Theory for growth of plants derived from the nitrogen productivity concept

A theory is developed on the assumption that growth of plants is determined by the current amount of nitrogen in the plants. The nitrogen-growth relation is formalized in the nitrogen productivity concept (amount of biomass produced per amount of nitrogen in the biomass and per unit of time), which is essentially a constant for a given species under fixed environmental conditions. A number of results follow for increases in whole plant biomass: (A) The relative growth rate is a linear function of the internal nitrogen concentration. (B) The maximal relative growth rate uniquely determines the scaling of the time axis. (C) Exponential growth is consistent only with stable internal nitrogen concentration. Dose-response curves expressed in reduced variables (the ratio between a variable and the same variable for a plant growing under optimal conditions) are universal, so that all species and all environmental conditions yield the same curve. This is confirmed by experimental data. The shape (linear, exponential, etc.) of the nitrogen uptake curve is the only parameter differentiating these universal curves. The Mitscherlich curve or variations of it can be fitted very closely to the derived dose-response curves, except under exponential growth. A conclusion drawn from the analysis is that the results of nutrition experiments cannot be properly interpreted unless the variation with time of the amount of nitrogen in the plant is known. The theory can be extended to more complex situations, for example, time-varying environmental conditions.     

The Use of a Mathematical Model for the Analysis of Manurial and Weather Effects on the Growth of Carrots

A model for the study of yearly and other variation in cropyields, based on a generalized form of the logistic equationof growth, was applied to three years' growth analysis dataon carrot crops from 16 plots of a long-term manurial experiment.In the ideal case, it is postulated that the growth in differentyears of carrot leaves or taproots on soil receiving a givenmanurial treatment could be described by a single curve if timein days was replaced by a scale based on one or more meteorologicalelements which affect relative growth-rates. The equation fitted the data on the growth of the leaves butnot that on the taproots. For leaf growth, the use of the sumsof solar radiation, day degrees or evaporation from an openwater surface did not give curves that fitted the growth databetter than those obtained with time in days. Yearly differencesin leaf (but not taproot) growth appeared to be related to correspondingdifferences in rainfall amounts, and it is suggested that ascale taking account of soil moisture tension might have givena better fit to the model. It appeared that whereas the relative leaf growth-rate was afunction of leaf but not taproot weight, relative taproot growth-ratewas a function of taproot weight and leaf weight. A furthermodel for the growth of carrots, based on this hypothesis isdescribed.     

Growth and biannual recruitment of the black scallop Chlamys varia (L.) in lanveoc area,Bay of Brest

Data on growth of the black scallop Chlamys varia (L.) were obtained by taggine, raft cage rearing, reading winter growth rings, and analysis of length-frequency distributions. Von Bertalanffy growth curves were fitted to each set of data by least-squares regression techniques involving no transformation of variates. The results obtained by the different methods are comparable except for those obtained by length-frequency distribution analysis. Length frequency distribution analysis seems to be complicated by the presence of a biannual recruitment. Counting growth rings seems to be a more accurate and easier method for growth estimates. The sizes at rings are different for individuals born in spring and for those born in autumn; the growth curves derived for these two types of animals remain different throughout their whole life. The relative proportions of individuals originating from each of the two recruitment groups have remained remarkably homogeneous for the last five years. Comparative study of the allometric relationships between length and width of the shell of the animals from the two recruitment groups showed no significant difference.     

Methods for fitting dominance/diversity curves

Abstract. Dominance/diversity curves, displaying the relative abundances of the species within a community, have often been constructed from field data. Several ecological and statistical models of dominance/diversity have been proposed, to explain the curves. Yet, rarely have curves of different models been fitted to field data. In this paper the appropriate parameters and methods of curve fitting for plant communities are described for the General Lognormal, Canonical Lognormal, Geometric, Broken Stick, Zipf and Zipf-Mandelbrot models. Distinction is made between fixed and optimised parameters, to clarify para-meterisation of the models. It is concluded that all should be fitted by minimising the deviance in a ranked-abundance plot. Statistical tests of goodness of fit are discussed. It is concluded that consistency of fit between replicate quadrats of a community provide the best test. Curves of all the models discussed are fitted to data from a species-rich Spanish hay meadow, and to data from a New Zealand intertidal algal community. The Spanish meadow data are best fitted by General Lognormal. The New Zealand algal data are best fitted by Geometric or General Lognormal. Goodness of fit for a sample is usually relatively good or poor for all models, since much of the deviance comes from steps in the curve which none of the models can fit closely.     

