Fig 1.
Angles describing the rotation of monomers relative to each other.
A: Microtubule-bound coordinate system xyz. α-tubulins are shown in dark green, β-tubulins in light green. x-axis is radial to the microtubule axis, y-axis is tangential to microtubule, z-axis is parallel to microtubule axis. B: Schematic representation of a curved tubulin dimer and the angles that characterize the magnitude of upper subunit tilt, θ, the direction of this tilt relative to the radius of the microtubule, φ, and the twist angle δ of tubulin monomers with respect to one another. The coordinate system xyz associated with the bottom monomer is shown in blue, the system XYZ, associated with upper monomer, is red. Cyan is an auxiliary coordinate system x'y'z', which is produced by rotation of the coordinate system xyz, so the vector oz becomes aligned with OZ.
Fig 2.
Simulations of GDP- and GTP-tubulin dimers.
A: GDP- and GTP-tubulin dimers at the end of 1 μs simulation (green) aligned onto initial straight structures (pink). GTP is shown in orange, GDP is purple. B: Projections of the unit OZ-vector of the β-subunit of GDP-tubulin dimer onto xy-plane of the α-tubulin at every nanosecond of the simulation after the first 500 ns. Dashed line schematically shows the circumference of the microtubule. Horizontal axis is tangential to the microtubule, vertical axis is directed radially toward microtubule axis. Green and red data points correspond to two different independent simulation runs. C: Projections of the center of mass of β-tubulin onto xy-plane of the α-tubulin during the motions, corresponding to major bending mode of free tubulin dimer. Lines of different colors mark different tubulin structures. Schematic in the upper right corner of this panel illustrate approximate direction of bending motions relative to the microtubule. D: GDP-tubulin tetramer in the end of 1 μs simulation (green) aligned onto initial straight structure (pink), shown in two orthogonal views: in the plain of bending (left) and along the microtubule axis (right). In the latter view, the subunits of the two adjacent protofilaments are also shown. E: Analogous views of GTP-tubulin tetramer. F: Projections of the center of mass of the top β-tubulin onto xy-plane of the bottom α-tubulin during the motions, corresponding to major bending mode of a free tubulin tetramer (also see S3 Movie).
Fig 3.
Quantitative analysis of intra-dimer interface of GTP- and GDP-tubulin tetramers.
A: Time-dependence of GDP-tubulin intra-dimer bend and twist angles. Colors mark independent simulation runs. Only one of the two intra-dimer interfaces for each tetramer simulation run is shown. B: Time-dependence of GTP-tubulin intra-dimer bend and twist angles. Colors mark independent simulation runs. C: Projections of the unit OZ-vector of the β-subunit of GDP-tubulin dimer onto the xy-plane of the α-tubulin at every ns after the first 500 ns of the simulation. Data and color-coding correspond to panel A. Dashed line schematically shows the circumference of the microtubule. Horizontal axis is tangential to the microtubule, vertical axis is directed radially toward microtubule axis. D: Projections of the unit OZ-vector of the β-subunit of GTP-tubulin dimer onto xy-plane of the α-tubulin at every ns after the first 500 ns of the simulation. Data and color-coding correspond to panel B.
Table 1.
Quantification of conformational changes at intra- and inter-dimer interfaces in molecular dynamics simulations.
Fig 4.
Quantitative analysis of inter-dimer interface of GTP- and GDP-tubulins.
A: Time-dependence of GDP-tubulin inter-dimer bend and twist angles. Colors mark three independent simulation runs. B: Time-dependence of GTP-tubulin inter-dimer bend and twist angles. Colors mark three independent simulation runs. C: Projections of the unit OZ-vector of the α-subunit of the GDP-tubulin inter-dimer interface onto the xy-plane of the β-tubulin at every ns after the first 500 ns of the simulation. Data and color-coding correspond to panel A. Dashed line schematically shows the circumference of the microtubule. Horizontal axis is tangential to the microtubule, vertical axis is directed radially toward microtubule axis. D: Projections of the unit OZ-vector of the α-subunit of GTP-tubulin inter-dimer interface onto xy-plane of the β-tubulin at every ns after the first 500 ns of the simulation. Data and color-coding correspond to panel B.
Table 2.
Quantification of conformational angles at intra- and inter-dimer tubulin interfaces in published structures.
Fig 5.
Effects of attachment to mechanical support on tubulin oligomer conformation.
A: GDP- and GTP-tubulin hexamers at the end of 1 μs simulation (green) aligned onto initial straight structures (pink), viewed from the side and from top. GTP molecule and Mg2+ ion are shown in orange, GDP molecule is purple. Five inter-tubulin interfaces are numbered from bottom to top (in yellow circles). B: Projections of the unit OZ-vector of the upper tubulin subunit at each tubulin interface onto xy-plane of the lower tubulin at every nanosecond of the simulation after the first 500 ns. Dashed line schematically shows the circumference of the microtubule. Horizontal axis is tangential to the microtubule, vertical axis is directed radially toward microtubule axis. Green and red data points correspond to GDP- and GTP-tubulin data, respectively. Interfaces are numbered as in panel A.
Fig 6.
Effects of lateral protofilament neighbors on tubulin oligomer relaxation from straight conformation.
A: three GDP-tubulin hexamers after 0.8 μs simulation, viewed in diagonal direction. B: the same structure as in panel A, viewed from top. C: Smoothed projections of the centers of mass of terminal β-tubulin subunits onto XY plane of microtubule-bound coordinate system. D: Number of contacts between Cα-atoms of the terminal β-tubulin subunits during one the simulation run. Red curve corresponds to contacts between the left and the middle protofilaments, blue curve is for contacts between the middle and the right protofilaments.
Table 3.
Harmonic stiffness of tubulin conformational angles, kBT/rad2.
Table 4.
Bending stiffness, Kbend, and torsional rigidity, Ktwist, of intra- and inter-tubulin interfaces, inferred from NMA.
Fig 7.
Implications of unequal interface stiffness and non-radial bending for dynamic properties of plus- and minus- microtubule tips.
Three tubulin protofilaments are highlighted. α-tubulins are dark green, and β- tubulins are light green. Red outlines mark the second to terminal layers of tubulins. In the GTP-state of tubulins those layers are under lower bending stress, and so they rate limit assembly-disassembly at plus- and minus-ends of the microtubule.