A Strategy to Calculate the Patterns of Nutrient Consumption by Microorganisms Applying a Two-Level Optimisation Principle to Reconstructed Metabolic Networks

Abstract

Bacterial responses to environmental changes rely on a complex network of biochemical reactions. The properties of the metabolic network determining these responses can be divided into two groups: the stoichiometric properties, given by the stoichiometry matrix, and the kinetic/thermodynamic properties, given by the rate equations of the reaction steps. The stoichiometry matrix represents the maximal metabolic capabilities of the organism, and the regulatory mechanisms based on the rate laws could be considered as being responsible for the administration of these capabilities. Post-genomic reconstruction of metabolic networks provides us with the stoichiometry matrix of particular strains of microorganisms, but the kinetic aspects of in vivo rate laws are still largely unknown. Therefore, the validity of predictions of cellular responses requiring detailed knowledge of the rate equations is difficult to assert. In this paper, we show that by applying optimisation criteria to the core stoichiometric network of the metabolism of Escherichia coli, and including information about reversibility/irreversibility only of the reaction steps, it is possible to calculate bacterial responses to growth media with different amounts of glucose and galactose. The target was the minimisation of the number of active reactions (subject to attaining a growth rate higher than a lower limit) and subsequent maximisation of the growth rate (subject to the number of active reactions being equal to the minimum previously calculated). Using this two-level target, we were able to obtain by calculation four fundamental behaviours found experimentally: inhibition of respiration at high glucose concentrations in aerobic conditions, turning on of respiration when glucose decreases, induction of galactose utilisation when the system is depleted of glucose and simultaneous use of glucose and galactose as carbon sources when both sugars are present in low concentrations. Preliminary results of the coarse pattern of sugar utilisation were also obtained with a genome-scale E. coli reconstructed network, yielding similar qualitative results.

