Kinetic analysis of biphasic protein modification reactions cooperative effects

A mathematical treatment of protein modification reactions is presented, and it is shown thai 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 stoichiomelric coefficient of a 'wo-exponential-summation equation is Ak_a/(k_a-k_b). where k_b and k_b are the constants of the two exponentials of the equation, and A is a constant. The value most frequently arrived at for A is (n?1)/n, where n is the number of modifiable residues per protein molecule, while values such as 1/n, or a/n (where a is an integer, and also where a < n) are also possible. In most of the cooperative protein modification models worked out, k_a is identical with k_n, viz., k_a is identical with the rate constant for the first stoichiometric protein modification.