The ratio and allometric scaling of speed, power, and strength in elite male rugby union players

This study compared the effectiveness of ratio and allometric scaling for normalizing speed, power, and strength in elite male rugby union players. Thirty rugby players (body mass [BM] 107.1 ± 10.1 kg, body height [BH] 187.8 ± 7.1 cm) were assessed for sprinting speed, peak power during countermovement jumps and squat jumps, and horizontal jumping distance. One-repetition maximum strength was assessed during a bench press, chin-up, and back squat. Performance was normalized using ratio and allometric scaling (Y/X), where Y is the performance, X, the body size variable (i.e., BM or BH), and b is the power exponent. An exponent of 1.0 was used during ratio scaling. Allometric scaling was applied using proposed exponents and derived exponents for each data set. The BM and BH variables were significantly related, or close to, performance during the speed, power and/or strength tests (p < 0.001-0.066). Ratio scaling and allometric scaling using proposed exponents were effective in normalizing performance (i.e., no significant correlations) for some of these tests. Allometric scaling with derived exponents normalized performance across all the tests undertaken, thereby removing the confounding effects of BM and BH. In terms of practical applications, allometric scaling with derived exponents may be used to normalize performance between larger rugby forwards and smaller rugby backs, and could provide additional information on rugby players of similar body size. Ratio scaling may provide the best predictive measure of performance (i.e., strongest correlations).     

Scaling of maximal oxygen uptake by lower leg muscle volume in boys and men.

The aim of this study was to critically examine the influence of body size on maximal oxygen uptake (VO2 max) in boys and men using body mass (BM), estimated fat-free mass (FFM), and estimated lower leg muscle volume (Vol) as the separate scaling variables. VO2 max and an in vivo measurement of Vol were assessed in 15 boys and 14 men. The FFM was estimated after percentage body fat had been predicted from population-specific skinfold measurements. By using nonlinear allometric modeling, common body size exponents for BM, FFM, and Vol were calculated. The point estimates for the size exponent (95% confidence interval) from the separate allometric models were: BM 0.79 (0.53-1.06), FFM 1.00 (0.78-1.22), and Vol 0.64 (0.40-0.88). For the boys, substantial residual size correlations were observed for VO2 max/BM0.79 and VO2 max/FFM1.00, indicating that these variables did not correctly partition out the influence of body size. In contrast, scaling by Vol0.64 led to no residual size correlation in boys or men. Scaling by BM is confounded by heterogeneity of body composition and potentially substantial differences in the mass exponent between boys and men. The FFM is precluded as an index of involved musculature because Vol did not represent a constant proportion of FFM [Vol proportional, variantFFM1.45 (95% confidence interval, 1.13-1.77)] in the boys (unlike the men). We conclude that Vol, as an indicator of the involved muscle mass, is the most valid allometric denominator for the scaling of VO2 max in a sample of boys and men heterogeneous for body size and composition.     

The effectiveness of scaling procedures for comparing ground reaction forces

Various scaling methods are used when attempting to remove the influence of anthropometric differences on ground reaction forces (GRF) when comparing groups. Though commonly used, ratio scaling often results in an over-correction. Allometric scaling has previously been suggested for kinetic variables but its effectiveness in partialing out the effect of anthropometrics is unknown due to a lack of consistent application. This study examined the effectiveness of allometric scaling vertical, braking and propulsive GRF and loading rate for 84 males and 47 females while running at 4.0 m/s. Raw, unfiltered data were ratio scaled by body mass (BM), height (HT), and BM multiplied by HT (BM1HT). Gender specific exponents for allometric scaling were determined by performing a log-linear (for BM and HT individually) or log-multilinear regression (BMHT). Pearson productmoment correlations were used to assess the effectiveness of each scaling method. Ratio scaling by BM, HT, or BM1HT resulted in an over-correction of the data for most variables and left a considerable portion of the variance still attributable to anthropometrics. Allometric scaling by BM successfully removed the effect of BM and HT for all variables except for braking GRF in males and vertical GRF in females. However, allometric scaling for BMHT successfully removed the effect of BM and HT for all reactionary forces in both genders. Based on these results, allometric scaling for BMHT was the most appropriate scaling method for partialing out the effect of BM and HT on kinetic variables to allow for effective comparisons between groups or individuals.     

