Explaining the scaling of transport costs: the role of stride frequency and stride length

The mechanisms which enable large animals to transport a unit of body mass through a unit distance at a lower metabolic cost than smaller animals have been the subject of numerous studies. Recent investigations have concluded that stride frequency is a main determinant. We examine the role of both stride frequency and stride length in determining the scaling of the cost of transport.
Slopes for regressions between stride frequency and speed and stride length and speed were determined in four species of rodents. These data were pooled with literature values for the slopes of stride frequency, stride length and cost of locomotion (all vs. speed) for a total of 17 species ranging in size from 30 g to 250 kg. Interspecific equations were calculated for each of these slopes versus body mass, and residuals from these allometric lines were calculated. Residuals were compared to see if variation in the rate of cost increase at a given size is related to variation in the rates of stride frequency and/or stride length increase.
The residual analysis revealed that the variation in transport cost is explicable only in terms of the interaction of stride frequency and stride length slopes. The product of the scaling exponents for stride frequency slope and stride length slope is not significantly different from the scaling exponent for the cost of transport. A model seeking to explain the scaling of the cost of transport must therefore consider the influence of both stride length and stride frequency.
We propose that absolutely longer limbs allow large animals to minimize the rate of increase of stride frequency and stride length with speed, and that this allows utilization of muscles with lower intrinsic rates of contraction, which in turn results in a lower mass-specific cost of transport.     

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.     

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.     

Above- and below-ground biomass relationships across 1534 forested communities

BACKGROUND AND AIMS: Prior work has shown that above- and below-ground dry biomass across individual plants scale in a near isometric manner across phyletically and ecologically diverse species. Allometric theory predicts that a similar isometric scaling relationship should hold true across diverse forest-types, regardless of vegetational composition. METHODS: To test this hypothesis, two compendia for forest-level above- and below-ground dry biomass per hectare (M(A) and M(R), respectively) were examined to test the hypothesis that M(A) vs. M(R) scales isometrically and in the same manner as reported for data from individual plants. Model Type II regression protocols were used to compare the numerical values of M(A) vs. M(R) scaling exponents (i.e. slopes of log-log linear relationships) for the combined data sets (n =1534), each of the two data sets, and data sorted into a total of 17 data subsets for community- and biome-types as well as communities dominated by angiosperms or conifers. KEY RESULTS: Among the 20 regressions examined, 15 had scaling exponents that were indistinguishable from that reported for M(A) vs. M(R) across individual plants. The isometric hypothesis could not be strictly rejected on statistical grounds; four of these 15 exponents had broad 95% confidence intervals resulting from small sample sizes. Significant variation was observed in the y-intercepts of the 20 regression curves, because of absolute differences in M(A) or M(R). CONCLUSIONS: The allometries of forest- and individual plant-level M(A) vs. M(R) relationships share strikingly similar scaling exponents, but differ because of considerable variation in y-intercepts. These results support prior allometric theory and provide boundary conditions for the scaling of M(A) and M(R).     

Testing the Metabolic Scaling Theory of tree growth

1.  Metabolic Scaling Theory (MST) predicts a 'universal scaling law' of tree growth. Proponents claim that MST has strong empirical support: the size-dependent growth curves of 40 out of 45 species in a Costa Rican forest have scaling exponents indistinguishable from the MST prediction.
2.  Here, we show that the Costa Rican study has been misinterpreted. Using Standardized Major Axis (SMA) line-fitting to estimate scaling exponents, we find that four out of five species represented by more than 100 stems have scaling exponents that deviate significantly from the MST prediction. On the other hand, sample sizes were too small to make strong inferences in the cases of 33 species represented by fewer than 50 stems.
3.  Recently, it has been argued that MST is useful for predicting average scaling exponents, even if individual species do not conform to the theory. We find that the mean scaling exponent of the Costa Rican trees is greater than predicted (across-species mean  =  0.44), and hypothesize that scaling exponents in natural forests will generally be greater than predicted, because the theory fails to model asymmetric competition for light.
4.   Synthesis . We highlight shortcomings in the interpretation of data used in support of a key MST prediction. We recommend that future research into biological scaling should compare the merits of alternative models rather than focusing attention on tests of a single theory.     

