Allometry Exponent Research Articles

Allometric scaling and allocation patterns: Implications for predicting productivity across plant communities

As the application of allometry continues to expand, the variability in the allometry exponent has generated a great deal of debate in forest ecology. Some studies have reported counterintuitive values of the exponent, but the sources of such values have remained both unexplored and unexplained. Therefore, the objectives of our analyses were to: (1) uncover the global patterns of allometric variation in stem height with stem diameter, crown radius with stem diameter or stem height, crown depth with stem diameter, crown volume with stem diameter, crown depth with crown diameter, aboveground biomass with stem diameter or height, and belowground biomass with aboveground biomass; (2) assess variations in allometry parameters with taxonomic levels, climate zones, biomes and historical disturbance regimes; and (3) identify the sources of counterintuitive values of the allometry exponents. Here, we provide novel insights into the tight allometric co-variations between stem and crown dimensions and tree biomass. We also show a striking similarity in scaling across climate zones, biomes and disturbance regimes consistent with the allometry constraint hypothesis. We show that the central tendency of the exponent is toward 2/3 for the scaling of stem height with diameter, crown dimensions with stem diameter and height, 5/2–8/3 for the scaling of aboveground biomass with stem diameter, and 1 for the scaling of belowground biomass with aboveground biomass. This is indicative of an integrated growth regulation acting in tandem on growth in stem diameter, height, crown dimensions and biomass allocation. We also demonstrate that counterintuitive values of the exponent arise as artifacts of small sample sizes (N < 60), measurement errors, sampling biases and inappropriate regression techniques. We strongly recommend the use of larger sample sizes (N > 60) and representative samples of the target population when testing hypothesis about allometric variation. We also caution against conflation of statistical artifacts with violations of theoretical predictions.

Size and History Combine in Allometry Relation of Technology Systems

Complex systems often exhibit amazingly regular behavior through allometry relations (ARs) between their functional attributes and their size. An empirical allometry relation (EAR) between two properties of a complex system relates the average functionality [Formula: see text] and the average size [Formula: see text][Formula: see text] = a[Formula: see text] where the allometry coefficient a and allometry exponent b are empirical constants fit to data. This EAR is a static relation and is found in every sub-discipline of Natural History. Herein we establish, both empirically and theoretically, that for some classes of evolving technology systems, the empirical allometry coefficient a is not constant, but is strongly dependent on the historical time at which the technology system originated. Specifically, we construct an EAR with a time-dependent coefficient, using the evolution of a broad class of military systems, over the last ten centuries, ranging from a medieval bowman to a modern rifleman, from a horse-drawn cannon to a tank. This time-dependent EAR is derived from fundamental considerations involving complexity, scaling, and renormalization group theory. The theory entails an information as well as complexity generalization of the traditional allometry coefficient that combines system size with the technology knowledge of the new system’s developers (time-dependence). It relates technology ARs to technology evolution relations, such as Moore’s Law, with implications for technology drawn from biological evolution analysis.

Evaluation of the effects of ontogenetic or maturation functions and chronic heart failure on the model analysis for the dose-response relationship of warfarin in Japanese children.

We previously demonstrated that the rational pediatric dosage of warfarin can be well-described by a SIZE parameter that includes an allometry exponent of weight. On the other hand, allometry alone is considered to be insufficient to predict drug clearance in neonates and infants. The primary purpose of the present study was to evaluate the effects of incorporation of the maturation process into the analysis model for the dose-response relationship of warfarin in Japanese children. In addition, we evaluated the effect of chronic heart failure (CHF) on the response to warfarin as an independent risk factor for increased anticoagulant effects. Thirty-eight patients with stable anticoagulation by warfarin were enrolled. During a mean follow-up period of 4.74 ± 3.51years, 1092 data points including prothrombin time-international normalized ratio (PT-INR) were obtained. The data were subjected to multiple regression analysis to identify covariates related to the anticoagulant effects. Two different models describing the maturation process did not improve the predictive performance for the dose-response relationship in pediatric patients. In addition to the SIZE-normalized daily dose, the vitamin K epoxide reductase complex 1 (VKORC1) genotype, and concomitant use of bosentan, CHF was identified as a covariate increasing the anticoagulant effects of warfarin to 118%. The SIZE parameter was useful even without incorporation of maturation models to describe the response to warfarin in pediatric patients, and our longitudinal follow-up study design with multiple observations was beneficial to detect changes within individual subjects.

