A model for energy-sufficient culture growth

Pirt's maintenance model has been widely accepted for the effects of growth rate and maintenance on growth yield. However, the interpretation of parameters in Pirt's model as biological constants is difficult for energy-sufficient culture growth. In this study, a mechanistic model for the growth energetics of energy-sufficient chemostat cultures is proposed and verified with literature data. In the model, the overutilization of the energy substrate in energy-sufficient culture growth is attributed to the defective regulation of the energy substrate metabolism and energy uncoupling. The model also uses an "energy surplus" concept to collectively represent the effects of energy excessiveness. The proposed model provides a better quantitative understanding of the maximum growth yield and maintenance of energy-sufficient cultures. It also explains the glucose concentration effect reported in the literature.     

The estimation of growth yield and maintenance parameters for photoautotrophic growth

Methods are presented for examining the consistency of experimental data for microbial growth where light energy is converted to chemical energy through photosynthesis. True growth yield and maintenance parameters are estimated for several sets of available experimental data. Methods of parameter estimation are presented which allow all of the measured variables to be used simultaneously for parameter estimation. The results show that a wide range of values have been found for the true growth yield and maintenance parameters. Values of the true growth yield range from 0.04 to values above those predicted by the Z-scheme model for photosynthesis.     

Effect of incorrectly estimated parameters on the control of specific growth rate inE. coli fed-batch fermentation

An exponential feeding strategy has been frequently used in fed-batch fermentation of recombinantE. coli. In this feeding scheme, growth yield and initial cell concentration, which can be erroneously determined, are needed to calculate the feed rate for controlling specific growth rate at the set point. The effect of the incorrect growth yield and initial cell concentration on the control of the specific growth rate was theoretically analyzed. Insignificance of the correctness of those parameters for the control of the specific growth rate was shown theoretically and experimentally.     

Physiological description of multivariate interdependencies between process parameters,morphology and physiology during fed‐batch penicillin production

Optimization of productivity and economics of industrial bioprocesses requires characterization of interdependencies between process parameters and process performance. In the case of penicillin production, as in other processes, process performance is often closely interlinked with the physiology and morphology of the organism used for production. This study presents a systematic approach to efficiently characterize the physiological effects of multivariate interdependencies between bioprocess design parameters (spore inoculum concentration, pO₂ control level and substrate feed rate), morphology, and physiology. Method development and application was performed using the industrial model process of penicillin production. Applying traditional, statistical bioprocess analysis, multivariate correlations of raw bioprocess design parameters (high spore inoculum concentration, low pO₂ control as well as reduced glucose feeding) and pellet morphology were identified. A major drawback of raw design parameter correlation models; however, is the lack of transferability across different process scales and regimes. In this context, morphological and physiological bioprocess modeling based on scalable physiological parameters is introduced. In this study, raw parameter effects on pellet morphology were efficiently summarized by the physiological parameter of the biomass yield per substrate. Finally, for the first time to our knowledge, the specific growth rate per spore was described as time‐independent determinant for switching from pellet to disperse growth during penicillin production and thus introduced as a novel, scalable key process parameter for pellet morphology and process performance. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:689–699, 2014     

Efficient feeding profile optimization for recombinant protein production using physiological information

A multivariate study was performed aiming at the optimization of a recombinant rhamnose inducible E. coli induction system with alkaline phosphatase as target product. The effects of typical factors with impact on post- as well as pre-induction feeding rates were investigated with respect to the space–time yield of the target product. The goal was increased understanding as well as quantitative characterization of these factors with respect to their physiological impact on the model system. The optical density (OD) at which the culture was induced had a strong positive effect on the space–time yield. Pre-induction growth rate (k) had a second-order effect, while induction feed rate drop (J), a factor defining the linear post-induction feed rate, was interacting with (k). However, explanation of the observed effects to acquire more understanding regarding their effect on cell metabolism was not straight forward. Hence, the original process parameters were transformed into physiological more meaningful parameters and served as the basis for a multivariate data analysis. The observed variance with respect to observed volumetric activity was fully explained by the specific substrate uptake rate (q _s) and induction OD, merging the process parameters pre-induction growth rate (k) and feed rate drop (J) into the physiological parameter specific substrate uptake rate (q _s). After transformation of the response volumetric activity (U/ml) into the biomass specific activity (U/g_biomass), the observed variance was fully explained solely by the specific substrate uptake rate (q _s). Due to physiological multivariate data analysis, the interpretation of the results was facilitated and factors were reduced. On the basis of the obtained results, it was concluded that the physiological parameter q _s rather than process parameters (k, J, induction OD) should be used for process optimization with respect to the feeding profile.     

