Application of ecological principles to microbial physiology:
Description of mixed-substrate microbial growth
by Lotka-Volterra models.
Atul Narang
Amoco Research Center, H-7
150 W. Warrenville Rd
Naperville, IL 60563

Mathematical ecology is one of most fertile areas of theoretical biology. It provides a rich theory that is remarkably effective in capturing the growth of multiple microbial species growing on a single growth-limiting substrate. However, in most cases of practical interest, be it natural environments or manmade bioreactors, microbes grow on mixtures of several substrates. In these important cases, the theory has not been very fruitful because the dynamics are dictated by physiological (intracellular) variables that are not included in current ecological models. The goal of this article is to bring to the attention of mathematical ecologists the identity of the key physiological variables in mixed-substrate systems. It is also intended to convey the notion that these key physiological variables are governed by Lotka-Volterra dynamics. Thus, the full force of Lotka-Volterra theory can be brought to bear upon problems in microbial physiology. It is hoped that these ideas will induce mathematical ecologists to contribute to the rudimentary field of microbial physiology.

The point is best illustrated by describing a classical problem of mixed-substrate growth. When microbes are supplied with a mixture of two growth-limiting substrates, they often exhibit anthropomorphic choice. They preferentially consume the substrate that, by itself, supports a higher growth rate. It is only after this "preferred" substrate is almost completely consumed that they start consuming the other "less preferred" substrate. Although this preferential growth pattern is the more commonly observed growth pattern, there are two-substrate systems in which both substrates are consumed simultaneously. Molecular biology has yielded the biochemical basis of these phenomena. The preferential growth pattern occurs because synthesis of the enzymes that catalyze the transport/catabolism of the "less preferred" substrate is completely suppressed in the presence of the "preferred" substrate. On the other hand, simultaneous utilization of the substrates occurs when suppression of enzyme synthesis is incomplete. Evidently, the key physiological variables are the transport/catabolic enzymes. It turns out that a simple mathematical model taking due account of the mechanism by which these enzymes are synthesized gives further insight into the preferential and simultaneous growth patterns. Because synthesis of the enzymes is autocatalytic, the equations describing the evolution of the enzymes are dynamically analogous to the Lotka-Volterra model for competing species. Thus, the preferential growth patterns occurs because the enzyme catalyzing the uptake of the "less preferred" substrate becomes "extinct"; the simultaneous growth pattern occurs because the enzymes for both substrates "coexist". This dynamical analogy implies that the cell is an ecological microcosm with the enzymes playing the role of "competing species". By extending this picture to microbial systems with multiple species, one can envision competition between species as competition between their enzymes. The analogy thus provides a conceptual framework for obtaining the physiological basis of interspecific interactions.

The type of behavior alluded to above is characteristic of mixtures in which the two substrates in question are substitutable, i.e., satisfy identical nutrient requirement such as two carbon sources or two nitrogen sources. There are, of course, other types of mixtures. Complementary mixtures, for instance, consist of substrates that satisfy entirely distinct nutrient requirements. An example of such a mixture is a medium containing glucose (carbon source) and ammonia (nitrogen source) as the growth-limiting substrates. In ecological models, interactions between species have classified into various categories (competing, synergistic, predatory-prey, etc). As noted above, the dynamics of the enzymes in substitutable substrate mixtures are analogous to the dynamics of the Lotka-Volterra model for competing species. It seems pertinent to ask whether other types of mixtures also have analogs in ecology. There is a vast body of experimental literature on the various types of mixed-substrate growth (Egli, 1995), but almost no mathematical models that attempt to answer this question. It is hoped that this article will stimulate the interest of mathematical biologists in problems of microbial physiology. It seems likely that existing ecological theories can, with suitable modifications, provide fresh insights into microbial physiology.

1. Egli, T. (1995). The ecological and physiological significance of the growth of heterotrophic microorganisms with mixtures of substrates, Adv. Microb. Ecol., 14, 797-806.
A comprehensive review of all the experimental literature on mixed-substrate growth

2. Narang, A. (1998). The dynamical analogy between microbial growth on mixtures of substrates and population growth of competing species. Biotechnol. Bioeng., 59, 116-121.
Develops the gist of this article in a more rigorous fashion.

3. Ramakrishna et al (1997). Cybernetic modeling of growth in mixed, substitutable substrate environments. Biotechnol. Bioeng., 52, 141-151.
A control-theoretic approach that replaces the description of regulatory mechanisms by optimality principles.

4. van Dedem, G. & Moo-Young, M. (1975). A model for diauxic growth. Biotechnol. Bioeng., 17, 1301-1312.
One of the earliest attempts to capture the physiology of mixed-substrate growth

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