A simple model to describe transient differences between cell number and biomass growth rates of Escherichia coli

Information on the response of a microbial culture to dynamic environmental conditions is necessary for the design of transient operation processes. However, most attempts at modelling culture response have been directed at describing the steady-state behavior. Thus, there is a need for adequate dynamic models for process design. Simulations of nutrient shifts were completed using a "single-cell" model for Escherichia coli. It was discovered that the specific mass growth rate and the specific number of cells growth rate were different under transient conditions, whereas at steady state (balanced growth) these rates are equivalent. Using these observations, a simple delay model to describe the transient behavior of the two growth rates is formulated and tested. The model contains as state variables only the readily measurable macroscopic quantities (biomass, cell number, and limiting nutrient). This model agreed well with the predictions of the single-cell model.     

Cybernetic modeling and regulation of metabolic pathways in multiple steady states of hybridoma cells

Hybridoma cells utilize a pair of complementary and partially substitutable substrates, glucose and glutamine, for growth. It has been shown that cellular metabolism shifts under different culture conditions. When those cultures at different metabolic states are switched to a continuous mode, they reach different steady states under the same operating conditions. A cybernetic model was constructed to describe the complementary and partial substitutable nature of substrate utilization. The model successfully predicted the metabolic shift and multiple steady-state behavior. The results are consistent with the experimental observation that the history of the culture affects the resulting steady state.     

The relationship between kinetics of substrate-limited transitions and steady-state growth in continuous cultures ofAquaspirillum autotrophicum limited by pyruvate

Heterotrophic growth at steady state and during transient states caused by the sudden change of the concentration of the limiting factor in the feed medium was investigated experimentally for continuous cultures ofAquaspirillum autotrophicum limited by pyruvate. A model for describing the growth at steady state was selected from three unstructured models after statistical tests of the data. This model postulates that the growth yield increases linearly with the growth rate. Growth during transitions where the substrate remained limiting at all times was fitted with first-order kinetics. Theoretical predictions of these kinetics were derived from the unstructured models used to describe steady state. The predicted rate coefficients of the transients were compared to the experimental coefficients. It appeared that the model which best described steady-state growth also provided the best predictions for growth during the transient state. It is a widespread opinion that unstructured models are adequate to describe growth under steady-state conditions but not to predict transitions in continuous culture. However, for the particular case studied here, no higher degree of complexity was required to describe transitions, provided the growth of the culture was always limited by the substrate.     

Cell Growth and Division: III. Conditions for Balanced Exponential Growth in a Mathematical Model

In a previous paper, we proposed a model in which the volume growth rate and probability of division of a cell were assumed to be determined by the cell's age and volume. Some further mathematical implications of the model are here explored. In particular we seek properties of the growth and division functions which are required for the balanced exponential growth of a cell population. Integral equations are derived which relate the distribution of birth volumes in successive generations and in which the existence of balanced exponential growth can be treated as an eigenvalue problem. The special case in which all cells divide at the same age is treated in some detail and conditions are derived for the existence of a balanced exponential solution and for its stability or instability. The special case of growth rate proportional to cell volume is seen to have neutral stability. More generally when the division probability depends on age only and growth rate is proportional to cell volume, there is no possibility of balanced exponential growth. Some comparisons are made with experimental results. It is noted that the model permits the appearance of differentiated cells. A generalization of the model is formulated in which cells may be described by many state variables instead of just age and volume.     

Estimation of cell DNA synthesis rate from flow-cytometric histograms

The rate of DNA synthesis along S-phase can be estimated on the basis of a simple model of a cell population, from a single FCM histogram under conditions of balanced exponential growth, and from two histograms suitably spaced in time under conditions of locally exponential S-phase influx. An algorithm that constructs a piecewise constant approximation of the synthesis rate and estimates the phase fractions from a single histogram is presented. The results of the application of the algorithm to a set of computer generated histograms in balanced exponential growth are reported.     