Appendices

Appendix 1

1.1 A. Reactions

Abbreviation	Official reaction name	Equation
ACKr	Acetate kinase	[c] : ac + ATP ⇔ actp + ADP
ACONT	Aconitase	[c] : cit ⇔ icit
ACt2r	Acetate reversible transport via proton symport	ac[e] + h[e] ⇔ ac [c] + h[c]
ADHEr	Acetaldehyde dehydrogenase	[c] : accoa + (2)h + (2)NADH ⇔ coa + EtOH + (2)NAD
ADK1	Adenylate kinase	[c] : AMP + ATP ⇔ (2)ADP
AKGDH	2-Oxoglutarate dehydrogenase	[c] : akg + coa + NAD → CO2 + NADH + succoa
AKGt2r	2-Oxoglutarate reversible transport via symport	akg[e] + h[e] ⇔ akg[c] + h[c]
ATPM	ATP maintenance requirement	[c] : ATP + H2O → ADP + h + pi
ATPS4r	ATP synthase (four protons for one ATP)	adp[c] + (4)h[e] + pi[c] ⇔ ATP[c] + (3) h[c] + H2O[c]
CO2t	CO2 transporter via diffusion	CO2[e] ⇔ CO2[c]
CS	Citrate synthase	[c] : accoa + H2O + oaa → cit + coa + h
CYTBD	Cytochrome oxidase bd (ubiquinol-8: 2 protons)	(2) h[c] + (0.5) O2[c] + q8h2[c] → (2)h[e] + H2O[c] + q8[c]
d-LACt2	d-lactate transport via proton symport	h[e] + lac - D[e] ⇔ h[c] + lac − D[c]
ENO	Enolase	[c] : 2pg ⇔ H2O + pep
ETOHt2r	Ethanol reversible transport via proton symport	ETOH[e] + h[e] ⇔ ETOH[c] + h[c]
EX_ac(e)	Acetate exchange	[e] : ac ⇔
EX_akg(e)	2-Oxoglutarate exchange	[e] : akg ⇔
EX_co2(e)	CO2 exchange	[e] : CO2 ⇔
EX_etoh(e)	Ethanol exchange	[e] : ETOH ⇔
EX_for(e)	Formate exchange	[e] : for ⇔
EX_fum(e)	Fumarate exchange	[e] : fum ⇔
EX_glc(e)	d-Glucose exchange	[e] : glc − D ⇔
EX_h(e)	H+ exchange	[e] : h ⇔
EX_h2o(e)	H2O exchange	[e] : H2O ⇔
EX_lac-D(e)	d-lactate exchange	[e] : lac − D ⇔
EX_o2(e)	O2 exchange	[e] : O2 ⇔
EX_pi(e)	Phosphate exchange	[e] : pi ⇔
EX_pyr(e)	Pyruvate exchange	[e] : pyr ⇔
EX_succ(e)	Succinate exchange	[e] : succ ⇔
FBA	Fructose-bisphosphate aldolase	[c] : fdp ⇔ dhap + g3p
FBP	Fructose-bisphosphatase	[e] : fdp + H2O → f 6p + pi
FORt	Formate transport via diffusion	for[e] ⇔ for[c]
FRD	Fumarate reductase	[c] : fadh2 + fum → fad + succ
FUM	Fumarase	[c] : fum + H2O ⇔ mal − L
FUMt2_2	Fumarate transport via proton symport (2 H)	fum[e] + (2)h[e] → fum[c] + (2)h[c]
G6PDH2r	Glucose 6-phosphate dehydrogenase	[c] : g6p + NADP ⇔ 6pgl + h + NADPH
GALKr	Galactokinase	[c] : ATP + gal ← ADP + gal1p + h
GALabc	d-Galactose transport via ABC system	ATP[c] + gal[e] + H2O[c] → ADP[c] + gal[c] + h[c] + pi[c]
GALt2	d-Galactose transport in via proton symport	gal[e] + h[e] → gal[c] + h[c]
GAPD	Glyceraldehyde-3-phosphate dehydrogenase	[c] : g3p + NAD + pi ⇔ 13 dpg + h + NADH
GLCpts	d-Glucose transport via PEP:Pyr PTS	glc − D[e] + pep[c] → g6p[c] + pyr[c]
GND	Phosphogluconate dehydrogenase	[c] : 6pgc + NADP → co2 + NADPH + ru5p − D
H2Ot	H2O transport via diffusion	H2O[e] ⇔ H2O[c]
ICDHyr	Isocitrate dehydrogenase (NADP)	[c] : icit + NADP ⇔ akg + CO2 + NADPH
ICL	Isocitrate lyase	[c] : icit → glx + succ
LDH_D	d-Lactate dehydrogenase	[c] : lac − D + NAD ⇔ h + NADH + pyr
MALS	Malate synthase	[c] : accoa + glx + H2O → coa + h + mal − L
MDH	Malate dehydrogenase	[c] : mal − L + NAD ⇔ h + NADH + oaa
ME1	Malic enzyme (NAD)	[c] : mal − L + NAD → CO2 + NADH + pyr
ME2	Malic enzyme (NADP)	[c] : mal − L + NADP → CO2 + NADPH + pyr
NADH11	NADH dehydrogenase (ubiquinone-8 & 2 protons)	(3)h[c] + NADH[c] + q8[c] → (2)h[e] + NAD[c] + q8h2[c]
NADTRHD	NAD transhydrogenase	[c] : NAD + NADPH → NADH + NADP
O2t	O2 transport (diffusion)	O2[e] ⇔ O2[c]
PDH	Pyruvate dehydrogenase	[c] : coa + NAD + pyr → accoa + CO2 + NADH
PFK	Phosphofructokinase	[c] : ATP + f6p → ADP + fdp + h
PFL	Pyruvate formate lyase	[c] : coa + pyr → accoa + for
PGI	Glucose-6-phosphate isomerase	[c] : g6p ⇔ f6p
PGK	Phosphoglycerate kinase	[c] : 3pg + ATP ⇔ 13dpg + ADP
PGL	6-Phosphogluconolactonase	[c] : 6pgl + H2O → 6pgc + h
PGM	Phosphoglycerate mutase	[c] : 2pg ⇔ 3pg
PGMT	Phosphoglucomutase	[c] : g1p ⇔ g6p
Pit	Inorganic phosphate exchange, diffusion	pi[c] ⇔ pi[e]
PPC	Phosphoenolpyruvate carboxylase	[c] : CO2 + H2O + pep → h + oaa + pi
PPCK	Phosphoenolpyruvate carboxykinase	[c] : ATP + oaa → ADP + CO2 + pep
PPS	Phosphoenolpyruvate synthase	[c] : ATP + H2O + pyr → AMP + (2)h + pep + pi
PTAr	Phosphotransacetylase	[c] : accoa + pi ⇔ actp + coa
PYK	Pyruvate kinase	[c] : ADP + h + pep → ATP + pyr
PYRt2r	Pyruvate reversible transport via proton symport	h[e] + pyr[e] ⇔ h[c] + pyr[c]
RPE	Ribulose 5-phosphate 3-epimerase	[c] : ru5p − D ⇔ xu5p − D
RPI	Ribose-5-phosphate isomerase	[c] : r5p ⇔ ru5p − D
SUCCt2_2	Succinate transport via proton symport (2 H)	(2)h[e] + succ[e] →(2)h[c] + succ[c]
SUCCt2b	Succinate efflux via proton symport	h[c] + succ[c] → h[e] + succ[e]
SUCD1i	Succinate dehydrogenase	[c] : fad + succ → fadh2 + fum
SUCD4	Succinate dehyrdogenase	[c] : fadh2 + q8 ⇔ fad + q8h2
SUCOAS	Succinyl-CoA synthetase (ADP-forming)	[c] : ATP + coa + succ ⇔ ADP + pi + succoa
TALA	Transaldolase	[c] : g3p + s7p ⇔ e4p + 6p
THD2	NAD(P) transhydrogenase	(2)h[e] + NADH[c] + NADP[c] → (2)h[c] + NAD[c] + NADPH[c]
TKT1	Transketolase	[c] : r5p + xu5p − D ⇔ g3p + s7p
TKT2	Transketolase	[c] : e4p + xu5p − D ⇔ f6p + g3p
TPI	Triose-phosphate isomerase	[c] : dhap ⇔ g3p
UDPG4E	UDPglucose 4-epimerase	[c] : udpg ⇔ udpgal
UGLT	UDPglucose-hexose-1-phosphate uridylyltransferase	[c] : gal1p + udpg ⇔ g1p + udpgal