Laboratory evaluation of two bioenergetics models applied to yellow perch: identification of a major source of systematic error

Laboratory growth and food consumption data for two size classes of age 2 year yellow perch Perca flavescens , each fed on two distinct feeding schedules at 21° C, were used to evaluate the abilities of the Wisconsin (WI) and Karas–Thoresson (KT) bioenergetics models to predict fish growth and cumulative consumption. Neither model exhibited consistently better performance for predicting fish body masses across all four fish size and feeding regime combinations. Results indicated deficiencies in estimates of resting routine metabolism by both models. Both the WI and KT models exhibited errors for predicting growth rates, which were strongly correlated with food consumption rate. Consumption-dependent prediction errors may be common in bioenergetics models and are probably the result of deficiencies in parameter values or assumptions within the models for calculating energy costs of specific dynamic action, feeding activity metabolism or egestion and excretion. Inter-model differences in growth and consumption predictions were primarily the result of differences in egestion and excretion costs calculated by the two models. The results highlighted the potential importance of parameters describing egestion and excretion costs to the accuracy of bioenergetics model predictions, even though bioenergetics models are generally regarded as being insensitive to these parameters. The findings strongly emphasize the utility and necessity of performing laboratory evaluations of all bioenergetics models for assurance of model accuracy and for facilitation of model refinement.     

Fed-batch operation of recombinant Escherichia coli containing trp promoter with controlled specific growth rate

A feb-batch operation for the production of bovine somatotropin (bST) under the control of tryptophan promoter in Escherichia coli was investigated. The plasmid used contains a two-cistron system and altered codon usage for higher expression of bST. Specific growth rate is an important parameter in the fermentation, because it affects the production of growth-inhibitory organic acids and the expression of recombinant protein. The feeding rate was adjusted to keep the specific growth rate constant in this study. The variable growth yield expressed as a function of time was used for the calculation of the feeding rate. The growth yield decreases during the fermentation as product expression is induced. The specific growth rate was well controlled; however, intracellular bST concentration decreased at high cell concentrations. This is considered to be due to degradation by proteases. The decrease was prevented by an exponential feeding of the yeast extract as an organic nitrogen source. (c) 1994 John Wiley & Sons, Inc.     

Estimating Initial Epidemic Growth Rates

The initial exponential growth rate of an epidemic is an important measure of disease spread, and is commonly used to infer the basic reproduction number $\mathcal{R}_{0}$ . While modern techniques (e.g., MCMC and particle filtering) for parameter estimation of mechanistic models have gained popularity, maximum likelihood fitting of phenomenological models remains important due to its simplicity, to the difficulty of using modern methods in the context of limited data, and to the fact that there is not always enough information available to choose an appropriate mechanistic model. However, it is often not clear which phenomenological model is appropriate for a given dataset. We compare the performance of four commonly used phenomenological models (exponential, Richards, logistic, and delayed logistic) in estimating initial epidemic growth rates by maximum likelihood, by fitting them to simulated epidemics with known parameters. For incidence data, both the logistic model and the Richards model yield accurate point estimates for fitting windows up to the epidemic peak. When observation errors are small, the Richards model yields confidence intervals with better coverage. For mortality data, the Richards model and the delayed logistic model yield the best growth rate estimates. We also investigate the width and coverage of the confidence intervals corresponding to these fits.     

Application of bioenergetics to modelling the microbial conversion of D-xylose to 2,3-butanediol

During the oxygen limiting growth of Klebsiella oxytoca, the xylose metabolism may be considered as consisting of three components: conversion to 2,3-butanediol by "fermentation," oxidation to carbon dioxide by respiration, and assimilation to cell mass. The amount of energy required for the assimilation of cell mass is assumed to determine the extent to which the two energy producing reactions occur. The activity of each energy producing pathway is also determined by the availability of oxygen and by the energy yield of each pathway. These relationships can be quantified by equating the ATP required for growth and maintenance to the ATP produced by the energy producing reactions. The resulting equation for butanediol production appears similar to the Luedeking and Piret model where the parameters alpha and beta are related to the maximum cell yield from ATP and the maintenance energy requirement. These parameters were estimated from 14 batch fermentations, and the resulting simulation was used to describe the effects of the oxygen transfer rate and the initial xylose concentration on the yields and rates of the 2,3-butanediol fermentation.     

