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.     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

Spontaneous oscillations in two 2-component cells coupled by diffusion

Mathematical examples are presented of oscillators with two variables which do not oscillate in isolation, but which do oscillate stably when coupled with a twin via difiusion. Two examples are presented, the LefeverPrigogine Brusselator and a system used to model glycolytic oscillations. The mathematical method is not the usual bifurcation theory, but rather a type of singular perturbation theory combined with bifurcation theory. For both examples, it is shown that all stationary solutions are unstable for appropriate parameter settings. In the case of the Brusselator, it is further shown that there exist limit cycles; i.e. stable oscillations, in this parameter range. A numerical example is presented.Partially supported by NSF     

Nonlinear control for algae growth models in the chemostat

This paper deals with output feedback control of phytoplanktonic algae growth models in the chemostat. The considered class of model is of variable yield type, meaning that the ratio between the environmental nutrient absorption rate and the cells’ growth rate varies, which is different from classical bioprocesses assumptions. On the basis of weak qualitative hypotheses on the analytical expressions of the involved biological phenomena (which guarantee robustness of the procedure toward modeling uncertainties) we propose a nonlinear controller and prove its ability to globally stabilize such processes. Finally, we illustrate our approach with numerical simulations and show its benefits for biological laboratory experiments, especially for ensuring persistence of the culture facing classical experimental problems.     

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.Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC A4823Research carried out at the University of Alberta while on a Canada-China Scholarly Exchange Program     

Gain-induced oscillations in blood pressure

 “Mayer waves” are long-period (6 to 12 seconds) oscillations in arterial blood pressure, which have been observed and studied for more than 100 years in the cardiovascular system of humans and other mammals. A mathematical model of the human cardiovascular system is presented, incorporating parameters relevant to the onset of Mayer waves. The model is analyzed using methods of Liapunov stability and Hopf bifurcation theory. The analysis shows that increase in the gain of the baroreflex feedback loop controlling venous volume may lead to the onset of oscillations, while changes in the other parameters considered do not affect stability of the equilibrium state. The results agree with clinical observations of Mayer waves in human subjects, both in the period of the oscillations and in the observed age-dependence of Mayer waves. This leads to a proposed explanation of their occurrence, namely that Mayer waves are a gain-induced oscillation. Received: 15 September 1997/Revised version: 15 March 1998     

Ammonium-limited growth and uptake by Oscillatoria agardhii in chemostat cultures

Growth of Oscillatoria agardhil was studied in ammonium-limited chemostat cultures, at various dilution rates (=growth rates, μ). The uptake kinetics for ammonium of nitrogen (ammonium or nitrate)-limited chemostat cultures also was investigated. The kinetics of ammonium-limited growth could be adequately described by both the Monod and Droop equations, and were closely similar to the nitrate-limited growth kinetics of this species. The uptake kinetics for ammonium showed similarities as well as differences with the uptake kinetics for nitrate. The similarities were apparent in the uptake capacity values for ammonium and nitrate , which were identical, high and independent of μ. The differences were to be found in the half-saturation constants for ammonium uptake and nitrate uptake , the former being hardly influenced by μ. A consitutive, high affinity, system is likely to operate in the uptake and assimilation of ammonium by nitrogen-limited O. agardhii. The use of ammonium uptake parameters in studies of growth-limiting factors in nature can provide information as to whether a nitrogen-limitation prevails in natural habitats of this species.     

Autotrophic growth and inorganic sulphur compound oxidation by Sulfolobus sp. in chemostat culture

Sulfolobus strain LM was grown in tetrathionate and thiosulphate-limited continuous culture. CO₂ limitation resulted in a decrease of the steady-state biomass and an increase in the specific rate of thiosulphate oxidation so that substrate did not accumulate in the medium. The initial step in thiosulphate utilization appeared to be its conversion to tetrathionate. The affinity for tetrathionate oxidation appeared to increase with prolonged continuous culture giving an apparent K _m of about 6 M tetrathionate, a higher affinity than for thiosulphate oxidation and in the same range as values observed with acidophilic, sulphur-oxidizing eubacteria.     

