Acetic acid production by Acetogenium kivui in continuous culture — kinetic studies and computer simulations

Summary This work considers the continuous production of acetic acid by the homoacetogenic and thermophilic bacterium Acetogenium kivui. A mathematical model for the growth kinetics has been developed. The unstructured model for growth and product formation includes product and substrate inhibition as well as maintenance energy effects. The associated model parameters have been identified by non-linear optimization and evidenced experimentally in continuous culture as steady-state data. By using a mineral medium with glucose as the energy and carbon source for the bacteria proper carbon balances are available. The model permits good predictions of steady-state concentrations. Offprint requests to: J. von Eysmondt     

Growth kinetics of Escherichia coli with galactose and several other sugars in carbon-limited chemostat culture

Kinetic models for microbial growth describe the specific growth rate (mu) as a function of the concentration of the growth-limiting nutrient (s) and a set of parameters. A typical example is the model proposed by Monod, where mu is related to s using substrate affinity (Ks) and the maximum specific growth rate (mu max). The preferred method to determine such parameters is to grow microorganisms in continuous culture and to measure the concentration of the growth-limiting substrate as a function of the dilution rate. However, owing to the lack of analytical methods to quantify sugars in the microgram per litre range, it has not been possible to investigate the growth kinetics of Escherichia coli in chemostat culture. Using an HPLC method able to determine steady-state concentrations of reducing sugars, we previously have shown that the Monod model adequately describes glucose-limited growth of E. coli ML30. This has not been confirmed for any other sugar. Therefore, we carried out a similar study with galactose and found steady-state concentrations between 18 and 840 micrograms.L-1 for dilution rates between 0.2 and 0.8.h-1, respectively. With these data the parameters of several models giving the specific growth rate as a function of the substrate concentration were estimated by nonlinear parameter estimation, and subsequently, the models were evaluated statistically. From all equations tested, the Monod model described the data best. The parameters for galactose utilisation were mu max = 0.75.h-1 and Ks = 67 micrograms.L-1. The results indicated that accurate Ks values can be estimated from a limited set of steady-state data when employing mu max measured during balanced growth in batch culture. This simplified procedure was applied for maltose, ribose, and fructose. For growth of E. coli with these sugars, mu max and Ks were for maltose 0.87.h-1, 100 micrograms.L-1; for ribose 0.57.h-1, 132 micrograms.L-1, and for fructose 0.70.h-1, 125 micrograms.L-1.     

Packed bed dynamics during microbial treatment of wastewater: modelling and simulation

A mathematical model consisting of mass balance equations and accounting for bioreaction and mass transfer is presented to describe both unsteady and steady-state degradation of phenol in a biofilter. The model has been validated for the steady-state situation with literature work. The model has been able to predict the dynamics of the biofiltration process with variations in system and operating conditions as inlet substrate concentration, liquid phase mass transfer coefficients, particle size, Henry's constant, inlet velocity, growth and half saturation constants and bed void fraction. The results show that inlet substrate concentration, inlet velocity, growth and half saturation constants and liquid phase mass transfer coefficients significantly control the operational dynamics. It is also shown that inhibition effects can be neglected for low concentrations (<0.5 kg m(-3)) of phenol. Thus, the model can be used as a design tool for a biofilter.     

Shoot and Root Activities During Steady-state Plant Growth

A simple model for steady-state plant growth is described. Thegrowth constant, measured during steady-state exponential growth,is related to the specific activities of the shoot and the root,enabling the effects of certain environmental variables (light,carbon dioxide and nitrogen) on the growth constant to be examined.The model is used to interpret data on the growth kinetics ofwheat (Macdowall, 1972a, b, c).     

