  Microtubules are dynamic intracellular fibers that have been observed
experimentally to undergo spontaneous self-alignment. We formulate a 3D
mean-field theory model to analyze the nematic phase transition of microtubules
growing and interacting within a 3D space then make a comparison with
computational simulations. We identify a control parameter $G_\text{eff}$ and
predict a unique critical value $G_\text{eff}=1.56$ for which a phase
transition can occur. Furthermore, we show both analytically and using
simulations that this predicted critical value does not depend on the presence
of zippering. The mean-field theory developed here provides an analytical
estimate of microtubule patterning characteristics without running
time-consuming simulations and is a step towards bridging scales from
microtubule behavior to multicellular simulations.
