Modelling Bacterial Growth of Lactobacillus curvatus as a Function of Acidity and Temperature

Models that describe the effect of acidity, temperature, and the combined effect of these variables on the growth parameters of Lactobacillus curvatus are developed and validated. Growth parameters (lag time, specific growth rate, and maximum population density) were calculated from growth data at different temperature-acidity combinations. Experiments were set up to assess the quantitative effects of temperature and acidity on the growth parameters rather than for parameter estimation solely. The effect of acidity is monitored at several constant temperature values. Models are set up and fitted to the data. The same procedure is used at constant acidity values to model the effect of temperature. For lag time, specific growth rate, and maximum population density, the effect of temperature could be multiplied with the effect of acidity to obtain combinatory models that describe the effect of both controlling factors on the growth parameters. Lag time measurements showed large deviations, and therefore the lag time models developed can only be used to estimate the order of magnitude of lag time.     

Determinants and rate laws of growth and death of hybridoma cells in continuous culture

Experimental data from six hybridoma cell lines grown under diverse experimental conditions in both normal continuous and perfusion cultures are analyzed with respect to the significance of nutrients and products in determining the growth and death rates of cells and with respect to their mathematical modeling. It is shown that neither nutrients (glucose and glutamine) nor the common products lactic acid, ammonia, and monoclonal antibody can be generally assumed to be the clear-limiting or inhibiting factors for most of the cultures. Correspondingly, none of the unstructured models existing in the literature can be generally applied to describe the experimental data obtained over a relatively wide range of cultivation conditions as considered in this work. Surprisingly, for all cultures the specific growth rate (mu) almost linearly correlates with the ratio of the viable cell concentration (NV) to the dilution (perfusion) rate (D). Similarly, the specific death rate (kd) is a function of the ratio of the total cell concentration (Nt) to the dilution (perfusion) rate. These results strongly suggest the formation of not yet identified critical factors or autoinhibitors that determine both the growth and death rates of hybridoma cells. Based on these observations, simple kinetic models are developed for mu and kd which describe the experimental data satisfactorily. Analysis of the experimental data with the kinetic models reveals that under the current cultivation conditions the formation rate of the autoinhibitor(s) or the sensitivity of cell growth and death to the autoinhibitor(s) is mainly affected by the medium composition. Irrespective of the cell lines, cells grown on serum-containing media have almost the same model parameters, which are distinctively different from those of cells grown on serum-free media. Furthermore, in contrast to the prevailing view, kd is shown to positively correlate with mu if the effects of cell concentration and dilution (perfusion) rate are considered. Several important implications of these findings are discussed for the optimization and control of animal cell culture.     

Modeling and testing treated tumor growth using cubic smoothing splines

Human tumor xenograft models are often used in preclinical study to evaluate the therapeutic efficacy of a certain compound or a combination of certain compounds. In a typical human tumor xenograft model, human carcinoma cells are implanted to subjects such as severe combined immunodeficient (SCID) mice. Treatment with test compounds is initiated after tumor nodule has appeared, and continued for a certain time period. Tumor volumes are measured over the duration of the experiment. It is well known that untreated tumor growth may follow certain patterns, which can be described by certain mathematical models. However, the growth patterns of the treated tumors with multiple treatment episodes are quite complex, and the usage of parametric models is limited. We propose using cubic smoothing splines to describe tumor growth for each treatment group and for each subject, respectively. The proposed smoothing splines are quite flexible in modeling different growth patterns. In addition, using this procedure, we can obtain tumor growth and growth rate over time for each treatment group and for each subject, and examine whether tumor growth follows certain growth pattern. To examine the overall treatment effect and group differences, the scaled chi-squared test statistics based on the fitted group-level growth curves are proposed. A case study is provided to illustrate the application of this method, and simulations are carried out to examine the performances of the scaled chi-squared tests.     

