Mathematical ecology has been the subject of intensive research for several decades. It has matured into a rich theory that is remarkably effective in capturing the growth of multiple species growing on a single growth-limiting substrate. However, in cases of practical interest, be it natural environments or man made bioreactors, microbes grow on mixtures of several substrates. In these important cases, the theory invariably fails because the dynamics are dictated by physiological (intracellular) variables that are not included in current models. The goal of this article is to bring to the attention of mathematical ecologists the identity of the key physiological variables. More importantly, I also wish to communicate the crucial 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 a simple explanation of these phenomena. The preferential growth pattern occurs because synthesis of the enzymes that catalyse the transport/catabolism of the "less preferred" substrate is completely suppressed in the presence of the "preferred" substrate. On the other hand, simultaneous utilisation of the substrates occurs when suppression of enzyme synthesis is incomplete. A simple mathematical model taking due account of the mechanism by which the enzymes, the key physiological variables, are synthesised gives an elegant explanation of 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 catalysing the uptake of the "less preferred" substrate becomes "extinct"; the simultaneous growth pattern occurs because the enzymes for both substrates "coexist". The reader is referred to papers below for more details. Here it suffices to observe that the analogy implies that the cell is an ecological microcosm with the enzymes playing the role of "competing species". If one were to now extend this picture to mixed-culture systems (multiple species), one can envision competition between species as competition between their enzymes. One thus obtains a physiological basis for interspecific interactions.
The type of behaviour 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. A medium containing glucose (carbon source) and ammonia (nitrogen source) as the growth-limiting substrates is an example of such a mixture. In ecological models, interaction between species have classified into various categories (competing, synergistic, predatory-prey, etc). The behaviour of substitutable substrate mixtures is analogous to the dynamics of the Lotka-Volterra model for competing species. The question that then arises is 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. It is hoped that this article will stimulate the interest of mathematical biologists in problems of microbial physiology. It would seem that existing ecological theories can, with suitable modifications, provide fresh insights in 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. It establishes the analogy under very weak conditions by appealing to the qualitative theory of Lotka-Volterra models developed by Hirsch & Smale in their text on dynamical systems.
3. Narang et al (1997). The dynamics of microbial growth on mixtures of substrates in batch reactors. J. Theor. Biol., 184, 301-317.
Deals with growth patterns observed in batch cultures. The starting point of this paper is a complete model accounting for both environmental variables (substrate concentrations, cell density) and physiological variables (enzyme levels, inducer concentrations). Through a series of approximations (perturbation theory), it is then shown that the essential dynamics are determined by the motion of the enzymes, which, in turn, obey Lotka-Volterra dynamics.
4. Narang, A. (1998). The steady states of microbial growth on mixtures of substitutable substrates in a chemostat. J. Theor. Biol., 184, 241-261.
The model for batch cultures is extended to continuous cultures and analysed using bifurcation theory. It is shown that the switch from one substrate to another, frequently observed in continuous cultures, is reflected in the model by a transcritical bifurcation. The upshot of the analysis is the construction of bifurcation diagram that yields the substrate utilisation pattern at any given dilution rate and feed concentrations.
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Last Updated: 31th July 1998.