The kinetics of conjugative plasmid transmission: fit of a simple mass action model.

A mass action model for the infectious transmission of conjugative plasmids and procedures to estimate its parameters are presented. The suitability of this model as an analog of the kinetics of conjugative plasmid transmission is examined with batch and chemostat populations of Escherichia coli K-12 and three of its plasmids, F-lac-pro, R1 (Km-Cm-Ap), and R1-drd-19 (Km-Cm-Ap). Evidence is presented that this mass action model, with a unique and constant rate parameter, represents a reasonable analog of the kinetics of plasmid transfer for bacterial populations dividing at a constant rate in either exponentially growing cultures or at equilibrium in chemostats. As anticipated from this model magnitudes of the transfer rate constant for these plasmids appear to be relatively insensitive to both total cell density and the donor-recipient ratio. For all plasmids, the value of the transfer rate constant in rapidly dividing (exponentially growing) cultures is considerably greater than its corresponding value in slowly dividing, chemostat equilibrium cultures and the values of the transfer rate constant of the permanently derepressed plasmids F-lac-pro and R1-drd-19 are considerably greater than that of the wild-type, repressed transfer plasmid R1. The implications of this apparent fit to a mass action model are discussed and a recommendation is made to use the transfer rate constant as the measure of the fertility of conjugative plasmids.     

Successful invasion of a food web in a chemostat

A food web in a chemostat is considered in which an arbitrary number of competitor populations compete for a single, essential, nonreproducing, growth-limiting substrate, and an arbitrary number of predator populations prey on some or all of the competitor populations. Although any number of predator populations may prey on the same competitor population, each predator population preys on only one competitor population. The dynamics of substrate uptake is modeled by Lotka-Volterra or Michaelis-Menten (Holling type I or II), but the dynamics of competitor uptake is restricted to Lotka-Volterra. Based on certain parameters, the model predicts the asymptotic survival or extinction of each of the different populations and suggests how competitor and/or predator populations could successfully invade the chemostat with or without causing a diverse ecosystem to crash. Similarly, it suggests how the elimination of certain populations could result in a more diverse or less diverse system.     

Mathematical models for continuous culture growth dynamics of mixed populations subsisting on a heterogeneous resource base. II. Predation and trophic structure

Models are presented for the joint dynamics of predators and prey, maintained in continuous flow chemostat culture. The predators are visualized as subsisting on one or more prey organisms, which in turn are visualized as subsisting on one or more substrate resources supplied by the investigator. The dynamic equations are translated into an analogous Lotka-Volterra predation model, and the criteria for the existence and stability of various equilibria are indicated. Denoting the number of different predator organisms as N_H, the number of different prey organisms by N_I and the number of different substrates as N_J, it is shown that the joint coexistence of all components requires 0 ? N_I ? N_H ? N_J. The model is extended to more complex situations by including additional trophic layers and by allowing trophic layer “leap-frogging.” The model may always be translated into an approximately quadratic differential equation of the Lotka-Volterra type. The α- and β-coefficients of these latter are really variables, and become quite complex for some of the multi-layered models.     

Growth of Trichosporon cutaneum under oxygen limitation: kinetics of oxygen uptake

The relationship between oxygen concentration and growth rate in the yeast Trichosporon cutaneum was studied. In order to establish the conditions for purely oxygen-limited growth, the cells were first grown in a carbon-limited chemostat, and kinetic parameters determined. The cells were then grown in an oxygen-limited chemostat at different dilution rates yielding different oxygen uptake rates. The steady-state dissolved oxygen tension was found at each dilution rate and the corresponding equilibrium dissolved oxygen tension was found at each dilution rate and the corresponding equilibrium dissolved oxygen concentration determined in the effluent medium. The relationship between oxygen concentration and growth rate followed Monod-type kinetics with an apparent K(O) of 4.38 x 10(-6)M.     

In Silico Geobacter sulfurreducens Metabolism and Its Representation in Reactive Transport Models

Microbial activity governs elemental cycling and the transformation of many anthropogenic substances in aqueous environments. Through the development of a dynamic cell model of the well-characterized, versatile, and abundant Geobacter sulfurreducens, we showed that a kinetic representation of key components of cell metabolism matched microbial growth dynamics observed in chemostat experiments under various environmental conditions and led to results similar to those from a comprehensive flux balance model. Coupling the kinetic cell model to its environment by expressing substrate uptake rates depending on intra- and extracellular substrate concentrations, two-dimensional reactive transport simulations of an aquifer were performed. They illustrated that a proper representation of growth efficiency as a function of substrate availability is a determining factor for the spatial distribution of microbial populations in a porous medium. It was shown that simplified model representations of microbial dynamics in the subsurface that only depended on extracellular conditions could be derived by properly parameterizing emerging properties of the kinetic cell model.     

