A dynamic mathematical model of the chemostat

A number of experimental studies on the dynamic, behavior of the chemostat have shown that the specific growth rate does not, instantaneously adjust to changes in the concentration of limiting substrate in the chemostat following disturbances in the steady state input limiting substrate concentration or in the steady state dilution rate. Instead of an instantaneous response, as would be predicted by the Monod equation, experimental studies have shown that the specific growth rate experiences a dynamic lag in responding to the changes in the concentration of limiting substrate in the culture vessel. The observed dynamic lag has been recognized by researchers in such terms as an inertial phenomenon and as a hysteresis effect, but as yet a systems engineering approach has not been applied to the observed data. The present paper criticizes the use of the Monod equation as a dynamic relationship and offers as an alternative a dynamic equation relating specific growth rate to the limiting substrate concentration in the chemostat. Following the development of equations, experimental methods of evaluating parameters are discussed. Dynamic responses of analog simulations (incorporating the newly derived equations) are compared with the dynamic responses predicted by the Monod equation and with the dynamic responses of experimental chemostats.     

Chemostat cultivation as a tool for studies on sugar transport in yeasts.

Chemostat cultivation enables investigations into the effects of individual environmental parameters on sugar transport in yeasts. Various means are available to manipulate the specific rate of sugar uptake (qs) in sugar-limited chemostat cultures. A straightforward way to manipulate qs is variation of the dilution rate, which, in substrate-limited chemostat cultures, is equal to the specific growth rate. Alternatively, qs can be varied independently of the growth rate by mixed-substrate cultivation or by variation of the biomass yield on sugar. The latter can be achieved, for example, by addition of nonmetabolizable weak acids to the growth medium or by variation of the oxygen supply. Such controlled manipulation of metabolic fluxes cannot be achieved in batch cultures, in which various parameters that are essential for the kinetics of sugar transport cannot be controlled. In sugar-limited chemostat cultures, yeasts adapt their sugar transport systems to cope with the low residual sugar concentrations, which are often in the micromolar range. Under the conditions, yeasts with high-affinity proton symport carriers have a competitive advantage over yeasts that transport sugars via facilitated-diffusion carriers. Chemostat cultivation offers unique possibilities to study the energetic consequences of sugar transport in growing cells. For example, anaerobic, sugar-limited chemostat cultivation has been used to quantify the energy requirement for maltose-proton symport in Saccharomyces cerevisiae. Controlled variation of growth conditions in chemostat cultures can be used to study the differential expression of genes involved in sugar transport and as such can make an important contribution to the ongoing studies on the molecular biology of sugar transport in yeasts.     

Effect of Nutrient Concentration on the Growth of Escherichia coli

The relationship between specific growth rate of Escherichia coli and the concentration of limiting nutrient (glucose or phosphate or tryptophan) has been determined for populations in a steady state. At high concentrations the specific growth rate is independent of the concentration of nutrient, but at low concentrations the specific growth rate is a strong function of the nutrient concentration. Such a relationship was predicted by Monod; however, Monod's equation does not predict the relationship over the entire range of nutrient concentration. If parameters of the equation are estimated from the results obtained at low concentrations, then at high concentrations of nutrient, the specific growth rate is significantly higher than that predicted by Monod's equation. These results were interpreted on the basis that the rate of growth is controlled by at least two parallel reactions and that the affinities of the enzymes catalyzing these reactions are different. The relationship between specific growth rate and mean cell volume was also measured, and the results indicate that mean cell volume depends not only on the specific growth rate but also on the nature of the limiting nutrient. There are different mean cell volumes at the same specific growth rate established by different limiting nutrients. Therefore, the mean cell volume is not uniquely determined by the specific growth rate.     

Mathematical model of transitional processes in the chemostat culture of microorganisms

A mathematical model of the growth of the cell culture was developed. The model takes into account changes of the levels of the enzymes which define the metabolism rate, transport of the substrate into the cell, regeneration of the donors of energy. The model is based on the proposition that the rate of overall protein synthesis in the cell is defined by the concentration of a few aminoacids limiting the growth. The chemostat culture of the methanol-assimilating yeast was used as the object of modelling. The model allows to explain the experimental kinetics of alterations in cell number (biomass) and other measurable characteristics of the culture during the transient process when the dilution rate was changed.     

