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Lecture notes. Written from the material of MIT 8.592J, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

11. Dynamic Instability of Microtubules

What this covers

This chapter looks at one physical function carried out by the cell's cytoskeleton: the oscillatory growth-and-shrinkage behaviour of microtubules known as dynamic instability, and two models built for it in the lecture — a coarse two-state stochastic model of the whole filament tip, and a finer model of the GTP-bound "cap" thought to control the switch between the two states. It assumes familiarity with master equations for continuous-time Markov processes and with solving linear PDEs by Fourier transform; the lecture points to a problem-set assignment for the latter technique, which was not supplied with this lecture.

Cell function and the cytoskeleton

Proteins carry out a wide range of cell functions: enzymatic activity (breaking down sugars), transport (haemoglobin carrying oxygen), sensing and signalling (transcription factors), and biophysical functions tied to force and motion — moving organelles, holding cell shape, producing motion. This chapter is about the last kind, worked out through the example of microtubules and the motors that ride them.

The interior of a cell is not a fluid bag: it holds specific shapes and carries out specific mechanical tasks through the active dynamics of protein fibres, the cytoskeleton. These fibres assemble from protein subunits much as polymers do, but are considerably stiffer because the subunits themselves are larger.

  • Microfilaments, the thinnest, are built from the monomer actin and are roughly 8 nm in diameter. They have a built-in directionality — a plus end and a minus end — and are typically held in a dynamic equilibrium in which monomers dissociate from the minus end while new monomers assemble at the plus end, a process called treadmilling.
  • Microtubules, the subject of the rest of this chapter, are larger: 25 nm in diameter, built from dimers of \(\alpha\)- and \(\beta\)-tubulin. A microtubule is a hollow cylinder made of 13 protofilaments stacked side by side, directional like actin filaments, and of variable length, growing up to about 25,000 nm.
  • Intermediate filaments, 8–25 nm across, occur in a variety of cells — for example strengthening the long axons of neurons.

Dynamic instability of microtubules

Discovered in the 1980s (Mitchison and Kirschner, 1984), dynamic instability is the phenomenon in which a microtubule grows at a slow, steady rate and is then interrupted by a "catastrophe": a sudden switch to rapid shortening, which persists until a "rescue" event restores the original slow growth. The cycle repeats stochastically, on a timescale of minutes.

The behaviour depends on a constant influx of energy, as does the cytoskeleton generally, and here that energy comes from hydrolysis of GTP (guanosine triphosphate) to GDP (guanosine diphosphate). A tubulin dimer must be bound to GTP before it can be added to the growing end of a microtubule, so the growing (plus) end is capped by GTP-bound tubulin. Further back from the tip, individual subunits can spontaneously hydrolyse their bound GTP to GDP — and it is the fate of that GTP-bound region, the cap, that turns out to control the transition between growing and shrinking.

A two-state model of the filament tip

Dogterom and Leibler (1993) modelled the growth and switching of a microtubule at the coarsest level that reproduces catastrophe and rescue: treat the tip as being in one of two states, growing or shrinking, each with its own constant velocity, and let it switch between the states at fixed rates.

growing (+) tip speed v₊ shrinking (–) tip speed v₋ f₊₋ (catastrophe) f₋₊ (rescue)
The tip is in one of two states, growing or shrinking, each with its own constant velocity; it switches between them at rates f₊₋ and f₋₊.

Let \(P_+(z,t)\) and \(P_-(z,t)\) be the probability that a growing, respectively shrinking, microtubule has length \(z\) at time \(t\). Dropping diffusive spreading within a state (small compared to the drift), the evolution is

\[\frac{\partial}{\partial t}P_+(z,t) = -f_{+-}P_+ + f_{-+}P_- - \frac{\partial}{\partial z}\big(v_+P_+\big)\]
\[\frac{\partial}{\partial t}P_-(z,t) = +f_{+-}P_+ - f_{-+}P_- + \frac{\partial}{\partial z}\big(v_-P_-\big)\]

The first two terms on each right-hand side are a birth–death process on the two-state label — probability leaves the growing population at rate \(f_{+-}\) and arrives from the shrinking population at rate \(f_{-+}\), and conversely — and the last term is transport of probability along \(z\) at that state's own drift velocity.

Because the rates and velocities do not depend on \(z\) or \(t\), the system is linear and translation-invariant, so it is solved by Fourier modes \(P_\pm(z,t)\propto e^{i(kz-\omega t)}\): substituting \(\partial_t\to-i\omega\), \(\partial_z\to ik\) turns the pair of PDEs into a \(2\times2\) linear algebraic system,

\[\begin{pmatrix} i\omega - f_{+-} - ikv_+ & f_{-+} \\ f_{+-} & i\omega - f_{-+} + ikv_- \end{pmatrix} \begin{pmatrix} P_+(k,\omega) \\ P_-(k,\omega) \end{pmatrix} = 0.\]

A non-trivial mode exists only where the determinant vanishes; solving that condition for \(\omega(k)\) and expanding for small \(k\) gives

\[\omega(k) = vk - Dk^2 + \cdots\]

The linear term is an effective net drift velocity \(v\) of the tip once the fast switching between the two ballistic states is averaged over; the quadratic term is an effective diffusion coefficient \(D\) that appears even though neither state is individually diffusive — it is generated purely by the randomness of when the switches happen, the same mechanism that produces diffusive spreading from a dichotomous (telegraph) process alternating between two constant velocities.

