Fig 1.
Characterization of the optimal solution space of metabolic models.
The optimal solution space can be characterized by three topological features: vertices (purple), rays (green), and linealities (blue). Typically, optimal solution spaces of microbial genome-scale models are characterized by many vertices and only a few linealities and rays. Linealities do not exist when reversible reaction are split. Vertices can be described by a fixed active part (red) which is identical for each vertex and a variable part (orange), a few CoPE-FBA subnetworks [16]. We refer to S1 Fig. for examples of rays we found in the E.coli iAF1260 genome-scale metabolic model.
Fig 2.
The optimal solution space of a toy model with reversible reactions.
Metabolites (capital letters) are converted by reversible (two headed arrows) and irreversible (single headed arrows) reactions to achieve the conversion of X to Y (underlined metabolites are boundary species). The forward direction of reversible reactions is defined from left to right or from top to bottom, and a backwards flux is denoted by a minor sign (e.g. -R13 indicates conversion from K to J). We maximized the flux through R18 with FBA, subject to steady-state constraints and J1 ≤ 2, where J1 is the flux through reaction R1. The optimal solution space is characterized by (A) one lineality of reactions {R2–R4} (red) and (B) four vertices that arise from two branches at intersections D and I: V1 (blue), V2 (red), V3 (green) and V4 (purple) (C) Two CoPE-FBA subnetworks illustrate the alternatives that create the four vertices shown in (B); in subnetwork one (blue) these are {R6–R8} (V1 and V2) and {R9–R10} (V3 and V4), and in subnetwork two (red) {R12–R14} (V2 and V4) and {R15, -R13, -R14} (V1 and V3).
Fig 3.
Vertices correspond to optimal-yield EFMs or convex combinations of those EFMs.
Vertices correspond to optimal-yield EFMs (A) if they are restricted by one flux constraint and to a convex combination of EFMs if they are restricted by more than one flux constraint (B). Colors represent different flux values (red = 2, orange = 1.5, green = 1, and blue = 0.5). (A) visualization of EFM1, EFM2, and EFM3 (out of the twelve optimal-yield EFMs normalized to J18 = 1). Both EFM1 and EFM3 have a corresponding vertex with and without splitting, whereas EFM2 has only with splitting a corresponding vertex. (B) taking a convex combination of EFM1 and EFM2 or EFM1 and EFM3 (panel A) corresponds to a vertex when the constraints are J1 ≤ 2 and J15 ≤ 0.5.
Fig 4.
Reversible-reaction splitting guarantees finding all non-decomposable flux pathways in the optimum.
Metabolites (capital letters) are converted by irreversible reactions to achieve the conversion of X to Y (underlined metabolites are boundary species). Split reversible reactions are denoted as R3f and R3b. We maximized the flux through R18 with FBA, subject to the steady-state constraint and J1 ≤ 2. The optimal solution space is now characterized by seven rays (A) and twelve vertices which originate from three CoPE-FBA subnetworks (B). (A) the five split reversible reactions and {R2–R4} in forward and backward direction form together seven rays. (B) three subnetworks give rise to twelve vertices (2×2×3). The third subnetwork (red) now has a third alternative flux distribution {R12, R15} which was without reversible-reaction splitting a convex combination of the other two flux distributions, {R12, R13f, R14f} and {R15, R13b, R14b}.
Table 1.
Characterization of the optimal solution space with and without reversible-reaction splitting.
Fig 5.
E.coli vertex cost follows a multimodal distribution.
For three growth conditions—aerobic (red, circle), aerobic restricted (purple, triangle), and anaerobic (blue, square)—we analyzed the vertex cost (PC) and vertex length (PL) of each vertex. Each dot in the main panel represents a vertex with a specific cost and length. Our results indicate that for E.coli the vertex length follows approximately a Gaussian-shaped distribution (dashed lines are Gaussian distributions with sample mean and sample standard deviation). Vertex cost follows a multimodal distribution; vertices are clustered in distinct groups with a specific cost. Due to file size limitations we only show a subset (10.000) of vertices for all conditions in the scatter plot.
Table 2.
Secondary optimization can reduce the optimal solution space to a unique flux distribution.