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Statement

Consider a hierarchical branching transport network with \(N\) daughter branches per node, feeding \(N_c\) invariant terminal units (capillaries). Under two network constraints — area-preserving branching and space-filling delivery — the whole-organism metabolic rate scales with body mass as

\[B \propto M^{3/4}.\]

Assumptions

  1. [asserted] Space-filling: the network fills the body volume, forcing the branch-length ratio \(\ell_{k+1}/\ell_k = N^{-1/3}\) (a length scale set by the cube root of the volume served).
  2. [asserted] Area-preserving branching: total cross-sectional area is conserved across a node, forcing the radius ratio \(r_{k+1}/r_k = N^{-1/2}\).
  3. [asserted] Terminal units (capillaries) are size-invariant across species; metabolic rate is proportional to their number, \(B\propto N_c\). These three are biological modeling premises, not theorems.

Proof

  1. [verified] (Per-level volume scaling.) Level \(k\) has \(N^k\) tubes each of volume \(\propto r_k^2\ell_k\) with \(r_k=r_0 N^{-k/2}\), \(\ell_k=\ell_0 N^{-k/3}\). The per-level network volume therefore scales as $\(N^k\cdot r_k^2\ell_k \;\propto\; \big(N\cdot N^{-1}\cdot N^{-1/3}\big)^k = N^{-k/3}.\)$ SymPy simplifies the growth factor \(N\cdot(N^{-1/2})^2\cdot N^{-1/3}\) to \(N^{-1/3}\).
  2. [asserted] (Volume-to-terminal-count relation.) Summing the geometric series over levels, total network volume \(V\) is dominated by its scaling with the number of terminal units and yields \(V\propto N_c^{4/3}\); equivalently \(N_c\propto V^{3/4}\).
  3. [verified] (Kleiber's law.) With \(M\propto V\) (mass proportional to network/blood volume) and \(B\propto N_c\) (metabolic rate proportional to capillary count), \(B\propto N_c\propto V^{3/4}\propto M^{3/4}\). SymPy gives the inverse exponent \(1/(4/3)=3/4\).

Verification

  • Symbolic (SymPy): per-level volume growth factor \(N\cdot(N^{-1/2})^2\cdot N^{-1/3}\) simplifies to \(N^{-1/3}\) (confirmed), and the reciprocal of the WBE volume exponent, \(1/(4/3)=3/4\), was computed to confirm the final power (confirmed).
  • Asserted (not re-derived in code): the intermediate relation \(V\propto N_c^{4/3}\) itself is the WBE geometric-series result quoted from West-Brown-Enquist (1997); only the per-level scaling feeding it and the final reciprocal were machine-checked. Hence step 2 is tagged [asserted].
  • Asserted: the two network constraints (space-filling \(\ell\)-ratio \(=N^{-1/3}\); area-preserving \(r\)-ratio \(=N^{-1/2}\)) and the proportionalities \(B\propto N_c\), \(M\propto V\) are the WBE biological modeling assumptions — they are inputs, not derived results. Hence the document is partially-verified: the algebra is machine-checked, the premises are asserted.

Historical source

The model was published by West, Brown, and Enquist in Science (1997), giving a mechanistic origin for Kleiber's empirical 1932 observation that metabolic rate scales as roughly the ¾ power of body mass.

Related concepts

Citations

  1. West, G. B., Brown, J. H., Enquist, B. J. "A general model for the origin of allometric scaling laws in biology." Science 276 (1997): 122-126.
  2. Kleiber, M. "Body size and metabolism." Hilgardia 6 (1932): 315-353.