Growth of mixed cultures on mixtures of substitutable substrates: the operating diagram for a structured model

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Abstract

The growth of mixed microbial cultures on mixtures of substrates is a problem of fundamental biological interest. In the last two decades, several unstructured models of mixed-substrate growth have been studied. It is well known, however, that the growth patterns in mixed-substrate environments are dictated by the enzymes that catalyse the transport of substrates into the cell. We have shown previously that a model taking due account of transport enzymes captures and explains all the observed patterns of growth of a single species on two substitutable substrates (J. Theor. Biol. 190 (1998) 241). Here, we extend the model to study the steady states of growth of two species on two substitutable substrates. The model is analysed to determine the conditions for existence and stability of the various steady states. Simulations are performed to determine the flow rates and feed concentrations at which both species coexist. We show that if the interaction between the two species is purely competitive, then at any given flow rate, coexistence is possible only if the ratio of the two feed concentrations lies within a certain interval; excessive supply of either one of the two substrates leads to annihilation of one of the species. This result simplifies the construction of the operating diagram for purely competing species. This is because the two-dimensional surface that bounds the flow rates and feed concentrations at which both species coexist has a particularly simple geometry: It is completely determined by only two coordinates, the flow rate and the ratio of the two feed concentrations. We also study commensalistic interactions between the two species by assuming that one of the species excretes a product that can support the growth of the other species. We show that such interactions enhance the coexistence region.

Introduction

The growth of mixed cultures on mixtures of substrates is a phenomenon of considerable practical and theoretical interest. A fundamental understanding of this problem has repercussions for

  • Food processing: Cheese is manufactured by mixtures of Streptococci and Lactobacilli, and yogurt is the product of Lactobacillus bulgaricus and Streptococcus thermophilus. The production of sauerkraut, beer, wine, and vinegar also depends on mixed-culture systems (Harrison and Wren, 1976).

  • Production of ethanol from renewable resources: The feedstock for production of fuel-grade ethanol from plants consists of two streams derived from the cellulosic and the hemicellulosic fractions of lignocellulose. The cellulosic stream consists of hexoses; the hemicellulosic stream contains a mixture of pentoses and hexoses. In the current process, each of these streams is fermented separately by distinct recombinant strains. To make this process economically viable, it would be desirable to carry out both fermentations in a single reactor (Ingram et al., 1999).

  • Bioremediation: Xenobiotic contaminants degrade faster if they are attacked by microbial consortia, rather than pure species.

The problem also has profound implications for microbial ecology. Early studies were concerned with the growth of multiple species on a single growth-limiting substrate that is being fed at a constant rate into a well-stirred chemostat. It was shown both theoretically and experimentally that under these conditions, no more than one of the species survives, regardless of the dilution rate and the feed concentration of the growth-limiting substrate (Aris and Humphrey, 1977; Hansen and Hubbell, 1980; Powell, 1958). This result is in sharp contrast to what is observed in Nature. In large bodies of water, many phytoplankton species coexist, an observation referred to as the “paradox of the plankton” (Hutchinson, 1961). To resolve this paradox, the basic assumptions of the early studies have been questioned (Fredrickson and Stephanopoulos, 1981). It has been argued that in Nature
  • the supply of nutrients is not constant;

  • the nutrients are not homogeneously distributed;

  • multiple growth-limiting substrates are present in the environment.

Thus, the problem of mixed-culture growth on mixtures of substrates has attracted considerable interest as one possible resolution of the paradox of the plankton. This initial interest culminated in two seminal papers which recognized that in the presence of multiple growth-limiting substrates, it is important to specify the nutritional requirements satisfied by the substrates (León and Tumpson, 1975; Tilman, 1977). Thus, two growth-limiting substrates are substitutable if they satisfy identical nutritional requirements, so that growth persists in the absence of either one of the substrates. The two substrates are complementary if they satisfy distinct nutritional requirements, so that growth is impossible in the absence of either one of the substrates. For example, during so-called heterotrophic growth of microbes, glucose and galactose would be considered substitutable since both function as carbon and energy sources, but glucose and ammonia would be complementary since glucose is a carbon source, whereas ammonia is a nitrogen source. In both studies, unstructured models of mixed-culture growth on substitutable and complementary mixtures of substrates were formulated, and necessary conditions for coexistence of the species were derived. But the use of unstructured models ignores the fact that uptake of substrates is regulated by the activity and level of the transport enzymes. In mixtures of substitutable substrates, this regulation often leads to preferential utilization of only one of the substrates (Egli 1995; Harder and Dijkhuizen 1976, Harder and Dijkhuizen 1982). It seems desirable then to study the problem of mixed-culture growth on mixtures of substrates with the help of a structured model that takes due account of the transport enzymes and their regulation. In earlier work, we formulated such a model for the growth of a single species on mixtures of substitutable substrates, and showed that it captures and explains all the experimental data in the literature (Narang et al., 1997; Narang 1998a, Narang 1998b). Here, we extend this model to study the growth of two species on a mixture of two substitutable substrates.