Phenomenological modeling of tumor diameter growth based on a mixed effects model

Over the last few years, taking advantage of the linear kinetics of the tumor growth during the steady-state phase, tumor diameter-based rather than tumor volume-based models have been developed for the phenomenological modeling of tumor growth. In this study, we propose a new tumor diameter growth model characterizing early, late and steady-state treatment effects. Model parameters consist of growth rhythms, growth delays and time constants and are meaningful for biologists. Biological experiments provide in vivo longitudinal data. The latter are analyzed using a mixed effects model based on the new diameter growth function, to take into account inter-mouse variability and treatment factors. The relevance of the tumor growth mixed model is firstly assessed by analyzing the effects of three therapeutic strategies for cancer treatment (radiotherapy, concomitant radiochemotherapy and photodynamic therapy) administered on mice. Then, effects of the radiochemotherapy treatment duration are estimated within the mixed model. The results highlight the model suitability for analyzing therapeutic efficiency, comparing treatment responses and optimizing, when used in combination with optimal experiment design, anti-cancer treatment modalities.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Cell cycle model for growth rate and death rate in continuous suspension hybridoma cultures

As a result of recent advances in flow cytometry, renewed interest is shown in modeling the kinetic behavior of cells in culture on the basis of cell cycle parameters. An important but often overlooked kinetic variable in hybridoma cultures is the cell death rate. Not only the overall cell growth but also the kinetics of nutrient metabolism and monoclonal antibody production have been shown to depend on the cell death rate in continuous suspension hybridoma cultures. The present study shows that the death rate in hybridoma cultures is proportional to the fraction of cells arrested in the G(1) phase of the cell cycle. The steady-state cell age distributions in the various phases of the division cycle have been calculated analytically. A simple mathematical model has been used to produce the profiles of the cycling and arrested cell fractions with respect to the dilution rate. The calculated steady-state growth rate, death rate, and viability profiles are shown to be in agreement with recently published experimental data from continuous suspension hybridoma cultures. (c) 1992 John Wiley & Sons, Inc.     

Modelling of the protonophoric uncoupling by phenoxyacetic acid of the plasma membrane potential of Penicillium chrysogenum

Physiological effects of phenoxyacetic acid, the penicillin V side-chain precursor, on steady-state continuous cultures of Penicillium chrysogenum have been studied both theoretically and experimentally. Theoretical calculations show that at an extracellular pH of 6.50, phenoxyacetic acid has negligible influence on the growth energetics due to protonophoric uncoupling of membrane potentials by passive diffusive uptake. On the other hand, when the extracellular pH is lowered to 5.00, a severe maintenance-related uncoupling effect of phenoxyacetic acid is calculated. These findings were confirmed experimentally by steady-state continuous cultivations with a high-yielding penicillin strain of P. chrysogenum performed on a chemically defined and glucose-limited medium at pH 6.50 and pH 5.00, both with and without phenoxyacetic acid present. The yield and maintenance coefficients were determined from steady-state measurements of the specific uptake rates of glucose and oxygen and the specific production rate of carbon dioxide as functions of the specific growth rate. Combining these data with a simple stoichiometric model for the primary metabolism of P. chrysogenum allows quantitative information to be extracted on the growth energetics in terms of ATP spent in maintenance- and growth-related processes, i.e. mATP and YxATP. The increased maintenance-related ATP consumption when adding phenoxyacetic acid at pH 5.00 agrees with the theoretical calculations on the uncoupling effect of phenoxyacetic acid. When YxATP is compared with earlier reported values for the theoretical ATP requirement for biosynthesis of P. chrysogenum, i.e. YxATP, growth, it is found that YxATP,growth is only 40-50% of YxATP, which stresses that a large amount of ATP is wasted in turnover of macromolecules, leaks, and futile cycles.     

A new mathematical approach predicts individual cell growth behavior using bacterial population information

A theoretical methodology has been developed for studying the growth kinetics of bacterial cells. It utilizes the steady-state cell length distribution in a bacterial population to predict the dependency of growth and division rates on cell length and age. The mathematical model has been applied to the analysis of two bacterial populations, a wild-type strain of Bacillus subtilis, and a minicell-producing strain that carries the divIVB1 mutation. The results show that our model describes the wild-type population very well and that the assumptions typically used in traditional methods are unrealistic. In the case of the minicell-producing mutant we find evidence that the rate of cell division must be a function not only of cell size but also of cell age.     

