A statistical model for the genetic origin of allometric scaling laws in biology

Many biological processes, from cellular metabolism to population dynamics, are characterized by particular allometric scaling (power-law) relationships between size and rate. Although such allometric relationships may be under genetic determination, their precise genetic mechanisms have not been clearly understood due to a lack of a statistical analytical method. In this paper, we present a basic statistical framework for mapping quantitative genes (or quantitative trait loci, QTL) responsible for universal quarter-power scaling laws of organic structure and function with the entire body size. Our model framework allows the testing of whether a single QTL affects the allometric relationship of two traits or whether more than one linked QTL is segregating. Like traditional multi-trait mapping, this new model can increase the power to detect the underlying QTL and the precision of its localization on the genome. Beyond the traditional method, this model is integrated with pervasive scaling laws to take advantage of the mechanistic relationships of biological structures and processes. Simulation studies indicate that the estimation precision of the QTL position and effect can be improved when the scaling relationship of the two traits is considered. The application of our model in a real example from forest trees leads to successful detection of a QTL governing the allometric relationship of third-year stem height with third-year stem biomass. The model proposed here has implications for genetic, evolutionary, biomedicinal and breeding research.     

Joint multiple quantitative trait loci mapping for allometries of body compositions and metabolic traits to body weights in broiler

In order to map quantitative trait loci (QTLs) for allometries of body compositions and metabolic traits in chicken, we phenotypically characterize the allometric growths of multiple body components and metabolic traits relative to BWs using joint allometric scaling models and then establish random regression models (RRMs) to fit genetic effects of markers and minor polygenes derived from the pedigree on the allometric scalings. Prior to statistically inferring the QTLs for the allometric scalings by solving the RRMs, the LASSO technique is adopted to rapidly shrink most of marker genetic effects to zero. Computer simulation analysis confirms the reliability and adaptability of the so-called LASSO-RRM mapping method. In the F₂ population constructed by multiple families, we formulate two joint allometric scaling models of body compositions and metabolic traits, in which six of nine body compositions are tested as significant, while six of eight metabolic traits are as significant. For body compositions, a total of 14 QTLs, of which 9 dominant, were detected to be associated with the allometric scalings of drumstick, fat, heart, shank, liver and spleen to BWs; while for metabolic traits, a total of 19 QTLs also including 9 dominant be responsible for the allometries of T4, IGFI, IGFII, GLC, INS, IGR to BWs. The detectable QTLs or highly linked markers can be used to regulate relative growths of the body components and metabolic traits to BWs in marker-assisted breeding of chickens.     

How dispersal limitation shapes species-body size distributions in local communities

A critical but poorly understood pattern in macroecology is the often unimodal species-body size distribution (also known as body size-diversity relationship) in a local community (embedded in a much larger regional species pool). Purely neutral community models that assume functional equivalence among species are incapable of explaining this pattern because body size is the key determinant of functional differences between species. Several niche-based explanations have been offered, but none of them is completely satisfactory. Here we develop a simple model that unites a neutral community model with niche-based theory to explain the relationship. In the model, species of similar size are assumed to belong to the same size guild. Within a size guild, all individuals are equivalent in their competition for resources, sensu Hubbell's neutral community model; they have the same speciation rate and dispersal capacities. Between size guilds, however, the total number of individuals, the speciation rate, and the dispersal capacities differ, but using known allometric scaling laws for these properties, we can describe the differences between size guilds. Our model predicts that species richness reaches an optimum at an intermediate body size, in agreement with observations. The optimum at intermediate body size is basically the result of a trade-off between, on the one hand, allometric scaling laws for the number of individuals and the speciation rate that decrease with body size and, on the other hand, the scaling law for active dispersal that increases with body size.     

High-Throughput Tissue Bioenergetics Analysis Reveals Identical Metabolic Allometric Scaling for Teleost Hearts and Whole Organisms

