Modeling and simulation of the main metabolism in Escherichia coli and its several single-gene knockout mutants with experimental verification

Kadir, Tuty Asmawaty Abdul; Mannan, Ahmad A; Kierzek, Andrzej M; McFadden, Johnjoe; Shimizu, Kazuyuki

doi:10.1186/1475-2859-9-88

Table 1 Kinetic rate equations

From: Modeling and simulation of the main metabolism in Escherichia coli and its several single-gene knockout mutants with experimental verification

Reactions	Model equations		References
Cell growth	$μ = {\begin{matrix} μ_{m} (1 - \frac{[X]}{X_{m}}) (\frac{[G l c^{e x}]}{K_{s} + [G l c^{e x}]}) k_{A T P} v_{A T P} (.), ([G l c^{e x}] > 0) \\ \frac{μ_{m A} [A c e^{e x}]}{K_{s A} + [A c e^{e x}]} k_{A T P} v_{A T P} (.), ([G l c^{e x}] \leq 1 a n d [A c e^{e x}] > 0) \end{matrix}$	(1a) (1b)
Phosphotransferasesystem	$v_{P T S} = \frac{v_{P T S}^{\max} [G l c^{e x}] \frac{[P E P]}{[P Y R]}}{(K_{a 1} + K_{a 2} \frac{[P E P]}{[P Y R]} + K_{a 3} [G l c^{e x}] + [G l c^{e x}] \frac{[P E P]}{[P Y R]}) (1 + \frac{{[G 6 P]}^{n_{G 6 P}}}{K_{G 6 P}})}$	(2a)	[8, 33]
Non-PTS glucokinase	$v_{G l k} = \frac{v_{G l k}^{\max} [G l c] [A T P]}{(K_{m}^{G l c} + [G l c]) (K_{m}^{A T P} + [A T P])}$	(2b)	[49]
Phosphoglucoseisomerase	$v_{P g i} = \frac{v_{P g i}^{\max} ([G 6 P] - \frac{[F 6 P]}{K_{e q}})}{K_{G 6 P} (1 + \frac{F 6 P}{K_{F 6 P} (1 + \frac{[F 6 P]}{K_{6 p g i n h}^{F 6 P}})} + \frac{[6 P G]}{K_{6 p g i n h}^{G 6 P}}) + G 6 P}$	(3)	[8]
Phospho-fructokinase	$v_{P f k} = \frac{v_{P f k}^{\max} K_{A T P} [F 6 P]}{\begin{matrix} K_{(A T P, A D P)} ([F 6 P] + K_{S}^{F 6 P} \frac{K_{b (A D P, A M P)} + \frac{[P E P]}{K_{P E P}}}{K_{a (A D P, A M P)}}) \times \\ (1 + \frac{L_{P f k}}{{(1 + [F 6 P] (\frac{K_{a (A D P, A M P)}}{K_{s}^{F 6 P} (K_{b (A D P, A M P)} + \frac{[P E P]}{K_{P E P}})}))}^{n_{P f k}}}) \end{matrix}}$	(4)	[8]
Adolase	$v_{A l d o} = \frac{v_{A l d o}^{\max} ([F D P] - \frac{[G A P] [G A P]}{K_{e q}})}{(K_{F D P} + [F D P] + \frac{K_{G A P} [G A P]}{[K_{e q} V_{b l f}]} + \frac{K_{D H A P} [G A P]}{[K_{e q} V_{b l f}]} + \frac{[F D P] [G A P]}{K_{i n h}^{P E P}} + \frac{[G A P] [G A P]}{K_{e q} V_{b l f}})}$	(5)	[8]
G3P dehydrogenase	$v_{G A P D H} = \frac{v_{G A P D H}^{\max} ([G A P] - \frac{[P E P] [N A D H]}{K_{e q} [N A D]})}{(K_{G A P} (1 + \frac{[P E P]}{K_{P G P}}) + [G A P]) (\frac{K_{N A D}}{N A D} (1 + \frac{[N A D H] \|}{K_{N A D H}}) + 1)}$	(6)	[8]
