Feedback control for chemostat models

 It is shown that a chemostat with two organisms can be made coexistent by means of feedback control of the dilution rate. Remaining freedom in the feedback law can be used to guarantee robustness or improve particular performance indices. Unfortunately a topological property prevents coexistence by feedback control for chemostats with more than two organisms. We apply our results to control bioreactors aimed at producing commercial products through genetically altered organisms. In all our results the coexistence takes its simplest form: a global asymptotically stable equilibrium point in the interior of the non-negative orthant. Received: 1 November 2001 / Revised version: 19 August 2002 / Published online: 19 November 2002 Mathematics Subject Classification (2000): 34H05, 34D23, 93C10 Keywords or phrases: Chemostat – Feedback control – Coexistence     

Periodic solution of a chemostat model with Monod growth rate and impulsive state feedback control

In this paper, we develop a mathematical model concerning a chemostat with impulsive state feedback control to investigate the periodicity of bioprocess. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions under which the model has a periodic solution of order one are obtained. Furthermore, we estimate the position of the periodic solution of order one and discuss the existence of periodic solution of order two. The theoretical results and numerical simulations indicate that the chemostat system with impulsive state feedback control either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentrations of microorganisms and substrate.     

Symbiotic growth of Acetobacter suboxydans and Saccharomyces carlsbergensis in a chemostat

The bacterium Acetobacter suboyxdans and the yeast Saccharomyces carlsbergensis have been grown together on a synthetic medium in a chemostat. Mannitol, the only carbon source fed to the fermenter, is oxidized by the bacteria to fructose. The yeast, which cannot attack mannitol, breaks down the fructose nearly completely. Eight steady states and five transitory periods after changes in flow rate have been analyzed to study the kinetics of the mixed culture. Separate cell concentrations were determined by a modified Coulter counter apparatus. Both sugars were monitored. Both bacteria and yeast may be modeled using Monod's equation, the latter with some deviations. The yeast is unable to grow beyond the washout point of the bacteria, even though its maximum growth rate is much higher. The yield of both organisms decreases with increasing dilution rate, as does their average cell size. After step changes in dilution rate, repeated oscillations of both sugar and cell concentrations usually occur before steady-state conditions are reattained. They are generally in phase, with no definite sign of a lag. Oscillations of yeast and fructose concentrations are more pronounced. Periods average about 6 hr and are not correlated with fermentation conditions or equipment variables. Repeated oscillations are not found after step-downs in pure cultures of A. suboxydans, leading to the conclusion that the instability in mixed cultures may be caused by a feedback mechanism from the yeast to the bacteria.     

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.     

Practical coexistence of two species in the chemostat - A slow-fast characterization

We show that the chemostat model with two species having different but close break-even concentrations exhibits a slow-fast dynamics. Considering small perturbations about the dilution rate for which break-even concentrations are identical, we use the Fenichel theory to show the coexistence of species for large times. Then we determine the reduced dynamics, which is non-trivial and characterized by the slopes of the growth functions about their break-even concentrations.     

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.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

Dynamic analysis of a microbial process: A systems engineering approach

A general mathematical model of the chemostat system is developed in order to define an experimental program of dynamic testing. A glucose-limited culture ofSaccharomyces cerevisiae was grown in a chemostat using chemically defined medium. The chemostat was perturbed from an initial steady state by changes in input glucose concentration, dilution rate, pH, and temperature. Dynamic responses of cell mass, glucose, cell number, RNA, and protein concentrations were measured. A number of simulation techniques were used in developing a dynamic mathematical model and in comparing the developed model with experimental data as well as the Monod model. The resulting model was found to be quantitatively accurate and superior to the Monod model. The developed model was interpreted in the light of cell physiology. Adjustment of intracellular RNA fraction was found to be rate limiting in acceleration of cell specific growth rate.     

Do spatial effects appear at low dilution rate in chemostat?

