Theory for modeling the copolymerization of tubulin and tubulin-colchicine complex

Substoichiometric concentrations of tubulin-colchicine complex (TC) inhibits microtubule assembly through a copolymerization reaction between tubulin and TC. We have determined the rates and extent of TC incorporation into bovine brain microtubules and developed a theory that models copolymerization. Our analysis suggests that while the apparent association rate constants for tubulin and TC are similar, the apparent dissociation rate constants for TC are a factor of five or more larger than those of tubulin. Copolymer composition showed only slight changes during assembly despite changes in the solution phase and showed little dependence at high TC upon the initial tubulin concentration. The theory was based on coupled Oosawa-Kasai equations that allow for the co-assembly of two components, tubulin and TC. An expression was derived that relates copolymer composition to reaction mixture composition and to the affinity of microtubule ends for tubulin and TC. This expression predicts copolymer composition at TC concentrations less than 10 microM and correlates composition with assembly inhibition. We perceive copolymerization as a facilitated incorporation of TC requiring the presence of tubulin. TC incorporation was dependent on the ratio of total tubulin to the dissociation constant for TC bound to microtubule ends. The copolymerization reaction is thus characterized by an interplay of two effects (a) where tubulin facilitates the incorporation of TC into the microtubule, and (b) where TC inhibits the assembly of tubulin into microtubules.