Phenotypic and genetic variability of estimated growth curve parameters in mice

Summary Data from 1,919 outbred ICR mice were used to examine the potential usefulness of growth curve parameters as selection criteria for altering the relationship between body weight and age. A logistic growth function was used to model growth through 12 weeks of age. Estimates of asymptotic weight (A), maximum growth rate (r) and age at point of inflection (t^*) were obtained by nonlinear least-squares. A log transformation was also used to stabilize residual variance. Phenotypic and genetic parameters were estimated for the estimated growth curve parameters and for body weights at 2, 3, 4.5, 6, 8 and 12 weeks of age. Heritabilities of estimated growth curve parameters (obtained with and without a log transformation, respectively) were: A (0.28±0.07, 0.28±0.07), r (0.35±0.07, 0.53±0.09) and t^* (0.41±0.08, 0.44±0.08). Estimated genetic correlations suggest that t^* may be useful in selecting for rapid early growth without increasing mature weight.     

Kinetics and modeling of temperature effects on batch xanthan gum fermentation

Batch fermentation kinetics of xanthan gum production from glucose by Xanthomonas campestris at temperatures between 22 degrees C and 35 degrees C were studied to evaluate temperature effects on cell growth and xanthan formation. These batch xanthan fermentations were modeled by the logistic equation for cell growth, the Luedeking-Piret equation for xanthan production, and a modified Luedeking-Piret equation for glucose consumption. Temperature dependence of the parameters in this model was evaluated. Growth-associated rate constants increased to a maximum at approximately 30 degrees C and then decreased to zero at approximately 35 degrees C. This temperature effect can be modeled using a square-root model. On the contrary, non-growth-associated rate constants increased with increasing temperature, following the Arrhenius relationship, in the entire temperature range studied. The model developed in this work fits the experimental data very well and can be used in a simulation study. However, due to the empirical nature of the model, the parameter values need to be reevaluated if the model is to be applied to different growth conditions.     

Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics

Aims:  To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants.
Methods and Results:  The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics.
Conclusion:  It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant ( k ) is directly proportional to the maximum specific growth rate constant ( μ _m) and initial substrate concentration ( S ₀) and also inversely related to the saturation constant ( K _s).
Significance and Impact of the Study:  The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.     

A mathematical model of the food-consumer system I. A case without food replenishment

A simple mathematical model was proposed to describe the dynamics of a food-consumer system. The model was based on the Logistic Theory and consisted of Eqs. (4), (5) and (6). The model was divided into the following three cases for further analyss; i) without food supply except at the initial time, ii) with continuous food supply at a constant rate, and iii) with food supply at varying rates. Only the first model was dealth with in this paper. The assumptions of the model 1 are that a definite amount of food is given only once at the initial time and only the feeding by animals is responsible for the decrease of food, and that the rate of decrease is proportional to the amount of animals. It is also assumed that the growth of animal population is represented by the logistic curve, and that the upper limit of the population is proportional to the amount of food at that time. For simplicity the parameters of basic differential equations are assumed to be constant throughout the time course. Analytical solutions of this non-linear model were given by Eqs. (8), (9), (10) and (11). The properties of time course of the food amount and consumer population were discussed from the mathematical and biological points of view. The method of the estimation of the three constants λ,b, and c from the experimental data was also suggested. Since we had no available data for animal populations, we applied the model, regarding reserve substance as x and new plant body as y, to the data of the initial growth of Azuki bean plant in the dark. This model is very simple, but it may be useful for analyzing the behavior of food-consumer system. And it may give some clue to the analysis of the more complex systems.     

Modelling avian growth with the Unified‐Richards: as exemplified by wader‐chick growth

Postnatal growth in birds is traditionally modelled by fitting three‐parameter models, namely the logistic, the Gompertz, or the von Bertalanffy models. The purpose of this paper is to address the utility of the Unified‐Richards (U‐Richards) model. We draw attention to two forms of the U‐Richards and lay down a set of recommendations for the analysis of bird growth, in order to make this model and the methods more accessible. We examine the behaviour of the four parameters in each model form and the four derived measurements, and we show that all are easy to interpret, and that each parameter controls a single curve characteristic. The two parameters that control the inflection point, enable us to compare its placement in two dimensions, 1) inflection value (mass or length at inflection) and 2) inflection time (time since hatching), between data sets (e.g. between biometrics or between species). We also show how the parameter controlling growth rate directly presents us with the relative growth rate at inflection, and we demonstrate how one can compare growth rates across data sets. The three traditional models, where the inflection value is fixed (to a specific percentage of the upper asymptote), provide incompatible growth‐rate coefficients. One of the two forms of the U‐Richards model makes it possible to fix not only the upper asymptote (adult value), but also the intersection with the y‐axis (hatching value). Fitting the new model forms to data validates the usefulness of interpreting the inflection placement in addition to the growth rate. It also illustrated the advantages and limitations of constraining the upper asymptote (adult value) and the y‐axis intersection (hatching value) to fixed values. We show that the U‐Richards model can successfully replace some of the commonly used growth models, and we advocate replacing these with the U‐Richards when modelling bird growth.     

