F(m
_{
i
})

=

$\sum _{{r}_{\eta}\sum \mathcal{R},{a}_{i,\eta}>0}{a}_{i,\eta}{v}_{\eta}$ = $\sum _{{r}_{\eta}\sum \mathcal{R},{a}_{i,\eta}<0}\left{a}_{i,\eta}\right{v}_{\eta}$

Total metabolite rate involved in the production of an intermediary metabolite m
_{
i
}, before its degradation by other reactions

d
_{
k,i
}

=

$\left\{\begin{array}{ll}1& \text{if}\phantom{\rule{2.35982pt}{0ex}}k=i\\ \sum _{{r}_{j}\sum \mathcal{R},{a}_{i,j}<0}\frac{\left{a}_{i,j}\right{v}_{j}}{F\left({m}_{i}\right)}\ast I{n}^{{r}_{j}}\left({m}_{k}\right)& \text{otherwise.}\end{array}\right.$

Ratio of a product flux (${m}_{k}\sum \mathcal{P}\cup O$) on the production of ${m}_{i}\sum \mathcal{I}\cup P\cup O$

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]

=

$\left(\begin{array}{l}{x}_{I}^{\left(m\right)}\\ {x}_{P}[x,m]\\ {x}_{O}[x,m]\end{array}\right)$

Rates of fluxes of component brought by the minput flux appearing in the composition of each metabolites. ${x}_{I}^{\left(m\right)}=(0,\dots ,0,C(m\left){v}_{I}\right(m),0,\dots ,0)$.

${D}_{2}\left[v\right]{x}_{I}^{\left(m\right)}$

=

${D}_{1}\left[v\right]\left(\begin{array}{l}{x}_{P}[x,m]\\ {x}_{O}[x,m]\end{array}\right)$

Constraints on component fluxes deduced from thematterinvariance law, derived from Eq.(2) below.