Is there an influence of body mass on digesta mean retention time in herbivores? A comparative study on ungulates

The relation between body mass (BM) and digesta mean retention time (MRT) in herbivores was the focus of several studies in recent years. It was assumed that MRT scaled with BM(0.25) based on the isometric scaling of gut capacity (BM(1.0)) and allometric scaling of energy intake (BM(0.75)). Literature studies that tested this hypothesis produced conflicting results, arriving sometimes at higher or lower exponents than the postulated 0.25. This study was conducted with 8 ruminants (n=2-6 per species) and 6 hindgut fermenting species/breeds (n=2-6, warthog n=1) with a BM range of 60-4000 kg. All animals received a ration of 100% grass hay with ad libitum access. Dry matter intake was measured and the MRT was estimated by the use of a solute and a particle (1-2 mm) marker. No significant scaling of MRT(particle) with BM was observed for all herbivores (32 BM(0.04), p=0.518) and hindgut fermenters (32 BM(0.00), p=1.00). The scaling exponent for ruminants only showed a tendency towards significance (29 BM(0.12), p=0.071). Ruminants on average had an MRT(particle) 1.61-fold longer than hindgut fermenters. Whereas an exponent of 0.25 is reasonable from theoretical considerations, much lower exponents were found in this and other studies. The energetic benefit of increasing MRT is by no means continuous, since the energy released from a given food unit via digestion decreases over time. The low and non-significant scaling factors for both digestion types suggest that in ungulates, MRT is less influenced by BM (maximal allometric exponent ≤0.1) than often reported.     

Interspecific allometry of morphological traits among trematode parasites: selection and constraints

Developmental constraints and selective pressures interact to determine the strength of allometric scaling relationships between body size and the size of morphological traits among related species. Different traits are expected to relate to body size with different scaling exponents, depending on how their function changes disproportionately with increasing body size. For trematodes parasitic in vertebrate guts, the risk of being dislodged should increase disproportionately with body size, whereas basic physiological functions are more likely to increase in proportion to changes in body size. Allometric scaling exponents for attachment structures should thus be higher than those for other structures and should be higher for trematode families using endothermic hosts than for those using ectotherms, given the feeding and digestive characteristics of these hosts. These predictions are tested with data on 363 species from 13 trematode families. Sizes of four morphological structures were investigated, two associated with attachment (oral and ventral suckers) and the other two with feeding and reproduction (pharynx and cirrus sac). The scaling exponents obtained were generally low, the majority falling between 0.2 and 0.5. There were no consistent differences within families between the magnitude of scaling exponents for different structures. Also, there was no difference in the values of scaling exponents between families exploiting endothermic hosts and those using ectotherms. There were strong correlations across families between the values of the scaling exponents for the oral sucker, the ventral sucker and the pharynx: in families where the size of one trait increases relatively steeply as a function of body size, the same is generally true of the other traits. These results suggest either that developmental constraints link several morphological features independently of their specific roles or that similar selection pressures operate on different structures, leading to covariation of scaling exponents. © 2009 The Linnean Society of London, Biological Journal of the Linnean Society, 2009, 96 , 533–540.     

A case of non-scaling in mammalian physiology? Body size, digestive capacity, food intake, and ingesta passage in mammalian herbivores

As gut capacity is assumed to scale linearly to body mass (BM), and dry matter intake (DMI) to metabolic body weight (BM(0.75)), it has been proposed that ingesta mean retention time (MRT) should scale to BM(0.25) in herbivorous mammals. We test these assumptions with the most comprehensive literature data collations (n=74 species for gut capacity, n=93 species for DMI and MRT) to date. For MRT, only data from studies was used during which DMI was also recorded. Gut capacity scaled to BM(1.06). In spite of large differences in feeding regimes, absolute DMI (kg/d) scaled to BM(0.76) across all species tested. Regardless of this allometry inherent in the dataset, there was only a very low allometric scaling of MRT with BM(0.14) across all species. If species were divided according to the morphophysiological design of their digestive tract, there was non-significant scaling of MRT with BM(0.04) in colon fermenters, BM(0.08) in non-ruminant foregut fermenters, BM(0.06) in browsing and BM(0.04) in grazing ruminants. In contrast, MRT significantly scaled to BM(0.24) (CI 0.16-0.33) in the caecum fermenters. The results suggest that below a certain body size, long MRTs cannot be achieved even though coprophagy is performed; this supports the assumption of a potential body size limitation for herbivory on the lower end of the body size range. However, above a 500 g-threshold, there is no indication of a substantial general increase of MRT with BM. We therefore consider ingesta retention in mammalian herbivores an example of a biological, time-dependent variable that can, on an interspecific level, be dissociated from a supposed obligatory allometric scaling by the morphophysiological design of the digestive tract. We propose that very large body size does not automatically imply a digestive advantage, because long MRTs do not seem to be a characteristic of very large species only. A comparison of the relative DMI (g/kg(0.75)) with MRT indicates that, on an interspecific level, higher intakes are correlated to shorter MRTs in caecum, colon and non-ruminant foregut fermenters; in contrast, no significant correlation between relative DMI and MRT is evident in ruminants.     