Futile cycling of ammonium ions via the high affinity potassium uptake system (Kdp) of Escherichia coli

Escherichia coli Frag1 was grown under various nutrient limitations in chemostat culture at a fixed temperature, dilution rate and pH both in the presence and the absence of a high concentration of ammonium ions by using either ammonium chloride or dl-alanine as the sole nitrogen source. The presence of high concentrations of ammonium ions in the extracellular fluids of potassium-limited cultures of E. coli Frag1 caused an increase of the specific rate of oxygen consumption of these cultures. In contrast, under phosphate-, sulphate- or magnesium-limited growth conditions no such increase could be observed. The presence of high concentrations of ammonium ions in potassium-limited cultures of E. coli Frag5, a mutant strain of E. coli Frag1 which lacks the high affinity potassium uptake system (Kdp), did not increase the specific rate of oxygen consumption.These results indicate that ammonium ions, very similar to potassium ions both in charge and size, are transported via the K dp leading to a futile cycle of ammonium ions and ammonia molecules (plus protons) across the cytoplasmic membrane. Both the uptake of ammonium ions and the extrusion of protons would increase the energy requirement of the cells and therefore increase their specific rate of oxygen consumption. The involvement of a (methyl)ammonium transport system in this futile cycle could be excluded.     

Effects of glucose supply on myeloma growth and metabolism in chemostat culture

The effects of the glucose supply on growth and metabolism of an SP2/0 derived recombinant myeloma cell line were studied in chemostat culture during growth on IMDM medium at a fixed dilution rate of 0.032 h^?1. Lowering of the feed medium glucose concentration from 25.0 to 1.4 mmol/L resulted in a decrease of steady-state viable cell concentration from 1.9 × 10⁹ L^?1, whereas viability remained above 90%. Mass balances indicated that only a minor amount of glucose was utilized via the TCA cycle irrespective of the glucose concentration in the feed medium. The apparent biosynthetic yield of cells from ATP was independent of the ratio between the specific glucose and glutamine consumption rate. It is concluded that the primary role of glucose is the provision of intermediates for anabolic reactions. In addition, glucose may play an indirect catabolic role in the process of glutaminolysis by providing the pyruvate for the transamination of glutamate to alanine and α-ketoglutarate. At low glucose concentrations in the feed medium, glutamine is probably the sole energy source for this myeloma in chemostat culture. © 1995 Wiley-Liss, Inc.     

Growth characteristics and oxidative capacity of<Emphasis Type="Italic"> Acetobacter aceti</Emphasis> IFO 3281: implications for <Emphasis Type="SmallCaps">l</Emphasis>-ribulose production

We studied the growth characteristics and oxidative capacities of Acetobacter aceti IFO 3281 in batch and chemostat cultures. In batch culture, glycerol was the best growth substrate and growth on ethanol occurred only after 6 days delay, although ethanol was rapidly oxidized to acetic acid. In continuous culture, both glycerol and ethanol were good growth substrates with similar characteristics. Resting cells in a bioreactor oxidized ribitol to l-ribulose with a maximal specific rate of 1.2 g g^–1 h^–1). The oxidation of ribitol was inhibited by ethanol but not by glycerol. Biomass yield (Y_SX; C-mmol/C-mmol) on ethanol and glycerol was low (0.21 and 0.17, respectively). In the presence of ribitol the yield was somewhat higher (0.25) with ethanol but lower (0.13) with glycerol, with respectively lower and higher CO₂ production. In chemostat cultures the oxidation rate of ribitol was unaffected by ethanol or glycerol. Cell-free extract oxidized ethanol very slowly but not ribitol; the oxidative activity was located in the cell membrane fraction. Enzymatic activities of some key metabolic enzymes were determined from steady-state chemostat with ethanol, glycerol, or ethanol/glycerol mixture as a growth limiting substrate. Based on the measured enzyme activities, metabolic pathways are proposed for ethanol and glycerol metabolism.     

