I n r(m k ) = $a k , j C ( m k ) ∑ m i ∑ ℳ , a i , j < 0 | a i , j | C ( m i ) if a k , j > 0 0 otherwise.$ Ratio of the flux of component c provided as substrate to a reaction r j recovered in the composition of the product m k . It is the sum of individual substrate contributions.
F(m i ) = $∑ r η ∑ R , a i , η > 0 a i , η v η$ = $∑ r η ∑ R , a i , η < 0 | a i , η | v η$ Total metabolite rate involved in the production of an intermediary metabolite m i , before its degradation by other reactions
d k,i = $1 if k = i - ∑ r j ∑ R , a i , j < 0 | a i , j | v j F ( m i ) ∗ I n r j ( m k ) otherwise.$ Ratio of a product flux ($m k ∑P∪O$) on the production of $m i ∑I∪P∪O$
x[v,m] = $x I ( m ) x P [ x , m ] x O [ x , m ]$ Rates of fluxes of component brought by the m-input flux appearing in the composition of each metabolites. $x I ( m ) =(0,…,0,C(m) v I (m),0,…,0)$.
$D 2 [v] x I ( m )$ = $- D 1 [v] x P [ x , m ] x O [ x , m ]$ Constraints on component fluxes deduced from thematter-invariance law, derived from Eq.(2) below.
1. We are given a stoichiometric matrix A and a vector $c∑ R p + n + q$ which describes the component composition of all metabolites. If v is a fixed flux distribution which is compatible with the stoichiometry of the system, A I O[v] is a matrix whose (i,j) input describes the proportion of component quantity contained in m i which is recovered in the flux of the output m. 