Allee dynamics generated by protection mutualisms can drive oscillations in trophic cascades

Understanding the relative effect of top predators and primary producers on intermediate trophic levels is a key question in ecology. Most previous work, however, has not considered either realistic nonlinearities in feedback between trophic levels or the effect of mutualists on trophic cascades. Here, we develop a realistic model for a protection mutualism that explicitly includes interactions between a protected herbivore and both its food plant and generalist predators. In the absence of protection, herbivores and plant resources approach a stable equilibrium, provided that predation is not so high as to cause herbivore extinction. In contrast, adding protection by mutualists increases the range of dynamical outcomes to include unstable equilibria, stable and unstable limit cycles, and heteroclinic orbits. By reducing the impact of predators, protection by mutualists can allow herbivores to exert strong negative effects on their host plants, which in turn can lead to repeated cycles of overexploitation and recovery. Our results indicate that it may be essential to consider protection mutualisms to understand the dynamics of trophic cascades. Conversely, it may be essential to explicitly include dynamical feedback between plants and herbivores to fully understand the population and community dynamical consequences of protection mutualism.     

An analysis of the growth of Gluconobacter oxydans in chemostat cultures

Gluconobacter oxydans was grown successively in glucose and nitrogen-limited chemostat cultures. Construction of mass balances of organisms growing at increasing dilution rates in glucose-limited cultures, at pH 5.5, revealed a major shift from extensive glucose metabolism via the pentose phosphate pathway to the direct pathway of glucose oxidation yielding gluconic acid. Thus, whereas carbon dioxide production from glucose accounted for 49.4% of the carbon input at a dilution rate (D)=0.05 h^-1, it accounted for only 1.3% at D=0.26 h^-1. This decline in pentose phosphate pathway activity resulted in decreasing molar growth yields on glucose. At dilution rates of 0.05 h^-1 and 0.26 h^-1 molar growth yields of 19.5 g/mol and 3.2 g/mol, respectively, were obtained. Increase of the steady state glucose concentration in nitrogen-limited chemostat cultures maintained at a constant dilution rate also resulted in a decreased flow of carbon through the pentose phosphate pathway. Above a threshold value of 15–20 mM glucose in the culture, pentose phosphate pathway activity almost completely inhibited. In G. oxydans the coupling between energy generation and growth was very inefficient; yield values obtained at various dilution rates varied between 0.8–3.4 g/cells synthesized per 0.5 mol of oxygen consumed.     

Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations,bistability and sensitivity to parameters

We consider a two-trophic ecological model comprising of two predators competing for their common prey. We cast the model into the framework of a singular perturbed system of equations in one fast variable (prey population density) and two slow variables (predator population densities), mimicking the common observation that the per-capita productivity rate decreases from bottom to top along the trophic levels in Nature. We assume that both predators exhibit Holling II functional response with one of the predators (territorial) having a density dependent mortality rate. Depending on the system parameters, the model exhibits small, intermediate and/or large fluctuations in the population densities. The large fluctuations correspond to periodic population outbreaks followed by collapses (commonly known as cycles of “boom and bust”). The small fluctuations arise due to a singular Hopf bifurcation in the system, and are ecologically more desirable. However, more interestingly, the system exhibits mixed-mode oscillations (which are concatenations of the large amplitude oscillations and the small amplitude oscillations) that indicate the adaptability of the species to prolong the time gap between successive cycles of boom and bust. Numerical simulations are carried out to demonstrate the extreme sensitivity of the system to initial conditions (chaos and bistability of limit cycles are observed) as well as to the system parameters (here we only show the sensitivity to the density dependent mortality rate of the territorial predator). This model throws light at the uncertainties in long term behaviors that are associated with a real ecological system. We show that even very small changes in the system parameters due to natural or human-induced causes can lead to a complete different ecological phenomenon, thus affecting the predictability of the density of the prey population. In this paper, we explain the mechanisms behind the irregular fluctuations in the population sizes in an attempt to understand the dynamics occurring in a natural population and also comment on the inherent uncertainties associated with the system.     

