  We investigate the cooperative dynamics of an ensemble of N microtubules
growing against an elastic barrier. Microtubules undergo so-called
catastrophes, which are abrupt stochastic transitions from a growing to a
shrinking state, and rescues, which are transitions back to the growing state.
Microtubules can exert pushing or polymerization forces on an obstacle, such as
an elastic barrier if the growing end is in contact with the obstacle. We use
dynamical mean-field theory and stochastic simulations to analyze a model where
each microtubule undergoes catastrophes and rescues and where microtubules
interact by force sharing. For zero rescue rate, cooperative growth terminates
in a collective catastrophe. The maximal polymerization force before
catastrophes grows linearly with N for small N or a stiff elastic barrier, in
agreement with available experimental results, whereas it crosses over to a
logarithmic dependence for larger N or a soft elastic barrier. For a nonzero
rescue rate and a soft elastic barrier, the dynamics becomes oscillatory with
both collective catastrophe and rescue events, which are part of a robust limit
cycle. Both the average and maximal polymerization forces then grow linearly
with N, and we investigate their dependence on tubulin on-rates and rescue
rates, which can be involved in cellular regulation mechanisms. We further
investigate the robustness of the collective catastrophe and rescue
oscillations with respect to different catastrophe models.
