Large-scale preparation of ribulosebisphosphate carboxylase from a recombinant system in Escherichia coli characterized by extreme plasmid instability

An ampicillin-resistant, RecA- strain of Escherichia coli (HB101) harboring the multicopy pBR322 plasmid containing the structural gene for ribulosebisphosphate carboxylase from Rhodospirillum rubrum was used to prepare large quantities of the carboxylase protein. This recombinant system was characterized by extreme plasmid instability, which resulted in part from the 1.7-fold faster growth rate of plasmid-free cells and in part from very rapid rates of plasmid segregation. The plasmid-containing organisms produced and excreted a large amount of beta-lactamase activity, with the result that ampicillin selection could only be maintained for a very short period of time, after which the plasmid-containing (carboxylase-producing) cells were overgrown by plasmid-free cells. The instability was so severe that even isolated colonies prepared on ampicillin-containing plates were impure and contained plasmid-free cells. Nevertheless, large quantities of carboxylase protein could be obtained from this system by using a highly dilute inoculum which allows selection of ampicillin-resistant (carboxylase-producing) organisms for a sufficient period of time so that the period of growth under nonselective conditions was minimized, and cells harvested at high cell densities contained large amounts of the carboxylase protein. In the present instance, 300-liter fermentations were initiated with a 0.3-microliter inoculum of freshly grown cells. After 20 h of growth in rich medium containing ampicillin, the harvested cells contained 74 g of ribulosebisphosphate carboxylase protein (average of two separate cultures). These results are discussed in terms of the general nature of plasmid instability and protocols available to minimize the effects of such instability.     

Large-scale preparation of ribulosebisphosphate carboxylase from a recombinant system in Escherichia coli characterized by extreme plasmid instability.

An ampicillin-resistant, RecA- strain of Escherichia coli (HB101) harboring the multicopy pBR322 plasmid containing the structural gene for ribulosebisphosphate carboxylase from Rhodospirillum rubrum was used to prepare large quantities of the carboxylase protein. This recombinant system was characterized by extreme plasmid instability, which resulted in part from the 1.7-fold faster growth rate of plasmid-free cells and in part from very rapid rates of plasmid segregation. The plasmid-containing organisms produced and excreted a large amount of beta-lactamase activity, with the result that ampicillin selection could only be maintained for a very short period of time, after which the plasmid-containing (carboxylase-producing) cells were overgrown by plasmid-free cells. The instability was so severe that even isolated colonies prepared on ampicillin-containing plates were impure and contained plasmid-free cells. Nevertheless, large quantities of carboxylase protein could be obtained from this system by using a highly dilute inoculum which allows selection of ampicillin-resistant (carboxylase-producing) organisms for a sufficient period of time so that the period of growth under nonselective conditions was minimized, and cells harvested at high cell densities contained large amounts of the carboxylase protein. In the present instance, 300-liter fermentations were initiated with a 0.3-microliter inoculum of freshly grown cells. After 20 h of growth in rich medium containing ampicillin, the harvested cells contained 74 g of ribulosebisphosphate carboxylase protein (average of two separate cultures). These results are discussed in terms of the general nature of plasmid instability and protocols available to minimize the effects of such instability.     

Effect of growth rate on stability and gene expression of a recombinant plasmid during continuous culture ofEscherichia coli in a non-selective medium

Summary Experimental results were obtained withEscherichia coli C600 galK (GAPDH), a genetically engineered strain that synthetizes a large quantity of glyceraldehyde-3-phosphate dehydrogenase (GAPDH), (80 % of the soluble proteins). Data concerning the stability of plasmid-containing cells and gene expression as a function of dilution rate have been obtained in continuous cultures. Contrary to other studies, our results show a clear indication that the rate of the recombinant activity was dependent on dilution rate. The results support the finding that the apparent stability of the plasmid decreases with dilution rate.     

Persistence of the pBR 322 plasmid in Escherichia coli K 12 grown in chemostat cultures

Populations of a Escherichia coli K 12 strain, containing the vector plasmid p BR 322, were grown in chemostat culture under glucose- and phosphatelimited conditions. Resistance to tetracycline and ampicillin were lost after prolonged cultivation, resulting in the production of apparent plasmid-free populations which were more competitive than the original population. This competitiveness between plasmid-free and plasmid-containing populations was greatest in environments where the nutrient restriction was severe. Also during sequential subcultivation in batch cultures loss of plasmid was observed.     

