= Monod equation
{c}
{title2=$K(C)=K_{\max}C/(K_0+C)$}
{wiki}
The <Monod equation> is an empirical saturating law for the <specific growth rate> of a culture limited by one nutrient. Write $s\ge0$ for its residual nutrient <concentration>, $\mu_{\max}>0$ for the limiting <specific growth rate>, and $K_s>0$ for the nutrient <concentration> at half that rate:
$$
\mu(s)=\mu_{\max}\frac{s}{K_s+s},\qquad \mu(K_s)=\frac{\mu_{\max}}2.
$$
Here $\mu$ and $\mu_{\max}$ have units of inverse time, whereas $K_s$ has the same units as $s$. The notation $K(C)$ in the title denotes the same growth function, with $C=s$, $K_{\max}=\mu_{\max}$ and $K_0=K_s$. For $s\ll K_s$, $\mu(s)\approx(\mu_{\max}/K_s)s$; for $s\gg K_s$, $\mu(s)$ approaches $\mu_{\max}$. Its <derivative> $\mu'(s)=\mu_{\max}K_s/(K_s+s)^2$ is positive: more nutrient increases the <specific growth rate>, with progressively smaller increments.
For example, take $\mu_{\max}=1\,\mathrm{h}^{-1}$ and $K_s=1\,\mathrm{g}\,\mathrm{L}^{-1}$. Nutrient <concentrations> of $0$, $1$ and $9\,\mathrm{g}\,\mathrm{L}^{-1}$ give <specific growth rates> $0$, $0.5$ and $0.9\,\mathrm{h}^{-1}$. In a <chemostat> with <dilution rate> $D=0.5\,\mathrm{h}^{-1}$, the intersection $\mu(s)=D$ selects the residual nutrient <concentration> $s=1\,\mathrm{g}\,\mathrm{L}^{-1}$, provided the feed can supply enough nutrient for a positive <biomass concentration>.
\Image[/monod-growth-and-dilution.png]
{title=<Monod equation> growth curve and the <dilution rate> selecting a <chemostat steady state>}
The <Monod equation> is useful because two measurable parameters describe both nutrient scarcity and saturation. It assumes the other conditions affecting growth remain controlled and does not represent inhibition at high nutrient <concentration>. Although its formula resembles the <Michaelis-Menten equation>, a whole culture's <specific growth rate> combines many processes; the resemblance alone does not derive it from <Michaelis-Menten kinetics> or identify $K_s$ with a <Michaelis constant>.
In his 1949 review, Monod presents the hyperbolic rate law as an empirical fit in equation (2), pp. 383-384, and discusses the difficulty of identifying a single growth-controlling reaction. His reported division rates use doublings per unit time; the natural-exponential <specific growth rate> used here is that division rate multiplied by $\ln2$. https://garcialab.berkeley.edu/courses/papers/Monod1949.pdf[Jacques Monod, The Growth of Bacterial Cultures, Annual Review of Microbiology 3 (1949), 371-394, DOI: 10.1146/annurev.mi.03.100149.002103]{ref}.
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