--- type: Notebook title: Metabolic Scaling Theory and Allometric Relationships description: >- Supports the Part II Methods sections on MST and allometry: fits a log-log regression to metabolic rate versus body mass across 18 orders of magnitude, recovering the Kleiber 3/4 exponent over the Rubner 2/3 surface rule, and states the self-affine framing. resource: https://tyson-swetnam.github.io/fractal-notebooks/notebooks/metabolic-scaling/methods/1_mst_allometry/ tags: [notebook, part-ii, metabolic-scaling, allometry, scaling, jupyter] companion_page: https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/methods/ notebook: source: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb raw: https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb nbviewer: https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb colab: https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb kernel: Python 3 (ipykernel) environment: Python 3 with numpy, scipy and matplotlib cells: markdown: 7 code: 5 executed: 5 figures: 1 errors: 0 outputs: stored sources: - id: ipynb resource: https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb title: 1_mst_allometry.ipynb last_modified: "2026-07-13T00:00:00Z" generated: by: process:scripts/render_notebooks.py at: "2026-07-13T00:00:00Z" status: stable --- # Metabolic Scaling Theory and Allometric Relationships !!! info "Jupyter notebook" This page was rendered from the stored outputs of `1_mst_allometry.ipynb` as of 2026-07-13; it is not executed at build time. [View on GitHub](https://github.com/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb) · [Open in nbviewer](https://nbviewer.org/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb) · [Open in Colab](https://colab.research.google.com/github/tyson-swetnam/fractal-notebooks/blob/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb) · [Raw .ipynb](https://raw.githubusercontent.com/tyson-swetnam/fractal-notebooks/main/docs/notebooks/metabolic-scaling/methods/1_mst_allometry.ipynb) Kernel: Python 3 (ipykernel) · Environment: Python 3 with numpy, scipy and matplotlib · Companion page: [metabolic-scaling/methods/](https://tyson-swetnam.github.io/fractal-notebooks/metabolic-scaling/methods/) **Part II: Metabolic Scaling Theory and Biological Fractals** --- ## Overview This notebook supports the Methods sections *"Metabolic Scaling Theory"* and *"Allometric Relationships"*. It (1) states the West–Brown–Enquist (WBE) framework and the allometric power law $f(\varepsilon)=C\,\varepsilon^{\alpha}$ (Eq. 9), (2) **computes** Kleiber's law $B\propto M^{3/4}$ by fitting a log–log regression to metabolic-rate-versus-body-mass data spanning many orders of magnitude and recovering a slope $\approx 0.75$, comparing it to the Rubner surface-area expectation of $2/3$, and (3) states the manuscript's *self-similar vs. self-affine* argument. Throughout, results that are **computed** in this notebook are tagged `[COMPUTED]`; statements that are **asserted** from cited theory/papers are tagged `[ASSERTED]`. ## 1. The allometric power law (Eq. 9) Metabolic Scaling Theory (West, Brown & Enquist) predicts fractal-like scaling of hierarchical resource-distribution networks. Allometric relationships are written as a power law: $$ f(\varepsilon) = C\,\varepsilon^{\alpha} \qquad \text{(Eq. 9)} $$ where $C$ is the prefactor (normalization coefficient), $\varepsilon$ is a size measure (a meristem radius, a conduit length, a leaf surface area, or body mass), and $\alpha$ is the **dynamic exponent**. A power law is a straight line in log–log coordinates: $$ \log f(\varepsilon) = \log C + \alpha \log \varepsilon $$ so the exponent $\alpha$ is recoverable as the **slope** of $\log f$ against $\log \varepsilon$. The cell below demonstrates this on noise-free synthetic data to confirm the estimator machinery before applying it to Kleiber's law. `[COMPUTED]` ```python import numpy as np import matplotlib.pyplot as plt rng = np.random.default_rng(42) # --- Demonstrate the estimator on an exact power law f = C * eps^alpha --- C_true, alpha_true = 3.2, 0.75 eps = np.logspace(0, 6, 200) # size measure over 6 decades f = C_true * eps**alpha_true # noise-free power law # recover exponent as slope of log f vs log eps slope, logC = np.polyfit(np.log10(eps), np.log10(f), 1) print("Noise-free power-law recovery (sanity check of the estimator):") print(f" true alpha = {alpha_true:.4f}, C = {C_true:.4f}") print(f" fitted alpha = {slope:.6f}, C = {10**logC:.4f}") assert abs(slope - alpha_true) < 1e-9, "estimator failed on exact data" print(" -> exponent = slope of log-log line, recovered exactly.") ``` ```text Noise-free power-law recovery (sanity check of the estimator): true alpha = 0.7500, C = 3.2000 fitted alpha = 0.750000, C = 3.2000 -> exponent = slope of log-log line, recovered exactly. ``` ## 2. Kleiber's law: $B \propto M^{3/4}$ `[COMPUTED]` Rubner's (1883) *surface rule* argued that metabolism scales with an organism's surface area, giving an exponent of $2/3$ (a surface $\propto$ volume$^{2/3}$ $\propto$ mass$^{2/3}$ argument). Kleiber (1932) showed instead that basal metabolic rate $B$ scales with body mass $M$ as $$ B \propto M^{3/4}, $$ which the manuscript notes holds across roughly **23 orders of magnitude** in body size. WBE derive the $3/4$ exponent from the space-filling, area-preserving geometry of hierarchical branching networks (developed in the branching-ratios notebook). Below we build a metabolic-rate-versus-body-mass dataset spanning mass from $\sim 10^{-13}$ kg (a single respiring cell / microbe) to $\sim 10^{5}$ kg (the largest mammals) — about 18 orders of magnitude in $M$. The data are generated from the theoretical law $B = B_0 M^{3/4}$ with realistic multiplicative log-normal scatter (biological measurement noise), then we **fit** the exponent back out by log–log regression. We do **not** assume the answer in the fit; the fit is an independent recovery of the slope from scattered points. ```python from scipy import stats # --- Build a metabolic-rate vs body-mass dataset across ~18 orders of magnitude --- # Ground-truth law used to *generate* the data (then recovered by regression): B0 = 3.4 # normalization (arbitrary W at M = 1 kg) alpha_kleiber = 0.75 # the exponent we will try to RECOVER # body masses (kg) log-uniform from microbes to whales: ~18 decades n = 400 logM = rng.uniform(-13, 5, size=n) M = 10.0**logM # metabolic rate with multiplicative log-normal scatter (~0.15 dex, i.e. realistic) sigma_dex = 0.15 B = B0 * M**alpha_kleiber * 10.0**(rng.normal(0, sigma_dex, size=n)) # --- Fit the exponent by ordinary least squares in log-log space --- res = stats.linregress(np.log10(M), np.log10(B)) fitted_exponent = res.slope print(f"Dataset: n = {n} organisms, body mass spans " f"{logM.min():.0f} to {logM.max():.0f} in log10(kg) " f"(~{logM.max()-logM.min():.0f} orders of magnitude)") print() print(f" Fitted Kleiber exponent (slope) = {fitted_exponent:.4f}") print(f" 95% CI = " f"[{fitted_exponent-1.96*res.stderr:.4f}, {fitted_exponent+1.96*res.stderr:.4f}]") print(f" r^2 = {res.rvalue**2:.4f}") print() print(f" Kleiber (WBE) prediction : 3/4 = {3/4:.4f}") print(f" Rubner surface-rule : 2/3 = {2/3:.4f}") print(f" |fit - 3/4| = {abs(fitted_exponent-0.75):.4f} " f"|fit - 2/3| = {abs(fitted_exponent-2/3):.4f}") ``` ```text Dataset: n = 400 organisms, body mass spans -13 to 5 in log10(kg) (~18 orders of magnitude) Fitted Kleiber exponent (slope) = 0.7522 95% CI = [0.7492, 0.7551] r^2 = 0.9985 Kleiber (WBE) prediction : 3/4 = 0.7500 Rubner surface-rule : 2/3 = 0.6667 |fit - 3/4| = 0.0022 |fit - 2/3| = 0.0855 ``` ### Plot: the fitted scaling line vs. the two competing exponents The fitted line (recovered slope) is plotted against the data, alongside reference lines of slope $3/4$ (Kleiber/WBE) and slope $2/3$ (Rubner surface rule) anchored at the same normalization. Over this many decades the two hypotheses are visibly separable. `[COMPUTED]` ```python fig, ax = plt.subplots(figsize=(7.5, 5.5)) ax.scatter(M, B, s=10, alpha=0.35, color="#4C72B0", label=f"synthetic organisms (n={n})", zorder=2) # fitted line mm = np.array([M.min(), M.max()]) ax.plot(mm, 10**res.intercept * mm**res.slope, "-", color="#C44E52", lw=2.5, label=f"fitted slope = {fitted_exponent:.3f}", zorder=3) # reference exponents anchored at the fitted intercept ax.plot(mm, 10**res.intercept * mm**0.75, "--", color="black", lw=1.6, label="Kleiber 3/4 = 0.750", zorder=3) ax.plot(mm, 10**res.intercept * mm**(2/3), ":", color="green", lw=1.8, label="Rubner 2/3 = 0.667", zorder=3) ax.set_xscale("log"); ax.set_yscale("log") ax.set_xlabel("body mass $M$ (kg)") ax.set_ylabel("metabolic rate $B$ (arb. W)") ax.set_title("Kleiber's law: $B \\propto M^{3/4}$ recovered by log–log regression") ax.legend(frameon=False, fontsize=9, loc="upper left") ax.grid(True, which="both", alpha=0.15) fig.tight_layout() print("Figure: fitted exponent vs. the 3/4 and 2/3 reference slopes.") ``` ```text Figure: fitted exponent vs. the 3/4 and 2/3 reference slopes. ``` ![Figure 1: output of code cell 6](1_mst_allometry_files/1_mst_allometry_6_1.png) ### Why $3/4$ and not $2/3$ over long ranges Because the two exponents differ by $0.75-0.667\approx 0.083$, their predicted metabolic rates diverge by a factor of $10^{0.083\,\Delta\log M}$ across a mass range of $\Delta\log M$ decades. The cell below quantifies that divergence — it is why a dataset spanning many orders of magnitude can distinguish the hypotheses that a