Stability fluctuations of plasmid-bearing cells: immobilization effects

The maintenance of the plasmid vectors pTG201 and pTG206 (which both carry the Pseudomonas putida xylE gene) and pB lambda H3 in Escherichia coli hosts was studied in free and immobilized continuous cultures. pTG201, containing the strong lambda PR promoter, was more quickly lost than plasmid pTG206, containing the tetracycline resistance gene promoter. The instability of pTG201 seems to be related to high expression of the cloned xylE genet. Fluctuations in the proportion of pTG201-containing cells were observed in the free system, suggesting the appearance of adaptive descendants (with and without plasmid) from the initial strains. The loss of plasmid vectors from E. coli cells and the fluctuations in the proportion of plasmid-containing cells could be prevented by immobilizing plasmid-containing bacteria in carrageenan gel beads.     

Metabolic analysis of the synthesis of high levels of intracellular human SOD in Saccharomyces cerevisiae rhSOD 2060 411 SGA122

The synthesis of human superoxide dismutase (SOD) in batch cultures of a Saccharomyces cerevisiae strain using a glucose-limited minimal medium was studied through metabolic flux analysis. A stoichiometric model was built, which included 78 reactions, according to metabolic pathways operative in these strains during respirofermentative and oxidative metabolism. It allowed calculation of the distribution of metabolic fluxes during diauxic growth on glucose and ethanol. Fermentation profiles and metabolic fluxes were analyzed at different phases of diauxic growth for the recombinant strain (P+) and for its wild type (P-). The synthesis of SOD by the strain P+ resulted in a decrease in specific growth rate of 34 and 54% (growth on glucose and ethanol respectively) in comparison to the wild type. Both strains exhibited similar flux of glucose consumption and ethanol synthesis but important differences in carbon distribution with biomass/substrate yields and ATP production 50% higher in P-. A higher contribution of fermentative metabolism, with 64% of the energy produced at the phosphorylation level, was observed during SOD production. The flux of precursors to amino acids and nucleotides was higher in the recombinant strain, in agreement with the higher total RNA and protein levels. Lower specific growth rates in strain P+ appear to be related to the decrease in the rate of synthesis of nonrecombinant protein, as well as a decrease in the activities of the pentose phosphate (PP) pathway and TCA cycle. A very different way of entry into the stationary phase was observed for each strain: in the wild-type strain most metabolic fluxes decreased and fluxes related to energy reserve synthesis increased, while in the P+ strain the flux of 22 reactions (including PP pathway and amino acids biosynthesis) related to SOD production increased their fluxes. Changes in SOD production rates at different physiological states appear to be related to the differences in building blocks availability between respirofermentative and oxidative metabolism. Using the present expression system, ideal conditions for SOD synthesis are represented by either active growth during respirofermentative metabolism or transition from a growing to a nongrowing state. An increase in SOD flux could be achieved using an expression system nonassociated to growth and potentially eliminating part of the metabolic burden.     

Effect of growth conditions on the rate of loss of the plasmid pAT153 from continuous cultures of Escherichia coli HB101

Abstract The plasmid pAT153 was lost less rapidly from carbon, nitrogen, phosphorous or sulphur-limited continuous cultures of Escherichia coli HB101 as the dilution rate increased. At a fixed dilution rate of 0.3 h⁻¹, the plasmid was maintained longer as the growth-limiting nutrient was changed from glucose to casamino acids (nitrogen-limited), phosphate or sulphate. These differences in the stability of maintenance were not due to parallel changes in the plasmid copy number. We propose that the rate of loss of pAT153 from E. coli HB101 is determined primarily by the ratio of growth rates of plasmid-containing bacteria and plasmid-free bacteria. This ratio increases with increasing growth rate and depends markedly on the growth-limiting nutrient, sulphate-limited growth being particularly suitable for the maintenance of this host-plasmid combination.     

