ axes forms a Kagome lattice; the repeated motif is highlighted in blue. The GTP-tubulin dissociation rate $\gamma_\text{ex}^\text{CB}$ can increase with cap length (horizontal axis). (e) Schematic of three stochastic trajectories on a background where darker circles represent sites that are visited more frequently. Catastrophes proceed via cap growth along the bottom edge, followed by two-step hydrolysis through the bulk and left edge. Hydrolysis starts at the encircled points denoted by $(x_\text{hydr},0)$.
• Figure 2: The topological model reproduces key features of catastrophe observed in experiments. (a) Microtubule length as a function of time, from model simulations. Catastrophes occur over a large range of length scales. Right: microtubule growth stutters briefly before catastrophe (yellow shaded region). (b) Distribution of catastrophe length and microtubule lifetime, fitted to experimental data from Ref gardner2011depolymerizing at $12\mu$M tubulin. Our model generates peaked distributions for both quantities. (c) Average catastrophe length and lifetime for different tubulin concentrations. Consistent with experiments from gardner2011depolymerizing, both quantities increase with concentration. Error bars represent one standard error.
• Figure 3: Bottom edge dynamics give an analytical condition for a peaked length distribution. (a) The cap length at the onset of hydrolysis, $x_\text{hydr}$, increases monotonically with catastrophe length. Error bars represent one standard error. (b) $P(x)$, the distribution for $x_\text{hydr}$, is peaked for our model parameters; here $r=220, r_\text{P}=200$. An analytical calculation (blue curve) agrees with simulation results. (c) On the bottom edge, a stochastic trajectory can either enter the bulk at $x$ (blue) or continue along the edge past $x$ (orange). (d) The bulk entry probability $p(x)$ takes contributions from a family of trajectories, which undergo different rounds of association/dissociation. The corresponding probabilities are shown on the right of each trajectory. (e) Our model provides an analytical condition for a peaked catastrophe length distribution, which arises when $P(x)$ has a maximum at $x^*>1$. This occurs when the dissociation rate $\gamma_\text{ex}^\text{CB}\propto l^n$ has exponent $n\geq 0.1$. Note that for $n<0.1$ (shaded area), $P(x)$ does not have a peak ($x^*=1$).
• Figure 4: A reduced 1D model with a single-component cap fails to capture catastrophe behavior. (a) State space for the reduced 1D model, where GDP-P$_\text{i}$-tubulin is coarse-grained out. $x$ denotes the cap length. Curved arrows denote transitions that remain after coarse-graining the 2D model. (b) Microtubule length as a function of time, from the 1D model. Microtubules do not regrow immediately after catastrophe. (c) Distributions of catastrophe length and lifetime give poor agreement with the same experimental data compared to the 2D model.