Mathematical models of microbial growth and competition in the chemostat regulated by cell-bound extracellular enzymes

A mathematical model of growth and competitive interaction of microorganisms in the chemostat is analyzed. The growth-limiting nutrient is not in a form that can be directly assimilated by the microorganisms, and must first be transformed into an intermediate product by cell-bound extracellular enzymes. General monotone functions, including Michaelis-Menten and sigmoidal response functions, are used to describe nutrient conversion and growth due to consumption of the intermediate product. It is shown that the initial concentration of the species is an important determining factor for survival or washout. When there are two species whose growth is limited by the same nutrient, three different modes of competition are described. Competitive coexistence steady states are shown to be possible in two of them, but they are always unstable. In all of our numerical simulations, the system approaches a steady state corresponding to the washout of one or both of the species from the chemostat.Research supported by NSF grant DMS-90-96279Research supported by NSERC grant A-9358     

How flocculation can explain coexistence in the chemostat

We study a chemostat model in which two microbial species grow on a single resource. We show that species coexistence is possible when the species which would normally win the exclusive competition aggregates in flocs. Our mathematical analysis exploits the fact that flocculation is fast compared to biological growth, a common hypothesis in floc models. A numerical study shows the validity of this approach in a large parameter range. We indicate how our model yields a mechanistic justification for the so-called density-dependent growth.     

How flocculation can explain coexistence in the chemostat

We study a chemostat model in which two microbial species grow on a single resource. We show that species coexistence is possible when the species which would normally win the exclusive competition aggregates in flocs. Our mathematical analysis exploits the fact that flocculation is fast compared to biological growth, a common hypothesis in floc models. A numerical study shows the validity of this approach in a large parameter range. We indicate how our model yields a mechanistic justification for the so-called density-dependent growth.     

Competitive yeast fermentation with selective flocculation and recycle

Continuous fermentations were carried out involving competition between two strains of Saccharomyces cerevisiae. One of the strains has a lower specific growth rate and is very flocculent, whereas the fastergrowing strain is nonflocculent. The product stream from the chemostat was fed into an inclined settler where the flocculent strain was partially separated from the nonflocculent strain as a result of the higher sedimentation rate of the flocculent cells. The underflow from the inclined settler, which was concentrated and enriched with flocculent cells, was recycled to the chemostat. When no recycle was used, the fastergrowing, nonflocculent yeast rapidly overtook the culture. With selective recycle, however, the experiments demonstrated that the slower-growing flocculent yeast could be maintained as the dominant species. A theoretical development is also presented in order to describe the competition between two strains in the bioreactor-settler system. The concept of selective recycle via selective flocculation and sedimentation offers a possible means of maintaining unstable recombinant microorganisms in continuous fermentations.     

Advantage of storage in a fluctuating environment

We will elaborate the evolutionary course of an ecosystem consisting of a population in a chemostat environment with periodically fluctuating nutrient supply. The organisms that make up the population consist of structural biomass and energy storage compartments. In a constant chemostat environment a species without energy storage always out-competes a species with energy reserves. This hinders evolution of species with storage from those without storage. Using the adaptive dynamics approach for non-equilibrium ecological systems we will show that in a fluctuating environment there are multiple stable evolutionary singular strategies (ss's): one for a species without, and one for a species with energy storage. The evolutionary end-point depends on the initial evolutionary state. We will formulate the invasion fitness in terms of Floquet multipliers for the oscillating non-autonomous system. Bifurcation theory is used to study points where due to evolutionary development by mutational steps, the long-term dynamics of the ecological system changes qualitatively. To that end, at the ecological time scale, the trait value at which invasion of a mutant into a resident population becomes possible can be calculated using numerical bifurcation analysis where the trait is used as the free parameter, because it is just a bifurcation point. In a constant environment there is a unique stable equilibrium for one species following the "competitive exclusion" principle. In contrast, due to the oscillatory dynamics on the ecological time scale two species may coexist. That is, non-equilibrium dynamics enhances biodiversity. However, we will show that this coexistence is not stable on the evolutionary time scale and always one single species survives.     

