Figure 1.
Model of a sheet-ended microtubule.
(a) The equilibrium structure of a sheet-ended microtubule revealed in Ref. [35]. (b) The partial enlarged view of the microtubule structure marked in the red box in (a), showing (1) the longitudinal tension or compression interaction, (2) the lateral tension or compression interaction, (3) the diagonal tension or compression interaction, (4) the longitudinal bending interaction, and (5) the lateral bending interaction. (c) The longitudinal bending and the dihedral bending interactions. The bending interaction considers the supplementary angle of the angle A–B–C, and the dihedral angle is defined between the plane A–B–C and A–A′–B. (d) The lateral bending and dihedral bending interactions. The bending interaction considers the supplementary angle of the angle D–E–F, and the dihedral angle is defined between the plane D–D′–E and E–E′–F.
Table 1.
Interaction definitions in the model.
Figure 2.
Sites for a coming tubulin dimer to assemble: (a) the dimer inserting into a gap, (b) the dimer associating a single-sided neighbor, and (c) the dimer falling upon the crest.
Figure 3.
Snapshots of the conformation and potential energy evolutions during a sheet-to-tube transition process.
The equilibrium states before assembly and after three consecutive closures are shown: (a) the evolution of the microtubule structure, and (b) the corresponding evolution of the total potential energy.
Figure 4.
Potential energy evolution during the sheet-to-tube transition process.
A continuous zipping of the seam counting three pairs of monomers is characterized. The consecutive closure happens when the microtubule is in an equilibrium conformation. (a) The evolution of the total potential energy, from which the energy barrier and energy difference between two equilibrium states are clearly detected. (b) The evolution of the seven energy components. Respecting the differences of orders of magnitude, a semilogarithmic coordinate is adopted.
Figure 5.
Comparison of the energy barriers and energy differences during sheet-to-tube transitions under different sheet structures: (a) the intrinsic curvatures of GTP-tubulins are of three different values, and (b) the sheets are in two different nucleotide states.
In both panels, the red lines represent the result for the standard model shown in Fig. 4. For clarity, the three sets of data have been offset horizontally, but not vertically.
Figure 6.
Potential energy evolution during a sheet-to-tube transition with randomly distributed GTPs.
(a) Snapshots of the longitudinal bending potential distribution during a time span of three closures of a monomer pair. The illustrated states are in equilibrium. (b) Evolution of the total potential energy.
Figure 7.
Energy barriers and energy steppings induced by a monomer pair closure for ten sheets of different lengths, varying from 1 to 10 monomers in the longitudinal direction.
The ten microtubule models are composed of protofilaments of the same length. The total potential energies are at different levels since the length of closed parts of the ten models are different.
Figure 8.
Harmonized assembly and closure during microtubule growth.
(a) Snapshots of the sheet structure during a time span of two subsequent closures of a monomer pair. When a layer at the end of the sheet has been fully filled by newly added tubulins, the seam will be zipped a same length. The blue and red-highlighted structures show two processes of the assembled dimers fulfilling a complete layer and the subsequent seam zipping up. (b) The microtubule end development at the two closures in (a). The two yellow-colored dimers are those finally fulfill a layer and trigger a corresponding closure.
Figure 9.
Influences of interaction constants on the energy barrier and energy difference between two equilibrium states during closure.
(a–c) Influences of the interaction constants of lateral dihedral, longitudinal dihedral, and diagonal tension or compression, respectively.