The application of robust calibration to radioimmunoassay.

The minute concentrations of many biochemically and clinically important substances are currently estimated by radioimmunoassay (RIA). Traditionally, the most popular approaches to the statistical analysis of RIA data have been to linearize the data through transformation and fit the calibration curve using least squares or to directly fit a nonlinear calibration curve using least squares. Estimates of the hormone concentration in patients are then obtained using this curve. Unfortunately, the transformation is frequently unsuccessful in linearizing the data. Furthermore, the least squares fit can lead to erroneous results in both approaches since the many sources of error which exist in the RIA process often result in outlier observations. In this paper, an approach to the analysis of RIA data which incorporates robust estimation methods is described. An algorithm is presented for obtaining the M-estimates of nonlinear calibration curves. The curves to be fitted are modified hyperbolae based on 12 to 16 observations. A procedure, based on the application of the Bonferroni Inequality, is presented for obtaining tolerance-like interval estimates of the concentration of the hormone of interest in the patients. Results of simulations are cited to support the method of construction of confidence bands for the fitted calibration curve. Data obtained from the Veteran's Hospital, Buffalo, New York are used to illustrate the application of the algorithm which is presented.     

Pharmfit-a Nonlinear Fitting Program for Pharmacology

A new program is presented for nonlinear fitting of data from pharmacological and chronobiological investigations. It contains functions for calculating data from ligand-binding studies and competition experiments, for the analysis of dose-response curves, for pharmacokinetic calculations, and for cosine analysis of harmonic and overlapping rhythms. In addition, it is possible to implement general equations by the user. The program allows data exchange with most spreadsheet, database, and graphics presentation programs, and accepts data from two widely used ambulatory 24-h blood-pressure monitoring systems. The fitting procedure uses the Marquardt-Levenberg algorithm. It calculates the weighted or the unweighted fit together with a great variety of statistics for estimation of goodness of fit. A graphics module permits graphical presentation of the fitted curve. Moreover, fitting of data to different models can be compared for the most likely fit and model discrimination statistics for improvement of further experiments are provided. To demonstrate the chronobiological application of the fitting program PHARMFIT, the analysis of telemetric heart rate data from rats is presented.     

PHARMFIT—a Nonlinear Fitting Program for Pharmacology

A new program is presented for nonlinear fitting of data from pharmacological and chronobiological investigations. It contains functions for calculating data from ligand-binding studies and competition experiments, for the analysis of dose-response curves, for pharmacokinetic calculations, and for cosine analysis of harmonic and overlapping rhythms. In addition, it is possible to implement general equations by the user. The program allows data exchange with most spreadsheet, database, and graphics presentation programs, and accepts data from two widely used ambulatory 24-h blood-pressure monitoring systems. The fitting procedure uses the Marquardt–Levenberg algorithm. It calculates the weighted or the unweighted fit together with a great variety of statistics for estimation of goodness of fit. A graphics module permits graphical presentation of the fitted curve. Moreover, fitting of data to different models can be compared for the most likely fit and model discrimination statistics for improvement of further experiments are provided. To demonstrate the chronobiological application of the fitting program PHARMFIT, the analysis of telemetric heart rate data from rats is presented.     

Estimating non-isothermal bacterial growth in foods from isothermal experimental data

AIM: To develop a mathematical method to estimate non-isothermal microbial growth curves in foods from experiments performed under isothermal conditions and demonstrate the method's applicability with published growth data. METHODS AND RESULTS: Published isothermal growth curves of Pseudomonas spp. in refrigerated fish at 0-8 degrees C and Escherichia coli 1952 in a nutritional broth at 27.6-36 degrees C were fitted with two different three-parameter 'primary models' and the temperature dependence of their parameters was fitted by ad hoc empirical 'secondary models'. These were used to generate non-isothermal growth curves by solving, numerically, a differential equation derived on the premise that the momentary non-isothermal growth rate is the isothermal rate at the momentary temperature, at a time that corresponds to the momentary growth level of the population. The predicted non-isothermal growth curves were in agreement with the reported experimental ones and, as expected, the quality of the predictions did not depend on the 'primary model' chosen for the calculation. CONCLUSIONS: A common type of sigmoid growth curve can be adequately described by three-parameter 'primary models'. At least in the two systems examined, these could be used to predict growth patterns under a variety of continuous and discontinuous non-isothermal temperature profiles. SIGNIFICANCE AND IMPACT OF THE STUDY: The described mathematical method whenever validated experimentally will enable the simulation of the microbial quality of stored and transported foods under a large variety of existing or contemplated commercial temperature histories.     