Pyruvate kinase	$v_{P y k} = \frac{v_{P y k}^{\max} [P E P] {(\frac{[P E P]}{K_{P E P}} + 1)}^{n_{P y k} - 1} [A D P]}{K_{P E P} (L_{P y k} {(\frac{1 + \frac{[A T P]}{K_{A T P}}}{\frac{[F D P]}{K_{F D P}} + \frac{[A M P]}{K_{A M P}} + 1})}^{n_{P y k}} + {(\frac{[P E P]}{K_{P E P}} + 1)}^{n_{P y k}}) ([A D P] + K_{A D P})}$	(7)	[8]
PEP carboxylase	$v_{P p c} = \frac{K_{1} + K_{2} [A c C o A] + K_{3} [F D P] + K_{4} [A c C o A] [F D P]}{1 + K_{5} [A c C o A] + K_{6} [F D P]} (\frac{[P E P]}{K_{m} + [P E P]})$	(8)	[38]
PEP carboxykinase	$v_{P c k} = v_{P c k}^{\max} (\frac{[O A A] \frac{[A T P]}{[A D P]}}{\begin{matrix} K_{m}^{O A A} \frac{[A T P]}{[A D P]} + [O A A] \frac{\| [A T P]}{[A D P]} + \frac{K_{i}^{A T P} K_{m}^{O A A}}{K_{i}^{A D P}} + \frac{K_{i}^{A T P} K_{m}^{O A A}}{K_{m}^{P E P} K_{i}^{^{A D P}}} [P E P] + \\ \frac{K_{i}^{A T P} K_{m}^{O A A}}{K_{i}^{P E P} K_{I}^{A T P}} \frac{[A T P] [P E P]}{[A D P]} + \frac{K_{i}^{A T P} K_{m}^{O A A}}{K_{i}^{A D P} K_{I}^{O A A}} [O A A] \end{matrix}})$	(9)	[15]
Pyruvate dehydrogenasecomplex	$v_{P D H} = \frac{\frac{v_{P D H}^{max}}{[N A D]} (\frac{1}{1 + K_{i} \frac{[N A D H]}{[N A D]}}) (\frac{[P Y R]}{K_{m}^{P Y R}}) (\frac{1}{K_{m}^{N A D}}) (\frac{[C O A]}{K_{m}^{C O A}})}{(1 + \frac{[P Y R]}{K_{m}^{P Y R}}) (\frac{1}{N A D} + \frac{1}{K_{m}^{N A D}} + \frac{[N A D H]}{K_{m}^{N A D H} [N A D]}) (1 + \frac{[C O A]}{K_{m}^{C O A}} + \frac{[A c C o A]}{K_{m}^{A c C o A}})}$	(10)	[40]
Pta	$v_{P t a} = \frac{v_{P t a}^{max} (\frac{1}{K_{i}^{A c C o A} K_{m}^{P}}) ([A c C o A] [P] - \frac{[A c P] [C o A]}{K_{e q}})}{(1 + \frac{[A c C o A]}{K_{i}^{A c C o A}} + \frac{[P]}{K_{i}^{P}} + \frac{A C P}{K_{i}^{A C P}} + \frac{[C o A]}{K_{i}^{C o A}} + (\frac{[A c C o A] [P]}{K_{i}^{A c C o A} K_{m}^{P}}) + (\frac{[A c P] [C o A]}{K_{m}^{A C P} K_{i}^{C o A}}))}$	(11)	[40, 41]
Ack	$v_{A c k} = \frac{v_{A c k}^{\max} (\frac{1}{K_{m}^{A D P} K_{m}^{A C P}}) ([A c P] [A D P] - \frac{[A C E] [A T P]}{K_{e q}})}{(1 + \frac{[A c P]}{K_{m}^{A C P}} + \frac{A C E}{K_{m}^{A C E}}) (1 + \frac{[A D P]}{K_{m}^{A D P}} + \frac{[A T P]}{K_{m}^{A T P}})}$	(12)	[40]
Acs	$v_{A c s} = \frac{v_{A c s}^{max} [A C E] [N A D P]}{(K_{m} + [A C E]) (K_{e q} + [N A D P])}$	(13)	[42]