A semicontinuous culture model that links cell growth to extracellular nutrient concentration

During semicontinuous culture, a sample of fixed volume is removed at regular time intervals to make measurements and/or harvest culture components, and an equal volume of fresh medium is immediately added to the culture, thereby instantaneously enhancing nutrient concentrations and diluting cell concentration. The resulting cell concentration versus time curve (i.e., the actual cell growth curve) has a saw-toothed appearance because of the periodic dilution of cell concentration. The observed cell concentrations correspond to the peaks of the saw-toothed curve. Cell growth rates are estimated from the locus of observed cell concentrations (i.e., from the apparent growth curve obtained by connecting the peaks of the saw-toothed curve). The sole preexisting model (Fencl's mode) for estimating cell growth rate is valid only when the cells are growing exponentially at a constant rate between samplings. This model has limited validity: despite the periodic enhancement of nutrient concentration, cell growth between samplings eventually causes nutrient depletion, and the cells cease to grow exponentially. Failure to recognize the limits of validity for Fencl' model has resulted in many erroneous applications of the model and, consequently, many incorrect estimates of cell growth rates. To provide a means for correctly estimating cell growth rates, Fencl's exponential model was extended, and a new model that describes the effects of nutrient depletion on cell growth in semi-continuous culture was obtained. The new model shows that exhaustion of a single growth-limiting nutrient in semicontinuous culture causes the locus of cell concentrations observed at time intervals of Deltat to follow a logistic growth curve. The actual cell growth rate was shown to equal the apparent logistic growth rate plus the effective dilution rate -Deltat(-1) In (1 - f), where f is the ratio of sample volume to total culture volume. Moreover, the model predicts that both the apparent logistic growth rate and the apparent steady-state cell concentration should rise linearly with the concentration of growth-limiting nutrient in the input medium, but fall linearly with increases in the effective dilution rate. The new logistic model for nutrient-limited cell growth in semicontinuous culture was successfully tested using published data for Asterionella formosa, Cyclotella meneghiniana, Daucus carota, and strain L mouse cells.     

A possible mechanism of metabolic regulation in Gibberella fujikuroi using a mixed carbon source of glucose and corn oil inferred from analysis of the kinetics data obtained in a stirrer tank bioreactor

A nonstructured model was used to study the dynamics of gibberellic acid production in a stirred tank bioreactor. Experimental data were obtained from submerged batch cultures of Gibberella fujikuroi (CDBB H‐984) grown in varying ratios of glucose‐corn oil as the carbon source. The nitrogen depletion effect was included in mathematical model by considering the specific kinetic constants as a linear function of the normalized nitrogen consumption rate. The kinetics of biomass growth and consumption of phosphate and nitrogen were based on the logistic model. The traditional first‐order kinetic model was used to describe the specific consumption of glucose and corn oil. The nitrogen effect was solely included in the phosphate and corn oil consumption and biomass growth. The model fit was satisfactory, revealing the dependence of the kinetics with respect to the nitrogen assimilation rate. Through simulations, it was possible to make diagrams of specific growth rate and specific rate of substrate consumptions, which was a powerful tool for understanding the metabolic interactions that occurred during the various stages of fermentation process. This kinetic analysis provided the proposal of a possible mechanism of regulation on growth, substrate consumptions, and production of gibberellic acid (GA₃) in G. fujikuroi. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29:1169–1180, 2013     

ROC analysis with multiple classes and multiple tests: methodology and its application in microarray studies

The accuracy of a single diagnostic test for binary outcome can be summarized by the area under the receiver operating characteristic (ROC) curve. Volume under the surface and hypervolume under the manifold have been proposed as extensions for multiple class diagnosis (Scurfield, 1996, 1998). However, the lack of simple inferential procedures for such measures has limited their practical utility. Part of the difficulty is that calculating such quantities may not be straightforward, even with a single test. The decision rule used to generate the ROC surface requires class probability assessments, which are not provided by the tests. We develop a method based on estimating the probabilities via some procedure, for example, multinomial logistic regression. Bootstrap inferences are proposed to account for variability in estimating the probabilities and perform well in simulations. The ROC measures are compared to the correct classification rate, which depends heavily on class prevalences. An example of tumor classification with microarray data demonstrates that this property may lead to substantially different analyses. The ROC-based analysis yields notable decreases in model complexity over previous analyses.     

