Chemostat Environment Research Articles

Competitive exclusion and coexistence in a general two-species stochastic chemostat model with Markov switching

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results.

Automated design of synthetic microbial communities

Microbial species rarely exist in isolation. In naturally occurring microbial systems there is strong evidence for a positive relationship between species diversity and productivity of communities. The pervasiveness of these communities in nature highlights possible advantages for genetically engineered strains to exist in cocultures as well. Building synthetic microbial communities allows us to create distributed systems that mitigate issues often found in engineering a monoculture, especially as functional complexity increases. Here, we demonstrate a methodology for designing robust synthetic communities that include competition for nutrients, and use quorum sensing to control amensal bacteriocin interactions in a chemostat environment. We computationally explore all two- and three- strain systems, using Bayesian methods to perform model selection, and identify the most robust candidates for producing stable steady state communities. Our findings highlight important interaction motifs that provide stability, and identify requirements for selecting genetic parts and further tuning the community composition.

The dynamics of a two host-two virus system in a chemostat environment

<p style='text-indent:20px;'>The coevolution or coexistence of multiple viruses with multiple hosts has been an important issue in viral ecology. This paper is to study the mathematical properties of the solutions of a chemostat model for two host species and two virus species. By virtue of the global dynamics of its submodels and the theories of uniform persistence and Hopf bifurcation, we derive sufficient conditions for the coexistence of two hosts with two viruses and coexistence of two hosts with one virus, as well as occurrence of Hopf bifurcation.

Evolution of chemotactic hitchhiking.

Bacteria typically reside in heterogeneous environments with various chemogradients where motile cells can gain an advantage over nonmotile cells. Since motility is energetically costly, cells must optimize their swimming speed and behaviour to maximize their fitness. Here, we investigate how cheating strategies might evolve where slow or nonmotile microbes exploit faster ones by sticking together and hitching a ride. Starting with physical and biological first principles, we computationally study the effects of sticking on the evolution of motility in a controlled chemostat environment. We find that stickiness allows for slow cheaters to dominate when chemoattractants are dispersed at intermediate distances. In this case, slow microbes exploit faster ones until they consume the population, leading to a tragedy of commons. For long races, slow microbes do gain an initial advantage from sticking, but eventually fall behind. Here, fast microbes are more likely to stick to other fast microbes and co-operate to increase their own population. We therefore conclude that whether the nature of the hitchhiking interaction is parasitic or mutualistic, depends on the chemoattractant distribution.

Geometric analysis of a model for cross‐feeding in the chemostat

A model for competition between bacterial strains in a chemostat is studied using a model reduction argument and phase portrait analysis for the specific case of trophic chain. The first strain feeds on glucose but secretes a metabolic intermediate (acetate), which the second strain consumes. However, the metabolic intermediate imposes a cost to growth to both strains. The geometric and asymptotic analysis of the reduced model allows a rigourous treatment of the role of this assumption in the emergence of coexistence (cross‐feeding) equilibria. Conditions for non‐existence of cross‐feeding equilibria are given for certain parameter ranges. In the case of trophic chains, existence of cross‐feeding equilibria does not rely only on the presence of the acetate specialist scavenging for the intermediate metabolite inside the resource‐limited chemostat environment. The glucose specialist must, in addition, possess a certain degree of tolerance to the metabolic intermediate. Relaxing the assumption of the cost of growth helps establish a cross‐feeding equilibrium.

Different adaptive strategies in E. coli populations evolving under macronutrient limitation and metal ion limitation

