Method

We considered 25 enzymatic reactions that affect the urea cycle as shown in Table 1 to build the model.

Table 1 25 enzymatic reactions we considered when we built the model.

Enzyme	Reaction formula	Enzyme number
OTC	carbamoyl phosphate + L-ornithine → L-citrulline + Pi	EC: 2.1.3.3 (argF,argI)
ASS	L-citrulline + L-aspartate + ATP → N-(L-arginino)succinate + AMP + PPi	EC: 6.3.4.5 (argG)
ASL	N-(L-arginino)succinate → fumarate + L-arginine	EC: 4.3.2.1 (argH)
ARG	L-arginine + H2O → L-ornithine + urea	EC: 3.5.3.1 (rocF)
CPS	L-glutamine + HCO3- + H2O + 2ATP → L-glutamate + carbamoyl phosphate + 2ADP + Pi	EC: 6.3.5.5
GOGAT	L-glutamine + 2-oxoglutarate + NADPH + H+ → 2 L-glutamate + NADP+	EC: 1.4.1.13
GS	L-glutamate + NH3 + ATP → L-glutamine + ADP + Pi	EC: 6.3.1.2
GLU	L-glutamine + H2O = L-glutamate + NH3	EC: 3.5.1.2
GDH	2-oxoglutarate + NH3 + NADPH + H+ = L-glutamate + NADP+ + H2O	EC: 1.4.1.4
AAA	acetyl-CoA + L-glutamate → CoASH + N-acetyl-L-glutamate	EC: 2.3.1.1 (argA)
AGK	N-acetyl-L-glutamate + ATP → N-acetyl-L-glutamate 5-phosphate + ADP	EC: 2.7.2.8 (argB)
AGPR	N-acetyl-L-glutamate 5-phosphate + NADPH + H+ → N-acetyl-L-glutamate 5-semialdehyde + NADP+ + Pi	EC: 1.2.1.38 (argC)
AOT	N-acetyl-L-glutamate 5-semialdehyde + L-glutamate → N-acetylornithine + 2-oxoglutarate	EC: 2.6.1.11 (argD)
AO	N-acetylornithine + H2O → L-ornithine + acetate	EC: 3.5.1.16 (argE)
FH	fumarate + H2O → L-malate	EC: 4.2.1.2
MDH	L-malate + NAD+ → oxaloacetate + NADH + H+	EC: 1.1.1.37
AST	oxaloacetate + L-glutamate = L-asparate + 2-oxoglutarate	EC: 2.6.1.1
CS	acetyl-CoA + oxaloacetate + H2O → citrate + CoASH	EC: 2.3.3.1
AH	citrate → isocitrate	EC: 4.2.1.3
IDH	isocitrate + NADP+ → 2-oxoglutarate + NADPH + H+ + CO2	EC: 1.1.1.42
OGDH	2-oxoglutarate + Enzyme N6-(lipoyl)lysine → [dihydrolipoyllysine-residue succinyl transferase]S-succinyldihydrolopiyllysine + CO2	EC: 1.2.4.2
DST	[dihydrolipoyllysine-residue succinyl transferase]S-succinyldihydrolopiyllysine +CoASH → succinyl-CoA + Enzyme N6-(dihydrolipoyl)lysine	EC: 2.3.1.61
E3	Enzyme N6-(dihydrolipoyl)lysine + NAD+ = Enzyme N6-(lipoyl)lysine + NADH + H	EC: 1.8.1.4
SCS	succinate + ADP + Pi → succinate + CoASH + ATP	EC: 6.2.1.5
SDH	succinate + FAD → fumarate + CoASH + FADH2	EC: 1.3.99.1

*Abbreviations of enzymes: OTC, ornithine transcarbamoylase; ASS, argininosuccinate synthase; ASL, argininosuccinate lyase; ARG, arginase; CPS, carbamoyl phosphate synthetase; GOGAT, glutamate synthase; GS, glutamine synthetase; GLU, glutaminase; GDH, glutamate dehydrogenase; AAT, amino-acid N-acetyltransferase; AGK, acetylglutamate kinase; AGPR, N-acetyl-γ-glutamylphosphate reductase; AOT, acetylornithine transaminase; AO, acetylornithine deacetylase; FH, fumarate hydratase; MDH, malate dehydrogenase; AST, asparate aminotransferase; CS, citrate (Si)-synthase; AH, aconitate hydratase; IDH, isocitrate dehydrogenase; OGDH, oxoglutarate dehydrogenase; DST, dihydrolipoyllysine-residue succinyltransferase; E3, dihydrolipoyl dehydrogenase; SCS, succinyl-CoA synthetase; SDH, succinate dehydrogenase Reversible and irreversible reactions are indicated, in the reaction equations, by the symbols = and →, respectively.
We classified the metabolites in these 25 enzymatic reactions into internal substances and external substances. Internal substances are the metabolites which are rarely supplied from another reaction. External substances are those which are already enough in the cell or are produced by other reactions, for example TCA cycle products or coenzymes. If we want to get consumption or production modes, we set the substances which can be input or output of the modes as external substances. At first, we set internal and external substances as shown in Table 2.

We combined the reactions catalyzed by AAA, AGK, AGRP, ATO and AO to simplify the reaction scheme. We also combined CS, AH, IDH, OGDH, DST, E3 SCS and SDH and represented them using one arrow. These simplifications are reasonable because a sequence of reactions can be seen as one reaction. See Table 1 for the names that correspond to the abbreviation of the enzymes and reaction formulas. To increase the amount of the components of the urea cycle, we first assumed that the intermediates of TCA cycle can be external substances because TCA cycle can supply them by adding glucose to the growth media under aerobic conditions. We then redefined internal and external substances as shown in Table 3.

To increase the amount of the L-aspartate and L-glutamate, we redefined the internal and external substances as we done before.

The calculation to determine elementary modes

We calculated based on (Schuster et al. 2000). We combined two reactions to cancel internal substances. To make calculation easy, we use the matrixes. For example, reactions which catalyzed by OTC and CPS are shown below.
L-ornithine + carbamoyl phosphate → L-citrulline + Pi L-glutamine + HCO3- + H2O + 2ATP → carbamoyl phosphate + L-glutamate + Pi + 2ADP

If we combine them, we get total reaction formula as below.

L-ornithine + L-glutamine + HCO3- + H2O + 2ATP →L-citrulline + L-glutamate + 2ADP + Pi

This operation means we canceled carbamoyl phosphate (internal substance) by combining two reactions. If we do this operation in matrixes like, the operation is following.