Stability studies of recombinant Saccharomyces cerevisiae in the presence of varying selection pressure

A recombinant yeast plasmid carrying the Ieu2 gene for auxotrophic complementation and a reporter gene for beta-galactosidase under the control of Gal10 promoter was studied in Saccharomyces cerevisiae. Growth, product formation, and plasmid stability were studied in defined, semi-defined, and complex media. The biomass concentration and specific activity were higher in complex medium than in defined medium, which was selective for the growth of plasmid-containing cells, leading to a 10-fold increase in volumetric activity. However, plasmid instability was very high in complex media with 50% plasmid-free cells emerging in the culture within 75 h of cultivation. In order to control instability, the growth rates of the plasmid-containing and plasmid-free cells were determined in semi-defined media, which consisted of defined medium supplemented with different concentrations of yeast extract. Below a critical concentration of yeast extract (0.05 g/L), the plasmid-containing cells had a growth rate advantage over the plasmid-free cells. This was possibly because, at this concentration of yeast extract, the availability of leucine became the rate-determining factor in the specific growth rate of plasmid-free cells. A feeding strategy was designed which maintained a low concentration of the residual yeast extract in the medium and thus continuously provided the plasmid-containing cells with a competitive advantage over the plasmid-free cells. This resulted in high stability as well as high cell density under non-selective conditions, which led to a 10-fold increase in the volumetric activity compared to that achieved in defined selective media. A simple mathematical model was formulated to verify the experimental data. The important state variables and process parameters, i.e., biomass concentration, beta-galactosidase expression, sucrose consumption, yeast extract consumption, and specific growth rates of the two cell populations, were evaluated. These variables and parameters along with the differential equations based on material balances as well as the experimental results obtained were used in a mathematical model for the fed-batch cultivation. These correctly verified the experimental data and clearly illustrated the concept behind the success of the fed-batch strategy under yeast extract starvation.     

Metabolic flux analysis of Escherichia coli expressing the Bacillus subtilis acetolactate synthase in batch and continuous cultures.

Metabolically engineered Escherichia coli expressing the B. subtilis acetolactate synthase has shown to be capable of reducing acetate accumulation. This reduction subsequently led to a significant enhancement in recombinant protein production. The main focus of this study is to systematically examine the effect of ALS in the metabolic patterns of E. coli in batch and continuous culture. The specific acetate production rate of a strain carrying the B. subtilis als gene is 75% lower than that of the control strain (host carrying the control plasmid pACYC184) in batch cultures. The ALS strain is further demonstrated to be capable of maintaining a reduced specific acetate production rate in continuous cultures at dilution rates ranging from 0.1 to 0.4 h-1. In addition, this ALS strain is shown to have a higher ATP yield and lower maintenance coefficient. The metabolic flux analysis of carbon flux distribution of the central metabolic pathways and at the pyruvate branch point reveals that this strain has the ability to channel excess pyruvate to the much less toxic compound, acetoin.     

Cometabolic oxidation of polychlorinated biphenyls in soil with a surfactant-based field application vector.

Polychlorinated biphenyl (PCB)-degradative genes, under the control of a constitutive promoter, were cloned into a broad-host-range plasmid and a transposon. These constructs were inserted into a surfactant-utilizing strain, Pseudomonas putida IPL5, to create a field application vector (FAV) in which a surfactant-degrading organism cometabolizes PCB. By utilizing a surfactant not readily available to indigenous populations and a constitutive promoter, selective growth and PCB-degradative gene expression are decoupled from biphenyl. Since PCB degradation via the biphenyl degradation pathway is nonadaptive in the absence of biphenyl, there is no selective pressure for PCB gene maintenance. The recombinant strains exhibited degradative activity against 25 of 33 PCB congeners in Aroclor 1248 in the absence of biphenyl. Whole-cell enzyme assays indicated that PCB-degradative activity of a recombinant strain carrying the PCB genes on a plasmid was approximately twice that of the same strain carrying the PCB genes on a transposon. Plasmid loss rates in the absence of antibiotic selection averaged 7.4% per cell division and were highly variable between experiments. Surfactant-amended slurries of PCB-contaminated electric power plant substation soil were inoculated with approximately 10(5) recombinant cells per ml. The populations of the added strains increased to greater than 10(9) cells per ml in 2 days, and cell growth coincided with PCB degradation. By 15 days, 50 to 60% of the indicator congener 2,3,2',5'-tetrachlorobiphenyl was degraded. The effectiveness of PCB degradation by the plasmid-containing strain depended on plasmid stability. The transposon-encoded PCB genes were much more stable, and in surfactant-amended soil slurries, PCB degradation was more consistent between experiments.     

