  The centrosomal aster is a mobile cellular organelle that exerts and
transmits forces necessary for nuclear migration and spindle positioning.
Recent experimental and theoretical studies of nematode and human cells
demonstrate that pulling forces on asters by cortical force generators are
dominant during such processes. We present a comprehensive investigation of a
first-principles model of aster dynamics, the S-model (S for stoichiometry),
based solely on such forces. The model evolves the astral centrosome position,
a probability field of cell-surface motor occupancy by centrosomal microtubules
(under an assumption of stoichiometric binding), and free boundaries of
unattached, growing microtubules. We show how cell shape affects the centering
stability of the aster, and its transition to oscillations with increasing
motor number. Seeking to understand observations in single-cell nematode
embryos, we use accurate simulations to examine the nonlinear structures of the
bifurcations, and demonstrate the importance of binding domain overlap to
interpreting genetic perturbation experiments. We find a rich dynamical
landscape, dependent upon cell shape, such as internal equatorial orbits of
asters that can be seen as traveling wave solutions. Finally, we study the
interactions of multiple asters and demonstrate an effective mutual repulsion
due to their competition for cortical force generators. We find, amazingly,
that asters can relax onto the vertices of platonic and non-platonic solids,
closely mirroring the results of the classical Thomson problem for
energy-minimizing configurations of electrons constrained to a sphere and
interacting via repulsive Coulomb potentials. Our findings both explain
experimental observations, providing insights into the mechanisms governing
spindle positioning and cell division dynamics, and show the possibility of new
nonlinear phenomena in cell biology.
