Genetic mapping of allometric scaling laws

Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling relationships between rate and size (power laws). A statistical model for mapping specific quantitative trait loci (QTLs) that are responsible for allometric scaling laws has been developed. We present an improved model for allometric mapping of QTLs based on a more general allometry equation. This improved model includes two steps: (1) use model II regression analysis to estimate the parameters underlying universal allometric scaling laws, and (2) substitute the estimated allometric parameters in the mixture-based mapping model to obtain the estimation of QTL position and effects. This model has been validated by a real example for a mouse F2 progeny, in which two QTLs were detected on different chromosomes that determine the allometric relationship between growth rate and body weight.     

Allometric changes during growth and reproduction in Eleutheria dichotoma (hydrozoa,athecata) and the problem of estimating body size in a microscopic animal

Changes in body morphology during growth and reproduction in the hydromedusa Eleutheria dichotoma are described in terms of variations in eight different characters: umbrella diameter, total surface area, tentacle area, umbrella area, tentacle knob diameter, number of embryos, and diameter and area of buds. Sexually (sex) and vegetatively (veg) reproducing medusae differ significantly in their body morphometrics. Statistically significant allometric relations exist between umbrella diameter and (1) central area (sex and veg); (2) tentacle area (veg); (3) total area (veg); (4) tentacle knob diameter (veg); (5) bud diameter; and (6) number of embryos. A significant correlation between umbrella diameter and area is also found in undetached buds. During sexual reproduction, umbrella area shows positive allometry and loses its correlations to total area, tentacle area, and tentacle knob diameter. Linear and nonlinear bivariate allometric coefficients allow estimation of total body size from only one or two easily measurable attributes, e.g., umbrella and tentacle knob diameter. Curve fitting by the classic allometric equation (y = bx^c) is only negligibly worse than that obtained with a “full” equation (y = a + ^c), and statistical confidence is better. Chemical analyses for carbon and nitrogen content allow estimation of biomass from the projection area of the body surface. The relation factors are 1.06 μgC mm^?2 (sex) and 1.14 μgC mm^?2 (veg) for carbon and 0.293 μgN mm^?2 (sex) and 0.287 μgN mm^?2 (veg) for nitrogen. The C:N ratios are 3.6 and 4.0 for sexual and vegetative medusae, respectively. The use of allometric regression formulas to calculate surface areas and to relate these to carbon content provides quick estimations of body size in a microscopic animal.     

Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.     

Growth Characteristics of the Common Mussel Mytilus edulis from the White Sea

We studied the following growth indices of the White Sea mussels Mytilus edulis: shell length, total weight, soft tissue weight, and shell weight. The coefficients of allometric relationships between the indices were determined. Age-related changes in the indices could be approximated by the Bertalanffy equation. The maximum age of mollusks in the studied population equaled 13 years (with the maximum shell length of 66.2 mm). Growth rate of littoral mussels in the region of Umba Settlement (Northern Kandalaksha Bay) was lower as compared to those published for other littoral White Sea populations (Chupa Bay).     

Numbats and aardwolves—how low is low? A re-affirmation of the need for statistical rigour in evaluating regression predictions

Many comparative physiological studies aim to determine if a particular species differs from a prediction based on a linear allometric regression for other species. However, the judgment as to whether the species in question conforms to this allometric relationship is often not based on any formal statistical analysis. An appropriate statistical method is to compare the new species’ value with the 95% confidence limits for predicting an additional datum from the relationship for the other species. We examine the basal metabolic rate (BMR) of the termitivorous numbat (Myrmecobius fasciatus) and aardwolf (Proteles cristatus) to demonstrate the use of the 95% prediction limits to determine statistically if they have a lower-than-expected BMR compared to related species. The numbat’s BMR was 83.6% of expected from mass, but fell inside the 95% prediction limits for a further datum; a BMR < 72.5% of predicted was required to fall below the one-tail 95% prediction limits. The aardwolf had a BMR that was only 74.2% of predicted from the allometric equation, but it also fell well within the 95% prediction limits; a BMR of only 41.8% of predicted was necessary to fall below the one-tail 95% prediction limits. We conclude that a formal statistical approach is essential, although it is difficult to demonstrate that a single species statistically differs from a regression relationship for other species.     