Effects of growth environment on recombinant plasmid stability in Saccharomyces cerevisiae grown in continuous culture

A recombinant strain of Saccharomyces cerevisiae, containing a 2-m-fragment-based plasmid (pYEa4) was grown under non-selective conditions in continuous culture. The decrease in the population carrying the plasmid-encoded auxotrophic marker, LEU2, was examined under different physiological conditions. The difference in growth rate (µ) between plasmid-free and plasmid-containing cells and the rate of plasmid segregation (R) were determined using a non-linear regression technique. Loss rates were greater in defined glucose-limited cultures than in complex glucose-limited cultures. Plasmid loss was µ-dominated in cultures grown on defined media whereas µ and R were co-dominant in cultures grown on complex medium. Loss rates increased with increasing dilution rate in complex glucose-limited cultures. The reverse was found in defined glucose-limited cultures. Plasmid retention and loss kinetics determined from defined magnesium-limited cultures were not significantly different from those observed in defined glucose-limited cultures. Although plasmid retention in defined phosphate-limited culture was not significantly different from that in defined glucose-limited culture, reduced R and increased µ indicated an alternative physiological effect of phosphate limitation on plasmid stability.     

Steady-state and transient-state analysis of growth and metabolite production in a Saccharomyces cerevisiae strain with reduced pyruvate-decarboxylase activity

Pyruvate decarboxylase is a key enzyme in the production of low-molecular-weight byproducts (ethanol, acetate) in biomass-directed applications of Saccharomyces cerevisiae. To investigate whether decreased expression levels of pyruvate decarboxylase can reduce byproduct formation, the PDC2 gene, which encodes a positive regulator of pyruvate-decarboxylase synthesis, was inactivated in the prototrophic strain S. cerevisiae CEN. PK113-7D. This caused a 3-4-fold reduction of pyruvate-decarboxylase activity in glucose-limited, aerobic chemostat cultures grown at a dilution rate of 0.10 h(-1). Upon exposure of such cultures to a 50 mM glucose pulse, ethanol and acetate were the major byproducts formed by the wild type. In the pdc2Delta strain, formation of ethanol and acetate was reduced by 60-70%. In contrast to the wild type, the pdc2Delta strain produced substantial amounts of pyruvate after a glucose pulse. Nevertheless, its overall byproduct formation was ca. 50% lower. The specific rate of glucose consumption after a glucose pulse to pdc2Delta cultures was about 40% lower than in wild-type cultures. This suggests that, at reduced pyruvate-decarboxylase activities, glycolytic flux is controlled by NADH reoxidation. In aerobic, glucose-limited chemostat cultures, the wild type exhibited a mixed respiro-fermentative metabolism at dilution rates above 0.30 h(-1). Below this dilution rate, sugar metabolism was respiratory. At dilution rates up to 0.20 h(-1), growth of the pdc2Delta strain was respiratory and biomass yields were similar to those of wild-type cultures. Above this dilution rate, washout occurred. The low micro(max) of the pdc2Delta strain in glucose-limited chemostat cultures indicates that occurrence of respiro-fermentative metabolism in wild-type cultures is not solely caused by competition of respiration and fermentation for pyruvate. Furthermore, it implies that inactivation of PDC2 is not a viable option for reducing byproduct formation in industrial fermentations.     

Enhanced plasmid maintenance in a CSTR upon square-wave oscillations in the dilution rate

Summary The effects of forced square-wave perturbations in the dilution rate on plasmid maintenance and gene expression of a population ofEscherichia coli K12 strain carrying the vector plasmid pBR322 grown in a chemostat with a non-selective medium were studied. It was observed that in the control experiments, where the dilution rates were kept constant, the percentage of plasmid-containing cells decreased after a period of time. Eventually, the culture was displaced by the plasmid-free cells. However, when the cells were exposed to forced oscillations in the dilution rate, the reactor culture was able to maintain a mixed population of plasmid-free and plasmid-containing cells for a longer period of time. The above observation seems to be independent of the source of the host cells. That is, the same results were obtained when the plasmid-free cells were generated from the culture itself due to defective partitioning of the plasmids or introduced externally.     