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.     

Kinetic study of hybridoma cell growth in continuous culture. I. A model for non-producing cells

The kinetic behavior of a nonproducing hybridoma clone AFP-27-NP was investigated in continuous culture under glucose-limited conditions. A total of more than 21, 000 h of cultures were operated at dilution rates ranging from 0.01 to 0.06 h(-1). The viable cell concentrations, dead cell concentrations, and cell volumes all varied with the dilution rate. A steady-state model was developed based on the biomass concentration and the glucose concentration. The specific growth rate as a function of glucose concentration is described by a model similar to the Monod model with a threshold glucose concentration and a minimum specific growth rate incorporated; the model is meaningful only at glucose concentrations and specific growth rates above these levels. A death rate is included in the model which is described by an inverted Monod-type function of glucose concentration. The yield coefficient based on glucose is constant in the lower range of specific growth rates and changes to a new constant value in the upper region of specific growth rates. No maintenance term for glucose consumption was needed; in the plot of specific glucose consumption rate vs. specific growth rate, the line intercepted the specific growth rate axis at a value close to the minimum growth rate. The values for the model parameters were determined from regression analysis of the steady-state data. The model predictions and experimental results fit very well.     

Mathematical modelling and simulation of aqueous two-phase continuous protein extraction.

A mathematical model has been developed to describe the continuous, steady-state operation of an aqueous two-phase system for protein extraction. The model is based on steady-state mass balances of the main components and phase equilibrium data. Experimental data on the separation of thaumatin from contaminant proteins of an homogenate of E. coli in a PEG4000/Phosphate system was used. The data shows the effect of the presence and absence of NaCl which was used to carry out the extraction of thaumatin into the PEG phase and back into the PO4(-3) phase. Simulation results showing the sensitivity to key process parameters, and the effect of process variables on performance are presented and discussed. The model can be used to predict performance and thus 'robustness' of process conditions as well as predict protein recovery yield and purity. This model can also be used to implement a suitable control strategy to maintain process stability.     

Cybernetic modeling of iron-limited growth and siderophore production

The cybernetic modeling framework developed by Ramkrishna and co-workers has been applied to a case of bacterial metabolite production, namely the production of siderophores (iron-chelating agents) associated with iron-limiting fermentation conditions. Experimental growth data showed that, even though final biomass levels were controlled by exhaustion of the carbon source, iron-limiting conditions also affected the biomass yield. A structured model which includes the process of an iron-limiting energy resource production was able to quantitatively account for this apparent dual-substrate limitation over a wide range of batch and continuous operating conditions. The experiments data also showed quite large difference in iron uptake over the wide range of operating condition and iron levels investigated. The inclusion in the model of the processes of low and high (siderophore-mediated) affinity iron transport, and siderophore production led to simulation results that were in good quantitative agreement with the siderophore, medium and cell iron levels, in both batch and steady-state continuous culture operating conditions.     