BackgroundAdaptive responses to nutrient limitation involve mutations that increase the efficiency of usage or uptake of the limiting nutrient. However, starvation of different nutrients has contrasting effects on physiology, resulting in different evolutionary responses. Most studies performed to understand these evolutionary responses have focused only on macronutrient limitation. Hence our understanding of adaptation under limitation of other forms of nutrients is limited. In this study, we compared the evolutionary response in populations evolving under growth-limiting conditions for a macronutrient and a major cation.ResultsWe evolved eight populations of E. coli in nutrient-limited chemostats for 400 generations to identify the genetic basis of the mechanisms involved in efficient usage of two nutrients: nitrogen and magnesium. Our population genomic sequencing work, based on this study and previous work, allowed us to identify targets of selection under these nutrient limiting conditions. Global transcriptional regulators glnGL were targets of selection under nitrogen starvation, while proteins involved in outer-membrane biogenesis (genes from the lpt operon) were targets of selection under magnesium starvation. The protein involved in cell-cycle arrest (yhaV) was a target of selection in both environments. We re-constructed specific mutants to analyze the effect of individual mutations on fitness in nutrient limiting conditions in chemostats and in batch cultures. We further demonstrated that adaptation to nitrogen starvation proceeds via a nutrient specific mechanism, while that to magnesium starvation involves a more general mechanism.ConclusionsOur results show two different forms of adaptive strategies under limitation of nutrients that effect cellular physiology in different ways. Adaptation to nitrogen starvation proceeds by upregulation of transcriptional regulator glnG and subsequently of transporter protein amtB, both of which results in increased nitrogen scavenging ability of the cell. On the other hand, adaptation to magnesium starvation proceeds via the restructuring of the cell outer-membrane, allowing magnesium to be redistributed to other biological processes. Also, adaptation to the chemostat environment involves selection for loss of function mutations in genes that under nutrient-limiting conditions interfere with continuous growth.

Statistical error model comparison for logistic growth of green algae (Raphidocelis subcapitata)

We validate a model for the population dynamics, as they occur in a chemostat environment, of the green algae Raphidocelis subcapitata, a species that is often used as a primary food source in toxicity experiments for the fresh water crustacean Daphnia magna. We collected longitudinal data from 4 replicate population experiments with R. subcapitata. This data was fit to a logistic growth model to reveal patterns of the algae growth in a continuous culture. Overall, our results suggest that a proportional error statistical model is the most appropriate for logistic growth modeling of R. subcapitata continuous population growth.

Continuous cultures of phytoplankton

The development of cultures of phytoplankton adapting throughout several days in an axenic, continuous-flow chemostat to yield a steady kinetic state of competing species is described mathematically. The adaptation of the growth rate to the chemostat environment inhibits integration of the equation of conservation of phytoplankton populations, though eventually when a steady state is reached the growth rate becomes equal to the rate of flow through the chemostat. Representation of species growth rates by a Verhuls formulation utilising experimentally determinable intra- and interspecies interaction constants permits the rapid prediction of the adaptation and alteration in the populations of competing phytoplankton species with changes in the chemostat environment. Illustrations of the behaviour of two and three competing species are extended to consideration of the stabilities of cultures of many competing species. Stable steady states of phytoplankton in a continuous-flow chemostat comprise a classic thermodynamic system and consequently the utilisation of light energy by the cells varies inversely with their growth rate. It is probable that when growth is nutrient limited, intra-and interspecies interaction parameters diminish as the demands of consumption are more nearly matched by the ratios of the limiting nutrients.

Competition and allelopathy with resource storage: Two resources

Allelopathy is added to a familiar mathematical model of competition between two species for two essential resources in a chemostat environment. Both species store the resources, and each produces a toxin that induces mortality in the other species. The corresponding model without toxins displays outcomes of competitive exclusion independent of initial conditions, competitive exclusion that depends on initial conditions (bistability), and globally stable coexistence, depending on tradeoffs between competitors in growth requirements and consumption of the resources. Introducing toxins that act only between, and not within species, can destabilize coexistence leading to bistability or other multiple attractors. Invasibility of the missing species into a resident׳s semitrivial equilibrium is related to competitive outcomes. Mutual invasibility is necessary and sufficient for a globally stable coexistence equilibrium, but is not necessary for coexistence at a locally stable equilibrium. Invasibility of one semitrivial equilibrium but not the other is necessary but not sufficient for competitive exclusion independent of initial conditions. Mutual non-invasibility is necessary but not sufficient for bistability. Numerical analysis suggests that when competitors display bistability in the absence of toxin production, increases in the overall magnitude of resource supply cause bistability to arise over a larger range of supply ratios between the two resources. When competitors display coexistence in the absence of toxin production, increases in overall resource supply destabilize coexistence and produce bistability or other configurations of multiple attractors over large ranges of supply ratios. The emergence of multiple attractors at high resource supplies suggests that blooms of harmful algae producing allelopathic toxins could be difficult to predict under such rich conditions.