Organismal metabolic rate, a fundamental metric in biology, demonstrates an allometric scaling relationship with body size. Fractal-like vascular distribution networks of biological systems are proposed to underlie metabolic rate allometric scaling laws from individual organisms to cells, mitochondria, and enzymes. Tissue-specific metabolic scaling is notably absent from this paradigm. In the current study, metabolic scaling relationships of hearts and brains with body size were examined by improving on a high-throughput whole-organ oxygen consumption rate (OCR) analysis method in five biomedically and environmentally relevant teleost model species. Tissue-specific metabolic scaling was compared with organismal routine metabolism (RMO₂), which was measured using whole organismal respirometry. Basal heart OCR and organismal RMO₂ scaled identically with body mass in a species-specific fashion across all five species tested. However, organismal maximum metabolic rates (MMO₂) and pharmacologically-induced maximum cardiac metabolic rates in zebrafish Danio rerio did not show a similar relationship with body mass. Brain metabolic rates did not scale with body size. The identical allometric scaling of heart and organismal metabolic rates with body size suggests that hearts, the power generator of an organism’s vascular distribution network, might be crucial in determining teleost metabolic rate scaling under routine conditions. Furthermore, these findings indicate the possibility of measuring heart OCR utilizing the high-throughput approach presented here as a proxy for organismal metabolic rate—a useful metric in characterizing organismal fitness. In addition to heart and brain OCR, the current approach was also used to measure whole liver OCR, partition cardiac mitochondrial bioenergetic parameters using pharmacological agents, and estimate heart and brain glycolytic rates. This high-throughput whole-organ bioenergetic analysis method has important applications in toxicology, evolutionary physiology, and biomedical sciences, particularly in the context of investigating pathogenesis of mitochondrial diseases.     

Scaling in the animal kingdom

Several of the known scaling laws in the animal kingdom are based on a so-called allometric correlation in which some physical quantity is presumed to scale as some power of the mass of the animal. Such a simple correlation, when deduced purely as an empirical result, often hides the physical balances that fix the relevant scaling law. In particular, the emphasis on a simple allometric scaling has often masked the fundamental role played by time scales associated with the physical balances being struck. In this paper I have concentrated on three different attributes to which the use of dimensional analysis, scaling arguments and some judicious guesswork have led to new results and an understanding of some balances that occur in the animal kingdom. The running speed of animals is examined and a rationale deduced for the resolution of a conundrum first posed by A.V. Hill of why it is that many animals appear to have approximately the same maximum speed. A complete dimensional analysis for scaling the basal metabolic rate for a class of animals suggests that a detailed understanding of the physical balances that fix the metabolic rate could be quite subtle. However, the use of such an analysis has led to the discovery of a new correlation for mammals, relating the metabolic rate to the mass and the pulse rate of the animal. At the heart of many scaling laws for animal motion is the provision of an estimate of how the skeletal structure depends on the mass of the animal. It has been known for some time that the assumption of isometry between the builds of animals is too constrictive to describe the observed scaling laws. It is shown here how to relax the isometric assumption and deduce scaling laws in good agreement with observation. Thus, it appears that the skeletal dimensions of many animals with exoskeletons are fixed by the need to support static rather than dynamical loads. The scaling laws associated with endoskeletons are more complex, apparently, though the analysis does suggest that it is dynamical loading which is decisive for the skeletal design of land mammals.     

Precision Mapping of Quantitative Trait Loci

Adequate separation of effects of possible multiple linked quantitative trait loci (QTLs) on mapping QTLs is the key to increasing the precision of QTL mapping. A new method of QTL mapping is proposed and analyzed in this paper by combining interval mapping with multiple regression. The basis of the proposed method is an interval test in which the test statistic on a marker interval is made to be unaffected by QTLs located outside a defined interval. This is achieved by fitting other genetic markers in the statistical model as a control when performing interval mapping. Compared with the current QTL mapping method (i.e., the interval mapping method which uses a pair or two pairs of markers for mapping QTLs), this method has several advantages. (1) By confining the test to one region at a time, it reduces a multiple dimensional search problem (for multiple QTLs) to a one dimensional search problem. (2) By conditioning linked markers in the test, the sensitivity of the test statistic to the position of individual QTLs is increased, and the precision of QTL mapping can be improved. (3) By selectively and simultaneously using other markers in the analysis, the efficiency of QTL mapping can be also improved. The behavior of the test statistic under the null hypothesis and appropriate critical value of the test statistic for an overall test in a genome are discussed and analyzed. A simulation study of QTL mapping is also presented which illustrates the utility, properties, advantages and disadvantages of the method.     