Length–weight relationships of mesopelagic fishes from the equatorial and tropical Atlantic waters: influence of environment and body shape

Length–weight relationships (LWRs) were estimated for 36 mesopelagic fish species collected from the equatorial and tropical Atlantic encompassing several oceanographic regions: oligotrophic, equatorial, Cape Blanc, Cape Verde and the Canary Islands. The sample was composed of myctophids (25 species), gonostomatids (5), sternoptychids (3), stomiids (2) and phosichthyids (1). The species were clustered according to body shape: “short-deep” (sternoptychids), “elongate” (gonostomatids, stomiids and some phosichthyids) and “fusiform” (myctophids and some phosichthyids). Three types of weight and LWRs were considered: wet weight (WW), eviscerated wet weight (eWW) and eviscerated dry weight (eDW). The study demonstrated that most species present a positive allometric growth, independent of the weight used. However, the allometric value varied in 40–50% of species depending on the type of weight considered. Significant variations linked to fish morphology were found in the relationship between the slope and intercept of the LWR equation. Significant differences were also noted in the water content linked to fish body shape. Based on the distributions of several species we compare their fitness between oceanographic regions using the relative condition factor (K_rel). Except for Diaphus brachycephalus (oligotrophic vs. equatorial waters) and Lampanyctus alatus (equatorial, Cape Blanc, Cape Verde and the Canary Islands), no regional significant differences were observed in the species analysed.     

Incisor size and diet revisited: The view from a platyrrhine perspective

Allometric relationships between incisor size and body size were determined for 26 species of New World primates. While previous studies have suggested that the incisors of Old World primates, and anthropoids in general, scale isometrically with body size, the data presented here indicate a negative allometric relationship between incisor size and body size among New World species. This negative allometry was exhibited by platyrrhines when either upper or lower incisor row length was regressed against body weight, and when either least-squares or bivariate principal axis equations were used. When upper incisor length was plotted against skull length, negative allometry could be sustained using both statistical techniques only when the full sample of 26 species was plotted. The choice of variables to represent incisor size and body size, and the choice of a statistical technique to effect the allometric equation, had a more pronounced impact on the location of individual species with regard to lines of best fit. Platyrrhines as a group have smaller incisors relative to body size than do catarrhines, regardless of diet. Among New World primates, small incisors represent a plausible primitive condition; species with relatively large incisors manifest a phyletic change associated with a dietary shift to foods that require increased incisal preparation. The opposite trend characterizes Old World primates. In spite of the taxonomic differences in relative incisor size between platyrrhine and catarrhine primates, inferences about diet derived from an allometric equation for all anthropoids should prove reliable as long as the species with unknown diet does not lie at the upper end of the body size range for platyrrhines or catarrhines.     

A common allometric equation for predicting stem weight of mangroves growing in secondary forests

The allometric relationship for stem weight Ws is usually expressed as a function of stem diameter and height, similar to the variable d.b.h.²H, which equals the squared diameter at breast height multiplied by tree height. However, this relationship often differs between tree species, and this segregation of the relationship by species forces the researcher to do a tremendous amount of field work to determine a series of allometric equations for all tree species in the forest. In this study, we examined the segregation in the d.b.h.²H–Ws allometric relationship for five mangrove species. We examined the overall stem shape and the specific gravity of stem relating to the allometric relationships. The difference in the specific gravity was found to be the main cause of the segregation in the d.b.h.²H–Ws relationship. By taking into consideration the specific gravity of stem, we established a common equation for the five mangrove species.     

Relative growth and sexual dimorphism in the hermit crab Clibanarius signatus Heller, 1861 (Anomura,Diogenidae) from the northern coast of the Persian Gulf,Iran