The tubular loop fermentor: Oxygen transfer,growth kinetics and design

Oxygen transfer measurements using a dynamic method and evaluated with an appropriate mathematical model have been made on a tubular loop bioreactor. Correlations of the type used in tank systems are used to describe the influence of power and aeration rate on the mass transfer coefficient. Yeast cultures grown on hydrocarbon and glucose substrates show growth characteristics similar to conventional tank results. Model considerations for large-scale tubular fermentors allow for the prediction of the steady-state oxygen profiles and maximum reactor length. Combination with two-phase flow and oxygen transfer correlations yields a design procedure for commercial scale tubular loop fermentors.     

Mapping quantitative trait loci for longitudinal traits in line crosses

Quantitative traits whose phenotypic values change over time are called longitudinal traits. Genetic analyses of longitudinal traits can be conducted using any of the following approaches: (1) treating the phenotypic values at different time points as repeated measurements of the same trait and analyzing the trait under the repeated measurements framework, (2) treating the phenotypes measured from different time points as different traits and analyzing the traits jointly on the basis of the theory of multivariate analysis, and (3) fitting a growth curve to the phenotypic values across time points and analyzing the fitted parameters of the growth trajectory under the theory of multivariate analysis. The third approach has been used in QTL mapping for longitudinal traits by fitting the data to a logistic growth trajectory. This approach applies only to the particular S-shaped growth process. In practice, a longitudinal trait may show a trajectory of any shape. We demonstrate that one can describe a longitudinal trait with orthogonal polynomials, which are sufficiently general for fitting any shaped curve. We develop a mixed-model methodology for QTL mapping of longitudinal traits and a maximum-likelihood method for parameter estimation and statistical tests. The expectation-maximization (EM) algorithm is applied to search for the maximum-likelihood estimates of parameters. The method is verified with simulated data and demonstrated with experimental data from a pseudobackcross family of Populus (poplar) trees.     

Testing differences in relative growth rate: A method avoiding curve fitting and pairing

A method is discussed to test differences in relative growth rates. This method is based on an analysis of variance, with In-transformed plant weight as dependent variable. A significant Group × Time interaction indicates differences in relative growth rates between groups. The advantages over the "classical" and "functional" growth analyses are: (1) No pairing procedure is required. (2) More than two groups may be evaluated in one analysis. (3) No decision is required about the polynomial used to fit the data. (4) By partitioning the interaction effect using orthogonal polynomials insight is gained into the nature of differences in relative growth rate. (5) By concentrating attention on the lower order terms of the polynomials, the influence of extraneous variation on conclusions may be minimized.     

Kinetics and stoichiometry of growth of plant cell cultures of Catharanthus roseus and Nicotiana tabacum in batch and continuous fermentors

Plant cell suspension cultures of Catharanthus roseus and Nicotiana tabacum were grown in stirred tank bioreactors operated in batch and continuous mode. The stoichiometry of growth of both species in steady-state glucose limited chemostats was studied at a range of different dilution rates. A linear relation was applied to describe specific glucose uptake, oxygen consumption, and carbon dioxide production as a function of the growth rate. Specific respiration deviated greatly from the linear relation. An unstructured mathematical model, based on the observed stoichiometry in the glucose limited chemostats, was applied to describe the growth in batch culture. From a comparison between the observed growth pattern in batch fermentors and computer simulations it appeared that the stoichiometry of growth of the C. roseus culture was different under steady-state and dynamic conditions. It was concluded that a mathematical model for the growth of suspension culture plant cells in which the biomass is considered to be a single compound with an average chemical composition is of limited value because large changes in the conmposition of the biomass may occur. (c) 1992 John Wiley & Sons, Inc.     

Adjusted Prediction of Plant Height with Application to Greenhouse Grown Poinsettias

The objective is to predict future plant growth using data from greenhouse grown poinsettias. A population‐mean growth trajectory can be predicted from growth conditions using previously published results. Four different predictors are used to identify the local group‐effects that come in addition to the population‐mean response to growth conditions, in order to improve the predictions of the final plant height for specific plant groups. A search for optimal model complexity and use of growth history during calibration is conducted using cross‐validation.     

