A general model for the origin of allometric scaling laws in biology

Self-generated power-law tails in probability distributions

We consider random processes characterized by the presence of correlations in their variance, or more generally in some of their moments. Typical examples are constituted by autoregressive conditional heteroskedasticity (ARCH) processes which are known to display power-law tails in the associated probability distributions. Here, we determine the corresponding exponents exactly and extend these results to relaxation phenomena which can be expected to play a role in natural sciences.

Similitude in the cardiovascular system of mammals

Scaling laws governing the cardiovascular system of mammals are discussed in the present review in a manner emphasizing their experimental basis. Specific attention is given to the well-known experimental laws requiring the rate of oxygen consumption and the heart rate of mammals to vary with body mass raised to the powers 3/4 and −1/4, respectively. This review involves reconsideration and further discussion of the previous work of the writer in which these and other scaling relationships were developed from fundamental considerations. The predicted scaling laws remain unchanged from the earlier work, but alternative assumptions leading to the laws are used so as to provide additional insight. The scaling laws are shown to have their origin in the basic design of the cardiovascular system and in the basic processes involved in its working. Modification of the design assumptions of the system to account for known differences in the relative heart masses of mammals and birds is shown to lead to the scaling laws for rate of oxygen consumption and heart rate of birds.

Low metabolic rate in scorpions: implications for population biomass and cannibalism

Scorpions are abundant in arid areas, where their population biomass may exceed that of vertebrates. Since scorpions are predators of small arthropods and feed infrequently across multi-year lifespans, a parsimonious explanation for their observed, anomalously high biomass may be a depressed metabolic rate (MR). We tested the hypothesis that scorpion MR is significantly depressed compared with that of other arthropods, and we also measured the temperature-dependence of the MR of scorpions to quantify the interaction between large seasonal variations in desert temperatures and MR and, thus, long-term metabolic expenditure. Scorpion MR increased markedly with temperature (mean Q(10)=2.97) with considerable inter-individual variation. At 25 degrees C, the MRs of scorpions from two genera were less than 24 % of those of typical terrestrial arthropods (spiders, mites, solpugids and insects) of the same mass. It is likely, therefore, that the low MR of scorpions contributes to their high biomass in arid areas. The combination of high biomass and high production efficiency associated with low MR may also favor a density-dependent ‘transgenerational energy storage’ strategy, whereby juveniles are harvested by cannibalistic adults that may be closely related to their juvenile prey.

Directionality theory and the evolution of body size

Directionality theory, a dynamic theory of evolution that integrates population genetics with demography, is based on the concept of evolutionary entropy, a measure of the variability in the age of reproducing individuals in a population. The main tenets of the theory are three principles relating the response to the ecological constraints a population experiences, with trends in entropy as the population evolves under mutation and natural selection. (i) Stationary size or fluctuations around a stationary size (bounded growth): a unidirectional increase in entropy; (ii) prolonged episodes of exponential growth (unbounded growth), large population size: a unidirectional decrease in entropy; and (iii) prolonged episodes of exponential growth (unbounded growth), small population size: random, non-directional change in entropy. We invoke these principles, together with an allometric relationship between entropy, and the morphometric variable body size, to provide evolutionary explanations of three empirical patterns pertaining to trends in body size, namely (i) Cope's rule, the tendency towards size increase within phyletic lineages; (ii) the island rule, which pertains to changes in body size that occur as species migrate from mainland populations to colonize island habitats; and (iii) Bergmann's rule, the tendency towards size increase with increasing latitude. The observation that these ecotypic patterns can be explained in terms of the directionality principles for entropy underscores the significance of evolutionary entropy as a unifying concept in forging a link between micro-evolution, the dynamics of gene frequency change, and macro-evolution, dynamic changes in morphometric variables.

