Tubulin polymers, microtubules, can switch abruptly from the assembly to shortening. These infrequent transitions, termed "catastrophes", affect numerous cellular processes but the underlying mechanisms are elusive. We approached this complex stochastic system using advanced coarse-grained molecular dynamics modeling of tubulin-tubulin interactions. Unlike in previous simplified models of dynamic microtubules, the catastrophes in this model arise owing to fluctuations in the composition and conformation of a growing microtubule tip, most notably in the number of protofilament curls. In our model, dynamic evolution of the stochastic microtubule tip configurations over a long timescale, known as the system's "aging", gives rise to the nonexponential distribution of microtubule lifetimes, consistent with experiment. We show that aging takes place in the absence of visible changes in the microtubule wall or tip, as this complex molecular-mechanical system evolves slowly and asymptotically toward the steady-state level of the catastrophe-promoting configurations. This new, to our knowledge, theoretical basis will assist detailed mechanistic investigations of the mechanisms of action of different microtubule-binding proteins and drugs, thereby enabling accurate control over the microtubule dynamics to treat various pathologies.