Fig 1.
A simple example network.
Fig 2.
The nullspace of the example network.
The red lines indicate that the nullspace is unbounded (in all directions). The blue arrows are basis vectors, which are also depicted as flux distributions.
Fig 3.
The flux cone of the example network.
The flux cone (dark grey) is obtained from the nullspace in Fig 2 by removing the halfspaces corresponding to the backward directions of the irreversible reactions R1 and R3 (removed part shown in light grey). The red dotted lines indicate the unbounded directions of the cone. The cone is bounded by its extreme rays (bold black dotted lines), which are represented by the extreme vectors y1 and y2(full blue arrows). Both extreme vectors are also elementary flux modes, and a third elementary flux mode exists in the interior of the cone (dashed blue arrow). The extreme vectors and elementary flux modes, respectively, are also depicted as flux distributions. Note that the flux cone is unbounded, however, it appears as a rectangle because it is cropped by the bounding box.
Fig 4.
A bounded flux polyhedron in the example network.
The two additional inhomogeneous constraints (an upper flux bound for reaction R1 and a lower flux bound for reaction R2) give rise to two hyperplanes r1 = 2 and r2 = −1 (green and yellow). These hyperplanes cut out the bounded flux polyhedron (dark grey) from the unbounded flux cone of Fig 3 (light grey). The polyhedron has five elementary flux vectors (full/dashed blue arrows and the zero vector), four of which correspond to vertices (full blue arrows and zero). The vertices and elementary flux vectors of the polyhedron are also depicted as flux distributions.
Fig 5.
Relationships between (flux) polyhedra and (flux) cones and their Elementary Vectors (EVs)/Elementary Flux Vectors (EFVs)/Elementary Flux Modes (EFMs).
Fig 6.
Comparison of Elementary Flux Modes (EFMs) (no flux bounds) and of Elementary Flux Vectors (EFVs) obtained by setting inhomogeneous flux constraints.
The latter include (i) maximal substrate (glucose) uptake rate, (ii) ATP maintenance demand, and (iii) certain levels of oxygen availability. Glucose (Glc) was used as substrate in all scenarios. (a) maximal biomass yield (gDW/mmol Glc); (b) maximal growth-rate (h-1); (c) maximal acetate yield (mmol/mmol Glc); (d) maximal acetate production rate (mmol/gDW/h); (e) maximal lysine yield (mmol/mmol Glc); and (f) maximal lysine production rate (mmol/gDW/h). Maximal production rates are not given for the EFMs because EFMs can be scaled to infinity. The number of optimal EFMs/EFVs is displayed in each bar. The white circles in (c) and (e) represent the maximal guaranteed product yields for growth-coupled product synthesis (minimal demanded biomass yield is 0.01 gDW/mmol Glc).