Friday, June 24, 2016

Chemostat Homework Problems

In the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a section on the chemostat.
2.6  The Chemostat
The chemostat is 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 volume V. It is well stirred and contains y bacteria with concentration C = y/V. Some of the nutrient solution is removed at rate Q and replaced by fresh nutrient. The bacteria in the solution are reproducing at rate b. The rate of change of y is
An equation governing the number of bacteria in a chemostat.
Therefore the growth rate is slowed to
A mathematical expression for the bacteria growth rate in a chemostat.
and can be adjusted by varying Q.
However, Russ and I didn’t write any new homework problems for this section. If a topic is worth discussing in the text, then it’s worth creating homework problems to reinforce and extend that discussion. So, here are some new problems about the chemostat.
Problem 21.1.  Often a chemostat is operated in steady state.
(a) Determine the solution removal rate Q required for steady state, as a function of the bacteria reproduction rate b and the container volume V, using Eq. 2.22. Determine the units of b, Q, and V, and verify that your expression for Q has the correct dimensions.
(b) If the rate Q is larger than the steady-state value, what is happening physically?
(c) Sometimes b varies with some external parameter (for example, temperature or glucose concentration), and you want to determine b as a function of that parameter. Suppose you can control Q and you can measure the number of bacteria y. Qualitatively design a way to determine b as 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 the voltage clamp used in electrophysiology.)
Problem 21.2.  Consider an experiment using a chemostat in which the bacteria's reproduction rate b slows as the number of bacteria y increases.
(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 of y once the chemostat reaches steady state, as a function of Q, V, b, and y.
(c) Suppose your chemostat has a volume of 1.7 liters. You measure the steady state value of y (arbitrary units) for different values of Q (liters per hour), as shown in the table below. Plot y versus Q, and determine b and y.

 Q    y
 0.2 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 rate Q in terms of a, the number of bacteria y, and the chemostat volume V, 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).
Screenshot of Exponential Growth of Bacteria: Constant Multiplication Through Division, by Stephen Hagen (American Journal of Physics, 78:1290–1296, 2010).
“Exponential Growth of Bacteria:
Constant Multiplication Through Division,”
by Stephen Hagen.
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’s what Hagen says about the chemostat.
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 rate 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 volume V. Thus, the bacterial population is continuously diluted at a rate D = Q/V. If this dilution rate exceeds the growth rate k [our b], the population is diluted, which allows its growth to accelerate until it matches the dilution rate, k = Q/V. (If D is too large, the culture will be diluted away entirely.) Therefore, the chemostat allows the experimenter to select the growth rate by selecting Q. 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."
I 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’s less dangerous).

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