A theoretical interpretation of length-biomass allometry of predominantly bidimensional seaweeds

Experimental studies on bidimensional seaweeds revealed a scaling exponent of 0.472 for their length-biomass allometry. This was significantly higher than the value 0.25, which was proposed earlier as universal for all primary producers, based on the data for unicellular microalgae and vascular plants. Later, an exponent of 0.5 was theoretically derived, which agreed, to some extent, with experimental findings. Here, it is shown that there exists a power-law relation between the two perpendicular length parameters along the directions of growth of a bidimensional organism. The length-biomass allometric parameters can be expressed in terms of this power index. A relation between the allometric scaling exponent and allometric constant, involving the mass per unit area, has been obtained analytically. A method is proposed to determine the power index experimentally. Some mathematical expressions, relating mass, length and other parameters, have been formulated and these would be useful for experimental purposes in allometric studies. Analyzing images from an experimental study, a lot of parameters, regarding flat seaweeds, have been determined by analytical and numerical techniques.     

Allometric scaling of wall shear stress from mice to humans: quantification using cine phase-contrast MRI and computational fluid dynamics

Allometric scaling laws relate structure or function between species of vastly different sizes. They have rarely been derived for hemodynamic parameters known to affect the cardiovascular system, e.g., wall shear stress (WSS). This work describes noninvasive methods to quantify and determine a scaling law for WSS. Geometry and blood flow velocities in the infrarenal aorta of mice and rats under isoflurane anesthesia were quantified using two-dimensional magnetic resonance angiography and phase-contrast magnetic resonance imaging at 4.7 tesla. Three-dimensional models constructed from anatomic data were discretized and used for computational fluid dynamic simulations using phase-contrast velocity imaging data as inlet boundary conditions. WSS was calculated along the infrarenal aorta and compared between species to formulate an allometric equation for WSS. Mean WSS along the infrarenal aorta was significantly greater in mice and rats compared with humans (87.6, 70.5, and 4.8 dyn/cm(2), P < 0.01), and a scaling exponent of -0.38 (R(2) = 0.92) was determined. Manipulation of the murine genome has made small animal models standard surrogates for better understanding the healthy and diseased human cardiovascular system. It has therefore become increasingly important to understand how results scale from mouse to human. This noninvasive methodology provides the opportunity to serially quantify changes in WSS during disease progression and/or therapeutic intervention.     

An ecophysiologically based mapping model identifies a major pleiotropic QTL for leaf growth trajectories of Phaseolus vulgaris

Crop modeling, a widely used tool to predict plant growth and development in heterogeneous environments, has been increasingly integrated with genetic information to improve its predictability. This integration can also shed light on the mechanistic path that connects the genotype to a particular phenotype under specific environments. We implemented a bivariate statistical procedure to map and identify quantitative trait loci (QTLs) that can predict the form of plant growth by estimating cultivar‐specific growth parameters and incorporating these parameters into a mapping framework. The procedure enables the characterization of how QTLs act differently in response to developmental and environmental cues. We used this procedure to map growth parameters of leaf area and mass in a mapping population of the common bean (Phaseolus vulgaris L.). Different sets of QTLs are responsible for various aspects of growth, including the initiation time of growth, growth rate, inflection point and asymptotic growth. A major QTL of a large effect was identified to pleiotropically affect trait expression in distinct environments and different traits expressed on the same organism. The integration of crop models and QTL mapping through our statistical procedure provides a powerful means of building a more precise predictive model of genotype‐phenotype relationships for crops.     

Computational identification of genes modulating stem height–diameter allometry

The developmental variation in stem height with respect to stem diameter is related to a broad range of ecological and evolutionary phenomena in trees, but the underlying genetic basis of this variation remains elusive. We implement a dynamic statistical model, functional mapping, to formulate a general procedure for the computational identification of quantitative trait loci (QTLs) that control stem height–diameter allometry during development. Functional mapping integrates the biological principles underlying trait formation and development into the association analysis of DNA genotype and endpoint phenotype, thus providing an incentive for understanding the mechanistic interplay between genes and development. Built on the basic tenet of functional mapping, we explore two core ecological scenarios of how stem height and stem diameter covary in response to environmental stimuli: (i) trees pioneer sunlit space by allocating more growth to stem height than diameter and (ii) trees maintain their competitive advantage through an inverse pattern. The model is equipped to characterize ‘pioneering’ QTLs (piQTLs) and ‘maintaining’ QTLs (miQTLs) which modulate these two ecological scenarios, respectively. In a practical application to a mapping population of full‐sib hybrids derived from two Populus species, the model has well proven its versatility by identifying several piQTLs that promote height growth at a cost of diameter growth and several miQTLs that benefit radial growth at a cost of height growth. Judicious application of functional mapping may lead to improved strategies for studying the genetic control of the formation mechanisms underlying trade‐offs among quantities of assimilates allocated to different growth parts.     

