Kinetics of protein modification reactions: analysis of modification-induced protein unfolding

A mathematical treatment of a two-sited, modification-induced protein unfolding model is presented, and it is shown that the dependence of the concentration of modified protein groups on reaction time is described by a linear, second-order, differential equation with nonzero right hand side. The analytic solution of this equation consists of a summation of exponential functions of reaction time. By assigning arbitrary values to the modification and isomerization rate constants of these equations, simulated cases of protein modification are presented, and the apparent end-point of the reaction is determined graphically. It is found that the apparent end-point of the reaction is, in most cases studied, different from the true value of two groups modified per protein molecule, and is a function of both the modification, and isomerization rate constants of the model. The first derivative of the protein modification reaction, at the start of the reaction, [E]'mod (0), is determined, for the same simulated cases of protein modification, by two different analytical methods. It is found that the [E]'mod(0) value, obtained from graphical and numerical analysis data, is in most cases in good agreement with the value expected from first principles. Finally, the different irreversible enzyme inhibition forms, contingent upon the different kinds of the enzyme inactivation-protein modification relationships of the protein modification model under consideration, are presented and discussed.