Section: Scientific Foundations

Bioprocess engineering and mathematical ecology

The chemostat is a laboratory device which goes back to the second world war, with the work of Monod and Szilard. It is used to study the growth of micro-organisms. The principle is simple: a continuous flow rate through a constant volume reactor provides nutrients to a population or a community of micro-organisms. At equilibrium the growth-rate must equal the artificial mortality induced by the outflow of the reactor. A simple model, for the case where the reactor is perfectly stirred, is given by a set of two differential equations, one for the variations of the nutrient concentration, the other one for the variations of the biomass concentration. This model is based on the classical law of mass action used in the modelling of chemical kinetics: the rate of a reaction is proportional to the product of the concentrations of the two reactants. In the case of population growth, this means that the growth-rate of a population depends on the nutrient concentration. This system of two equations has been perfectly understood for more than half a century.

The chemostat model is a good first approximation of a waste-water treatment plant. From this simple model one can develop models which incorporate more realistic assumptions like:

• existence of a complicated trophic chain in the digestion process,

• consideration of non-perfect mixing inducing diffusion processes,

• consideration of mass transport in plug-flow reactors,

• parallel or cascade connections of reactors,

• re-circulation of the biomass,

• aggregation of micro-organisms in flocks,

• constitution of bio-films,

which lead to complicated systems of coupled partial differential equations of transport-diffusion type. Due to the presence of non-monotonic kinetics the theory of equations of this type is not yet perfectly understood. Determination of stable stationary solutions is often a question of current research and numerical simulations are used. Moreover, the control of industrial plants addresses new questions in the domain of robust control and observers.

Since a waste-water treatment plant is a microbial ecosystem, microbial ecology is fundamental for the understanding of our processes. Microbial ecology in “perfectly stirred bio-reactors” is certainly the field of ecology, in which the representation of species by concentrations changing over time and governed by ordinary differential equations (ODE) is the most justified; this is the point of view we adopt in this objective.

In a chemostat (or bioreactor) the classical model of growth of a species on one limiting substrate is well known, widely used and very efficient. The growth model of several species competing for the same substrate built upon the same assumptions, predicts the extinction of all species except one. This is known as the Competitive Exclusion Principle (CEP). This predictions has caused much debate, as it turns out that, in nature as well as in the laboratory, species coexist in a very great number. This paradox of a model that seems to be valid for one species but not for many, requires clarification. We address it in the specific case of the chemostat.

To our knowledge this question is not studied in France, at least in terms of modelling. In the United States, beyond the “historical researchers” (Waltman, Armstrong and McGehee, Hansen and Hubbell ...), we are aware of the work of Wolkowicz and her collaborators. However, we do not know any teams that have addressed this subject as systematically as we do in conjunction with biologists.

An ecosystem is a system in which various populations of different species are interacting between them and reacting to the environmental abiotic parameters. Concepts of competition, predation, symbiosis are used to describe these interactions and try to understand important questions like the biodiversity and the productivity of the ecosystem. The biodiversity is related to the number of species which is supported by the ecosystem. There are many ways of quantifying the biodiversity of a microbial ecosystems. The most intuitive measurement of diversity consists in evaluating the richness, which simply is the number of species. The productivity measures the rate at which abiotic resources are transformed into biomass. An old prediction of theoretical population models says that, in a constant environment, an ecosystem with n different kinds of resources can support at most n different species (different means that the ways two species use resources are different). This prediction is not realized in waste-water treatment plants where it was demonstrated, using tools of molecular biology (fingerprinting techniques such as SSCP, see below), that a small number of resources (maintained at a constant level) is able to maintain a huge number of species. This shows that the classical model of the perfectly stirred reactor is no longer valid if one wants to model the biodiversity in the reactor. We explore alternative models based on the consideration of growth-rates which are not solely nutrient-dependent, but are also density-dependent, which means that the growth rate may depend not only on the nutrient concentration but also on the density of the biomass. More specifically, based on physical arguments, we currently work with models where the growth rates decrease with the biomass concentration. A special case of density-dependence is so-called ratio-dependence which was much discussed recently.

Since a density-dependent model is a macroscopic model, it is important to understand how the density-dependence is a consequence of the microscopic behaviors of individuals. Since direct observation of the behavior of bacteria is difficult, mathematical modelling is of great help. The hypotheses, at the microscopic level, are expressed in terms of partial differential equations or in terms of individually based models so that macroscopic consequences are derived, either by using mathematical reasoning or computer simulations. Finally, mathematical analysis is the starting point for the design of new experiments which could validate hypotheses of the theoretical models. But conducting biological experiments requires time, energy and qualified people for rigorous validation (many protocols have to be checked for ensuring that contamination or side-effects do not degrade the results).