Figure 1. Balanced growth conditions. An intracellular metabolite M_A is in steady state if the sum of reactions rates (in molar units) v₁ and v_3,rev is equal to the sum of reaction rates v₂, v₃ and v₄. Metabolite M_A could be glucose 6-phosphate, a central metabolite synthesized from glucose via glucokinase (or hexokinase, v₁), and further metabolized through the Embden–Meyerhof–Parnas pathway via glucose-6-phosphate isomerase (v₃, reversible), through the pentose phosphate pathway via glucose-6-phosphate dehydrogenase (v₂), and through biosynthetic pathways via phosphoglucomutase (v₄).

Scheme 1. Calculating the drain of a precursor metabolite for biomass formation.

Figure 2. Application of linear metabolic flux analysis. A metabolic network, including the Embden–Meyerhof–Parnas pathway, the tricarboxylic acid cycle, the ethanol formation pathway, and the pyruvate carboxylase-catalysed reaction. Reaction v₈ represents the electron chain, with an assumed P/O ratio of 1.2. In order to include ATP and  CO₂ as intracellular metabolites that are balanced, reactions v₉ and v₁₀, representing the excess of ATP produced by the cells and the carbon dioxide evolution rate (CER), respectively, are included in the model.

For the sake of clarity, the chosen metabolic network contains a number of simplifications in relation to real metabolic networks. Thus, it is assumed that cell growth (biomass formation) is negligible, which means that there is no drain of precursor metabolites for biomass synthesis. Also, it is considered that ATP – instead of GTP – is the cofactor in the succinate thiokinase-catalysed reaction, and that NADH – instead of FADH₂ – is the cofactor in the succinate dehydrogenase-catalysed reaction. Reactions around metabolites that are not part of any branch point in the network are lumped together. Squared metabolites are those considered as branch point metabolites.

AcCOA, acetyl-coenzyme A; AKG, -ketoglutarate; ATP, adenosine triphosphate; CIT, citrate; ETH, ethanol; FADH₂, flavin–adenine dinucleotide, reduced form; FUM, fumarate; GTP, guanosine triphosphate; G6P, glucose 6-phosphate; ICIT, isocitrate; MAL, malate; NADH, nicotinamide–adenine dinucleotide, reduced form; OAA, oxaloacetate; PYR, pyruvate; P3G, 3-phosphoglycerate; SUC, succinate; SucCOA, succinyl-coenzyme A.

Figure 3. Metabolic network structures for which linear MFA fails. (a) It is impossible to calculate the fluxes in the two branches of the network depicted on the left without measuring at least one flux between metabolite M_A and metabolite M_G. In some cases, there is a link between the two branches, via a cofactor such as NADH. In this case, as indicated in the figure for reactions converting M_D into M_E and M_C into M_G, it might be possible to calculate the fluxes v₁ and v₂ through the two parallel branches of the network. (b) In the case of reversible reactions or reactions carrying fluxes in opposite directions, it is only possible to calculate the net flux, i.e. the difference between the two opposite fluxes. The degree of reversibility or the flux carried out by each enzyme, in the case of fluxes in opposite directions, cannot be calculated.

Figure 4. General idea of a labelling experiment. In a labelling experiment, a labelled substrate is fed to the organism. In this example, a three-atom substrate is used (S_ext) labelled in the first position. Fully labelled atoms are shaded in dark blue, whereas partially labelled positions are shaded in blue to light-blue. The labelling pattern in the product (P_ext), as well as in the intracellular metabolite A, directly reflects the relative magnitudes of the fluxes in the metabolic network. Thus, by measuring the amount of label in the product and/or in metabolite A, and employing the appropriate mathematical tools, it is possible to get qualitative and quantitative information on the relative utilization of pathways in a given metabolic network.

Scheme 2. Classification of ¹³C-labelled molecules. In this example, a three-carbon molecule is shown, but the principles can be extended to any molecule or molecule fragment obtained in the mass spectrometer. Filled circles correspond to ¹³C atoms, whereas nonfilled circles represent ¹²C atoms. The nomenclature I_i can be given with the index either on a decimal or on a binary basis (in the latter case, 0 corresponds to a ¹²C atom and 1 to a ¹³C atom).

Figure 5. Mass spectrum of alanine. In a mass spectrum, the abundance of the detected ions is represented as a function of the mass-to-charge ratio (m/z). In this example, two (N, N)-dimethylformamide dimethyl acetal-derivatized alanine fragments are shown: one containing all three carbon atoms of the amino acid, the molecular ion (cluster starting at m/z 158), and one containing only carbon atoms 2 and 3 (cluster starting at m/z 99). In each cluster, the peak corresponding to the lowest mass is that representing the fraction of all molecules that do not contain any ¹³C atoms (m). The other peaks represent the fraction of molecules containing one, two or three ¹³C atoms (m+1, m+2, and m+3, respectively), the mass isotopomers.

Figure 6. Absorption spectrum of a three-carbon-atom fragment. In an NMR spectrum, the absorption intensity of a central ¹³C atom in a three-carbon-atom fragment is represented as a function of the chemical shift in the ¹³C dimension. The multiplet signal will present nine peaks: the area of the singlet peak S is proportional to the amount of ¹³C atoms that have only ¹²C neighbouring atoms; the area of the doublet peaks D1 are proportional to the amount of ¹³C atoms that have a ¹³C neighbouring atom to the right; the area of the doublet peaks D2 are proportional to the amount of ¹³C atoms that have a ¹³C neighbouring atom to the left; and the area of the double doublet peaks DD are proportional to the amount of ¹³C atoms that have ¹³C neighbouring atoms to the right and to the left. J₁ is the scalar coupling constant between the central ¹³C atom and its neighbouring atom to the left, whereas J₂ is the scalar coupling constant between the central ¹³C atom and its neighbouring atom to the right. The coupling constants J₁ and J₂ are usually different, as the chemical environment of the neighbouring atoms are different owing to the different atoms to which they are bound. When the analysed ¹³C atom is at the extremity of a fragment, only three peaks appear in the spectrum: one singlet peak and two doublet peaks, as the analysed ¹³C atom only has one neighbour atom in this case. From the measurements, it is possible to calculate the fractional labelling of the two neighbouring atoms according to eqns [i] and [ii].

Scheme 3. Principles of fractional labelling and isotopomer models.

Scheme 4. Principles of the calculation of metabolic fluxes in nonlinear MFA. A labelling experiment is carried out, from which some of the fluxes can be measured (e.g. extracellular fluxes and fluxes to formation of biomass), as well as the labelling in some intracellular metabolites. Flux calculation is an iterative process that starts with an initial guess on a subset of all fluxes, which are called free fluxes. From the set of free fluxes, the remaining fluxes can be calculated using the metabolic model, and with the complete set of fluxes the labelling of all the intracellular metabolites can be calculated using a model describing the carbon transitions in the individual biochemical reactions (either a fractional labelling or an isotopomer model). The calculated labelling of intracellular metabolites and the set of fluxes for which the corresponding measurements are available are used as input for a minimization routine that aims at minimizing the differences between measured and calculated data. These differences are divided by the standard deviation of the corresponding measurement, so that they become dimensionless and normalized. If the calculated error at the end of the minimization routine is appropriately small, and the metabolic model is physiologically adequate, the corresponding set of fluxes is the one that best describes the metabolism under investigation.