Kinetics

We modeled our enzyme reactions in the propane pathway with Michaelis-Menten enzyme kinetics. It is widely used in metabolical modeling of enzymes. Michaelis-Menten kinetics assumes that the reaction an 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 and $$V_{max}$$ tells us the maximum speed of the enzyme. $$K_{M}$$ 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. $$K_{cat}$$ is the turnover number (unit: $$\tfrac{1}{min}$$ ), which describes the speed at which an enzyme processes the substrate to a 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 [1].

We understand and accept the fact that the kinetic data we have used in our model is very rough, due to varying measurement conditions and the fact that the measurements have been done in vitro, whereas our system functions in vivo. However, we believe our model can give us more reliable information about the bottlenecks of the pathway than mere educated guesses.

AtoB

2$$\cdot$$Acetyl-CoA $$\rightarrow$$ Acetoacetyl-CoA + CoA

AtoB (acetyl-CoA C-acetyltransferase) 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: [2]) 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:

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

To note

$$K_{cat}^{AtoB}$$

10653 1/min

[3]

Forward reaction

$$K_{M}^{AtoB:Acetyl\text{-}CoA}$$

0.00047 mol/l

[2]

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

FadB2 (3-hydroxybutyryl-CoA dehydrogenase) 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 [1] 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.

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

To note

$$K_{cat1}^{FadB2}$$

0.677 1/min

[4]

Forward reaction

$$K_{cat2}^{FadB2}$$

0.723 1/min

[4]

Reverse reaction

$$K_{M}^{FadB2:Acetoacetyl\text{-}CoA}$$

65.6 mmol/l

[4]

Forward reaction

$$K_{M}^{FadB2:NADPH}$$

50 mmol/l

[4]

Forward reaction

$$K_{M}^{FadB2:3\text{-}Hydroxybutyryl\text{-}CoA}$$

43.5 mmol/l

[4]

Reverse reaction

$$K_{M}^{FadB2:NADP^+}$$

29.5 mmol/l

[4]

Reverse reaction

Hbd

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

The enzyme used in the propane pathway, Hbd (3-hydroxybutyryl-CoA dehydrogenase), is from Clostridium acetobutylicum, but only values to be found were for Clostridium Kluyveri. However, since the species are very close relatives, we can assume the values to be close enough for comparison.

The reaction is reversible, but according to [5], the specific activity of 3-hydroxybutyryl-CoA dehydrogenase (forward) as measured in the direction of acetoacetyl-CoA reduction is 478.6 U/mg protein. The rate of the oxidation reaction (reverse) proceeded with 36.6 U / mg protein. Because of the disparity between these rates we 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. Based on these pieces of information we can assume that the reaction is either random or ordered Bi Bi -reaction so the rate equation is as follows.

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

To note

$$K_{cat}^{Hbd}$$

336.4 1/min

[5]

Forward reaction, Clostridium Kluyveri

$$K_{M}^{Hbd:Acetoacetyl\text{-}CoA}$$

5e-5 mol/l

[5]

Clostridium Kluyveri

$$K_{M}^{Hbd:NADPH}$$

7e-5 mol/l

[5]

Clostridium Kluyveri

Crt

3-hydroxybutyryl-CoA $$\rightarrow$$ Crotonyl-CoA + H$$_2$$O

Crt (3-hydroxybutyryl-CoA dehydratase) is found from Clostridium acetobutylicum. Since there is only one substrate in the reaction, we can form the rate equation from basic Michaelis-Menten kinetic model. We assumed the reaction to be irreversible since the enzyme is quite efficient.

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

To note

$$K_{cat}^{Crt}$$

1279.8 1/min

[6]

Forward reaction

$$K_{M}^{Crt:3\text{-}Hydroxybutyryl\text{-}CoA}$$

3e-5 mol/l

[6]

Ter

Crotonyl-CoA + NADH + H$$^+$$ $$\rightarrow$$ Butyryl-CoA + NAD$$^+$$

Ter (trans-2-enoyl-CoA reductase) is from Treponema denticola. Its reaction without H$$^+$$ is an ordered bi-bi reaction mechanism with NADH binding ﬁrst [7]. Since we found no references for the reaction to be reversible, we modeled it as irreversible.

