由微分方程所描述的微生物连续培养动力系统(Ⅰ)

陆续介绍微生物连续培养(Chemostat)的基本原理,以单种微生物连续培养模型为基础,较详细地介绍几类由微分方程所描述的微生物连续培养动力系统模型,涉及的问题有解的稳定性,系统的持久性,周期解和Hopf分支等．     

Model for continuous culture which considers the viability concept

The effect of the rate of ‘natural death’ of bacteria on the steady state behaviour of continuous culture has been studied. A model which has no real biochemical basis but which gives good experimental correlation has been proposed. Populations of Acrobacter Acrogenes harvested from the chemostat at dilution rates greater than 0.1 HR^?1 were found to be over 95% viable. The effect of the rate of death on the steady state yields becomes significant only at very low dilution rates. The experimental work of other workers has also been simulated to test the validity of the model. Theoretically it is also shown that an ‘apparent’ lag in batch cultures will be observed if the innoculum is of very low viability.     

The ideal free distribution and bacterial growth on two substrates

A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.     

Bacterivorous grazers facilitate organic matter decomposition: a stoichiometric modeling approach

There is widespread empirical evidence that protist grazing on bacteria reduces bacterial abundances but increases bacteria-mediated decomposition of organic matter. This paradox has been noted repeatedly in the microbiology literature but lacks a generally accepted mechanistic explanation. To explain this paradox quantitatively, we develop a bacteria-grazer model of organic matter decomposition that incorporates protozoa-driven nutrient recycling and stoichiometry. Unlike previous efforts, the current model includes explicit limitation, via Liebig's law of minimum, by two possible factors, nutrient and carbon densities, as well as their relative ratios in bacteria and grazers. Our model shows two principal results: (1) when the environment is carbon limiting, organic matter can always be decomposed completely, regardless of the presence/absence of grazers; (2) when the environment is nutrient (such as nitrogen) limiting, it is possible for organic matter to be completely decomposed in the presence, but not absence, of grazers. Grazers facilitate decomposition by releasing nutrients back into the environment, which would otherwise be limiting, while preying upon bacteria. Model analysis reveals that facilitation of organic matter decomposition by grazers is positively related to the stoichiometric difference between bacteria and grazers. In addition, we predict the existence of an optimal density range of introduced grazers, which maximally facilitate the decomposition of organic matter in a fixed time period. This optimal range reflects a trade-off between grazer-induced nutrient recycling and grazer-induced mortality of bacteria.     

Role of toxin and nutrient for the occurrence and termination of plankton bloom--results drawn from field observations and a mathematical model

The termination of harmful algal blooms (HABs) and coexistence of phytoplankton-zooplankton populations are of great importance to human health, ecosystem, environment, tourism and fisheries. In this paper we propose a three-component model consisting of dissolved limiting nutrients (N) supplied at constant rate and partially recycled after the death of plankton by bacterial decomposition, phytoplankton (P) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. Our analysis leads to different thresholds which are expressible in terms of model parameters and determine the existence and stability of various states of the system. We observe that phytoplankton-zooplankton persist if the maximal zooplankton ingestion rate exceeds a lower threshold value. It is shown that the coexistence equilibrium loses its stability when the dilution rate of the nutrient concentration passes through a critical value and Hopf bifurcation occurs that induces oscillations of the population. Our results indicate that the occurrence of bloom increases when the nutrient concentration is very high, and in that case toxin produced by the phytoplankton plays a very crucial role towards the termination of the planktonic bloom.     

Can phosphorus limitation inhibit dissolved organic carbon consumption in aquatic microbial food webs? A study of three food web structures in microcosms

Microcosms with three different food web structures and phosphorus (P) limited growth medium were used to study the interactions between P and organic carbon (C) fractions in pelagic food webs. The cultures were run with low dilution to allow the biological processes to determine the outcome. A double isotope technique was used to follow the C and P compartments. In all systems the primary production was P limited. The measured P:C ratios and the observed accumulation of degradable dissolved organic carbon (DOC) indicated that the growth of heterotrophic bacteria was also P limited. The presence of neither algal grazers nor flagellates feeding on bacteria altered the limitation pattern. A net loss of P from the bacterial fraction was observed after the bloom. Different strategies for nutrient aquisition and growth are proposed as mechanisms enabling simultaneous P limitation of algae and bacteria, and a concomitant accumulation of degradable DOC. The ability of the algae to grow with low P:C ratio keeps the regeneration of P through grazers low enough to cause sustained P limitation of both algae and bacteria. The grazers were important producers of DOC when present. This implies that the usual assumption of carbon limited bacterial growth may lead to wrong conclusions regarding the dynamics of plankton communities and the DOC pool.     

