The kinetic equations describing the reactions above can be derived from the GEBIK equations[3] and are written as
| | (3a) |
| | (3b) |
| | (3c) |
| | (3d) |
| | (3e) |
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.
Equations 3 can be simplified by using the quasi-steady-state (QSS) approximation, that is, for ;[4] under the QSS, the kinetic equations describing the MMM problem become
| | (4a) |
| | (4b) |
| | (4c) |
| | (4d) |
where is the Michaelis–Menten constant (also known as the half-saturation concentration and affinity).
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
| | (4a) |
| | (4b) |
| | (4c) |
| | (4d) |
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[5] can be obtained if is chosen as the independent variable and , , and ) are rewritten as functions of so to obtain
| | (5a) |
| | (5b) |
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
| | (6) |
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
| | (7) |
Substituting Eq. (7) into Eq. (5a} and solving for with the initial value , one obtains the implicit solution for as
| | (8) |
with the constants
| | (9a) |
| | (9b) |
| | (9c) |
| | (9d) |
| | (9e) |
For any chosen value of , the biomass concentration can be calculated with Eq. (7) at a time given by Eq. (8). The corresponding values of and can be determined using the mass balances introduced above.
Michaelis, L.; Menten, M. L. (1913). "Die Kinetik der Invertinwirkung". Biochem Z. 49: 333–369
Monod J. (1949) The growth of bacterial cultures. Annu. Rev. Microbial. 3, 371–394
Maggi F. and W. J. Riley, (2010), Mathematical treatment of isotopologue and isotopomer speciation and fractionation in biochemical kinetics, Geochim. Cosmochim. Acta, doi:10.1016/j.gca.2009.12.021 Briggs G.E.; Haldane, J.B.S., "A note on the kinetics of enzyme action", \textit{Biochem J.} \textbf{1925}, \textit{19(2)}, 338–339.
Maggi F. and La Cecilia D., (2016), "An implicit analytic solution of Michaelis–Menten–Monod kinetics", American Chemical Society, ACS Omega 2016, 1, 894−898, doi:10.1021/acsomega.6b00174