Mathematical representation for flux balance analysis

Metabolic fluxes are responsible for maintaining the homeostatic state of the cell. This condition may be translated into the assumption that the metabolic network functions in or near a non-equilibrium steady state (NESS). That is, all of the concentrations are treated as constant in time. Under this assumption, the biochemical fluxes are balanced to maintain constant concentrations of all internal metabolic species. If the stoichiometry of a system made up of M species and N fluxes is known, then the stoichiometric numbers can be systematically tabulated in a 

M x N matrix, known as the stoichiometric matrix [31, 32], The S,7 entries of the stoichiometric matrix are determined by the stoichiometric numbers appearing in the reactions in the network. For example, if the j 111 reaction has the form  

The fundamental law of conservation of mass dictates that the vector of steady state fluxes, J, satisfies 

consider that species A is transported into the system at rate bA and species B is transported out at rate Then the mass-balance equations J = b can be expressed  

Algebraic analysis of this equation reveals that mass-balanced solutions exist if and only if bA = bs- Equation (9.5) can be simplified to J2 = h = J —bA. Thus, mass balance does not provide unique values for the internal reaction fluxes. In fact, for this example, solutions exist for 

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