Form and function of fetal and neonatal pulmonary arterial bifurcations

Bifurcation is a basic form of vascular connection. It is composed of a parent vessel of diameter d(0), and two daughter vessels, d(1) and d(2), where d(0) > d(1) >/= d(2). Optimal values for the bifurcation area ratio, beta = (d(1)(2) + d(2)(2))/d(0)(2), and the junction exponent, x, in d(0)(x) = d(1)(x) + d(2)(x), are postulated to be universal in nature. However, we have hypothesized that the perinatal pulmonary arterial circulation is an exception. Arterial diameters were measured in pulmonary vascular casts of a fetal lamb (140 days gestation/145 days term) and a neonatal lamb (1 day old). The values for beta and x were evaluated in 10,970 fetal and 846 neonatal bifurcations sampled from the proximal and intermediate arterial regions. Mean values and confidence intervals (CI) for the fetus were beta = 0.890 (0.886-0.895 CI) and x = 1.75 (1.74-1.76 CI); and for the newborn were beta = 0.913 (0.90-0.93 CI) and x = 1. 79 (1.75-1.82 CI). These values are significantly different from Murray's law (beta > 1, x = 3) or the West-Brown-Enquist law (beta = 1, x = 2). Therefore, perinatal pulmonary bifurcation design appears to be distinctive and exceptional. The decreasing cross-sectional area with branching leads to the hemodynamic consequence of shear stress amplification. This structural organization may be important for facilitating vascular development at low flow rates; however, it may be the origin of unstable reactivity if elevated blood flow and pressure occurs.

A review of porcine pathophysiology: a different approach to disease

Diseases are often thought to result from a single cause. Although this is sometimes the case, e.g. with a highly virulent infection such as Classical Swine Fever (CSF), more often clinical disease in swine herds results from multiple predisposing factors. This is especially true in modern intensive pig husbandry, in which the role of highly infectious diseases is limited to (nonetheless devastating) outbreaks. More important nowadays are diseases, although associated with an agent, without a clear pathogenesis. The emphasis in disease control thus far has been on treatment, eradication and prevention. This has been achieved by focusing attention on husbandry factors, such as climate, housing, hygiene, management, and nutrition. Although this approach has been successful for a number of diseases, several health problems are persistent. There are strong indications that in the latter, intrinsic animal factors are important. Successful handling of these problems requires knowledge of the (patho)physiology of the pig. In this article, several characteristics of pig physiology associated with the occurrence of disease are described. It appears that the modern (fattening) pig is exceptional among other animal species in that its cardiovascular system is mismatched to its body weight. It is argued that this particular disposition causes relatively minor disturbances to have major consequences in the pig. This concept of pig physiology is central to the understanding of the hitherto poorly understood pathogenesis of several diseases, such as oedema disease.

Physiology of angiogenesis

Angiogenesis is a key prerequisite for growth in all vertebrate embryos and in many tumors. Rapid growth requires efficient transport of oxygen and metabolites. Hence, for a better understanding of tissue growth, biophysical properties of vascular systems, in addition to their molecular mechanisms, need to be investigated. The purpose of this article is twofold: (1) to discuss the biophysics of growing and perfused vascular systems in general, emphasizing non-sprouting angiogenesis and remodeling of vascular plexuses; and (2) to report on cellular details of sprouting angiogenesis in the initially non-perfused embryonic brain and spinal cord. It is concluded that (1) evolutionary optimization of the circulatory system corresponds to highly conserved vascular patterns and angiogenetic mechanisms; (2) deterministic and random processes contribute to both extraembryonic and central nervous system vascularization; (3) endothelial cells interact with a variety of periendothelial cells during angiogenesis and remodeling; and that (4) mathematical models integrating molecular, morphological and biophysical expertise improve our understanding of normal and pathological angiogenesis and account for allometric relations.

Is the sheet-flow design a 'frozen core' (a Bauplan) of the gas exchangers? Comparative functional morphology of the respiratory microvascular systems: illustration of the geometry and rationalization of the fractal properties

The sheet-flow design is ubiquitous in the respiratory microvascular systems of the modern gas exchangers. The blood percolates through a maze of narrow microvascular channels spreading out into a thin film, a "sheet". The design has been convergently conceived through remarkably different evolutionary strategies. Endothelial cells, e.g. connect parallel epithelial cells in the fish gills and reptilian lungs; epithelial cells divide the gill filaments in the crustacean gills, the amphibian lungs, and vascular channels on the lung of pneumonate gastropods; connective tissue elements weave between the blood capillaries of the mammalian lungs; and in birds, the blood capillaries attach directly and in some areas connect by short extensions of the epithelial cells. In the gills, skin, and most lungs, the blood in the capillary meshwork geometrically lies parallel to the respiratory surface. In the avian lung, where the blood capillaries anastomose intensely and interdigitate closely with the air capillaries, the blood occasions a 'volume' rather than a 'sheet.' The sheet-flow design and the intrinsic fractal properties of the respiratory microvascular systems have produced a highly tractable low-pressure low-resistance region that facilitates optimal perfusion. In complex animals, the sheet-flow design is a prescriptive evolutionary construction for efficient gas exchange by diffusion. The design facilitates the internal and external respiratory media to be exposed to each other over an extensive surface area across a thin tissue barrier. This comprehensive design is a classic paradigm of evolutionary convergence motivated by common enterprise to develop corresponding functionally efficient structures. With appropriate corrections for any relevant intertaxa differences, use of similar morphofunctional models in determining the diffusing capacities of various gas exchangers is warranted.

