Growth probability and formation time of the individual Oosawa-Kasai protein fibril

Protein fibrils are currently of great academic and practical interest because of their involvement in scores of severe human diseases and their promising use in various high-technology devices. The Oosawa-Kasai (OK) model of protein self-assembly into fibrils has been widely used to gain mechanistic insight into the process of fibril formation and growth. Here this model is employed to obtain exact and mathematically simple expressions for the probability P_{n} of an individual fibril of n protein monomers to grow to a macroscopically large size and for the mean time τ_{n} that such a fibril needs for its formation. These expressions quantify the increase of P_{n} and the decrease of τ_{n} with increasing the concentration of monomeric protein in the solution. When used for analysis of experimental P_{n} and τ_{n} data, they make it possible to determine the parameters characterizing fibril nucleation and growth in the framework of the OK model. Finally, an expression is found for the mean time of the first appearance of an n-sized fibril in the protein solution. The results obtained are applicable to the formation of other aggregates corresponding to the OK fibrils, such as the one-dimensional Kossel-Stranski crystals and Ising ferromagnets.