Selection Affects Genes Involved in Replication during Long-Term Evolution in Experimental Populations of the Bacteriophage φX174

Observing organisms that evolve in response to strong selection over very short time scales allows the determination of the molecular mechanisms underlying adaptation. Although dissecting these molecular mechanisms is expensive and time-consuming, general patterns can be detected from repeated experiments, illuminating the biological processes involved in evolutionary adaptation. The bacteriophage φX174 was grown for 50 days in replicate chemostats under two culture conditions: Escherichia coli C as host growing at 37°C and Salmonella typhimurium as host growing at 43.5°C. After 50 days, greater than 20 substitutions per chemostat had risen to detectable levels. Of the 97 substitutions, four occurred in all four chemostats, five arose in both culture conditions, eight arose in only the high temperature S. typhimurium chemostats, and seven arose only in the E. coli chemostats. The remaining substitutions were detected only in a single chemostat, however, almost half of these have been seen in other similar experiments. Our findings support previous studies that host recognition and capsid stability are two biological processes that are modified during adaptation to novel hosts and high temperature. Based upon the substitutions shared across both environments, it is apparent that genome replication and packaging are also affected during adaptation to the chemostat environment, rather than to temperature or host per se. This environment is characterized by a large number of phage and very few hosts, leading to competition among phage within the host. We conclude from these results that adaptation to a high density environment selects for changes in genome replication at both protein and DNA sequence levels.

Resource Competition: A Bifurcation Theory Approach

We develop a framework for analysing the outcome of resource competition based on bifurcation theory. We elaborate our methodology by readdressing the problem of competition of two species for two resources in a chemostat environment. In the case of perfect-essential resources it has been extensively discussed using Tilman’s representation of resource quarter plane plots. Our mathematically rigorous analysis yields bifurcation diagrams with a striking similarity to Tilman’s method including the interpretation of the consumption vector and the resource supply vector. However, our approach is not restricted to a particular class of models but also works with other trophic interaction formulations. This is illustrated by the analysis of a model considering interactively-essential or complementary resources instead of prefect-essential resources. Additionally, our approach can also be used for other ecosystem compositions: multiple resources–multiple species communities with equilibrium or oscillatory dynamics. Hence, it gives not only a new interpretation of Tilman’s graphical approach, but it constitutes an extension of competition analyses to communities with many species as well as non-equilibrium dynamics.

Individual-based modelling of adaptation in marine microbial populations using genetically defined physiological parameters

Recent advances in genomics have led to a dramatic upwards revision of marine microbial diversity and a greater appreciation of the important role evolutionary dynamics play in structuring microbial communities. This has presented a significant challenge to marine ecosystem models, which are traditionally diversity poor, and often do not include adaptive/evolutionary processes. Here we explore the use of evolutionary individual-based models (IBMs) as a means of addressing some of these issues. In the IBM, we associate a digital ‘genome’ with each agent, which codes for the phenotypic traits of simulated organisms. Random mutations at the point of reproduction then allow adaptation in response to changing environmental conditions. Trade-offs between different physiological parameters result in different growth strategies emerging under different forcing scenarios. As an idealised test-case we consider resource competition in a chemostat environment, and compare the individual-based approach to a more traditional population-level model. When run in a non-evolutionary context using a clonal population of agents, the IBM reproduces the results of the population-level model. With evolutionary processes enabled, optimally adapted agents are observed to rise to prominence within the agent population. Their physiological trait values are shown to compare well with theoretical optimal trait combinations derived using resource competition theory. In more variable environments, the model is also shown to capture adaptation in response to changed environmental conditions. We conclude that IBMs represent a useful framework for building detailed models linking (sub-)individual-level processes to emergent ecosystem-level behaviour in simplified 0- or 1-D representations of the environment, which should complement global marine ecosystem models of high spatial complexity but necessarily simple process representation.

