Allometries and scaling laws interpreted as laws: a reply to Elgin

I analyze here biological regression equations known in the literature as allometries and scaling laws. My focus is on the alleged lawlike status of these equations. In particular I argue against recent views that regard allometries and scaling laws as representing universal, non-continent, and/or strict biological laws. Although allometries and scaling laws appear to be generalizations applying to many taxa, they are neither universal nor exceptionless. In fact there appear to be exceptions to all of them. Nor are the constants in allometries and scaling laws truly constant, stable, or universal in character, but vary in value across different taxa and background conditions. Moreover, these equations represent evolutionary, strongly contingent generalizations, which threatens their lawlike status. Lastly, allometries and scaling laws do not offer stable probabilities to which they hold in different backgrounds. I further suggest that many allometries and scaling laws function to elucidate explananda rather than explanantia or covering laws.     

Validation of Normalizations,Scaling, and Photofading Corrections for FRAP Data Analysis

Fluorescence Recovery After Photobleaching (FRAP) has been a versatile tool to study transport and reaction kinetics in live cells. Since the fluorescence data generated by fluorescence microscopy are in a relative scale, a wide variety of scalings and normalizations are used in quantitative FRAP analysis. Scaling and normalization are often required to account for inherent properties of diffusing biomolecules of interest or photochemical properties of the fluorescent tag such as mobile fraction or photofading during image acquisition. In some cases, scaling and normalization are also used for computational simplicity. However, to our best knowledge, the validity of those various forms of scaling and normalization has not been studied in a rigorous manner. In this study, we investigate the validity of various scalings and normalizations that have appeared in the literature to calculate mobile fractions and correct for photofading and assess their consistency with FRAP equations. As a test case, we consider linear or affine scaling of normal or anomalous diffusion FRAP equations in combination with scaling for immobile fractions. We also consider exponential scaling of either FRAP equations or FRAP data to correct for photofading. Using a combination of theoretical and experimental approaches, we show that compatible scaling schemes should be applied in the correct sequential order; otherwise, erroneous results may be obtained. We propose a hierarchical workflow to carry out FRAP data analysis and discuss the broader implications of our findings for FRAP data analysis using a variety of kinetic models.     

Beyond the '3/4-power law': variation in the intra- and interspecific scaling of metabolic rate in animals

In this review I show that the '3/4-power scaling law' of metabolic rate is not universal, either within or among animal species. Significant variation in the scaling of metabolic rate with body mass is described mainly for animals, but also for unicells and plants. Much of this variation, which can be related to taxonomic, physiological, and/or environmental differences, is not adequately explained by existing theoretical models, which are also reviewed. As a result, synthetic explanatory schemes based on multiple boundary constraints and on the scaling of multiple energy-using processes are advocated. It is also stressed that a complete understanding of metabolic scaling will require the identification of both proximate (functional) and ultimate (evolutionary) causes. Four major types of intraspecific metabolic scaling with body mass are recognized [based on the power function R=aMb, where R is respiration (metabolic) rate, a is a constant, M is body mass, and b is the scaling exponent]: Type I: linear, negatively allometric (b<1); Type II: linear, isometric (b=1); Type III: nonlinear, ontogenetic shift from isometric (b=1), or nearly isometric, to negatively allometric (b<1); and Type IV: nonlinear, ontogenetic shift from positively allometric (b>1) to one or two later phases of negative allometry (b<1). Ontogenetic changes in the metabolic intensity of four component processes (i.e. growth, reproduction, locomotion, and heat production) appear to be important in these different patterns of metabolic scaling. These changes may, in turn, be shaped by age (size)-specific patterns of mortality. In addition, major differences in interspecific metabolic scaling are described, especially with respect to mode of temperature regulation, body-size range, and activity level. A 'metabolic-level boundaries hypothesis' focusing on two major constraints (surface-area limits on resource/waste exchange processes and mass/volume limits on power production) can explain much, but not all of this variation. My analysis indicates that further empirical and theoretical work is needed to understand fully the physiological and ecological bases for the considerable variation in metabolic scaling that is observed both within and among species. Recommended approaches for doing this are discussed. I conclude that the scaling of metabolism is not the simple result of a physical law, but rather appears to be the more complex result of diverse adaptations evolved in the context of both physico-chemical and ecological constraints.     

