Kinetics

We modeled our enzyme reactions in propane pathway with Michaelis-Menten enzyme kinetics. It is widely used in metabolical modeling of enzymes and assumes that the reaction enzyme catalyses is rapid compared to the enzyme and substrate joining together and leaving each other. The archetypical Michaelis-Menten equation for a reaction with one substrate and one product, i.e. \(S \rightarrow P; E \) is \[ \frac{d[P]}{dt} = \frac{V_{max}[S]}{K_{M}+[S]}, \] 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 \( V_{max} \), also called the Michaelis constant. Usually we need to calculate \( V_{max} \) by \( K_{cat}\cdot [E] \) where \([E]\) is enzyme concentration and \( K_{cat} \) is the turnover number (unit: \( \tfrac{1}{min} \) ), which describes the speed at which an enzyme processes substrate to product. 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 highlighted enzymes in every subcategory?

AtoB

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

AtoB is native to 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 we get the following rate equation:

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

FadB2

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

FadB2 is found from Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv). The reaction it catalyzes is reversible and we have assumed it to follow random bi bi reaction model.

The equilibrium constant \(K_{eq}\) in reversible random bi bi model is from Haldane relationship \[ K_{eq} = \frac{V_1\cdot K_{M}^{FadB2:3\text{-}Hydroxybutyryl\text{-}CoA}\cdot K_{M}^{FadB2:NADP^+}}{V_2\cdot K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot K_{M}^{FadB2:NADPH}}.\] See Enzyme Kinetics: Principals and Methods by Hans Bisswanger (2002) for reference. We have not taken H\(^+\) concentration into account in this calculation which is justified because it needs to be fairly constant in the cell or otherwise the cell will die off. This yields us the following as our reaction rate equation.

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

Hbd

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 Clostridium 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.

This needs some clarification, right now it is quite fast in its conclusions.

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

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 ...

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

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 .....

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

YciA

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

YciA is found in 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.

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

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 is 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 can be modeled using Bi Uni Uni Bi Ping Pong mechanism. Then the rate equation becomes

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

Sfp

Sfp does not directly process the substrates in our pathway, but instead acts as an activating enzyme for Car. We have modeled the reactions concerning Sfp here.

Ado

We need to write here why we model these so crudely...

Aldehyde deformylating oxygenase is the final enzyme in the propane pathway, turning butyraldehyde into propane. We are using and ADO mutant (A134F) that has an increased activity towards short-chained aldehydes, such as butyraldehyde. Furthermore, we are enhancing the electron supply to ADO by overexpressing its presumed natural electron acceptor/donor ferredoxin. To reduce ferredoxin under aerobic conditions, we co-express NADPH/ferredoxin/flawodoxin-oxidoreductase(Fpr).

Using A134F mutant and a ferredoxin reducing system including Fpr improve propane production. Combining all these improvements is challenging from the modeling point of view, as there are no kinetic parameters available for the reaction where both the ADO A134F mutant and a ferredoxin reducing system are used. As no sufficient data is available, we cannot model the ADO reaction like we have modeled the other reactions in the propane pathway.

We know that the wild-type ADO together with PMS/NADH reducing system has kcat value 0.0031±0.0001 min−1 and Km value 10.1±0.9 mM for the reaction from butyraldehyde to propane. A134F mutant has been shown to be more efficient than wild-type ADO and ferredoxin reducing system more efficient for ADO than a PMS/NADH reducing system. Therefore we can rather safely assume 10.1±0.9 mM to be the maximum Km possible and 0.0031±0.0001 min−1 to be the minimum kcat possible for estimating ADO reaction kinetics in our system.

Since we could not model the reactions that govern ADO's function, we approximated these reactions by simplifying the enzyme kinetics that govern ADO to the simplest case of Miclaelis-Menten kinetics. While this is not ideal, with current data and within these time limitations we can not do better.

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

Other Constants

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