Continued growth of Escherichia coli after stopping medium addition to a K+-limited chemostat culture

The steady-state bacterial dry wt of Escherichia coli, growing under K+-limited conditions in the chemostat, was inversely dependent on the growth rate. This phenomenon was more carefully investigated in medium-flow stop experiments. Growth did not stop immediately but continued for a time, initially at the same rate as before. The dry wt increased to a value corresponding to a steady-state growth rate near zero, independent of the initial specific growth rate. This was observed in both the wild-type strain and a mutant that lacked the high-affinity K+ uptake system. The wild-type strain maintained a low extracellular K+ concentration both in the chemostat under steady-state conditions and after stopping the medium flow. The mutant, on the other hand, maintained a much higher extracellular K+ concentration in the steady state, which decreased to much lower values after stopping the medium flow. From the increase in bacterial dry wt and the low external K+ concentration after stopping the medium flow it is concluded that the intracellular K+ is redistributed among the cells, including new cells. The growth yield on K+ was highest in the stationary growth phase of a batch culture and all steady-state cultures converged ultimately to this yield value after the medium flow had been stopped. It is proposed that the growth rate of E. coli under K+-limited conditions is determined by the intracellular K+ concentration.     

Control of cell length in Bacillus subtilis.

During inhibition of deoxyribonucleic acid synthesis in Bacillus subtilis 168 Thy-minus Tryp-minus, the rate of length extension is constant. A nutritional shift-up during thymine starvation causes an acceleration in the linear rate of length extension. During a nutritional shift-up in the presence of thymine, the rate of length extension gradually increases, reaching a new steady state at about 50 min before the new steady-state rate of cell division is reached. The steady-state rates of nuclear division and length extension are reached at approximately the same time. The ratio of average cell length to numbers of nuclei per cell in exponential cultures is constant over a fourfold range of growth rates. These observations are consistent with: (i) surface growth zones which operate at a constant rate of length extension under any one growth condition, but which operate at an absolute rate proportional to the growth rate of the culture, (ii) a doubling in number of growth zones at nuclear segregation, and (iii) a requirement for deoxyribonucleic acid replication for the doubling in a number of sites.     

Kinetic, dynamic, and pathway studies of glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous culture: IV. Enzymes and fluxes of pyruvate metabolism

The activities of pyruvate kinase (PK), pyruvate: formate-lyase (PFL), pyruvate dehydrogenase (PDH), and citrate synthase (CS) involved in the anaerobic glycerol conversion by Klebsiella pneumoniae were studied in continuous culture under conditions of steady states and sustained oscillations. Both the in vitro and in vivo activities of PK, PFL, and PDH are strongly affected by the substrate concentration and its uptake rate, as is the in vitro activity of CS. The flux from phosphoenolpyruvate to pyruvate is found to be mainly regulated on a genetic level by the synthesis rate of PK, particularly at low substrate concentration and low growth rate. In contrast, the conversion of pyruvate to acetyl-CoA is mainly regulated on a metabolic level by the in vivo activities of PFL and PDH. The ratio of in vitro to in vivo activities is in the range of 1 to 1.5 for PK, 5 to 17 for PFL and 5 to 80 for PDH under the experimental conditions. The regulation of in vivo activity and synthesis of these enzymes is sensitive to fluctuations of culture conditions, leading to oscillations of both the in vitro and in vivo activities. In particular, PFL is strongly affected during oscillations; its average in vitro activity is only about half of its corresponding steady-state value under similar environmental conditions. The average in vitro activities of PDH and PK under oscillations are close to their corresponding steady-state values. In contrast to all other enzymes measured for the glycerol metabolism by K. pneumoniae PFL and PDH are more effectively in vivo utilized under oscillations than under steady state, underlining the peculiar role of pyruvate metabolism in the dynamic responses of the culture.     

Relationship between chromosome replication and cell division in a thymineless mutant of Escherichia coli B/r.