A new mathematical model for combining growth and energy intake in animals: The case of the growing pig

A simultaneous model for analysis of net energy intake and growth curves is presented, viewing the animal's responses as a two dimensional outcome. The model is derived from four assumptions: (1) the intake is a quadratic function of metabolic weight; (2) the rate of body energy accretion represents the difference between intake and maintenance; (3) the relationship between body weight and body energy is allometric and (4) animal intrinsic variability affects the outcomes so the intake and growth trajectories are realizations of a stochastic process. Data on cumulated net energy intake and body weight measurements registered from weaning to maturity were available for 13 pigs. The model was fitted separately to 13 datasets. Furthermore, slaughter data obtained from 170 littermates was available for validation of the model. The parameters of the model were estimated by maximum likelihood within a stochastic state space model framework where a transform-both-sides approach was adopted to obtain constant variance. A suitable autocorrelation structure was generated by the stochastic process formulation. The pigs’ capacity for intake and growth were quantified by eight parameters: body weight at maximum rate of intake (149-281 kg); maximum rate of intake (25.7-35.7 MJ/day); metabolic body size exponent (fixed: 0.75); the daily maintenance requirement per kg metabolic body size (0.232-0.303 MJ/(day×kg^0.75)); reciprocal scaled energy density ; a dimensional exponent, θ₆ (0.730-0.867); coefficient for animal intrinsic variability in intake (0.120-0.248 MJ^0.5) and coefficient for animal intrinsic variability in growth (0.029-0.065 kg^0.5). Model parameter values for maintenance requirements and body energy gains were in good agreement with those obtained from slaughter data. In conclusion, the model provides biologically relevant parameter values, which cannot be derived by traditional analysis of growth and energy intake data.     

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     

Estimation of true growth and product yields in aerobic cultures

In the microbial production of useful products, it is important to understand the allocation of substrate energy for maintanance, growth, and product formation. Methods are presented to obtain point and 95% confidence interval estimates for the true growth yield parameter, true product yield parameter, and the maintenance parameter. Methods are presented which allow all data to be used simultaneously for those cases where more than the minimum number of measurements are made at each specific growth rate (or dilution rate). Three estimation methods and two forms of the energy allocation equations are investigated. Point estimates are similar for the three methods, but interval estimates are considerably larger for one of the three methods. The results depend on the form of the equations.     

Kinetics of chitinase production. II. Relationship between bacterial growth, chitin hydrolysis and enzyme synthesis

A comprehensive model for chitinase production during growth of Serratia marcescens QMB 1466 on chitin was developed taking into account the rate of chitin hydrolysis in order to estimate the rate of bacterial growth. In relating growth with enzyme synthesis the total enzyme concentration was used as the sum of the enzyme present in the bulk of the fermentation broth and the enzyme adsorbed on the chitin particles. The equations constituting the proposed model were fitted to the experimental results from both continuous and batch fermentation to obtain parameters describing substrate yield, metabolic maintenance, and enzyme yields.     

Plant growth and respiration re-visited: maintenance respiration defined - it is an emergent property of, not a separate process within, the system - and why the respiration : photosynthesis ratio is conservative

Background and Aims

Plant growth and respiration still has unresolved issues, examined here using a model. The aims of this work are to compare the model''s predictions with McCree''s observation-based respiration equation which led to the ‘growth respiration/maintenance respiration paradigm’ (GMRP) – this is required to give the model credibility; to clarify the nature of maintenance respiration (MR) using a model which does not represent MR explicitly; and to examine algebraic and numerical predictions for the respiration:photosynthesis ratio.

Methods

A two-state variable growth model is constructed, with structure and substrate, applicable on plant to ecosystem scales. Four processes are represented: photosynthesis, growth with growth respiration (GR), senescence giving a flux towards litter, and a recycling of some of this flux. There are four significant parameters: growth efficiency, rate constants for substrate utilization and structure senescence, and fraction of structure returned to the substrate pool.

