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Table 10 Modeling the quantitative allocation of input nutrients in output products

From: Exploring metabolism flexibility in complex organisms through quantitative study of precursor sets for system outputs

I n r(m k ) = a k , j C ( m k ) m i , 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 PO) on the production of m i IPO
D 1[v] = (d k,i ) p<k,ip+n+q Linear transformation of matter components contained in intermediary or output metabolites
D 2[v] = (d k,i ) p<kp+n+q,1≤ip Linear transformation of matter component contained in input metabolites
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.