Prediction of Clearance and Dose of Midazolam in Preterm and Term Neonates: A Comparative Study Between Allometric Scaling and Physiologically Based Pharmacokinetic Modeling

Children are not small adults because besides size there are subtle physiological and biochemical differences between children and adults. Like adults, children also require medicine for the management or cure for the underlying diseases. To select a right dose in children, pharmacokinetic (PK) information is warranted. However, in many instances, a PK study in neonates and infants may not be possible. Therefore, various methods are used to predict PK parameters in this group of population, and these predicted parameters may help to calculate a safe dose for the very young children. Allometry is widely used for the prediction of PK parameters in children and subsequently one can predict dose from these predicted PK parameters. Physiologically based pharmacokinetic modeling (PBPK) has also become a useful tool to achieve these goals. Therefore, the objective of this study was to compare the predictive performance of allometry and PBPK for a test compound, midazolam in preterm, and term neonates. In this study, there were 5 preterm neonates (gestational age ranging from 34 to 37 weeks) and 5 term neonates (gestational age ranging from 38 to 41 weeks). PBPK modeling was performed using PK-Sim 6.0 and clearance, as well as midazolam dose in neonates was predicted. Clearance and midazolam dose in neonates was also predicted by allometric scaling. In this study, the allometric exponents for the prediction of midazolam clearance in preterm neonates and term neonates were 1.2 and 1.1, respectively. Similarly, for the prediction of midazolam neonatal dose, the exponent of allometry was either 0.9 or 1.0. The predicted midazolam clearance and dose by both methods were then compared with observed midazolam clearance and dose in neonates. The results of the study showed a slightly better prediction of midazolam clearance in neonates by PBPK than allometric scaling. However, the projected dose of midazolam in neonates was comparable between the 2 methods. Overall, it was noted that both PBPK and allometric model can be used to predict clearance and dose of midazolam in neonates.

Simple Methodology to Estimate Bioavailability in Early Clinical Studies: Theory and Reality

Aim: We propose a simple methodology to estimate oral bioavailability from Phase I clinical trials, which can be broadly phrased as ‘calculated bioavailability’ (FCAL). Materials & methods: FCAL is estimated by obtaining the product of circulating maximal observed drug concentration (Cmax’) and the estimated volume of distribution at steady state (VSS’) followed by dividing this product by the administered dose (d). VSS’ is estimated by allometric scaling based on the preclinical intravenous data. Results & conclusion: Overall, these analyses indicate that when the absorption is faster than elimination process and the VSS’ allometry exponent is between 0.9 and 1.1, the Fcal is a good estimate of the F.

Population Pharmacokinetics: Some Observations in Pediatric Modeling for Drug Clearance.

The objective of this study is to evaluate the predictive performance of several models to predict drug clearance in preterm and term neonates. Five models using different types of allometric and linear models were developed. Two sets of data were used to develop these models (data from preterm neonates to adults and data from preterm and term neonates). Models were also developed with (normalized to 70kg) or without body weight normalization (body weight 1kg). From the literature, clearance values for four drugs from neonates to adults were obtained. External data were used to evaluate the predictive performance of these models in preterm and term neonates. The results of the study indicated that (1) normalization to a standard body weight had no impact on the predictive performance of the models, (2) the model developed from preterm neonates to adults using fixed exponent 0.75 provided inaccurate estimate (overestimation) of drug clearance in neonates, (3) a far superior prediction of clearance was observed with the model when the exponents of allometry were estimated than the model using exponent 0.75, (4) linear models with the exception of the model with intercept provided comparable results to the estimated exponent model and were superior in their predictive performance to the model using exponent 0.75, and (5) when the models were developed from neonate data, the predictive performance of all models were similar. Overall, the study indicated that body weight normalization had no impact on the performance of model prediction, the exponents of allometry in pharmacostatistical models should be estimated rather than fixed, and more studies are needed to evaluate the suitability of linear models for the prediction of drug clearance in neonates.

Dominance of the suppressed: Power-law size structure in tropical forests.

Tropical tree size distributions are remarkably consistent despite differences in the environments that support them. With data analysis and theory, we found a simple and biologically intuitive hypothesis to explain this property, which is the foundation of forest dynamics modeling and carbon storage estimates. After a disturbance, new individuals in the forest gap grow quickly in full sun until they begin to overtop one another. The two-dimensional space-filling of the growing crowns of the tallest individuals relegates a group of losing, slow-growing individuals to the understory. Those left in the understory follow a power-law size distribution, the scaling of which depends on only the crown area-to-diameter allometry exponent: a well-conserved value across tropical forests.