Kinetics of benzoate-induced loss of the TOL plasmid from Pseudomonas putida MT15 during growth in chemostat culture

Potassium-limited chemostat cultures of Pseudomonss putida MT15, grown on excess glucose, displayed approximately 100% plasmid loss after 60 generations of growth in the presence of 5 mM benzoate. The kinetics of plasmid loss indicated that plasmid-free cells displayed a growth rate advantage, which we attribute to selective inhibition of the growth of plasmid-containing cells by benzoate. However, stable, mixed populations of plasmid-free cells, deletants and plasmid-containing cells were selected during growth under glucose limitation in the presence of benzoate. This behaviour indicated that the plasmid-free cells in these cultures displayed a growth rate disadvantage and that their appearance was due entirely to benzoate-induced segregational instability of the plasmid.     

Utility of an Escherichia coli strain engineered in the substrate uptake system for improved culture performance at high glucose and cell concentrations: an alternative to fed-batch cultures

Overflow metabolism is an undesirable characteristic of aerobic cultures of Escherichia coli. It results from elevated glucose consumption rates that cause a high substrate conversion to acetate, severely affecting cell physiology and bioprocess performance. Such phenomenon typically occurs in batch cultures under high glucose concentration. Fed-batch culture, where glucose uptake rate is controlled by external addition of glucose, is the classical bioprocessing alternative to prevent overflow metabolism. Despite its wide-spread use, fed-batch mode presents drawbacks that could be overcome by simpler batch cultures at high initial glucose concentration, only if overflow metabolism is effectively prevented. In this study, an E. coli strain (VH32) lacking the phosphoenolpyruvate: carbohydrate phosphotransferase system (PTS) with a modified glucose transport system was cultured at glucose concentrations of up to 100 g/L in batch mode, while expressing the recombinant green fluorescence protein (GFP). At the highest glucose concentration tested, acetate accumulated to a maximum of 13.6 g/L for the parental strain (W3110), whereas a maximum concentration of only 2 g/L was observed for VH32. Consequently, high cell and GFP concentrations of 52 and 8.2 g/L, respectively, were achieved in VH32 cultures at 100 g/L of glucose. In contrast, maximum biomass and GFP in W3110 cultures only reached 65 and 48%, respectively, of the values attained by the engineered strain. A comparison of this culture strategy against traditional fed-batch culture of W3110 is presented. This study shows that high cell and recombinant protein concentrations are attainable in simple batch cultures by circumventing overflow metabolism through metabolic engineering. This represents a novel and valuable alternative to classical bioprocessing approaches.     

Use of copper sulphate and Paraquat as selectable resistance markers for stable maintenance of yeast recombinant plasmids

Saccharomyces cerevisiae SOD2and CUP1genes were used to maintain high-copy number plasmids (YEp) in laboratory and industrial yeast strains. The plasmid, YEpS , containing the SOD2 gene was unstable in a sod2° mutant. However when Paraquat (0.5 mM) was used as a selective agent, the plasmid was maintained in the sod2° mutant but lost in the wild-type strain. When the CUP1 gene was inserted into YEpS 1 , the resulting plasmid (YEpCuS ) was 100% stable in the sod2° mutant grown in Cu -containing medium. In the absence of Cu , the proportion of plasmid-containing cells fell to 20%. YEpS was also transformed into an industrial strain, transformants could be selected in Paraquat-containing medium but showed poor stability.     

Incidence of Plasmids in Marine Vibrio spp. Isolated from an Oil Field in the Northwestern Gulf of Mexico

Presumptive marine Vibrio spp. were collected from an operational oil field and control site located in the northwestern Gulf of Mexico. Of 440 isolates analyzed for the presence of extrachromosomal deoxyribonucleic acid elements or plasmids by using the cleared lysate and agarose gel techniques, 31% showed distinct plasmid bands on agarose gels. A majority of the plasmids detected were estimated to have molecular masses of 10 × 10⁶ or less. Multiple plasmids were observed in approximately half of the plasmid-containing strains. A number of isolates contained plasmids with similar banding and mobility patterns. The oil field area had noticeably more plasmid-containing strains (35 versus 23% in the control site) and a greater number of plasmids per plasmid-containing strain (an average of 2.5 plasmids, versus 1.5 in the control site). Oil field discharges might have resulted in increased plasmid incidence and diversity.     

