Every microtubule is either growing or collapsing, and switches at random. The population sits at a steady state that no individual in it occupies — which means a constant average is not evidence that anything has settled.
Before 1984 the assumption about a polymer in solution was that it approaches a steady state: monomers on, monomers off, length settling toward whatever the concentration supports. Tim Mitchison and Marc Kirschner looked at individual microtubules and found that no microtubule does this.
Each one is either growing or shrinking, at very different speeds, and switches between the two abruptly and at random. Growth is slow, around $2\ \mu\mathrm{m}/\mathrm{min}$; collapse is fast, an order of magnitude faster. The switch from growth to collapse they named catastrophe; the switch back, rescue. CITED
The population is at steady state. No member of it is. The mean length of the ensemble sits still while every individual is doing something violent, and the stillness is a statistical fact about a collection, not a property of anything in it.
A corpus that reasons about attractors should take that seriously. An average that is constant is not evidence that anything has settled.
The mechanism they proposed is a threshold, and it is the cleanest one in cell biology.
Tubulin arrives carrying GTP. It is hydrolysed to GDP some time after it is incorporated, and GDP-tubulin in the lattice is strained — it wants to curl outward and would peel the tubule apart if it were exposed. So long as hydrolysis lags behind addition, the tip carries a layer of GTP-tubulin holding the strained body together.
While the cap exists the microtubule grows. If addition falters, hydrolysis catches the tip, the cap is lost, and the strained lattice below unpeels at full speed. Not gradually weaker — there, then not there.
The fold, in the corpus's own vocabulary: a control quantity (cap size) is driven toward zero, and at zero the system does not become slightly less stable. It changes branch, and the branch it changes to is travelling twenty times faster in the other direction.
The two-state model has an exact threshold, and it decides whether a cell can build anything.
$$J = v_g f_{\mathrm{res}} - v_s f_{\mathrm{cat}}$$
$J < 0$: lengths reach a stationary distribution with mean $\langle L\rangle = v_g v_s / (v_s f_{\mathrm{cat}} - v_g f_{\mathrm{res}})$. $J > 0$: the mean length grows without bound.
Simulated, with $v_g = 2$, $v_s = 20\ \mu\mathrm{m}/\mathrm{min}$, $f_{\mathrm{cat}} = 0.3/\mathrm{min}$:
fres=0.1 J=−5.80 ⟨L⟩= 7.19 theory 6.90
fres=1.0 J=−4.00 ⟨L⟩= 10.42 theory 10.00
fres=2.0 J=−2.00 ⟨L⟩= 20.19 theory 20.00
fres=2.9 J=−0.20 ⟨L⟩=292.41 theory 200.00
fres=3.0 J= 0.00 ⟨L⟩=711 and rising
fres=5.0 J=+4.00 ⟨L⟩=7954 and rising COMPUTED
In the bounded phase $\langle L\rangle$ is the same at $T = 2\,000$ and $T = 20\,000$ minutes — 9.48, 9.46, 10.12, 10.42. In the unbounded phase it doubles when the observation doubles — 842, 1955, 4005, 7954.
A single long run cannot tell the two apart; a mean length of 700 looks like a large number either way. Only the dependence on observation time does, and this is the general form of a mistake this corpus has made twice today in other chapters: a statistic reported from a run that had not reached the regime it was measuring. SHOWN
The microtubule has thirteen protofilaments — thirteen parallel tracks of tubulin closing into a tube. This is not the only number the lattice can take: assemblies with eleven to sixteen occur in vitro and in some organisms, and the number is set by the nucleating template. Thirteen is the number in almost every animal cell, and it is the number for which the protofilaments run parallel to the tube axis rather than winding around it. A kinesin walking a wound track spirals; on thirteen it goes straight. CITED
The index card for this chapter reads 13 protofilaments (Fibonacci), and the temptation is to connect thirteen to the ladder that runs through Book VI. This page does not make that connection, because no argument for it exists here. Thirteen is a Fibonacci number; so are eight and twenty-one, and the microtubule is not eight or twenty-one for reasons that are about lattice geometry and the γ-tubulin ring, not about recurrences.
The established fact is the supertwist: thirteen is where it vanishes. Whether that has anything to do with the corpus's ladder is OPEN, and writing it down as though it were settled would be the exact failure this series keeps auditing itself for.
What the chapter does establish is the shape. A cell holds a structure in place by running every element of it at a threshold, letting individuals fail constantly, and reading only the population. Bak's pile does the same thing with sand. The microtubule does it with a nucleotide, and it does it in every dividing cell in your body, right now, about once a minute.
| Operator | In this chapter | In dm³ |
|---|---|---|
| C | tubulin-GTP arriving — the monomer pool, fixed | compression: the constraint |
| K | hydrolysis catching up with addition; the cap thinning | curvature driven to $\kappa^*$ |
| F | catastrophe — the cap gone, the strained lattice unpeeling at 20 µm/min | the fold SHOWN |
| U | rescue, or the population mean — whichever is the observable | the branch, read statistically COMPUTED |
Every number on this page is produced by book7/ch-mitchison-kirschner-verify.py. It records in its own closing block what it establishes and what it does not.
T. Mitchison and M. Kirschner, “Dynamic instability of microtubule growth”, Nature 312, 1984, 237–242.
T. Mitchison and M. Kirschner, “Microtubule assembly nucleated by isolated centrosomes”, Nature 312, 1984, 232–237.
M. Dogterom and S. Leibler, “Physical aspects of the growth and regulation of microtubule structures”, Phys. Rev. Lett. 70, 1993 — the bounded/unbounded criterion used here.
D. Chrétien and R. H. Wade, “New data on the microtubule surface lattice”, Biol. Cell 71, 1991 — protofilament number and supertwist.
H. V. Goodson and E. M. Jonasson, “Microtubules and microtubule-associated proteins”, Cold Spring Harb. Perspect. Biol. 10, 2018.