Kinetics of substrate utilization in adenine-limited chemostat cultures of Bacillus subtilis KYA741.

The specific rates of limiting substrate utilization were investigated in adenine- or glucose-limited chemostat cultures of Bacillus subtilis KYA741, an adenine-requiring strain, at 37 degrees C. With the glucose-limited cultures, the specific rate of glucose consumption versus dilution rate gave a linear relationship from which the true growth yield and maintenance coefficient were determined to be 0.09 mg of bacteria per mg of glucose and 0.2 mg of glucose per mg of bacteria per h, respectively. With the adenine-limited cultures, adenine as the limiting substrate was not completely consumed at lower dilution rates (e.g., D less than 0.1), unlike in the glucose-limited cultures. When a linear relationship of specific rate of adenine consumption versus dilution rate was extrapolated to zero dilution rate, a negative value for the specific rate of adenine consumption, -0.01 mg of adenine per mg of bacteria per h, was obtained, giving a true growth yield for adenine of 5.2 mg of bacteria per mg of adenine. On the other hand, the maintenance coefficient of oxygen uptake gave a positive value of 8.1 x 10(-3) mmol/mg of bacteria per h. Based on previous results showing that adenine is resupplied by lysing cells, we developed kinetic models of adenine utilization and cell growth that gave a good estimation of the peculiar behavior of cell growth and adenine utilization in adenine-limited chemostat cultures.     

由微分方程所描述的微生物连续培养动力系统(II)

该文为论文 [1 ]的继续 ,主要介绍具有时间滞后的微生物连续培养模型     

Symposium on “Protozoan Ecology”: The Role of Protozoa in Litters and Soils1

Soil is the focus of organic matter turnover in terrestrial ecosystems and is an interstitial mosaic of microsites composed of particle aggregates and pore spaces, where transformation, decomposition, mineralization, and humification of organic matter takes place. Microorganisms and animals are scattered discontinuously in these microsites. Microarthropods and larger fauna increase the rate and amount of mineralization by comminution of organic matter and by redistribution of microsites through movements of earthworms and large arthropods; however, mineralization and return of nutrients to plants occurs in the community of bacteria, fungi, protozoa, and nematodes living in the water films covering aggregates and filling pore spaces. Protozoa, especially small amoebae, are important bacterial grazers because they can enter tiny spaces unavailable to nematodes. The latter graze bacteria, fungi, and protozoa. Protozoan and nematode predation increase the amounts of soluble nutrients and decrease the competitive abilities of bacteria, thus making these nutrients more available to plants. Protozoa enhance nutrient recycling out of proportion to their biomass.     

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.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Effect of biostimulants on 2,4,6-trinitrotoluene (TNT) degradation and bacterial community composition in contaminated aquifer sediment enrichments

2,4,6-Trinitrotoluene (TNT) is a toxic and persistent explosive compound occurring as a contaminant at numerous sites worldwide. Knowledge of the microbial dynamics driving TNT biodegradation is limited, particularly in native aquifer sediments where it poses a threat to water resources. The purpose of this study was to quantify the effect of organic amendments on anaerobic TNT biodegradation rate and pathway in an enrichment culture obtained from historically contaminated aquifer sediment and to compare the bacterial community dynamics. TNT readily biodegraded in all microcosms, with the highest biodegradation rate obtained under the lactate amended condition followed by ethanol amended and naturally occurring organic matter (extracted from site sediment) amended conditions. Although a reductive pathway of TNT degradation was observed across all conditions, denaturing gradient gel electrophoresis (DGGE) analysis revealed distinct bacterial community compositions. In all microcosms, Gram-negative γ- or β-Proteobacteria and Gram-positive Negativicutes or Clostridia were observed. A Pseudomonas sp. in particular was observed to be stimulated under all conditions. According to non-metric multidimensional scaling analysis of DGGE profiles, the microcosm communities were most similar to heavily TNT-contaminated field site sediment, relative to moderately and uncontaminated sediments, suggesting that TNT contamination itself is a major driver of microbial community structure. Overall these results provide a new line of evidence of the key bacteria driving TNT degradation in aquifer sediments and their dynamics in response to organic carbon amendment, supporting this approach as a promising technology for stimulating in situ TNT bioremediation in the subsurface.     

