# Scaling of Metabolic Scaling within Physical Limits

## Abstract

**:**

## 1. Introduction

^{b}, where a is the scaling coefficient and b is the scaling exponent or slope in log-log space [1,2,3,4,5]). According to this 3/4-power law (also called Kleiber’s law) as log body mass increases 4-fold, log metabolic rate should increase 3-fold. As a result, smaller organisms tend to have higher rates of metabolism and of other energy-dependent processes per unit body mass than larger organisms. This law, one of the few that appears to be well established in biology, has attracted much attention from both biological and physical scientists. Not surprisingly, frequent attempts have been made to use the quantitative methods of physics, a field which focuses largely on natural laws, to explain Kleiber’s law (e.g., [6,7,8,9,10,11,12,13,14,15,16,17]). However, these mostly deterministic explanations (but see [18,19,20]) have failed to explain fully the marked diversity of metabolic scaling relationships that actually exists in the living world (b ranging between ~0 to >1, but mostly between 2/3 and 1 [21,22,23,24,25]). Thus, there has been a need for new theoretical approaches to explain this diversity. This is an important objective because typically most of the variation in metabolic rate observed in various groups of organisms is related to body size (see e.g., [26,27]), and the rates of many other kinds of biological processes are in turn related to metabolic rate [3,28,29].

## 2. Overview of the Metabolic-Level Boundaries Hypothesis (MLBH)

#### 2.1. Conceptual Foundation of the MLBH

#### 2.1.1. Body Volume and Surface Area as Physical Boundary Limits

^{2/3}, so should metabolic rate. However, when several interspecific analyses in mammals and other ectothermic organisms with variable body temperatures revealed near 3/4 (not 2/3) power scaling, the death of SA theory was proclaimed (e.g., [3,5]), although not by Kleiber [1,33], who was among the first to report 3/4-power metabolic scaling in mammals (see [23]). A major problem with this premature obituary is that it did not recognize that the application of SA theory may be context dependent, including the specific value of b that it predicts. First, body SA may not always scale to the 2/3-power as expected in isomorphic organisms, but may scale differently with b values as high as 1 in organisms that grow mainly in one or two dimensions ([23,42,43]; also see Section 4). Second, the relevant SA may not be external but internal (e.g., respiratory and alimentary surfaces), which may scale with various b values (not just 2/3), as well [21,42]. Third, the relative influence of SA on metabolic scaling may vary among taxa and physiological states [22,23]. For example, the thermoregulation of homeothermic (endothermic) animals is more affected by SA-related heat loss than that of poikilothermic (ectothermic) animals, and thus as expected, recent interspecific analyses show that b is often near 2/3 in the former, but >2/3 (sometimes approaching 1) in the latter (reviewed in [21,22,23]). In addition, as postulated by the MLBH, SA effects on b may be stronger at high resting L when resource and waste fluxes are more influential, than at low L when V-related RD effects are stronger. In short, SA and RD effects on metabolic scaling should not be considered as being deterministic in a simple all-or-none sense, but as being contingent on various internal and external contexts.

#### 2.1.2. Covariation of the Elevation (L) and Slope (b) of Metabolic Scaling Relationships

#### 2.1.3. Internal and External Influences on L and b

#### 2.2. Predictive Power of the MLBH

## 3. Evidence Appearing to Support or Contradict the MLBH

#### 3.1. Effects of Physiological or Developmental State

**Figure 1.**Mean scaling exponents (b) and elevations (L) (±95% confidence intervals) for resting and active (maximal) metabolism of 10 teleost fishes: Carassius auratus [51], Coregonus albula ([54]; and additional data from J. Ohlberger), Ctenopharyngodon idellus [63], Cyclopterus lumpus [52], Esox lucius [62], Macrozoarces americanus [52], Myoxocephalus scorpius [52], Oncorhynchus nerka [59], Salmo gairdneri [60] and Stizostedion vitreum [61]. As predicted by the metabolic-level boundaries hypothesis (MLBH), b is significantly higher (approaching 1) for fishes in an active high-L state, than for those in an inactive low-L state (results of t-tests are shown). Note also that b for resting metabolic rate is quite high (significantly greater than 3/4) as would be expected in ectotherms with a low inactive L, according to the metabolic-level boundaries hypothesis (MLBH).

#### 3.2. Effects of Taxonomic or Ecological Lifestyle Differences

**Figure 2.**Scaling exponents (b ± 95% confidence intervals) in relation to metabolic level (L) for standard (resting) metabolism at three temperatures (open circles: 4, 8 and 15 °C from left to right in each panel [54]) and for active metabolism at five swimming speeds and three temperatures (solid circles:

**A**: 15 °C;

**B**: 8 °C;

**C**: 4 °C; calculated from [54] and additional data provided by J. Ohlberger, personal communication). The scaling exponent b varies mainly between 2/3 and 1, and shows concave upward relationships with L that become narrower at lower temperatures, as predicted by the metabolic-level boundaries hypothesis. The dotted horizontal lines indicate scaling in relation to volume (b = 1) and surface area (b = 2/3).