An empirical assessment of tree branching networks and implications for plant allometric scaling models

Several theories predict whole‐tree function on the basis of allometric scaling relationships assumed to emerge from traits of branching networks. To test this key assumption, and more generally, to explore patterns of external architecture within and across trees, we measure branch traits (radii/lengths) and calculate scaling exponents from five functionally divergent species. Consistent with leading theories, including metabolic scaling theory, branching is area preserving and statistically self‐similar within trees. However, differences among scaling exponents calculated at node‐ and whole‐tree levels challenge the assumption of an optimised, symmetrically branching tree. Furthermore, scaling exponents estimated for branch length change across branching orders, and exponents for scaling metabolic rate with plant size (or number of terminal tips) significantly differ from theoretical predictions. These findings, along with variability in the scaling of branch radii being less than for branch lengths, suggest extending current scaling theories to include asymmetrical branching and differential selective pressures in plant architectures.     

How far do animals go? Determinants of day range in mammals

Day range (daily distance traveled) is an important measure for understanding relationships between animal distributions and food resources. However, our understanding of variation in day range across species is limited. Here we present a day range model and compare predictions against a comprehensive analysis of mammalian day range. As found in previous studies, day range scales near the 1/4 power of body mass. Also, consistent with model predictions, taxonomic groups differ in the way day range scales with mass, associated with the most common diet types and foraging habitats. Faunivores have the longest day ranges and steepest body mass scaling. Frugivores and herbivores show intermediate and low scaling exponents, respectively. Day range in primates did not scale with mass, which may be consistent with the prediction that three-dimensional foraging habitats lead to lower exponents. Day ranges increase with group size in carnivores but not in other taxonomic groups.     

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.     

Testing Foundations of Biological Scaling Theory Using Automated Measurements of Vascular Networks

Scientists have long sought to understand how vascular networks supply blood and oxygen to cells throughout the body. Recent work focuses on principles that constrain how vessel size changes through branching generations from the aorta to capillaries and uses scaling exponents to quantify these changes. Prominent scaling theories predict that combinations of these exponents explain how metabolic, growth, and other biological rates vary with body size. Nevertheless, direct measurements of individual vessel segments have been limited because existing techniques for measuring vasculature are invasive, time consuming, and technically difficult. We developed software that extracts the length, radius, and connectivity of in vivo vessels from contrast-enhanced 3D Magnetic Resonance Angiography. Using data from 20 human subjects, we calculated scaling exponents by four methods—two derived from local properties of branching junctions and two from whole-network properties. Although these methods are often used interchangeably in the literature, we do not find general agreement between these methods, particularly for vessel lengths. Measurements for length of vessels also diverge from theoretical values, but those for radius show stronger agreement. Our results demonstrate that vascular network models cannot ignore certain complexities of real vascular systems and indicate the need to discover new principles regarding vessel lengths.     

Assessing the Jarman–Bell Principle: Scaling of intake,digestibility, retention time and gut fill with body mass in mammalian herbivores

