Analysis of the fractional modification of the monocyclic enzyme cascades, defined in an alternative way involving the two forms of the modified protein

This paper presents the discussion of a more complete definition of the fractional modification of an monocyclic enzyme cascade suggested, but not discussed, by Varon et al. (Bull Math Biol 68(7):1461-1493, 2006) as the quotient of the sum of all forms of the modified protein, i.e. the free one and the intermediate complex converter enzyme of the original protein-modified protein, between the initial concentration of the target protein. From this general equation, obtained under three assumptions necessary to linearize the set of differential equations describing the kinetic of the system, we derive, as particular cases, other simpler expressions, by applying additional simplifying assumptions, which are, therefore, of a smaller range of validity. We discuss the relationships between the kinetic parameters and concentrations needed for the fulfillment of both the necessary and unnecessary assumptions. The goodness of the analysis was tested by using the shape in the steady state of the simulated time progress curves obtained by numerical integration. Seven arbitrarily chosen examples differing in one or more of the values of the concentrations and/or kinetic parameters have been used to support the results.