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

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

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 $m i ∑ I ∪ P ∪ O$
D ₁[v]	=	(d _k,i)_{p<k,i≤p+n+q}	Linear transformation of matter components contained in intermediary or output metabolites
D ₂[v]	=	(d _k,i)_{p<k≤p+n+q,1≤i≤p}	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.

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.

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