Search

In this paper we extend to enzyme systems the results previously obtained in paper I of this series for linear compartmental systems. We obtain the time course equations for both the enzyme and ligand species involved in the reaction mechanisms, which fit a general enzyme system model when the connections between the different enzyme species are of first or pseudofirst order. The kinetic equations obtained here for a given species, enzyme or ligand have the advantage over all previous equations described in the literature, in that they are in the most simplified form possible, since they only contain the kinetic parameters and initial concentrations of the enzymatic reaction which really have some influence on the time progress curves of the species under study. These kinetic equations are denominated optimized equation to distinguish them from the others, which shall call non-optimized equations. We discuss those cases when both types of equation coincide and we show how, when they do not coincide, the non-optimized equations can be simplified to the optimized ones. Therefore, we show that the optimized equations could be used in all cases to avoid the need of subsequent simplifications to eliminate the parameters that play no role in the corresponding time equations. To illustrate the use of this procedure we will apply it to two simple examples of enzymatic reactions.

Software application is implemented in this work to take full advantage of the characteristics of current operating systems and to provide the optimized symbolic kinetic equations for both enzyme and ligand species involved in enzyme reactions. This software, called SKEE-w2013, is implemented using C# language and runs under all operating systems from Windows XP up to Windows 8. It is applicable to any enzyme reaction mechanism that fits the general reaction scheme proposed previously by our group. It can be downloaded, free of charge, from http://oretano.iele-ab.uclm.es/~BioChem-mg/software.php . Besides the optimized equations, the software can provide non-optimized equations, so that the user can compare the advantage of using optimized equations rather than the non-optimized ones, whenever they do not coincide. Moreover, the software circumvents the limitations of other existing previous software tools implemented with what are nowadays obsolete programming languages and that, moreover, are limited to non-optimized kinetic equations.

The kinetic study carried out in paper I of this series (preceding article) on enzyme systems involving zymogen activation and the simultaneous action of two different, mutually exclusive inhibitors (Scheme 1) allows, new dimensionless kinetic parameters to be suggested. These parameters furnish quantitative information about the relative weight of the activation and inhibition routes, i.e. the conditions under which the activation prevails over the inhibition and vice versa, as well as the absolute and relative contributions to overall inhibition of each of the inhibition routes and their synergistic effect. These results can be easily and directly applied to any of the thousands of particular cases of the model. Examples are given for different particular cases.

In this paper, a general mechanism of zymogen activation and the simultaneous action of two different, mutually exclusive, two-step inhibitors acting on both the activating and the activated enzymes is proposed and kinetically analyzed. This generalization offers the advantage of being applicable to a high number of real cases, since most mechanisms of zymogen activation involving reversible or irreversible, one or two step, equal or different inhibitors are particular cases of the general model here studied. Analogously, from the equations corresponding to the general model or from any of their particular cases, it is possible to obtain the equations corresponding to cases in which one or more of the reversible steps are in rapid equilibrium. The number and type of the particular cases arising from the general model are obtained in a systematic way.