The purpose of this study was to investigate the relative growth and sexual dimorphism in the hermit crab Clibanarius signatus. The evaluation was done with 955 specimens (494 males, 251 females, and 210 intersexes) captured in Persian Gulf (Iran) during January to December 2015. Animals were submitted to measurements related to weight (BW, total wet weight) and body size related to cephalic shield (SW, width; and SL, length) and propodus of both chelipeds (CPL, length; and CPW, width). Males were larger and heavier than females and intersexes. Both males and females showed a negative allometric growth for the SL–BW and SL–SW relationships, but a positive allometric growth to intersex specimens. To SL–CPL relationship, a negative allometric growth was confirmed in males and females independent of the laterality of the CPL, whereas a contrast was verified in intersexes, with a positive allometric growth occurred for both hands. To SL–CPW relationship, a negative allometric growth (b < 1) occurred in females, independent of the laterality of the CPW, while in males, a positive allometric pattern was confirmed. In intersexes, this relationship was positive except for the right CPW which was isometric. Sexual dimorphism was evident in Clibanarius signatus, with males being the largest and females the smallest specimens in the population.     

Individual growth of <Emphasis Type="Italic">Lymnaea stagnalis</Emphasis> (Lymnaeidae,Gastropoda): II. Late postlarval ontogeny

Our investigation on the growth of 14 individuals of the great pond snail Lymnaea stagnalis was performed in an aquatic culture at 18°C beginning from the 10th week after hatching until death. It has been demonstrated that the increase in the mollusk mass follows an S-like curve during the whole studied period. Linear growth (conch height) follows a parabolic (convex) curve until the age of 39 weeks. Both weight and linear growth during studied period significantly approximate to the Bertalanffy equation, while the interrelation between mass and conch height corresponds to the allometric equation. The meanings of the coefficients of these equations do not differ significantly in different individuals. At the age of 38 to 39 weeks, all mollusks demonstrate breakage in the curve of linear growth, then followed with abrupt slowing of growth until stopping or even decreasing in size in some cases. Neither the Bertalanffy equation nor the allometric relation describe the linear growth of individuals with ages exceeding 39 weeks.     

The Allometry of primate skeletal weight

There is a widely held assumption that skeletal weights of mammals increase disproportionately with increased body size. Recent empirical studies have supported this assumption, and it has been suggested that this might account for the fact that metabolic rate scales to body weight with a negative allometry. Other studies, however, have suggested that skeletal weight in primates is directly proportionate to body weight. The results of this study support this latter interpretation and also indicate that the same is true for two other orders of mammals that were a part of the earlier allometric studies. The evidence suggests that skeletal weight scales isometricallywith body weight within individual mammalian orders. From this it is concluded that skeletal weight does not play any part in determining the negatively allometric scaling of metabolic rate.     

Age and growth of Atlantic bluefin tuna, Thunnus thynnus (Osteichthyes: Thunnidae), in the Mediterranean Sea

The objective of the study was to describe the biometry of Mediterranean bluefin tuna, Thunnus thynnus, the biology of which is not yet well understood. A total of 504 specimens was collected from 1998 to 2005 in the central part of the Mediterranean basin. They were sexed and measured; fork lengths (FL) ranged from 51.0 to 255.0 cm while body weights (W) ranged from 2.6 to 247.0 kg. The first spiniform ray (spine) of the first dorsal fin was removed and cross‐sectioned near the condyle base in order to count annuli for age estimation. The regression coefficient (b) of the female FL–W relationship was significantly higher than that of the male, and both sexes displayed a negatively allometric growth (b < 3); male regression equation: ln W = ?2.942 + 2.730 ln FL; female regression equation: ln W = ?3.660 + 2.878 ln FL. Based on counts of the translucent zones in the sections of the first ray of the first dorsal fin, estimated ages ranged from 1 to 15 years for males and 1 to 14 years for females. The correlation between the spine ray (R) and FL fit the allometric model best; the R–FL regression equations of the two sexes did not differ significantly and the overall equation was: ln FL = 3.721 + 0.851 ln R. Due to the R–FL allometric correlation, estimates of fork lengths at previous ages, FL_i, were back‐calculated with a body proportional hypothesis. Von Bertalanffy growth equations were derived from both observed and back‐calculated FLs‐at‐age, which did not differ significantly. Moreover, no significant difference was found between the growth equations of the two sexes; the overall equation was FL_t = 373.08 [1?e^{?0.07(t + 1.76)}]. Weight‐at‐age values were derived from the von Bertalanffy predicted FLs‐at‐age by the FL–W correlation equations for males and females. The paper represents the first comprehensive study on the biometry, including age and growth, of bluefin tuna captured in the Mediterranean Sea.     