Fractional dynamics of allometry

Abstract Allometry relations (ARs) in physiology are nearly two hundred years old. In general if X ij is a measure of the size of the i th member of a complex host network from species j and Y ij is a property of a complex subnetwork embedded within the host network an intraspecies AR exists between the two when Y ij = aX ijb. We emphasize that the reductionist models of AR interpret X ij and Y ij as dynamic variables, albeit the ARs themselves are explicitly time independent. On the other hand, the phenomenological models of AR are based on the statistical analysis of data and interpret 〈X i〉 and 〈Y i〉 as averages over an ensemble of individuals to yields the interspecies AR 〈Y i〉 = a〈X i〉b. Modern explanations of AR begin with the application of fractal geometry and fractal statistics to scaling phenomena. The detailed application of fractal geometry to the explanation of intraspecies ARs is a little over a decade old and although well received it has not been universally accepted. An alternate perspective is given by the interspecies AR based on linear regression analysis of fluctuating data sets. We emphasize that the intraspecies and interspecies ARs are not the same and show that the interspecies AR can only be derived from the intraspecies one for a narrow distribution of fluctuations. This condition is not satisfied by metabolic data as is shown separately for aviary and mammal data sets. The empirical distribution of metabolic allometry coefficients is shown herein to be Pareto in form. A number of reductionist arguments conclude that the allometry exponent is universal, however herein we derive a deterministic relation between the allometry exponent and the allometry coefficient using the fractional calculus. The co-variation relation violates the universality assumption. We conclude that the interspecies physiologic AR is entailed by the scaling behavior of the probability density, which is derived using the fractional probability calculus.

Response to the comments of Professors Anderson & Holford

The following are my responses to the comments of Professors Anderson & Holford (henceforth mentioned as A & H in the text) on my manuscript entitled: ‘Evaluation of a morphine maturation model for the prediction of morphine clearance in children’[1].

Are allometry and macroevolution related?

The allometric relationship between the basal metabolic rate B and total body mass M is B = aMb, where the allometry coefficient a and the allometry exponent b have been fit to various data sets for over 150 years. The best fit of the allometry exponent to 391 mammalian species is given by Heusner [7] to be midway between the leading theoretical values of 2/3 and 3/4. Most theoretical investigations have focused on determining the proper value of b entailed by an appropriate biological model and with some notable exceptions ignored the allometry coefficient a altogether. Herein, we shift the focus and use the above data to settle on an empirical value of b that gives rise to an allometry coefficient with random variability described by a Pareto distribution. This new perspective suggests an interesting biological interpretation of the statistical fluctuations of the allometry coefficient. The time distribution of the intermittent fluctuations in a are determined to be of the same statistical form as those of speciation found in the punctuated equilibrium theory of macroevolution (Eldredge and Gould [3], Sneppen et al. [5] and Rikvold and Zia [17]).

Theoretical Versus Empirical Allometry: Facts Behind Theories Application to Pharmacokinetics

Although, no credible data are available, it is widely believed that the pharmacokinetic parameters such as clearance, volume of distribution, and elimination half-life can be scaled with a fixed exponent of 0.75, 1.0, and 0.25, respectively. However, logically it is difficult to perceive that the exponents of allometry for a given parameter will revolve around a fixed number. Over the last 80 years, theoretical biologists and physiologists are entangled in an argument whether the exponent of basal metabolic rate against body weight is 0.67 or 0.75. There are evidence for and against these two exponents. However, in-depth investigative works suggest that there is not really any universal exponent for any allometrically related parameter and the exponent of allometry depends on several factors. The main focus of this report is to show the readers that there is enough evidence from experimental data that negate the notion of fixed exponents in biology, physiology, and pharmacokinetics. In short, the notion of a fixed exponent is theoretical and there is no evidence that the exponent of a physiological or pharmacokinetic parameter revolves around a fixed number.

The allometry of energy reserve depletion: test of a mechanism for size-dependent winter mortality.