Energy conservation by pyrroloquinoline quinol-linked xylose oxidation in Pseudomonas putida NCTC 10936 during carbon-limited growth in chemostat culture

Abstract When grown in carbon source-limited chemostat cultures with lactate or glucose as the carbon and energy source and xylose as an additional source of reducing equivalents, Pseudomonas putida NCTC 10936 oxidized xylose to xylonolactone and xylonate. No other products were formed from this pentose, nor was it incorporated into biomass. The presence of xylose in these cultures resulted in higher Y_glucose and Y_lactate values as compared to cultures without xylose indicating that biologically useful energy was conserved during the periplasmic oxidation of xylose. As the Y₀ values for growth on glucose or on lactate alone were equal to the Y₀ values with xylose as co-substrate, it is concluded that for flucose- or lactate-limited growth energy conservation by PQQH₂ oxidation is as efficient as by NADH₂ oxidation.     

Calcium oscillations induced by gambierol in cerebellar granule cells

Gambierol is a marine polyether ladder toxin derived from the dinoflagellate Gambierdiscus toxicus. To date, gambierol has been reported to act either as a partial agonist or as an antagonist of sodium channels or as a blocker of voltage‐dependent potassium channels. In this work, we examined the cellular effect of gambierol on cytosolic calcium concentration, membrane potential and sodium and potassium membrane currents in primary cultures of cerebellar granule cells. We found that at concentrations ranging from 0.1 to 30 µM, gambierol‐evoked [Ca²⁺]c oscillations that were dependent on the presence of extracellular calcium, irreversible and highly synchronous. Gambierol‐evoked [Ca²⁺]c oscillations were completely eliminated by the NMDA receptor antagonist APV and by riluzole and delayed by CNQX. In addition, the K⁺ channel blocker 4‐aminopyridine (4‐AP)‐evoked cytosolic calcium oscillations in this neuronal system that were blocked by APV and delayed in the presence of CNQX. Electrophysiological recordings indicated that gambierol caused membrane potential oscillations, decreased inward sodium current amplitude and decreased also outward IA and IK current amplitude. The results presented here point to a common mechanism of action for gambierol and 4‐AP and indicate that gambierol‐induced oscillations in cerebellar neurons are most likely secondary to a blocking action of the toxin on voltage‐dependent potassium channels and hyperpolarization of sodium current activation. J. Cell. Biochem. 110: 497–508, 2010. © 2010 Wiley‐Liss, Inc.     

Metabolic fluxes during the growth of thermotolerant methylotrophic Bacillus strains in methanol-sufficient chemostat cultures

Growth of the thermotolerant methylotrophic Bacillus strain TS1 in methanol-limited chemostat culture showed that the substrate was oxidized solely to biomass and CO₂. When a pulse of methanol was added to the growth vessel anabolism could be shown to be dissociated from catabolism for a transient period of time. Present data shows that when the organism was grown with a limitation other than carbon, some of the substrate was channelled into metabolite over-production. When the organism was grown under N-limitation 2-oxoglutarate accumulated in the culture medium in small amounts whilst acetate accumulated under all carbon excess conditions. Although the average carbon recovery was 92%, analysis of the culture filtrates for other metabolites failed to show significant amounts of any individual product above those detected in carbon-limited growth comditions. The results are discussed in relation to published data.     

Long-term oscillations in grassland productivity induced by drought

Disturbances such as drought have immediate impacts on ecosystem functioning, but little is known about long-term dynamic consequences of disturbance. Here, we show that a major drought perturbed prairie grassland from nearly constant average annual production, and induced 9 years of oscillations with a 2-year period. This pattern occurred in unmanipulated plots in many widely separated fields that were part of two different long-term studies. Using direct and indirect tests, we could reject the hypothesis that the oscillations were externally imposed by climate or herbivores. Weak evidence supported the hypothesis that oscillations were internally generated, caused by a litter and nitrogen dependent feedback on productivity. This hypothesis was supported by the results of two other long-term studies, in which burning and, separately, nitrogen addition eliminated the 2-year oscillations in plant production. However, more direct evidence failed to support the litter hypothesis. A final, but untested hypothesis was suggested by the observation that production tended to be synchronized amongst dominant plants. Drought may have synchronized production dynamics amongst plants with biennially greater allocation to above-ground growth. Regardless of the specific mechanism, our results show that a single disturbance may have long-lasting effects on the dynamics of plant production.     