A model for size and age changes in the pharmacokinetics of paracetamol in neonates, infants and children

AIMS

The aims of this study were to describe paracetamol pharmacokinetics in neonates and infants.

METHODS

Infants in their first 3 months of life (n = 30) were randomised to sequentially receive one of three paracetamol formulations (dose 30-40 mg kg-1) over a 2 day period. The formulations were (a) elixir, (b) glycogelatin capsule suppository and (c) triglyceride base suppository. Approximately six blood samples were taken after each dose over the subsequent 10-16 h. Data were analysed using a nonlinear mixed effect model. These neonatal and infant data were then included with data from four published studies of paracetamol pharmacokinetics (n = 221) and age-related pharmacokinetic changes investigated.

RESULTS

Population pharmacokinetic parameter estimates and their coefficients of variation (CV%) for a one compartment model with first order input, lag time and first order elimination were volume of distribution 69.9 (18%) l and clearance 13.0 (41%) l h-1 (standardized to a 70 kg person). The volume of distribution decreased exponentially with a half-life of 1.9 days from 120 l 70 kg-1 at birth to 69.9 l 70 kg-1 by 14 days. Clearance increased from birth (4.9 l h-1 70 kg-1) with a half-life of 3.25 months to reach 12.4 l h-1 70 kg-1 by 12 months. The absorption half-life (tabs) for the oral preparation was 0.13 (154%) h with a lag time (tlag) of 0.39 h (31%). Absorption parameters for the triglyceride base and capsule suppositories were tabs 1.34 (90%) h, tlag 0.14 h (31%) and tabs 0.65 (63%) h, tlag 0.54 h (31%), respectively. The tabs for elixir and capsule suppository in children under 3 months were 3.68 and 1.51 times greater than children over 3 months. The relative bioavailability of rectal formulations compared with elixir were 0.67 (30%) and 0.61 (23%) for the triglyceride base and capsule suppositories, respectively.

CONCLUSIONS

Total body clearance of paracetamol at birth is 62% and volume of distribution 174% that of older children. A target concentration above 10 mg l-1 in approximately 50% subjects can be achieved by a dose from 45 mg kg-1 day-1 at birth and up to 90 mg kg-1 day-1 in 5-year-old children. A reduced dose of 75 mg kg-1 day-1 in an 8-year-old child is sufficient because clearance is a nonlinear function of weight.

A class of flow bifurcation models with lognormal distribution and fractal dispersion

We report a quantitative analysis of a simple dichotomous branching tree model for blood flow in vascular networks. Using the method of moment-generating function and geometric Brownian motion from stochastic mathematics, our analysis shows that a vascular network with asymmetric branching and random variation at each bifurcating point gives rise to an asymptotic lognormal flow distribution with a positive skewness. The model exhibits a fractal scaling in the dispersion of the regional flow in the branches. Experimentally measurable fractal dimension of the relative dispersion in regional flow is analytically calculated in terms of the asymmetry and the variance at local bifurcation; hence the model suggests a powerful method to obtain the physiological information on local flow bifurcation in terms of flow dispersion analysis. Both the fractal behavior and the lognormal distribution are intimately related to the fact that it is the logarithm of flow, rather than flow itself, which is the natural variable in the tree models. The kinetics of tracer washout is also discussed in terms of the lognormal distribution.

