Michaelis-Menten¶

Module: multiphasic

Category: materialprop

Type string: "Michaelis-Menten"

Parameters¶

Name	Description	Default	Units
`override_vbar`	override_vbar	false	[]
`Vbar`	Vbar	0	[]
`Km`	Km	0	[]
`c0`	c0	0	[]
`vR`			[]
`vP`			[]
`forward_rate`			[]

Description¶

The material type for the Michaelis-Menten reaction is Michaelis-Menten.

The Michaelis-Menten reaction may be used to model enzyme kinetics where the enzyme $\mathcal{E}^{e} $triggers the conversion of the substrate $\mathcal{E}^{s}$ into the product $\mathcal{E}^{p}$. The product molar supply is given by

\[ \hat{c}^{p}=\begin{cases} \frac{V_{\max}c^{s}}{K_{m}+c^{s}} & c^{s}\geqslant c_{0}\\ 0 & c^{s}<c_{0} \end{cases}\,, \]

where $c^{s}$ is the substrate concentration. The default value of $c_{0}$ is $0$. This relation may be derived, with some simplifying assumptions, by applying the law of mass action to the combination of two reactions,

\[ \mathcal{E}^{e}+\mathcal{E}^{s}\rightleftharpoons\mathcal{E}^{es}\to\mathcal{E}^{e}+\mathcal{E}^{p} \]

Since the enzyme is not modeled explicitly in the Michaelis-Menten approximation to these two reactions, this reaction model is effectively equivalent to

\[ \mathcal{E}^{s}\to\mathcal{E}^{p}. \]

Therefore, this reaction accepts only one reactant parameter vR and one product parameter vP. For consistency with the formulation of this reaction, the stoichiometric coefficients should be set to $\nu_{R}^{s}=\nu_{P}^{p}=1$, so that $\hat{c}^{p}=\hat{\zeta}$.

Example:

<reaction name="enzyme kinetics" type="Michaelis-Menten">
  <Vbar>0</Vbar>
  <vR sol="1">1</vR>
  <vP sol="2">1</vP>
  <forward_rate type="constant">
    <k>1.0</k>
  </forward_rate>
  <Km>5.0</Km>
</reaction>