Testing metabolic scaling theory using intraspecific allometries in Antarctic microarthropods

Quantitative scaling relationships among body mass, temperature and metabolic rate of organisms are still controversial, while resolution may be further complicated through the use of different and possibly inappropriate approaches to statistical analysis. We propose the application of a modelling strategy based on the theoretical approach of Akaike's information criteria and non‐linear model fitting (nlm). Accordingly, we collated and modelled available data at intraspecific level on the individual standard metabolic rate of Antarctic microarthropods as a function of body mass (M), temperature (T), species identity (S) and high rank taxa to which species belong (G) and tested predictions from metabolic scaling theory (mass‐metabolism allometric exponent b = 0.75, activation energy range 0.2–1.2 eV). We also performed allometric analysis based on logarithmic transformations (lm). Conclusions from lm and nlm approaches were different. Best‐supported models from lm incorporated T, M and S. The estimates of the allometric scaling exponent linking body mass and metabolic rate resulted in a value of 0.696 ± 0.105 (mean ± 95% CI). In contrast, the four best‐supported nlm models suggested that both the scaling exponent and activation energy significantly vary across the high rank taxa (Collembola, Cryptostigmata, Mesostigmata and Prostigmata) to which species belong, with mean values of b ranging from about 0.6 to 0.8. We therefore reached two conclusions: 1, published analyses of arthropod metabolism based on logarithmic data may be biased by data transformation; 2, non‐linear models applied to Antarctic microarthropod metabolic rate suggest that intraspecific scaling of standard metabolic rate in Antarctic microarthropods is highly variable and can be characterised by scaling exponents that greatly vary within taxa, which may have biased previous interspecific comparisons that neglected intraspecific variability.     

PHYLOGENETICALLY INFORMED ANALYSIS OF THE ALLOMETRY OF MAMMALIAN BASAL METABOLIC RATE SUPPORTS NEITHER GEOMETRIC NOR QUARTER-POWER SCALING

The form of the relationship between the basal metabolic rate (BMR) and body mass (M) of mammals has been at issue for almost seven decades, with debate focusing on the value of the scaling exponent ( b , where BMR ∝ M^b ) and the relative merits of b = 0.67 (geometric scaling) and b = 0.75 (quarter-power scaling). However, most analyses are not phylogenetically informed (PI) and therefore fail to account for the shared evolutionary history of the species they consider. Here, we reanalyze the most rigorously selected and comprehensive mammalian BMR dataset presently available, and investigate the effects of data selection and phylogenetic method (phylogenetic generalized least squares and independent contrasts) on estimation of the scaling exponent relating mammalian BMR to M. Contrary to the results of a non-PI analysis of these data, which found an exponent of 0.67–0.69, we find that most of the PI scaling exponents are significantly different from both 0.67 and 0.75. Similarly, the scaling exponents differ between lineages, and these exponents are also often different from 0.67 or 0.75. Thus, we conclude that no single value of b adequately characterizes the allometric relationship between body mass and BMR.     

Changes in body temperature influence the scaling of VO2max and aerobic scope in mammals

Debate on the mechanism(s) responsible for the scaling of metabolic rate with body size in mammals has focused on why the maximum metabolic rate (VO2max ) appears to scale more steeply with body size than the basal metabolic rate (BMR). Consequently, metabolic scope, defined as VO2max/BMR, systematically increases with body size. These observations have led some to suggest that VO2max, and BMR are controlled by fundamentally different processes, and to discount the generality of models that predict a single power-law scaling exponent for the size dependence of the metabolic rate. We present a model that predicts a steeper size dependence for VO2max than BMR based on the observation that changes in muscle temperature from rest to maximal activity are greater in larger mammals. Empirical data support the model's prediction. This model thus provides a potential theoretical and mechanistic link between BMR and VO2 max.     

