The role of subunit entropy in cooperative assembly. Nucleation of microtubules and other two-dimensional polymers.

The self-assembly and nucleation of two-dimensional polymers is described by a theory based on a model of rigid subunits and bonds and simple principles of thermodynamics. The key point in the theory is to separate as an explicit parameter the free energy, primarily attributed to the entropy of the free subunit, that is required to immobilize a subunit in the polymer. Quantitative relations for the association of a subunit forming a longitudinal bond, a lateral bone, or both together are obtained, which demonstrate the basis and magnitude of cooperativity. The same formalism leads to a quantitative estimate for th concentration of the small polymers that are important intermediates in nucleation. It is shown that, if the concentration of free subunits is below a certain "critical supersaturation," the concentration of some essential intermediates is too low to support any significant assembly and nucleation is blocked. If the subunit concentration is above the critical supersaturation, all of the small intermediates are sufficiently stable to form and grow spontaneously. The theory predicts a critical supersaturation of 3.5 to 7 (the ratio of subunit concentration to the equilibrium solubility) for parameters appropriate to assembly of the microtubule wall. Experimentally, nucleation and assembly of microtubules is obtained at somewhat lower concentrations, 1.5 to 3 times the equilibrium solubility. Special mechanisms that could stabilize small polymers and facilitate nucleation of microtubule assembly are suggested.