Table 2.

Cellerator user defined, regulatory, and enzymatic arrow forms.

Arrow forma,b	Typical ODE Termc
{e · X ⇒ f · Y, g(X)}	[Xi]′ = −eig(X), [Yi]′ = fig(X)
$\left\{X\stackrel{ℰ}{\mapsto }\mathrm{Y},\mathrm{\text{ Hill}}[v,n,K,a,T]\right\}$	${[\mathrm{Y}]}^{\prime }\text{=}\frac{v[ℰ]{\left(a\text{ + }T·[X]\right)}^n}{K^n\text{ + }{\left(a\text{ + }T·[X]\right)}^n}$
$\left\{X\stackrel{ℰ}{\mapsto }\mathrm{Y},\mathrm{\text{ GRN}}[v,T,n,h]\right\}$	${[\mathrm{Y}]}^{\prime }\text{=}\frac{v[ℰ]}{\text{1+}\exp \left(-h-T·{[X]}^n\right)}$
$\left\{X\stackrel{ℰ}{\mapsto }\mathrm{Y},\mathrm{\text{ SSystem}}[\tau ,C_{\text{+}},C_{-},n^{+},n^{-}]\right\}$	${[\mathrm{Y}]}^{\prime }=\frac{[ℰ]}{\tau }\left\{C_{+}{\displaystyle {\prod }_i{X}_i^{n_i^{+}}}-C_{-}{\displaystyle {\prod }_i{X}_i^{n_i^{-}}}\right\}$
$\left\{X\stackrel{ℰ}{\mapsto }\mathrm{Y},\mathrm{\text{ NHCA}}[v,\{T^{+},T^{-}\},n,m,k]\right\}$	${[\mathrm{Y}]}^{\prime }\text{=}\frac{v[ℰ]{\displaystyle {\prod }_i{\left(\text{1+}{T}_i^{+}{[\mathrm{X}]}_i^{n_i}\right)}^m}}{k{\displaystyle {\prod }_i{\left(\text{1+}{T}_i^{-}{[\mathrm{X}]}_i^{n_i}\right)}^m}\text{+}{\displaystyle {\prod }_i{\left(\text{1+}{T}_i^{+}{[\mathrm{X}]}_i^{n_i}\right)}^m}}$
{{X, Y} ⇒ rational[a, d, m, n]}	${[\mathrm{Y}]}^{\prime }d\text{=}\frac{a_{\text{0}}\mathrm{\text{ + rest}}\left(a\right)·{[X]}^n}{d_{\text{0}}\mathrm{\text{ + rest}}\left(d\right)·{[X]}^n}$
{X ↦ Y, USER[v, T, n, h, f]}	[Y]′ = vf(h − T · [X]n)
$\left\{\mathrm{X}\stackrel{ℰ}{\Rightarrow }\mathrm{Y},\mathrm{\text{ MM}}[\text{K},\text{ v}]\right\}$	${[\mathrm{Y}]}^{\prime }\text{ = }-{[\mathrm{X}]}^{\prime }\text{ = }\frac{v[ℰ][\mathrm{X}]}{K\text{ + }[\mathrm{X}]}$
$\left\{\mathrm{X}\stackrel{ℰ}{\Rightarrow }\mathrm{Y},\mathrm{\text{ MWC}}[k,n,c,L,K]\right\}$	${[\mathrm{Y}]}^{\prime }e\text{=}-{[\mathrm{X}]}^{\prime }\text{= }k[ℰ]\frac{\alpha {\left(\text{1+}\alpha \right)}^{n-\text{1}}\text{+}L\alpha \text{c}{\left(\text{1+}\alpha c\right)}^{n-\text{1}}}{{\left(\text{1+}\alpha \right)}^n\text{+}L{\left(\text{1+}\alpha c\right)}^{n-\text{1}}}$

^aThe catalyst species ℰ is optional and may be omitted.

^bBoldface quantities may be either scalars or vectors. Vectors enclosed by curly brackets. The notation v = p^q means a vector v with components v_i = p^qi_i.

^cIf the catalyst is omitted replace[ℰ] with 1.

^dIn rational, rest(x) is the vector x with its first element removed.

^eIn MWC, α = [X]/K.