Growth curve models of repeated binary response

Experimental designs that include repeated measures of binary response variables over time and under different conditions are common in biology. In such settings, it is often desirable to characterize the response pattern over time. When response variables are continuous, this characterization can be made in terms of a growth model such as the Potthoff-Roy growth curve model. We illustrate how a similar growth curve modeling strategy can be implemented using weighted least squares (WLS) methods for binary response data. The growth models are constructed in terms of polynomial functions across marginal response. However, when growth models are fit to repeated binary response, the nonsignificant higher-order polynomial functions are dropped from the model, rather than used as covariates. Dropping the nonsignificant polynomials from the model will reduce the number of response functions, and help avoid small-sample problems that can occur when the number of correlated response functions is large and sample sizes are small. The reduced set of response functions are then modeled using WLS methods. We illustrate such models with an example of binary fly oviposition response (accept or reject) exhibited by two populations of flies at four ages to two types of fruit.     

用遗传算法进行重组大肠杆菌发酵动力学的分析

重组大肠杆菌在诱导表达人表皮生长因子的过程促使细菌的生长受到抑制,一部分重组菌丧失了分裂能力,但仍保持着一定的代谢活力,分离成为存活但不能培养的细菌,根据大肠杆菌在表达外源蛋白过程中细胞生理状态的不同将细菌分为三类,提出一个描述诱导表达过程中重组大肠杆菌分离、生长的动力学模型．应用遗传算法对不同底物浓度的细胞生长、分离和产物合成的动力学参数进行了有效地估计,避免了传统算法可能陷于局部最优的问题,模型计算结果与实验结果吻合良好．分离模型在初始糖浓为5-20g/L的范围内可以较好地描述发酵过程中细胞生长、分离和目标产物表达的过程并具有一定的预测能力．     

Grain Growth Dynamics in Winter Wheat Crops

A procedure for analysing the dynamics of grain growth is developedand applied to data from winter wheat crops. The algorithm estimatesgrowth rate in relation to local data and preserves a shapethrough the whole growth curve by optimization subject to physiologically-justifiedconstraints: that grain growth rate decreases monotonically,and that there is a discontinuity to zero rate at physiologicalmaturity. It is assumed that effects of temperature can be allowedfor by expressing growth in relation to thermal time, accumulatedtemperature above a base temperature. Conditions are derivedunder which an optimal base temperature for grain growth rateexists under these physiological constraints. When applied todata from crops with different sowing dates and nitrogen treatmentsin different years, weak optimal base temperatures of 2.0 and7.5 °C were obtained for growth rate and duration, respectively.Initial growth rates were similar for all crops, but there wereconsiderable variations in rate later in grain growth. Thesevariations were, in part, related to the overall crop growthrate divided by the number of grains. Grain growth, modelling, winter wheat     

Modelling zinc uptake by rice crop using a Barber-Cushman approach

The Barber-Cushman mechanistic nutrient uptake model which has been utilized extensively to describe and predict nutrient uptake by crop plants at different stages of crop growth was evaluated for its ability to predict the Zn uptake by rice seedlings. Uptake of the nutrient is, therefore, determined by the rate of nutrient supply to the root surface by mass flow and diffusion. Inter root competition and time dependent root density are accounted for by soil volume that delivers nutrients. The radii of these cylinders decline with increasing density. Since mass flow and diffusion each supply zinc to the root, the process can be described mathematically using the model of Barber-Cushman (1984). The 11 parameters of the model for the uptake by rice cultivars were measured by established experimental techniques. Zinc uptake at different growth stages predicted by the model was compared to measured zinc uptake by rice cultivars grown on sandy loam soil in a green house. Predicted zinc uptake was significantly correlated with observed uptake r ²=0.99^**. Sensitivity analysis was also used to investigate the impact of changes in soil nutrient supply, root morphological and root uptake kinetic parameters on simulated nutrient uptake. Overall results of sensitivity analysis indicate that the half distance between root axes, rate of root growth and water flux affect the uptake of zinc particularly at their higher values rather than at lower values and DaZn is the most sensitive parameter for zinc uptake at its lower values.     