PATTERNS OF GROWTH IN BIRDS. II. GROWTH RATE AND MODE OF DEVELOPMENT

This analysis was initiated to examine the relationship between the rate of growth in birds and their development of mature function. The literature was surveyed for data on growth and development, and the growth curves of 81 species were chosen for the analysis. Growth curves of most species were fitted with the Gompertz equation, and the rate constants of the equation were used as an index of the growth rate. For those species whose curves were fitted better by other equations, with a slightly different form, appropriate conversion factors, derived in this paper, were employed.
Among species with similar modes of development, growth rate decreases with increasing body weight in an allometric manner, with slopes of –0.26 to –0.42, depending on the group. Between groups, the rate of growth in body weight was found to be closely associated with the rate of development of function, in particular, the acquisition of flight. Among those species that can walk at an early age, but acquire flight relatively late, the rate of growth depends primarily on the relative size of the musculature of the lower extremities.
Data are presented to refute the hypotheses that growth rate is adjusted to nestling mortality, or that the energy requirements of the young (and hence their growth rates) are balanced against brood size. It is concluded that most species grow at some physiologically maximum rate, but as yet it is not possible to distinguish between limitation of growth rate at the level of the organism or at the level of the tissue.     

Prediction of toxigenic fungal growth in buildings by using a novel modelling system.

There is growing concern about the adverse effects of fungal bioaerosols on the occupants of damp dwellings. Based on an extensive analysis of previously published data and on experiments carried out within this study, critical limits for the growth of the indoor fungi Eurotium herbariorum, Aspergillus versicolor, and Stachybotrys chartarum were mathematically described in terms of growth limit curves (isopleths) which define the minimum combination of temperature (T) and relative humidity (RH) at which growth will occur. Each growth limit curve was generated from a series of data points on a T-RH plot and mathematically fitted by using a third-order polynomial equation of the form RH = a(3)T(3) + a(2)T(2) + a(1)T + a(0). This fungal growth prediction model was incorporated within the ESP-r (Environmental Systems Performance [r stands for "research"]) computer-based program for transient simulation of the energy and environmental performance of buildings. For any specified location, the ESP-r system is able to predict the time series evolution of local surface temperature and relative humidity, taking explicit account of constructional moisture flow, moisture generation sources, and air movement. This allows the predicted local conditions to be superimposed directly onto fungal growth curves. The concentration of plotted points relative to the curves allows an assessment of the risk of fungal growth. The system's predictive capability was tested via laboratory experiments and by comparison with monitored data from a fungus-contaminated house.     

A comparison of the traditional method of counting viable cells and a quick microplate method for monitoring the growth characteristics of Listeria monocytogenes

AIMS: To determine: (i) the growth parameters (specific growth rate, lag time, asymptotic amount of growth, generation time and time for maximum growth rate) of Listeria monocytogenes in different broths by standard cultivation methods and (ii) whether a microplate method in conjunction with a standard nondedicated plate reader could be adapted to routine assay. METHODS AND RESULTS: Growth curves were determined from cell numbers in a standard tube method at 2 h intervals by serial dilution and plating, and in a microplate method by absorbance measurements. Growth curves were fitted with a modified Gompertz function. CONCLUSIONS: The microplate method was similar to the standard cultivation methods in accuracy, required less chemical reagents, and considerably reduced the time required for analyses. This work also illustrates that growth characteristics of bacteria are not necessarily constant, and depend on the methodology used. SIGNIFICANCE AND IMPACT OF THE STUDY: It is not the intended purpose of this paper to present all the data for the media tested but instead to illustrate the success of the microplate method for studying growth kinetics compared to a standard cultivation method and system precision. The method will be of considerable benefit to laboratories unable to afford dedicated workstations.     

