Figure 10

Spontaneous (entropically driven) segregation (demix) of straight and curved microtubules and final orientational ordering under sufficiently crowded conditions. Here we illustrate the self-organization of microtubule system which is initially homogeneous, i.e., it is constituted by approximately equal amount of straight and curved microtubules. In addition, the system contains minority of curved microtubules () whose curvature is at the critical threshold for alignment (we call them ‘slightly curved microtubules’). Under crowded conditions, the distance between neighboring microtubules becomes rapidly shorter than the average microtubule length; the system is unstable and undergoes spontaneous phase segregation (demix) of coexisting phases (curved and straight), subsequently the separate final patterns are formed by spontaneous orientational ordering. A. Homogenous mix of straight and curved microtubules. B1. Demix of straight microtubule phase with minority of slightly curved microtubules. (∆ Ssd , ∆ Ssm - entropy of demix and mix, respectively). B2. Demix of curved microtubules phase. (∆ Scd , ∆ Scm entropy of demix, and mix, respectively). C1 Orientational ordering of straight microtubules with slightly curved microtubules. (∆ Sso , ∆ Ssdo - entropy of orientation and disorientation, respectively). C1.1 To be finally accommodated the curvature of curved microtubule should be made smaller than the critical threshold. This could be done by forced microtubule depolymerisation/repolymerisation that may be induced by mechanism of summation of pairwise electrostatic forces of neighboring microtubules. C1.2 If curvature of the curved microtubule cannot be reduced, curved microtubule is rejected. C2. Curved microtubules orientational partial ordering. (∆ Sco , ∆ scdo - entropy of partial orientation and disorientation, respectively). Spontaneity of demix and orientational ordering of either population is entropically driven, i.e.: $\begin{array}{l}\mathit{\Delta Ssd}+\mathit{\Delta Sso}>\mathit{\Delta Ssm}+\mathit{\Delta Ssdo}\\ \mathit{\Delta Scd}\mathit{\Delta Sco}>\mathit{\Delta Scm}+\mathit{\Delta Scdo}\end{array}$ (1)