Chromosomal mapping of a major quantitative trait locus regulating compensatory renal growth in the rat

Despite extensive research conducted over the past century, the mechanisms of compensatory renal growth (CRG) remain a mystery. Insight into the mechanisms that regulate CRG might be gained by identifying genetic factors that influence this complex phenotype. In a large set of recombinant inbred strains derived from the spontaneously hypertensive rat and the Brown Norway rat, a genome scan for quantitative trait loci (QTL) that regulate CRG was performed. The CRG score was expressed as a ratio of the weight of the remnant right kidney at 8 wk of age to the weight of the left kidney at 5 wk of age, both adjusted for body weight. QTL mapping was performed using Map Manager QT and the strain distribution patterns of more than 600 genetic markers. It was found that CRG after unilateral nephrectomy is a multifactorially determined trait with a substantial genetic component. The heritability of CRG approached 40%. Genome wide scan analysis revealed significant evidence of linkage to a region of rat chromosome 4 designated Crg 1 that accounted for more than 50% of the additive genetic variance of CRG in the recombinant inbred strains. The detection of a major QTL influencing CRG in the rat should provide new opportunities for identifying mechanisms that regulate this historically enigmatic phenomenon and may also have implications for research on the pathogenesis of end-stage kidney disease.

Allometric respiration/body mass data for animals to be used for estimates of inhalation toxicity to young adult humans

The relationship between body weight (BW) and respiratory minute volume (V(m)) was reviewed by collecting a database from the literature. The data were separated into anaesthetized and non-anaesthetized groups. Only young adult terrestrial mammals were included in the final data set. This database is the largest to be reported to date, is the first to separate the anaesthetized and non-anaesthetized groups and is matched to the target population of young, fit adult humans. The data set of non-anaesthetized animals contained 142 studies representing 2616 animals and 18 species from mice at 12 g body weight to horses and a giraffe at ca. 500 kg body weight. Analysis of the data indicated a power law (allometric) relationship between the minute volume and body weight. The resulting allometric equations for the empirical relationship between minute volume and body weight are: log(10)V()(m)= -0.302 + 0.809 log(10)BW and V(m) = 0.499 BW(0.809)where V(m) is the minute volume (l min(-1)) and BW is the body weight (kg). From these equations, a minute volume of 15.5 lmin(-1)was obtained for a 70 kg human in the same physiological and/or emotional state as the animals. The results of the analyses were compared to other empirical studies in the literature, the more recent of which also indicated a scaling factor of 0.8. The relationship between minute volume and body weight is recommended for use in estimating the inhalation toxicity to young adult humans (military personnel), because this is the first study to use a large database focused exclusively upon non-anaesthetized young adult terrestrial mammals.

Intertidal mesograzers in field microcosms: linking laboratory feeding rates to community dynamics

Mesograzers (herbivores <2.5 cm) are both diverse and abundant, but their relative effects on intertidal communities have rarely been quantified. Here I examine the effects of crustacean and polychaete mesograzers on two intertidal resources, the red alga Odonthalia floccosa Esp. (Falkenb.) and the epiphytic diatom Isthmia nervosa Kütz. The mesograzers were hermit crabs (Pagurus hirsutiusculus (Dana) and P. granosimanus (Benedict)), amphipods (Hyale frequens Stout and H. pugettensis (Dana)), isopod (Idotea wosnesenskii (Brandt)), juvenile kelp crab (Pugettia producta (Randall)), and polychaete worm (Platynereis bicanaliculata (Baird)). Feeding rates on Isthmia, measured in the laboratory for different consumer species and size classes, scaled allometrically with body mass. Consumption rates were 2-23% of body mass daily on a fresh weight basis. However, feeding rates on Odonthalia did not scale, suggesting that size will not always indicate per capita effect. Mesograzer densities were measured on Tatoosh Island, Washington, USA. The mesograzer predicted to have the largest total effect (P. hirsutiusculus), based on densityxfeeding rate, was neither the most abundant nor the most voracious. The validity of these sorts of predictions depends on how well feeding rates measured in the laboratory approximate per capita effects under field conditions. Predictions were compared to observed effects in field microcosms. Given known numbers of mesograzers, predictions were made about the amount of Isthmia biomass that should disappear over 2 weeks from microcosms (9x9x12 cm) anchored in tidepools. Average per capita effects in field microcosms were correlated with laboratory feeding rates, but, for three species with significant feeding on Isthmia, effects were lower than feeding rates predicted. Feeding trials may overestimate community impact because they fail to account for alternative food, search times, resource productivity and stimulation of growth, or interference from other consumers. Nevertheless, densities of mesograzers can reach sufficiently high levels so that even feeble per capita effects combine to alter biomass of epiphytes and perhaps other small algae.

