  Polymerization of microtubules is ubiquitous in biological cells and under
certain conditions it becomes oscillatory in time. Here simple reaction models
are analyzed that capture such oscillations as well as the length distribution
of microtubules. We assume reaction conditions that are stationary over many
oscillation periods, and it is a Hopf bifurcation that leads to a persistent
oscillatory microtubule polymerization in these models. Analytical expressions
are derived for the threshold of the bifurcation and the oscillation frequency
in terms of reaction rates as well as typical trends of their parameter
dependence are presented. Both, a catastrophe rate that depends on the density
of {\it guanosine triphosphate} (GTP) liganded tubulin dimers and a delay
reaction, such as the depolymerization of shrinking microtubules or the decay
of oligomers, support oscillations. For a tubulin dimer concentration below the
threshold oscillatory microtubule polymerization occurs transiently on the
route to a stationary state, as shown by numerical solutions of the model
equations. Close to threshold a so--called amplitude equation is derived and it
is shown that the bifurcation to microtubule oscillations is supercritical.
