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 \[ \text{rate} = \frac{V_{max}[S]}{K_{M}+[S]}, \] where \([S]\) is substrate concentration. \( V_{max} \) tells us the maximum speed of the enzyme and \( K_{M} \) is the substrate concentration at which the reaction rate is half of \( V_{max} \). Usually we need to calculate \( V_{max} \) by \( K_{cat}\cdot [E] \) where \([E]\) is enzyme concentration and \( K_{cat} \) is specific activity (unit: \( \tfrac{\mu mol}{min \cdot mg} \) ). Only few of our reactions follow this very basic equation, and for the most of them we need to use multisubstrate reaction kinetics. For more information, see for example Enzyme Kinetics: Principals and Methods by Hans Bisswanger (2002).

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

AtoB is found from Escherichia Coli. The reaction shown above is reversible, but since the ratio of forward and reversible reaction favores strongly the forward one (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) we can approximate is as irreversible.

Based on this article, we know that the reaction follows Ping Pong Bi Bi -model and so get the following rate equation.

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

Constant	Value	Source	To note
\( K_{cat}^{AtoB} \)	10653 1/min	needs checking	Forward reaction
\( K_{M}^{AtoB:Acetyl\text{-}CoA} \)	0.00047 mol/l	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	Is there something special about this?

FadB2

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

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

Constant	Value	Source	To note
\( K_{cat1}^{FadB2} \)	0.677 1/min	needs to be checked	Forward reaction
\( K_{cat2}^{FadB2} \)	0.723 1/min	needs to be checked	Reverse reaction
\( K_{M}^{FadB2:Acetoacetyl\text{-}CoA} \)	65.6 mol/l	needs to be checked	Forward reaction
\( K_{M}^{FadB2:NADPH} \)	50 mol/l	needs to be checked	Forward reaction
\( K_{M}^{FadB2:3\text{-}Hydroxybutyryl\text{-}CoA} \)	43.5 mol/l	needs to be checked	Reverse reaction
\( K_{M}^{FadB2:NADP^+} \)	29.5 mol/l	needs to be checked	Reverse reaction

Hdb

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

The enzyme is from Clostridium acetobutylicum, but only values to be found were for Clostridium Kluyveri. This is not a problem however since the species are very close relatives and so the values ought to be close enough for comparison.

The reaction is reversible, but according to Purification and Properties of NADP-Dependent L( +)-3-Hydroxybutyryl-CoA Dehydrogenase from Clostridiurn kluyveri; Eur. J. Biochem. 32,51-56 (1973), the specific activity of the 3-hydroxybutyryl-CoA dehydrogenase (forward) as measured in the direction of acetoacetyl-CoA reduction is 478.6 U/mg protein and the rate of the oxidation reaction (reverse) proceeded with 36.6 U / mg protein so we can again approximate the reaction as irreversible.

We don’t consider how \(H^+\) affects the reaction which is justified by knowing that its concentration in the cell should always be quite constant; otherwise the cell will die. This is why we can assume that the reaction is either random or ordered Bi Bi -reaction and so the rate equation is as follows.

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

Constant	Value	Source	To note
\( K_{cat}^{Hdb} \)	336.4 1/min	Purification and Properties of NADP-Dependent L( +)-3-Hydroxybutyryl-CoA Dehydrogenase from Clostridiurn kluyveri; Eur. J. Biochem. 32,51-56 (1973)	Forward reaction, Clostridium Kluyveri
\( K_{M}^{Hdb:Acetoacetyl\text{-}CoA} \)	5e-5 mol/l	Purification and Properties of NADP-Dependent L( +)-3-Hydroxybutyryl-CoA Dehydrogenase from Clostridiurn kluyveri; Eur. J. Biochem. 32,51-56 (1973)	Clostridium Kluyveri
\( K_{M}^{Hdb:NADPH} \)	7e-5 mol/l	Purification and Properties of NADP-Dependent L( +)-3-Hydroxybutyryl-CoA Dehydrogenase from Clostridiurn kluyveri; Eur. J. Biochem. 32,51-56 (1973)	Clostridium Kluyveri

Crt

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

Crt is found from Clostridium acetobutylicum. Since there is only one substrate in the reaction, we can form the rate equation by basic Michaelis-Menten. We have assumed really reversible reaction as irreversible, because of ...