The stabilizing cap: what sets the switching rate

The model above takes \(f_{+-}\) and \(f_{-+}\) as given constants. The question the lecture poses next is where they come from: how does the transition between growing and shrinking actually happen, and how should it depend on the concentration of free tubulin (and GTP) in solution?

Flyvbjerg, Holy and Leibler (1994) answered this by modelling the GTP cap directly. A microtubule grows by adding tubulin-GTP dimers at its tip; behind the tip, individual subunits spontaneously hydrolyse their bound GTP to GDP. Track two positions along the filament: \(T\), the growing tip, and \(E\), the leading edge of hydrolysis — the point up to which some subunit has already converted to GDP. The unbroken stretch of GTP-tubulin between \(E\) and \(T\) is the stabilizing cap, of length \(x\).

GDP-tubulin lattice GTP cap (stable) cap length x E T v_T: growth v_E: hydrolysis front
The stable cap is the unbroken stretch of GTP-tubulin between the hydrolysis front E and the growing tip T. T advances by addition of tubulin-GTP; E advances because hydrolysis converts GTP to GDP somewhere within the cap. A catastrophe is the event that E catches T.

The rules of the model:

  • \(T\) advances at a growth rate that depends on the concentration of tubulin-GTP in solution: more subunits available, faster growth.
  • \(E\) advances because hydrolysis, occurring spontaneously anywhere within the cap, converts some GTP-bound subunit to GDP; the position of \(E\) is set by the frontmost such conversion.
  • A catastrophe is the event that \(E\) catches \(T\): the cap length reaches zero, no GTP-bound subunits remain to stabilize the tip, and the microtubule switches into rapid shrinkage.

This picture is supported by the observation that catastrophes are rarer when growth is faster, at high tubulin concentration — a fast-advancing \(T\) stays ahead of a hydrolysis front \(E\) whose speed does not depend on tubulin concentration, keeping the cap long. The lecture also poses, without resolving in the material at hand, a puzzle for the model to answer: if a microtubule that had been growing is suddenly transferred to a tubulin-poor solution, the time until catastrophe is observed to depend little on the concentration it had been growing in beforehand.

Writing \(p(x,t)\) for the probability that the cap has length \(x\) at time \(t\), Flyvbjerg, Holy and Leibler propose the master equation

\[\partial_t p(x,t) = -v\,\partial_x p + D\,\partial_x^2 p - r\,x\,p + r\int_x^\infty dy\, p(y,t),\]

with \(v \equiv v_T - v_E\) and \(D \equiv D_T + D_E\). Reading it term by term:

  • \(-v\,\partial_x p\): drift of the cap length at the net rate \(v\), tip growth minus the advance of the hydrolysis front.
  • \(D\,\partial_x^2 p\): diffusion of the cap length, with \(D_T\) and \(D_E\) adding because the two boundaries move independently and each contributes its own fluctuations.
  • \(-r\,x\,p\): loss — hydrolysis can strike at random at any of the \(x\) subunits currently in the cap, each at rate \(r\), so a cap of length \(x\) is destroyed (cut down to something shorter) at total rate proportional to its own length.
  • \(r\int_x^\infty dy\,p(y,t)\): gain — a cap of length \(x\) is created whenever a longer cap, of length \(y>x\), suffers a hydrolysis event exactly a distance \(x\) behind its own tip, cutting it down to length \(x\). Summing over all longer caps \(y\) replenishes the population at \(x\), matching the loss term above.

Sources

  • Lecture slides (paraphrased text for §§3, 3.1, 3.1.1, cell function, cytoskeletal filaments, dynamic instability): computational-biology/mit-ocw/8592j/lectures/20-slides.md, MIT 8.592J (Statistical Physics in Biology), from sources/ocw-8592j/lectures/20-slides.pdf.
  • The two-state model, its master equations, the Fourier-transform solution, and the Flyvbjerg–Holy–Leibler cap model and its master equation are read directly from the lecture's handwritten slide images, dated 4/11/07 and marked "Lec. 17": 20-slides/figures/p002-1.jpeg ("Dynamic Instability of Microtubules", citing Mitchison and Kirschner (1984) and Dogterom and Leibler (1993)) and 20-slides/figures/p004-1.jpeg (citing Flyvbjerg, Holy and Leibler (1994)). The auto-generated slide text breaks off mid-sentence before reaching this material, which is why the images rather than the paraphrase are cited for it.
  • Referenced by the lecture but not supplied with it: "Assignment 7", pointed to for the Fourier-transform method used to solve the coupled linear master equations.
  • No transcript, written notes, or exercise set was supplied for this lecture.

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