The relationship between chromosome replication and cell division was investigated in a thymineless mutant of Escherichia coli B/r. Examination of the changes in average cell mass and DNA content of exponential cultures resulting from changes in the thymine concentration in the growth medium suggested that as the replication time (C) is increased there is a decrease in the period between termination of a round of replication and the subsequent cell division (D). Observations on the pattern of DNA synthesis during the division cycle were consistent with this relationship. Nevertheless, the kinetics of transition of exponential cultures moving between steady states of growth with differing replication velocities provided evidence to support the view that the time of cell division is determined by termination of rounds of replication under steady-state conditions.     

Cell elongation and division probability during the Escherichia coli growth cycle.

A new method is presented for determining the growth rate and the probability of cell division (separation) during the cell cycle, using size distributions of cell populations grown under steady-state conditions. The method utilizes the cell life-length distribution, i.e., the probability that a cell will have any specific size during its life history. This method was used to analyze cell length distributions of six cultures of Escherichia coli, for which doubling times varied from 19 to 125 min. The results for each culture are in good agreement with a single model of growth and division kinetics: exponential elongation of cells during growth phase of the cycle, and normal distributions of length at birth and at division. The average value of the coefficient of variation was 13.5% for all strains and growth rates. These results, based upon 5,955 observations, support and extend earlier proposals that growth and division patterns of E. coli are similar at all growth rates and, in addition, identify the general growth pattern of these cells to be exponential.     

Studies of fungal growth and intermediary carbon metabolism under steady and non-steady state conditions

The physiology of Aspergillus nidulans strain 224 has been studied under conditions of batch- and glucose-limited chemostat-culture and the effect of different steady state growth rates and dissolved oxygen tensions (DOT) examined. Measurements of the specific activities of selected glucose enzymes, the extent of oxygen uptake inhibition by glycolytic inhibitors, and radiorespirometric analyses were made in order to follow the variations in glucose catabolism, which occurred under these conditions. Greatly increased activity of the hexosemonophosphate (HMP) pathway was found during: (i) exponential growth of batch cultures; (ii) at near maximum specific growth rates (μ = 0.072 hr^?1) (DOT = 156 mm Hg); and (iii) at low DOT levels (<30 mm Hg) (μ = 0.050 hr^?1) in chemostat cultures. These changes in glucose eatabolism have been discussed in terms of the biosynthetic demands of the fungus under the influence of changing growth pressures. Preliminary studies also have been made of transition state behavior following stepwise alteration of the DOT. A new steady state was established after 4–5 culture doublings during which period an “overshoot” in HMP pathway activity occurred; these kinetics are indicative of a derepression of certain glucose enzymes. Low molecular weight phenols are synthesized during the exponential phase in batch cultures and these are further metabliized to a major secondary metabolite, melanin, at the onset of stationary phase conditions. The kinetics of tyrosinase production in steady state chemostats differs from those that might be predicted for an enzyme associated solely with secondary metabolism. A primary physiological role for this oxidase in Aspergillus nidulans has been postulated.     

Quantitative aspects of cellular turnover

Living organisms do not just grow by synthesizing cellular components. As part of the necessary steps for existence, some components are degraded after synthesis. Even for bacteria in balanced, exponential growth some substances, under some conditions, are turned over. In other phases of growth turnover can be much more extensive, but it is still selective. This review covers studies with animals as a way to put the studies on microorganisms in perspective. The history, the mathematics, and experimental design of turnover experiments are reviewed. The important conclusion is that most of the proteins during balanced growth are very stable in bacteria, although ribosomal proteins are degraded under starvation conditions. Another generalization is that the process of wall enlargement in general is associated with obligatory turnover of the peptidoglycan.     