Analyzing the dynamics of cell growth and protein production in mammalian cell fed-batch systems using logistic equations

The logistic modeling approach was used to describe experimental viable cell density (X) and product concentration (P) data from two industrial fed-batch mammalian cell culture processes with maximum product concentrations in the 3.0-9.4 g/l range. In both cases, experimental data were well described by the logistic equations and the resulting specific growth rate and protein productivity profiles provided useful insights into the process kinetics. Subsequently, sensitivity equations for both the X and P models were analyzed which helped characterize the influence of model parameters on X and P time courses. This was augmented by conventional sensitivity analyses where five values of each model parameter, 25% apart, were used to generate X and P time courses. Finally, results from sensitivity analysis were used to simulate X and P time courses that were reflective of typical early- and late-stage fed-batch cell culture processes. Different combinations of the logistic model parameters were used to arrive at the same final product concentration demonstrating the ability of the logistic approach to describe the multitude of process paths that result in the same final product concentration. Overall, the capability of the logistic equations to well describe X and P data from fed-batch cultures, coupled with their ability to simulate the multitude of paths leading up to the desired cell density and product concentration profiles, make them a useful tool during mammalian cell fed-batch process development.     

Analyzing population growth curves

Assessing animal population growth curves is an essential feature of field studies in ecology and wildlife management. We used five models to assess population growth rates with a number of sets of population growth rate data. A 'generalized' logistic curve provides a better model than do four other popular models. Use of difference equations for fitting was checked by a comparison of that method and direct fitting of the analytical (integrated) solution for three of the models. Fits to field data indicate that estimates of the asymptote, K, from the 'generalized logistic' and the ordinary logistic agree well enough to support use of estimates of K from the ordinary logistic on data that cannot be satisfactorily fitted with the generalized logistic. Akaike's information criterion is widely used, often with a small sample version AIC_c. Our study of five models indicated a bias in the AIC_c criterion, so we recommend checking results with estimates of variance about regression for fitted models. Fitting growth curves provides a valuable supplement to, and check on computer models of populations.     

QTL methodology for response curves on the basis of non-linear mixed models, with an illustration to senescence in potato

The improvement of quantitative traits in plant breeding will in general benefit from a better understanding of the genetic basis underlying their development. In this paper, a QTL mapping strategy is presented for modelling the development of phenotypic traits over time. Traditionally, crop growth models are used to study development. We propose an integration of crop growth models and QTL models within the framework of non-linear mixed models. We illustrate our approach with a QTL model for leaf senescence in a diploid potato cross. Assuming a logistic progression of senescence in time, two curve parameters are modelled, slope and inflection point, as a function of QTLs. The final QTL model for our example data contained four QTLs, of which two affected the position of the inflection point, one the senescence progression-rate, and a final one both inflection point and rate.     

Global parameter optimization for cardiac potassium channel gating models.

Quantitative ion channel model evaluation requires the estimation of voltage dependent rate constants. We have tested whether a unique set of rate constants can be reliably extracted from nonstationary macroscopic voltage clamp potassium current data. For many models, the rate constants derived independently at different membrane potentials are not unique. Therefore, our approach has been to use the exponential voltage dependence predicted from reaction rate theory (Stevens, C. F. 1978. Biophys. J. 22:295-306; Eyring, H., S. H. Lin, and S. M. Lin. 1980. Basic Chemical Kinetics. Wiley and Sons, New York) to couple the rate constants derived at different membrane potentials. This constrained the solution set of rate constants to only those that also obeyed this additional set of equations, which was sufficient to obtain a unique solution. We have tested this approach with data obtained from macroscopic delayed rectifier potassium channel currents in voltage-clamped guinea pig ventricular myocyte membranes. This potassium channel has relatively simple kinetics without an inactivation process and provided a convenient system to determine a globally optimized set of voltage-dependent rate constants for a Markov kinetic model. The ability of the fitting algorithm to extract rate constants from the macroscopic current data was tested using "data" synthesized from known rate constants. The simulated data sets were analyzed with the global fitting procedure and the fitted rate constants were compared with the rate constants used to generate the data. Monte Carlo methods were used to examine the accuracy of the estimated kinetic parameters. This global fitting approach provided a useful and convenient method for reliably extracting Markov rate constants from macroscopic voltage clamp data over a broad range of membrane potentials. The limitations of the method and the dependence on initial guesses are described.     