Fractal dimensions of cranial sutures and waveforms.

Two quite different shapes of cranial sutures ostensibly yield fractal dimensions. The rare, intricate sutures yield the more valid fractal dimensions because self-similar scaling provides a double-log plot of negative slope. These sutures are fractals over a range of several r values. Some of the highly folded, wavy sutures in humans also fill space except at very tiny values of r, but are nonfractal. A great deal depends on whether the dimension D is > 1 and by how much, whether the curve yields a false fractal dimension, whether the curve scales and shows self-similarity, and whether the scaling occurs regularly in the same pattern. We suggest careful attention to the inverse power law equations, which when misused can yield false fractal values. Cranial sutures vary from the simple wavy sutures to the complex folded ones, and, in rare instances, evolve and develop to the self-similar, scaling, elaborate ones called intricate sutures. The main thing is to express the biology precisely, whether waveform regularity or irregularity or scaling elaboration conserving space and the original shape. D values may not in themselves reliably allow such a distinction, by whatever method used.     

Functional blueprints: an approach to modularity in grown systems

The engineering of grown systems poses fundamentally different system integration challenges than ordinary engineering of static designs. On the one hand, a grown system must be capable of surviving not only in its final form, but at every intermediate stage, despite the fact that its subsystems may grow unevenly or be subject to different scaling laws. On the other hand, the ability to grow offers much greater potential for adaptation, either to changes in the environment or to internal stresses developed as the system grows. I observe that the ability of subsystems to tolerate stress can be used to transform incremental adaptation into the dynamic discovery of viable growth trajectories for the system as a whole. Using this observation, I propose an engineering approach based on functional blueprints, under which a system is specified in terms of desired performance and means of incrementally correcting deficiencies. I explore how manifold geometric programming can support such an approach by simplifying the construction of distortion-tolerant programs, then demonstrate the functional blueprints approach by applying it to integrate simplified models of tissue growth and vascularization, and further show how the composed system may itself be modulated for use as a component in a more complex design.     

Exploitation dynamics of fish stocks

I address the question of the fluctuations in fishery landings. Using the fishery statistics time-series collected by the Food and Agriculture Organization of the United Nations since the early 1950s, I here analyze fishing activities and find two scaling features of capture fisheries production: (i) the standard deviation of growth rate of the domestically landed catches decays as a power–law function of country landings with an exponent of value 0.15 and (ii) the average number of fishers in a country scales to the 0.7 power of country landings. I show how these socio-ecological patterns may be related, yielding a scaling relation between these exponents. The predicted scaling relation implies that the width of the annual per capita growth-rate distribution scales to the 0.2 power of country landings, i.e. annual fluctuations in per capita landed catches increase with increased per capita catches in highly producing countries. Beside the scaling behavior, I report that fluctuations in the annual domestic landings have increased in the last 30 years, while the mean of the annual growth rate declined significantly after 1972.     

A Scaling Rule for Landscape Patches and How It Applies to Conserving Soil Resources in Savannas