The ratio of RNA to total nucleic acid content as a quantitative measure of unbalanced cell growth

Use of the metachromatic dye, acridine orange, to stain cells in suspension for flow cytometry allows for the simultaneous measurement of DNA and RNA content in individual cells. The relative RNA content as a function of total cellular nucleic acid content [alpha r = RNA/(RNA + DNA)] is a constant value, characteristic for particular cell lines during their exponential growth under optimal conditions. This ratio can be estimated for the G1A, G1B, S, and G2 + M cell cycle compartments. Changes in growth rate or the addition of antitumor drugs induces characteristic changes in the ratio either evenly throughout or at a particular phase of the cell cycle. Under such conditions, measurement of cellular DNA and RNA content provides a sensitive assay of any deviation from balanced cell growth. Unbalanced growth caused by suboptimal culture conditions or as a result of incubation with various antitumor agents is illustrated. Examples of unbalanced growth which are not correlated with cell viability as measured by cell clonogenicity are discussed.     

NON-STEADY-STATE ANALYSIS OF CELLULAR DEVELOPMENT

A discrete, compartmental, representation is proposed for cellular development. Equations describing the kinetics of cellular development (with or without cell proliferation) are derived in both differential and integral form. Using an integral formulation, a method is proposed for the computation of compartment transit times (maturation times) which does not require the use of radioactive or other tracers. Although some restrictive conditions are imposed, the applicability of the method does not depend on strictly steady-state conditions, nor on purely exponential growth or decay. The method permits analysis of cellular development in certain non-steady cases for which existing analytical methods are inadequate. It is possible that patterns of cellular development during embryogenesis, recovery from insults, or growth in culture (e.g. of normal and leukaemic human granulocytes) could be examined using this method of analysis.     

Invertase activity of intact cells of Saccharomyces cerevisiae growing on sugar cane molasses. I. Steady-state continuous culture tests

During the steady-state continuous culture of Saccharomyces cerevisiae on sugar cane blackstrap molasses under different experimental conditions, oscillatory variations of the invertase activity of the intact yeast cells were observed. The continuous morphological changes of the cells wall and of the periplasmic space affecting the interaction between invertase and sucrose molecules could be responsible by the observed oscillatory phenomena. The average invertase activity at the steady state is linearly correlated to the cell's growth rate.     

Timing is everything: optimizing crop yield for <Emphasis Type="Italic">Thalassiosira pseudonana</Emphasis> (Bacillariophyceae) with semi-continuous culture

There is relatively little choice in cultivation methods for growing algae outdoors, either in open pond systems or closed photobioreactors—as batch, continuous, or semi-continuous culture. Algal batch culture grown in a nutrient replete environment with adequate sunlight will become self-shaded with sufficient cell density and enter a stage in the growth dynamic known as the “phase of linear growth.” It is during this phase of linear growth that primary production is at maximum and that the highest biomass is harvested. The inherent problem with batch culture is that the exponential (and possibly lag) phases necessary to achieve densities required prior to the phase of linear growth consume time and waste surface area, and thereby make this an inefficient method to grow algae. Semi-continuous culture can be forced into shade-limiting conditions by reducing growth rate from maximum through dilution, whereby phases of lag and exponential growth are skipped, and culture growth is put into a state similar to a perpetual phase of linear growth with an appropriate culture harvest/dilution cycle. Importantly, semi-continuous culture can increase net growth efficiency over batch culture when compared by shade-limited growth rate. However, scientific study and theory covering shade-limited algal growth under semi-continuous culture conditions are nearly non-existent, which currently makes its application to phycological technologies impractical through “hit and miss” strategies. This laboratory study compares shade-limited growth dynamics for batch and semi-continuous cultures of Thalassiosira pseudonana (small-sized, marine diatom). Theory for optimizing production of mass algal culture with semi-continuous culture technique through cycle period and harvest volume is developed, and guidelines to practical industrial applications are provided.     