Key Results

The model can simulate McCree''s data on respiration, providing an alternative interpretation to the GMRP. The model''s parameters are related to parameters used in this paradigm. MR is defined and calculated in terms of the model''s parameters in two ways: first during exponential growth at zero growth rate; and secondly at equilibrium. The approaches concur. The equilibrium respiration:photosynthesis ratio has the value of 0·4, depending only on growth efficiency and recycling fraction.

Conclusions

McCree''s equation is an approximation that the model can describe; it is mistaken to interpret his second coefficient as a maintenance requirement. An MR rate is defined and extracted algebraically from the model. MR as a specific process is not required and may be replaced with an approach from which an MR rate emerges. The model suggests that the respiration:photosynthesis ratio is conservative because it depends on two parameters only whose values are likely to be similar across ecosystems.     

Modeling of microbial substrate conversion,growth and product formation in a recycling fermentor

Paracoccus denitrificans and Bacillus licheniformis were grown in a carbon- and energy source-limited recycling fermentor with 100% biomass feedback. Experimental data for biomass accumulation and product formation as well as rates of carbon dioxide evolution and oxygen consumption were used in a parameter optimization procedure. This procedure was applied on a model which describes biomass growth as a linear function of the substrate consumption rate and the rate of product formation as a linear function of the biomass growth rate. The fitting procedure yielded two growth domains for P. denitrificans. In the first domain the values for the maximal growth yield and the maintenance coefficient were identical to those found in a series of chemostat experiments. The second domain could be described best with linear biomass increase, which is equal to a constant growth yield. Experimental data of a protease producing B. licheniformis also yielded two growth domains via the fitting procedure. Again, in the first domain, maximal growth yield and maintenance requirements were not significantly different from those derived from a series of chemostat experiments. Domain 2 behaviour was different from that observed with P. denitrificans. Product formation halts and more glucose becomes available for biomass formation, and consequently the specific growth rate increases in the shift from domain 1 to 2. It is concluded that for many industrial production processes, it is important to select organisms on the basis of a low maintenance coefficient and a high basic production of the desired product. It seems less important that the maximal production becomes optimized, which is the basis of most selection procedures.     

The effects of group size, leaf size, and density on the performance of a leaf-mining moth

1. The effects of two factors, leaf size and group size, on the performance of the Tupelo leafminer, Antispila nysaefoliella (Lepidoptera: Heliozelidae), were examined by fitting growth models to mine expansion data using nonlinear mixed-effects models. 2. The rate of mine expansion served as a proxy for larval performance because of its correlation with both feeding activity and growth rate and is also the means by which a larva achieves its final mine size (or total consumption). 3. Leaf size was used as a measure of resource availability, and was expected to reduce the impact of resource competition and enhance larval performance. 4. In contrast to the unidirectional effects expected for leaf size (i.e. more resources should enhance performance), the direction for the effects of group size was expected to depend on the mechanism(s) driving the effect. For example, if there is resource competition among larvae in a group, then this could increase the feeding rates of some larvae or reduce the total consumption of others. However, if leaf mining induces host plant chemical defences, then larger groups might elicit a greater defensive response by the host plant (at the leaf), and hence, be characterized by reduced feeding and growth rates. 5. To investigate these interactions, two growth models, the Gompertz model and a modified version of the von Bertalanffy growth equation, were fitted to time series of the sizes of individual leaf mines using nonlinear mixed-effects models. Linear and nonlinear associations of each factor (group size or leaf size) with model parameters were then evaluated using a hierarchical testing procedure by determining: (i) whether inclusion of the factor produced a better-fit model, and (ii) if it did, the form of that relationship (i.e. linear or nonlinear). 6. Three patterns were detected with these analyses. (i) Leaf size had a significant positive, linear relationship with mine expansion rate. (ii) Group size had a significant quadratic relationship with mine expansion rate. (iii) The effects of leaf and group size on the maximum mine size were opposite to those found with growth rate.     