Scaling issues are complex, yet understanding issues such as scale dependencies in ecological patterns and processes is usually critical if we are to make sense of ecological data and if we want to predict how land management options, for example, are constrained by scale. In this article, we develop the beginnings of a way to approach the complexity of scaling issues. Our approach is rooted in scaling functions, which integrate the scale dependency of patterns and processes in landscapes with the ways that organisms scale their responses to these patterns and processes. We propose that such functions may have sufficient generality that we can develop scaling rules—statements that link scale with consequences for certain phenomena in certain systems. As an example, we propose that in savanna ecosystems, there is a consistent relationship between the size of vegetation patches in the landscape and the degree to which critical resources, such as soil nutrients or water, become concentrated in these patches. In this case, the features of the scaling functions that underlie this rule have to do with physical processes, such as surface water flow and material redistribution, and the ways that patches of plants physically “capture” such runoff and convert it into plant biomass, thereby concentrating resources and increasing patch size. To be operationally useful, such scaling rules must be expressed in ways that can generate predictions. We developed a scaling equation that can be used to evaluate the potential impacts of different disturbances on vegetation patches and on how soils and their nutrients are conserved within Australian savanna landscapes. We illustrate that for a 10-km² paddock, given an equivalent area of impact, the thinning of large tree islands potentially can cause a far greater loss of soil nitrogen (21 metric tons) than grazing out small grass clumps (2 metric tons). Although our example is hypothetical, we believe that addressing scaling problems by first conceptualizing scaling functions, then proposing scaling rules, and then deriving scaling equations is a useful approach. Scaling equations can be used in simulation models, or (as we have done) in simple hypothetical scenarios, to collapse the complexity of scaling issues into a manageable framework. Received 8 December 1998; accepted 17 August 1999.     

Predicting trophic relations in ecological networks: A test of the Allometric Diet Breadth Model

Few food web theory hypotheses/predictions can be readily tested using likelihoods of reproducing the data. Simple probabilistic models for food web structure, however, are an exception as their likelihoods were recently derived. Here I test the performance of a more complex model for food web structure that is grounded in the allometric scaling of interactions with body size and the theory of optimal foraging (Allometric Diet Breadth Model—ADBM). This deterministic model has been evaluated by measuring the fraction of trophic relations it correctly predicts. I contrasted this value with that produced by simpler models based on body sizes and found that the quantitative information on allometric scaling and optimal foraging does not significantly increase model fit. Also, I present a method to compute the p-value for the fraction of trophic interactions correctly predicted by the ADBM, or any other model, with respect to three probabilistic models. I find that the ADBM predicts significantly more links than random graphs, but other models can outperform it. Although optimal foraging and allometric scaling may improve our understanding of food webs, the ADBM needs to be modified or replaced to find support in the data.     

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.     

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.     

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.     

A modified cable formalism for modeling neuronal membranes at high frequencies

Intracellular recordings of cortical neurons in vivo display intense subthreshold membrane potential (V_m) activity. The power spectral density of the V_m displays a power-law structure at high frequencies (>50 Hz) with a slope of ∼−2.5. This type of frequency scaling cannot be accounted for by traditional models, as either single-compartment models or models based on reconstructed cell morphologies display a frequency scaling with a slope close to −4. This slope is due to the fact that the membrane resistance is short-circuited by the capacitance for high frequencies, a situation which may not be realistic. Here, we integrate nonideal capacitors in cable equations to reflect the fact that the capacitance cannot be charged instantaneously. We show that the resulting nonideal cable model can be solved analytically using Fourier transforms. Numerical simulations using a ball-and-stick model yield membrane potential activity with similar frequency scaling as in the experiments. We also discuss the consequences of using nonideal capacitors on other cellular properties such as the transmission of high frequencies, which is boosted in nonideal cables, or voltage attenuation in dendrites. These results suggest that cable equations based on nonideal capacitors should be used to capture the behavior of neuronal membranes at high frequencies.     

