The Chemostat with an Inhibitor

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. 

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An interesting part of the technique was the way in which the equal lambdas were obtained. One strain of E. coli had a growth rate that was inhibited by a chemical (Nalidixic acid) while the other strain was essentially immune to the compound (see the third graph in Figure 6.1). By adding the proper amount of the chemical, it was possible to alter the growth rate so as to make the resulting lambda values equal. The chemostat with an inhibitor will be studied in Chapter 4. 

This simply states the biologically intuitive fact that if one of the competitors could not survive in the simple chemostat, that competitor will not survive in the chemostat with an inhibitor. Thus we may assume w, > 1 and 0 < A, < 1, / = 1, 2. There are some other simple cases that can also be easily dispatched. 

As noted in [LH], the ability of Xi to consume the inhibitor (8 > 0) is of crucial importance. Lenski and Hattingh refer to this ability of Xi to detoxify the environment and note that without it (i.e., with 6 = 0 in (2.2)) p t) tends to unity as t tends to infinity. Therefore, the limiting system obtained by dropping the p equation and replacing p by 1 describes the dynamics of (2.2) on the omega limit set. This limiting system is just the equations for competition in the chemostat without an inhibitor and where is replaced by w,/(l). Competitive exclusion must then result. 

In the preceding chapter we saw how the chemostat could be modified to account for a new phenomenon - the presence of an inhibitor. In this chapter we extend the idea behind the simple chemostat to a new apparatus in order to model a property of ecological systems that it is not possible to model in the simple chemostat. The idea is to capture the essentials of the new phenomenon without destroying the tractability of the chemostat either as a mathematical model or as an experimental one. A very simple situation will be described here a more complicated one -with a less explicit (in the sense of less computable) analysis - will be discussed in the next chapter. Just as the chemostat is a basic model for competition in the simplest situation, the apparatus here shows promise of being the model for competition along a nutrient gradient. 

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 hypothesis of Proposition 3.2 excludes the case Ai = A2. This is ordinarily not biologically important because the A, are computed from measured quantities it is unlikely that they would be exactly the same (or the same with respect to this environment). However, an interesting potential application is the case where the organisms are indeed the same (mutants of the same organism) except for their sensitivity to the antibiotic. Intuitively, if the organisms are the same except for sensitivity to the inhibitor, one expects the X population to lose the competition when the inhibitor is present. However, establishing this mathematically cannot be done directly from the comparison theorem as used before, since if A] = A2 then coexistence occurs with the chemostat equations used for comparison purposes. In order to use the comparison principle, one needs a better estimate than f(p) <1, p>0. This is the purpose of the following lemma. 

The theorem shows clearly that plasmid loss is detrimental (or fatal) to the production of the chemostat. To compensate for this possibility, in commercial production a plasmid that codes for resistance to an antibiotic is added to the DNA that codes for the item to be produced. Thus, if the plasmid is lost then the wild type is susceptible to (inhibited by) the antibiotic. The antibiotic is introduced into the feed bottle along with the nutrient. The dynamics produced by adding an inhibitor to the chemostat was modeled in Chapter 4. A new direction for research on chemostat models would be to include the inhibitor, as in Chapter 4, and the plasmid model of this section (or one of the more general models) into the same model. This is a mathematically more difficult problem to analyze, since the reduced system will not be planar. Moreover, because the methods of monotone dynamical systems do not apply, other techniques would need to be found in order to obtain global results. The model also assumes extremely simple behavior for the plasmid more could be included in a model. 

Limit cycles also appear in Chapter 4, but no bifurcation theorems were used (although the Hopf bifurcation theorem could have been used). Uniqueness of these cycles is a question of major interest and importance. To more accurately model the chemostat as it is used in commercial production, the plasmid model discussed in Chapter 10 should be combined with the inhibitor model of Chapter 4. More specifically, consider two organisms - differing only by the presence or absence of a plasmid that confers immunity to the inhibitor - competing in a chemostat (equation (4.2) of Chapter 9 modified for the presence of the inhibitor). The techniques of analysis used in Chapter 4 do not apply, since the system is not competitive in the mathematical sense. Yet an understanding of this system would be very important. 

The multi-stream multi-stage system is a valuable means for obtaining steady-state growth when, in a simple chemostat, the steady-state is unstable eg when the growth-limiting substrate is also a growth inhibitor. This system can also be used to achieve stable conditions with maximum growth rate, an achievement that is impossible in a simple chemostat (substrate-limited continuous culture). 

The rest point Eq always exists, and 2 exists with xj = 1 — A2 and p the root of (3.5) if 0 < A2 < 1, which is contained in our basic assumption (3.6). The existence of 1 is a bit more delicate. In keeping with the definitions in (3.3), define Aq = fli/(mi/(l)-l). Then 0< Aq < 1 corresponds to the survivability of the first population in a chemostat under maximal levels of the inhibitor. Easy computations show that 1 = (1 — Aq, 0,1) will exist if Aq > 0 and will have positive coordinates and be asymptotically stable in the X -p plane if 0 < Aq < 1. If 1 — Aq is negative then [ is neither meaningful nor accessible from the given initial conditions, since thex2-p plane is an invariant set. The stability of either 1 or 2 will depend on comparisons between the subscripted As. The local stability of each rest point depends on the eigenvalues of the linearization around those points. The Jacobian matrix for the linearization of (3.2) at i = 1,2, takes the form... 

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