Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals

Abstract

The fact that metabolic rate scales as the three-quarter power of body mass (M) in unicellular, as well as multicellular, organisms suggests that the same principles of biological design operate at multiple levels of organization. We use the framework of a general model of fractal-like distribution networks together with data on energy transformation in mammals to analyze and predict allometric scaling of aerobic metabolism over a remarkable 27 orders of magnitude in mass encompassing four levels of organization: individual organisms, single cells, intact mitochondria, and enzyme molecules. We show that, whereas rates of cellular metabolism in vivo scale as M(-1/4), rates for cells in culture converge to a single predicted value for all mammals regardless of size. Furthermore, a single three-quarter power allometric scaling law characterizes the basal metabolic rates of isolated mammalian cells, mitochondria, and molecules of the respiratory complex; this overlaps with and is indistinguishable from the scaling relationship for unicellular organisms. This observation suggests that aerobic energy transformation at all levels of biological organization is limited by the transport of materials through hierarchical fractal-like networks with the properties specified by the model. We show how the mass of the smallest mammal can be calculated (approximately 1 g), and the observed numbers and densities of mitochondria and respiratory complexes in mammalian cells can be understood. Extending theoretical and empirical analyses of scaling to suborganismal levels potentially has important implications for cellular structure and function as well as for the metabolic basis of aging.

Fig 1.

Basal metabolic rate for mammals as a function of body mass on a logarithmic scale (blue circles). The solid blue line represents the predicted three-quarter power scaling law, covering over six orders of magnitude in mass from a shrew to an elephant. Values for cells in vivo for these same mammals are shown as a vertical blue band at a cellular mass of 3 × 10⁻⁹ g. These are related to the corresponding whole mammal values by a linear relationship, Eq. 2, as shown by the dashed blue lines. The upper dashed blue line is predicted to intercept the solid blue line at M = μ, close to the mass of a shrew, and to extrapolate to the value for an isolated cell in vitro (red data point; see Fig. 2). Also shown (red dots) are in vivo values for a mitochondrion, the respiratory complex, and a cytochrome oxidase molecule.

Fig 2.

Metabolic power of single mammalian cells as a function of body mass on a logarithmic scale. Blue circles represent cells in vivo calculated for the same mammals as described in Fig. 1. Red circles represent cultured cells in vitro of six mammalian species: mouse, hamster, rat, rhesus monkey, human, and pig (32). The solid blue line is the M^−1/4 prediction for cells in vivo from Eq. 3, and the solid red line is the predicted constant for cells in vitro from Eq. 5. The two lines are predicted to intersect at M = μ ≈ 1 g, at which they have the value B ≈ 3 × 10⁻¹¹ W.

Fig 3.

Metabolic power of an isolated mammalian cell, mitochondrion, respiratory complex, and cytochrome oxidase molecule (red dots) as a function of their mass on a logarithmic scale. The solid red line is the M^3/4 prediction (Eq. 6). Also shown are data for unicellular organisms (green dots), which, when adjusted to mammalian body temperature, closely follow the same scaling relationship.

Fig 4.

A logarithmic plot of metabolic power as a function of mass, which summarizes Figs. 1–3. The entire range is shown, covering 27 orders of magnitude from a cytochrome oxidase molecule and respiratory complex through a mitochondrion and a single cell in vitro (red dots), up to whole mammals (blue dots). The solid red and blue lines through the corresponding dots are the M^3/4 predictions. The dashed blue line is the linear extrapolation from M = μ, the approximate mass predicted and observed for the smallest mammal to an isolated mammalian cell, as shown in Eq. 4. The open circles represent the cellular data shown in Fig. 2: red indicates cells in vitro, and blue indicates cells in vivo.