三参数增长模型拟合:以季风常绿阔叶林中两个优势乔木种群为例

Logistic、Mitscherlich、Gompertz方程是一类三参数饱和增长曲线模型,广泛地应用于许多学科领域．本文基于logistic方程饱和值K估计的三点法、四点法,推导出Mitscherlich、Gompertz方程K值的三点法、四点法估计公式,并以南亚热带季风常绿阔叶林中两种优势乔木厚壳桂、黄果厚壳桂种群为例,先用三点法或四点法估计出K值,再通过线性回归与非线性回归相结合的方法,可获得三个增长模型中三个参数的最优无偏估计．实例研究表明,两个优势种群增长数据均符合三个增长模型,但更符合增长曲线呈S形的logistic、Gompertz方程,且以logistic方程最适合于观察;黄果厚壳桂种群增长快于厚壳桂种群．     

Integrated steady state rate equations and the determination of individual rate constants.

Integrated steady state rate equations have been used to determine the kinetic constants (Vs, Ks, Vp, and Kp) and rate constants (k1, k2, k3, and k4) of the reversible enzyme mechanism: (see article). The fumarase reaction has been used as a model to illustrate the procedures for determining these constants. In contrast to initial velocity studies, the values of the constants have been obtained by examining the enzyme reaction in only one direction rather than in both forward and reverse directions. To accomplish this, a new procedure is described for fitting data to integrated rate equations which eliminates problems encountered when data are analyzed graphically. The advantages of examining on enzyme reaction in one direction with these new procedures allow this method to be extended to the examination of enzymes with simple mechanisms where initial velocities are difficult to measure because either the substrate or product is not readily available, or because the reaction is not readily reversible.     

Influence of the growth rate calculation on the relationship between growth rate and temperature

Three calculations of the growth rate (e.g. slope of a plot of the log10 of cfu ml-1 vs time, mum of the Gompertz equation and the reciprocal of time to obtain 108 cfu ml-1) were compared for Escherichia coli TG1 growing in tryptone soy broth medium at temperatures ranging from 14 to 39 degrees C. Up to now, the influence of using such different definitions on the relationship between microbial growth rate and temperature has never been investigated. In order to compare these calculation procedures, a dimensionless analysis based on the following normalized variables, mudim = mu/muopt and Tdim = [T-Tmin]/[Topt-Tmin], was used (Dantigny 1998). The influence of suboptimal temperatures on the growth rate was represented by means of a Belehràdek-type model based on a power function law: [mudim] = [Tdim]alpha. The influence of the different growth rate calculations on the model constants was assessed. Despite the great dependence of the raw growth rate values on the calculation procedure, the dimensionless analysis demonstrated that the alpha-value is independent of the growth rate definition. This result suggests that any definition for the growth rate can be utilized in studies aimed at determining the influence of temperature on microbial growth and highlights the interest of using dimensionless variables to overcome differences in the order of magnitude of the growth rate data and to avoid confusion between definitions.     

Photosynthetic characteristics and photosynthesis-light response curve models of summer maize

Aim The photosynthesis-light response curve is the most commonly used method to explore the relationship between photosynthetically active radiation and the net photosynthetic rate, because it is more effective to reflect the plant photosynthetic characteristics. And it is very meaningful for researchers to choose a suitable summer corn (Zea mays) photosynthetic model and optimal the models of photosynthesis-light response curve by analyzing the differences between simulation and observation in each growth period of some plant. So the object of this paper was to propose some useful suggestions for the choice of summer corn photosynthetic modes and the optimization of the photosynthetic light response curves model in further.Methods In this paper, three typical photosynthetic models were used to fitting the photosynthetic light response curve for upper leaf, leaf at ear of grain and lower leaf of summer corn during bell and milk period. And then the fitting degree of each model was compared to the measured data. Photosynthetic active radiation was divided into three parts, and the fitting residual errors of these three models were analyzed individually.Important findings The photosynthetic characteristic parameters such as maximum net photosynthetic rate (P_nmax), saturated light intensity (I_sat) and dark respiration rate (R_d) decreased constantly with a top-down leaf position and the parameters at milk stage were generally lower than the bell stage. Each growth period and leaf position could fit the curve, but some deviation exists for the P_nmax and I_sat in the rectangular hyperbolic model and the non-rectangular hyperbolic model. The results of curve fitting residual showed that the simulation values from Ye Zi-Piao model were closest to the actual values, and especially for the high photosynthetically active radiation section.     

