Simulation of Density Gradients of Astral Microtubules at Cell Surface in Cytokinesis of Sea Urchin Eggs

Abstract

Astral microtubules extend close to the cell surface just before cytokinesis in sea urchins eggs. At this time, a small region with a constant area is considered around a point on the egg surface. To calculate the number of microtubules that reach the surface region, i.e. the microtubule density at the point, a simple mathematical model was set up. The density was estimated at many surface points in multipolar and distorted eggs by using the model. A contour map was drawn to investigate the density gradients. The gradient patterns were compared with the distributions of contractile-ring microfilaments. The simulated cases were: (1) an unusual distribution of contractile-ring microfilaments in an egg that had polyasters and was compressed by a coverslip; (2) formation of contractile-ring microfilaments at the equatorial region in compressed eggs with a centrally-located mitotic apparatus; (3) normal furrowing in the plane of the spindle midpoint in eggs inserted into a glass loop or confined in a capillary; (4) failure of furrow formation in spherical eggs treated with ethyl urethane and revival of furrowing by pushing the equatorial surface closer to the spindle. These simulations proposed the hypothesis that contractile-ring microfilaments form at the surface regions where the microtubule has a local minimum, not a local maximum. In addition, it was suggested that the probability of the formation of contractile-ring microfilaments is dependent on how abruptly the density gradient changes at the local-minimum point. These results support the idea that the gradient pattern of the microtubule density determines whether and where contractile-ring microfilaments appear.