Eqs. (C3) and (C4) imply that the volume of each capillary is (C6) The number of capillaries is (C7) Inserting our expressions for β>, β<, γ and N into Eq. (C6) gives (C8) By making use of the formula for a geometric series (Spiegel 1994), the first sum in Eq. (C2) becomes (C9) By defining N̅ N − ̅, the second expression in Eq. (C2) becomes (C10) Combining the expressions for these two sums and some algebra gives us the blood volume: (C11) Remember that, according the first assumption of the WBE theory, the volume of a capillary V is the same regardless of the mass of the organism. Hence, it is a mass-independent quantity. The same is true for the branching ratio n and the number N̅. Thus the only mass-dependent variable on the right-hand side of Eq. (C11) is N . Defining the mass-independent constants (C12) gives us (C13) In Appendix A it was shown that body mass M is proportional to blood volume V . Therefore (C14) Intuitively, we expect the total basal metabolic rate B to equal the sum of the metabolic rates of the individual service volumes. Since there are N such service volumes, B ∝ N , or equivalently, N ∝ B. So we have (C15) As the mass M becomes infinite, an infinite number of capillaries N is needed to provide blood to the service volumes. Hence B becomes infinite, as well. Thus, in the limit as M → ∞, the second term in parentheses becomes vanishingly small, and we obtain Kleiber’s Law: (C16) APPENDIX D: Derivation of the Sigmoid Growth Curve The WBE ontogenetic growth model partitions metabolic energy use between the energy needed to maintain existing tissue and the energy needed to produce new tissue. The total basal metabolic rate B is the sum of the individual metabolic rates of the body cells, plus the rate at which energy is used to form new cells: (D1) The summation is over the different tissue types within the body. For each tissue type, there are Nc cells, each having a cellular metabolic rate of Bc, with Ec the energy needed to form a new cell for that tissue type. To simplify the analysis, we will treat all Nc of the body’s cells as having an average cellular metabolic rate Bc, with Ec being the energy required to create an average body cell: (D2) The organism’s total body mass m is (D3) Differentiating both sides of Eq. (D3) with respect to time yields (D4) 1 0 4 4 1 3 3 3 0 1 0 1 C b cap cap cap C cap V C N CN C N N − = + = + 1 0 4 1 3 3 0 1 C b cap C cap M AV AC N N − = = + 4 1 3 3 0 1 1 M AB AB − = + HEBERT Allometric and metabolic scaling 2023 ICC 226
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