Exponential growth and allometric equations

Earlier the simple exponential growth law has been used when attempts have been made to explain theoretically the existence of allometric equations in biology (Bertalanffy, 1968; Giinther, 1975). We show here that the exponential growth law in which the proportion coefficient also depends on time is a better approximation. We consider especially the metabolic rate and the body mass of man during the growth process.     

Techniques for estimating allometric equations.

Morphologists have long been aware that differential size relationships of variables can be fo great value when studying shape. Allometric patterns have been the basis of many interpretations of adaptations, biomechanisms, and taxonomies. It is of importance that the parameters of the allometric equation be as accurate estimates as possible since they are so commonly used in such interpretations. Since the error term may come into the allometric relation either exponentially or additively, there are at least two methods of estimating the parameters of the allometric equation. That most commonly used assumes exponentiality of the error term, and operates by forming a linear function by a logarithmic transformation and then solving by the method of ordinary least squares. On the other hand, if the rrror term comes into the equation in an additive way, a nonlinear method may be used, searching the parameter space for those parameters which minimize the sum of squared residuals. Study of data on body weight and metabolism in birds explores the issues involved in discriminating between the two models by working through a specific example and shows that these two methods of estimation can yield highly different results. Not only minimizing the sum of squared residuals, but also the distribution and randomness of the residuals must be considered in determing which model more precisely estimates the parameters. In general there is no a priori way to tell which model will be best. Given the importance often attached to the parameter estimates, it may be well worth considerable effort to find which method of solution is appropriate for a given set of data.     

Estimation of parameters of allometric equations.

Accurate parameter estimation of allometric equations is a question of considerable interest. Various techniques that address this problem exist. In this paper it is assumed that the measured values are normally distributed and a maximum likelihood estimation approach is used. The computations involved in this procedure are reducible to relatively simple forms, and an efficient numerical algorithm is used. A listing of the computer program is included as an appendix.     

On the allometric growth of tissues in fruits

This paper deals with the relative growth of three different fruit tissues. Their morphogenetic periods and the mathematical constraints involved are described, and more precisely, the paper shows an allometric relationship (Y=nX ^m ) between the widths (X, Y) of the main tissues in stone fruits such as cherries, peaches and prunes. The mathematical relationships between the growth of the mesocarp and of the endocarp of somePrunus fruits are described, and it is proved that before the formation of the embryo, growth is allometric, in agreement with conclusions drawn from some experimental data. However, according to another study, the growth of the mesocarp and of the endocarp are ruled by autocatalytic and monomolecular functions, before as well as after the formation of the embryo. In this case, it is proved that if allometry exits in stone fruits, it can only be anantiometry (m=−1). To solve the dilemma, two main alternatives are proposed and discussed. We conclude that, while allometry is established on reasonable grounds before the formation of the embryo, after the formation of the embryo the mesocarp and endocarp evolve independently since a center for the coordination of growth no longer exists, and each tissue can grow according to its own independent rules.     

Pitfalls and challenges of estimating population growth rate from empirical data: consequences for allometric scaling relations

The intrinsic rate of increase is a fundamental concept in population ecology, and a variety of problems require that estimates of population growth rate be obtained from empirical data. However, depending on the extent and type of data available (e.g. time series, life tables, life history traits), several alternative empirical estimators of population growth rate are possible. Because these estimators make different assumptions about the nature of age‐dependent mortality and density‐dependence of population dynamics, among other factors, these quantities capture fundamentally different aspects of population growth and are not interchangeable. Nevertheless, they have been routinely commingled in recent ecoinformatic analyses relating to allometry and conservation biology. Here we clarify some of the confusion regarding the empirical estimation of population growth rate and present separate analyses of the frequency distributions and allometric scaling of three alternative, non‐interchangeable measures of population growth. Studies of allometric scaling of population growth rate with body size are additionally sensitive to the statistical line fitting approach used, and we find that different approaches yield different allometric scaling slopes. Across the mix of population growth estimators and line fitting techniques, we find scattered and limited support for the key allometric prediction from the metabolic theory of ecology, namely that log₁₀(population growth rate) should scale as ?0.25 power of log₁₀(body mass). More importantly, we conclude that the question of allometric scaling of population growth rate with body size is highly sensitive to previously unexamined assumptions regarding both the appropriate population growth parameter to be compared and the line fitting approach used to examine the data. Finally, we suggest that the ultimate test of allometric scaling of maximum population growth rates with body size has not been done and, moreover, may require data that are not currently available.     