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.     

短暂的糖氧剥夺导致小鼠黑质核团的即时性损伤

为研究运动控制关键核团黑质在代谢负荷下发生的即时性损伤,研制了多通道组织灌流槽,改进了以MTT、甲酚紫染色考察脑组织损伤的方法,获得了20 min的短时程糖氧剥夺所引起的小鼠黑质区即时性功能和形态损伤的组织化学依据。1)MTT检测显示,糖氧剥夺使黑质区域MTT与线粒体琥珀酸脱氢酶的反应产物甲臜晶体密度减少,并且这种减少在黑质的中部到吻侧的背部区域最为明显(约42.5%);2)甲酚紫染色结果表明,糖氧剥夺不影响脑片黑质区面积和着色斑块的数密度,而使斑块的平均面积减低(6.9%)。上述研究结果提示短时程糖氧剥夺可导致黑质区空间依存性的线粒体功能损伤,并且具有与凋亡相似的形态学特征。     

The effect of carbon dioxide on the growth kinetics of fructose-limited chemostat cultures of Acetobacterium woodii DSM 1030

The growth of the anaerobic acetogenic bacterium Acetobacterium woodii DSM 1030 was investigated in fructose-limited chemostat cultures. A defined medium was developed which contained fructose, mineral salts, cysteine · HCl and Ca pantothenate (1 mg · 1^–1) supplied in a vitamin supplement. Growth at high dilution rates was dependent on the presence of CO₂ in the gas phase. The ^max was found to be 0.16 h^–1 and the fructose maintenance requirement was 0.1 to 0.13 mmol fructose · (g dry wt)^–1 · h^–1. A growth yield of 61 g dry wt · (mol fructose)^–1, corrected for the cell maintenance requirement and for incorporation of fructose carbon into cell biomass, was determined from the fructose consumption. A corresponding growth yield of 69 g dry wt · (mol fructose)^–1 was calculated from the acetate production assuming that fructose fermentation was homoacetogenic. A Y_ATP of 12.2 to 13.8 g dry wt · (mol ATP)^–1 was calculated from these growth yields using a value of 5 mol ATP · (mol fructose)^–1 as an estimate of the amount of ATP synthesised from fructose fermentation. The addition of yeast extract (0.5 g · 1^–1) to the medium did not influence the ^max or cell yield. After prolonged growth under fructose-limited conditions the requirement of the culture for CO₂ in the gas phase was reduced.Abbreviations YE yeast extract - IC inorganic carbon - D fermenter dilution rate : h^–1 - M_X maintenance requirement for X: mmol X · (g dry wt)^–1 · h^–1 - X may be fructose (Fruct), fructose consumed in energy metabolism (Fruct [E]), acetate (Ac) - ATP CO₂, NH _inf4 ^sup+ or Pi - qX specific rate of utilisation or consumption of X: mmol X · (g dry wt)^–1 · h^–1 - V fermenter volume: litre - rC · Cell, fermenter cell carbon production: mmol C · h^–1 - Y_X yield of cells on X: g dry wt · (mol X)^–1 - Y _infx ^supmax the yield corrected for cell maintenance: g dry wt · (mol X)^–1 - S_ATP stoichiometry of ATP synthesis from fructose: mol ATP · (mol frucose)^–1 - x cell concentration: g dry wt · 1^–1 - specific growth rate : h^–1 - ^max maximum specific growth rate: h^–1     

Resonant properties in the paddlefish electrosensory system caused by delayed feedback

Introduction The paddlefish electrosensory system consists of receptor cells in the skin that sense minute electric fields from their prey, small water fleas. The receptors thereby measure the difference of the voltage at the skin surface against the voltage inside the animal. Due to a high skin impedance, this internal voltage is considered to be relatively fixed. Results We found, however, that this internal voltage can fluctuate. It shows damped oscillations to a single short electric field pulse and changes, with some time delay, according to the previous history of stimulation, and shows resonance at a certain frequency. Conclusions Computer simulations show that these phenomena can be explained by the presence of delayed feedback where the internal voltage is part of the feedback loop.