2.6 The Chemostat

Thechemostatis used by bacteriologists to study the growth of bacteria (Hagen 2010). It allows the rapid growth of bacteria to be observed over a longer time scale. Consider a container of bacterial nutrient of volumeV. It is well stirred and containsybacteria with concentrationC=y/V. Some of the nutrient solution is removed at rateQand replaced by fresh nutrient. The bacteria in the solution are reproducing at rateb. The rate of change ofyis

Therefore the growth rate is slowed to

and can be adjusted by varyingHowever, Russ and I didn't write any new homework problems for this section. If a topic is worth discussing in the text, then it is worth creating homework problems to reinforce and extend that discussion. So, here are some new problems about the chemostat.Q."

Problem 21.1. Often a chemostat is operated in steady state.

(a) Determine the solution removal rateQrequired for steady state, as a function of the bacteria reproduction rateband the container volumeV, using Eq. 2.22. Determine the units ofb,Q, andV, and verify that your expression forQhas the correct dimensions.

(b) If the rateQis larger than the steady-state value, what is happening physically?

(c) Sometimesbvaries with some external parameter (for example, temperature or glucose concentration), and you want to determinebas a function of that parameter. Suppose you can controlQand you can measure the number of bacteriay. Qualitatively design a way to determinebas your external parameter changes, assuming that for each value of the parameter your chemostat reaches steady state. (If unsure how to begin, take a look at Sec. 6.13.1 about thevoltage clampused in electrophysiology.)

Russ and I cite an American Journal of Physics article about the exponential growth of bacteria, written by Stephen Hagen (volume 78, pages 1290-1296, 2010). Here is what Hagen says about the chemostat.Problem 21.2. Consider an experiment using a chemostat in which the bacteria's reproduction ratebslows as the number of bacteriayincreases.

(a) Modify Eq. 2.22 so that "b" becomes "b(1 -y/y_{∞})", analogous to the logistic model (Sec. 2.10).

(b) Determine the value ofyonce the chemostat reaches steady state, as a function ofQ,V,b, andy_{∞}.

(c) Suppose your chemostat has a volume of 1.7 liters. You measure the steady state value ofy(arbitrary units) for different values ofQ(liters per hour), as shown in the table below. PlotyversusQ, and determinebandy_{∞}.

Qy0.2 11.64 0.4 9.47 0.6 7.31 0.8 5.14 1.0 2.98

Problem 21.3. Let the growth rate of the bacteria in your chemostat be limited by a small, constant amount of some essential metabolite, so the term "by" in Eq. 2.22 is replaced by a constant "a".

(a) Find an expression for the solution removal rateQin terms ofa, the number of bacteriay, and the chemostat volumeV, when the chemostat is in steady state.

(b) Determine the time constant governing how quickly the chemostat reaches steady state (Hint: see Sec. 2.8).

"Because the growth rate of the cell determines its size and chemical composition, a device that allows us to fine tune the growth rate will select the physiological properties of the cells. The bacterial chemostat is such a machine. In the chemostat a bacterial culture grows in a well-stirred vessel while a supply of fresh growth medium is fed into the vessel at a fixed flow rateI like the analogy to the nuclear reactor. Adjusting the flow rate in a chemostat is like pulling the cadmium control rods in and out of an atomic pile (except it is less dangerous).Q(volume/time). At the same time, the medium (containing bacteria) is continuously withdrawn from the vessel at the same rate so as to maintain constant volumeV. Thus, the bacterial population is continuously diluted at a rateD=Q/V. If this dilution rate exceeds the growth ratek[ourb], the population is diluted, which allows its growth to accelerate until it matches the dilution rate,k=Q/V. (IfDis too large, the culture will be diluted away entirely.) Therefore, the chemostat allows the experimenter to select the growth rate by selectingQ. Because it harnesses an exponential growth process to produce a tunable, steady output, we might think of the chemostat as the microbiological analog of a nuclear fission reactor. Interestingly, the chemostat reactor was first described by the physicist Leo Szilard (with Aaron Novick), who also (with Enrico Fermi) patented the nuclear reactor."

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