Non-competitive product inhibition in lactic acid fermentation from glucose

Summary A kinetic study regarding product inhibition in lactic acid fermentation by Streptococcus faecalis, which produces l-lactic acid, was performed in a chemostat at various feed concentrations of glucose (10, 20, and 30 g/l) at pH 7.0. Steady-state kinetic constants for the specific consumption rate of glucose and the specific production rate of lactic acid were determined at a residual glucose concentration below 2 g/l, which was accomplished in a chemostat. All the parameters, the specific growth rate, the specific consumption rate of glucose, and the specific production rate of lactic acid, were definitely related to non-competitive inhibition with regard to the concentration of the product, lactic acid.Offprint requests to: K. Hiyama     

The impulse response of Chlamydomonas reinhardii in nitrite-limited chemostat culture

This article presents a clear experimental determination of the behavior of nutrient uptake rate, mean cell nutrient content, and specific growth rate following the injection of pulses of additional limiting nutrient into a chemostat culture of Chlamydomonas reinhardii. The uptake rate per cell is a hyperbolic function of external nutrient concentration. The specific growth rate is related to the mean cell nutrient content by a hysteresis loop. The data obtained is used to test the performance of the Caperon-Droop mean cell quota model. It is demonstrated that this model cannot be used under severe transient conditions, even when modified by the introduction of a discrete time delay, a simple memory function, or time-dependent intracellular nutrient processing.     

Legume-Rhizobium interactions: host induced alterations in capsular polysaccharides and infectivity of cowpea Rhizobia

Klebsiella aerogenes NCTC418 was cultured anaerobically under glucose-limited conditions in chemostat cultures at various growth rates, ranging from 0.13 h^-1 to 0.82 h^-1. It was found that the specific uptake rate of glucose varied linearly with the growth rate and that under these conditions glucose was fermented solely to acetate and ethanol plus CO₂+H₂ and formate.When steady-state cultures were pulsed with cell saturating concentrations of glucose, the specific glucose aptake rate increased immediately and substantially. However, at steady-state growth rates lower than 0.5 h^-1, this increase was not accompanied by a change in the growth rate, in contrast to cultures growing at higher rates. It was found that relief of the glucose limitation resulted in a shift in fermentation pattern: at the lower growth rates 50% or more of the extra glucose taken up was fermented to D-lactate.Incubation experiments with sonified cells revcaled that K. aerogenes possessed all the enzymes needed to convert dihydroxyacetone phosphate to methylglyoxal and subsequently to D-lactate, and that the rate at which this overall conversion occurred in vitro was in close agreement with the production rate of D-lactate in vivo. Moreover, it was found that the activities of the enzymes of the methylglyoxal bypass were dependent on the imposed growth rate. At higher growth rates, where cells possessed the potential to increase their growth rate immediately, the activity of methylglyoxal synthase was relatively low.it could be shown that, under low growth rate conditions, the uncoupling effect of the methylglyoxal bypass was highly effective and that, as a consequence thereof, a significant increase in the uptake rate of the energy source was accompanied by only a marginal increase in the rate at which ATP was synthesized.     

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.     

Dynamics of chemostat culture:the effect of a delay in cell response

The effect of introducing a delay between a change in concentration of limiting nutrient and the response in division rate of the population is investigated. It is shown that this might or might not introduce a delay in the control of the chemostat system, depending upon how the uptake rate is assumed to double during the cell cycle. An area in the parameter space where autonomous oscillations will occur for one specific model is found analytically. This result is confirmed by computer simulations.     

Effects of temperature and phosphorous concentration on microbial sulfate reduction by Desulfovibrio desulfuricans

The effects of temperature and phosphorous concentration on the rate and the extent of microbial sulfate reduction with lactate as carbon and energy source were investigated for Desulfovibrio desulfuricans. The continuous culture experiments (chemostat) were conducted at pH 7.0 from 12 to 48 degrees C. The maximum specific growth rate (mu(max)) was relatively constant in the range 25 degrees C-43 degrees C and dramatically decreased outside this temperature range. The half-saturation coefficient was minimum at 25 degrees C. Cell yield was highest in the optimum temperature range (35 degrees C-43 degrees C) for growth. Maintenance energy requirements for D. desulfuricans were not significant. Two moles of lactate is consumed for every mole of sulfate reduced, and this stoichiometric ratio is not temperature dependent. Steady state rate and stoichiometric coefficients accurately predicted transient behavior during temperature shifts. The extent of extracellular polymeric substance (EPS) is related to the concentration of phosphorous in the medium. EPS production rate increased with decreased phosphorous loading rate. Failure to discriminate between cell and EPS formation by D. desulfuricans leads to significant overestimates of the cell yield. The limiting C:P ratio for D. desulfuricans was in the range of 400:1 to 800:1.     

