  Assembly and stability of mitotic spindle is governed by the interplay of
various intra-cellular forces, e.g. the forces generated by motor proteins by
sliding overlapping anti-parallel microtubules (MTs) polymerized from the
opposite centrosomes, the interaction of kinetochores with MTs, and the
interaction of MTs with the chromosomes arms. We study the mechanical behavior
and stability of spindle assembly within the framework of a minimal model which
includes all these effects. For this model, we derive a closed--form analytical
expression for the force acting between the centrosomes as a function of their
separation distance and we show that an effective potential can be associated
with the interactions at play. We obtain the stability diagram of spindle
formation in terms of parameters characterizing the strength of motor sliding,
repulsive forces generated by polymerizing MTs, and the forces arising out of
interaction of MTs with kinetochores. The stability diagram helps in
quantifying the relative effects of the different interactions and elucidates
the role of motor proteins in formation and inhibition of spindle structures
during mitotic cell division. We also predict a regime of bistability for
certain parameter range, wherein the spindle structure can be stable for two
different finite separation distances between centrosomes. This occurrence of
bistability also suggests mechanical versatility of such self-assembled spindle
structures.