**Figure 3.**(

**A**) Scaling of log resting (endogenous) metabolic rate in relation to log wet body mass in various groups of uni- and multicellular organisms (based on data from [22,74]). The symbols at the ends of each least squares regression line denote the minimum and maximum body masses for each sample (◊ unicellular prokaryotes; ♦ unicellular algae and protozoans; ● invertebrates: winged insects, wingless insects, crustaceans, spiders, gastropods, bivalves and polychaetes; ○ diapausing insects; ■ vertebrates: birds, mammals, reptiles, amphibians and fishes; □ hibernating mammals; ▲ vascular plants (tree saplings): from approximately left to right, and for each symbol in approximate descending order of metabolic level. The slopes (scaling exponents, b) of thick solid lines are not significantly different from 1, whereas the slopes of thick dashed lines are not significantly different from 2/3 (b values are indicated). (

**B**) Whole organism metabolic scaling exponents (b ± 95% confidence limits) vs. metabolic level (L = mass-specific metabolic rate at the midpoint of each log-log regression shown in panel A among all of the groups of organisms analyzed; following [22,44]). The solid diagonal line represents the regression for multicellular organisms (b = 0.665–0.167 (L), r = −0.901, P < 0.00001, N = 15), whereas the dashed diagonal line represents the regression for unicellular organisms (b = 0.949–0.271 (L), r = −0.846, P = 0.357, N = 3). These negative relationships are predicted by the metabolic-level boundaries hypothesis, which may also explain why multicellular organisms show metabolic scaling slopes approaching 2/3 at lower metabolic levels than do unicellular organisms because multicellular organisms are much larger and thus typically have significantly smaller body surface area to volume ratios. Dotted lines in panels A and B represent scaling in proportion to body-surface area (b = 2/3) and body mass (b = 1), and according to the 3/4-power law. The elevations of the dotted lines in panel A are arbitrary; these lines are meant only to show specific theoretical slopes for visual comparison with those of the empirical scaling relationships.

#### 3.3. Effects of Environmental Conditions

**Figure 5.**The metabolic scaling exponent (b ± 95% confidence intervals) of the fish Galaxias maculatus shows a concave downward relationship with oxygen level (data from [122] and M. Urbina, personal communication), as predicted by the metabolic-level boundaries hypothesis (see text). The dotted horizontal lines indicate scaling in relation to volume (b = 1) and surface area (b = 2/3).

## 4. Reassessing Predictions of the MLBH in the Context of the CMT

## 5. Outlook for the MLBH: Future Research Directions

#### 5.1. Factors Affecting L and b and Their Mechanistic Basis

#### 5.2. Quantitative Extensions of the MLBH

#### 5.3. Hierarchical Expansion of the MLBH

#### 5.4. Synthesizing the MLBH with Other Models and Theories

**Figure 6.**Relationships among resting metabolic level (L), the metabolic scaling exponent (b), cell size (CS: mean red blood cell area) and genome size (GS: DNA content) in 22 non-polyploid teleost fishes (data from Table S1 of [44] and the Animal Genome Size DataBase and Cell Size Database compiled by [141]). Cell size was indexed as the elliptical area (μm

^{2}) per red blood cell measured by image analysis or calculated as π(CLD/2)(CSD/2), where CLD is cell long diameter, and CSD is cell short diameter. Genome size was estimated as the haploid DNA contents (C-value, pg) per red blood cell. When multiple values for cell size or genome size were available, the mean of these values was used. In addition, in a few cases when species values for cell size or genome size were not available, values from conspecific species were used. (

**A**) L is significantly negatively correlated with CS; (

**B**) L is unrelated to GS, even though GS is significantly positively correlated with CS (r = 0.602, p = 0.0050); (

**C**) The scaling exponent b is significantly positively correlated with CS, as would be expected because b is inversely related to L [44], a major prediction of the metabolic-level boundaries hypothesis.

#### 5.5. Applying the MLBH to Other Biological Processes

^{−1}·year

^{−1}) scale with slopes that are inversely related to L, which increases successively in plants (b = 0.834), lizards (b = 0.805), mammals (b = 0.63 or 0.685, based on two estimates) and birds (b = 0.48 to 0.77 in a single curvilinear relationship) (data from [146,147,148,149]). Therefore, I believe that the MLBH can contribute to the formulation of a general metabolic theory of biology (also see [23]). However, in doing so, it should be recognized that metabolic rate may not necessarily drive the rates of other biological processes, but may also respond to them in a supportive, co-adjusted way [29].

## 6. Conclusions

## Acknowledgements

## Conflicts of Interest

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Glazier, D.S.
Scaling of Metabolic Scaling within Physical Limits. *Systems* **2014**, *2*, 425-450.
https://doi.org/10.3390/systems2040425

**AMA Style**

Glazier DS.
Scaling of Metabolic Scaling within Physical Limits. *Systems*. 2014; 2(4):425-450.
https://doi.org/10.3390/systems2040425

**Chicago/Turabian Style**

Glazier, Douglas S.
2014. "Scaling of Metabolic Scaling within Physical Limits" *Systems* 2, no. 4: 425-450.
https://doi.org/10.3390/systems2040425