Comparing the Effects of Interactive and Noninteractive Complementary Nutrients on Growth in a Chemostat

We compare the effects of interactive and noninteractive complementary nutrients on the growth of an organism in the chemostat. We also compare these two situations to the case when the nutrients are substitutable. In previous studies, complementary nutrients have been assumed to be noninteractive. However, more recent research indicates that some complementary nutrient relationships are interactive. We show that interactive complementary and substitutable nutrients can lead to higher population densities than do noninteractive complementary nutrients. We numerically illustrate that if the washout rate is high, an organism can persist at higher densities when the complementary nutrients are interactive than when they are noninteractive, which can result in the extinction of the organism. Finally, we present an example by making a small adjustment to the model that leads to a single nutrient model with an intermediate metabolite of the original substrate as the nutrient for the organism.


Introduction
We consider a basic, resource-based model of growth in the chemostat. Such models have applications in ecology to model a simple lake and in biotechnology to model the commercial bio-reactor. Experimental verification of the match between theory and experiment in the chemostat can be found in [1]. Basic growth in the chemostat is described by the dimensionless system For a detailed discussion of growth in the chemostat and a description of the constants (input of the nutrient), y S (yield constant), and D (dilution (washout) rate), see Smith and Waltman [2].
Two nutrients are complementary if they meet different needs for an organism. For example, ammonia provides nitrogen while glucose provides carbon [3] (building blocks of protein). Similarly, two nutrients are substitutable if they meet the same needs for an organism. For example, glucose, galactose, maltose, ribose, arabinose, and fructose all provide energy (sugar) [4].
See Stroot et al. [5], for a recent study of Acinetobac-ter spp. bacteria in an activated sludge bioreactor system using the noninteractive Monod model for multiple nutrients. However, other research [3,4,[6][7][8][9] indicates that a model of interactive multiple limiting nutrients may be more appropriate for some situations. Of particular interest, Lendenmann and Egli [4] discuss several growth models appropriate for substitutable interactive nutrients and compare them to the growth of E. coli with sugar nutrients glucose, galactose, maltose, ribose, arabinose, and fructose. Whang et al. [9] perform a similar study using bacteria from the wastewater of a food-processing plant. On the other hand, Bapat et al. [10] use a Monod model to study the growth of A. mediterranei S699 with multiple interactive complementary nutrients. Champagne et al. [11] form a model of cometabolism with two interactive complementary nutrients in a well-mixed system. Bae and Rittmann [12] develop a dual-limiting model, compare the results to experimental data, and observe that they agree, which provides further evidence that a multiple nutrient limiting model might be more appropriate in some situations than a single nutrient limiting model. From [12], having an accurate kinetic model for dual limitation is essential for a proper design of treatment operations such as in bioremediation or contaminated groundwater... Dual limitation also can be critical for predicting the fate of pollutants in certain natural environments, such as a deep lake or an ocean...
To consider a single organism's growth in the chemostat for two nutrients, we study the dimensionless system (2) can be rewritten as Because the solutions of and as and 2 2 as , in the limit as , System (6) is asymptotic to the equation Depending on whether the nutrients are complementary (noninteractive or interactive) or substitutable and assuming Monod (or Michaelis-Menten) kinetics typical choices of f take the forms given by Equations (3)- (5). For all three situations, x = 0 is a boundary rest point. If f is given by (4), we the have the additional rest points: as the Equation (8) Similarly, if f is given by (5), we obtain the additional rest points: as the Equation (9 Numerical results using f given by (3) (noninteractive complementary), (4) (interactive complementary), and (5) (substitutable) using the parameter values given in Table  2 are illustrated in Figure 1. In the figure, we can observe that substitutable nutrients lead to higher population densities than do complementary nutrients, which is not surprising. However, the differences in the densities from noninteractive and interactive nutrients are more surprising. When the nutrients are interactive, Rows 1 and 2 show that there can be a second (unstable) equilibrium population density. The stable population density is attained more quickly when the nutrients are interacttive than when they are noninteractive. Finally, the stable population density is generally higher when the nutrients are interactive than when they are noninteractive.
In fact, the population densities can be quite large as illustrated in Figure 2. Consider the values listed in Row 2 of Table 2 (corresponding to Row 2 of Figure 1). In Figure 2, we increase D (the washout rate) from D = 0.25 to D = 0.75 (row 1), D = 1.25 (row 2), and D = 1.75 (row 3). Observe that as D increases, interactive complementary nutrients lead to higher population densities than do noninteractive complementary nutrients. In fact, when the washout rate is sufficiently high, extinction occurs when the nutrients are noninteractive complementary while stable persistence occurs when the nutri-  Table 2.

An Intermediate Metabolite
A particularly interesting situation occurs when one substance degrades to a nutrient for the growth of an organism. Specifically, Sanchez et al. [18], study the particular situation in which phenol degrades to an intermediate metabolite that is then the primary nutrient for the organism (bacterium Pseudomonas putida Q5). This situation is particularly interesting because a "harmful" substance degrades to a state in which it is a nutrient for the organism under consideration that is growing in the chemostat, rendered harmless, and eliminated. To model this situation, System (2) is adjusted to where a > 0 is a positive constant. Sanchez et al. [18], successfully fit a model of the form of System (10) using     (2), we can reduce it to a system of two equations. To do so, let Then, and System (10) can rewritten as Because the solution of is as , in the limit as , System (12) is asymptotic to the system For the problem to be biologically meaningful, the feasible region is Assuming Monod kinetics, we now assume that f 1 takes the form given by (11) and that n = p = 1 and that     , where Evaluated at 2 0 , the Jacobian has eigenvalues λ 1,2 = −D, so is always stable. If we eliminated S 2 rather than S 1 , the limiting system is Again, assuming that f 1 takes the form given by (11) and that n = p = 1 and that      E is always stable. Finally, if we had eliminated x rather than S or S , the limiting system is with feasible region Using the same assumptions regarding f 1 and m     Table 3.  Table 5. However, in the first two rows of Table 5, (x, S 1 , S 2 ) = (6.23, 2.76, 1) and (8.46, 2.54, 1) are stable spirals while in the third row, (x, S 1 , S 2 ) = (2.92, 1.93, 0.143) is an unstable spiral.
Thus, depending on the parameter values equilibrium states may be stable or unstable. Moreover, slight adjustments to the parameter values might make a stable state unstable and vice-versa.

Conclusions
In this paper, we have formed a model of g h in a We have used numerical results to graphically illustrate le than do complementta inte mentary nutrients. It is possible that if the nutrients are