Differences in allometric scaling of physiological characters have the appeal to explain species diversification and niche differentiation along a body mass (BM) gradient — because they lead to different combinations of physiological properties, and thus may facilitate different adaptive strategies. An important argument in physiological ecology is built on the allometries of gut fill (assumed to scale to BM^1.0) and energy requirements/intake (assumed to scale to BM^0.75) in mammalian herbivores. From the difference in exponents, it has been postulated that the mean retention time (MRT) of digesta should scale to BM^1.0–0.75 = BM^0.25. This has been used to argue that larger animals have an advantage in digestive efficiency and hence can tolerate lower-quality diets. However, empirical data does not support the BM^0.25 scaling of MRT, and the deduction of MRT scaling implies, according to physical principles, no scaling of digestibility; basing assumptions on digestive efficiency on the thus-derived MRT scaling amounts to circular reasoning. An alternative explanation considers a higher scaling exponent for food intake than for metabolism, allowing larger animals to eat more of a lower quality food without having to increase digestive efficiency; to date, this concept has only been explored in ruminants. Here, using data for 77 species in which intake, digestibility and MRT were measured (allowing the calculation of the dry matter gut contents (DMC)), we show that the unexpected shallow scaling of MRT is common in herbivores and may result from deviations of other scaling exponents from expectations. Notably, DMC have a lower scaling exponent than 1.0, and the 95% confidence intervals of the scaling exponents for intake and DMC generally overlap. Differences in the scaling of wet gut contents and dry matter gut contents confirm a previous finding that the dry matter concentration of gut contents decreases with body mass, possibly compensating for the less favorable volume–surface ratio in the guts of larger organisms. These findings suggest that traditional explanations for herbivore niche differentiation along a BM gradient should not be based on allometries of digestive physiology. In contrast, they support the recent interpretation that larger species can tolerate lower-quality diets because their intake has a higher allometric scaling than their basal metabolism, allowing them to eat relatively more of a lower quality food without having to increase digestive efficiency.     

江西常绿阔叶林木本植物不同冠层高度当年生小枝茎构型对叶生物量的影响

以江西阳际峰自然保区69种木本植物为对象,采用标准化主轴回归(SMA)的方法,对不同冠层高度当年生小枝的构件生物量(叶生物量、茎生物量、小枝生物量)与茎构型特征(茎直径、茎长度、茎宽长比、茎体积和茎密度)进行了分析,研究当年生小枝茎构型对叶生物量的影响.结果表明:不同冠层高度及生活型间叶生物量、茎生物量、小枝生物量、茎直径、茎长度、茎宽长比及茎体积均无显著差异,而茎密度差异显著.不同冠层高度及生活型间,叶生物量与茎生物量及总生物量之间均呈显著等速生长关系.小枝叶生物量分别与茎直径、茎体积呈显著异速生长关系,且该异速生长指数在不同冠层高度无显著差异.茎长度、茎宽长比及茎密度对当年生小枝叶生物量变异的解释力较小(<24%).冠层高度和生活型对小枝叶-茎生物量的等速分配关系影响不显著.小枝茎构型中,相比于茎长度、茎宽长比以及茎密度,茎的直径与体积对当年生小枝叶生物量的影响更大,冠层高度对当年生小枝茎构型与叶生物量间的异速分配模式无显著影响.     

Metabolic scaling in insects supports the predictions of the WBE model

The functional association between body size and metabolic rate (BS-MR) is one of the most intriguing issues in ecological physiology. An average scaling exponent of 3/4 is broadly observed across animal and plant taxa. The numerical value of 3/4 is theoretically predicted under the optimized version of West, Brown, and Enquist's vascular resource supply network model. Insects, however, have recently been proposed to express a numerically different scaling exponent and thus application of the WBE network model to insects has been rejected. Here, we re-analyze whether such variation is indeed supported by a global deviation across all insect taxa at the order and family levels to assess if specific taxa influence insect metabolic scaling. We show that a previous reported deviation is largely due to the effect of a single insect family (Termitidae). We conclude that the BS-MR relationship in insects broadly supports the core predictions of the WBE model. We suggest that the deviation observed within the termites warrants further investigation and may be due to either difficulty in accurately measuring termite metabolism and/or particularities of their life history. Future work on allometric scaling should assess the nature of variation around the central tendencies in scaling exponents in order to test if this variation is consistent with core assumptions and predictions of the WBE model that stem by relaxing its secondary optimizing assumptions that lead to the 3/4 exponent.     