A rapid-equilibrium model for the control of the Calvin photosynthesis cycle by cytosolic orthophosphate

1. A simple model based on rapid-equilibrium assumptions is derived which relates the steady-state activity of the Calvin cycle for photosynthetic carbohydrate formation in C3 plants to the kinetic properties of a single cycle enzyme (fructose bisphosphatase) and of the phosphate translocator which accounts for the export of photosynthate from the chloroplast. Depending on the kinetic interplay of these two catalysts, the model system may exhibit a single or two distinct modes of steady-state operation, or may be unable to reach a steady state. 2. The predictions of the model are analysed with regard to the effect of external orthophosphate on the steady-state rate of photosynthesis in isolated chloroplasts under conditions of saturating light and CO2. Due to the possible existence of two distinct steady states, the model may account for the stimulatory as well as the inhibitory effects of external phosphate observed in experiments with intact chloroplasts. Stability arguments indicate, however, that only the steady-state case corresponding to phosphate inhibition of the rate of photosynthesis could be of physiological interest. 3. It is concluded that chloroplasts under physiological conditions most likely operate in a high-velocity steady state characterized by a negative Calvin cycle flux control coefficient for the phosphate translocator. This means that any factor enhancing the export capacity of the phosphate translocator can be anticipated to decrease the actual steady-state rate of photosynthate export due to a decreased steady-state rate of cyclic photosynthate production.     

Inhibition kinetics of phenol degradation from unstable steady-state data

Multiplicity of steady states of a continuous culture with an inhibitory substrate was used to estimate kinetic parameters under steady-state conditions. A continuous culture of Pseudomonas cepacia G4, using phenol as the sole source of carbon and energy, was overloaded by increasing the dilution rate above the critical dilution rate. The culture was then stabilized in the inhibitory branch by a proportional controller using the carbon dioxide concentration in the reactor exhaust gas as the controlled variable and the dilution rate as the manipulated variable. By variation of the set point, several unstable steady states in the inhibitory branch were investigated and the specific phenol conversion rates calculated. In addition, phenol degradation was investigated under substrate limitation (chemostat operation).The results show that the phenol degradation by P. cepacia can be described by the same set of inhibition parameters under substrate limitation and under high substrate concentrations in the inhibitory branch. Biomass yield and maintenance coefficients were identical. Fitting of the data to various inhibition models resulted in the best fit for the Yano and Koga equation. The well-known Haldane model, which is most often used to describe substrate inhibition by phenol, gave the poorest fit. The described method allows a precise data estimation under steady-state conditions from the maximum of the biological reaction rate up to high substrate concentrations in the inhibitory branch. Inhibition parameter estimation by controlling unstable steady states may thus be useful in avoiding discrepancies between data generated by batch runs and their application to continuous cultures which have been often described in the literature. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 54: 567-576, 1997.     

Diameter of cells of a thermosensitive dnaA mutant of Escherichia coli cultivated at intermediate temperatures.

Strains of Escherichia coli K-12 carrying the dnaA46 mutation exhibited a progressively decreasing DNA concentration and a progressively increasing cell size as the temperature was raised from 31 to 37 degrees C. Above 37 degrees C, steady-state exponential growth could not be maintained. The increase in average cell size with increasing growth temperature was due to an increase in cell length. There was no change in cell width. This seems to rule out the hypothesis that the increase in cell width in thy strains cultivated on low concentrations of thymine is due to the decrease in DNA concentration which also occurs under these conditions.     

Analysis of the dynamic and steady-state responses of growth rate and turgor pressure to changes in cell parameters

The physical analysis of plant cell enlargment is extended to show the dependence of turgor pressure and growth rate under steady-state conditions on the parameters which govern cell wall extension and water transport in growing cells and tissues, and to show the dynamic responses of turgor and growth rate to instantaneous changes in one of these parameters. The analysis is based on the fact that growth requires simultaneous water uptake and irreversible wall expansion. It shows that when a growing cell is perturbed from its steady-state growth rate, it will approach the steady-state rate with exponential kinetics. The half-time of the transient adjustment depends on the biophysical parameters governing both water transport and irreversible wall expansion. When wall extensibility is small compared to hydraulic conductance, the growth rate is controlled by the yielding properties of the cell wall, while the half-time for changes in growth rate is controlled by the water transport parameters. The reverse situation occurs when hydraulic conductance is lower than wall extensibility. The analysis also shows explicitly that turgor pressure is tightly coupled with growth rate when growth is controlled by both water transport and wall yielding parameters.