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Equilibria of idealized confined astral microtubules and coupled spindle poles.

Maly IV - PLoS ONE (2012)

Bottom Line: It is found that depending on parameters, the symmetric position of the spindle can be stable or unstable.If they are not, then it is necessary to ask what forces external to the microtubule cytoskeleton counteract the bending effects sufficiently to actively establish symmetry.Conversely, regarding the cases with asymmetry, it is now necessary to investigate whether the cell controls the microtubule parameters so that the bending favors asymmetry apart from any forces that are external to the microtubule cytoskeleton.

View Article: PubMed Central - PubMed

Affiliation: Department of Computational and Systems Biology, University of Pittsburgh School of Medicine, Pittsburgh, Pennsylvania, United States of America. ivanvmaly@gmail.com

ABSTRACT
Positioning of the mitotic spindle through the interaction of astral microtubules with the cell boundary often determines whether the cell division will be symmetric or asymmetric. This process plays a crucial role in development. In this paper, a numerical model is presented that deals with the force exerted on the spindle by astral microtubules that are bent by virtue of their confinement within the cell boundary. It is found that depending on parameters, the symmetric position of the spindle can be stable or unstable. Asymmetric stable equilibria also exist, and two or more stable positions can exist simultaneously. The theory poses new types of questions for experimental research. Regarding the cases of symmetric spindle positioning, it is necessary to ask whether the microtubule parameters are controlled by the cell so that the bending mechanics favors symmetry. If they are not, then it is necessary to ask what forces external to the microtubule cytoskeleton counteract the bending effects sufficiently to actively establish symmetry. Conversely, regarding the cases with asymmetry, it is now necessary to investigate whether the cell controls the microtubule parameters so that the bending favors asymmetry apart from any forces that are external to the microtubule cytoskeleton.

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Limiting case of θmax = π, short microtubules.(A) Pole force function. (B) Spindle force function. (C) Equilibrium conformation. L = 0.8 R, S = 0.65 R. For clarity, few microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. Their values of θ are sampled uniformly between 0 and θmax.
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pone-0038921-g003: Limiting case of θmax = π, short microtubules.(A) Pole force function. (B) Spindle force function. (C) Equilibrium conformation. L = 0.8 R, S = 0.65 R. For clarity, few microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. Their values of θ are sampled uniformly between 0 and θmax.

Mentions: The opposite extreme case is also revealing–the special case of complete, intrinsically spherical asters at each pole (θmax = π). Now the behavior depends on whether the astral microtubules are longer or shorter than the cell radius. The case of short microtubules is simple. After the aster comes in contact with the boundary, the force exerted on the pole increases gradually with xp. The graduality is due to the number of the microtubules in contact with the boundary increasing gradually in this case, as compared with the case of θmax = 0. In addition, only the axial microtubules (θ = 0), whose contribution to an aster with θmax≠0 is infinitesimal, go through developing the buckling force during the axial movement of the spindle; all others deflect on contact in a continuous manner. The bending leads to the decrease in stiffness, as can be seen in Fig. 3. The decrease in stiffness of the aster (the progressively shallower slope of the force curve) is, however, different from the above-considered case, where the very magnitude of the elastic force decreased with the progressing deformation. The numerical results (Fig. 3) indicate that the softening effect of the deformation (Fig. 2) in the case of the complete aster is more than offset by the increasing numbers of microtubules that come in contact with the boundary: Although each progressively bending microtubule exerts a progressively lower force, the number of the bending microtubules grows so rapidly that the total force is incresing. Even though the total force is a nonlinear function of the pole position, the force resisting the outward movement of the pole is monotonic. In this sense it is similar to simple linear (Hookean) elasticity. Thus, a spindle with two complete asters of short microtubules exhibit stability of symmetry. In addition, the monotonicity means that there is only one equilibrium conformation of the mitotic microtubule cytoskeleton, insofar as the latter is large enough to maintain contact with the cell boundary.