Fractal nature of regional ventilation distribution

High-resolution measurements of pulmonary perfusion reveal substantial spatial heterogeneity that is fractally distributed. This observation led to the hypothesis that the vascular tree is the principal determinant of regional blood flow. Recent studies using aerosol deposition show similar ventilation heterogeneity that is closely correlated with perfusion. We hypothesize that ventilation has fractal characteristics similar to blood flow. We measured regional ventilation and perfusion with aerosolized and injected fluorescent microspheres in six anesthetized, mechanically ventilated pigs in both prone and supine postures. Adjacent regions were clustered into progressively larger groups. Coefficients of variation were calculated for each cluster size to determine fractal dimensions. At the smallest size lung piece, local ventilation and perfusion are highly correlated, with no significant difference between ventilation and perfusion heterogeneity. On average, the fractal dimension of ventilation is 1.16 in the prone posture and 1. 09 in the supine posture. Ventilation has fractal properties similar to perfusion. Efficient gas exchange is preserved, despite ventilation and perfusion heterogeneity, through close correlation. One potential explanation is the similar geometry of bronchial and vascular structures.

Effect of increasing blood flow on distribution of pulmonary transit times in man

The architecture of the vascular network is a major determinant of the distribution of transit times in the organ. Changes in pulmonary blood volumes, pulmonary transit times and the heterogeneity (coefficient of variation) of pulmonary transit times in man were measured using the first-pass radiotracer technique both at rest and during muscular exercise. Dichotomously branching and dispersive flow models of the lung vasculature were applied. The results showed that total pulmonary blood volume increased by 29%, total transit time decreased by 55% and the heterogeneity of pulmonary transit times decreased by 31% when blood flow increased threefold. All changes obeyed power law functions. The changes were greatest when blood flow increased from baseline to intermediate levels. Further increases in total blood flow were associated with decreases in transit times but not with more homogeneous distribution of transit times. The simulated results suggest that the transit time through the arterial vessels and the large veins have a negligible heterogeneity. Most of the variability of transit times through the lungs is due to the heterogeneity of the transit times through the capillaries and venules.

On the fractal properties of natural human standing

We analyzed the temporal evolution of the displacement of the center of pressure (COP) during prolonged unconstrained standing (30 min) in non-impaired human subjects. The COP represents the collective outcome of the postural control system and the force of gravity and is the main parameter used in studies on postural control. Our analysis showed that the COP displacement during human standing displays fractal properties that were quantified by the Hurst exponent obtained from the classical rescaled adjusted range analysis. The average fractal or Hurst exponent (H) was 0.35+/-0.06. The presence of long-range correlations from a few seconds to several minutes due to the fractal characteristics of the postural control system has several important implications for the analysis of human balance.

Prospective allometric scaling: does the emperor have clothes?

We are not suggesting that an allometric relationship between pharmacokinetic parameters in animals and humans does not exist; we believe that it does. We are suggesting that using prospective AS to select doses in an FTIM study may lead to a false sense of security given the large publication bias in the literature. There are a number of unrecognized pitfalls to this approach, including (1) prediction intervals so wide as to be of limited use, (2) prediction error is often no better than arbitrarily chosen constants, and (3) it is not possible to determine which drugs will fail a priori. We encourage journals to publish studies in which prospective AS has failed so as scientists we may begin to see what makes these compounds different, with a goal toward better prediction of human pharmacokinetics.

Heat transfer in a microvascular network: the effect of heart rate on heating and cooling in reptiles (Pogona barbata and Varanus varius)

