Table 5 Open in new tab Definition of... | Oxford Academic

Table 5

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Definition of functions appearing in process rate expressions and stoichiometric factors from Table 3.

Symbol	Description
monod	Monod model with the first argument (x) being a concentration or abundance and the second argument (h) being a half saturation constant. The non-standard third argument (z) is a horizontal shift parameter. The latter represents a minimum concentration / abundance below which the function returns zero. $m o n o d (x, h, z) = \{\begin{matrix} 0 & for x - z \leq 0 \\ \frac{x - z}{x - z + h} & else \end{matrix})$
inh	Standard linear inhibition model. The first argument is the concentration of the inhibitor (x), the second argument is the minimum inhibitory concentration (mic). $i n h (x, m i c) = \{\begin{matrix} 0 & for x \geq m i c \\ 1 - \frac{x}{m i c} & else \end{matrix})$
on	Steep sigmoidal function to represent an “upward” step. Returns zero if the first argument (x) falls below a threshold defined by the second argument (t), otherwise returns one. The coefficients a and b controlling the steepness were chosen to be a = 0.9 t and b = 1.1 t. $o n (x, t) = \{\begin{matrix} 0 \\ \frac{1}{2} \\ 1 \\ 1 \end{matrix} \begin{matrix} for x < a \\ {(\frac{x - a}{t - a})}^{2} & for a \leq x < t \\ - \frac{1}{2} {(\frac{b - x}{b - t})}^{2} & for t \leq x \leq b \\ for x > b \end{matrix})$
off	Similar to the “on” function but the step is downward from 1 to 0 as the first argument exceeds the threshold defined by the second argument. $o f f (x, t) = 1 - o n (x, t)$

Symbol	Description
monod	Monod model with the first argument (x) being a concentration or abundance and the second argument (h) being a half saturation constant. The non-standard third argument (z) is a horizontal shift parameter. The latter represents a minimum concentration / abundance below which the function returns zero. $m o n o d (x, h, z) = \{\begin{matrix} 0 & for x - z \leq 0 \\ \frac{x - z}{x - z + h} & else \end{matrix})$
inh	Standard linear inhibition model. The first argument is the concentration of the inhibitor (x), the second argument is the minimum inhibitory concentration (mic). $i n h (x, m i c) = \{\begin{matrix} 0 & for x \geq m i c \\ 1 - \frac{x}{m i c} & else \end{matrix})$
on	Steep sigmoidal function to represent an “upward” step. Returns zero if the first argument (x) falls below a threshold defined by the second argument (t), otherwise returns one. The coefficients a and b controlling the steepness were chosen to be a = 0.9 t and b = 1.1 t. $o n (x, t) = \{\begin{matrix} 0 \\ \frac{1}{2} \\ 1 \\ 1 \end{matrix} \begin{matrix} for x < a \\ {(\frac{x - a}{t - a})}^{2} & for a \leq x < t \\ - \frac{1}{2} {(\frac{b - x}{b - t})}^{2} & for t \leq x \leq b \\ for x > b \end{matrix})$
off	Similar to the “on” function but the step is downward from 1 to 0 as the first argument exceeds the threshold defined by the second argument. $o f f (x, t) = 1 - o n (x, t)$

All functions return a unitless result in range 0–1.

Table 5

Open in new tab

Definition of functions appearing in process rate expressions and stoichiometric factors from Table 3.

Symbol	Description
monod	Monod model with the first argument (x) being a concentration or abundance and the second argument (h) being a half saturation constant. The non-standard third argument (z) is a horizontal shift parameter. The latter represents a minimum concentration / abundance below which the function returns zero. $m o n o d (x, h, z) = \{\begin{matrix} 0 & for x - z \leq 0 \\ \frac{x - z}{x - z + h} & else \end{matrix})$
inh	Standard linear inhibition model. The first argument is the concentration of the inhibitor (x), the second argument is the minimum inhibitory concentration (mic). $i n h (x, m i c) = \{\begin{matrix} 0 & for x \geq m i c \\ 1 - \frac{x}{m i c} & else \end{matrix})$
on	Steep sigmoidal function to represent an “upward” step. Returns zero if the first argument (x) falls below a threshold defined by the second argument (t), otherwise returns one. The coefficients a and b controlling the steepness were chosen to be a = 0.9 t and b = 1.1 t. $o n (x, t) = \{\begin{matrix} 0 \\ \frac{1}{2} \\ 1 \\ 1 \end{matrix} \begin{matrix} for x < a \\ {(\frac{x - a}{t - a})}^{2} & for a \leq x < t \\ - \frac{1}{2} {(\frac{b - x}{b - t})}^{2} & for t \leq x \leq b \\ for x > b \end{matrix})$
off	Similar to the “on” function but the step is downward from 1 to 0 as the first argument exceeds the threshold defined by the second argument. $o f f (x, t) = 1 - o n (x, t)$

Symbol	Description
monod	Monod model with the first argument (x) being a concentration or abundance and the second argument (h) being a half saturation constant. The non-standard third argument (z) is a horizontal shift parameter. The latter represents a minimum concentration / abundance below which the function returns zero. $m o n o d (x, h, z) = \{\begin{matrix} 0 & for x - z \leq 0 \\ \frac{x - z}{x - z + h} & else \end{matrix})$
inh	Standard linear inhibition model. The first argument is the concentration of the inhibitor (x), the second argument is the minimum inhibitory concentration (mic). $i n h (x, m i c) = \{\begin{matrix} 0 & for x \geq m i c \\ 1 - \frac{x}{m i c} & else \end{matrix})$
on	Steep sigmoidal function to represent an “upward” step. Returns zero if the first argument (x) falls below a threshold defined by the second argument (t), otherwise returns one. The coefficients a and b controlling the steepness were chosen to be a = 0.9 t and b = 1.1 t. $o n (x, t) = \{\begin{matrix} 0 \\ \frac{1}{2} \\ 1 \\ 1 \end{matrix} \begin{matrix} for x < a \\ {(\frac{x - a}{t - a})}^{2} & for a \leq x < t \\ - \frac{1}{2} {(\frac{b - x}{b - t})}^{2} & for t \leq x \leq b \\ for x > b \end{matrix})$
off	Similar to the “on” function but the step is downward from 1 to 0 as the first argument exceeds the threshold defined by the second argument. $o f f (x, t) = 1 - o n (x, t)$

All functions return a unitless result in range 0–1.