Limits...
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 = 0.(A) Pole force function. (B) Spindle force function. (C) Symmetric equilibrium. (D) Asymmetric equilibrium. B–D: L = 0.8 R, S = 0.65 R. For clarity, only two microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper.
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pone-0038921-g002: Limiting case of θmax = 0.(A) Pole force function. (B) Spindle force function. (C) Symmetric equilibrium. (D) Asymmetric equilibrium. B–D: L = 0.8 R, S = 0.65 R. For clarity, only two microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper.

Mentions: It is expedient to consider first some limiting cases that set the theoretical context for the mechanics of the more biologically relevant regimes. The simplest behavior is exhibited by the structural case of astral microtubules that emanate from the pole exclusively along the axis of the spindle (θmax = 0). As the pole moves collinearly with the spindle axis, the microtubules will abut on the cell boundary, buckle, and bend, exerting force on the pole. A sample calculation is presented in Fig. 2. When the axial distance of the pole from the boundary is greater than the length of the astral microtubules, the force is zero. When the distance is equal to the length, the magnitude of the force can take any value between zero and the buckling force. For shorter distances, the force decreases, as the increasingly bent microtubules become less efficient at resisting the displacement of the pole. When the movement of two such poles is coupled through the spindle proper, three regimes are possible. If the spindle proper is short, the astral microtubules may not come in contact with the boundary, and the symmetric position will be a neutral equilibrium. This is only possible with short microtubules, whose length L satisfies the following relation to the spindle length S and cell radius R: L+S/2<R. The special case of the entire structure just fitting in the cell without deformation (L+S/2 = R) is unlikely to be realized. When L+S/2 is greater than R, microtubule emanating from one pole or from both must be bent.


Equilibria of idealized confined astral microtubules and coupled spindle poles.

Maly IV - PLoS ONE (2012)

Limiting case of θmax = 0.(A) Pole force function. (B) Spindle force function. (C) Symmetric equilibrium. (D) Asymmetric equilibrium. B–D: L = 0.8 R, S = 0.65 R. For clarity, only two microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0038921-g002: Limiting case of θmax = 0.(A) Pole force function. (B) Spindle force function. (C) Symmetric equilibrium. (D) Asymmetric equilibrium. B–D: L = 0.8 R, S = 0.65 R. For clarity, only two microtubule forms are plotted. These microtubules lie in the (x,y) plane that passes through the spindle axis. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper.
Mentions: It is expedient to consider first some limiting cases that set the theoretical context for the mechanics of the more biologically relevant regimes. The simplest behavior is exhibited by the structural case of astral microtubules that emanate from the pole exclusively along the axis of the spindle (θmax = 0). As the pole moves collinearly with the spindle axis, the microtubules will abut on the cell boundary, buckle, and bend, exerting force on the pole. A sample calculation is presented in Fig. 2. When the axial distance of the pole from the boundary is greater than the length of the astral microtubules, the force is zero. When the distance is equal to the length, the magnitude of the force can take any value between zero and the buckling force. For shorter distances, the force decreases, as the increasingly bent microtubules become less efficient at resisting the displacement of the pole. When the movement of two such poles is coupled through the spindle proper, three regimes are possible. If the spindle proper is short, the astral microtubules may not come in contact with the boundary, and the symmetric position will be a neutral equilibrium. This is only possible with short microtubules, whose length L satisfies the following relation to the spindle length S and cell radius R: L+S/2<R. The special case of the entire structure just fitting in the cell without deformation (L+S/2 = R) is unlikely to be realized. When L+S/2 is greater than R, microtubule emanating from one pole or from both must be bent.

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