Citrate Synthase	$v_{C S} = \frac{v_{C S}^{\max} [A c C o A] [O A A]}{\begin{matrix} (K_{d}^{A c C o A} K_{m}^{O A A} + K_{m}^{A c C o A} [O A A]) + ([A c C o A] K_{m}^{O A A} (1 + \frac{[N A D H]}{K_{i 1}^{N A D H}})) + \\ ([A c C o A] [O A A] (1 + \frac{[N A D H]}{K_{i 2}^{N A D H}})) \end{matrix}}$	(14)	[43]
ICDH	$v_{I C D H} = \frac{[I C D H] \frac{K_{f}}{K_{m}^{i C i T}} K_{d}^{N A D P} ([I C I T] - \frac{[N A D P H] [2 K G]}{K_{e q}^{I C D H} [N A D P]})}{(\begin{array}{l} \frac{1}{[N A D P]} + \frac{[I C I T] K_{m}^{N A D P}}{K_{m}^{i C i T} K_{d}^{N A D P} [N A D P]} + \frac{1}{K_{d}^{N A D P}} + \frac{[I C I T]}{K_{m}^{i C i T} K_{d}^{N A D P}} + \\ \frac{[I C I T]}{K_{d}^{i C i T} [N A D P]} \frac{[N A D P H] K_{m}^{N A D P}}{K_{m}^{i C i T} K_{d}^{N A D P} K_{e i n h}^{N A D P H}} + \frac{[N A D P H] K_{e k n h}^{2 K G}}{K_{m}^{2 K G} K_{e n h e}^{N A D P H} [N A D P]} + \frac{[2 K G] K_{m}^{N A D P H}}{K_{m}^{2 K G} K_{e n h e}^{N A D P H} [N A D P]} + \\ \frac{[2 K G]}{K_{m}^{2 K G}} \frac{[N A D P H]}{K_{e n h e}^{N A D P H} [N A D P]} + \frac{[2 K G] K_{m}^{N A D P H}}{K_{m}^{2 K G} K_{e n h e}^{N A D P H}} \frac{[N A D P H]}{K_{e k n}^{N A D P} [N A D P]} \end{array})}$	(15)	[44]
Isocitrate lyase	$v_{I c l} = \frac{v_{I c l_f}^{max} \frac{[I C I T]}{K_{m}^{i C i T}}}{(1 + \frac{[I C I T]}{K_{m}^{i C i T}} + \frac{[S U C]}{K_{m}^{S U C}} + \frac{[G O X]}{K_{m}^{G O X}} + \frac{[I C I T]}{K_{m}^{i C i T}} \frac{[S U C]}{K_{m}^{S U C}} + \frac{[S U C]}{K_{m}^{S U C}} \frac{[G O X]}{K_{m}^{G O X}} + \frac{I}{K_{I}})}$	(16)	[46]
Malate synthase	$v_{M S} = \frac{v_{M S_f}^{max} \frac{[G O X]}{K_{m}^{G O X}} \frac{[A c C o A]}{K_{m}^{A c C o A}} - v_{M S_r}^{\max} \frac{[M A L]}{K_{m}^{M A L}}}{(1 + \frac{[G O X]}{K_{m}^{G O X}} + \frac{[M A L]}{K_{m}^{M A L}} + (1 + \frac{[A c C o A]}{K_{m}^{A c C o A}}))}$	(17)	[46]
αKG dehydrogenase	$v_{α K G D H} = \frac{v_{2 K G D H}^{\max} [α K G] [C o A]}{{\begin{matrix} \frac{K_{m}^{N A D} [α K G] [C o A]}{[N A D]} + K_{m}^{C o A} [α K G] + K_{m}^{2 K G} [C o A] + [α K G] [C o A] + \\ \frac{K_{m}^{2 K G} K_{z} [S U C] [N A D H]}{K_{I}^{^{S U C}} [N A D]} + \frac{K_{m}^{N A D} [α K G] [C o A] [N A D H]}{K_{I}^{N A D H} [N A D]} + \\ \frac{K_{m}^{C o A} [α K G] [S U C]}{K_{I}^{S U C}} + \frac{K_{m}^{2 K G} K_{Z} [α K G] [S U C] [N A D H]}{K_{I}^{^{2 K G}} K_{I}^{^{S U C}} [N A D]} \end{matrix}}}$	(18)	[45]
Succinate dehydrogenase	$v_{S D H} = \frac{v_{S D H 1} v_{S D H 2} ([S U C] - \frac{[F U M]}{K_{e q}})}{K_{m}^{S U C} v_{S D H 2} + v_{S D H 2} [S U C] + \frac{V_{S D H 1} [F U M]}{K_{e q}}}$	(19)	[45]
