Dynamics of chemostat in which one microbial population grows on multiple complementary nutrients

The equations of a chemostat in which one microbial population grows on multiple rate-limiting nutrients are formulated. The dynamics of a chemostat involving growth on complementary nutrients is studied through stability analysis of the system of equations. Some conditions are derived that relate the dynamic behavior of the chemostat to its operating conditions and can be applied to any model for the specific growth rate of the population. It is shown that, if maintenance of the population is neglected, the system exhibits no sustained or damped oscillations. If maintenance of the population is considered, damped oscillations are observed for some operating conditions.     

Population growth regulated by intraspecific competition for energy or time: Some simple representations

An examination is made of some of the ways populations can grow in response to changes in their own density. Under two different assumptions on birth and death rates, models for single-species population growth that incorporate intraspecific competition by interference but not exploitation are of logistic form. Where an individual's net energy input from feeding is inversely proportional to population size, population growth follows a convex curve, whether interference is included or not. Data of Smith (1963) on Daphnia populations are fit well by this kind of curve. Combination of the two kinds of growth can produce S-shaped curves whose inflection is displaced from that value—half the carrying capacity—given by the logistic; an upward displacement is favored by a high ratio of metabolic and replacement costs to feeding input. Inflection points from real curves are much more often higher than expected from the logistic. Nonmonotonic growth curves can arise when there is instantaneous feedback between consumers and resource availability; certain of these equations are of logistic or convex form at equilibrium. The possible effect of r- and K-selection on the biological parameters, such as feeding efficiency, used to construct the monotonie equations is discussed, and the equations are extended to 2-species competition. Table III characterizes some simple single-species growth curves.     

Formulating variable carrying capacity by exploring a resource dynamics-based feedback mechanism underlying the population growth models

Most of the population growth models comprise the concept of carrying capacity presume that a stable population would have a saturation level characteristic. This indicates that the population growth models have a common implicit feature of resource-limited growth, which contributes at a later stage of population growth by forming a numerical upper bound on the population size. However, a general underlying resource dynamics of the models has not been previously explored, which is the focus of present study. In this paper, we found that there exists a conservation of energy relationship comprising the terms of available resource and population density, jointly interpreted here as total available vital energy in a confined environment. We showed that this relationship determines a density-dependent functional form of relative population growth rate and consequently the parametric equations are in the form depending upon the population density, resource concentration, and time. Thus, the derived form of relative population growth rate is essentially a feedback type, i.e., updating parametric values for the corresponding population density. This resource dynamics-based feedback approach has been implemented for formulating variable carrying capacity in a confined environment. Particularly, at a constant resource replenishment rate, a density-dependent population growth equation similar to the classic logistic equation is derived, while one of the regulating factors of the underlying resource dynamics is that the resource consumption rate is directly proportional to the resource concentration. Likewise two other population growth equations similar to two known popular growth equations are derived based on this resource dynamics-based feedback approach. Using microcosm-derived data of fungus T. virens, we fitted one derived population growth model against the datasets, and concluded that this approach is practically implementable for studying a single population growth regulation in a confined environment.     

Adaptive advantages of patterns of growth and asexual reproduction of the sea anemone Metridium senile (L.) in intertidal and submerged populations

Individual size, rate of growth, and mode and frequency of asexual reproduction are life-history traits of primary importance for sea anemones. These traits determine sexual reproductive output, affect an individual's probability of survival, and are crucial in adapting an individual to its environmental surroundings. The sea anemone Metridium senile (L.) is highly variable in ecological distribution and life history, including rate of growth, individual size, and rate of asexual reproduction. Gonad size (measured as cross-sectional area of gonadal tissue) increases with body weight, so individuals should grow as large and as rapidly as possible to maximize individual sexual reproductive output. Cessation of growth and small body size in intertidal populations suggest that growth is constrained by genetic or environmental conditions. The growth of intertidal individuals transplanted to harbor-float panels demonstrated that growth limits are imposed by environmental factors, most probably limited food and feeding time and damage from wave exposure (which stimulates fragmentation). Individuals in harbor-float populations, which are continuously immersed, grow much larger, and large individuals comprise a greater proportion of the population than in the intertidal zone. The highest rate of fragmentation observed was on harbor-float panels. Patterns of growth and asexual reproduction provide adaptive advantages for M. senile. For harborfloat individuals, large individual size increases gamete production and may increase feeding efficiency. For intertidal individuals, asexual reproduction allows growth despite individual size constraints and rapid population growth, with specific advantages resulting from clone formation.     

Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations

We develop and analyze a mathematical model, in the form of a system of ordinary differential equations (ODEs), governing cancer growth on a cell population level with combination immune, vaccine and chemotherapy treatments. We characterize the ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and identifying basins of attraction. These system characteristics are useful not only to gain a broad understanding of the specific system dynamics, but also to help guide the development of combination therapies. Numerical simulations of mixed chemo-immuno and vaccine therapy using both mouse and human parameters are presented. We illustrate situations for which neither chemotherapy nor immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire tumor.     

The relative influence of temperature and food on the metabolism of a marine invertebrate

Many benthic marine invertebrates exhibit a seasonal cycle in activities such as feeding, growth and reproduction. In temperate regions, this seasonality is typically correlated with coincident cycles in photoperiod, temperature and food availability and it can be difficult to determine which of these environmental factors is the key driver. Polar regions are characterised by greatly reduced seasonal variation in temperature, and an enhanced seasonality of food availability; they therefore form a natural laboratory for distinguishing the ecological effects of food from those of temperature. Here, we report a study of the common shallow water urchin Sterechinus neumayeri from Rothera Point, Antarctica. This species exhibits a marked seasonal variation in metabolic rate and feeding activity (which ceases completely in winter). In this study the metabolic rate of urchins collected in late winter and held in the laboratory without food was compared with that of wild urchins undertaking the transition to summer feeding and growth. Starved urchins showed a small rise in metabolic rate in summer which could be explained entirely by the small increase in temperature (Q(10)=2.5). At the same time, the wild population showed a much larger increase in metabolic rate related largely to the costs of feeding and growth. Rates of nitrogen excretion were also much larger in wild urchins, and the O:N atomic ratio indicated that starved urchins were depending to a greater extent on lipid and carbohydrate. Gut mass and test organic content showed no change in starved urchins, indicating that metabolic substrate was being provided by the gonad. The data suggest that in wild S. neumayeri only 15-20% of the summer increase in metabolism is caused directly by the temperature rise whereas 80-85% is caused by increased physiological activity associated with feeding, growth and spawning.     

Enhancement of Candida rugosa lipase production by using different control fed-batch operational strategies

Simulation studies have predicted that maximum lipase activity is reached with fed-batch operation strategies. In this work, two different fed-batch operational strategies have been studied: constant substrate feeding rate and specific growth rate control. A constant substrate feeding rate strategy showed that maximum aqueous lipolytic activity (55 U/mL) was reached at low substrate feeding rates, whereas lipase tends to accumulate inside the cell at higher rates of substrate addition. In the second fed-batch strategy studied, a feedback control strategy has been developed based on the estimation of state variables (X and mu) from the measurement of indirect variables such as CER by means of mass spectrometry techniques. An on-off controller was then used to maintain the specific growth rate at the desired value by adjusting the substrate feeding rate. A constant specific growth rate strategy gave higher final levels of aqueous lipolytic activity (117 U/mL) at low specific growth rates. At higher specific growth rates the enzyme remained accumulated inside the cell, as was observed with a constant feeding fed-batch strategy. With a constant specific growth rate strategy, lipase production by Candida rugosa was enhanced 10-fold compared to a batch operation. Purification studies have demonstrated that lipolytic and esterasic specific activity ratios of Candida rugosa isoenzymes can be modified by using different operational conditions. These studies have also showed that the isoenzymes obtained in a controlled growth rate strategy are around three- to four-fold more active than those obtained in a constant feeding rate strategy.     

Macrokinetic model for methylotrophic Pichia pastoris based on stoichiometric balance

A macrokinetic model for Pichia pastoris expressing recombinant human serum albumin is proposed. The model describes the balances of some key metabolites, ATP and NADH, during glycerol and methanol metabolism. In the glycerol growth phase, the metabolic pathways mainly include phosphorylation, glycolysis, tricarboxylic acid cycle, and respiratory chain. In the methanol growth phase, methanol is oxidized to formaldehyde at first. Then, while a part of formaldehyde is oxidized to formate, the rest is condensed with xylulose-5-monophosphate to form glyceraldehyde-3-phosphate, and further assimilated to form cell constituents. The metabolic pathways following glyceraldehyde-3-phosphate were assumed to be similar to those in the glycerol growth phase. Based on the model, the macrokinetic bioreaction rates such as the specific substrate consumption rate, the specific growth rate, the specific acetyl-CoA formation rate as well as the specific oxygen uptake rate are obtained. The specific substrate consumption rate and the specific growth rate are then coupled into a bioreactor model such that the relationship between substrate feeding rates and the main state variables, i.e., the medium volume, the concentrations of the biomass, the substrate, and the product, is set up. Experimental results demonstrate that the model can describe the cell growth and the protein production with reasonable accuracy.     

