TABLE III.

Commonly used process representations in biochemistry and biology (continued).

Process type	Mathematical form	Symmetric	Additional comments
Mass-action	$\alpha {\prod }_i{x}_i^{n_i}$	yes	$\alpha$ is the rate constant, and $n_i$ is the stoichiometric coefficient of species $i$ [5].
Competitive product inhibition	$\frac{V_{\text{max}}S}{S+K_{\mathrm{m}}\left(1+\frac{P}{K_{\mathrm{m}}^P}\right)}$	no	$V_{\text{max}}$ is the maximum reaction rate, $K_{\mathrm{m}}$ is the Michaelis–Menten constant for the substrate, and $K_{\mathrm{m}}^P$ the product concentration at which the forward reaction rate is half-maximal [2, 3, 66].
Substrate inhibition (Haldane equation)	$\frac{V_{\text{max}}S}{S\left(1+\frac{S}{K_{\mathrm{i}}}\right)+K_{\mathrm{m}}}$	n/a	$V_{\text{max}}$ is the maximum reaction rate, $K_{\mathrm{m}}$ is the Michaelis–Menten constant for the substrate, and $K_{\mathrm{i}}$ is the substrate inhibition constant [67].
Hill	$\frac{V_{\mathrm{m}\mathrm{a}\mathrm{x}}{S}^h}{S^h+K_{\mathrm{m}}^h}$	n/a	$V_{\text{max}}$ is the maximum reaction rate, $K_{\mathrm{m}}$ is the Michaelis–Menten constant for substrate $S$, and $h$ is the Hill coefficient representing the cooperativity of the binding sites.
Reversible Hill	$\frac{V_{\mathrm{m}\mathrm{a}\mathrm{x}}\frac{S}{K_{\mathrm{m}}^S}{\left(\frac{S}{K_{\mathrm{m}}^S}+\frac{P}{K_{\mathrm{m}}^P}\right)}^{h-1}}{1+{\left(\frac{S}{K_{\mathrm{m}}^S}+\frac{P}{K_{\mathrm{m}}^P}\right)}^h}$	no	$V_{\text{max}}$ is the maximum reaction rate, $K_{\mathrm{m}}^S$ is the Michaelis–Menten constant for substrate $S$, $K_{\mathrm{m}}^P$ is the Michaelis–Menten constant for product $P$, and $h$ is the Hill coefficient representing the cooperativity of the binding sites [68].
Saturable and cooperative formalism	$V{\prod }_i\frac{x_i^{n_i}}{K_i+x_i^{n_i}}$	yes	$V$ represents the saturated reaction rate for large values of all $x_i$, and $K_i$ and $n_i$ are parameters associated with variable $x_i$, encoding sensitivity to and saturation with respect to $x_i$ [25].
Monod	${\mu }_{\mathrm{m}\mathrm{a}\mathrm{x}}\frac{S}{S+K_{\mathrm{m}}}X$	no	${\mu }_{\text{max}}$ is the maximum specific growth rate under saturating substrate concentrations, $K_{\mathrm{m}}$ is the Michaelis–Menten constant for substrate $S$, and $X$ is the biomass concentration [27, 28].
Holling type I	$a{T}_{\mathrm{s}}XP$	no	$a$ is a proportionality constant, $T_{\mathrm{s}}$ is the prey-searching time, $X$ is the prey density, and $P$ is the predator density.
Holling type II	$\frac{a{T}_{\mathrm{t}}X}{1+abX}P$	no	$a$ is a proportionality constant, $T_{\mathrm{t}}$ is the total time available for searching and handling prey, $b$ is the handling time per prey item, $X$ is the prey density, and $P$ is the predator density [32, 33].