A stoichiometric matrix, S (m × n), was constructed for the M. barkeri metabolic network where m is the number of metabolites and n is the number of reactions. The corresponding entry in the stoichiometric matrix, Sij, represents the stoichiometric coefficient for the participation of the ith metabolite in the jth reaction. FBA was then used to solve the linear programming problem under steady-state criteria (Varma and Palsson, 1994 (link); Kauffman et al, 2003 (link); Price et al, 2004 (link)). The linear steady-state problem can be represented by the equation:
where v(n × 1) is a vector of reaction fluxes. Because the linear problem is normally an underdetermined system for genome-scale metabolic models, there exists multiple solutions for v that satisfy equation (1). To find a solution for v, the cellular objective of producing the maximal amount of biomass constituents, represented by the ratio of metabolites in the BOF, is optimized in the linear system. This is achieved by adding an additional column vector to S, Si,BOF, containing the stoichiometric coefficients for the metabolites in the BOF and then subsequently maximizing the reaction flux through the corresponding element in v, vBOF, under the steady-state criteria. Additionally, constraints that are imposed on the system are in the form of:
where αi and βi are the lower and upper limits placed on each reaction flux, vi, respectively. For reversible reactions, −∞⩽vi⩽∞, and for irreversible reactions, 0⩽vi⩽∞.
The constraints on the reactions that allow metabolites entry to the extracellular space were set to 0⩽vi⩽∞ if the metabolite was not present in the medium, meaning that the compounds could leave, but not enter the system. For the metabolites that were in the medium, the constraints were set to −∞⩽vi⩽∞ for all except the limiting substrate and cysteine. When cysteine was a media component, it was allowed only for use as a source of sulfur by restricting hydrogen sulfide from exiting the system. Artificial transhydrogenase cycles in the network (Reed et al, 2006 (link)) were avoided by only allowing the net flux through a set of potential NAD(H)/NADP(H) cycling reactions in one direction. The reaction flux through the BOF was constrained from 0⩽vBOF⩽∞ and the BOF was generated as a linear equation consisting of the molar amounts of metabolic constituents that make up the dry weight content of the cell (Table III) and a GAM (mmol ATP gDW−1) reaction to account for nonmetabolic growth activity,
The full BOF is included in Supplementary information 1.
Aside from the BOF, an NGAM (mmol ATP gDW−1 h−1) value was used as an energy ‘drain' on the system during the linear programming calculations and accounts for nongrowth cellular activities (Pirt, 1965 (link)). The NGAM was represented as a set flux in the reaction flux vector, vNGAM. The corresponding reaction vector in the stoichiometric matrix, Si,NGAM, was in the form of an ATP maintenance reaction identical to equation (3).
Linear programming calculations were performed using the previously mentioned SimPheny™ software platform and the MATLAB®, version 7.0.0.19920 (The MathWorks Inc., Natick, MA) software platform on which the linear programming package LINDO (Lindo Systems Inc., Chicago, IL) was used as a solver.