Chemostat‐based transcriptional analysis of growth rate change and BCL‐2 over‐expression in NS0 cells

We investigated the transcriptional response of NS0 cells undergoing controlled nutrient growth change in continuous chemostat culture using mouse microarrays. A 50% reduction in growth rate resulted in detectable alterations in the expression of 29 genes in NS0 cells. Notably, expression of genes in three major biological processes, namely transcriptional, translational, and protein processing functions, were modified. To further elucidate the advantage of the chemostat environment for establishment of "omic" data sets, an expression profile of the over-expressed gene bcl-2 in NS0 cells was probed. Functional analysis revealed that the underlying altered molecular mechanism was particularly associated with G1 cell cycle progression, protein synthesis, and apoptosis. Importantly, these findings agreed with the physical function of the cells. Despite an increase in survival rate, bcl-2 over-expression resulted in a decrease of specific productivity, glucose consumption, oxygen uptake rate and intracellular protein content, indicating a lower energy generating metabolism. Further, a prolongation of G1 cell cycle phase was evident on lowering the growth rate. Overall, the application of microarray analysis to chemostat-grown cultures offers an excellent combination for the interpretation of transcriptomic profiles to elucidate the molecular mechanisms during nutrient growth change and bcl-2 over-expression.

Toxic action and antibiotic in the chemostat: permanence and extinction of a model with functional response

A system of periodic coefficients functional differential equations is used to model the single microorganism in the chemostat environment with a periodic nutrient and antibiotic input. Furthermore, the total toxic action on the microorganism expressed by an integral term is considered in our system. Based on the technique of analysis, we obtain sufficient conditions which guarantee the permanence of the system and extinction of the microorganism.

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.

Evolutionary genomics of ecological specialization.

We used a combination of genomic techniques to monitor chromosomal evolution across hundreds of generations as Escherichia coli adapted to growth-limiting concentrations of either lactulose, methyl-galactoside, or a 72:28 mixture of the two. DNA microarrays identified 8 unique duplications and 16 unique deletions among 42 evolvants from 23 chemostat experiments. Each mutation was confirmed by sequencing PCR-amplified flanking genomic DNA and, except for one deletion, an insertion sequence was found at the break point. vPCR of insertion sequences identified these same mutations and 16 additional insertions (all confirmed by sequencing). The pattern of genomic evolution is highly reproducible. Statistical analyses show that duplications at lac and mutations in mgl are adaptations specific to lactulose and to methyl-galactoside, respectively. Adaptation to mixed sugars is characterized by similar mutations, but lac duplications and mgl mutations usually arise in different backgrounds, producing ecological specialists for each sugar. This suggests that an antagonistic pleiotropic tradeoff between duplications at lac and mutations in mgl retards the evolution of generalists. Other mutations that repeatedly appear in replicate experiments are adaptations to the chemostat environment and are not specific to one or the other sugar.

Uniform persistence and periodic solution of chemostat-type model with antibiotic

A system of functional differential equations is used to modelthe single microorganism in the chemostat environment with aperiodic nutrient and antibiotic input. Based on the techniqueof Razumikhin, we obtain the sufficient condition for uniformpersistence of the microbial population. For general periodicfunctional differential equations, we obtain a sufficientcondition for the existence of periodic solution, therefore,the existence of positive periodic solution to the chemostat-typemodel is verified.

Evaluation of Saccharomyces cerevisiae grown in a multistage chemostat environment under increasing levels of glucose

The concept of net consumption and/or production rates of fermentation metabolites has been proposed and applied to evaluate the performance of Saccharomyces cerevisiae grown in a multistage chemostat environment under increasing glucose concentrations (increased gravity). Results show that both rates are strongly affected by high glucose doses (osmotic pressure) and that ethanol inhibition has a profound influence on the performance of yeast cells.

Chemostat type equations modelling a polluted environment

The effects of toxic substances on two microorganisms community in a polluted chemostat environment is studied. The stability analysis of the nonnegative equilibria has been performed. A threshold for the survival or the extinction of populations has been determined.

Models of competition in the chemostat with instantaneous and delayed nutrient recycling

Two models for competition of two populations in a chemostat environment with nutrient recycling are considered. In the first model, the recycling is instantaneous, whereas in the second, the recycling is delayed. For each model an equilibrium analysis is carried out, and persistence criteria are obtained. This paper extends the work done by Beretta et al. (1990) for a single species.