Advantage of storage in a fluctuating environment

We will elaborate the evolutionary course of an ecosystem consisting of a population in a chemostat environment with periodically fluctuating nutrient supply. The organisms that make up the population consist of structural biomass and energy storage compartments. In a constant chemostat environment a species without energy storage always out-competes a species with energy reserves. This hinders evolution of species with storage from those without storage. Using the adaptive dynamics approach for non-equilibrium ecological systems we will show that in a fluctuating environment there are multiple stable evolutionary singular strategies (ss's): one for a species without, and one for a species with energy storage. The evolutionary end-point depends on the initial evolutionary state. We will formulate the invasion fitness in terms of Floquet multipliers for the oscillating non-autonomous system. Bifurcation theory is used to study points where due to evolutionary development by mutational steps, the long-term dynamics of the ecological system changes qualitatively. To that end, at the ecological time scale, the trait value at which invasion of a mutant into a resident population becomes possible can be calculated using numerical bifurcation analysis where the trait is used as the free parameter, because it is just a bifurcation point. In a constant environment there is a unique stable equilibrium for one species following the “competitive exclusion” principle. In contrast, due to the oscillatory dynamics on the ecological time scale two species may coexist. That is, non-equilibrium dynamics enhances biodiversity. However, we will show that this coexistence is not stable on the evolutionary time scale and always one single species survives.     

Coexistence in the chemostat as a result of metabolic by-products

Classical chemostat models assume that competition is purely exploitative and mediated via a common, limiting and single resource. However, in laboratory experiments with pathogens related to the genetic disease Cystic Fibrosis, species specific properties of production, inhibition and consumption of a metabolic by-product, acetate, were found. These assumptions were implemented into a mathematical chemostat model which consists of four nonlinear ordinary differential equations describing two species competing for one limiting nutrient in an open system. We derive classical chemostat results and find that our basic model supports the competitive exclusion principle, the bistability of the system as well as stable coexistence. The analytical results are illustrated by numerical simulations performed with experimentally measured parameter values. As a variant of our basic model, mimicking testing of antibiotics for therapeutic treatments in mixed cultures instead of pure ones, we consider the introduction of a lethal inhibitor, which cannot be eliminated by one of the species and is selective for the stronger competitor. We discuss our theoretical results in relation to our experimental model system and find that simulations coincide with the qualitative behavior of the experimental result in the case where the metabolic by-product serves as a second carbon source for one of the species, but not the producer.     

Study of the dynamic behavior of the chemostat system

This paper deals with a theoretical study on the dynamic, character of the chemostat system. It. is primarily based on the Monod model for growth limitation, although certain more complex models are considered. Since the Monod model is described in terms of two variables, an analysis by use of a phase plane plot will show the various possible types of behavior theoretically expected for transient conditions of the system. In this paper it will be shown that the chemostat system might show an overshoot (or an underswing) with respect to changes in cell and substrate concentrations, depending on the extent to which the system might be disturbed from steady-slate conditions. Other types of transient behavior ran also be expected when one of the system parameters such as dilution rate or input substrate concentration is disturbed in a stepwise manner. The simple Monod chemostat model was found never to oscillate in either a damped or a sustained manner as has been experimentally reported. Discussion is included about the transient behavior of other chemostat models such as that involving a variable yield coefficient, i.e., including the effect of cell maintenance requirements.     

A multiple chemostat system for consortia studies on microbially influenced corrosion

A multiple chemostat system has been developed in which metal specimens can be exposed to a consortium of bacteria. The system comprises a single test chemostat containing the test specimen operated at a high dilution rate to facilitate the wash out of planktonic bacteria, selecting for attached or biofilm growth. This chemostat is fed at a steady low rate by a number of separate chemostats each of which contains a pure axenic culture of one member of the consortium being tested. This system has the advantage of providing a continual inoculum of the test species to the test specimen allowing both aerobic and anaerobic bacteria to be grown in the same system. Constant levels of three bacterial types were maintained in the system: Pseudomonas aeruginosa, Thiobacillus ferrooxidans and Desulfovibrio vulgaris. Exposure of 316L stainless steel electrodes to this system resulted in increased corrosion of coupons exposed biotically, as compared to those exposed abiotically. A current monitoring technique and electrochemical impedance spectroscopy were used to evaluate effects of bacteria on metallic corrosion.     

The formation of mixed culture biofilms of oral species along a gradient of shear stress

A chemostat mixed culture system was used to produce two distinct ecological states, state-1 (caries-like microcosm) and state-2 (periodontal-like microcosm). Eleven bacterial species (Streptococcus gordonii, Strep. mitis I, Strep. mutans, Strep. oralis, Actinomyces naeslundii, Lactobacillus casei, Neisseria subflava, Fusobacterium nucleatum, Porphyromonas gingivalis, Prevotella nigrescens, Veillonella dispar) were used to inoculate the planktonic system. A flow cell, designed to produce convergent flow with increasing shear stress, was attached to the chemostat system, and the resultant biofilms developed from the state-1 and state-2 microcosms along the shear stress gradient were examined and compared using image analysis and viable counts. The biofilm produced from state-1 showed a lower shear stress tolerance (0.146 Pa) than the state-2 biofilm (0.236 Pa). The biofilm compositions did not vary along the gradient of shear stress and were dependent on the initial inoculum conditions. Gram-positive species were predominant in the state-1 biofilm, while Gram-negative species were predominant in state-2.     