Despite the importance of the problem, review articles show that the experimental data are sparse (Fredrickson, 1977; Gottschal 1986, Gottschal 1993). This reflects the difficulty of measuring the population densities of multiple species. In the past, this was done by exploiting morphological differences between the species, or by selective plating techniques in which inhibitors are added to block the growth of all but one of the species. These methods are tedious and prone to error. Recent developments in flow cytometry and 16S RNA-based probes permit quick and accurate measurements of multiple population densities (Porter and Pickup, 2000). These technological advances are likely to foster rapid growth of the experimental literature on mixed cultures (for recent applications, see Muller et al., 2000; Rogers et al., 2000). The goal of this paper is to submit conclusions derived from a structured model that can be subjected to the test of these experiments. We shall be concerned, in particular, with the following questions

  • 1.

    What are the flow rates and feed concentrations of the two substitutable substrates at which both species coexist?

  • 2.

    How is their coexistence affected if one of the species excretes a product that influences the growth of the other species?

The first question is the crux of the ecological problem referred to above. The second question is important because pure competition between species is an idealization that is difficult to realize in practice. Most microbial species excrete metabolic products that can stimulate or inhibit the specific growth rate of the other species; this results in the establishment of commensalistic or amensalistic, rather than competitive, interactions (Fredrickson and Tsuchiya, 1977). It is, therefore, of practical interest to assess the effect of excretion on the phenomenon of coexistence.

The paper is organized as follows. In Section 2, we extend our earlier model of mixed-substrate growth to mixed cultures. In Section 3, we compute the operating diagram delineating the region of the parameter space in which coexistence is feasible (Pavlou and Fredrickson, 1989). This is done for two types of inter-specific interactions—pure competition and commensalism. Finally, the conclusions are summarized in Section 4. The key results are as follows:

  • 1.

    If the interaction between the two species is purely competitive, then:

    • (a)

      At any given flow rate, coexistence is possible only if the ratio of the two feed concentrations lies within a certain interval. Excessive supply of either one of the two substrates results in extinction of one of the species.

    • (b)

      The operating diagram delineating the flow rate and feed concentrations at which the two species coexist has a particularly simple geometry.

  • 2.

    If, however, one of the species excretes a product that can support the growth of the other species, the coexistence region is significantly enhanced.

Section snippets

Model

The kinetic scheme of our model is shown in Fig. 1. As a notational convention for the rest of the paper, the index i will denote the species number, and the index j will denote the substrate number. Thus, Ci denotes the ith species, Sj denotes the jth substrate, Eij denotes the “lumped” system of inducible enzymes catalysing the uptake and peripheral catabolism of Sj by Ci, Xij denotes the inducer for Eij, and Ci denotes all intracellular constituents in the ith species, except Eij and Xij.

Simulations

The simulations were done using Mathematica (Wolfram, 1999) and CONTENT (Kuznetsov, 1998). The parameter values used in the simulations are shown in Table 1. Appendix C shows the rationale for order-of-magnitude estimates of the parameters. The parameter values were then adjusted to ensure that C1 and C2 have opposite substrate preferences. Specifically, the parameter values for C1 were chosen so that S1 is the preferred substrate for C1; that is, s1 approaches s1f at a rate slower than the

Conclusions

We extended our earlier structured model for growth of a single species on two substitutable substrates to accommodate the growth of two species on two substitutable substrates. Our goal was to determine

  • 1.

    The dilution rates and feed concentrations at which both species coexist.

  • 2.

    The influence of excretion on this region of coexistence.

In the course of our studies, we found that:
  • 1.

    If the interaction between the two species is purely competitive, the coexistence region is completely determined by the

Acknowledgements

During the course of this research, Gregory Reeves was partially supported by the University Scholars Program at the University of Florida.

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