A physiological analogy of the niche for projecting the potential distribution of plants

Aim  To develop a physiologically based model of the plant niche for use in species distribution modelling. Location  Europe. Methods  We link the Thornley transport resistance (TTR) model with functions which describe how the TTR’s model parameters are influenced by abiotic environmental factors. The TTR model considers how carbon and nutrient uptake, and the allocation of these assimilates, influence growth. We use indirect statistical methods to estimate the model parameters from a high resolution data set on tree distribution for 22 European tree species. Results  We infer, from distribution data and abiotic forcing data, the physiological niche dimensions of 22 European tree species. We found that the model fits were reasonable (AUC: 0.79–0.964). The projected distributions were characterized by a false positive rate of 0.19 and a false negative rate 0.12. The fitted models are used to generate projections of the environmental factors that limit the range boundaries of the study species. Main conclusions  We show that physiological models can be used to derive physiological niche dimensions from species distribution data. Future work should focus on including prior information on physiological rates into the parameter estimation process. Application of the TTR model to species distribution modelling suggests new avenues for establishing explicit links between distribution and physiology, and for generating hypotheses about how ecophysiological processes influence the distribution of plants.     

Thermophilic bacterium Geobacillus uralicus growth is a function of temperature and pH: a synthetic chemostat model-based kinetic analysis

The synthetic chemostat model (SCM), originally developed to describe nonstationary growth under widely varying concentrations of the limiting substrate, was modified to account for the effects of nontrophic factors such as temperature and pH. The bacterium Geobacillus uralicus, isolated from an ultradeep well, was grown at temperatures ranging from 40 to 75 degrees C and at pH varying from 5 to 9. The biomass kinetics was reasonably well described by the SCM, including the phase of growth deceleration observed in the first hours after a change in the cultivation temperature. In an early stage of batch growth in a neutral or alkalescent medium, bacterial cells showed reversible attachment to the glass surface of the fermentation vessel. The temperature dependence of the maximum specific growth rate (micron) was fitted using the equation micron = Aexp(lambda T)/[1 + expB[1-C/(T + 273)]], where A, lambda, B, and C are constants. The maximum specific growth rate of 2.7 h-1 (generation time, 15.4 min) was attained on a complex nutrient medium (peptone and yeast extract) at 66.5 degrees C and pH 7.5. On a synthetic mineral medium with glucose, the specific growth rate declined to 1.2 h-1 and the optimal temperature for growth decreased to 62.3 degrees C.     

A Population-Based Morphologically Structured Model for Hyphal Growth and Product Formation in Streptomycin Fermentation

Summary A population model discriminating the hyphae according to the hyphal length and a morphologically structured model considering the specific function of different morphological forms of a hypha are combined together to describe mycelial growth, substrate consumption and secondary metabolite formation in streptomycin fermentation. In the population model, the growth modes of hyphae with different age or length are considered, while in the morphologically structured model, the morphological forms of hyphae and their functions in growth and metabolism are described. The population model and the morphologically structured model are interrelated by a branching function and a differentiation function. In the model, the growth rate of immature apical compartment is distinguished from those of matured ones, branching is proposed to occur only in the subapical region, and the hyphal compartment is assumed to synthesize secondary metabolites. The model is successfully applied to simulate the batch fermentation process of streptomycin production. The growth characteristics of filamentous microorganisms are also discussed using the model predictions.     

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.     

On the use of apparent kinetic parameters for immobilized enzyme with noncompetitive substrate inhibition

The use of a simple rate equation with apparent parameters to describe the kinetic behavior of an immobilized enzyme with noncompetitive substrate inhibition was assessed. To do so, the reaction rate was calculated as a function of the interfacial substrate concentration, and the results were used to identify the apparent kinetic parameters by nonlinear regression. This procedure was repeated for different values of the diffusional constraints and of the inhibition constant. The equation using apparent parameters can describe the global kinetic behavior, provided that the diffusional and inhibitory constraints are not too high. When the constraints are high, a Michaelis-Menten equation can be used to model the kinetics for interfacial concentrations lower than the concentration leading to the maximum reaction rate.     

Modelling conjugation with stochastic differential equations

Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared to the model without plate conjugation. The modelling approach described in this article can be applied generally when modelling dynamical systems.