Mathematical description of holometabolous life cycles

The deterministic continuous equations are developed for the population levels of the various life stages of holometabolous species. Special attention is paid to the larval stage in which the variable of weight, in addition to the traditional variables of age and time, is included; growth rate and the genetic variability of the growth rate are allowed for. A pair of equations is derived that permits computation of the larval population level without regard to the levels of the other life stages. Finite difference equations are developed, simple analytical forms for vital rates are adopted, and a numerical example is given.     

A rational approach to improving productivity in recombinant Pichia pastoris fermentation

A Mut(S) Pichia pastoris strain that had been genetically modified to produce and secrete sea raven antifreeze protein was used as a model system to demonstrate the implementation of a rational, model-based approach to improve process productivity. A set of glycerol/methanol mixed-feed continuous stirred-tank reactor (CSTR) experiments was performed at the 5-L scale to characterize the relationship between the specific growth rate and the cell yield on methanol, the specific methanol consumption rate, the specific recombinant protein formation rate, and the productivity based on secreted protein levels. The range of dilution rates studied was 0. 01 to 0.10 h(-1), and the residual methanol concentration was kept constant at approximately 2 g/L (below the inhibitory level). With the assumption that the cell yield on glycerol was constant, the cell yield on methanol increased from approximately 0.5 to 1.5 over the range studied. A maximum specific methanol consumption rate of 20 mg/g. h was achieved at a dilution rate of 0.06 h(-1). The specific product formation rate and the volumetric productivity based on product continued to increase over the range of dilution rates studied, and the maximum values were 0.06 mg/g. h and 1.7 mg/L. h, respectively. Therefore, no evidence of repression by glycerol was observed over this range, and operating at the highest dilution rate studied maximized productivity. Fed-batch mass balance equations, based on Monod-type kinetics and parameters derived from data collected during the CSTR work, were then used to predict cell growth and recombinant protein production and to develop an exponential feeding strategy using two carbon sources. Two exponential fed-batch fermentations were conducted according to the predicted feeding strategy at specific growth rates of 0.03 h(-1) and 0.07 h(-1) to verify the accuracy of the model. Cell growth was accurately predicted in both fed-batch runs; however, the model underestimated recombinant product concentration. The overall volumetric productivity of both runs was approximately 2.2 mg/L. h, representing a tenfold increase in the productivity compared with a heuristic feeding strategy.     

Comparison of logistic equations for population growth.

Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity.     

Influence of specific growth rate on specific productivity and glycosylation of a recombinant avidin produced by a Pichia pastoris Mut+ strain

A recombinant avidin-producing Mut+ Pichia pastoris strain was used as a model organism to study the influence of the methanol feeding strategy on the specific product productivity (q(p)) and protein glycosylation. Fed-batch cultivations performed at various specific growth rates (micro) and residual methanol concentrations showed that the specific avidin productivity is growth-dependent. The specific productivity increases strongly with the specific growth rate for micro ranging from 0 to 0.02 h(-1), and increases only slightly with the specific growth rate above this limit. N-terminal glycosylation was also found to be influenced by the specific growth rate, since 9-mannose glycans were the most abundant form at low growth rates, whereas 10-mannose carbohydrate chains were favored at higher micro. These results show that culture parameters, such as the specific growth rate, may significantly affect the activity of glycoproteins produced in Pichia pastoris. In terms of process optimization, this suggests that a compromise on the specific growth rate may have to be found, in certain cases, to work with an acceptable productivity while avoiding the addition of many mannoses.     

On Intraspecies Forms of the Mongolian grayling (<Emphasis Type="Italic">Thymallus brevirostris</Emphasis>Kessl.) from Hoton Nur Lake (Western Mongolia)

Morphological and ecological traits of the Mongolian grayling Thymallus brevirostrisfrom Hoton Nur Lake are investigated with consideration of the assumption for a possible presence in this species of two morpho-ecological forms. The data on growth, feeding, sex, and age composition are presented. Plastic and meristic characters, distribution of fish in the water body, and feeding are analyzed. Within the age range of 3+ to 6+, in the population there are two groups of specimens differing in growth rate, diet, and some plastic characters of the head. These two groups may be designated as “predatory” and “benthos-feeding” forms. The predatory form lacks specimens of junior age groups; the benthos_feeding form lacks specimens older than seven years. Validity of these morpho-ecological groups is discussed.     