Fumarase	$v_{F u m} = \frac{v_{F u m 1} v_{F u m 2} ([F U M] - \frac{[M A L]}{K_{F u m e q}})}{K_{m}^{F u m} v_{F u m 1} + v_{F u m 2} [F U M] + \frac{V_{F u m 1} [M A L]}{K_{e q}}}$	(20)	[45]
Malate dehydrogenase	$v_{M D H} = \frac{v_{M D H}_{1} v_{M D H}_{2} ([M A L] - \frac{[O A A]}{K_{e q}})}{(\begin{array}{l} \frac{K_{I}^{N A D} K_{m}^{M A L} v_{M D H 2}}{[N A D]} + K_{m}^{M A L} v_{M D H 2} + \frac{K_{m}^{N A D} V_{M D H 2} [M A L]}{[N A D]} + v_{M D H 2} [M A L] + \\ \frac{K_{m}^{O A A} v_{M D H 1} [N A D H]}{K_{e q} [N A D]} + \frac{K_{m}^{N A D H} v_{M D H 1} [O A A]}{K_{e q} [N A D]} + \frac{v_{M D H 1} [N A D H] [O A A]}{K_{e q} [N A D]} + \\ \frac{v_{M D H 1} K_{m}^{O A A} [N A D H]}{K_{e q} K_{I}^{N A D}} + \frac{v_{M D H 2} K_{m}^{N A D} [M A L] [O A A]}{K_{I}^{O A A} [N A D]} + \frac{v_{M D H 2} [M A L] [N A D H]}{K_{I}^{N A D H}} + \\ \frac{v_{M D H 1} [M A L] [N A D H] [O A A]}{K_{e q} K_{I}^{M A L} [N A D]} + \frac{v_{M D H 2} [M A L] [O A A]}{K_{I I}^{O A A}} + \frac{v_{M D H 1} [N A D H] [O A A]}{K_{I I}^{N A D} K_{e q}} + \\ \frac{K_{I}^{N A D} v_{M D H 2} [M A L] [N A D H] [O A A]}{K_{I I}^{N A D} K_{m}^{O A A} K_{I}^{N A D H}} \end{array})}$	(21)	[45]
Malic enzyme	$v_{M e z} = \frac{v_{M e z}^{\max} [M A L] [N A D P]}{(K_{M A L} + [M A L]) (K_{e q} + [N A D P])}$	(22)	[45]
G6PDH	$v_{G 6 P D H} = \frac{v_{G 6 P D H}^{\max} [G 6 P]}{([G 6 P] + K_{G 6 P}) (1 + \frac{[N A D P H]}{K_{N A D P H}^{G 6 P}}) (\frac{K_{N A D P}}{[N A D P]} (1 + \frac{N A D P H}{K_{N A D P H}^{N A D P}}) + 1)}$	(23)	[8]
PGDH	$v_{G 6 P D H} = \frac{v_{G 6 P D H}^{\max} [G 6 P]}{([6 P G] + K_{6 P G}) (1 + \frac{K_{N A D P}}{N A D P} (1 + \frac{[N A D P H]}{K_{N A D P H}^{i n h}}) (1+ \frac{[A T P]}{K_{A T P}^{i n h}}))}$	(24)	[8]
Rpe	$v_{R p e} = v_{R p e}^{max} ([R u 5 P] - \frac{[X u 5 P]}{K_{e q}^{R p e}})$	(25)	[8]
Rpi	$v_{R p i} = v_{R p i}^{\max} ([R u 5 P] - \frac{[R 5 P]}{K_{e q}^{R p i}})$	(26)	[8]
TktA	$v_{T k t A} = v_{T k t A}^{max} ([R 5 P] [X u 5 P] - \frac{[S 7 P] [G A P]}{K_{e q}^{T k t A}})$	(27)	[8]
TktB	$v_{T k t B} = v_{T k t B}^{max} ([X u 5 P] [E 4 P] - \frac{[F 6 P] [G A P]}{K_{e q}^{T k t B}})$	(28)	[8]
Tal	$v_{T a l} = v_{T a l}^{max} ([G A P] [S 7 P] - \frac{[E 4 P] [F 6 P]}{K_{e q}^{T k t B}})$	(29)	[8]

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