Michaelis–Menten–Monod kinetics

The kinetic equations describing the reactions above can be derived from the GEBIK equationsMaggi 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}} and are written as

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; =\; -k\_1[S][E]\; +\; k\_\{-1\}[C],$|{{EquationRef|3a}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[C]\}\{\backslash text\{d\}t\}\; =\; k\_1[S][E]\; -\; k\_\{-1\}[C]\; -\; k[C],$|{{EquationRef|3b}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[P]\}\{\backslash text\{d\}t\}\; =\; k[C],$|{{EquationRef|3c}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[B]\}\{\backslash text\{d\}t\}\; =\; -\; Y\; \backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; -\; \backslash mu\_B\; [B],$|{{EquationRef|3d}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[E]\}\{\backslash text\{d\}t\}\; =\; -\; zY\; \backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; -\; \backslash frac\{\backslash text\{d\}[C]\}\{\backslash text\{d\}t\}\; -\; \backslash mu\_E\; [E],$|{{EquationRef|3e}}}}

where $\backslash mu\_B$ is the biomass mortality rate and $\backslash mu\_E$ 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 $\backslash frac\{\backslash text\{d\}[C]\}\{\backslash text\{d\}t\}\; =\; 0$;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. under the QSS, the kinetic equations describing the MMM problem become

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; =\; -k\; [E]\backslash frac\{[S]\}\{K+[S]\}\; ,$|{{EquationRef|4a}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[P]\}\{\backslash text\{d\}t\}\; =\; -\backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\},$|{{EquationRef|4b}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[B]\}\{\backslash text\{d\}t\}\; =\; -\; Y\; \backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; -\; \backslash mu\_B\; [B],$|{{EquationRef|4c}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[E]\}\{\backslash text\{d\}t\}\; =\; -\; zY\; \backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; -\; \backslash mu\_E\; [E],$|{{EquationRef|4d}}}}

where $K\; =\; (k\_\{-1\}\; +k)/k\_1$ 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

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[P]\}\{\backslash text\{d\}t\}\; =\; k\; [E]\backslash frac\{[S]\}\{K+[S]\}\; ,$|{{EquationRef|4a}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[S]\}\{\backslash text\{d\}t\}\; =\; -\backslash frac\{\backslash text\{d\}[P]\}\{\backslash text\{d\}t\},$|{{EquationRef|4b}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[B]\}\{\backslash text\{d\}t\}\; =\; Y\; \backslash frac\{\backslash text\{d\}[P]\}\{\backslash text\{d\}t\}\; -\; \backslash mu\_B\; [B],$|{{EquationRef|4c}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}[E]\}\{\backslash text\{d\}t\}\; =\; -\; z\; \backslash frac\{\backslash text\{d\}[B]\}\{\backslash text\{d\}t\}\; ,$|{{EquationRef|4d}}}}

where $S$, $P$, $E$, $B$ are explicit function of time $t$. 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 solutionMaggi 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}} can be obtained if $P$ is chosen as the independent variable and $t(P)$, $S(P)$, $E(P)$ and $B(P$) are rewritten as functions of $P$ so to obtain

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}t(P)\}\{\backslash text\{d\}P\}\; =\; \backslash left(k\; z\; B(P)\backslash frac\{S\_0\; -P\}\{K\; +S\_0\; -P\}\; \backslash right)^\{-1\},$|{{EquationRef|5a}}}}

{{NumBlk|:|$\backslash frac\{\backslash text\{d\}B(P)\}\{\backslash text\{d\}P\}\; =\; Y\; -\; \backslash mu\_B\; B(P)\; \backslash frac\{\backslash text\{d\}t(P)\}\{\backslash text\{d\}P\},$|{{EquationRef|5b}}}}

where $S(t)$ has been substituted by $S(P)=\; S\_0\; -P$ as per mass balance $S\_0\; +P\_0\; =\; S+P$, with the initial value $S\_0\; =\; S(P)$ when $P=P\_0=0$, and where $E(t)$ has been substituted by $zB(P)$ as per the linear relation $E\; =\; z\; B$ expressed by Eq. (4d). The analytic solution to Eq. (5b) is

{{NumBlk|:|$B(P)\; =\; B\_0+\; \backslash left(Y\; -\backslash frac\{\backslash mu\_B\; \}\{k\; z\}\backslash right)\; P\; +\; \backslash frac\{\; \backslash mu\_BK\; \}\{k\; z\}\backslash ln\backslash left(\; 1-\backslash frac\{P\}\{S\_0\}\; \backslash right)\; ,$|{{EquationRef|6}}}}

with the initial biomass concentration $B\_0\; =\; B(P)$ when $P=0$. To avoid the solution of a transcendental function, a polynomial Taylor expansion to the second-order in $P$ is used for $B(P)$ in Eq. (6) as

{{NumBlk|:|$B(P)\; =\; B\_0\; +\; \backslash left(Y\; -\backslash frac\{\backslash mu\_B\; \}\{zk\}\; -\; \backslash frac\{\backslash mu\_B\; K\}\{zkS\_0\}\; \backslash right)P\; -\; \backslash frac\{\backslash mu\_B\; K\}\{2\; zk\; S\_0^2\; \}\; P^2\; +O(P^3)$|{{EquationRef|7}}}}

Substituting Eq. (7) into Eq. (5a} and solving for $t(P)$ with the initial value $t(P=0)\; =\; 0$, one obtains the implicit solution for $t(P)$ as

{{NumBlk|:|$$

t(P) = -\frac{F'}{2} \ln \left( \frac{ \sqrt{Q}+2HP+M }{ \sqrt{Q}-2HP-M } \right) - \frac{ K }{ 2N'}\ln\left(\frac{(S_0-P)^2}{HP^2 +MP+N }\right) + \frac{F'}{2} \ln\left( \frac{ \sqrt{Q}+M }{ \sqrt{Q} -M } \right) - \frac{ K }{ 2N'}\ln\left(\frac{S_0^2}{N }\right).

|{{EquationRef|8}}}}

with the constants

{{NumBlk|:||{{EquationRef|9a}}}}

{{NumBlk|:|$M\; =\; k\; zY\; -\backslash mu\_B\; -\; \backslash frac\{\backslash mu\_B\; K\}\{S\_0\},\; N\; =\; k\; z\; B\_0\; >\; 0,$|{{EquationRef|9b}}}}

{{NumBlk|:|$M\text{'}\; =\; -(M+2HS\_0),\; N\text{'}\; =\; N\; +\; M\; S\_0\; +\; H\; S\_0^2\; \backslash neq\; 0,$|{{EquationRef|9c}}}}

{{NumBlk|:|$-Q\; =4HN-M^2\; \backslash ne\; 0,$|{{EquationRef|9d}}}}

{{NumBlk|:|$F\; =\; \backslash frac\{2\}\{\backslash sqrt\{-Q\}\}\; -\; \backslash frac\{KM\text{'}\}\{N\text{'}\backslash sqrt\{-Q\}\},\; F\text{'}\; =\; \backslash frac\{2\}\{\backslash sqrt\{Q\}\}\; -\; \backslash frac\{KM\text{'}\}\{N\text{'}\backslash sqrt\{Q\}\},$|{{EquationRef|9e}}}}

For any chosen value of $P$, the biomass concentration can be calculated with Eq. (7) at a time $t(P)$ given by Eq. (8). The corresponding values of $S(P)\; =\; S\_0-P$ and $E(P)\; =\; zB(P)$ can be determined using the mass balances introduced above.

• Enzyme kinetics
• Michaelis–Menten kinetics
• Monod
• GEBIK equations

{{reflist}}

Category:Enzyme kinetics