Aerobic nitrate and nitrite reduction in continuous cultures of Escherichia coli E4

Nitrate and nitrite was reduced by Escherichia coli E4 in a l-lactate (5 mM) limited culture in a chemostat operated at dissolved oxygen concentrations corresponding to 90–100% air saturation. Nitrate reductase and nitrite reductase activity was regulated by the growth rate, and oxygen and nitrate concentrations. At a low growth rate (0.11 h^–1) nitrate and nitrite reductase activities of 200 nmol · mg^–1 protein · min^–1 and 250 nmol · mg^–1 protein · min^–1 were measured, respectively. At a high growth rate (0.55 h^–1) both enzyme activities were considerably lower (25 and 12 nmol mg^–1 · protein · min^–1). The steady state nitrite concentration in the chemostat was controlled by the combined action of the nitrate and nitrite reductase. Both nitrate and nitrite reductase activity were inversely proportional to the growth rate. The nitrite reductase activity decreased faster with growth rate than the nitrate reductase. The chemostat biomass concentration of E. coli E4, with ammonium either solely or combined with nitrate as a source of nitrogen, remained constant throughout all growth rates and was not affected by nitrite concentrations. Contrary to batch, E. coli E4 was able to grow in continuous cultures on nitrate as the sole source of nitrogen. When cultivated with nitrate as the sole source of nitrogen the chemostat biomass concentration is related to the activity of nitrate and nitrite reductase and hence, inversely proportional to growth rate.     

A mechanism of microbial cell growth

Presently empirical expressions, especially the Monod equation, are used to quantitatively relate microbial growth rate to limiting substrate concentration in the solution. In this paper microbial growth is postulated to occur by a mechanism involving a mass transfer or assimilation process. The assimilation process is assumed to be substrate mass transfer limited and hence proportional to the limiting substrate concentration. The ingestion is assumed independent of limiting substrate concentration and only dependent upon internal reaction rates. The quantitative relationship between limiting substrate and microbial growth rate resulting from this mechanism is developed. Under certain limiting conditions this expression is shown to reduce to the Monod equation and under other conditions it reduces to the Lotka-Volterra relationship. This mechanism is applied to batch and continuous cultures and the results obtained are compared quantitatively with experiment.     

Models for the continuous culture of microorganisms under both oxygen and carbon limiting conditions

Two models for predicting the behavior of cultures of microorganisms under both oxygen and carbon limiting conditions have been evlauated on a chemostat growing Candida utilis on a glycolysis suppressing glycerol medium. The work indicated that parameter values obtained under wholly oxygen limiting or wholly carbon limiting conditions successfully predict the behavior of the chemostat under the wide range of flow and substrate concentration conditions tested. Both models are satisfactory and hence it is deduced that the simpler one may be used with confidence. It was found that Monod kinetics were applicable to the growth rate dependence on oxygen concentration but that Contois kinetics were superior for the corresponding dependence on carbon substrate concentration.     

The effect of growth conditions on production and excretion of extracellular antigens by three ascomycetous yeasts

Ascomycetous yeasts produce extracellular antigens that are almost specific for the species. The antigen production by Hansenula wickerhamii and Stephanoascus ciferrii was independent of the carbon source and was proportional to the final cell density of the cultures. The same was true of chemostat cultures of Stephanoascus ciferrii, irrespective of the dilution rate and whether glucose or ammonia was the limiting nutrient. In cultures of Saccharomyces cerevisiae, however, antigen excretion mainly took place in the late exponential growth phase. Large amounts of antigen were extracted from the cell wall of Saccharomyces cerevisiae. A small amount was detected in the cytoplasm.     

The ferrous iron oxidation kinetics of Thiobacillus ferrooxidans in batch cultures

The ferrous iron oxidation kinetics of Thiobacillus ferrooxidans in batch cultures was examined, using on-line off-gas analyses to measure the oxygen and carbon dioxide consumption rates continuously. A cell suspension from continuous cultures at steady state was used as the inoculum. It was observed that a dynamic phase occurred in the initial phase of the experiment. In this phase the bacterial ferrous iron oxidation and growth were uncoupled. After about 16 h the bacteria were adapted and achieved a pseudo-steady state, in which the specific growth rate and oxygen consumption rate were coupled and their relationship was described by the Pirt equation. In pseudo-steady state, the growth and oxidation kinetics were accurately described by the rate equation for competitive product inhibition. Bacterial substrate consumption is regarded as the primary process, which is described by the equation for competitive product inhibition. Subsequently the kinetic equation for the specific growth rate, μ, is derived by applying the Pirt equation for bacterial substrate consumption and growth. The maximum specific growth rate, μ _max, measured in the batch culture agrees with the dilution rate at which washout occurs in continuous cultures. The maximum oxygen consumption rate, q _O2,max, of the cell suspension in the batch culture was determined by respiration measurements in a biological oxygen monitor at excess ferrous iron, and showed changes of up to 20% during the course of the experiment. The kinetic constants determined in the batch culture slightly differ from those in continuous cultures, such that, at equal ferric to ferrous iron concentration ratios, biomass-specific rates are up to 1.3 times higher in continuous cultures. Received: 8 February 1999 / Accepted: 17 February 1999     

