Size distribution of protein polymers

Theoretical analyses were made on the size distribution of tubular or helical polymers of globular protein molecules in the reversible polymerization. In the true final equilibrium, polymerization approximates to a kind of crystallization and the average degree of polymerization becomes very large; nevertheless, the size distribution is of a simple exponential type, the same as found in macromolecular chemistry (Flory, 1953). Even when spontaneous nucleation is inhibited and the number of polymers is given, the final distribution must tend towards the exponential type. Free energy due to deviation of the size distribution from the true equilibrium was calculated. It is very much smaller than the main free energy coming from the monomer-polymer equilibrium. Kinetics showed that polymerization having characteristics of crystallization usually consists of three stages—nucleation, growth and redistribution of polymer size. In the first and second stages where the rate of depolymerization is negligible, the concentration of monomers approaches closely to the equilibrium value. In the third stage, where both polymerization and depolymerization take place nearly at the same rate, the size distribution is slowly transformed into the exponential type. The relaxation time for such redistribution was estimated as a function of rate constants and the average degree of polymerization under various conditions. All of the theoretical results are quantitatively in good agreement with experimental data on polymerization of globular proteins such as actin and flagellin. Brief analyses were added on the size distribution of two-dimensional membraneous polymers and distorted polymers.