For Michaelis–Menten–Monod (MMM) 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 (QSS) approximation, that is, for ; under the QSS, the kinetic equations describing the MMM problem become where is the Michaelis–Menten constant (also known as the half-saturation concentration and affinity). 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. (4b) and (4d) are linearly dependent on Eqs. (4a) and (4c), which are the two differential equations that can be used to solve the MMM problem. An implicit analytic solution can be obtained if is chosen as the independent variable and , , 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. (4d). The analytic solution to Eq. (5b) 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. (6) as Substituting Eq. (7)