A modified chemostat system to study the ecology of oral biofilms

Previously, we developed a chemostat system to study the behaviour and properties of a community of up to 10 species of oral bacteria. The present study describes modification of this system to incorporate removable and replaceable hydroxyapatite (the major mineral in human dental enamel) disks on which biofilms could develop. Hydroxyapatite disks were immersed in the chemostat for known time periods, and the bacterial content of biofilms determined by viable counting. Initial deposition rates were rapid, with all 10 species detected after 1 h, and the numbers of bacteria in biofilms continued to increase for 21 d. The species composition of biofilms reflected that of the surrounding fluid phase, and showed only limited signs of the type of 'species succession' which is observed in developing dental plaque in vivo , although anaerobic species increased in proportion in older biofilms. Four-day biofilms showed the least variability and were chosen as the 'standard biofilm' for more detailed study. Variability in the bacterial composition of 4-d biofilms was comparable both within a single chemostat run and between independent chemostat runs. Glucose pulsing in the absence of pH control resulted in the selection of cariogenic species; the disruption of the biofilm community was less marked than that of the equivalent planktonic culture. The model system has considerable potential in studying the effects of a variety of factors on biofilm development, as well as in comparing the efficacy of antimicrobial systems against biofilms.     

Growth of mixed cultures ofParacoccus denitrificans andDesulfovibrio desulfuricans in homogeneous and in heterogeneous culture systems

Obligate aerobes and anaerobes coexist closely in natural ecosystems. One species representative of each class (Desulfovibrio desulfuricans andParacoccus denitrificans) were selected for investigation in different laboratory growth systems. When incubated together, the aerobe protected the anaerobe against oxygen poisoning allowing the latter to grow and to reduce sulfate. When gas phase oxygen tension was systematically altered in stirred batch cultures, both species grew over a broad range of oxygen concentrations. Similar experiments in a chemostat indicated that conditions were far more exacting. No steady state was possible; however, at one critical oxygen tension the two species entered a stable oscillating cycle that lasted for 12 days. When the two organisms were grown in a gradostat in opposing gradients of sulfate plus oxygen and lactate, they coexisted but at different positions in the system. The actual position of the sulfate reducer was determined by the sulfate concentration in the reservoir.     

Characterization of a three bacteria mixed culture in a chemostat: evaluation and application of a quantitative terminal-restriction fragment length polymorphism (T-RFLP) analysis for absolute and species specific cell enumeration

Growth dynamics of Pseudomonas aeruginosa, Burkholderia cepacia, and Staphylococcus aureus in a batch and chemostat, were investigated as a laboratory model system for persistent infections in cystic fibrosis. Most species-specific enumeration methods for mixed cultures are laborious or only qualitative, and therefore impede generation of quantitative data required for validation of mathematical models. Here, a quantitative T-RFLP method was evaluated and applied for specific and absolute cell number enumerations. The method was tested to be unbiased by quantitative sample composition and allowed reproducible enumerations of mixed cultures. For assay validation, samples of defined concentration containing one, two or three species were quantified. Logarithmically transformed absolute cell numbers of single-species dilutions were linear within a lower working range of 10(4)-10(6) cfu/mL (species-dependent) and an upper working range of 10(10) cfu/mL. Quantifications of single species (10(6)-10(10) cfu/mL) spiked with one or two other species agreed well with single species controls. Differences between slopes of first order linear regression of spiked and pure dilution series were insignificant. Coefficient of variation of defined mixed replicates was maximum 4.39%, of a three-species chemostat it was maximum 1.76%. T-RFLP monitoring of pure cultures in parallel shake flasks and of a three-species mixed chemostat gave very consistent results. Coexistence of at least two species after a time period equivalent to more than 33 volume exchanges was found. This result was not predicted from pure cultures clearly indicating the need for quantitative mixed culture experiments to better understand microbial growth dynamics and for mathematical model validation.     