Allometric Convergence in Savanna Trees and Implications for the Use of Plant Scaling Models in Variable Ecosystems

Theoretical models of allometric scaling provide frameworks for understanding and predicting how and why the morphology and function of organisms vary with scale. It remains unclear, however, if the predictions of ‘universal’ scaling models for vascular plants hold across diverse species in variable environments. Phenomena such as competition and disturbance may drive allometric scaling relationships away from theoretical predictions based on an optimized tree. Here, we use a hierarchical Bayesian approach to calculate tree-specific, species-specific, and ‘global’ (i.e. interspecific) scaling exponents for several allometric relationships using tree- and branch-level data harvested from three savanna sites across a rainfall gradient in Mali, West Africa. We use these exponents to provide a rigorous test of three plant scaling models (Metabolic Scaling Theory (MST), Geometric Similarity, and Stress Similarity) in savanna systems. For the allometric relationships we evaluated (diameter vs. length, aboveground mass, stem mass, and leaf mass) the empirically calculated exponents broadly overlapped among species from diverse environments, except for the scaling exponents for length, which increased with tree cover and density. When we compare empirical scaling exponents to the theoretical predictions from the three models we find MST predictions are most consistent with our observed allometries. In those situations where observations are inconsistent with MST we find that departure from theory corresponds with expected tradeoffs related to disturbance and competitive interactions. We hypothesize savanna trees have greater length-scaling exponents than predicted by MST due to an evolutionary tradeoff between fire escape and optimization of mechanical stability and internal resource transport. Future research on the drivers of systematic allometric variation could reconcile the differences between observed scaling relationships in variable ecosystems and those predicted by ideal models such as MST.     

Modeling the influence of body size on weightlifting and powerlifting performance

The purpose of this study was to examine 1) if lifting performance in both the weightlifting (WL) and powerlifting (PL) scale with body mass (M) in line with theory of geometric similarity, and 2) whether there are any gender differences in the allometric relationship between lifting performance and body size. This was performed by analyzing ten best WL and PL total results for each weight class, except for super heavyweight, achieved during 2000-2003. Data were analysed with the allometric and second-order polynomial model, and detailed regression diagnostics was applied in order to examine appropriateness of the models used. Results of the data analyses indicate that 1) women's WL and men's PL scale for M in line with theory of geometric similarity, 2) both WL and PL mass exponents are gender-specific, probably due to gender differences in body composition, 3) WL and PL results scale differently for M possibly due to their structural and functional differences. However, the obtained mass exponents does not provide size-independent indices of lifting performances since the allometric model exhibit a favourable bias toward middleweight lifters in most lifting data analyzed. Due to possible deviations from presumption of geometric similarity among lifters, future studies on scaling lifting performance should use fat-free mass and height as indices of body size.     

代谢异速生长理论及其在微生物生态学领域的应用

新陈代谢是生物的基本生理过程,影响生物在不同环境中参与物质循环和能量转化的过程.代谢速率作为生物体重要的生命过程指标,几乎影响所有的生物活性速率,且在很多研究中均表现出异速生长现象.所谓代谢异速是指生物体代谢速率与其个体大小(或质量)之间存在的幂函数关系.代谢异速生长理论的提出,从机制模型角度解释了代谢异速关系这一普遍存在的生命现象.该理论利用分形几何学及流体动力学等原理,从生物能量学角度阐释了异速生长规律的机理,证实了3/4权度指数的存在;但同时有研究表明,权度指数因环境因素等影响处于2/3-1范围之间而非定值.随着研究工作的深入,代谢异速生长理论研究从起初的宏观动植物领域拓展到了微生物领域,在研究微生物的代谢异速生长理论时,可将微生物的可操作分类单元(Operational taxonomic unit,OTU)或具有特定功能的功能群视为一个微生物个体,基于其遗传多样性和功能多样性特征进行表征,以便于将微生物群落多样性与其生态功能性联系起来,使该理论在微生物生态学领域得到有效的补充和完善.尽管细菌具有独特的生物学特性,但与宏观生物系统中观测到的现象表现出明显的一致性.有研究表明,3个农田土壤细菌基于遗传多样性的OTU数的平均周转率分别为0.71、0.80和0.84,介于2/3与1之间,可能与生物代谢异速指数有一定关联,为微生物代谢异速指数的研究提出了一个参考解决方案.鉴于微生物个体特征和生物学特性,在分析代谢速率与个体大小关系中,从微生物单位个体的定义、个体大小表征到计量单位的统一,仍需更多的理论支持.分析了代谢异速生长理论在微生物与生态系统功能关系研究中的可能应用,延伸了该理论的应用范围,并对尚待加强的研究问题进行了评述和展望.     