Biomass production

0.2 G6P + 0.071 F6P + 0.898 R5P + 0.361 E4P + 0.129 T3P + 1.4996 3PG + 0.519 PEP + 2.833 PYR + 3.748 AcCoA + 1.787 OAA + 1.079 alfa-KG + 42.703 ATP + 18.22 NADPH ➔ 3.748 CoA + 18.22 NADP + 42.703 ADP + 42.703 Pi + BIOMASS

1.2 B. Upper and Lower Bounds of the Fluxes

• 1) Thirty-six reversible reactions: ACKr, ACONT, ACt2r, ADHEr, ADK1, AKGt2r, ATPS4r, CO2t, D-LACt2, ENO, ETOHt2r, FBA, FORt, FUM, G6PDH2r, GAPD, H2Ot, ICDHyr, LDH_D, MDH, PGI, PGK, PGM, PIt, PTAr, PYRt2r, RPE, RPI, SUCD4, SUCOAS, TALA, TKT1, TKT2, TPI, GALKr and PGMT are bounded by −1000 and 1000.

• 2) Twenty-seven irreversible reactions: AKGDH, CS, CYTBD, FBP, FRD, FUMt2_2, GND, ICL, MALS, ME1, ME2, NADH11, NADTRHD, PDH, PFK, PFL, PGL, PPC, PPCK, PPS, PYK, SUCCt2_2, SUCCt2b, SUCD1i, THD2, UDPG4E and UGLT are bounded by 0 and 1000.

• 3) O2t (Oxygen transport) is bounded by −1000 and 10 (reversible).

• 4) ATPM (ATP maintenance requirement) is bounded by 7.6 and 1000 (irreversible).

• 5) GROWTH (biomass production) is bounded by 0.1 and 1000 (irreversible).

• 6) GALabc (d-galactose transport via ABC system) and GALt2 (d-galactose transport in via proton symport): their sum is bounded by 0 and 20 (irreversible).

• 7) GLCpts (d-glucose transport via PEP:Pyr PTS) is bounded by 0 and 20 (irreversible).

Appendix 2: The Model

Let J be the set of all the reactions in the stoichiometry matrix N (including the 69 internal and the 15 external reactions). We then define two subsets of J: J _int including the 69 internal reactions and J _ext including the 15 external reactions.

Master optimisation problem

Objective function:

$$Z = {Max} \quad v_{growth} $$

Subject to:

$$\begin{array}{*{20}l} {{\mathbf{N}} \cdot {\mathbf{\nu }} = o} \hfill \\ {\begin{array}{*{20}l} {y_j \cdot \nu _j^1 \leqslant \nu _j \leqslant y_j \cdot \nu _j^{\text{u}} } \hfill&{\forall \;j \in J_{{\text{int}}} } \hfill \\ {\nu _j^1 \leqslant \nu _j \leqslant \nu _j^{\text{u}} } \hfill&{\forall \;j \in J_{{\text{ext}}} } \hfill \\ \end{array} } \hfill \\ {\sum\limits_{j \in J_{{\text{int}}} } {y_j = Z*} } \hfill \\ {y_j \in \left\{ {0,1} \right\}} \hfill \\ {\nu _j \in \Re } \hfill \\ \end{array} $$

where Z* is given by the lower level optimisation problem:

$$Z^* = {Min} \;\sum\limits_{j \in J_{{int} } } {y_j } $$

Subject to:

$$\begin{array}{*{20}l} {{\mathbf{N}} \cdot {\mathbf{\nu }} = o} \hfill \\ {\begin{array}{*{20}l} {y_j \cdot \nu _j^1 \leqslant \nu _j \leqslant y_j \cdot \nu _j^{\text{u}} } \hfill&{\forall \;j \in J_{{\text{int}}} } \hfill \\ {\nu _j^1 \leqslant \nu _j \leqslant \nu _j^{\text{u}} } \hfill&{\forall \;j \in J_{{\text{ext}}} } \hfill \\ \end{array} } \hfill \\ {\nu _{{\text{growth}}}^{\text{1}} \leqslant \nu _{{\text{growth}}}^{\text{u}} } \hfill \\ {y_j \in \left\{ {0,1} \right\}} \hfill \\ {\nu _j \in \Re } \hfill \\ \end{array} $$