Global dynamics of microbial competition for two resources with internal storage

We study a chemostat model that describes competition between two species for two essential resources based on storage. The model incorporates internal resource storage variables that serve the direct connection between species growth and external resource availability. Mathematical analysis for the global dynamics of the model is carried out by using the monotone dynamical system theory. It is shown that the limiting system of the model basically exhibits the familiar Lotka-Volterra alternatives: competitive exclusion, coexistence, and bi-stability, and most of these results can be carried over to the original model. B. Li’s research was partially supported by NSF grants DMS 0211614 and DMS 0616445. H. L. Smith’s research was supported in part by NSF grant DMS 0414270.     

Growth kinetics of Pseudomonas putida G7 on naphthalene and occurrence of naphthalene toxicity during nutrient deprivation

The objectives of this work were (1) to demonstrate how the chemostat approach could be modified to allow determination of kinetic parameters for a sparingly soluble, volatile substrate such as naphthalene and (2) to examine the influence of the interactions of various nutrients on possible growth-inhibitory effects of naphthalene. Pseudomonas putida G7 was used as a model naphthalene-degrading microorganism. Naphthalene was found to be toxic to P. putida G7 in the absence of a nitrogen source or oxygen. The death rate of cells grown on minimal medium plus naphthalene and then exposed to naphthalene under anoxic conditions was higher than that observed under oxic conditions in the absence of a nitrogen source. The presence of necessary nutrients for the biodegradation of PAH compounds is indicated to be important for the survival of microorganisms that are capable of PAH degradation. The amounts of ammonia and oxygen necessary for naphthalene biodegradation and for suppression of naphthalene toxicity were calculated from growth yield coefficients. A chemostat culture of P. putida G7 using naphthalene as a carbon and energy source was accomplished by using a feed augmented with a methanol solution of naphthalene so as to provide sufficient growth to allow accurate evaluation of kinetic parameters. When naphthalene was the growth-limiting substrate, the degradation of naphthalene followed Monod kinetics. Maximum specific growth rate (micrometer) and Monod constant (Ks) were 0.627 +/- 0.007 h-1 and 0.234 +/- 0.0185 mg/L, respectively. The evaluation of biodegradation parameters will allow a mathematical model to be applied to predict the long-term behavior of PAH compounds in soil when combined with PAH transport parameters.     

Microbial Ecophysiology of Whey Biomethanation: Comparison of Carbon Transformation Parameters, Species Composition, and Starter Culture Performance in Continuous Culture

Changes in lactose concentration and feed rate altered bacterial growth and population levels in a whey-processing chemostat. The bacterial population and methane production levels increased in relation to increased lactose concentrations comparable to those in raw whey (6%) and converted over 96% of the substrate to methane, carbon dioxide, and cells. Sequential increases in the chemostat dilution rate demonstrated excellent biomethanation performance at retention times as low as 25 h. Retention times shorter than 25 h caused prevalent bacterial populations and methane production to decrease, and intermediary carbon metabolites accumulated in the following order: acetate, butyrate, propionate, lactate, ethanol, and lactose. Bacterial species dominated in the chemostat as a function of their enhanced substrate uptake and growth kinetic properties. The substrate uptake kinetic properties displayed by the mixed chemostat population were equivalent to those of individual species measured in pure culture, whereas the growth kinetic properties of species in mixed culture were better than those measured in pure culture. A designed starter culture consisting of Leuconostoc mesenteroides, Desulfovibrio vulgaris, Methanosarcina barkeri, and Methanobacterium formicicum displayed biomethanation performance, which was similar to that of a diverse adapted mixed-culture inoculum, in a continuous contact digestor system to which 10 g of dry whey per liter was added. Preserved starter cultures were developed and used as inocula for the start-up of a continuous anaerobic digestion process that was effective for biomethanation of raw whey at a retention time of 100 h.     