Analytical solution for a hybrid Logistic-Monod cell growth model in batch and continuous stirred tank reactor culture

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis–Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y_x/s) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology.     

Effect of dilution rate and methanol‐glycerol mixed feeding on heterologous Rhizopus oryzae lipase production with Pichia pastoris Mut+ phenotype in continuous culture

The induction using substrate mixtures is an operational strategy for improving the productivity of heterologous protein production with Pichia pastoris. Glycerol as a cosubstrate allows for growth at a higher specific growth rate, but also has been reported to be repressor of the expression from the AOX1 promoter. Thus, further insights about the effects of glycerol are required for designing the induction stage with mixed substrates. The production of Rhizopus oryzae lipase (ROL) was used as a model system to investigate the application of methanol‐glycerol feeding mixtures in fast metabolizing methanol phenotype. Cultures were performed in a simple chemostat system and the response surface methodology was used for the evaluation of both dilution rate and methanol‐glycerol feeding composition as experimental factors. Our results indicate that productivity and yield of ROL are strongly affected by dilution rate, with no interaction effect between the involved factors. Productivity showed the highest value around 0.04–0.06 h^?1, while ROL yield decreased along the whole dilution rate range evaluated (0.03–0.1 h^?1). Compared to production level achieved with methanol‐only feeding, the highest specific productivity was similar in mixed feeding (0.9 UA g‐biomass^?1 h^?1), but volumetric productivity was 70% higher. Kinetic analysis showed that these results are explained by the effects of dilution rate on specific methanol uptake rate, instead of a repressor effect caused by glycerol feeding. It is concluded that despite the effect of dilution rate on ROL yield, mixed feeding strategy is a proper process option to be applied to P. pastoris Mut⁺ phenotype for heterologous protein production. © 2015 American Institute of Chemical Engineers Biotechnol. Prog., 31:707–714, 2015     

Modeling and optimization of hairy root growth in fed‐batch process

This article proposes a feeding strategy based on a kinetic model to enhance hairy roots growth. A new approach for modeling hairy root growth is used, considering that there is no nutrient limitation thanks to an appropriate feeding, and the intracellular pools are supposed to be always saturated. Thus, the model describes the specific growth rate from extracellular concentration of the major nutrients and nutrient uptakes depend on biomass growth. An optimized feeding strategy was determined thanks to the model to maintain the major nutrient levels at their optimum assuming optimal initial concentrations. The optimal feed rate is computed in open loop using kinetic model prediction or in closed loop using conductivity measurements to estimate biomass growth. Datura innoxia was chosen as the model culture system. Shake flask cultures were used to calibrate the model. Finally, cultures in bioreactor were performed to validate the model and the control laws. © 2010 American Institute of Chemical Engineers Biotechnol. Prog., 2010     

Diffusion models for population dynamics incorporating individual behavior at boundaries: applications to refuge design.

We construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The models are based on random walks where there may be a bias in the direction an individual moves when it encounters an interface. This sort of dispersal process is called skew Brownian motion. Our models take the form of diffusion equations with matching conditions across the interface between regions for population densities and fluxes. We combine the dispersal models with linear population growth models which assume that the population growth rate differs between regions of different habitat types. We use those models to study issues of refuge design. We specifically consider how the effectiveness of buffer zones depends on their size, quality, and the population's response to the interface between the buffer zone and the refuge.     

Modeling suppression of cell death by Bcl-2 over-expression in myeloma NS0 6A1 cells

A novel population-balance model was employed to evaluate the suppression of cell death in myeloma NS0 6A1 cells metabolically engineered to over-express the apoptotic suppressor Bcl-2. The model is robust in its ability to simulate cell population dynamics in batch suspension culture and in response to thymidine-induced growth inhibition: 89% of simulated cell concentrations are within two standard deviations of experimental data. Kinetic rate constants in model equations suggest that Bcl-2 over-expression extends culture longevity from 6 days to at least 15 days by suppressing the specific rate of early apoptotic cell formation by more than 6-fold and necrotic cell formation by at least 3-fold, despite nearly a 3-fold decrease in initial cell growth rate and no significant change in the specific rate of late apoptotic cell formation. This computational analysis supports a mechanism in which Bcl-2 is a common mediator of early apoptotic and necrotic events occurring at rates that are dependent on cellular factors accumulating over time. The model has current application to the rational design of cell cultures through metabolic engineering for the industrial production of biopharmaceuticals.