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

To note

$$K_{cat}^{Ter}$$

5460 1/min

[7]

Forward reaction

$$K_{M}^{Ter:Crotonyl\text{-}CoA}$$

70 µmol/l

[7]

$$K_{M}^{Ter:NADH}$$

5.2e-06 mol/l

[7]

$$K_{I}^{Ter:Butyryl\text{-}CoA}$$

1.98e-07 mol/l

[7]

YciA

Butyryl-CoA + H$$_2$$O $$\rightarrow$$ Butyrate + CoA

YciA (acyl-CoA thioester hydrolase) is found in Haemophilus influenzae. When searching for information about this enzyme no references for it being reversible were found. Because of this we modeled it as irreversible. 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.

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

To note

$$K_{cat}^{YciA}$$

1320 1/min

[8]

Forward reaction

$$K_{M}^{YciA:Butyryl\text{-}CoA}$$

3.5e-06 mol/l

[8]

CAR

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

CAR-enzyme (carboxylic acid reductase) 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. Thus, the rate equation will be

$\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]}$

To note

$$K_{cat}^{CAR}$$

150 1/min

[9]

Forward reaction, calculated from a plot

$$K_{M}^{CAR:Butyrate}$$

0.013 mol/l

[9]

Calculated from a plot

$$K_{M}^{CAR:NADPH}$$

4.8e-05 mol/l

[9]

$$K_{M}^{CAR:ATP}$$

0.000115 mol/l

[9]

Sfp

Sfp (4'-phosphopantetheinyl transferase) does not directly affect to the intermediates in our pathway, but instead acts as an activating enzyme for CAR. We have modeled the reactions concerning Sfp here.

Aldehyde deformylating oxygenase is the final enzyme in the propane pathway, turning butyraldehyde into propane. We are using an 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/flavodoxin-oxidoreductase (Fpr).

Using an A134F mutant and a ferredoxin reducing system including Fpr improves 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't make better assumptions.

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

To note

$$K_{cat}^{ADO}$$

0.03 1/min

[10]

Forward reaction

$$K_{M}^{ADO:Butyraldehyde}$$

0.0101 mol/l

[10]

Other Constants

The following table provides information about typical concentrations in a cell that we use in our model.

To note

[Acetyl-CoA]

0.00061 mol/l

[11]

glucose-fed, exponentially growing E. coli

[Acetoacetyl-CoA]

2.2e-05 mol/l

[11]

glucose-fed, exponentially growing E. coli

[CoA]

0.00014 mol/l

[11]

glucose-fed, exponentially growing E. coli

0.00012 mol/l

[11]

glucose-fed, exponentially growing E. coli

[NADP$$^+$$]

2.1e-06 mol/l

[11]

glucose-fed, exponentially growing E. coli

8.3e-05 mol/l

[11]

glucose-fed, exponentially growing E. coli

[NAD$$^+$$]

0.0026 mol/l

[11]

glucose-fed, exponentially growing E. coli

[ATP]

0.0096 mol/l

[11]

glucose-fed, exponentially growing E. coli

[AMP]

0.00028 mol/l

[11]

glucose-fed, exponentially growing E. coli

[H$$_2$$O]

38.85 mol/l

Concentration of water in water is $$\frac{\frac{m}{V}}{M}$$. E.coli is about 70% water. Thus, the water concentration in E.coli is $$70\% \cdot \frac{1000 \frac{g}{l}}{18.01 g/mol} = 38.85 \frac{mol}{l}$$

Sources

[1] Enzyme Kinetics: Principals and Methods by Hans Bisswanger (2002)

[2] 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

[3] Thiolases of Escherichia coli: purification and chain length specificities Feigenbaum, J.; Schulz, H.; Journal of Bacteriology, Volume 122, Issue 2, May 1975, Pages 407-411

[4] Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982

[5] Purification and Properties of NADP-Dependent L(+)-3-Hydroxybutyryl -CoA Dehydrogenase from Clostridium kluyveri; Eur. J. Biochem. 32,51-56 (1973)

[6] Purification and Characterization of Crotonase from Clostridium acetobutylicum; The journal of Biological Chemistry, Volume 247, Number 16, August 1972, Pages 5266-5271

[7] Biochemical and Structural Characterization of the trans-Enoyl-CoA Reductase from Treponema denticola; Biochemistry 2012, 51, 6827−6837

[8] Divergence of Function in the Hot Dog Fold Enzyme Superfamily: The Bacterial Thioesterase YciA; Biochemistry 2008, 47, 2789–2796

[9] 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

[10] Production of Propane and Other Short-Chain Alkanes by Structure-Based Engineering of Ligand Specificity in Aldehyde-Deformylating Oxygenase, Khara et al (2013)

[11] Absolute Metabolite Concentrations and Implied Enzyme Active Site Occupancy in Escherichia coli, Bennett et al, 2009