Team:Aalto-Helsinki/Kinetics
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 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 highlited 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:
\[ \frac{K_{cat}^{AtoB} \cdot [AtoB] \cdot [Acetyl\text{-}CoA]^2}{[Acetyl\text{-}CoA]^2+2\cdot K_{M}^{AtoB:Acetyl\text{-}CoA}\cdot [Acetyl\text{-}CoA]} \]
Constant |
Value |
Source |
To note |
\( K_{cat}^{AtoB} \) |
10653 1/min |
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 |
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\(^+\)
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 we need to have that in the cell or otherwise it will die off. This yields us the following as our reaction rate equation.
\[ \frac{[Acetoacetyl\text{-}CoA]\cdot [NADPH]-\frac{[3\text{-}hydroxybutyryl\text{-}CoA]\cdot [NADP^+]}{K_{eq}}} {\frac{K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot K_{M}^{FadB2:NADPH}}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{K_{M}^{FadB2:NADPH}\cdot [Acetoacetyl\text{-}CoA]}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{ K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot [NADPH]}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{K_{M}^{FadB2:Acetoacetyl\text{-}CoA}\cdot [NADP^+]}{K_{eq}\cdot K_{cat2}^{FadB2}\cdot [FadB2]}+} \] \[ \cdots \frac{}{+\frac{K_{M}^{FadB2:NADP^+}\cdot [3\text{-}hydroxybutyryl\text{-}CoA]}{K_{eq}\cdot K_{cat2}^{FadB2}\cdot [FadB2]}+\frac{[Acetoacetyl\text{-}CoA]\cdot [NADPH]}{K_{cat1}^{FadB2}\cdot [FadB2]}+\frac{[NADP^+]\cdot [3\text{-}hydroxybutyryl\text{-}CoA]}{K_{eq}\cdot K_{cat2}^{FadB2}\cdot [FadB2]}}\]
Constant |
Value |
Source |
To note |
\( K_{cat1}^{FadB2} \) |
0.677 1/min |
Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982 |
Forward reaction |
\( K_{cat2}^{FadB2} \) |
0.723 1/min |
Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982 |
Reverse reaction |
\( K_{M}^{FadB2:Acetoacetyl\text{-}CoA} \) |
65.6 mmol/l |
Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982 |
Forward reaction |
\( K_{M}^{FadB2:NADPH} \) |
50 mmol/l |
Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982 |
Forward reaction |
\( K_{M}^{FadB2:3\text{-}Hydroxybutyryl\text{-}CoA} \) |
43.5 mmol/l |
Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982 |
Reverse reaction |
\( K_{M}^{FadB2:NADP^+} \) |
29.5 mmol/l |
Characterization of a b-hydroxybutyryl-CoA dehydrogenase from Mycobacterium tuberculosis; Microbiology,Volume 156, July 2010, Pages 1975-1982 |
Reverse reaction |
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.
\[ \frac{K_{cat}^{Hbd}\cdot [Hbd] \cdot [Acetoacetyl\text{-}CoA]\cdot [NADPH]}{[Acetoacetyl\text{-}CoA]\cdot [NADPH] + K_{M}^{Hbd:NADPH}\cdot [Acetoacetyl\text{-}CoA]+K_{M}^{Hbd:Acetoacetyl\text{-}CoA}\cdot [NADPH]} \]
Constant |
Value |
Source |
To note |
\( K_{cat}^{Hbd} \) |
336.4 1/min |
Purification and Properties of NADP-Dependent |
Forward reaction, Clostridium Kluyveri |
\( K_{M}^{Hbd:Acetoacetyl\text{-}CoA} \) |
5e-5 mol/l |
Purification and Properties of NADP-Dependent |
Clostridium Kluyveri |
\( K_{M}^{Hbd:NADPH} \) |
7e-5 mol/l |
Purification and Properties of NADP-Dependent |
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 |
\( K_{cat}^{Crt} \) |
1279.8 1/min |
Purification and Characterization of Crotonase from Clostridium acetobutylicum; The journal of Biological Chemistry, Volume 247, Number 16, August 1972, Pages 5266-5271 |
Forward reaction |
\( K_{M}^{Crt:3\text{-}Hydroxybutyryl\text{-}CoA} \) |
3e-5 mol/l |
Purification and Characterization of Crotonase from Clostridium acetobutylicum; The journal of Biological Chemistry, Volume 247, Number 16, August 1972, Pages 5266-5271 |
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 |
\( K_{cat}^{Ter} \) |
5460 1/min |
Biochemical and Structural Characterization of the trans-Enoyl-CoA Reductase from Treponema denticola; Biochemistry 2012, 51, 6827−6837 |
Forward reaction |
\( K_{M}^{Ter:Crotonyl\text{-}CoA} \) |
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 |
Biochemical and Structural Characterization of the trans-Enoyl-CoA Reductase from Treponema denticola; Biochemistry 2012, 51, 6827−6837 |
Is there something special about this? |
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.
\[ \frac{K_{cat}^{YciA}\cdot [YciA]\cdot [Butyryl\text{-}CoA]}{K_{M}^{YciA:Butyryl\text{-}CoA} +[Butyryl\text{-}CoA]} \]
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 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
\[\frac{K_{cat}^{Car}\cdot [Car]\cdot [Butyrate]\cdot [NADPH]\cdot [ATP]}{K_{M}^{Car:Butyrate}\cdot K_{M}^{Car:NADPH}\cdot [ATP]+K_{M}^{Car:ATP}\cdot [Butyrate]\cdot [NADPH]+K_{M}^{Car:NADPH}\cdot [Butyrate]\cdot [ATP]}\]\[\cdots \frac{}{+K_{M}^{Car:Butyrate}\cdot [NADPH]\cdot [ATP]+ [Butyrate]\cdot [NADPH]\cdot [ATP]}\]
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 |
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 |
Is there something special about this? |
\( K_{M}^{Car:ATP} \) |
0.000115 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 |
Is there something special about this? |
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
\[ \frac{K_{cat}^{Ado}\cdot [Ado]\cdot [Butyrate]}{K_{M}^{Ado:Butyrate} +[Butyrate]} \]
Constant |
Value |
Source |
To note |
\( K_{cat}^{Ado} \) |
0.03 1/min |
Production of Propane and Other Short-Chain Alkanes by Structure-Based Engineering of Ligand Specificity in Aldehyde-Deformylating Oxygenase, Khara et al (2013) |
Forward reaction |
\( K_{M}^{Ado:Butyraldehyde} \) |
0.0101 mol/l |
Production of Propane and Other Short-Chain Alkanes by Structure-Based Engineering of Ligand Specificity in Aldehyde-Deformylating Oxygenase, Khara et al (2013) |
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] |
0.00012 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 |
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 \( 70\% \cdot \frac{1000 \frac{g}{l}}{18.01 g/mol} = 38.85 \frac{mol}{l} \) |