Dynamic effects related to steady-state multiplicity in continuous Saccharomyces cerevisiae cultivations

The behavioral differences between chemostat and productostat cultivation of aerobic glucose-limited Saccharomyces cerevisiae were investigated. Three types of experiments were conducted: a chemostat, where the dilution rate was shifted up or down in stepwise manner; and a productostat, with either stepwise changed or a rampwise increased ethanol setpoint, i.e., an accelero-productostat. The transient responses from chemostat and productostat experiments were interpreted using a simple metabolic flux model. In a productostat it was possible to obtain oxido-reductive steady states at dilution rates far below Dcrit due to a strong repression of the respiratory system. However, these steady states could not be obtained in a chemostat, since a dilution rate shift-down from an oxido-reductive steady state led to a derepression of the respiratory system. It can therefore be concluded that the range of dilution rates where steady-state multiplicity can be obtained differs depending on the operation mode and that this dilution rate multiplicity range may appear larger in a productostat than in a chemostat. A more narrow multiplicity range, however, was obtained when the productostat was operated as an accelero-productostat.     

Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation

Microbial populations compete for nutrient resources, and the simplest mathematical models of competition neglect differences in the nutrient content of individuals. The simplest models also assume a spatially uniform habitat. Here both of these assumptions are relaxed. Nutrient content of individuals is assumed proportional to cell size, which varies for populations that reproduce by division, and the habitat is taken to be an unstirred chemostat where organisms and nutrients move by simple diffusion. In a spatially uniform habitat, the size-structured model predicts competitive exclusion, such that only the species with lowest break-even concentration persists. In the unstirred chemostat, coexistence of two competitors is possible, if one has a lower break-even concentration and the other can grow more rapidly. In all habitats, the calculation of competitive outcomes depends on a principal eigenvalue that summarizes relationships among cell growth, cell division, and cell size.     

Association between competition and obligate mutualism in a chemostat

In this paper, we consider a simple chemostat model involving two obligate mutualistic species feeding on a limiting substrate. Systems of differential equations are proposed as models of this association. A detailed qualitative analysis is carried out. We show the existence of a domain of coexistence, which is a set of initial conditions in which both species survive. We demonstrate, under certain supplementary assumptions, the uniqueness of the stable equilibrium point which corresponds to the coexistence of the two species.     

Asymptotic Behavior of a Chemostat Model with Constant Recycle Sludge Concentration

In this work, we study a several species aerobic chemostat model with constant recycle sludge concentration in continuous culture. We reduce the number of parameters by considering a dimensionless model. First, the existence of a global positive uniform attractor for the model with different removal rates is proved using the theory of dissipative dynamical systems. Hence, we investigate the asymptotic behavior of the model under small perturbations using methods of singular perturbation theory and we prove that, in the case of two species in competition, the unique equilibrium which is positive is globally asymptotically stable. Finally, we establish the link between the open problem of the chemostat with different removal rates and monotone functional responses, and our model when two species compete on the same nutrient. We give some numerical simulations to illustrate the results.     

In vitro gene expression dissected: chemostat surgery for mycobacterium tuberculosis

A unique approach, combining defined and reproducible in vitro models with DNA microarrays, has been developed to study environmental modulation of mycobacterial gene expression. The gene expression profiles of samples of Mycobacterium tuberculosis, from independent chemostat cultures grown under defined and reproducible conditions, were found to be highly correlated. This approach is now being used to study the effect of relevant stimuli, such as limited oxygen availability, on mycobacterial gene expression. A modification of the chemostat culture system, enabling largevolume controlled batch culture, has been developed to study starvation survival. Cultures of M. tuberculosis have been maintained under nutrient-starved conditions for extended periods, with 10(6) - 10(7) bacilli surviving in a culturable state after 100 days. The design of the culture system has made it possible to control the environment and collect multiple time-course samples to study patterns of gene expression. These studies demonstrate that it is possible to perform long-term studies and obtain reproducible expression data using controlled and defined in vitro models.     

A new method for studying microaerobic fermentations. I. A theoretical analysis of oxygen programmed fermentation

An experimental method for studying microaerobic fermentation, called oxygen programmed fermentation, is introduced. The method if based on a chemostat. The mathematical equations governing the dynamics of the system are derived and simulations are made for two principally different cases: a purely respirative organism, and an organism capable of fermentation during oxygen limitation. It is shown that at a suitably chosen ramp rate, the dissolved oxygen concentration in the broth can be made to decrease almost linearly. It is suggested that the greatest use of oxygen programmed fermentation will be in initial experiments. Compared with chemostat studies, a scan of different oxygenation rates will provide a time-saving method of finding the interacting regions for metabolic transitions. Furthermore it is shown that the methods makes it possible to study cell physiology at condition which would normally lead to washout. (c) 1994 John Wiley & Sons, Inc.