Precision of allometric scaling equations for trees can be improved by including the effect of ecological interactions

Allometric scaling relationships of the form Y = aX ^b are widely utilized in many types of models and analyses of tree structure. They are often viewed as static relationships where both the scaling exponent (b) and the normalization constant (a) obtain empirical values that are fixed within a single set of data. Among different sets of data, their values can show environmental variability. However, there have been only few attempts to give a mechanistic interpretation for this variability. We used field data to demonstrate how the scaling relationships in trees can be modified by ecological interactions. Moreover, we show how such processes can be incorporated into the scaling models to improve the fit and the information content of the scaling equations. When fixed theoretical scaling exponents were used instead of empirical exponents and when the effect of competitive interactions between trees was described by separate submodels that predicted the value of the normalisation constant in the scaling equations, it was possible to obtain 4–10% improvement in the model fit of three different structural scaling relationships. Our results suggest that unexplained variation in the values of the scaling parameters can be substituted by an identified effect of ecological factors on the value of the normalisation constant. This agrees with recent theoretical suggestions stating that ecological factors can directly influence the value of normalisation constants.     

Structure and volume of the hoki, Macruronus novaezelandiae (Merlucciidae), swimbladder

The structure of the hoki, Macruronus novaezelandiae , swimbladder is similar to well-developed swimbladders in other deep water fish that undergo extensive diel vertical migrations. The preponderance of a globular mass of submucosal tissue in the posterio-ventral floor of the swimbladder in large individuals is an unusual feature of unknown function, but has characteristics similar to regressed swimbladders that occur in some species of the Myctophidae. Two out of three different estimates of swimbladder volume were significantly lower than 'neutral buoyancy volume' estimates, probably because the swimbladders were inflated artificially without accounting for their natural compliancy and internal pressure. Volume estimates from such swimbladders may reduce substantially the accuracy of target strength estimates from morphometry-based models of the swimbladder. However, this may be a trivial consideration in acoustic surveys compared to the probable occurrence of diel and seasonal variation in swimbladder shape and volume.     

Allometric scaling of foraging rate with trail dimensions in leaf-cutting ants

Leaf-cutting ants (Atta spp.) create physical pathways to support the transport of resources on which colony growth and reproduction depend. We determined the scaling relationship between the rate of resource acquisition and the size of the trail system and foraging workforce for 18 colonies of Atta colombica and Atta cephalotes. We examined conventional power-law scaling patterns, but did so in a multivariate analysis that reveals the simultaneous effects of forager number, trail length and trail width. Foraging rate (number of resource-laden ants returning to the nest per unit time) scaled at the 0.93 power of worker numbers, the -1.02 power of total trail length and the 0.65 power of trail width. These scaling exponents indicate that individual performance declines only slightly as more foragers are recruited to the workforce, but that trail length imposes a severe penalty on the foraging rate. A model of mass traffic flow predicts the allometric patterns for workforce and trail length, although the effect of trail width is unexpected and points to the importance of the little-known mechanisms that regulate a colony's investment in trail clearance. These results provide a point of comparison for the role that resource flows may play in allometric scaling patterns in other transport-dependent entities, such as human cities.     

The space-lifetime hypothesis: viewing organisms in four dimensions, literally

Much of the debate about alternative scaling exponents may result from unawareness of the dimensionality appropriate for different data and questions; in some cases, analysis has to include a fourth temporal dimension, and in others, it does not. Proportional scaling simultaneously applied to an organism and its generation time, treating the latter as a natural fourth dimension, produces a simple explanation for the 3/4 power in large-scale interspecies comparisons. Analysis of data sets of reduced dimensionality (e.g., data sets constructed such that one or more of the four dimensions are fixed), results in predictably lower metabolic exponents of 2/3 and 1/2 under one and two constraints, respectively. Our space-lifetime view offers a predictive framework that may be useful in developing a more complete mechanistic theory of metabolic scaling.     

