Michaelis–Menten–Monod kinetics
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 formula_1 and formula_2 are the forward and backward equilibrium rate constants, formula_3 is the reaction rate constant for product release, formula_4 is the biomass yield coefficient, and formula_5 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 formula_6 is the biomass mortality rate and formula_7 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 formula_8; under the QSS, the kinetic equations describing the MMM problem become where formula_9 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 formula_10, formula_11, formula_12, formula_13 are explicit function of time formula_14. 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 formula_11 is chosen as the independent variable and formula_16, formula_17, formula_18 and formula_19) are rewritten as functions of formula_11 so to obtain where formula_21 has been substituted by formula_22 as per mass balance formula_23, with the initial value formula_24 when formula_25, and where formula_26 has been substituted by formula_27 as per the linear relation formula_28 expressed by Eq. (4d). The analytic solution to Eq. (5b ) is with the initial biomass concentration formula_29 when formula_30. To avoid the solution of a transcendental function, a polynomial Taylor expansion to the second-order in formula_11 is used for formula_32 in Eq. (6) as Substituting Eq. (7) into Eq. (5a} and solving for formula_16 with the initial value formula_34, one obtains the implicit solution for formula_16 as with the constants For any chosen value of formula_11, the biomass concentration can be calculated with Eq. (7) at a time formula_16 given by Eq. (8). The corresponding values of formula_38 and formula_39 can be determined using the mass balances introduced above