The General Chemostat

In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincare-Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis-Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis-Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing death rates (replacing the parameter D by D, in the equations) does not change the competitive exclusion result. 

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Several special terms are used to describe traditional reaction engineering concepts. Examples include yield coefficients for the generally fermentation environment-dependent stoichiometric coefficients, metabolic network for reaction network, substrate for feed, metabolite for secreted bioreaction products, biomass for cells, broth for the fermenter medium, aeration rate for the rate of air addition, vvm for volumetric airflow rate per broth volume, OUR for 02 uptake rate per broth volume, and CER for C02 evolution rate per broth volume. For continuous fermentation, dilution rate stands for feed or effluent rate (equal at steady state), washout for a condition where the feed rate exceeds the cell growth rate, resulting in washout of cells from the reactor. Section 7 discusses a simple model of a CSTR reactor (called a chemostat) using empirical kinetics. 

The biological conclusion is, of course, that differing removal rates do not alter competitive exclusion in the chemostat. One anticipates that a similar conclusion is true if the Michaelis-Menten dynamics is replaced by the general monotone term f,(S) used in Section 3. However, the Liapunov calculations depend on this form and the general question is still unresolved. 

We take as the model that of the simple chemostat of Chapter 1, with input nutrient S(t) and organism x t) growing on that nutrient, and add two predators on x which we label y and z- It is assumed that both the nutrient uptake from the lowest level and the predation from the highest level follow Michaelis-Menten or Monod kinetics. The use of the Monod formulation has already been discussed for the consumption of the nutrient. That the same format should apply in the case of a predator feeding on prey is not immediately clear. This formulation is one of a general class known as a Holling s type-II functional responses [Hoi]. A nice discussion can be found in [MD, p. 5], which we repeat here. 

Armed with the preceding discussion, we can now state the main theorem of this chapter. It shows that, in contrast to the basic chemostat, coexistence can occur if the competition is at a higher trophic level. (We remind the reader of the general assumption of hyperbolicity of limit cycles.)... 

In the first two chapters the general theory of the chemostat was developed, and it was shown that competitive exclusion is the expected outcome. In Chapter 3, coexistence was shown to occur when the competition was at a higher trophic level the mechanism was simply the oscillation of the object of the competition - the prey in the case being considered. In this chapter, we return to the basic chemostat model but add another factor, the presence of an inhibitor. The inhibitor affects the nutrient uptake rate of one of the competitors but is taken up by the other without ill effect. The use of Nalidixic acid in the experiments of Hansen and Hubbell [HH], discussed in Chapter 1, is an example. Its effect on one strain of E. coli was essentially nil while the growth rate of the other was severely diminished. 

As noted in the discussion section of Chapter 2, there remains a gap in our knowledge of the basic chemostat model in the case of differing removal rates for the competitors. The principal open problem is to extend the result of [Hsul], described in Section 4 of Chapter 2, to general monotone uptake functions. It would also be desirable to include the class of not necessarily monotone uptake functions identified in [BWol], The recent work [WLu] represents a major step in this direction. 

In the pharmaceutical industry, the reactors are closed to maintain aseptic conditions, and agitation and aeration are provided as needed. The classic example is the Monod chemostat, often found in laboratories. A representative chemostat, fermentation vessel, or bioreactor, is shown in Fig. 2 with an air sparger, agitator, and feed or exit streams. The continuous flow system is usually replaced by a batch process for production in the pharmaceutical and biotechnology industries. The general relationships for a constant volume system are developed in terms of cell concentration (A), often the component of interest, and substrate concentration (5), such as a sugar or oxygen. 

Another specific but important application of the CSTR is in biochemical reactor systems, for both evaluation of kinetic parameters in the laboratory and in commercial operation. The CSTR in such applications is most often referred to as a chemostat. Let us consider, then, the analysis for a typical unstructured culture of micro-organisms. Recall the general form of mass balance, equation (4-51),... 

An informative example of the general patterns of behavior of the chemostat upon variation of D has been given by Bailey and Ollis [J.E. Bailey and D.F. Ollis, Biochemical Engineering Fundamentals, McGraw-FIill Book Co., New York, NY, (1977)] and is shown in Figure 4.11 for a typical set of parameters. Note that the reactor is very sensitive to changes in D when operation is at conditions near washout. This sensitivity becomes particularly important if one wishes to maximize the amount of biomass effluent from the reactor, CD. Flere of course the requirement is... 

In this case we assume that we know the dilution rate (D-F/V) precisely as a function of time. In a chemostat D is often constant since the feed flowrate and the volume are kept constant. In a fed-batch culture the volume is continuously increasing. The dilution rate generally varies with respect to time although it could also be kept constant if the operator provides an exponentially varying feeding rate. 

As noted in the introduction, the model is that of a standard chemostat with two competitors, but with the added feature that an inhibitor is also input from an external source. The nutrient (and inhibitor) uptake and conversion (in the case of nutrient) are assumed to follow Michaelis-Menten dynamics. The results are probably valid for general monotone dynamics, although this has not been established. 

The purpose of this chapter is to present a model of competition in the chemostat for a single resource that takes these factors into account, and to determine the extent to which these factors influence the outcome of competition. The model presented here is a special case of a more general class of models formulated in [DMKH] and treated in [MD, sec. 1.3]. This simplified model is one of several considered in an elegant paper by Cushing [Cu2]. Most of the results of this chapter are taken from [Cu2]. 

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