Basic concepts and principles of stoichiometric modeling of metabolic networks

Abstract

Metabolic networks supply the energy and building blocks for cell growth and maintenance. Cells continuously rewire their metabolic networks in response to changes in environmental conditions to sustain fitness. Studies of the systemic properties of metabolic networks give insight into metabolic plasticity and robustness, and the ability of organisms to cope with different environments. Constraint-based stoichiometric modeling of metabolic networks has become an indispensable tool for such studies. Herein, we review the basic theoretical underpinnings of constraint-based stoichiometric modeling of metabolic networks. Basic concepts, such as stoichiometry, chemical moiety conservation, flux modes, flux balance analysis, and flux solution spaces, are explained with simple, illustrative examples. We emphasize the mathematical definitions and their network topological interpretations.

Keywords: Constraint-based modeling; Flux balance analysis; Flux modes; Metabolism; Optimal solution space.

Figure 1

A simplified metabolic pathway to illustrate the concept of flux modes. (A) A network diagram of a simplified metabolic network. Arrows indicate reactions and are labeled as Rn. Double-headed arrows indicate reversible reactions. Irreversible reactions are indicated by single-headed arrows, which point in the thermodynamically preferred direction. Underlined metabolites are considered to be fixed in concentration to allow for a steady state. Note that all reactions are uni–uni reactions, except R25, which has stoichiometry of A + T → U + 1 / 2 S. We can rewrite this stoichiometry as A₂ + P → AP + A to illustrate that there is no stoichiometric inconsistency with the isomerization reactions. To deal with thermodynamic inconsistencies, imagine adding fixed metabolites V and W to R24 to drive this reaction forward. A description of this model in the SBML level 3 package can be found in the Supporting information. (B) An overview of the seven flux modes. Colors correspond to flux values.

Figure 2

FBA and flux variability analysis (FVA) of E. coli model iAF1260 in a defined mineral medium. (A) FVA performed on the toy metabolic model. Resulting spans are shown for all reactions in which zero span is for fixed (in-)active reactions (gray, R25 is inactive), a span of one is for active but variable reactions (purple), and large spans are for reactions (red) in cycles. (B) Flux distribution resulting from FBA on the genome-scale model predicted that 82% of the reactions were inactive (J_i = 0) and only 18% of the fluxes carried a non-zero flux (J_i ≠ 0). (C) Analysis of the results of FVA revealed that 94% of the metabolic network was fixed (fixed fluxes) and only 6% of all fluxes (variable fluxes) could vary without changing the growth rate. 49% of these variable fluxes have a finite span, while 51% have an infinite span, suggesting their involvement in infeasible cycles. Out of those fixed fluxes, 84% never carry any mass (J_i = 0) and 16% are active (J_i ≠ 0). (D) Absolute spans of some reactions, resulting from FVA, are presented. All reaction names are taken from the model itself.

Figure 3

Topological characterization of all EFMs. (A)–(X) Type I EFMs. (Y) Type II EFM. (Z)-(AB) Type IV EFM. Visualizing ExPas requires decoupling of all reversible reactions into two irreversible reactions. Because all exchange reactions are irreversible, the set of relevant ExPas match this set of EFMs. Colors correspond to reaction values (red = 1, blue = 1/2).

Figure 4

Topological characterization of the optimal FBA solution space. (A) This FBA program contains one ray (blue; R23 and R24) and two linealities (green; R02–R04 and R14, R19–R21). (B)–(E) Visualization of the four vertices this FBA program contains. Each vertex represents a route from substrate to product with a maximum yield. The values indicate the predicted flux values. Reaction R01 was bounded between zero and one. (F) The two sub-networks detected with CoPE-FBA. Both sub-networks contain two alternative flux distributions, resulting in 2 x 2 possible vertices shown in (B)–(E).