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.     

Trade-offs between the metabolic rate and population density of plants

The energetic equivalence rule, which is based on a combination of metabolic theory and the self-thinning rule, is one of the fundamental laws of nature. However, there is a progressively increasing body of evidence that scaling relationships of metabolic rate vs. body mass and population density vs. body mass are variable and deviate from their respective theoretical values of 3/4 and -3/4 or -2/3. These findings questioned the previous hypotheses of energetic equivalence rule in plants. Here we examined the allometric relationships between photosynthetic mass (M(p)) or leaf mass (M(L)) vs. body mass (beta); population density vs. body mass (delta); and leaf mass vs. population density, for desert shrubs, trees, and herbaceous plants, respectively. As expected, the allometric relationships for both photosynthetic mass (i.e. metabolic rate) and population density varied with the environmental conditions. However, the ratio between the two exponents was -1 (i.e. beta/delta = -1) and followed the trade-off principle when local resources were limited. Our results demonstrate for the first time that the energetic equivalence rule of plants is based on trade-offs between the variable metabolic rate and population density rather than their constant allometric exponents.     

Improved approximations to scaling relationships for species, populations, and ecosystems across latitudinal and elevational gradients

Historically, allometric equations relate organismal traits, such as metabolic rate, individual growth rate, and lifespan, to body mass. Similarly, Boltzmann or Q(10) factors are used to relate many organismal traits to body temperature. Allometric equations and Boltzmann factors are being applied increasingly to higher levels of biological organization in an attempt to describe aggregate properties of populations and ecosystems. They have been used previously for studies that analyse scaling relationships between populations and across latitudinal gradients. For these kinds of applications, it is crucial to be aware of the "fallacy of the averages", and it is often problematic or incorrect to simply substitute the average body mass or temperature for an entire population or ecosystem into allometric equations. We derive improved approximations to allometric equations and Boltzmann factors in terms of the central moments of body size and temperature, and we provide tests for the accuracy of these approximations. This framework is necessary for interpreting the predictions of scaling theories for large-scale systems and grants insight into which characteristics of a given distribution are important. These approximations and tests are applied to data for body size for several taxonomic groups, including groups with multiple species, and to data for temperature at locations of varying latitude, corresponding to ectothermic body temperatures. Based on these results, the accuracy and utility of these approximations as applied to biological systems are assessed. We conclude that approximations to allometric equations at the species level are extremely accurate. However, for systems with a large range in body size, evaluating the skewness and kurtosis is often necessary, so it may be advantageous to calculate the exact form for the averaged scaling relationships instead. Moreover, the improved approximation for the Boltzmann factor, which uses the average and standard deviation of temperature, is quite accurate and represents a significant improvement over previous approximations.     

Exponential mapping of quantitative trait loci governing allometric relationships in organisms

Allometric scaling relationships or quarter-power rules, as a universal biological law, can be viewed as having some genetic component, and the particular genes (or quantitative trait loci, QTL) underlying these allometric relationships can be mapped using molecular markers. We develop a mathematical and statistical model for mapping allometric QTL on the basis of nonlinear power functions using Taylors approximation theory. Simulation studies indicate that the QTL position and effect can be estimated using our model, but the estimation precision can be improved from the higher- over lower-order approximation when the sample size used and gene effects are small. The application of our approach in a real example from forest trees leads to successful detection of a QTL governing the allometric relationship between 3rd-year stem height and 3rd-year stem biomass. It is expected that our model will have broad implications for genetic, evolutionary, biomedical and breeding research.     

异速生长模型研究概述

最近,关于异速生长模型的讨论再次成为焦点,讨论热点为异速生长指数的取值及其理论解释.本文综述了WBE 97、BMR(99)模型的相关研究,重点介绍了MGL模型及由此模型得到的结果:个体整体的新陈代谢率与个体的质量没有明显依赖关系,其标度指数不是一个固定的值,而是一个区间[2/3,1].考虑的视角从个体整体的新陈代谢率转到单位质量的新陈代谢率,通过对不同物种、不同环境的单位质量新陈代谢率的研究,发现对大多数物种,其值落在一个具有普适性的上、下界的区间内;认为存在单位质量的新陈代谢率最小值确定了个体的大小,并建立基于该最小值的描述个体大小与温度关系的数学模型,该模型得到实验数据验证.     