Thermally-induced changes in heart rate and blood flow in reptiles are believed to be of selective advantage by allowing animal to exert some control over rates of heating and cooling. This notion has become one of the principal paradigms in reptilian thermal physiology. However, the functional significance of changes in heart rate is unclear, because the effect of heart rate and blood flow on total animal heat transfer is not known. I used heat transfer theory to determine the importance of heat transfer by blood flow relative to conduction. I validated theoretical predictions by comparing them with field data from two species of lizard, bearded dragons (Pogona barbata) and lace monitors (Varanus varius). Heart rates measured in free-ranging lizards in the field were significantly higher during heating than during cooling, and heart rates decreased with body mass. Convective heat transfer by blood flow increased with heart rate. Rates of heat transfer by both blood flow and conduction decreased with mass, but the mass scaling exponents were different. Hence, rate of conductive heat transfer decreased more rapidly with increasing mass than did heat transfer by blood flow, so that the relative importance of blood flow in total animal heat transfer increased with mass. The functional significance of changes in heart rate and, hence, rates of heat transfer, in response to heating and cooling in lizards was quantified. For example, by increasing heart rate when entering a heating environment in the morning, and decreasing heart rate when the environment cools in the evening a Pogona can spend up to 44 min longer per day with body temperature within its preferred range. It was concluded that changes in heart rate in response to heating and cooling confer a selective advantage at least on reptiles of mass similar to that of the study animals (0. 21-5.6 kg).

Mouse to elephant: biological scaling and Kt/V

The construct Kt/V is used by the nephrology community in prescribing dialysis dose. The concerns that have been raised as to what value of V to use in the calculation of Kt/V touch on the more central question of whether filtration rate should be normalized by a parameter other than V. Within the animal kingdom, a number of physiological variables scale to body size according to an equation of the form Y = YoMb, where Yo is a constant, M is body mass, and b is a scaling exponent. Glomerular filtration rate (GFR) in mammals weighing from 30 g to 503 kg scales to body weight with an exponent of 0.77. Hence, GFR per unit body weight (or Kt/V) decreases significantly with increasing body size. Metabolic rate also scales to body size in a wide range of mammals according to the same general equation and with a scaling exponent of 0.75. Because GFR and metabolic rate scale to body mass with virtually the same exponent, a ratio of the two yields a constant independent of body size. We propose that the ratio (filtration rate/metabolic rate) replace Kt/V. Such a ratio would underscore the linkage between filtration rate (and dialysis therapy) and the metabolic demands of the body.

Nanoscale topography of the corneal epithelial basement membrane and Descemet's membrane of the human

PURPOSE

Quantitatively define and compare the nanoscale topography of the corneal epithelial basement membrane (anterior basement membrane) and Descemet's membrane (posterior basement membrane) of the human.

METHODS

Human corneas not suitable for transplantation were obtained from the Wisconsin Eye Bank. The corneas were placed in 2.5 mM EDTA for 2.5 h or 30 min. for removal of the epithelium or endothelium, respectively. After removal of the overlying cells, specimens were fixed in 2% glutaraldehyde and either examined in this state by atomic force microscopy only or dehydrated through an ethanol series and prepared for transmission electron microscopy (TEM), scanning electron microscopy (SEM), and atomic force microscopy (AFM).

RESULTS

The subepithelial and subendothelial basement membrane surfaces have a similar appearance that consists of an interwoven meshwork of fibers and pores. Topographic feature sizes were found to be in the nanometer size range with the epithelial basement membrane features larger and less densely packed than Descemet's membrane features. The topographic features are fractile in nature and increase surface area for cell contact.

CONCLUSION

With the use of the TEM, SEM, and AFM, a detailed description of the surface topography of corneal epithelial basement membrane and Descemet's membrane of the human cornea are provided. The significance of differences in corneal basement membrane topography may reflect differences in function of the overlying cells or may be related to differences in cell migration and turnover patterns between the epithelium and endothelium.

Empirical laws of survival and evolution: their universality and implications

Presented analysis of human and fly life tables proves that with the specified accuracy their entire survival and mortality curves are uniquely determined by a single point (e.g., by the birth mortality q(0)), according to the law, which is universal for species as remote as humans and flies. Mortality at any age decreases with the birth mortality q(0). According to life tables, in the narrow vicinity of a certain q(0) value (which is the same for all animals of a given species, independent of their living conditions), the curves change very rapidly and nearly simultaneously for an entire population of different ages. The change is the largest in old age. Because probability to survive to the mean reproductive age quantifies biological fitness and evolution, its universal rapid change with q(0) (which changes with living conditions) manifests a new kind of an evolutionary spurt of an entire population. Agreement between theoretical and life table data is explicitly seen in the figures. Analysis of the data on basic metabolism reduces it to the maximal mean lifespan (for animals from invertebrates to mammals), or to the maximal mean fission time (for bacteria), and universally scales them with the total number of body atoms only. Phenomenological origin of this unification and universality of metabolism, survival, and evolution is suggested. Their implications and challenges are discussed.