Synchronous cultures from the baby machine. A model for animal cells

A baby-machine system that produces newborn Escherichia coli cells from cultures immobilised on a membrane was developed many years ago in an attempt to attain optimal synchrony with minimal disturbance of steady-state growth, and a model designed to characterise the nature and quality of the synchrony of such cells in a quantitative manner has been published. The baby machine has now been adapted for animal cells, and the present article is an attempt to modify the model to include these cells as well. The model consists of five elements, giving rise to five adjustable parameters (and a proportionality constant): a major, essentially synchronous group of cells with ages distributed normally about zero; a minor, random component from a steady-state population on the membrane that had undergone only very little age selection during the elution process; a fixed background count, to allow for the signals recorded by the electronic particle counter produced by debris and electronic noise; a time-shift, to account for differences between time of cell division and end of sample collection; and the coefficient of variation of the interdivision-time distribution, taken to be reciprocal-normal. It is this last feature, a reciprocal-normal rather than a Pearson type III interdivision-time distribution, that distinguishes this version of the model from its predecessor. The model is fitted by unconstrained non-linear least-squares to data from three different leukemia cell lines. The standard errors of the parameters are quite small in all cases, making their estimates highly significant; the quality of the fit is striking. The five parameters of the model can be divided into two nuisance parameters, two that are associated with the methodology and one that describes an inherent property of the cell itself; it turns out that both methodology parameters are zero in all three data sets studied. We also discuss the partition of the transition-time dispersion between the age distribution of the newborn cells and the age distribution of dividing cells and show that a reliable estimate of the corresponding parameters requires an experiment that extends over at least two and a half doubling times.     

Model of steady-state-biofilm kinetics

A steady-state biofilm is defined as one that has neither net growth nor decay over time. The model, developed for steady-state-biofilm kinetics with a single substrate, couples the flux of substrate into a biofilm to the mass (or thickness) of biofilm that would exist at steady-state for a given bulk substrate concentration. Based on kinetic and energetic constraints, this model predicts for a single substrate that a steady-state bulk concentration, S_min, exists below which a steady-state biofilm cannot exist. Thus, in the absence of adsorption of bacteria from the bulk water and for substrate concentration below S_min, substrate flux and biofilm thickness are zero. Equations are provided for calculating the steady-state substrate flux and biofilm thickness for S greater than S_min. An example is provided to demonstrate the use of the steadystate model.     

Periodic operation of immobilized live cell bioreactor for the production of candicidin

A lumped model for cell growth and secondary metabolite production in an immobilized live cell bioreactor has been developed. This model is applied here to simulate the performance of an immobilized bioreactor under steady-state conditions and under conditions of periodically varying concentration of a growth-limiting substrate. The results of the simulation study were experimentally verified in the case of the production of the antibiotic candicidin by Streptomyces griseus in an immobilized bioreactor with forced periodic operation. The results of the studies suggest that periodically operated immobilized live cell bioreactors can provide a potent alternative for the production of non-growth-associated biochemicals, as compared to free cell fermentations, pulsed fermentations with process cycle regeneration, and nonregenerated bioreactors. This work has demonstrated that by frequent pulsing of the growth limiting nutrient, stable extended production can be obtained at high specific cellular productivities.     

Cat lung hemodynamics: comparison of experimental results and model predictions

Commonly, attempts have been made to learn about the structure and function of the pulmonary vascular bed from measurements of arterial and venous pressures and blood flow rate under steady-state conditions (e.g., from pressure vs. flow data) or dynamic conditions (e.g., from vascular occlusion data). Zhuang et al. (J. Appl. Physiol. 55: 1341-1348, 1983) have presented a detailed model of steady-state cat lung hemodynamics based on direct measurements of anatomical and elasticity data. This model provides an opportunity to better understand the information content of the hemodynamic data. Therefore, in the present study we carried out a series of steady-state and dynamic experiments on isolated cat lungs. We then compared the results with those predicted by the model. We found that the model provided a good fit to the steady-state data. However, to fit the dynamic data, some modifications were necessary to account for the viscous behavior of the vessel walls and to move the first moment of the distribution of vascular resistance toward the arterial end of the vascular bed relative to that of the distribution of vascular compliance. Due to the sensitivity of the vascular resistance to small changes in vessel diameters and branching ratio, the modifications in morphometry represent small changes in morphometric data and are probably within the range of uncertainty in such data. The modifications had little effect on the steady-state model simulations but substantially improved the dynamic model simulations, suggesting that the dynamic data are quite sensitive to small changes in the relative distributions of vessel diameters and elasticity.