Dichotomy of eutherian reproduction and metabolism

How anatomical, physiological and ecological (life history) features scale with body mass is a fundamental question in biology. There is an ongoing debate in the scientific literature whether allometric scaling follows a universal pattern that can be described in a single model, or differs between groups. However, recently some analyses were published demonstrating a change in scaling across the body mass range: brain‐size allometry of mammals indicates that scaling follows a curvilinear pattern in double‐logarithmic space, and a quadratic pattern in double‐logarithmic space was found in one of the largest physiological datasets, on basal metabolic rate (MR) in mammals. Here, we analysed a variety of independent datasets on anatomical, physiological and ecological characteristics in mammals, birds and reptiles to answer the question whether the quadratic scaling is a universal biological law, or a pattern unique to mammals. The pattern was present in mammalian basal and field MR, brain size, and reproduction parameters, but neither in other organ allometries in mammals, nor in the scaling of MR in birds and reptiles. However, the curvature was better explained by separate allometric scaling of three different mammalian reproduction strategies: marsupials, and eutherian mammals with one and with many offspring. The two latter strategies are distributed unequally over the body mass range in eutherian mammals. Our findings show that a quadratic model, as well as a traditional allometric model with a universal scaling exponent (such as 0.67 or 0.75), may be inappropriate in mammals as they are a result of different scalings within these three reproductive groups. We propose that the observed distribution pattern is the result of the eutherian mammal clade's uniquely pronounced dichotomy of reproductive strategies.     

Aspects of line-fitting in bivariate allometric analyses

One of the fundamental problems involved in analyses of the scaling effects of body size (allometric analysis is the choice of an appropriate best-fit line in bivariate logarithmic plots. Following a discussion of some basic aspects of allometric analysis, the tow mai procedures for the determination of a best-fit line - the least-squares regression and the major axis - are examined with respect to their different properties and underlying models. It is important to distinguish intraspecific from interspecific scaling and to recognize the distinction between use of a best-fit line to define a relationship and use of the line for prediction. An alternative model to the bivariate normal distribution, referred to as the 'extruded normal distribution', is presented and its implications are examined with respect to two test cases (scaling of basal metabolic rate in human males; scaling of population density in mammals).     

Predicting root biomass from branching patterns of Douglas-fir root systems

There are many examples of branching networks in nature, such as tree crowns, river systems, arteries and lungs. These networks have often been described as being self-similar, or following scale-invariant branching rules, and this property has been used to derive several scaling laws. In this paper we model root systems of Douglas-fir ( Pseudotsuga menziesii var. glauca (Beissn.) Franco) as branching networks following several simple branching rules. Our objective is to establish a relationship between trunk diameter and root biomass. We explore the effect of the self-similar branching assumption on this relationship. Using data collected from a mature stand in British Columbia, we find that branching asymmetry and the rate of root taper change with root size, thereby violating the assumption of self-similarity. However, the data are in general agreement with Leonardo da Vinci's area-preserving branching hypothesis. We use the field data to parameterize two models, one assuming self-similar branching and a second incorporating the measured size dependencies of branching parameters. The two models differ by only a small amount (≈8%) in their predictions. For both models, the predicted relationship between trunk diameter and root biomass is in good concordance with previously published empirical data. We conclude that the assumption of self-similar branching, although violated by the data, nevertheless provides a useful tool for predicting the allometric relationship between trunk diameter and root biomass. Finally, we use our models to show that the geometric properties of individual bifurcations fundamentally change the root biomass cost of different root topologies.     

WBE 模型及其在生态学中的应用：研究概述

介绍了WBE模型,综述了该模型在生态学中的应用进展。WBE模型,以及以该模型为基础的MTE模型,假设生物体为自相似分形网络结构,提出代谢速率和个体大小之间存在3/4指数关系,分别预测了从个体到生物圈多个尺度上的生物属性之间的异速生长关系,而且部分得到了验证。WBE模型的应用涵盖了个体组织生物量、年生长率,种群密度和生态系统单位面积产量、能量流动率等多个方面;即使在生物圈大尺度上,WBE模型也可用来预测试验中无法直接测量的特征变量的属性,如全球碳储量的估算等。至今,关于WBE和MTE模型仍然存在各种褒贬争论,讨论焦点主要集中于模型建立的前提假设以及权度指数的预测。今后的研究工作应规范试验技术和方法,考虑物种多样性和环境等因素的影响,提出符合各类生物的模型结构体系,使其具有更广泛的应用性和预测性。