Biological similarity theories: a comparison with the empirical allometric equations.

Twelve biological variables were submitted to dimensional analysis in accordance with the MLT-system of physics (M, mass; L, length; T, time). Each of these variables has a characteristic numerical value for the exponents alpha for mass, beta for length, and gamma for time. By means of Newton's reduction coefficient (chi), the three dimensions (MLT) can be expressed as power functions of body mass (Mb); the exponent (b) is the result of the combination of the three dimensional exponents (alpha, beta, gamma). By linear regression analysis of 203 allometric exponents (betaE) obtained from the literature, the following equation was found for the regression exponent (bR) (equation: see text). The estimated numerical coefficients (ki) for the three exponents (alpha, beta, gamma) of the basic dimensions (MLT) do not agree with those of the prevailing theories of biological similarity.     

Some theoretical considerations on the inhibition of tumor growth by ultrastrong magnetic fields

The forces acting on the microstructure of a biological system have been derived in a systematic way starting from Maxwell's stress tensor. It is shown that with a magnetic field above 100 kilogauss and with a magnetic field gradient above 10 kilogauss/cm, the magnetic forces acting on a cell may become strong enough to disrupt cellular membranes and distort the ordered process of cellular division. Since cancer cells seem to have a smaller mechanical strength than normal cells, it might be possible to selectively damage cancer cells by this method. Strong magnetic fields can be most economically generated by a superconducting coil submerged in liquid helium. A simple experiment in vitro with a tissue culture could test the theoretical predictions.     

Some electrophysiological consequences of electrogenic sodium and potassium transport in cardiac muscle: a theoretical study

A previous paper described a kinetic model for electrogenic sodium-potassium transport in cardiac muscle, combining a thermodynamically-constrained transport model with simple passive permeabilities for sodium and potassium to generate a cardiac action potential (Chapman, Kootsey & Johnson, 1979). The present paper explores the extent to which this simplest of active-passive transport models can account (without further modification) for the electrophysiological behavior of cardiac muscle. The long term (several minutes) changes in the duration of the action potential observed following a change in stimulation rate are predicted by the model through a shift in the steady-state current-voltage relationship caused by small changes in inside ion concentrations. The diastolic hyperpolarization observed following an increase in rate is also predicted, including the linear relationship between the maximum diastolic depolarization and the rate of stimulation. Varying the outside potassium concentration in the model produces changes in the rest potential and current-voltage relationship similar to published data. Deviations from ideal potassium electrode behavior occur at both high and low concentrations because of effects on the pump. The model not only predicts the observed shift of the current-voltage curve in the depolarizing direction with increasing [K⁺]₀, but also the crossing of the curve in normal [K ⁺]₀ without having to assume a variation in g_K. Anoxia was introduced into the model by changing the concentrations of ATP and ADP, thereby enabling the model to account for the rapid diastolic depolarization observed in myocardial ischemia.     

How to derive biological information from the value of the normalization constant in allometric equations

Allometric equations are widely used in many branches of biological science. The potential information content of the normalization constant b in allometric equations of the form Y = bX(a) has, however, remained largely neglected. To demonstrate the potential for utilizing this information, I generated a large number of artificial datasets that resembled those that are frequently encountered in biological studies, i.e., relatively small samples including measurement error or uncontrolled variation. The value of X was allowed to vary randomly within the limits describing different data ranges, and a was set to a fixed theoretical value. The constant b was set to a range of values describing the effect of a continuous environmental variable. In addition, a normally distributed random error was added to the values of both X and Y. Two different approaches were then used to model the data. The traditional approach estimated both a and b using a regression model, whereas an alternative approach set the exponent a at its theoretical value and only estimated the value of b. Both approaches produced virtually the same model fit with less than 0.3% difference in the coefficient of determination. Only the alternative approach was able to precisely reproduce the effect of the environmental variable, which was largely lost among noise variation when using the traditional approach. The results show how the value of b can be used as a source of valuable biological information if an appropriate regression model is selected.     

