Examples for problems in elementary mode analysis. (A) Condensed network of glycolysis (dark reactions) and Entner–Doudoroff pathway (gray reactions). The pentose phosphate pathway has been omitted for clarity. In most analyses of glycolysis, the Entner–Doudoroff pathway is not considered. Hence, not all pathways from glucose into the TCA cycle are found, and the wrong conclusion might be drawn that only the pentose–phosphate pathway (not shown) can be used to bypass the knockout of one of the enzymes converting G6P to FDP. In vivo the knockout of the corresponding reactions is partially bypassed by a flux through the Entner–Doudoroff pathway (Fischer and Sauer 2003). (B) Modeling the Entner–Doudoroff pathway by adding an outflow of G6P and an inflow of G3P and Pyr only partially resolves the problem (thick arrows). Such an approach is often used in elementary mode analysis to avoid the consideration of some pathways in detail (e.g., the outflow of succinyl-CoA from the TCA cycle in Schuster et al. 1999, analyzed in Results). However, this can lead to fluxes like in C, which are not part of any feasible pathway when the entire network in A is considered. This is because the coupling of the influx of Pyr with the outflow reactions of G6P and the inflow reaction of G3P are neglected in B. (D) Elementary mode analysis does not allow one to analyze the dependencies between the subsystems S₁ and S₂ unless the reactions connecting them are taken into account. Thus, it is not possible to deduce that a zero flux from G6P to FDP in S₁ would imply that there cannot be a positive flux from DHAP to G3P in S₂. A list of abbreviations can be found in Supplemental material S2.

Scheme of part of the central metabolism of E. coli, which is here studied as a subnetwork of a genome-scale network. A list of abbreviations can be found in Supplemental materials S2 and S3.

Amino acid producing elementary flux patterns. (A) Alanine; (B–D) aspartate; (E–H) glutamate. Black reactions belong to the elementary flux pattern; gray reactions are the remaining reactions of the subsystem. A list of abbreviations can be found in Supplemental materials S2 and S3.

Determining minimal media for the production of amino acids. (A) Network under consideration for computation of elementary flux patterns. Dark reactions belong to the subsystem. (B) Elementary flux patterns of the subsystem. The inflow reaction that is used and the amino acid that can be produced from this input medium are indicated. Examples of minimal and nonminimal media are provided by the synthesis of histidine, which can be produced from glutamate alone as well as from ammonia and glutamate, respectively. The latter elementary flux pattern is more efficient in terms of glutamate consumed. A list of abbreviations can be found in Supplemental material S2.

Elementary flux patterns in a condensed model of the TCA cycle and adjacent reactions. (A) Model of the entire network. The TCA cycle is chosen as a subsystem (dark reactions). (B–D) Elementary flux patterns of the system are indicated by thick black arrows. The associated pathway through the entire system is indicated by thick gray arrows. The thickness of the arrows corresponds to the relative flux through each reaction. (E) Alternative pathway through the entire system using the same reactions of the flux pattern depicted in C. While the pathway in C corresponds to the glyoxylate cycle that can be used for growth on fatty acids, E corresponds to the phosphoenolpyruvate-glyoxylate cycle used as a catabolic pathway during growth on low glucose concentrations (Fischer and Sauer 2003). A list of abbreviations can be found in Supplemental material S2.

Alternative pathways in the central metabolism of E. coli. Dashed arrows represent condensed reactions. (A) Alternative glyoxylate producing pathway; (B) Entner–Doudoroff pathway; (C) glycerate pathway; (D) GABA-shunt. The lower pathway represents the standard route from oxoglutarate to succinate in the TCA cycle, and the upper pathway depicts the GABA shunt. A list of abbreviations can be found in Supplemental materials S2 and S3.

Elementary flux pattern belonging to an aspartate-producing pathway. (Bold arrows) The reactions of the elementary flux pattern (i.e., they belong to the subsystem). The species entering the subsystem are acetyl-CoA and glyoxylate via the alternate glyoxylate-producing pathway presented in Figure 4A. (Dashed lines) Condensed reaction. Selected fluxes are given. A list of abbreviations can be found in Supplemental materials S2 and S3.

Scenario for the application of minimal cut sets. (A) Reaction network under consideration, which is supplied with acetate and glutamate. It is assumed that the production of histidine should be prevented. Histidine is produced from the central metabolism intermediate ribose-5 phosphate (Umbarger 1978). Assuming that it is not possible to entirely knock out the reactions r₁₇ and r₁₈ that directly produce ribose-5 phosphate, the reactions r₁₀–r₁₅ are defined as alternative targets (black reactions). (B) In order to determine minimal cut sets, r₁₀–r₁₅ are added to the subsystem containing the outflow reaction of histidine. From the elementary flux patterns, the minimal cut sets can be obtained. (C) Minimal sets of reactions that need to be knocked out to suppress the production of histidine: (black crosses) MCS1; (dark gray crosses) MCS2; (light gray crosses) MCS3. Subsequently, the minimal cut sets can be ranked according to different criteria, like side effects or required effort to knock out the corresponding genes, to determine the optimal knockout strategy. A list of abbreviations can be found in Supplemental material S2.