Why Microtubules Run in Circles: Mechanical Hysteresis of the Tubulin Lattice

The fate of every eukaryotic cell subtly relies on the exceptional mechanical properties of microtubules. Despite significant efforts, understanding their unusual mechanics remains elusive. One persistent, unresolved mystery is the formation of long-lived arcs and rings, e.g., in kinesin-driven gliding assays. To elucidate their physical origin we develop a model of the inner workings of the microtubule’s lattice, based on recent experimental evidence for a conformational switch of the tubulin dimer. We show that the microtubule lattice itself coexists in discrete polymorphic states. Metastable curved states can be induced via a mechanical hysteresis involving torques and forces typical of few molecular motors acting in unison, in agreement with the observations.

Figure 1

(a)–(c) Mysterious ring formation of microtubules gliding on a kinesin motor carpet (from [18], [22], and [24]).

Figure 2

(a) The bistable tube model with internal prestrains leading to three different lattice states: long and straight (“$L$”), short and straight (“$S$”), and curved (“$C$”). (b) A typical polymorphic energy landscape, cf.  Eq. (3), with three metastable states and the associated polymorphic signature (with $C$ the ground state, and $L$ and $S$ “excited” states).

Figure 3

The polymorphic state diagram for a MT cross section as a function of effective force $f$ and torque $m$, cf. Eq. (3). The diagram depends on the parameter $c_p$, estimated from the experimental measurements of protofilament curvature and MT lattice shortening (cf. main text).

Figure 4

Dynamic shape evolutions of gliding MTs that become temporarily stuck and buckle. All parameters are identical in (a)–(c), except for $\mathrm{\Delta }G$ resulting in different effective forces $f$, cf. Fig. 3 and panel (d). (a) A MT far from the polymorphic region ($\mathrm{\Delta }G=+1.7\mathrm{kT}$ corresponding to $\gamma =1$) behaves like a gliding WLC upon buckling and release. (b) A MT for which the buckling induces a switch to the curved state ($\mathrm{\Delta }G=-5.2\mathrm{kT}$, $\gamma =0.4$). The finally formed ring is metastable; i.e., $L$ has lower energy than $C$. (c) A MT displaying multiple buckles that persist after unsticking and roll-up ($\mathrm{\Delta }G=-7.3\mathrm{kT}$, $\gamma =0.215$). (d) Trajectories in the polymorphic state diagram corresponding to the dynamics shown in (b) (blue curve, circles) and (c) (red curve, diamonds), resulting in a hysteretic loop upon torque creation and release, cf. the sketch in (e). Scale bars in (a)–(c) are $1\mu \mathrm{m}$, gliding velocity $v=1\mu \mathrm{m}/\mathrm{s}$. All parameter values can be found in [41].