The growth and energy metabolism of <Emphasis Type="Italic">Lymnaea stagnalis</Emphasis> (Lymnaeidae,Gastropoda): I. Early postlarval period

The growth and the oxygen consumption rate of Lymnaea stagnalis were studied during the first ten weeks after hatching. It is shown that these processes are atypical during early ontogenesis in comparison with adult mollusks. The obtained data on linear (height of shell) and weight growth can be equally well approximated with the von Bertalanffy equation or exponential and power equations. Both linear and weight growth are characterized by an approximately constant specific rate associated with synchronous oscillations of a week period. The oscillations were observed also for the oxygen consumption rate, but of another period (about 2.6 weeks). On average the metabolic rate after the initial triple increase during the first three days remains stable. The power coefficient of the allometric dependence of the total weight on the shell height is significantly less than that of the adult.     

Cephalisation bei Caniden

Cephalisation in Canidae Studied were the intraspecific and interspecific relationships between brain weight and body weight in Canidae. Inclination values of the allometric lines: intraspecific – 0,25, interspecific 0,571. Most of the studied species have nearly the same degree of cephalisation, deviations from the interspecific allometric line are small; the cephalisation of Nyctereutes and Otocyon is less developed than in the other canid species. There is no difference in cephalisation between most canids and felids. Mustelids show a high variability in cephalisation. Criticized were the data of Bauchot (1985) and Gittleman (1986) on brain weight and body weight.     

扎龙湿地不同生境芦苇分株的形态特征及生物量分配的异速关系

不同环境条件下的植物个体可以表现出形态特征的分异和物质分配的权衡与调整。采用大样本抽样调查与统计分析方法,比较研究扎龙湿地不同生境芦苇(Phragmites Australis)生殖株和营养株的形态特征以及生物量分配的异速关系。结果表明：在9月末,盐碱生境、旱生生境、湿生生境和水生生境芦苇分株的生长表现出较大的生态可塑性,株高和株重均以盐碱生境最小,水生生境最大,最大值与最小值的比值分别为1.3—3.3和1.8—5.1,分株生长在种群间的变异度高于种群内,与营养株相比,生殖株的变异度较低;分株的支持分配与生产分配的比值为1.8—4.2,生产分配以盐碱生境最高,以水生生境最低,而支持分配和生殖分配表现与生产分配相反的序位;生殖株的花序长和花序重与株高间呈直线函数形式增长,株高和株重低于种群平均值的20%和35%的分株不进行有性生殖;叶重、叶鞘和茎重以及分株重与株高间呈幂函数形式的异速生长关系。植物通过改变个体的形态特征以及调整构件间生物量分配适应不同环境,而受遗传因素控制的构件间生长关系却相对稳定。     

Bias in interspecific allometry: examples from morphological scaling in varanid lizards

The traditional approach to allometric analysis entails the fitting of a straight line to logarithmic transformations of the data, after which parameters in a two-parameter allometric equation are estimated by back-transformation to the original scale. We re-examined published data for dimensions of the limbs in 22 species of varanid lizards to illustrate the biases that can be introduced into allometric analyses by applying the aforementioned protocol. Statistical models fit to the original data by linear and nonlinear regression conformed better with underlying assumptions than did models obtained by back-transformation from logarithms, and the former generally were better than the latter for describing limb dimensions over the full range in body size. Allometric exponents estimated by the traditional method therefore were based on inappropriate and inaccurate statistical models and, consequently, were biased and misleading. Investigators can avoid problems such as these by performing preliminary graphical and statistical analyses on data in their original scale and by validating the fitted model. Logarithmic transformations should be used sparingly and only for cause.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 96 , 296–305.     

Minimizing Bias in Biomass Allometry: Model Selection and Log‐Transformation of Data

Nonlinear regression is increasingly used to develop allometric equations for forest biomass estimation (i.e., as opposed to the traditional approach of log‐transformation followed by linear regression). Most statistical software packages, however, assume additive errors by default, violating a key assumption of allometric theory and possibly producing spurious models. Here, we show that such models may bias stand‐level biomass estimates by up to 100 percent in young forests, and we present an alternative nonlinear fitting approach that conforms with allometric theory.     