Cardiac growth in staged human fetuses: an allometric approach

The human cardiac growth was studied on 30 staged fetuses (gestational age ranged from 10 to 38 weeks post conception, WPC). The hearts were quantitatively evaluated considering their weight and the following linear parameters: the pulmonary (da.p.) and aortic (dA) internal diameters, the right (VR) and left (VL) ventricular wall thickness, the ventricular width (dV), and the length "sulcus terminalis-apex cordis" (Lst-apex). The data were correlated with the gestational age by using the allometric method (Y = bX alpha). The equations and the growth curves are presented. In the fetal life, we observe that the body weight is 134 times larger than the heart weight, the Lst-apex is 1.07 times larger than the dV, the da.p. is 1.50 times larger than the dA, and the VL is 1.19 times larger than the VR.     

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

Sexual size dimorphism and allometric growth of Morelet's crocodiles in captivity

Few studies have conducted morphological analyses of crocodilians, and little information exists on differences between size-classes and sexes in Neotropical crocodilians. In this study, we measured nine morphological traits in 121 captive Morelet's crocodiles Crocodylus moreletii (81 females and 40 males). Our results revealed that individuals < 2 m total length do not exhibit sexual dimorphism in morphometric characteristics. However, for crocodiles over 2 m in length, males were significantly larger than females in terms of dorsal-cranial length, cranial width, snout width and snout-ventral length. In general, morphological traits demonstrated a strongly significant relationship with total length at the smaller size class of 150-200 cm length. However, in the highest size class of 250-300 cm length (large adult males), morphological traits were no longer significantly related with total length. Male crocodiles demonstrated allometric growth of cranial morphology with significantly greater increase in cranial width, snout width, and mid-snout width relative to total length at higher size classes. Morphological dimorphism and allometric growth may be associated with adaptive strategies for reproductive success.     

Some consequences of the hierarchical approach to phyllotaxis

This paper presents some of the consequences of the introduction of a concept of information and of a principle of optimal design in a representation of phyllotaxis by means of hierarchies. Our approach, which we call phylogenetic, has experimental supports and is based on the concept of an algorithm: growth matrices generate rhythms and vortices of growth among which is the hierarchy of Fibonacci representing the phyllotaxis of Fibonacci. It gives a necessary and sufficient condition which produces this type of phyllotaxis, and in the light of the propositions obtained, it can explain why Fibonacci's phyllotaxis is the most omnipresent in Nature.     

Some allometric and non-allometric relationships between chlorophyll-a and abundance variables of phytoplankton

We analysed data from three Bulgarian reservoirs, with trophic status from meso to eutrophic. Two kinds of relations (non-allometric and allometric) of chlorophyll with phytoplankton density, biovolume and surface area were investigated. The non-allometric relationships compare these phytoplankton variables with chlorophyll-a concentration, while the allometric ones include comparisons of average individual volume (AIV) of phytoplankters with chlorophyll-a content per individual (Chl.N^–1), per unit biovolume (Chl.BV^–1) and per unit surface area (Chl.SA^–1).Maximum values of the rank correlation of the three non-allometric relationships were found in the eutrophic reservoir. The highest allometric correlation coefficients were observed in the mesotrophic reservoir. Two non-allometric relationships, Chl with N and Chl ith BV calculated on unified data from all three reservoirs showed lower significance and a non-linear character. Chl-SA relationship was always statistically significant and varied from linear to slightly non-linear when calculated on weighed values. Two allometric relationships, AIV with Chl.BV^–1 and AIV with Chl.N^–1 seem to be linear. A third AIV and Chl.SA^–1 was described by a polinomial of the second degree, indicating that the smallest and largest phytoplankton individuals have a similar chlorophyll content per unit surface area. At the highest trophy, the Chl.SA^–1 seemed lowest.The frequently investigated relationship between chlorophyll-a and biovolume ranged from statistically not significant in the mesotrophic to highly significant in the eutrophic reservoir. This tendency was generalized by obtaining a statistically significant rank correlation between the levels of significance of chlorophyll-biovolume correlations and the corresponding chlorophyll-a concentrations. The non-linear character of the chlorophyll-biovolume relationship over a wide range of trophy was probably caused by changes in surface area-biovolume ratio.Deceased.