= Chemostat
{wiki}
A <chemostat> maintains a well-mixed culture at constant volume $V$ by feeding fresh nutrient solution and removing culture at the same volumetric flow $Q$. Its <dilution rate> is $D=Q/V$. Mixing makes the outflow's nutrient <concentration> and <biomass concentration> equal to those inside the vessel; continuous replacement supplies nutrient and also removes growing cells.
The simplest one-nutrient <chemostat> uses nutrient <concentration> $s(t)$, <biomass concentration> $x(t)$, constant feed nutrient <concentration> $s_{\mathrm{in}}>0$, and constant <biomass yield> $Y>0$. Assume a cell-free feed, one growth-limiting nutrient, constant volume, and no cell retention, death or additional nutrient consumption for maintenance. With <specific growth rate> $\mu(s)$, its <ordinary differential equations> are
$$
\frac{\mathrm{d}s}{\mathrm{d}t}=D(s_{\mathrm{in}}-s)-\frac{\mu(s)x}{Y},\qquad
\frac{\mathrm{d}x}{\mathrm{d}t}=(\mu(s)-D)x.
$$
The nutrient balance contains inflow $Ds_{\mathrm{in}}$, outflow $Ds$, and consumption $\mu(s)x/Y$; the biomass balance contains production $\mu(s)x$ and removal $Dx$. If $s$ is in grams of nutrient per litre and $x$ in grams of dry biomass per litre, $Y$ is grams of dry biomass produced per gram of nutrient consumed. Thus both balances have units of their respective <concentration> per time. The <Monod equation> supplies an empirical choice of $\mu(s)$; the balances themselves express the stated production, consumption and transport assumptions.
The appeal of a <chemostat> is that, at a positive <chemostat steady state>, adjusting the pump sets the <specific growth rate> to $D$. The residual nutrient <concentration> then adjusts to that rate, and the available nutrient together with <biomass yield> sets the <biomass concentration>. This gives a controlled way to compare cultures at selected growth rates and a simple quantitative warning of <chemostat washout>.
Two 1950 publications on continuous culture are Novick and Szilard's Description of the Chemostat, Science 112(2920), 715-716, dated 15 December, and Monod's La technique de culture continue. Théorie et applications, Annales de l'Institut Pasteur 79, 390-410. The latter's original bibliographic details are reproduced in the contents of Monod's collected papers. These references identify contemporary publications without establishing priority between them. https://doi.org/10.1126/science.112.2920.715[Aaron Novick and Leo Szilard, Description of the Chemostat (1950)]{ref}. https://api.pageplace.de/preview/DT0400.9780323142632_A23647608/preview-9780323142632_A23647608.pdf[Monod's 1950 original, listed in Selected Papers in Molecular Biology by Jacques Monod, contents p. vi]{ref}.
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