Dynamics of microbial propagation: Models considering inhibitors and variable cell composition

Mathematical models for microbial growth in batch and continuous cultures are formulated. The models have been referred to as distributed models since the microbial population in a culture is looked upon as protoplasmic mass distributed uniformly throughout the culture. Growth is regarded as the increase in this mass by conversion of medium components into biological mass and metabolic products. Two sets of models have been presented. The first arise from introducing additional considerations into the model proposed by Monod to account for the stationary phase and the phase of decline in a batch culture. These have been referred to as unstructured, distributed models since they do not recognize any form of structure in the protoplasmic mass. The models in the second set are referred to as structured, distributed models. Structure is introduced by considering the protoplasmic mass to be composed of two groups of substances which interact with each other and with substances in the environment to produce growth. The structured models account for the dependence of growth on the past, history of the cells; thus they predict all growth phases observed in batch cultures, whereas the unstructured models do not predict a lag phase. The full implications of the models for continuous propagation, as determined by the method of stability analysis and transient calculations, are discussed. The models prediet a number of new results and should be confronted with experiments.     

Cell cycle operation during batch growth of fission yeast populations

Batch cultivation provides a continuous sequence of different environments useful for studying responses of cell cycle controls. Flow cytometry measurements have been made of the frequency functions for protein, RNA, and DNA at different times during batch growth of the fission yeast Schizosaccharomyces pombe. The mean cellular protein and RNA contents and their variances tend to increase with increasing population specific growth rates. Analysis of the mid-exponential phase DNA frequency function data indicates that DNA synthesis occupies 12% of the total cell cycle time and is completed at the same time as cell separation. Coordination of DNA synthesis and cell separation is less precise when population growth rate is low in late lag and early stationary phases.     

Modeling of typical microbial cell growth in batch culture

A mathematical model was developed, based on the time dependent changes of the specific growth rate, for prediction of the typical microbial cell growth in batch cultures. This model could predict both the lag growth phase and the stationary growth phase of batch cultures, and it was tested with the batch growth ofTrichoderma reesei andLactobacillus delbrucckii.     

Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

Individual‐based and stochastic modeling of cell population dynamics considering substrate dependency

In this article, we propose an individual‐based and stochastic modeling approach that is capable of describing the bacterial cell population dynamics during a batch culture. All stochastic nature inherent in intracellular molecular level reactions and cell division processes were considered in a single model framework by embedding a sub‐model describing individual cell's growth kinetics in a discrete event simulation algorithm. The resultant unique feature of the model is that the effects of the stochasticities on the cell population dynamics can be investigated for different substrate‐dependent cell growth kinetics. When Monod kinetics was used as the sub‐model, the stochasticities only slightly affected the cell mass increase and substrate consumption profiles during the batch culture although they were still important in describing the changes of cell population distributions. When Andrews substrate inhibition kinetics was used, however, it was revealed that the overall cell population dynamics could be seriously influenced by the stochasticities. Under a critical initial substrate level, the cell population could proliferate against the substrate inhibition only when the stochasticities were considered. Biotechnol. Bioeng. 2009;103: 891–899. © 2009 Wiley Periodicals, Inc.     

Cell size distribution as a parameter for the predetermination of exponential growth during repeated batch cultivation of CHO cells

The routine measurement of the cell size distribution of a Chinese hamster ovary (CHO) cell population during a repeated batch process enables the predetermination of exponential growth even 24 h before the population enters the log phase, due to a short but significantly increased cell size during the lag phase. A prolongation of the stationary phase causes to progressive limitation in asparagine, serine, and ethanolamine. Such extended limitation influences the duration of the following lag phase and obviously induces a synchronization of the cell population that can be monitored easily by a fast cell size analyzing technique. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 793-797, 1997.     

Effect of the development phase and growth rate of a Shigella sonnei population on the reproduction of homologous bacteriophage

This study showed that the minimum latent period (20 minutes) of the intracellular multiplication of dysentery bacteriophage S-9 in the population of S. sonnei substrate strain under the conditions of static heterogeneous surface batch cultivation was observed at the end of the lag phase and at the growth acceleration phase, in the first and second thirds of the exponential curve, while the maximum latent period (35-40 minutes) was observed at the stationary phase. The maximum yield of phage S-9 from one infected bacterial cell (628.3 +/- 116.8) was registered during the first third of the phase of the exponential growth of the bacterial population and the minimum yield (18.66 +/- 6.6), at the beginning of the lag phase. The significant direct correlation between the specific growth rate of the bacterial population and the yield of the phage from one infected bacterial cell at the end of the lag phase, at the growth acceleration and deceleration phases, as well as the significant inverse correlation between the yield of the phage and the time of the generation of the bacterial population at the growth acceleration phase were established.     

Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures.

Growth of Saccharomyces cerevisiae on glucose in aerobic batch culture follows the well-documented diauxic pattern of completely fermenting glucose to ethanol during the first exponential growth phase, followed by an intermediate lag phase and a second exponential growth phase consuming ethanol. In continuous cultures over a range of intermediate dilution rates, the yeast bioreactor exhibits sustained oscillations in all the measured concentrations, such as cell mass, glucose, ethanol, and dissolved oxygen, the amounts of intracellular storage carbohydrates, such as glycogen and trehalose, the fraction of budded cells as well as the culture pH. We present here a structured, unsegregated model for the yeast growth dynamics developed from the 'cybernetic' modeling framework, to simulate the dynamic competition between all the available metabolic pathways. This cybernetic model accurately predicts all the key experimentally observed aspects: (i) in batch cultures, duration of the intermediate lag phase, sequential production and consumption of ethanol, and the dynamics of the gaseous exchange rates of oxygen and carbon dioxide; and (ii) in continuous cultures, the spontaneous generation of oscillations as well as the variations in period and amplitude of oscillations when the dilution rate or agitatin rate are changed.     

Modelling temporal decoupling between biomass and numbers during the transient nitrogen-limited growth of a marine phytoflagellate

A mathematical model of single-nutrient-limited algal growthis presented in which carbon fixation and cell division aredifferent functions of assimilated nutrient. The model successfullydescribes key features of the growth of Isochrysis galbana inammonium-limited batch culture under continuous illuminationand in light/dark cycles. The incorporation of a nutrient processingtime allows the simulation of a time lag between net carbonfixation and cell division, and enables the model to describechanges in the mean carbon content and carbon/nitrogen ratioof the cells. The model can be completely parameterized fromstandard batch culture experiments.     

A new modeling technique and computer simulation of bacterial growth

A mathematical model that describes substrate utilization and cell growth in terms of two potentially rate-limiting enzyme systems has been developed. Consideration of substrate inhibition and enzyme repression have been incorporated. The model provides a rational approach for characterizing non-steady-state phenomena. The model has been used to analyze batch test data to illustrate the effects of inhibition, repression, and concurrent substrate utilization. Its utility lies in the fact that it provides a quantitative framework for describing changes in the activity levels of cells that result from changes in substrate concentration and/or substrate type. The lag phase resulting from exposure to a new substrate can be modeled.     

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.     

Mathematical model of cell growth and phosphatase biosynthesis in Saccharomyces carlsbergensis under phosphate limitation.

The rate kinetics of growth and acid phosphate formation in the batch culture of Saccharomyces carlsbergensis LAM 1068 was studied under varying degrees of phosphate limitation. The mathematical model that was developed is concerned with the time lag for exponential growth, the biphasic growth on a substrate (glucose) and its product (ethanol), sustained growth on conservative phosphate, and the derepression of acid phosphatase. The numerical calculations using appropriate parametric constants successfully described the variation in the cell mass, glucose, ethanol, and inorganic phosphate concentrations, and the enzyme activity of acid phosphatase during aerobic growth of S. carlsbergensis under five different conditions of phosphate starvation. A simulation study revealed that the optimum initial phosphate concentration in the medium giving a high productivity of acid phosphatase was 2.0 mg phosphorus/g glucose liter.     

Survival Probability of Beneficial Mutations in Bacterial Batch Culture

The survival of rare beneficial mutations can be extremely sensitive to the organism’s life history and the trait affected by the mutation. Given the tremendous impact of bacteria in batch culture as a model system for the study of adaptation, it is important to understand the survival probability of beneficial mutations in these populations. Here we develop a life-history model for bacterial populations in batch culture and predict the survival of mutations that increase fitness through their effects on specific traits: lag time, fission time, viability, and the timing of stationary phase. We find that if beneficial mutations are present in the founding population at the beginning of culture growth, mutations that reduce the mortality of daughter cells are the most likely to survive drift. In contrast, of mutations that occur de novo during growth, those that delay the onset of stationary phase are the most likely to survive. Our model predicts that approximately fivefold population growth between bottlenecks will optimize the occurrence and survival of beneficial mutations of all four types. This prediction is relatively insensitive to other model parameters, such as the lag time, fission time, or mortality rate of the population. We further estimate that bottlenecks that are more severe than this optimal prediction substantially reduce the occurrence and survival of adaptive mutations.     