The chemostat theory on two species competition has shown that the dilution rate where transition of dominance occurs – transition-dilution rate – is independent of limiting-nutrient concentration. However, we obtained the experimental data indicating that the transition-dilution rate changed with variations in limiting-ammonium concentrations, using the chemostat mixed-culture of the cyanobacterium Microcystis novacekii and the green alga Scenedesmus quadricauda. The transition-dilution rate was dependent on the concentration of limiting ammonium in the influx culture medium. We tried to simulate the experimental results. Though the dilution rate has been considered independent of nutrient concentration, we introduce the effective dilution rate that depends on nutrient concentration (ammonium concentration in this study). A hyperbolic Monod-type function is used to represent the effective dilution rate for each species. The maximum dilution rate of the function is set to be the mechanical dilution rate (nominal dilution rate) of the chemostat culture. The calculation shows that the nominal transition-dilution rate where transition of dominance occur decreases with increased concentration. This simulation is well consistent with our experimental data. These results may suggest that the species-specificity of limiting nutrients, here nitrogen. Or they may imply that the depreciation of nitrogen becomes critical when both dilution rate and concentration are very low, especially for the green algae. In the latter case, spatial effects are induced internally in the ecosystem.     

Stability of commensalistic systems

The response of two-species commensalistic systems in a chemostat has been investigated after perturbations in steady state conditions and after step changes in dilution rate. The system is inherently stable with not more than three overshoots and undershoots possible. More complicated commensalistic systems are less stable, with limit cycle response occurring after dilution rate changes when feedback inhibition and feedforward activation occurs. In general variation of feedback parameters is more effective in changing the behavior of the systems than variation of feedforward parameters. Limited agreement with the experimental data of Chao and Reilly was obtained.     

Kinetics of biphasic growth of yeast in continuous and fed-batch cultures

The influence of dilution rate on the production of biomass, ethanol, and invertase in an aerobic culture of Saccharomyces carlsbergensis was studied in a glucose-limited chemostat culture. A kinetic model was developed to analyze the biphasic growth of yeast on both the glucose remaining and the ethanol produced in the culture. The model assumes a double effect where glucose regulates the flux of glucose catabolism (respiration and aerobic fermentation) and the ethanol utilization in yeast cells. The model could successfully demonstrate the experimental results of a chemostat culture featuring the monotonic decrease of biomass concentration with an increase of dilution rate higher than 0.2 hr^?1 as well as the maximum ethanol concentration at a particular dilution rate around 0.5 hr^?1. Some supplementary data were collected from an ethanol-limited aerobic chemostat culture and a glucose-limited anaerobic chemostat culture to use in the model calculation. Some parametric constants of cell growth, ethanol production, and invertase formation were determined in batch cultures under aerobic and anaerobic states as summarized in a table in comparison with the chemostat data. Using the constants, a prediction of the optimal control of a glucose fed-batch yeast culture was conducted in connection with an experiment for harvesting a high yield of yeast cells with high invertase activity.     

Aggregate-formation by Clostridium butyricum

Summary Clostridium butyricum was grown in a glucose-limited chemostat culture at a dilution rate of 0.1 h^–1 at pH 6.0. With 0.9% w/v input glucose in the medium the cells were found to grow in suspension and glucose was fermented completely to acetate and butyrate. An increase in the input concentration of glucose resulted in increased concentrations of end-products, but not all extra glucose was consumed. It could be demonstrated that this was due to a lowering of the maximal growth rate by elevated levels of butyric acid. However, prolonged growth in the presence of high glucose concentrations led to an increase in biomass. This was caused by the selection of a variant that was less sensitive to butyrate. This variant was able to form aggregates in an anaerobic gas-lift reactor at high dilution rates. Inoculation of these aggregates in a conventional chemostat culture with high glucose input resulted in an aggregated culture that remained stable for at least 6 months, and in which all glucose was consumed. Whether the organisms grew in suspension or in aggregates was found to be determined by the concentration of butyrate. The isolation of aggregate-forming variants from chemostat cultures leads to a very simple and new type of immobilization technique.Offprint requests to: G. R. Zoutberg     

Time delay in simple chemostat models.

The models of Monod and Williams, for the growth of unicellular organisms in chemostats, give strongly damped transients in the biomass and cell number when the flow rate of the chemostat is changed. A simple trick is used to incorporate time delay in these models while still allowing a conventional stability analysis. For long enough time delays the equilibrium point is unstable and limit cycles can be compared. Results obtained using Williams' model, with weakly damped transients as a result of using moderately long time delay, are compared with his data in which cell numbers show weak damping but biomass shows strong damping.     