Activity-driven synapse elimination leads paradoxically to domination by inactive neurons

In early postnatal life, multiple motor axons converge at individual neuromuscular junctions. However, during the first few weeks after birth, a competitive mechanism eliminates all the inputs but one. This phenomenon, known as synapse elimination, is thought to result from competition based on interaxonal differences in patterns or levels of activity (for review, see Lichtman,1995). Surprisingly, experimental data support two opposite views of the role of activity: that active axons have a competitive advantage (Ribchester and Taxt, 1983; Ridge and Betz, 1984; Balice-Gordon and Lichtman, 1994) and that inactive axons have a competitive advantage (Callaway et al., 1987, 1989). To understand this paradox, we have formulated a mathematical model of activity-mediated synapse elimination. We assume that the total amount of transmitter released, rather than the frequency of release, mediates synaptic competition. We further assume that the total synaptic area that a neuron can support is metabolically constrained by its activity level and size. This model resolves the paradox by showing that a competitive advantage of higher frequency axons early in development is overcome at later stages by greater synaptic efficacy of axons firing at a lower rate. This model both provides results consistent with experiments in which activity has been manipulated and an explanation for the origin of the size principle (Henneman, 1985).

The maximum oxygen consumption and aerobic scope of birds and mammals: getting to the heart of the matter

Resting or basal metabolic rates, compared across a wide range of organisms, scale with respect to body mass as approximately the 0.75 power. This relationship has recently been linked to the fractal geometry of the appropriate transport system or, in the case of birds and mammals, the blood vascular system. However, the structural features of the blood vascular system should more closely reflect maximal aerobic metabolic rates rather than submaximal function. Thus, the maximal aerobic metabolic rates of birds and mammals should also scale as approximately the 0.75 power. A review of the literature on maximal oxygen consumption and factorial aerobic scope (maximum oxygen consumption divided by basal metabolic rate) suggests that body mass influences the capacity of the cardiovascular system to raise metabolic rates above those at rest. The results show that the maximum sustainable metabolic rates of both birds and mammals are similar and scale as approximately the 0.88 +/- 0.02 power of body mass (and aerobic scope as approximately the 0.15 +/- 0.05 power), when the measurements are standardized with respect to the differences in relative heart mass and haemoglobin concentration between species. The maximum heart beat frequency of birds and mammals is predicted to scale as the -0.12 +/- 0.02 power of body mass, while that at rest should scale as -0.27 +/- 0.04.

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

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

The fourth dimension of life: fractal geometry and allometric scaling of organisms

Fractal-like networks effectively endow life with an additional fourth spatial dimension. This is the origin of quarter-power scaling that is so pervasive in biology. Organisms have evolved hierarchical branching networks that terminate in size-invariant units, such as capillaries, leaves, mitochondria, and oxidase molecules. Natural selection has tended to maximize both metabolic capacity, by maximizing the scaling of exchange surface areas, and internal efficiency, by minimizing the scaling of transport distances and times. These design principles are independent of detailed dynamics and explicit models and should apply to virtually all organisms.

An investigation of the flow dependence of temperature gradients near large vessels during steady state and transient tissue heating

Temperature distributions measured during thermal therapy are a major prognostic factor of the efficacy and success of the procedure. Thermal models are used to predict the temperature elevation of tissues during heating. Theoretical work has shown that blood flow through large blood vessels plays an important role in determining temperature profiles of heated tissues. In this paper, an experimental investigation of the effects of large vessels on the temperature distribution of heated tissue is performed. The blood flow dependence of steady state and transient temperature profiles created by a cylindrical conductive heat source and an ultrasound transducer were examined using a fixed porcine kidney as a flow model. In the transient experiments, a 20 s pulse of hot water, 30 degrees C above ambient, heated the tissues. Temperatures were measured at selected locations in steps of 0.1 mm. It was observed that vessels could either heat or cool tissues depending on the orientation of the vascular geometry with respect to the heat source and that these effects are a function of flow rate through the vessels. Temperature gradients of 6 degrees C mm(-1) close to large vessels were routinely measured. Furthermore, it was observed that the temperature gradients caused by large vessels depended on whether the heating source was highly localized (i.e. a hot needle) or more distributed (i.e. external ultrasound). The gradients measured near large vessels during localized heating were between two and three times greater than the gradients measured during ultrasound heating at the same location, for comparable flows. Moreover, these gradients were more sensitive to flow variations for the localized needle heating. X-ray computed tomography data of the kidney vasculature were in good spatial agreement with the locations of all of the temperature variations measured. The three dimensional vessel path observed could account for the complex features of the temperature profiles. The flow dependences of the transient temperature profiles near large vessels during the pulsed experiments were consistent with the temperature distributions measured in the steady state experiments and provided unique insights into the process of convective heat transfer in tissues. Finally, it was shown that even for very short treatment times (3-20 s), large vessels had significant effects on the tissue temperature distributions.

