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Sizing Up Allometric Scaling Theory

Figure 2

Schematic scaling relation for finite-size corrections in networks with only area-preserving branching.

The dashed line schematically depicts the 3/4 power law that relates the number of capillaries, Ncap, to the blood volume Vblood. This scaling relationship is a straight line in logarithmic space (ln Ncap versus ln Vblood) and represents the leading-order behavior in the limit of infinite blood volume and organism size. The solid line dramatizes the curvature for the scaling relation for finite-size networks obtained when vessel radii are determined solely by area-preserving branching. The dotted line illustrates the consequences of a linear regression on the curve for finite-size organisms (solid line). Since the solid line depicts the predicted curvilinear relationship that deviates above and away from the infinite-size asymptote, Equation 16, the WBE model predicts that fits to data for organisms whose vascular networks are built only with area-preserving branching will yield scaling exponents smaller than 3/4.

Figure 2

doi: https://doi.org/10.1371/journal.pcbi.1000171.g002