A chemostat study on proline uptake and metabolism of Leishmania donovani.

Leishmania donovani grew in the chemostat on proline as its sole carbon and energy source at a maximum growth rate of 1.39 divisions per day. The efficiency of proline metabolism decreased with increasing external proline concentration. The internal concentration of proline and its intracellular metabolites was low when proline was the growth rate limiting substrate and high when proline was available in excess. In time-course experiments proline uptake leveled off after 30 min, independent of the culture conditions prior to the experiment. Proline uptake depended on the external proline concentration in a manner that is best described as the combination of an enzymatic and a diffusion component. Adaptation to different proline concentrations did not occur and no evidence was found that proline is actively transported by L. donovani.     

Growth yields and the efficiency of oxidative phosphorylation of Paracoccus denitrificans during two- (carbon) substrate-limited growth

Paracoccus denitrificans was grown aerobically during two-(carbon)substrate-limitation on mannitol and methanol in chemostat cultures. Theoretical growth parameters were calculated based on the presence of 2 or 3 sites in the electron-transport chain of Paracoccus denitrificans. Experimental growth parameters determined during two-(carbon)substrate growth were conform to the presence of 3 sites of oxidative phosphorylation, while cells grown only on mannitol possessed 2 sites. The maximum growth yield on adenosine triphosphate (ATP), corrected for maintenance requirements, determined in chemostat experiments in which the methanol concentration is less than 2.11 times the mannitol concentration was 8.6 g of biomass. When the methanol concentration was more than 2.11 times the mannitol concentration the maximum growth yield on adenosine triphosphate decreased due to the more energy consuming process of CO₂-assimilation. Cells use methanol only as energy source to increase the amount of mannitol used for assimilation purposes. When the methanol concentration in chemostat experiments was more than 2.11 times the mannitol concentration, all mannitol was used for assimilation and excess energy derived from methanol was used for CO₂-assimilation via the ribulose-bisphosphate cycle. The synthesis of ribulosebisphosphate carboxylase was repressed when the methanol concentration in chemostat experiments was less than 2.11 times the mannitol concentration or when Paracoccus denitrificans was grown in batch culture on both methanol and mannitol. When in chemostat experiments the methanol concentration was more than 2.11 times the mannitol concentration ribulose-bisphosphate carboxylase activity could be demonstrated and CO₂-assimilation will occur. It is proposed that energy produced in excess activates or derepresses the synthesis of the necessary enzymes of the ribulose-bisphosphate cycle in Paracoccus denitrificans. Consequently growth on any substrate will be carbonas well as energy-limited. When methanol is present in the nutrient cells of Paracoccus denitrificans synthesize a CO-binding type of cytochrome c, which is essential for methanol oxidase activity.The reason for the increase in efficiency of oxidative phosphorylation from 2 to 3 sites is most probably the occurrence of this CO-binding type of cytochrome c in which presence electrons preferentially pass through the a-type cytochrome region of the electron-transport chain.Non Standard Abbreviations X prosthetic group of methanol dehydrogenase - q _substrate specific rate of consumption of substrate (mol/g biomass. h.) - Y _substrate, Y _substrate ^MAX are respectively the growth yield and the maximum growth yield corrected for maintenance requirements (g biomass/mol) - m _substrate maintenance requirement (mol substrate/g biomass) - specific growth rate (h^-1) - M [methanol]/[mannitol] ratio in the nutrient - N part of mannitol that is assimilated when M=o - R _m amount of methanol-equivalents that has the same energy content as 1 mannitol-equivalent - P/O _N , P/O _F , P/O _X is the amount of ATP produced during electron-transport of two electrons from respectively NADH+H⁺, FADH₂ and XH₂ to oxygen