Predicting risk of decompression sickness in humans from outcomes in sheep

In animals, the response to decompression scales as a power of species body mass. Consequently, decompression sickness (DCS) risk in humans should be well predicted from an animal model with a body mass comparable to humans. No-stop decompression outcomes in compressed air and nitrogen-oxygen dives with sheep (n = 394 dives, 14.5% DCS) and humans (n = 463 dives, 4.5% DCS) were used with linear-exponential, probabilistic modeling to test this hypothesis. Scaling the response parameters of this model between species (without accounting for body mass), while estimating tissue-compartment kinetic parameters from combined human and sheep data, predicts combined risk better, based on log likelihood, than do separate sheep and human models, a combined model without scaling, and a kinetic-scaled model. These findings provide a practical tool for estimating DCS risk in humans from outcomes in sheep, especially in decompression profiles too risky to test with humans. This model supports the hypothesis that species of similar body mass have similar DCS risk.

Size and form in efficient transportation networks

Many biological processes, from cellular metabolism to population dynamics, are characterized by allometric scaling (power-law) relationships between size and rate. An outstanding question is whether typical allometric scaling relationships--the power-law dependence of a biological rate on body mass--can be understood by considering the general features of branching networks serving a particular volume. Distributed networks in nature stem from the need for effective connectivity, and occur both in biological systems such as cardiovascular and respiratory networks and plant vascular and root systems, and in inanimate systems such as the drainage network of river basins. Here we derive a general relationship between size and flow rates in arbitrary networks with local connectivity. Our theory accounts in a general way for the quarter-power allometric scaling of living organisms, recently derived under specific assumptions for particular network geometries. It also predicts scaling relations applicable to all efficient transportation networks, which we verify from observational data on the river drainage basins. Allometric scaling is therefore shown to originate from the general features of networks irrespective of dynamical or geometric assumptions.

Unified view of scaling laws for river networks

Scaling laws that describe the structure of river networks are shown to follow from three simple assumptions. These assumptions are (1) river networks are structurally self-similar, (2) single channels are self-affine, and (3) overland flow into channels occurs over a characteristic distance (drainage density is uniform). We obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent. We further demonstrate that the two predominant descriptions of network structure (Tokunaga's law and Horton's laws) are equivalent in the case of landscapes with uniform drainage density. The results are tested with data from both real landscapes and a special class of random networks.

Criticality and scaling in evolutionary ecology

Fluctuations in ecological systems are known to involve a wide range of spatial and temporal scales, often displaying self-similar (fractal) properties. Recent theoretical approaches are trying to shed light on the nature of these complex dynamics. The results suggest that complexity in ecology and evolution comes from the network-like structure of multispecies communities that are close to instability. If true, these ideas might change our understanding of how complexity emerges in the biosphere and how macroevolutionary events could be decoupled from microevolutionary ones.

A fractal approach for evaluation of pulmonary circulation in man at rest and during exercise

Changes in pulmonary circulation caused by muscular exercise and body position are usual in daily life. By using first-pass radiocardiography and fractal analysis, pulmonary circulation in man was evaluated at rest and during muscular exercise. At rest, pulmonary circulation was heterogeneous as described by the relative dispersion (which is the coefficient of variation, i.e. the standard deviation of the pulmonary transit times divided by the mean transit time; RD = 0.51 +/- 0.06) and fractal in nature. During exercise, pulmonary circulation became more homogeneous (RD = 0.35 +/- 0.04; P < 0.001). The calculated fractal dimension decreases from 1.09 at rest to 1.06 during exercise. The model identifies a cubic-law response for circulation heterogeneity and a quarter-power law for resistance during muscular exercise.