How to record a million synaptic weights in a hippocampal slice

A key step toward understanding the function of a brain circuit is to find its wiring diagram. New methods for optical stimulation and optical recording of neurons make it possible to map circuit connectivity on a very large scale. However, single synapses produce small responses that are difficult to measure on a large scale. Here I analyze how single synaptic responses may be detectable using relatively coarse readouts such as optical recording of somatic calcium. I model a network consisting of 10,000 input axons and 100 CA1 pyramidal neurons, each represented using 19 compartments with voltage-gated channels and calcium dynamics. As single synaptic inputs cannot produce a measurable somatic calcium response, I stimulate many inputs as a baseline to elicit somatic action potentials leading to a strong calcium signal. I compare statistics of responses with or without a single axonal input riding on this baseline. Through simulations I show that a single additional input shifts the distribution of the number of output action potentials. Stochastic resonance due to probabilistic synaptic release makes this shift easier to detect. With ~80 stimulus repetitions this approach can resolve up to 35% of individual activated synapses even in the presence of 20% recording noise. While the technique is applicable using conventional electrical stimulation and extracellular recording, optical methods promise much greater scaling, since the number of synapses scales as the product of the number of inputs and outputs. I extrapolate from current high-speed optical stimulation and recording methods, and show that this approach may scale up to the order of a million synapses in a single two-hour slice-recording experiment.     

Energetic inequivalence in eusocial insect colonies

The energetic equivalence rule states that population-level metabolic rate is independent of average body size. This rule has been both supported and refuted by allometric studies of abundance and individual metabolic rate, but no study, to my knowledge, has tested the rule with direct measurements of whole-population metabolic rate. Here, I find a positive scaling of whole-colony metabolic rate with body size for eusocial insects. Individual metabolic rates in these colonies scaled with body size more steeply than expected from laboratory studies on insects, while population size was independent of body size. Using consumer-resource models, I suggest that the colony-level metabolic rate scaling observed here may arise from a change in the scaling of individual metabolic rate resulting from a change in the body size dependence of mortality rates.     

A unifying explanation for diverse metabolic scaling in animals and plants

The scaling of metabolic rate with body mass has long been a controversial topic. Some workers have claimed that the slope of log-log metabolic scaling relationships typically obeys a universal 3/4-power law resulting from the geometry of resource-transport networks. Others have attempted to explain the broad diversity of metabolic scaling relationships. Although several potentially useful models have been proposed, at present none successfully predicts the entire range of scaling relationships seen among both physiological states and taxonomic groups of animals and plants. Here I argue that our understanding may be aided by three shifts in focus: from explaining average tendencies to explaining variation between extreme boundary limits, from explaining the slope and elevation (metabolic level) of scaling relationships separately to showing how and why they are interrelated, and from focusing primarily on internal factors (e.g. body design) to a more balanced consideration of both internal and external (ecological) factors. By incorporating all of these shifts in focus, the recently proposed metabolic-level boundaries hypothesis appears to provide a useful way of explaining both taxonomic and physiological variation in metabolic scaling relationships. This hypothesis correctly predicts that the scaling slope should vary mostly between 2/3 and 1 and that it should be related to metabolic (activity) level according to an approximately U-shaped function. It also implies that the scaling of other energy-dependent biological processes should be related to the metabolic level of the organisms being examined. Some data are presented that support this implication, but further research is needed.     

Adjustments to McConville et al. and Young et al. body segment inertial parameters

Body segment inertial parameters (BSIPs) are important data in biomechanics. They are usually estimated from predictive equations reported in the literature. However, most of the predictive equations are ambiguously applicable in the conventional 3D segment coordinate systems (SCSs). Also, the predictive equations reported in the literature all include two assumptions: the centre of mass and the proximal and distal endpoints are assumed to be aligned, and the inertia tensor is assumed to be principal in the segment axes. These predictive equations, restraining both position of the centre of mass and orientation of the principal axes of inertia, become restrictive when computing 3D inverse dynamics, when analyzing the influence of BSIP estimations on joint forces and moments and when evaluating personalized 3D BSIPs obtained from medical imaging. In the current study, the extensive data from McConville et al. (1980. Anthropometric relationships of body and body segment moments of inertia. AFAMRL-TR-80-119, Aerospace Medical Research Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio) and from Young et al. (1983. Anthropometric and mass distribution characteristics of the adults female. Technical Report AFAMRL-TR-80-119, FAA Civil Aeromedical Institute, Oklaoma City, Oklaoma) are adjusted in order to correspond to joint centres and to conventional segment axes. In this way, scaling equations are obtained for both males and females that provide BSIPs which are directly applicable in the conventional SCSs and do not restrain the position of the centre of mass and the orientation of the principal axes. These adjusted scaling equations may be useful for researchers who wish to use appropriate 3D BSIPs for posture and movement analysis.     

