Allometric Scaling in Biology

The design of living beings is not only a matter of molecular biology but also of geometry and physics. This was once more demonstrated with a biophysical model by G. B. West et al. (1) that had been fine-tuned to predict the famous scaling law of metabolism (2), namely a 3/4 power relation between body mass and energy consumption. While properties and predictions of the model are remarkable, as N. Williams points out in a related Perspective (3), we would like to question the view that fractal geometry is essential for real life. We rather believe that fractal scaling of a branched transport system may not be an unequivocal prerequisite, but merely may simplify calculations. West et al. have not evaluated their “fractal geometry” approach against an approach without assumptions about a “fractal” structure. Consequently, their conclusion that “fractal” design is essential for their predictions is not valid because a “control experiment” is lacking. Our arguments are based on observations during embryonic angiogenesis (4), on properties and predictions of a nonfractal blood vessel growth model (5), and on the original report (1). (i) During embryonic development, blood vessels do not follow a fractal branching pattern, that is, in segmental arteries, capillary plexus, and vascular rings (4, 5). (ii) Even in adult life, nonfractal branching is constantly observed, for example in vessels branching off the aorta; moreover, arterial anastomoses, in addition to interconnected venous networks, are common and represent a situation that is not covered by the fractal approach (4, 5). (iii) The scaling law of metabolism, and other putative “fractal” properties like flow heterogeneity and so forth, can also be derived with a nonfractal model, as we have discussed recently (5). (iv) The West et al. model (1) is—strictly speaking—not fractal, because of the finite number of bifurcations, and definitely is not self-similar throughout the entire system, because the diameter relations vary from large vessels (where area-preserving branching is assumed) to smaller vessels (where cubic branching is postulated). We acknowledge West’s et al. “zeroth order” approach and their cautious interpretation of the model. Nevertheless, in order to assess the influence of a (nearly) fractal as opposed to a nonfractal design, they would have to evaluate the difference. They did not include a (3D) visualization of the simulated system, and the geometric properties were not evaluated. Their model does not appear to comply with real patterns, whose branch length distributions indicate nongeometric increase of the number of small branches (5). Moreover, evolutionary constraints on the design of a vascular system (except in mammals) may come from a developmental bottleneck: eggs are (nearly) closed systems, where parsimony of blood and vessel material, but not of pumping energy, may be required to shape embryonic organs and to maximize fitness (5). Finally, comparable approaches (6) were inspired by Mandelbrot’s famous book (7).