Analysis of a simple metabolic network.

**A. Problem statement**. The system is confined by the dashed line and consisted of internal metabolites A, B, C, D, and P that are linked through internal reactions r

_{2}, r

_{3}, r

_{5}, r

_{6r}, r

_{7}. External metabolites A

_{ext}, B

_{ext}, D

_{ext}, and P

_{ext} can either enter or exit the system by exchange reactions r

_{1}, r

_{8r}, r

_{4}, r

_{9}, respectively. Two reversible reactions r

_{6r} and r

_{8r} allow the reactions to proceed in either forward or backward directions. By definition reversible reactions can have either positive or negative fluxes but irreversible reactions only have non-negative fluxes. From the stoichiometric reactions, a stoichiometric matrix

for internal metabolites can be set up where rows correspond to internal metabolites and columns represent reactions. Each element s

_{ij} represents the stoichiometric coefficient of a metabolite i in reaction j. The coefficient is positive if the metabolite (product) is produced and negative if the metabolite (reactant) is consumed.

**B. Metabolic Flux Analysis.** The stoichiometric matrix

is partitioned into

and

r is partitioned into

where subscripts u, m are referred to “unmeasured” and “measured”, respectively. The calculation of

r_{u} is feasible if and only if

r_{m} is known.

**C. Flux Balance Analysis.** The objective function is to maximize flux through the secretion of desired product P when only A is considered the only substrate with r

_{1} = 1 unit.

**D. Metabolic Pathway Analysis.** By using METATOOL, elementary mode analysis identifies 8 unique elementary modes listed in the matrix form

where rows correspond to reactions and columns represent elementary modes. The asterisks indicate that these elementary modes are also extreme pathways.

**E. Geometric interpretation.** The admissible flux cone represents all possible pathways that can exist. The cone is spanned by four extreme pathways that represent the edge of the cone. Some elementary modes lie on the face and inside the cone. Metabolic Flux Analysis identifies only a pathway that lies anywhere in the cone (star in purple). For instance, metabolic flux vector in

**B** can be expressed as a nonnegative linear combination of extreme pathways or elementary modes in

**D** as follows: r = 0.3

EM^{*}_{1} + 0.75

EM^{*}_{2} + 0.7

EM^{*}_{3} + 0.25

EM^{*}_{5} (the asterisks refer to elementary modes that are also extreme pathways) or r = 0.2169

EM_{4} + 0.1669

EM_{6} + 0.0831

EM_{7} + 0.5331 EM

_{8}. Similarly, Flux Balance Analysis represents only a pathway that lies anywhere in the cone (triangle in orange) and satisfies the defined objective function. For instance, metabolic flux vector in

**C** can be expressed as a non-negative linear combination of extreme pathways or elementary modes in

**D** as follows:

r = 0.35

EM^{*}_{1} + 0.65

EM^{*}_{3} + 1.0

EM^{*}_{5} or

r = 0.65

EM_{6} + 0.35

EM_{7}. Metabolic Pathway Analysis identifies all genetically independent pathways with extreme pathways shown in blue circle and with elementary modes shown in red square.

**F.** Graphical representation of extreme pathways and elementary modes for the simple metabolic network.

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