Bi-trophic food chain dynamics with multiple component populations

Food web models describe the patterns of material and energy flow in communities. In classical food web models the state of each population is described by a single variable which represents, for instance, the biomass or the number of individuals that make up the population. However, in a number of models proposed recently in the literature the individual organisms consist of two components. In addition to the structural component there is an internal pool of nutrients, lipids or reserves. Consequently the population model for each trophic level is described by two state variables instead of one. As a result the classical predator-prey interaction formalisms have to be revised. In our model time budgets with actions as searching and handling provide the formulation of the functional response for both components. In the model, assimilation of the ingested two prey components is done in parallel and the extracted energy is added to a predators reserve pool. The reserves are used for vital processes; growth, reproduction and maintenance. We will explore the top-down modelling approach where the perspective is from the community. We will demonstrate that this approach facilitates a check on the balance equations for mass and energy at this level of organization. Here it will be shown that, if the individual is allowed to shrink when the energy reserves are in short to pay the maintenance costs, the growth process has to be 100% effective. This is unrealistic and some alternative model formulations are discussed. The long-term dynamics of a microbial food chain in the chemostat are studied using bifurcation analysis. The dilution rate and the concentration of nutrients in the reservoir are the bifurcation parameters. The studied microbial bi-trophic food chain with two-component populations shows chaotic behaviour.     

Novel mathematical modeling on pilot-scale of plug-flow aerated submerged biofilm reactor for dissolved organic matter and nitrogen removal

The investigation aimed to present mathematical models for describing the dynamic behavior of the dissolved organic matter removal and nitrification in the Aerated Submerged Bio-Film (ASBF) for a plug-flow reactor. Based on the experimental data from the batch system of the ASBF pilot plant, mathematical models for the plug-flow reactor were developed to predict dissolved organic matter and ammonia nitrogen removal rates as a function of heterotrophic and autotrophic bacteria populations, dissolved organic matter concentrations, ammonia nitrogen concentrations, dissolved oxygen concentrations, and temperature. The mathematical models for dissolved organic matter and ammonia nitrogen removal in ASBF include two differential equations reflecting heterotrophic and autotrophic bacteria populations, and a number of kinetic parameters. Consequently, the results present a better insight into the dynamics behavior of heterotrophic and autotrophic biofilm growth and their practical application to wastewater for dissolved organic matter and ammonia nitrogen removal process. The mathematical model for ammonia nitrogen and dissolved organic matter removals present good results for the plug-flow reactor.     

Use of coimmobilized biological systems to degrade toxic organic compounds

The concept of coimmobilizing cell mass (and/or enzyme) and adsorbent in a hydrogel matrix for biodegradation of toxic organic chemicals was introduced. Under defined experimental conditions, the coimmobilized system using activated carbon and Phanerochaete chrysosporium was compared with nonimmobilized systems for the degradation of pentachlorophenol (PCP). It was demonstrated that the coimmobilized system degraded PCP more effectively than the nonimmobilized system. A solid substrate included in the coimmobilized system could support the biodegradation. Isolation of the degrading agents from a model interrupting microorganism by the coimmobilized capsule membrane reduced the interference on the biodegradation. In simulated contaminated soil extract and sand, the coimmobilized system also exhibited higher degradative ability and stability than the nonimmobilized systems.     

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient–two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebigs minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.Revised version: 21 July 2003     

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.     

Delayed feedback control for a chemostat model

In this paper, we discuss asymptotic properties and numerical simulations of a chemostat model with delayed feedback control. A chemostat model with two organisms can be made coexistent by feedback control of the dilution rate which depends affinely on the concentrations of two organisms [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48]. Then the coexistence takes its simplest form; the equilibrium point in the non-negative orthant is globally asymptotically stable. We show that stability of the equilibrium point is changed by 'time-delay' caused in controlling the dilution rate after measuring the concentrations of two organisms.     

Stoichiometry of carbohydrate fermentation and microbial growth efficiency in a continous culture of mixed rumen bacteria

Summary A population of mixed rumen bacteria was maintained in a chemostat at four different dilution rates, with glocose as the growth limiting carbon and energy substrate. Increasing the dilution rate shifted the proportions of end products: methane decreased and propionate increased. Fermentation and hydrogen balances were calculated from the fermentation end products. Values were similar to earlier ones from batch incubations of rumen contents. This suggests that theoretical overall reaction schemes for carbohydrate fermentation in the rumen, proposed earlier, are also valid in continuous culture.A positive correlation between dilution rate and microbial growth efficiency (gN_inc./kg OM_f was observed, confirming earlier work.Apparently conflicting results of chemostat work and recent in vivo experiments are discussed.     

A size-structured model of bacterial growth and reproduction

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.