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Flux Caps
IntroductionIn the typical formulation of a flux balance analysis (FBA) problem, the fluxes of the internal chemical reactions are constrained to the interval [0, ∞). This does not mean that internal fluxes are unbounded, but rather, the true upper bound is not known a priori. Using the constraint [0, ∞) simply ensures that the flux space is not artificially restricted. If the exchange reactions of a stoichiometric network are constrained to be finite, then for fundamental thermodynamic reasons, the flux of each internal reaction should be finite as well [1]. This is not always the case in the typical FBA formulation: the fluxes of reactions participating in steady state cycles (type III extreme pathways) are unbounded in the positive direction [2]. The application of flux caps is one way to prohibit these physically unrealistic unbounded fluxes. What are flux caps?Flux caps are linear combinations of fluxes that are used to bound other fluxes. They contain a single flux from each cycle in which an unbounded reaction participates. By constraining the value of a flux cap, an unbounded flux will become "capped". This will be illustrated for the stoichiometric network below.
The dotted line indicates the system boundary, and the table on the right denotes the defined flux constraints. The reactions corresponding to fluxes V1-4 are internal, and V5 and V6 represent the fluxes of exchange reactions. This stoichiometric network contains two cycles, V1 + V2 + V4 and V1 + V3, causing each of the internal fluxes to be unbounded. Flux caps can be determined for each flux of this stoichiometric network by visual inspection.
Notice that flux caps are minimal in the sense that they only contain a single flux from each cycle in which their corresponding reaction participates. Also notice that multiple flux caps can exist for a single reaction. If this stoichiometric network had contained "redundant" reactions (i.e. multiple reactions that are stoichiometrically equivalent), they could appear together in the same flux caps as well. In other words, it is unnecessary to create separate flux caps for redundant reactions. An algorithm for determining all the possible flux caps in a stoichiometric network is described below. Identifying all possible flux capsStoichiometric networks often contain stoichiometrically equivalent (i.e. "redundant") reactions. Let ei denote any particular set of stoichiometrically equivalent reactions in a network S. Let E denote the collection of all sets ei in S. Designate an arbitrary reaction from each set ei in E as "primary" and the other reactions as "secondary". The following steps are executed for each reaction R that participates in a cycle:
Using flux capsFlux caps are used to compute Mahadevan-Schilling flux intervals. This will be illustrated for the stoichiometric network and flux caps described above. The first step is to determine finite upper bounds for each of the reactions participating in cycles. For V1:
For V3:
The upper bounds for V2 and V4 are computed in similar ways, resulting in the following values.
These values are used to define a new set of flux constraints:
Now that a finite upper bound is defined for each flux, the Mahadevan-Schilling flux intervals can be computed. Each flux is individually minimized and maximized, resulting in the following:
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