Sexual dimorphism in the otolith shape of shi drum,Umbrina cirrosa (L.), in the eastern Mediterranean Sea: Fish size–otolith size relationships

Little is known about possible differences in sagitta otolith size and shape between sexes of the shi drum, Umbrina cirrosa, and relationships between their body and otolith size. Thus, this study aimed to fill this knowledge gap via examination of 414 sagittal otoliths from 108 male (total length 13.8–26.8 cm) and 99 female (13.5–26.7 cm) U. cirrosa caught between May 2017 and April 2018 in gillnets set at a depth of ~15 m in Mersin Bay, Eastern Mediterranean Sea. No statistical differences were observed between the shape indices of the left-sided and right-sided sagitta. However, there were significant differences in the size and shape of otoliths between males and females. The slopes of allometric power functions from otolith width × fish sizes gave significant differences between males and females (ANCOVA, P < 0.05). The relationship for length × weight of otoliths from both males and females showed isometric growth, whereas the relationship of otolith width × otolith weight showed positive allometry. Negative allometric growth was observed for the relationship otolith length × otolith width. In summary, this study revealed the presence of sexual dimorphism in the otolith shape of U. cirrosa, and the data on regression relationships of fish-otolith sizes can be used to estimate fish size from U. cirrosa otolith sizes.     

Cube law, condition factor and weight–length relationships: history, meta-analysis and recommendations

This study presents a historical review, a meta‐analysis, and recommendations for users about weight–length relationships, condition factors and relative weight equations. The historical review traces the developments of the respective concepts. The meta‐analysis explores 3929 weight–length relationships of the type W = aL^b for 1773 species of fishes. It shows that 82% of the variance in a plot of log a over b can be explained by allometric versus isometric growth patterns and by different body shapes of the respective species. Across species median b = 3.03 is significantly larger than 3.0, thus indicating a tendency towards slightly positive‐allometric growth (increase in relative body thickness or plumpness) in most fishes. The expected range of 2.5 < b < 3.5 is confirmed. Mean estimates of b outside this range are often based on only one or two weight–length relationships per species. However, true cases of strong allometric growth do exist and three examples are given. Within species, a plot of log a vs b can be used to detect outliers in weight–length relationships. An equation to calculate mean condition factors from weight–length relationships is given as K_mean = 100aL^b?3. Relative weight W_rm = 100W/(a_mL^bm) can be used for comparing the condition of individuals across populations, where a_m is the geometric mean of a and b_m is the mean of b across all available weight–length relationships for a given species. Twelve recommendations for proper use and presentation of weight–length relationships, condition factors and relative weight are given.     

Development of a Compatible Taper Function and Stand-Level Merchantable Volume Model for Chinese Fir Plantations

Chinese fir (Cunninghamia lanceolata [Lamb.] Hook) is one of the most important plantation tree species in China with good timber quality and fast growth. It covers an area of 8.54 million hectare, which corresponds to 21% of the total plantation area and 32% of total plantation volume in China. With the increasing market demand, an accurate estimation and prediction of merchantable volume at tree- and stand-level is becoming important for plantation owners. Although there are many studies on the total tree volume estimation from allometric models, these allometric models cannot predict tree- and stand-level merchantable volume at any merchantable height, and the stand-level merchantable volume model was not seen yet in Chinese fir plantations. This study aimed to develop (1) a compatible taper function for tree-level merchantable volume estimation, and (2) a stand-level merchantable volume model for Chinese fir plantations. This “taper function system” consisted in a taper function, a merchantable volume equation and a total tree volume equation. 46 Chinese fir trees were felled to develop the taper function in Shitai County, Anhui province, China. A second-order continuous autoregressive error structure corrected the inherent serial autocorrelation of different observations in one tree. The taper function and volume equations were fitted simultaneously after autocorrelation correction. The compatible taper function fitted well to our data and had very good performances in diameter and total tree volume prediction. The stand-level merchantable volume equation based on the ratio approach was developed using basal area, dominant height, quadratic mean diameter and top diameter (ranging from 0 to 30 cm) as independent variables. At last, a total stand-level volume table using stand basal area and dominant height as variables was proposed for local forest managers to simplify the stand volume estimation.