Growth-size scaling relationships of woody plant species differ from predictions of the Metabolic Ecology Model

The Metabolic Ecology Model predicts that tree diameter ( D ) growth ( dD/dt ) scales with D ^1/3. Using data on diameter growth and height–diameter relationships for 56 and 40 woody species, respectively, from forests throughout New Zealand, we tested one prediction and two assumptions of this model: (i) the exponent of the growth–diameter scaling relationship equals 1/3 and is invariant among species and growth forms, (ii) small and large individuals are invariant in their exponents and (iii) tree height scales with D ^2/3. We found virtually no support for any prediction or assumption: growth–diameter scaling exponents varied substantially among species and growth forms, correlated positively with species' maximum height, and shifted significantly with increasing individual size. Tree height did not scale invariantly with diameter. Based on a quantitative test, violation of these assumptions alone could not explain the model's poor fit to our data, possibly reflecting multiple, unsound assumptions, as well as unaccounted-for variation that should be incorporated.     

Evaluation of Scaling Invariance Embedded in Short Time Series

Scaling invariance of time series has been making great contributions in diverse research fields. But how to evaluate scaling exponent from a real-world series is still an open problem. Finite length of time series may induce unacceptable fluctuation and bias to statistical quantities and consequent invalidation of currently used standard methods. In this paper a new concept called correlation-dependent balanced estimation of diffusion entropy is developed to evaluate scale-invariance in very short time series with length . Calculations with specified Hurst exponent values of show that by using the standard central moving average de-trending procedure this method can evaluate the scaling exponents for short time series with ignorable bias () and sharp confidential interval (standard deviation ). Considering the stride series from ten volunteers along an approximate oval path of a specified length, we observe that though the averages and deviations of scaling exponents are close, their evolutionary behaviors display rich patterns. It has potential use in analyzing physiological signals, detecting early warning signals, and so on. As an emphasis, the our core contribution is that by means of the proposed method one can estimate precisely shannon entropy from limited records.     

Fat fractal and multifractals for protein and enzyme surfaces.

The roughness and irregularity of the surfaces in the protein and enzyme are fractal features that may be characterized by fractal dimensions and mass exponents. The surface fractal dimensions calculated by the variation method are different from those obtained by other methods, since the former is applicable to the self-affine system. Thus the results reported here are reliable for the surfaces. However, the fat fractal and multifractal features of proteins and enzymes are studied by simulation. The surface mass exponents are regarded as another kind of scaling exponent, and the spectrum f(alpha) provides further detailed information about the surfaces of enzyme and protein. The applications of the spectrum f(alpha) to the enzymatic reactions is also discussed.     

Phenotypic plasticity in the scaling of avian basal metabolic rate

Many birds exhibit short-term, reversible adjustments in basal metabolic rate (BMR), but the overall contribution of phenotypic plasticity to avian metabolic diversity remains unclear. The available BMR data include estimates from birds living in natural environments and captive-raised birds in more homogenous, artificial environments. All previous analyses of interspecific variation in BMR have pooled these data. We hypothesized that phenotypic plasticity is an important contributor to interspecific variation in avian BMR, and that captive-raised populations exhibit general differences in BMR compared to wild-caught populations. We tested this hypothesis by fitting general linear models to BMR data for 231 bird species, using the generalized least-squares approach to correct for phylogenetic relatedness when necessary. The scaling exponent relating BMR to body mass in captive-raised birds (0.670) was significantly shallower than in wild-caught birds (0.744). The differences in metabolic scaling between captive-raised and wild-caught birds persisted when migratory tendency and habitat aridity were controlled for. Our results reveal that phenotypic plasticity is a major contributor to avian interspecific metabolic variation. The finding that metabolic scaling in birds is partly determined by environmental factors provides further support for models that predict variation in scaling exponents, such as the allometric cascade model.