We experimentally tested the hypothesis that energy reserve depletion varies inversely with size in the fish Menidia menidia, an estuarine fish known to exhibit size-dependent winter mortality. Individuals in two size groups were starved at two winter temperatures (4°and 8°C) and sacrificed at a range of time intervals (up to 127 days). Lipid levels and lean tissue were analyzed to estimate somatic energy storage. As predicted, energy depletion was greater at high temperatures, and proportionally greater in small than in large fish. After 60 days of starvation at 4°C, small fish retained an average of 67% of their original energy reserves (vs 53% at 8°C), while large fish retained an average of 80% (vs 66% at 8°C). At 4°C, fish that were fed depleted their energy reserves as rapidly as unfed fish, but at 8°C, fish that were fed maintained reserves at higher levels than unfed fish. A high proportion of unfed fish (56% at 4°C, 27% at 8°C) died before they were to be sacrificed. Survival probability did not vary with size, nor was it influenced by the amount of energy reserves. The rate of energy depletion (equivalent to routine metabolic rate) decreased gradually over time, particularly in small fish. Routine metabolism did not conform to a single scaling relationship. Within each temperature-size group, the routine rate declined more rapidly than metabolically active mass (lean mass). At 8°C, the difference between size groups in energy depletion rate conformed closely to the expected allometry exponent of 0.8. In contrast, at 4°C, the estimated allometry exponent increased over the experiment (-0.19 to 2.5). We conclude that strategies to minimize energy loss may often modify bioenergetic scaling relationships.

Prenatal and postnatal growth and allometry of stature, head circumference, and brain weight in québec children.

AbstractA recently proposed four‐parameter generalization of the Gompertz curve (Jolicoeur et al., 1992a) is used to analyze chronological growth and allometry in the stature, head circumference, and brain weight of 101 male and 95 female fetuses and children autopsied at the Sainte‐Justine Hospital for children in Montréal. Although asymptotic stature cannot be determined accurately because the sample does not include adults, the data nevertheless suggest that head circumference and brain weight terminate most of their growth much earlier than stature. This conclusion is also supported by the fact that, whereas the log‐log relationship between head circumference and brain weight is almost straight and closely approximates simple allometry, the log‐log relationships between stature, on one hand, and head circumference or brain weight, on the other hand, are markedly curved, indicating complex allometry. Applied to 78 males and 73 females whose total age is between 20 weeks and 1.5 years, the confidence intervals of the slope of the major axis of logarithmically transformed data indicate that the allometry exponent of brain weight relative to head circumference is somewhat higher than 3.0, which may reflect changes in head proportions. © 1993 Wiley‐Liss, Inc.

Statistical methods for studying the relationship between maximum aerobic power and body weight. Comparison between Twa and Oto of the lake Tumba area (Zaire)

In neither 84 Twa Pygmoids of the Zairian equatorial forest, nor 60 Oto, their taller suzerains, are conditions fulfilled that allow for the (VO 2 max: body weight) ratio to express validly VO 2 max independent of body-mass. Neither population shows a significant departure from normality in the distribution of the two variables or from the rectilinearity of their relationship. The major axes of their bivariate distributions lead to an unbiased comparison of the two populations. The observed isometry contradicts some theoretical views concerning the relative dimensionality of the two variables, which led to the conclusion of allometry with a coefficient of 2/3 or 3/4. The use of the bilogarithmic equation for estimating the allometry exponent is criticized, and in its place the concept of bidimensional structural relationship (Jolicoeur & Heusner 1971) is advocated. Implications of this concept are developed. A wide range of the coefficients of variation of both VO 2 max and body weight is demonstrated in human populations. It is concluded that no departure from isometry has ever been validly proven in any adult population. The use of muscle mass instead of body weight in the ratio would not improve this lack of validity.

Variations in Respiratory Enzymes of Embryos in the Loach Fishes

1. Activities of catalase and of succinic dehydrogenase in unfertilized eggs and in embryos of four kinds of loach fishes and those of hybrids derived from the crosses between these fishes were measured.2. The activities per embryo as well as per unit volume of eggs of catalase were greater in embryos of the larger kind and less in those of the smaller kind. The allometry exponent b among the four kinds of embryos in the activity against the egg volume was 1.50.3. The activity of succinic dehydrogenase per unit volume of eggs in unfertilized eggs was alike with each other among the four kinds of fishes but that of embryos was greater in embryos of the smaller kind than in those of the larger kind. The exponent b in the allometry of the activity against the egg volume of unfertilized eggs among the four was 1.04, and that of embryos 0.64.4. The activities of catalase in the hybrid embryos did not differ from those of the maternal control. While the activities per unit volume of eggs of succinic dehydrogenase in the hybrid embryos differed from those of the maternal control as well as those of the paternal control, being intermediate between those in embryos of the parental fishes.5. The relationship among egg size, respiratory rates, activities of these enzymes and temperature adaptation in the loach fishes were discussed.