Allometric scaling of intraspecific space use

Allometric scaling relationships enable exploration of animal space-use patterns, yet interspecific studies cannot address many of the underlying mechanisms. We present the first intraspecific study of home range (HR) allometry relative to energetic requirements over several orders of magnitude of body mass, using as a model the predatory fish, pike Esox lucius. Analogous with interspecific studies, we show that space use increases more rapidly with mass (exponent = 1.08) than metabolic scaling theories predict. Our results support a theory that suggests increasing HR overlap with body mass explains many of these differences in allometric scaling of HR size. We conclude that, on a population scale, HR size and energetic requirement scale allometrically, but with different exponents.     

The scaling of body size and mass in a host-parasitoid association: influence of host species and stage

We describe the allometry of body mass and body size as measured by hind-tibia length in males of Monoctonus paulensis (Ashmead) (Hymenoptera: Braconidae, Aphidiinae), a solitary parasitoid of aphids. To assess the influence of host quality on allometric relationships, we reared parasitoids on second and fourth nymphal instars of four different aphid species, Acyrthosiphon pisum (Harris), Macrosiphum creelii Davis, Myzus persicae (Sulzer) and Sitobion avenae (F.), under controlled conditions in the laboratory. Dry mass was positively correlated with hind-tibia length, and could be predicted from it, in unparasitized aphids, in aphid mummies containing parasitoid pupae, and in the parasitoid. The reduced-major-axis scaling exponents for the regression of dry mass on hind-tibia length were species-specific in aphids, reflecting differences in volume and shape between species. In mummified aphids, the stage at death influenced the size/mass relationship. In males of M. paulensis, the allometric exponent varied between parasitoids developing in different kinds of host. Individuals developing in pea aphid were absolutely larger in dry mass as well as proportionately larger relative to their hind-tibia length. We discuss the allometry of body size and body mass in relation to parasitoid fitness.     

Modeling the influence of body size on V(O2) peak: effects of model choice and body composition.

This study examined the bivariate relationship between peak oxygen uptake (V(O2) peak); l/min) and body size in adult men (n = 1,314, age 17-66 yr), using both "simple" and "full" iterative nonlinear allometric models. The simple model was described by V(O2) peak = M(b) (or FFM(b)) exp(c SR-PA) exp(a + d age) epsilon (where M is body mass in kg; FFM is fat-free mass in kg; SR-PA is self-reported physical activity; epsilon is a multiplicative error term; and exp indicates natural antilogarithms). The full model was described by V(O2) peak = M(b) (or FFM(b)) exp(c SR-PA) exp(a + d age) + e (epsilon), where e is a permitted Y-intercept term. The M exponent obtained from simple allometry was 0.65 [95% confidence interval (CI), 0.59-0.71], suggestive of a curvilinear relationship constrained to pass through the origin. This "zero Y-intercept" assumption was examined via the full allometric model, which revealed an M exponent of 1.00 (95% CI, 0.7-1.31), together with a positive Y-intercept term (e) of 1.13 (95% CI, 0.54-1.73). The FFM exponents were not significantly different from unity in either the simple or full allometric models. It appears that the curvilinearity of the simple allometric model (using total M) is fictitious and is due to the inappropriate forcing of the regression line through the origin. Utilizing FFM as the body-size variable revealed a linear relationship between body size and V(O2) peak, irrespective of model choice. We conclude that the population mass exponent for V(O2) peak is close to unity.     

The link between respiratory capacity and changing metabolic demands during growth of northern pike, Esox lucius L.

The relationship between metabolic rate of pike (Y, mgO2) and body weight (X, g) over the range 40–1291 gat 15° C is of the form: Y=aX^b. For resting metabolic rate (V_o2, rest), the scaling coefficient, b , is 0.80 and for maximum metabolic rate measured after exhaustive swimming (V₀₂, max), b is 0.99. Factorial metabolic scope (V₀₂, max/ V₀₂, rest) increases with body weight. Peak postprandial oxygen consumption (V₀₂, ASDA) is a constant multiple of V₀₂ rest for any discrete meal (expressed as % of body weight) up to 10% body weight. V₀₂ASDA after a single meal can utilize the entire metabolic scope (V₀₂, max—V₀₂, rest) of juvenile but not adult pike.     