In situ pulse respirometric methods for the estimation of kinetic and stoichiometric parameters in aerobic microbial communities

In situ pulse respirometry was applied in an activated sludge bubble column treating synthetic wastewater for the estimation of the (i) maximum specific oxygen consumption rate, (ii) substrate affinity constant, (iii) biomass growth yield, (iv) maintenance coefficient, and (v) specific endogenous respiration rate. Parameters obtained from respirometry were compared to those obtained by the chemostat method, based on substrate and biomass measurements, under several dilution rates. The low sensitivity of substrate measurement methods and the difficulties of sampling heterogeneous biomass suspension are critical issues limiting the applicability of the chemostat method. Additionally, the extensive time consuming nature of this method allows concluding that chemostat method presents several disadvantages in comparison with in situ pulse respirometric techniques. Parameters were obtained from respirograms by fitting ASM1 and ASM3 models, and from experiments performed by injecting pulses of increasing substrate concentration. The injection of pulses of increasing concentration was the most adequate method, with several advantages such as a simpler experimental data interpretation, and results with better confidence.Considering the assessment and comparison of the experimental and calculation methods presented, it is recommended that the estimation of kinetic and stoichiometric parameters in mixed aerobic cultures should preferentially be performed by using in situ respirometric techniques.     

Dynamics of chemostat in which one microbial population grows on multiple complementary nutrients

The equations of a chemostat in which one microbial population grows on multiple rate-limiting nutrients are formulated. The dynamics of a chemostat involving growth on complementary nutrients is studied through stability analysis of the system of equations. Some conditions are derived that relate the dynamic behavior of the chemostat to its operating conditions and can be applied to any model for the specific growth rate of the population. It is shown that, if maintenance of the population is neglected, the system exhibits no sustained or damped oscillations. If maintenance of the population is considered, damped oscillations are observed for some operating conditions.     

Biodiversity,habitat area,resource growth rate and interference competition

For the majority of species, per capita growth rate correlates negatively with population density. Although the popular logistic equation for the growth of a single species incorporates this intraspecific competition, multi-trophic models often ignore self-limitation of the consumers. Instead, these models often assume that the predator-prey interactions are purely exploitative, employing simple Lotka-Volterra forms in which consumer species lack intraspecific competition terms. Here we show that intraspecific interference competition can account for the stable coexistence of many consumer species on a single resource in a homogeneous environment. In addition, our work suggests a potential mechanism for field observations demonstrating that habitat area and resource productivity strongly positively correlate to biodiversity. In the special case of a modified Lotka-Volterra model describing multiple predators competing for a single resource, we present an ordering procedure that determines the deterministic fate of each specific consumer. Moreover, we find that the growth rate of a resource species is proportional to the maximum number of consumer species that resource can support. In the limiting case, when the resource growth rate is infinite, a model with intraspecific interference reduces to the conventional Lotka-Volterra competition model where there can be an unlimited number of coexisting consumers. This highlights the crucial role that resource growth rates may play in promoting coexistence of consumer species.     

The regulatory effect of fermentable sugar levels on the production of leukotoxin by Actinobacillus actinomycetemcomitans

The relationship between sugar availability and RTX (repeats in toxin) cytotoxin (leukotoxin) production in the periodontopathic bacterium, Actinobacillus actinomycetemcomitans, was investigated using a chemostat. A. actinomycetemcomitans 301-b produced significant amounts of leukotoxin in anaerobic fructose-limited chemostat cultures at a dilution rate of 0.15 h⁻¹ and at pH 7.0. When the growth limitation was relieved by pulsing the cultures with 50 or 150 mM fructose (final concentrations), leukotoxin production immediately stopped and the amount of cellular leukotoxin decreased until the culture was returned to fructose-limited conditions. Leukotoxin synthesis was also repressed in the chemostat cultures by pulsing with glucose but not with the non-fermentable sugar analog, α-methyl-d-glucoside. Leukotoxin production was also repressed by fructose in chemostat cultures of ATCC 33384, which is generally recognized as a non-leukotoxin-producing or minimally leukotoxic strain.     

Modeling lipid accumulation in oleaginous fungi in chemostat cultures: I. Development and validation of a chemostat model for Umbelopsis isabellina

Lipid-accumulating fungi may be able to produce biodiesel precursors from agricultural wastes. As a first step in understanding and evaluating their potential, a mathematical model was developed to describe growth, lipid accumulation and substrate consumption of the oleaginous fungus Umbelopsis isabellina (also known as Mortierella isabellina) in submerged chemostat cultures. Key points of the model are: (1) if the C-source supply rate is limited, maintenance has a higher priority than growth, which has a higher priority than lipid production; (2) the maximum specific lipid production rate of the fungus is independent of the actual specific growth rate. Model parameters were obtained from chemostat cultures of U. isabellina grown on mineral media with glucose and NH₄ ⁺. The model describes the results of chemostat cultures well for D > 0.04 h⁻¹, but it has not been validated for lower dilution rates because of practical problems with the filamentous fungus. Further validation using literature data for oleaginous yeasts is described in part II of this paper. Our model shows that not only the C/N-ratio of the feed, but also the dilution rate highly influences the lipid yield in chemostat cultures.     

