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.

Show MeSH
Sample equilibrium conformations.For clarity, only 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. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper. In all examples, θmax = 0.6π. (A) L = 0.7 R, S = 1.2 R. (B) and (C) represent the alternative conformations that exist with L = 1.1 R and S = 0.4 R. (D) L = 0.95 R, S = 0.7 R.
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC3375304&req=5

pone-0038921-g009: Sample equilibrium conformations.For clarity, only 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. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper. In all examples, θmax = 0.6π. (A) L = 0.7 R, S = 1.2 R. (B) and (C) represent the alternative conformations that exist with L = 1.1 R and S = 0.4 R. (D) L = 0.95 R, S = 0.7 R.

Mentions: Among the theoretically possible structures and equilibria, several can be considered paradigmatic, based on qualitative examination of images of spindles in the experimental literature. Firstly, there is the case of a long spindle with short astral microtubules that radiate from the poles in a wide solid angle. The equilibrium conformation is plotted in Fig. 9A. According to the preceding analysis, in this regime (large S/R, small L/R, large θmax), the symmetric equilibrium is the only equilibrium, and it is stable. Awaiting measurements motivated by this theory, it appears that this regime is common among the variety of equally dividing cells. The HeLa cultured cells are one example [13].


Equilibria of idealized confined astral microtubules and coupled spindle poles.

Maly IV - PLoS ONE (2012)

Sample equilibrium conformations.For clarity, only 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. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper. In all examples, θmax = 0.6π. (A) L = 0.7 R, S = 1.2 R. (B) and (C) represent the alternative conformations that exist with L = 1.1 R and S = 0.4 R. (D) L = 0.95 R, S = 0.7 R.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0038921-g009: Sample equilibrium conformations.For clarity, only 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. The circumference is the section of the cell surface, and the thicker line segment depicts the spindle proper. In all examples, θmax = 0.6π. (A) L = 0.7 R, S = 1.2 R. (B) and (C) represent the alternative conformations that exist with L = 1.1 R and S = 0.4 R. (D) L = 0.95 R, S = 0.7 R.
Mentions: Among the theoretically possible structures and equilibria, several can be considered paradigmatic, based on qualitative examination of images of spindles in the experimental literature. Firstly, there is the case of a long spindle with short astral microtubules that radiate from the poles in a wide solid angle. The equilibrium conformation is plotted in Fig. 9A. According to the preceding analysis, in this regime (large S/R, small L/R, large θmax), the symmetric equilibrium is the only equilibrium, and it is stable. Awaiting measurements motivated by this theory, it appears that this regime is common among the variety of equally dividing cells. The HeLa cultured cells are one example [13].

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