Self-similarity in species–area relationship and in species abundance distribution

In an influential paper, Harte et al. highlighted the scaling or 'self-similar' character of the power law species–area relationship (SAR), and used this feature to derive a species abundance distribution (SAD) and an endemics–area relationship. Here I show that their analysis was incorrect and leads to unrealistic results. I develop a different approach and obtain different results, both for SAD and for endemism. In particular, I show that the power law SAR is naturally associated with the power law statistical distribution, which is the only self-similar distribution and closely matches empirical SADs. The results in this paper shed light on some of the main issues that have been discussed with regard to SARs: their relationship with the lognormal and with the neutral theory, the relative importance of sampling effects vs other mechanisms, and the deviations from a power law. The equations that I develop are simple and easy to apply to field studies.     

Scale dependence of immigration rates: models, metrics and data

1. We examine the relationship between immigration rate and patch area for different types of movement behaviours and detection modes. Theoretical models suggest that the scale dependence of the immigration rate per unit area (I/A) can be described by a power model I/A = i*Area(zeta), where zeta describes the strength of the scale dependence. 2. Three types of scaling were identified. Area scaling (zeta = 0) is expected for passively dispersed organisms that have the same probability of landing anywhere in the patch. Perimeter scaling (-0.30 > zeta > -0.45) is expected when patches are detected from a very short distance and immigrants arrive over the patch boundary, whereas diameter scaling (zeta = -0.5) is expected if patches are detected from a long distance or if search is approximately linear. 3. A meta-analysis of published empirical studies of the scale dependence of immigration rates in terrestrial insects suggests that butterflies show diameter scaling, aphids show area scaling, and the scaling of beetle immigration is highly variable. We conclude that the scaling of immigration rates in many cases can be predicted from search behaviour and the mode of patch detection.     

Why allometric scaling enhances stability in food web models

It has recently been shown that the incorporation of allometric scaling into the dynamic equations of food web models enhances network stability if predators are assigned a higher body mass than their prey. We investigate the underlying mechanisms leading to this stability increase. The dynamic equations can be written such that allometric scaling influences these equations at three places: the time scales of predator and prey dynamics become separated, the energy outflow to the predators is decreased, and intraspecific competition is increased relative to metabolic rates. For five food web topologies and various network sizes (i.e., species richness), we study the effect of each of these modifications on the percentage of surviving species separately and find that the decreased interaction strengths and the increased intraspecific competition are responsible for the enhanced stability. We also investigate the range of parameter values for which an enhanced stability is observed.     

The scaling of the size and stiffness of primary flight feathers

Birds encompass a large range of body sizes, yet the importance of body size on feather morphology and mechanical properties has not been characterized. In this study, I examined the scaling relationships of primary flight feathers within a phylogenetically diverse sample of avian species varying in body size by nearly three orders of magnitude. I measured the scaling relationships between body mass and feather linear dimensions as well as feather flexural stiffness. The resnlts of an independent contrasts analysis to test the effects of phylogenetic history on the characters measured had no effect on the scaling relationships observed. There was slight, but not significant, positive allometry in the scaling of shaft diameter with respect to feather length across a range of body masses. The scaling of feather length and diameter against body mass was not significantly different from isometry. Flexural stiffness, however, exhibited strong negative allometry. Therefore, larger birds have relatively more flexible feathers than smaller birds. The more flexible primary feathers of large birds may reduce stresses on the wing skeleton during take-off and landing and also make these feathers less susceptible to mechanical failure. Conversely, the greater flexibility of these feathers may also reduce their capacity to generate aerodynamic lift.