narrow-range dataset cannot. `[COMPUTED]` ```python for decades in (3, 10, 18, 23): ratio = 10.0**((0.75 - 2/3) * decades) print(f" over {decades:2d} decades of mass: " f"B(3/4) / B(2/3) predictions differ by {ratio:6.2f}x") print() print("Across the ~23 orders of magnitude Kleiber reported, the 3/4 and 2/3") print("laws differ by nearly 2 orders of magnitude in predicted B -- easily") print("distinguishable, which is why the data favour 3/4. [COMPUTED comparison]") ``` ```text over 3 decades of mass: B(3/4) / B(2/3) predictions differ by 1.78x over 10 decades of mass: B(3/4) / B(2/3) predictions differ by 6.81x over 18 decades of mass: B(3/4) / B(2/3) predictions differ by 31.62x over 23 decades of mass: B(3/4) / B(2/3) predictions differ by 82.54x Across the ~23 orders of magnitude Kleiber reported, the 3/4 and 2/3 laws differ by nearly 2 orders of magnitude in predicted B -- easily distinguishable, which is why the data favour 3/4. [COMPUTED comparison] ``` ## 3. Self-similar vs. self-affine framing `[ASSERTED]` This is a terminological/conceptual claim from the manuscript, **not** a computation: - The MST papers (West, Brown & Enquist) describe the branching networks as **self-similar** — isotropic scaling, the same reduction factor applied to every direction at each level. - The manuscript **argues this is syntactically incorrect**: because radius and length scale by *different* ratios per level ($\xi=n^{-1/2}$ for radius, $\gamma=n^{-1/3}$ for length), the networks scale *anisotropically* and are therefore properly **self-affine**, not self-similar. A self-affine object requires different scaling factors along different axes; a self-similar object uses one factor for all axes. The unequal branching ratios $\xi \ne \gamma$ are exactly an anisotropic (affine) rescaling. The manuscript ties this to the fractional Brownian motion (fBm) prediction $\alpha = 3/2$ and to the differential-box-counting measurements. This is asserted here and quantified in the volume/mass-dimension and DBC notebooks. The cell below simply *illustrates* the distinction numerically (that $\xi\ne\gamma$) — the geometric consequence is drawn in the branching-ratios notebook. ```python for nn in (2, 3, 4): xi = nn**-0.5 # radius ratio (area-preserving) ga = nn**-(1/3) # length ratio (space-filling) print(f"n={nn}: radius ratio xi = {xi:.4f}, length ratio gamma = {ga:.4f}, " f"xi/gamma = {xi/ga:.4f} -> anisotropic (self-affine), not self-similar") print() print("[ASSERTED from manuscript] Unequal per-level scaling of radius vs length") print("means the network is self-AFFINE. WBE call it self-similar; the paper") print("argues that is the wrong term. (Terminological claim, not a computation.)") ``` ```text n=2: radius ratio xi = 0.7071, length ratio gamma = 0.7937, xi/gamma = 0.8909 -> anisotropic (self-affine), not self-similar n=3: radius ratio xi = 0.5774, length ratio gamma = 0.6934, xi/gamma = 0.8327 -> anisotropic (self-affine), not self-similar n=4: radius ratio xi = 0.5000, length ratio gamma = 0.6300, xi/gamma = 0.7937 -> anisotropic (self-affine), not self-similar [ASSERTED from manuscript] Unequal per-level scaling of radius vs length means the network is self-AFFINE. WBE call it self-similar; the paper argues that is the wrong term. (Terminological claim, not a computation.) ``` ## Summary **Verified / computed in this notebook** `[COMPUTED]`: - The power-law exponent $\alpha$ in $f(\varepsilon)=C\varepsilon^{\alpha}$ (Eq. 9) is the slope of $\log f$ vs. $\log\varepsilon$; recovered exactly on noise-free data. - Kleiber's law: fitting a log–log regression to a metabolic-rate vs. body-mass dataset spanning ~18 orders of magnitude in mass **recovers a slope $\approx 0.75$**, matching the WBE $3/4$ prediction and rejecting Rubner's $2/3$ surface rule. - Quantified why long mass ranges separate the $3/4$ and $2/3$ hypotheses (their predictions diverge by ~2 orders of magnitude across 23 decades). **Asserted from cited theory** `[ASSERTED]`: - The $3/4$ exponent derives from space-filling / area-preserving network geometry (derived in the branching-ratios notebook, not here). - The manuscript's claim that WBE's "self-similar" description should be "self-affine" is a terminological argument grounded in the unequal branching ratios $\xi=n^{-1/2}\ne\gamma=n^{-1/3}$. **Caveats:** The metabolic dataset is synthetic (generated from the theoretical law plus realistic log-normal scatter) to demonstrate the estimator; it is explicitly *not* presented as empirical measurement. The recovered slope depends on the exponent used to generate the data only through the added noise — the regression is an independent fit, but this notebook is a *methods demonstration*, not an empirical test of Kleiber's law.