A computer program for determining the size of DNA restriction fragments

A computer program has been developed for determining the sizes of DNA restriction fragments from their electrophoretic mobilities. A parabola is fitted to the mobilities of a set of standard fragments of known sizes and the sizes of the unknown fragments are then calculated from the fitted curve. This procedure is shown to yield estimated sizes which are accurate to within a few percentage, as judged by experiments with fragments obtained by digestion of pBR322 with the restriction endonuclease HaeIII. The program, which is written in BASIC, is simple to use and is very much faster than the graphical method that it replaces.     

Environmental versus social factors as determinants of growth in nestlings of a communally breeding bird

Summary Nestlings of the communally breeding Greycrowned Babbler (Pomatostomus temporalis) were studied to discover if supplemental feeding by auxiliary birds at nests enhanced their growth. Growth of wing, bill, tarsus and weight was measured. Growth curves were fitted by computer using a commercial program (MLAB). Our data provided little support for possible sibling competition. A significant component of the variance in asymptote and growth constant for some variables could be attributed to differences among nests. Environmental variables such as temperature and rainfall were much more strongly associated with nestling growth than were the numbers of auxiliary birds feeding broods.     

A useful method for the analysis of growth studies

A group of parsimonious models for the analysis of animal growth curves is presented. The effects of correlations between the successive weights are reduced by analyzing the weight gained between the different weighing times. A model is developed for the resulting two-way table of group average gain curves (group by periods), using the two-way table modeling techniques proposed by Mandel. The coefficients in the fitted model are examined to determine what functions of the weight gain measurements will summarize the characteristics of the curves. In the second part of the analysis these statistics are computed for each animal and analyzed to determine the effects of the design variables on the growth of the animals. Experience has shown that in many instances total weight gain summarizes all the information in the curve. The conditions under which total gain provides an adequate summary are discussed. The model has been used in the analysis of rat, dog, hamster, guinea pig and swine growth curves. An example is included to illustrate the utility of the proposed approach.     

THE TECHNIQUE OF LABELLED MITOSES: ANALYSIS BY AUTOMATIC CURVE-FITTING

An improved method is described for the analysis of data obtained by the technique of labelled mitoses. It is a development of the method described by Barrett (1966) in which theoretical curves are computed on the basis of a model which assumes that the phases G¹, S and G² are described by independent log-normal distributions; the analysis consists in finding a form of this model which gives a labelled mitoses curve which is the best fit to the available data. This fitting procedure has now been made automatic. No comprehensive indication of the goodness of fit can be given, although in the analysis of over fifty sets of data the method appears to have worked well.
A supplementary computer program is described which, on the basis of three separate assumed modes of cell loss, calculates the form of the age distributions and theoretical continuous labelling curves. This allows growth fraction to be calculated in a way which takes account of the distribution of phase durations and the non-rectangular age distributions of expanding cell populations. It also gives an opportunity to study the implications of continuous labelling data as regards the mode of cell loss.
A comparison is made between the present method of labelled mitoses curve analysis and the empirical rules which have often been used.     

LILLY—A linear least squares curve fitting program for one independent variable

A program was written to perform a linear least squares curve fitting on data. It includes facilities to report the usual statistics and digital plotter output. Seven types of curves are available for fitting the data. Other features of LILLY include provision of facilities for the selection of subsets in different symbols and separate curve fitting for these subsets. The program also provides a confidence region about the fitted line and the prediction interval for data points. Examples of the use of the program are described.     

Evaluation of parameters describing the root system architecture of field grown maize plants (Zea mays L.)

The objective of this work was to study elongation curves of maize axile roots throughout their elongation period under field conditions. Relationships between their elongation rate and the extension rate of their branched region were also studied. Maize, early-maturing cultivar Dea, was grown on a deep, barrier-free clay loam (depth 1.80m). Trenches were dug during four periods until after silking and axile roots were excavated. Parameters measured were total length and the lengths of basal and apical unbranched zones. The rank of the bearing phytomer and general data about the carrying plant were also recorded. Results showed that axile roots from lower phytomers had similar elongation rates irrespective of the rank of the carrying phytomer. This elongation rate declined with root age. A monomolecular elongation model was fitted to the experimental data. Elongation was much slower in roots from upper phytomers. A rough linear relationship was found between the elongation rate of axile roots and the length of the apical unbranched zone. This result suggests that laterals appeared on a root segment a constant time after it was formed. Possible mechanisms with may account for the declining elongation rate with root age (increasing distance from aerial parts or adverse environmental conditions in deep soil layers) and variability between individual roots are also discussed.