Kinetics

We modeled our enzyme reactions in propane pathway with Michaelis-Menten enzyme kinetics. It is widely used in this kind of modeling and assumes that the reaction enzyme catalyses is rapid compared to the enzyme and substrate joining together and leaving each other. The very basic equation for irreversible one substrate reaction is \[ \text{rate} = \frac{V_{max}[S]}{K_{cat}+[S]}, \] where \([S]\) is substrate concentration. \( V_{max} \) tells us the maximum speed of the enzyme and \( K_{cat} \) ... (how calculated from specific activity!). Only few of our reactions follow this very basic equation, and for the most of them we need to use multisubstrate reaction kinetics. (reference for the book?)

pic of our pathway here to make things more clear. Do we want pictures with highlited enzymes in every subcategory?

AtoB

2\(\cdot\)Acetyl-CoA \(\rightarrow\) Acetoacetyl-CoA + CoA

Vf/Vr: 22,3. Source: Molecular and catalytic properties of the acetoacetyl-coenzyme A thiolase of Escherichia coli; Archives of Biochemistry and Biophysics Volume 176, Issue 1, September 1976, Pages 159–170

\[ \frac{K_{cat}^{AtoB} \cdot [AtoB] \cdot [Acetyl\text{-}CoA]^2}{[Acetyl\text{-}CoA]^2+2\cdot K_{M}^{AtoB:Acetyl\text{-}CoA}\cdot [Acetyl\text{-}CoA]} \]

FadB2

Acetoacetyl-CoA + NADPH + H\(^+\) \(\rightarrow\) 3-Hydroxybutyryl-CoA + NADP\(^+\)

\[ \frac{[Acetoacetyl\text{-}CoA]\cdot [NADPH]-\frac{[3\text{-}hydroxybutyryl\text{-}CoA]\cdot [NADP^+]}{K_{eq}}} {\frac{K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot K_{M}^{FadB2:NADPH}}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{K_{M}^{FadB2:NADPH}\cdot [Acetoacetyl\text{-}CoA]}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{ K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot [NADPH]}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot [NADP^+]}{K_{eq}\cdot K_{cat2}^{FadB2}\cdot [FadB2]}+} \] \[ \cdots \frac{}{+\frac{K_{M}^{FadB2:NADP^+}\cdot [3\text{-}hydroxybutyryl\text{-}CoA]}{K_{eq}\cdot K_{cat2}^{FadB2}\cdot [FadB2]}+\frac{[Acetoacetyl\text{-}CoA]\cdot [NADPH]}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{[NADP^+]\cdot [3\text{-}hydroxybutyryl\text{-}CoA]}{K_{eq}\cdot K_{cat2}^{FadB2}\cdot [FadB2]}}\]

Hdb

Acetoacetyl-CoA + NADPH + H\(^+\) \(\rightarrow\) 3-Hydroxybutyryl-CoA + NADP\(^+\)

Species is Clostridium acetobutylicum, but Clostridium Kluyveri ought to be close enough for comparison.

The specific activity of the 3-hydroxybutyryl-CoA dehydrogenase (forward) as measured in the direction of acetoacetyl-CoA reduction was 478.6 U/mg protein. The rate of the oxidation reaction (reverse) proceeded with 36.6 U / mg protein. Source: Purification and Properties of NADP-Dependent L( +)-3-Hydroxybutyryl-CoA Dehydrogenase from Clostridiurn kluyveri; Eur. J. Biochem. 32,51-56 (1973)

\[ \frac{K_{cat}^{Hdb}\cdot [Hbd] \cdot [Acetoacetyl\text{-}CoA]\cdot [NADPH]}{[Acetoacetyl\text{-}CoA]\cdot [NADPH] + K_{M}^{Hdb:NADPH}\cdot [Acetoacetyl\text{-}CoA]+K_{M}^{Hdb:Acetoacetyl\text{-}CoA}\cdot [NADPH]} \]

Crt

3-hydroxybutyryl-CoA \(\rightarrow\) Crotonyl-CoA + H\( _2\)O

\[ \frac{K_{cat}^{Crt}\cdot [Crt]\cdot [3\text{-}hydroxybutyryl\text{-}CoA]}{K_{M}^{Crt:3\text{-}Hydroxybutyryl\text{-}CoA} +[3\text{-}hydroxybutyryl\text{-}CoA]} \]

Ter

Crotonyl-CoA + NADH + H\( ^+\) \(\rightarrow\) Butyryl-CoA + NAD\( ^+\)

an ordered bi-bi reaction mechanism with NADH binding first. Source: Biochemical and Structural Characterization of the trans-Enoyl-CoA Reductase from Treponema denticola; Biochemistry 2012, 51, 6827−6837

\[ \frac{K_{cat}^{Ter}\cdot [Ter] \cdot [Crotonyl\text{-}CoA]\cdot [NADH]}{[Crotonyl\text{-}CoA]\cdot [NADH] + K_{M}^{Ter:NADH}\cdot [Crotonyl\text{-}CoA]+K_{M}^{Ter:Crotonyl\text{-}CoA}\cdot [NADH] + K_{I}^{Ter:Butyryl\text{-}CoA}\cdot K_{M}^{Ter:NADH}} \]

YciA

Butyryl-CoA + H\( _2\)O \(\rightarrow\) Butyrate + CoA

\[ \frac{K_{cat}^{YciA}\cdot [YciA]\cdot [Butyryl\text{-}CoA]}{K_{M}^{YciA:Butyryl\text{-}CoA} +[Butyryl\text{-}CoA]} \]

Car

Butyrate + NADPH + ATP \(\rightarrow\) Butyraldehyde + NADP\(^+\) + AMP + 2P\(_i\)

\[\frac{K_{cat}^{Car}\cdot [Car]\cdot [Butyrate]\cdot [NADPH]\cdot [ATP]}{K_{M}^{Car:Butyrate}\cdot K_{M}^{Car:NADPH}\cdot [ATP]+K_{M}^{Car:ATP}\cdot [Butyrate]\cdot [NADPH]+K_{M}^{Car:NADPH}\cdot [Butyrate]\cdot [ATP]}\]\[\cdots \frac{}{+K_{M}^{Car:Butyrate}\cdot [NADPH]\cdot [ATP]+ [Butyrate]\cdot [NADPH]\cdot [ATP]}\]

Sfp

Ado

\[ \frac{K_{cat}^{Ado}\cdot [Ado]\cdot [Butyrate]}{K_{M}^{Ado:Butyrate} +[Butyrate]} \]

Other Constants

This is a table of already known typical concentrations in a cell that we use in our model.