A robust fed-batch feeding strategy for optimal parameter estimation for baker's yeast production

Parameter identification of structured models is often a problem in biotechnology, because the poor data situation and the number of unknown parameters only allow for inaccurate estimates. But often only a subset of all kinetic parameters of the model are of interest for production purposes, e.g. for fed-batch cultivation. These parameters should be estimated with a given accuracy. In addition, the experiments for information acquisition with respect to these parameters should be as simple as possible and should consider some practical restrictions. In this contribution a fed-batch feeding strategy is proposed to allow for an accurate estimation of yield and of critical growth rate of baker's yeast. The feeding also allows for economic and stereotyped use of staff and equipment and is therefore suitable for routine use in screening of strains and media. The overall pattern is similar to that one, usually used in production scale to minimize errors by limited model validity. After an initial phase for achieving a reproducible state three different growth rates are adjusted to cover the range of possible critical growth rates. From biomass and ethanol measurements yield and critical growth rate can be estimated with an accuracy of about 2.1%. The fermentation pattern ends up with a constant feeding rate to simulate a limited oxygen transfer rate and to allow for an uptake of residual sugar and ethanol before a dough test can be carried out. Beside experimental results simulations and sensitivity analyses are shown.List of Symbols P ethanol concentration - S substrate concentration - S _f substrate concentration in feed - T fermentation time - V fermenter volume - X biomass concentration - C measurement error covariance matrix - F Fisher information matrix - X state variables - Y output variables - X _p state sensitivity functions with respect to parameters - Y _p output sensitivity functions - e eigenvectors - k vector of limitation and inhibition parameters - n number of observations - q _in feeding stream - q _b stream for samples and ammonia feed - r vector of specific turnover rates - y vector of yields - specific weight - eigenvalues - specific growth rate - _set exponent in exponential feeding - standard deviation Dedicated to the 65th birthday of Professor Fritz Wagner.A. O. Ejiofor and B. O. Solomon are grateful to the Alexander von Humboldt Stiftung for granting them fellowships and to GBF for providing all the materials necessary for their successful research stay in Germany.     

Maximum likelihood estimation of growth yields

This work is concerned with statistical methods to estimate yield and maintenance parameters associated with microbial growth. For a given dilution rate, an experimenter typically measures substrate concentration, oxygen utilization rate, the rate of carbon dioxide evolution, and biomass concentration. These correlated response variables each contain information about the maintenance and yield parameters of interest. A maximum likelihood estimator which combines this correlated information for the yield and maintenance parameters is proposed, evaluated, and tested on literature data. Both point and interval estimators are considered.     

What the hen can tell about her eggs: egg development on the basis of energy budgets

By a simple model involving the state variables size and storage, it is possible to describe a wide variety of observations on the feeding, growth, energy storage and reproduction of animals. The model is based on the assumption that reproduction, growth as well as maintenance depend on the stored energy only and not directly on feeding. If an egg is thought of as a non-feeding animal, the model predicts the respiration ontogeny and growth of the embryo inside the egg. These predictions seem to hold well for published data on the development of eggs of fish and ratite, precocial and altricial birds. The latter two are known to follow different respiration ontogenies, but both are described well, differing only in one (compound) parameter value. The model explains why the incubation times of eggs of different species tend to increase linearly with egg size to the power 1/4, and why kiwis and petrels, which lay relatively large eggs, have to brood them much longer than larger birds with eggs of the same size. Conversely, it explains why the small eggs of the (parasitic) European cuckoo, hatch earlier than the still smaller eggs of their tiny hosts.Furthermore, it has been shown how the maintenance rate constant, which frequently appears in the microbial literature, can be obtained from measurements on the respiration and weight ontogeny in embryos, so linking independent lines of research. Application of the model shows an increase of the maintenance rate constant from bacteria, crustaceans, up to fish and birds, and a decrease from bacteria to green algae, suggesting lines of evolutionary development.     

Feeding under predation hazard: testing models of adaptive behavior with stream fish

Many empirical studies support the premise that animals consider both the benefits of feeding and the cost of mortality when making behavioral decisions, and many theoretical studies predict animal behavior in the presence of a feeding-mortality trade-off. However, empirical work is lacking in studies that quantitatively assess alternative models. Using data from two sets of behavioral experiments examining stream minnows (bluehead chubs) foraging in the presence of sunfish predators (green sunfish), we assess, via statistical model fitting, the utility of four basic optimization models of foraging behavior. Our analysis of feeding and mortality of the minnows indicates that mortality is incurred so as to feed above maintenance requirements, that feeding rate is suppressed in response to the presence of predators, and that the balance of feeding against mortality can be estimated using a life-history parameter theta, interpreted theoretically as the marginal rate of substitution of mortality rate for growth rate. Our results indicate that both body size and age are probably involved in determining the value of theta, and we suggest that future studies should focus on estimating and understanding this parameter.