Spontaneous catastrophe and induced catastrophe in regulation of order–disorder transition. (a) The transition from N_I phase to less ordered N_II phase and disorder with the increase of spontaneous catastrophe rate k_sc (red square) or k_ic (blue dot). (b) Same data as (a) but the horizontal-axis is mapped to the time averaged frequency of real catastrophe events (RCE) for all the samples. The critical angle for collision induced dynamic catastrophe is α_c = 40°, the growth speed k_g = 70 nm/s, and effective microtubule width W = 25 nm. The simulations start from N_I state as obtained in Fig. 4(b) at k_g = 70 nm/s. Each point is averaged with 5 samples with time interval 1 s, and average simulation time longer than 10⁵s.