Kinetic analysis of biphasic protein modification reactions. Cooperative effects

A mathematical treatment of protein modification reactions is presented, and it is shown that in these cases protein modification is described by a summation of exponential functions of reaction time, the number of exponentials being equal to the number of modified protein species. It is shown that, in cases of protein modification cooperativity, there is a strict dependence of the coefficients of the multiexponential modification equation on the constants of the same equation. The conditions necessary for a reduction of a multiexponential protein modification equation to one of a summation of two exponentials only are examined. The possible formulae for the coefficients of a two-exponential-summation equation, used to describe the modification of protein models with two, three or four modifiable residues (as well as some aspects of models with five and six modifiable residues) per protein molecule are derived. It is seen that the number of such coefficients is severely limited. The most frequently obtained formula for the lower stoichiometric coefficient of a two-exponential-summation equation Is Aka/(ka-kb), where ka and kb are the constants of the two exponentials of the equation, and A is a constant.(ABSTRACT TRUNCATED AT 250 WORDS)