Global analysis of competition for perfectly substitutable resources with linear response

We study a model of the chemostat with two species competing for two perfectly substitutable resources in the case of linear functional response. Lyapunov methods are used to provide sufficient conditions for the global asymptotic stability of the coexistence equilibrium. Then, using compound matrix techniques, we provide a global analysis in a subset of parameter space. In particular, we show that each solution converges to an equilibrium, even in the case that the coexistence equilibrium is a saddle. Finally, we provide a bifurcation analysis based on the dilution rate. In this context, we are able to provide a geometric interpretation that gives insight into the role of the other parameters in the bifurcation sequence. Funding was provided by the National Science Foundation-funded ADVANCE Institutional Transformation Program at New Mexico State University, fund # NSF0123690. Research partially supported by the Natural Science and Engineering Research Council of Canada.     

Bacterial competition in serial transfer culture

A mathematical model of bacterial competition for a single growth-limiting substrate in serial transfer culture is formulated. Each bacterial strain is characterized by a growth response function, e.g. Monod function determined by a maximum growth rate and half-saturation nutrient concentration, and the length of its lag phase following the dilution event. The goal of our study is to understand what factors determine an organisms fitness or competitive ability in serial transfer culture. A motivating question is: how many strains can coexist in serial transfer culture? Unlike competition in the chemostat, coexistence of two strains can occur in serial transfer culture. Numerical simulations suggest that more than two may coexist.     

Study of the dynamic behavior of the chemostat system

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

Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor

The asymptotic behavior of solutions of a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with two distributed delays and an external inhibitor is considered. The model presents a refinement of the one considered by Lu and Hadeler [Z. Lu, K.P. Hadeler, Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. Biosci. 167 (2000) p. 177]. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. Furthermore, it is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Conditions for boundedness of solutions and existence of non-negative equilibrium are given. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the uniform persistence of the system are also carried out. By constructing appropriate Liapunov-like functionals, some sufficient conditions of global attractivity to the extinction equilibria are obtained and the combined effects of the delays and the inhibitor are studied.     

The periodically forced Droop model for phytoplankton growth in a chemostat

 It is proved that the periodically forced Droop model for phytoplankton growth in a chemostat has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the sub-threshold condition holds, the phytoplankton population is washed out of the chemostat. If the super-threshold condition holds, then there is a unique periodic solution, having the same period as the forcing, characterized by the presence of the phytoplankton population, to which all solutions approach asymptotically. Furthermore, this result holds for a general class of models with monotone growth rate and monotone uptake rate, the latter possibly depending on the cell quota. Received 10 October 1995; received in revised form 26 March 1996     

Estimation of steady-state culture characteristics during acceleration-stats with yeasts

Steady-state culture characteristics are usually determined in chemostat cultivations, which are very time-consuming. In contrast, acceleration-stat (A-stat) cultivations in which the dilution rate is continuously changed with a constant acceleration rate are not so time-consuming, especially at high acceleration rates. Therefore, the A-stat could be advantageous to use instead of the chemostat. However, the highest acceleration rate, meaning the fastest A-stat that can be applied for estimating steady-state culture characteristics, is not known yet. Experimental results obtained with Zygosaccharomyces rouxii, an important yeast in soy sauce processes, showed that the culture characteristics during the A-stat with an acceleration rate of 0.001 h(-2) were roughly comparable to those of the chemostat. For higher acceleration rates the deviation between the culture characteristics in the A-stat and those in the chemostat obtained at the same dilution rate generally started to increase. The source of these deviations was examined by simulation for Saccharomyces cerevisiae. The simulations demonstrated that this deviation was not only dependent on the metabolic adaptation rate of the yeast, but also on the rate of change in environmental substrate concentrations during A-stats. From this work, it was concluded that an A-stat with an acceleration rate of 0.001 h(-2) is attractive to be used instead of chemostat whenever a rough estimation of steady-state culture characteristics is acceptable.