Power and metabolic scope of bird flight: a phylogenetic analysis of biomechanical predictions

For flying animals aerodynamic theory predicts that mechanical power required to fly scales as P proportional, variant m (7/6) in a series of isometric birds, and that the flight metabolic scope (P/BMR; BMR is basal metabolic rate) scales as P (scope) proportional, variant m (5/12). I tested these predictions by using phylogenetic independent contrasts from a set of 20 bird species, where flight metabolic rate was measured during laboratory conditions (mainly in wind tunnels). The body mass scaling exponent for P was 0.90, significantly lower than the predicted 7/6. This is partially due to the fact that real birds show an allometric scaling of wing span, which reduces flight cost. P (scope) was estimated using direct measurements of BMR in combination with allometric equations. The body mass scaling of P (scope) ranged between 0.31 and 0.51 for three data sets, respectively, and none differed significantly from the prediction of 5/12. Body mass scaling exponents of P (scope) differed significantly from 0 in all cases, and so P (scope) showed a positive body mass scaling in birds in accordance with the prediction.     

Allometric cascade: a model for resolving body mass effects on metabolism

Expanding upon a preliminary communication (Nature 417 (2002) 166), we here further develop a "multiple-causes model" of allometry, where the exponent b is the sum of the influences of multiple contributors to control. The relative strength of each contributor, with its own characteristic value of b(i), is determined by c(i), the control contribution or control coefficient. A more realistic equation for the scaling of metabolism with body size thus can be written as BMR=MR(0)Sigmac(i)(M/M(0))(bi), where MR(0) is the "characteristic metabolic rate" of an animal with a "characteristic body mass", M(0). With M(0) of 1 unit mass (usually kg), MR(0) takes the place of the value a, found in the standard scaling equation, b(i) is the scaling exponent of the process i, and c(i) is its control contribution to overall flux, or the control coefficient of the process i. One can think of this as an allometric cascade, with the b exponent for overall energy metabolism being determined by the b(i) and c(i) values for key steps in the complex pathways of energy demand and energy supply. Key intrinsic factors (such as neural and endocrine processes) or ecological extrinsic factors are considered to act through this system in affecting allometric scaling of energy turnover. Applying this model to maximum vs. BMR data for the first time explains the differing scaling behaviour of these two biological states in mammals, both in the absence and presence of intrinsic regulators such as thyroid hormones (for BMR) and catecholamines (for maximum metabolic rate).     

The ontogenetic allometry of body morphology and chemical composition in dairy goat wethers

We studied the ontogenetic growth of goat wethers (castrated male goats) of the Saanen and Swiss Alpine breeds based on a large range of intraspecific body mass (BM). The body parts and the chemical constituents of the empty body were described by the allometric function by using BM and the empty body mass (EBM) as the predictors for morphological traits and chemical composition, respectively. We fitted the allometric scaling function by applying the SAS NLMIXED procedure, but to evaluate assumptions regarding variances in morphological and compositional traits, we combined the scaling function with homoscedastic (MOD1), and the heteroscedastic exponential (MOD2) and power-of-the-mean (MOD3) variance functions. We also predicted the ontogenetic growth by using the traditional log-log transformation and back-transformed results into the arithmetic scale (MOD4). We obtained predictions from MOD4 in the arithmetic scale by a two-step process, and evaluated MOD1, MOD2 and MOD3 by a model selection framework, and compared MOD4 with MOD1, MOD2 and MOD3 based on goodness-of-fit measures. Based on information criteria for model selection, heterogeneous variance functions were more likely to describe 10 over 36 traits with a low level of model selection uncertainty. One trait was predicted by averaging the MOD1 and MOD2 variance functions; and nine traits were better described by averaging the MOD2 and MOD3 variance functions. The predictions for other 16 traits were averaged from MOD1, MOD2 and MOD3. However, MOD4 better described 11 traits according to the goodness-of-fit measures. Depending on the variable being analyzed, the body parts and the chemical amounts exhibited the three types of allometric behavior with respect to BM and EBM, that is, positive, negative and isometric ontogenetic growth. Reference BMs, that is, 20, 27, 35 and 45 kg, were used to compute the net protein and energy requirements based on the first derivative of the scaling function, and the results were presented in reference to the EBM and EBM^0.75. Both the net protein and energy requirements scaled to EBM^0.75 increased from 20 to 45 kg of BM.