  We analyze experimental observations of microtubules undergoing small
fluctuations about a "balance point" when mixed in solution of two different
kinesin motor proteins, KLP61F and Ncd. It has been proposed that the
microtubule movement is due to stochastic variations in the densities of the
two species of motor proteins. We test this hypothesis here by showing how it
maps onto a one-dimensional random walk in a random environment. Our estimate
of the amplitude of the fluctuations agrees with experimental observations. We
point out that there is an initial transient in the position of the microtubule
where it will typically move of order its own length. We compare the physics of
this gliding assay to a recent theory of the role of antagonistic motors on
restricting interpolar microtubule sliding of a cell's mitotic spindle during
prometaphase. It is concluded that randomly positioned antagonistic motors can
restrict relative movement of microtubules, however they do so imperfectly. A
variation in motor concentrations is also analyzed and shown to lead to greater
control of spindle length.
