Michaelis–Menten–Monod kinetics


For Michaelis–Menten–Monod kinetics it is intended the coupling of an enzyme-driven chemical reaction of the Michaelis–Menten type with the Monod growth of an organisms that performs the chemical reaction. The enzyme-driven reaction can be conceptualized as the binding of an enzyme E with the substrate S to form an intermediate complex C, which releases the reaction product P and the unchanged enzyme E. During the metabolic consumption of S, biomass B is produced, which synthesizes the enzyme, thus feeding back to the chemical reaction. The two processes can be expressed as
where and are the forward and backward equilibrium rate constants, is the reaction rate constant for product release, is the biomass yield coefficient, and is the enzyme yield coefficient.

Transient kinetics

The kinetic equations describing the reactions above can be derived from the GEBIK equations and are written as
where is the biomass mortality rate and is the enzyme degradation rate. These equations describe the full transient kinetics, but cannot be normally constrained to experiments because the complex C is difficult to measure and there is no clear consensus on whether it actually exists.

Quasi-steady-state kinetics

Equations 3 can be simplified by using the quasi-steady-state approximation, that is, for ; under the QSS, the kinetic equations describing the MMM problem become
where is the Michaelis–Menten constant.

Implicit analytic solution

If one hypothesizes that the enzyme is produced at a rate proportional to the biomass production and degrades at a rate proportional to the biomass mortality, then Eqs. 4 can be rewritten as
where,,, are explicit function of time. Note that Eq. and are linearly dependent on Eqs. and are rewritten as functions of so to obtain
where has been substituted by as per mass balance, with the initial value when, and where has been substituted by as per the linear relation expressed by Eq.. The analytic solution to Eq. is
with the initial biomass concentration when. To avoid the solution of a transcendental function, a polynomial Taylor expansion to the second-order in is used for in Eq. as
Substituting Eq. into Eq. (5a