. 2020 Oct 22;117(45):27795–27804. doi: 10.1073/pnas.2013061117

Table 1.

Common examples of scalable and asymptotically scalable flux functions, where $x_{j}, x_{k}$ are upstream nodes of the flux, $x_{z}$ is a maintenance node of the flux, and $x_{p}$ is a regulatory node of the flux

Scalable flux functions		Asymptotically scalable flux functions
$r X_{j}$	$r > 0$	All scalable fluxes
$r Y_{j}^{a} Y_{k}^{b} N$	$r > 0; a, b \geq 1$	$X_{j} (\frac{b_{0} + b_{1} X_{p} + b_{2} X_{p}^{2}}{a_{0} + a_{1} X_{p} + a_{2} X_{p}^{2}})$	$a_{j}, b_{j} > 0$
$\frac{r X_{j}^{θ} X_{z}}{X_{j}^{θ} + K N^{θ}}$	$r, K > 0$ ; $θ \geq 1$	$(a + b e^{- c X_{p}^{m}}) X_{j}$	$a, b, c > 0$ , $m \geq 1$
$\frac{r X_{j} X_{k}}{a X_{j} + b X_{k} + c N}$	$r, a, b, c > 0$	$[a + \tan^{- 1} (b X_{p})] X_{j}$	$a > \frac{π}{2}$
$[a + b \sin (\frac{c X_{p}}{N})] X_{j}$	$a > b > 0; c > 0$	$[a + \sin (\frac{b + X_{p}}{c + X_{p}})] X_{j}$	$a > 1; b, c > 0$

The system size is $N = X_{1} + \dots + X_{n}$ .