A study of long-term effects on plasmid-containing Escherichia coli in carbon-limited chemostat using 2D-fluorescence spectrofluorimetry

Strain stability of plasmid-containing recombinant organisms is clearly important for industrial applications. Stability is normally assessed by methods such as selective colony forming units or by simply measuring the recombinant product. These methods are typically performed off-line, are time-consuming, and do not give detailed information on the changes in the metabolism. In the current work, long-term stability of a plasmid-containing strain of Escherichia coli (W3110.shik1) capable of shikimic acid overproduction was studied by means of a 2D-fluorescence sensor (BioView) able to emit and detect light in ranges of 260-560 nm and 300-600 nm, respectively. Long-term carbon-limited chemostat experiments were made under both selective (tetracycline-containing medium) and nonselective conditions. It is shown that the fluorescence spectra provide information about metabolic changes at an earlier stage, thereby giving a noninvasive method for monitoring of strain stability. Further, the fluorescence measurements showed that (i) the metabolic changes in the strain W3110.shik1 with time were qualitatively different in selective and nonselective environment, (ii) plasmid recombination resulted primarily in increased biomass yield, and (iii) a change in metabolism probably involving FAD/FMN and pyridoxal-5-P occurred in all experiments. It was concluded that the strain was not stable in any growth condition for more than about 25 growth generations and even less if plasmid recombination took place.     

Kinetics of loss of a recombinant plasmid in Bacillus subtilis

Recombinant plasmid pCEDS is structurally unstable in Bacillus subtilis cultures. We have previously shown that stability can be independently increased by changing from a complex medium supporting high growth rates to a chemically-defined medium supporting a lower growth rate and removal of a 4.77-kb EcoRI fragment from pCED3 to give plasmid YS1. Further stabilization was achieved by combining the two approaches. In the present work, we show that the stabilization of the plasmid-encoded LacZ(+) phenotype can be explained solely by the effect on the growth rate ratio between cells containing modified and parental plasmids. By using modified stability experiments (where a single cell rather than a suspended colony was used to initiate growth), independent growth rate measurements, and a simple mathematical model, we can describe the kinetics of the loss of the LacZ(+) phenotype in terms of two variables, alpha and p (where alpha is the ratio of growth rates between modified and parental cells, and p is the probability of obtaining modified cells from parental cells). Under the conditions tested, the average values of alpha were 1.52 for cultures growing in complex medium, 1.28 for cultures growing in defined medium, and 1.18 for cultures containing the modified plasmid pYS1 growing in complex medium. The calculated p values ranged between 10(-8) and 10(-10) under all conditions. Plasmid (pYS137) was used to directly estimate plasmid deletion rates in B. subtilis and it showed a rate between 5 x 10(-8) and 1.1 x 10(-9) deletions/cell/generation. In contrast to B. subtilis, there were no detectable differences in growth rates between Escherichia coli strains harboring plasmid pCEDS and plasmid-free cells. These results explain the observed stability of pCEDS in E. coli cultures and indicate that readily detected instability in B. subtilis cultures can be the result of rare deletion events.     

Variation and modeling of the probability of plasmid loss as a function of growth rate of plasmid-bearing cells of Escherichia coli during continuous cultures

A large number of models concerning cultures of genetically engineered bacteria have been described. Among them, some are specifically adapted to continuous cultures and lead to the determination of two variables: (i) the difference in the specific growth rates between plasmid-carrying cell and plasmid-free cells (deltamu) and (ii) the frequency of plasmid loss by plasmid-containing cells (p(r)mu(+)). Until now, studies have been performed on the global expression p(r)mu(+) and deltamu, whose value during continuous assays have been supposed approximately constant (mean value) and not on separate values of both terms p(r) and mu(+), respectively, probability of plasmid loss and specific growth rate of the plasmid-carrying cells. So far these studies do not allow examination of the relationship between these two last parameters. Experimental results were obtained with Escherichia coli C600 galk (GAPDH), a genetically engineered strain that synthetizes an elevated quantity of glyceraldehyde-3-phosphate dehydrogenase (GAPDH). From data obtained during continuous cultures, it is shown that during an assay, deltamu, and p(r)mu(+) do not remain constant. An appropriate mathematical analysis of the expression of mu(-) (specific growth rate of the plasmid-free cells) and mu(+) has been built up. This allows the evaluation of the values of mu(+) and mu(-) during the continuous cultures carried out at different dilution rates. Values of p(r) have been calculated from these data. Indeed our results show that p(r) increases with mu(+). A modeling approach which allows correct simulation of this variation is also proposed. This model is derived from the Hill equation regarding cooperative binding of enzymic type reaction. (c) 1993 John Wiley & Sons, Inc.