A steady state model for analyzing the cellular binding, internalization and degradation of polypeptide ligands

We demonstrate that the interaction of polypeptide ligands with cells under physiological conditions can be described by a set of steady state equations. These equations include four new rate constants: V_r, the rate of insertion of receptors into the cell membrane; K_e, the endocytotic rate constant of occupied receptors; K_t, the turnover rate constant of unoccupied receptors; and K_h, the rate constant of hydrolysis of internalized ligand. Several simple procedures are described for determining these constants. In experiments in which epidermal growth factor and human fibroblasts were used, the cell-ligand interactions followed the predictions of the steady state model. The utility of the steady state equations is demonstrated by establishing the kinetic basis of the commonly observed “down regulation” phenomenon and by quantitating the effect of methylamine on the endocytotic and degradation rates of epidermal growth factor. We also show that the slope of a “Scatchard plot” of steady state binding data is a complex constant including terms for the endocytotic rate of both occupied and unoccupied receptors. The X-intercept of such a plot is a function of the insertion rate of new receptors, the internalization rate of occupied receptors and the degradation rate of the internalized ligand. The steady state equations allow one to predict changes in cellular ligand binding resulting from alterations in the four rate constants. They also provide a foundation for computer simulations of ligand-cell interactions, which closely correspond to experimental data. These approaches should facilitate studies on the control of cellular activities by these polypeptide ligands.     

夏玉米光合特性及光响应曲线拟合

通过对光合-光响应曲线的研究来探索光合有效辐射与净光合速率的关系是一种非常重要的手段。合适的模型才能较好地反映植株的光合特性。分析由于生育期与叶位的不同而导致各模型拟合值与实测值差异的变化性以及不同光合有效辐射强度下各模型的适用性, 可为夏玉米(Zea mays)光合模型的选择和光合-光响应曲线模型的进一步优化提供参考。该研究运用3种典型的光合模型对夏玉米大喇叭口期与乳熟期上部叶、穗位叶与下部叶做光合-光响应曲线拟合, 对比各模型的拟合度以及对实测数据的反映情况, 并将光合有效辐射分为3段, 分析3种模型在每段的拟合残差。结果表明: 最大净光合速率(P_nmax)、饱和光强(I_sat)、暗呼吸速率(R_d)等光合特征参数值随叶位由上而下呈降低趋势, 乳熟期普遍小于大喇叭口期; 不同生育期和叶位3种模型均可以拟合, 直角双曲线模型、非直角双曲线模型对P_nmax、I_sat的拟合值与实测值有一定的偏差; 通过残差分析表明叶子飘模型的拟合值与实测值最为相符, 尤其对高光部分的拟合表现出独有的优越性。     

Dual control over microvillus elongation during enterocyte development

1. Previously determined logistic growth constants describing enterocyte microvillus development for a variety of species were analysed for possible interactions taking place between enterocyte migration rate (R) and the size of individual crypts (CD). 2. Microvillus elongation, the c-value of a logistic growth curve, was found to increase linearly with crypt depth and reciprocally with decreasing migration rate. The starting microvillus length of a basal crypt enterocyte, the a-value, also increased linearly with CD without being affected by R. 3. The mathematical equation describing the effects of CD and R on M, the maximal microvillus length, was M = 0.0016 CD + 0.073 CD/R, where M and CD are measured in micron and R in micron/hr. 4. The relationship found between R, CD and M is explained by suggesting that the crypt environment enables enterocytes to respond to an initiating signal imposed on cells as they begin to migrate onto villi. The possible nature of this putative signal is also discussed.     

Structured latent growth curves for twin data.

We describe methods to fit structured latent growth curves to data from MZ and DZ twins. The well-known Gompertz, logistic and exponential curves may be written as a function of three components - asymptote, initial value, and rate of change. These components are allowed to vary and covary within individuals in a structured latent growth model. Such models are highly economical, requiring a small number of parameters to describe covariation across many occasions of measurement. We extend these methods to analyse longitudinal data from MZ and DZ twins and focus on the estimation of genetic and environmental variation and covariation in each of the asymptote, initial and rate of growth factors. For illustration, the models are fitted to longitudinal Bayley Infant Mental Development Scale data published by McArdle (1986). In these data, all three components of growth appear strongly familial with the majority of variance associated with the shared environment; differences between the models were not great. Occasion-specific residual factors not associated with the curve components account for approximately 40% of variance of which a significant proportion is additive genetic. Though the growth curve model fit less well than some others, they make restrictive, falsifiable predictions about the mean, variance and twin covariance of other (not yet measured) occasions of measurement.