Equilibria of idealized confined astral microtubules and coupled spindle poles.

Maly IV - PLoS ONE (2012)

Limiting case of θmax = π, short microtubules.(A) Pole force function. (B) Spindle force function. (C) Equilibrium conformation. L = 0.8 R, S = 0.65 R. For clarity, few microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. Their values of θ are sampled uniformly between 0 and θmax.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3375304&req=5

pone-0038921-g003: Limiting case of θmax = π, short microtubules.(A) Pole force function. (B) Spindle force function. (C) Equilibrium conformation. L = 0.8 R, S = 0.65 R. For clarity, few microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. Their values of θ are sampled uniformly between 0 and θmax.
Mentions: The opposite extreme case is also revealing–the special case of complete, intrinsically spherical asters at each pole (θmax = π). Now the behavior depends on whether the astral microtubules are longer or shorter than the cell radius. The case of short microtubules is simple. After the aster comes in contact with the boundary, the force exerted on the pole increases gradually with xp. The graduality is due to the number of the microtubules in contact with the boundary increasing gradually in this case, as compared with the case of θmax = 0. In addition, only the axial microtubules (θ = 0), whose contribution to an aster with θmax≠0 is infinitesimal, go through developing the buckling force during the axial movement of the spindle; all others deflect on contact in a continuous manner. The bending leads to the decrease in stiffness, as can be seen in Fig. 3. The decrease in stiffness of the aster (the progressively shallower slope of the force curve) is, however, different from the above-considered case, where the very magnitude of the elastic force decreased with the progressing deformation. The numerical results (Fig. 3) indicate that the softening effect of the deformation (Fig. 2) in the case of the complete aster is more than offset by the increasing numbers of microtubules that come in contact with the boundary: Although each progressively bending microtubule exerts a progressively lower force, the number of the bending microtubules grows so rapidly that the total force is incresing. Even though the total force is a nonlinear function of the pole position, the force resisting the outward movement of the pole is monotonic. In this sense it is similar to simple linear (Hookean) elasticity. Thus, a spindle with two complete asters of short microtubules exhibit stability of symmetry. In addition, the monotonicity means that there is only one equilibrium conformation of the mitotic microtubule cytoskeleton, insofar as the latter is large enough to maintain contact with the cell boundary.

Bottom Line: It is found that depending on parameters, the symmetric position of the spindle can be stable or unstable.If they are not, then it is necessary to ask what forces external to the microtubule cytoskeleton counteract the bending effects sufficiently to actively establish symmetry.Conversely, regarding the cases with asymmetry, it is now necessary to investigate whether the cell controls the microtubule parameters so that the bending favors asymmetry apart from any forces that are external to the microtubule cytoskeleton.

View Article: PubMed Central - PubMed

Affiliation: Department of Computational and Systems Biology, University of Pittsburgh School of Medicine, Pittsburgh, Pennsylvania, United States of America. ivanvmaly@gmail.com

ABSTRACT
Positioning of the mitotic spindle through the interaction of astral microtubules with the cell boundary often determines whether the cell division will be symmetric or asymmetric. This process plays a crucial role in development. In this paper, a numerical model is presented that deals with the force exerted on the spindle by astral microtubules that are bent by virtue of their confinement within the cell boundary. It is found that depending on parameters, the symmetric position of the spindle can be stable or unstable. Asymmetric stable equilibria also exist, and two or more stable positions can exist simultaneously. The theory poses new types of questions for experimental research. Regarding the cases of symmetric spindle positioning, it is necessary to ask whether the microtubule parameters are controlled by the cell so that the bending mechanics favors symmetry. If they are not, then it is necessary to ask what forces external to the microtubule cytoskeleton counteract the bending effects sufficiently to actively establish symmetry. Conversely, regarding the cases with asymmetry, it is now necessary to investigate whether the cell controls the microtubule parameters so that the bending favors asymmetry apart from any forces that are external to the microtubule cytoskeleton.

Show MeSH