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

Constant	Value	Source	To note
\( K_{cat}^{Crt} \)	1310.8 1/min	needs checking	Forward reaction
\( K_{M}^{Crt:3\text{-}Hydroxybutyryl\text{-}CoA} \)	3e-5 mol/l	needs checking	Is there something special about this?

Ter

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

Ter is from Treponema denticola. Its reaction without H\( ^+\) is 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). Irreversibility we can justify .....

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

Constant	Value	Source	To note
\( K_{cat}^{Ter} \)	1881.6 1/min	needs checking	Forward reaction
\( K_{M}^{Ter:Crotonyl\text{-}CoA} \)	2.7e-06 mol/l	70 µmol/l Biochemical and Structural Characterization of the trans-Enoyl-CoA Reductase from Treponema denticola; Biochemistry 2012, 51, 6827−6837	Is there something special about this?
\( K_{M}^{Ter:NADH} \)	5.2e-06 mol/l	Biochemical and Structural Characterization of the trans-Enoyl-CoA Reductase from Treponema denticola; Biochemistry 2012, 51, 6827−6837	Is there something special about this?
\( K_{I}^{Ter:Butyryl\text{-}CoA} \)	1.98e-07 mol/l	needs checking	Is there something special about this?

YciA

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

YciA is found from Haemophilus influenzae. Irreversibility assumption because no mention about any other case? We know that there is abundance of water in the cell, so when considering rate equation we can safely assume that it doesn't have much effect to it. This is why we can again use the basic Michaelis-Menten rate equation.

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

Constant	Value	Source	To note
\( K_{cat}^{YciA} \)	1320 1/min	Divergence of Function in the Hot Dog Fold Enzyme Superfamily: The Bacterial Thioesterase YciA; Biochemistry 2008, 47, 2789–2796	Forward reaction
\( K_{M}^{YciA:Butyryl\text{-}CoA} \)	3.5e-06 mol/l	Divergence of Function in the Hot Dog Fold Enzyme Superfamily: The Bacterial Thioesterase YciA; Biochemistry 2008, 47, 2789–2796	Is there something special about this?

Car

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

Car-enzyme is originally from Mycobacterium marinum. We assumed that this reaction is irreversible, which justified because we have ATP in the reactants so we know that the possible reverse reaction can’t be very efficient. For the same reasons as mentioned before, we didn’t consider \(H^+\) in equations. We know that the reaction follows Bi Uni Uni Bi Ping Pong mechanism so the rate equation is

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

Constant	Value	Source	To note
\( K_{cat}^{Car} \)	150 1/min	Carboxylic acid reductase is a versatile enzyme for the conversion of fatty acids into fuels and chemical commodities; PNAS | January 2, 2013 | vol. 110 | no. 1 | 87–92	Forward reaction, calculated from a plot
\( K_{M}^{Car:Butyrate} \)	0.013 mol/l	Carboxylic acid reductase is a versatile enzyme for the conversion of fatty acids into fuels and chemical commodities; PNAS | January 2, 2013 | vol. 110 | no. 1 | 87–92	Calculated from a plot
\( K_{M}^{Car:NADPH} \)	4.8e-05 mol/l	needs checking	Is there something special about this?
\( K_{M}^{Car:ATP} \)	0.000115 mol/l	needs checking	Is there something special about this?

Sfp

Ado

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

Constant	Value	Source	To note
\( K_{cat}^{Ado} \)	0.215 1/min	needs checking	Forward reaction
\( K_{M}^{Ado:Butyraldehyde} \)	0.0101 mol/l	needs checking	Is there something special about this?

Other Constants

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

Constant	Value	Source	To note
[Acetyl-CoA]	0.00061 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[Acetoacetyl-CoA]	2.2e-05 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[CoA]	0.00014 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[NADPH]	00012 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[NADP\( ^+\)]	2.1e-06 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[NADH]	8.3e-05 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[NAD\( ^+\)]	0.0026 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[ATP]	0.0096 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[AMP]	0.00028 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli
[H\( _2\)O]	38.85 mol/l	Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009	glucose-fed, exponentially growing E. coli