Figure 1.
Method of approximation of electric properties of microtubules.
The MT (A) is divided into so called MT rings (B) forming the spiral, where corresponding dipoles were placed in the center of gravity of respective monomer (C). Center of gravity and dipoles were calculated from molecular structure of tubulin (D).
Figure 2.
Model of mitotic spindles and corresponding intensities of electric field.
The model of mitotic spindle of ellipsoidal cell with equivalent radius 3.3 µm (A). The position of equatorial plane is marked by red dashed line. Magnitude of intensity of electric field in three main plains of the ellipsoidal cell (boundary of the cell is depicted by white circle) in time ps after pulsed excitation is shown in (B–D). The plane shown in (D) is the equatorial plane. All pictures have the same spatial and intensity scale and the same quality factor of vibrations,
. All time locks were taken in
ps after pulsed excitation. Presented data correspond to boundary conditions with fixed ends.
Figure 3.
Spectrum of vibrations of mitotic spindle.
Spectrum as a function of the quality factor (A) and size of model cell (B). The value of quality factor is coded according to the colorbar in A. Solid line in B corresponds to boundary conditions with fixed ends, dashed line to free ends in equatorial plane.
Figure 4.
Time evolution of intensity of electric field in equatorial plane for different boundary conditions.
The magnitude of intensity of electric field depends on time and boundary conditions. It undergoes exponential decay due to damping after pulsed excitation. Map of electrical intensities (A) and amplitude of mechanical oscillations of all MTs (B) are shown for free ends of MTs in equatorial plane. Data for fixed boundary conditions are shown in (C) and (D). Data correspond to ellipsoidal cell with equivalent radius of 3.3 m. Note different time scales in (A, B) and (C, D). Quality factor of vibrations, , is the same for all examples.
Figure 5.
Statistical analysis of electric field generated by random and fixed undamped vibrations.
Data are shown for free-ends boundary conditions (A–C, G–I) and for fixed-ends in equatorial plane (D–F, J–L). Maximal values (left column A, D, G, J), mean values (middle column B, E, H, K) and minimum values (right column C, F, I, L) of the intensity of electric field in equatorial plane are shown. Data corresponding to random vibrations are shown in (A–F), results for pulse-driven vibrations are displayed in (G–L).
Figure 6.
Time evolution of intensity of electric field in equatorial plane for different feeding.
Time versus electric intensity plots are shown for fixed-ends and pulsed undamped (A), pulsed damped (B), random undamped (C) and random damped (D) excitation followed by free-ends and pulsed undamped (E), pulsed damped (F), random undamped (G) and random damped (H) excitation shown in points 1–4 of equatorial plane specified in time-lock (I). (J and K) show time versus electric intensity plots in central point of the equatorial plane. Fixed-end boundary condition is displayed in (J), free-end boundary condition is displayed in (K). Dashed curves show data for pulsed undamped, pulsed damped, random undamped and random damped conditions.
Table 1.
List of parameters.
Figure 7.
Detailed geometry of the model.
The overall structure of mitotic spindle in the model (A). Kinetochore and polar microtubules grow from MTOC along ellipsoidal trajectory towards the equatorial plane (B). The MTOC is devided into two parts. One serves as a base for kinetochore and polar MTs (orange), the second for astral MTs (green). Nucleation centers of MTs are distributed uniformly on the surface of the MTOC (C).
Figure 8.
Transformation of coordinates used for calculation of radiation of a dipole.
The schematics explains the geometrical meaning of parameters in Eqs. 4–7.
Figure 9.
Frequency of vibrations of MT as a function of its length.
Molecular dynamics and normal mode analysis models of MT (red) were extrapolated (black) for this purpose to the region of lengths relevant to this model (blue).
Figure 10.
Electrical parameters of the cytosol.
We used homogeneous electrical properties of the surroundings of the MTs in our model. The figure shows frequency versus complex permittivity plot. The real part of the complex permittivity (up) represents the value of the relative electrical permittivity, and therefore energy stored in the material, and the imaginary part (down) corresponds to dielectric losses.