Simultaneous versus sequential optimal experiment design for the identification of multi-parameter microbial growth kinetics as a function of temperature

Optimal experiment design for parameter estimation (OED/PE) has become a popular tool for efficient and accurate estimation of kinetic model parameters. When the kinetic model under study encloses multiple parameters, different optimization strategies can be constructed. The most straightforward approach is to estimate all parameters simultaneously from one optimal experiment (single OED/PE strategy). However, due to the complexity of the optimization problem or the stringent limitations on the system's dynamics, the experimental information can be limited and parameter estimation convergence problems can arise. As an alternative, we propose to reduce the optimization problem to a series of two-parameter estimation problems, i.e., an optimal experiment is designed for a combination of two parameters while presuming the other parameters known. Two different approaches can be followed: (i) all two-parameter optimal experiments are designed based on identical initial parameter estimates and parameters are estimated simultaneously from all resulting experimental data (global OED/PE strategy), and (ii) optimal experiments are calculated and implemented sequentially whereby the parameter values are updated intermediately (sequential OED/PE strategy).This work exploits OED/PE for the identification of the Cardinal Temperature Model with Inflection (CTMI) (Rosso et al., 1993). This kinetic model describes the effect of temperature on the microbial growth rate and encloses four parameters. The three OED/PE strategies are considered and the impact of the OED/PE design strategy on the accuracy of the CTMI parameter estimation is evaluated. Based on a simulation study, it is observed that the parameter values derived from the sequential approach deviate more from the true parameters than the single and global strategy estimates. The single and global OED/PE strategies are further compared based on experimental data obtained from design implementation in a bioreactor. Comparable estimates are obtained, but global OED/PE estimates are, in general, more accurate and reliable.     

A model for the dynamics of an annual plant population

A model for the dynamics of a single species population of plants is proposed and its use demonstrated by the analysis of a simple example. The model incorporates the effects of microsite variation by allowing for individual differences in growth and death rates within each season. We demonstrate that an increase in the variance in individual growth rates may increase both the chances that a plant population will persist and the equilibrium size of that population. We also show that even if size-dependent death is occurring, it may not have a significant effect on the shape of the size frequency distribution. An extension of the model to multispecies communities of plants suggests an experimental procedure to determine whether competition is responsible for excluding a particular plant species from a community that appears otherwise to be suitable. A more detailed analysis of the model for a two-species community produces conditions for competitive coexistence reminiscent of those from the Lotka-Volterra competition equations. Another extension suggests that selection will favor those genotypes that maximize the product of germination probability and mass of seeds produced, if survivorship and growth are not substantially altered. Finally, an analog to r- and K-selection theory for animal populations is developed. Selection in low-density populations favors increasing growth rate, and in high-density populations favors minimizing the effect of neighbors on one's own growth rate.     

Measurement of minimum substrate concentration (Smin) in a recycling fermentor and its prediction from the kinetic parameters of Pseudomonas strain B13 from batch and chemostat cultures.

The minimum substrate concentration required for growth, Smin, was measured for Pseudomonas sp. strain B13 with 3-chlorobenzoate (3CB) and acetate in a recycling fermentor. The substrates were provided alone or in a mixture. Smin values predicted with kinetic parameters from resting-cell batches and chemostat cultures differed clearly from the values measured in the recycling fermentor. When 3CB and acetate were fed as single substrates, the measured Smin values were higher than the individual Smin values in the mixture. The Smin in the mixture reflected the relative energy contributions of the two substrates in the fermentor feed. The energy-based maintenance coefficients during zero growth in the recycling fermentor were comparable for all influent compositions (mean +/- standard deviation, 0.34 +/- 0.07 J mg [dry weight]-1 h-1). Maintenance coefficient values for acetate were significantly higher in chemostat experiments than in recycling-fermentor experiments. 3CB maintenance coefficients were comparable in both experimental systems. The parameters for 3CB consumption kinetics varied remarkably with the experimental growth conditions in batch, chemostat, and recycling-fermentor environments. The results demonstrate that the determination of kinetic parameters in the laboratory for prediction of microbial activity in complex natural systems should be done under conditions which best mimic the system under consideration.     