Detachment ofPseudomonas fluorescens from biofilms on glass surfaces in response to nutrient stress

The effects of glucose and nitrogen depletion on the colonization of glass Petri plates byPseudomonas fluorescens were studied in batch culture. Colonization of the surfaces was initiated before colonization of the bulk phase, and biofilm formation was observed. This resulted in an apparent lag in the batch growth curve for the cell suspension. The lag phase was an artifact caused by the partitioning of cells between the bulk and solid phase of the culture and was not due to a reduction in the growth rate of unattached cells. The specific growth rate of the unattached cells (0.331 hour^–1) was almost twice that determined for the total population (0.171 hour^–1). Consequently the growth rate of biofilm-forming bacteria cannot be determined in batch culture unless the growth of both attached and unattached cells is monitored, and batch growth curves may contain artifacts due to the formation and dispersion of biofilms. The depletion of either glucose or nitrogen led to the active detachment of cells from the biofilm. An increase in the hydrophobicity of unattached cells was noted on depletion of carbon. This increase was the result of emigration of cells from the surface into the bulk phase.Paper contribution number 128, Centre de Rechcrches Alimentaires de St Hyacinthe.     

A deterministic model for monophasic growth of batch cultures of bacteria

Experimental observations of bacterial numbers employing high resolution electrical conductance measurements of the culture provide the basis for a proposed deterministic model of monophasic growth of populations in batch culture. The model postulates that the production and growth of each bacterium is accompanied by the generation of a constant mass of toxic end-products and that specific growth rate declines in proportion to the ratio of the accumulated mass of these substances to the dry mass of the nutrient medium when the substrate is non-limiting. The theoretical relationship is found to fit extensive data for Escherichia coli (NCIB 9132) very closely and offers an analytical basis for the logistic curve frequently observed to represent the time-dependence of growth. These data incidentally provide substantial evidence that lag time and generation time are each independent of both inoculum number and concentration of the medium.     

A simple, leaky cell growth model for plant cell aggregates

A simple growth model is proposed for plant cell aggregates which accounts for leakage of a single intermediate metabolite from the aggregates to the medium. This model predicts a lag phase in the growth curve whose extent is determined by the intermediate metabolite leakage coefficient and its equilibrium distribution coefficient between the medium and the cell aggregates, the size of the inoculum relative to the system total water content, and the initial intermediate metabolite content in the medium. The model thus provides for an interaction between growing plant cells and their environment in a way that has heretofore been unquantified. Preliminary validation of the model has been made against literature data of Dioscorea deltoidea grown in batch suspension cell culture on sucrose, yielding a correlation coefficient of 0.997. The predicted glucose + fructose concentration in the medium agrees reasonably well with experimental measurements after ca, 3.5 days of culture, although a discrepancy exists between model prediction and experiment immediately after startup. Further validation of the model is suggested on this and other plant species.     

A simple kinetic model for myeloma cell culture with consideration of lysine limitation

A simple kinetic model is developed to describe the dynamic behavior of myeloma cell growth and cell metabolism. Glucose, glutamine as well as lysine are considered as growth limiting substrates. The cell growth was restricted as soon as the extracellular lysine is exhausted and then intracellular lysine becomes a growth limiting substrate. In addition, a metabolic regulator model together with the Monod model is used to deal with the growth lag phase after inoculation or feeding. By using these models, concentrations of substrates and metabolites, as well as densities of viable and dead cells are quantitatively described. One batch cultivation and two fed-batch cultivations with pulse feeding of nutrients are used to validate the model.     

Probabilistic neural networks using Bayesian decision strategies and a modified Gompertz model for growth phase classification in the batch culture of Bacillus subtilis

Probabilistic neural networks (PNNs) were used in conjunction with the Gompertz model for bacterial growth to classify the lag, logarithmic, and stationary phases in a batch process. Using the fermentation time and the optical density of diluted cell suspensions, sampled from a culture of Bacillus subtilis, PNNs enabled a reliable determination of the growth phases. Based on a Bayesian decision strategy, the Gompertz based PNN used newly proposed definition of the lag and logarithmic phases to estimate the latent, logarithmic and stationary phases. This network topology has the potential for use with on-line turbidimeter for the automation and control of cultivation processes.     

Kinetics of growth of the ciliate Tetrahymena pyriformis on Escherichia coli

The growth of the ciliate Tetrahymena pyriformis on non-growing Escherichia coli has been studied by following the time courses of population densities and protozoan mean cell volume in batch cultures. Viable, non-encysted protozoa always stopped feeding before the bacterial density was reduced to zero and non-feeding ciliates tended to swim faster than feeding ciliates. In addition, the number of bacteria and other particles of bacterial size consumed in the formation of one new ciliate, when averaged over the lag and reproductive phases of a culture, declined toward a limiting value of about 1.6 x 10(4) particles per ciliate as the initial density of such particles was increased.