Module 4: Allometric Scaling

In 1932, Max Kleiber measured oxygen consumption in mammals from mice to steers and found:

Rubner (1883) had proposed \(B \propto M^{2/3}\) based on the idea that heat loss is proportional to surface area. But Kleiber's data clearly favoured 3/4. Hemmingsen (1960) extended the relation to unicellular organisms and plants, spanning 22 orders of magnitude of mass. The exponent 3/4 is the most robust single exponent in all of biology.

1.1 Why 3/4 and Not 2/3?

The surface-law argument fails because heat dissipation is not the binding constraint; resource delivery through a fractal vascular network is. The WBE derivation (Module 3 Section 4):

Body mass \(M \propto V_{\text{blood}} \propto N_{\text{cap}}^{4/3}\); basal rate\(B \propto N_{\text{cap}}\). Eliminating\(N_{\text{cap}}\) gives \(B \propto M^{3/4}\).

Once metabolic rate scales as \(M^{3/4}\), many other biological quantities derive automatically. Here are the main ones.

2.1 Heart Rate

Cardiac output \(\dot Q\) (litres/min) is proportional to metabolic rate:\(\dot Q \propto M^{3/4}\). Stroke volume \(V_s \propto M\) (ventricle volume scales with body size). Therefore heart rate:

Mouse: ~600 bpm; human: ~70 bpm; elephant: ~30 bpm; blue whale: ~8 bpm.

2.2 Lifespan

Biological time (development, gestation, lifespan) slows with body size as\(t_{\text{bio}} \propto M^{1/4}\). Lifespan, gestation time, time to maturity, and the period of the circadian-but-longer-scale dampings all follow this pattern.

2.3 The One-Billion-Heartbeat Rule

Total heartbeats in a lifetime:

Remarkably, a mouse and an elephant have similar total heartbeat counts. Humans, thanks to medicine, exceed this baseline (\(\approx 2.5\text{--}3\times 10^9\)) but the pattern persists across species. The same holds for breaths.

2.4 Other Exponents

Let the network have \(N\) branching levels; each level has a branching ratio\(n\) (usually 2). Denote level-\(k\) radius and length by\(r_k, L_k\).

3.1 Space-Filling

Each terminal unit (capillary) services a volume of tissue \(v_N\). Total volume is\(V = N_{\text{cap}} v_N\) and \(v_k \propto L_k^3\) because each branch fills its own cube:

3.2 Area-Preserving Pulsatile Flow

For large arteries where blood is pulsatile, minimising wave reflection (Womersley impedance matching) requires \(\sum r_{k+1}^2 = r_k^2\), i.e.\(n r_{k+1}^2 = r_k^2\), so:

(In the small viscous regime like capillaries, Murray's cubic law gives \(\beta = n^{-1/3}\).)

3.3 Blood Volume Scales as M

Total blood volume:

With \(\beta^2 = n^{-1}\) and \(\gamma = n^{-1/3}\) we have\(n\beta^2\gamma = n^{-1/3}\). Summing the geometric series:

More carefully, dominated by last level: \(V_b \propto n^N \pi r_N^2 L_N\). Given\(r_N, L_N\) fixed by invariant terminals, \(V_b \propto N_{\text{cap}}\). But also \(r_0 = r_N \beta^{-N}\), \(L_0 = L_N \gamma^{-N}\), so\(r_0^2 L_0 = r_N^2 L_N n^{N(1+1/3)} = r_N^2 L_N n^{4N/3}\). Combined with\(V_b \propto r_0^2 L_0\) (volume dominated by aorta segment):

Since mass \(M \propto V_b\) (isometry) and \(B \propto N_{\text{cap}}\)(each capillary supplies a fixed metabolic unit):

In Discorsi (1638), Galileo reasoned that an animal scaled up by a factor\(k\) in linear size has volume (and weight) \(\propto k^3\)but cross-sectional bone area \(\propto k^2\); bone stress therefore grows\(\propto k\). A simple upper bound:

For cortical bone yield stress \(\sigma_{\max} \approx 200\text{ MPa}\) and\(\rho \approx 1000\,\text{kg/m}^3\), the limiting length is:

An absurdly large upper bound because we assumed geometric similarity. In reality, land animals adopt elastic similarity (McMahon 1973): \(r \propto L^{3/2}\), giving \(\sigma \propto L^{1/2}\) — a much slower rise. The largest known land animals (sauropods ~90 tonnes) are near the practical bone-locomotor limit, while aquatic animals like blue whales (150 t) escape bone constraints thanks to buoyancy.

4.1 Circulatory Limits

WBE gives metabolic rate \(B \propto M^{3/4}\), but mass-specific rate\(B/M \propto M^{-1/4}\). Oxygen uptake rate drops with size, yet so does the cell-specific demand. The real limit arises from capillary density: \(\rho_{\text{cap}} \propto M^{-1/4}\). Below a critical density, diffusion cannot meet demand, setting a theoretical maximum near\(10^5\) kg — close to the blue whale.

Plants have no heart; they draw water upward via the cohesion-tension mechanism. The water column in xylem is under tension, and the tensile strength of liquid water (before cavitation and embolism) is the hard limit on tree height.

5.1 Hydrostatic Component

Lifting water of density \(\rho_w = 1000\,\text{kg/m}^3\) through height \(H\)requires a tension:

5.2 Frictional Losses

The Hagen-Poiseuille equation gives the frictional gradient:

For a 100 m redwood, friction adds another ~1-2 MPa, yielding total tension\(|P| \approx 2\text{--}3\,\text{MPa}\).

5.3 The ~130 m Limit

Koch et al. (2004) measured tension at the tops of the tallest Sequoia sempervirens(Hyperion, 115.9 m). They found tension limits near 2 MPa — the onset of cavitation for coniferous xylem. Extrapolating models gives a theoretical upper bound of\(H_{\max} \approx 122\text{--}130\) m, consistent with the tallest recorded tree, Eucalyptus regnans at ~133 m (historical records).