Aerobic Hydrogen Production via Nitrogenase in Azotobacter vinelandii CA6

The diazotroph Azotobacter vinelandii possesses three distinct nitrogenase isoenzymes, all of which produce molecular hydrogen as a by-product. In batch cultures, A. vinelandii strain CA6, a mutant of strain CA, displays multiple phenotypes distinct from its parent: tolerance to tungstate, impaired growth and molybdate transport, and increased hydrogen evolution. Determining and comparing the genomic sequences of strains CA and CA6 revealed a large deletion in CA6''s genome, encompassing genes related to molybdate and iron transport and hydrogen reoxidation. A series of iron uptake analyses and chemostat culture experiments confirmed iron transport impairment and showed that the addition of fixed nitrogen (ammonia) resulted in cessation of hydrogen production. Additional chemostat experiments compared the hydrogen-producing parameters of different strains: in iron-sufficient, tungstate-free conditions, strain CA6''s yields were identical to those of a strain lacking only a single hydrogenase gene. However, in the presence of tungstate, CA6 produced several times more hydrogen. A. vinelandii may hold promise for developing a novel strategy for production of hydrogen as an energy compound.     

Use of chemostat data for modelling extracellular-inulinase production by Kluyveromyces marxianus in a high-cell-density fed-batch process

Production of extracellular inulinase by low-cell-density (2 kg dry weight·m⁻³) sucrose-limited chemostat cultures of Kluyveromyces marxianus obeyed saturated kinetics at dilution rates ranging from 0.02 to 0.5 h⁻¹. A non-structured Monod-type equation, describing the relation between specific growth rate and specific extracellular-inulinase production rate, was used to fit experimental data. THis equation was subsequently incorporated in a model for the production of biomass and extracellular inulinase in a high-cell-density (> 100 kg dry weight·m⁻³) fed-batchculture of K. marxianus grown on sucrose. The model adequately described biomass production in the fed-batch culture. However, the production of extracellular inulinase in the fed-batch process was slightly higher than predicted by the model. This observation may be related to differences in growth conditions between in the chemostat and fed-batch cultures.     

Chemostat cultures of yeasts, continuous culture fundamentals and simple unstructured mathematical models.

Fundamental aspects of chemostat cultures are reviewed. Using yeast cultures as examples, it is shown that steady states in chemostats may be predicted quantitatively by combining the correct number of unstructured kinetic models with expressions for existing stoichiometric constraints. The necessary number of such kinetic models corresponds to the number of limiting substrates and increases with the number of different metabolic pathways available to the strain. This is demonstrated by an experimental comparison of yeast growth limited by glucose alone for which metabolism is oxidative, and growth doubly limited by both glucose and oxygen, which occurs according to an oxido-reductive metabolism. The steady state data for such experiments can in principle be predicted based on a minimal amount of information by a simple stoichiometric model. It represents the overall stoichiometry of growth by a superposition of a fully oxidative and a fully reductive growth reaction and uses the concept of "aerobicity" to characterize the relative importance of the two reactions.     

A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria

When the growth of bacteria in a chemostat is controlled by limiting the supply of a single essential nutrient, the growth rate is affected both by the concentration of this nutrient in the culture medium and by the amount of time that it takes for the chemical and physiological processes that result in the production of new biomass. Thus, although the uptake of nutrient by cells is an essentially instantaneous process, the addition of new biomass is delayed by the amount of time that it takes to metabolize the nutrient. Mathematical models that incorporate this "delayed growth response" (DGR) phenomenon have been developed and analysed. However, because they are formulated in terms of parameters that are difficult to measure directly, these models are of limited value to experimentalists. In this paper, we introduce a DGR model that is formulated in terms of measurable parameters. In addition, we provide for this model a complete set of criteria for determining persistence versus extinction of the bacterial culture in the chemostat. Specifically, we show that DGR plays a role in determining persistence versus extinction only under certain ranges of chemostat operating parameters. It is also shown, however, that DGR plays a role in determining the steady-state nutrient and bacteria concentrations in all instances of persistence. The steady state and transient behavior of solutions of our model is found to be in agreement with data that we obtained in growing Escherichia coli 23716 in a chemostat with glucose as a limiting nutrient. One of the theoretical predictions of our model that does not occur in other DGR models is that under certain conditions a large delay in growth response might actually have a positive effect on the bacteria's ability to persist.