Geometric Theory Predicts Bifurcations in Minimal Wiring Cost Trees in Biology Are Flat

The complex three-dimensional shapes of tree-like structures in biology are constrained by optimization principles, but the actual costs being minimized can be difficult to discern. We show that despite quite variable morphologies and functions, bifurcations in the scleractinian coral Madracis and in many different mammalian neuron types tend to be planar. We prove that in fact bifurcations embedded in a spatial tree that minimizes wiring cost should lie on planes. This biologically motivated generalization of the classical mathematical theory of Euclidean Steiner trees is compatible with many different assumptions about the type of cost function. Since the geometric proof does not require any correlation between consecutive planes, we predict that, in an environment without directional biases, consecutive planes would be oriented independently of each other. We confirm this is true for many branching corals and neuron types. We conclude that planar bifurcations are characteristic of wiring cost optimization in any type of biological spatial tree structure. Author Summary: Morphology is constrained by function and vice-versa. Often, intricate morphology can be explained by optimization of a cost. However, in biology, the exact form of the cost function is seldom clear. Previously, for many different natural trees authors have reported that most bifurcations are planar and we confirm this here for branching corals and mammalian neurons. In a three-dimensional space, where bifurcations can have many shapes, it is not clear why they are mostly planar. We show, using a geometric proof, that bifurcations that are part of an optimal wiring cost tree should be planar. We demonstrate this by proving that a bifurcation that is not planar cannot be part of an optimal wiring cost tree, using a very general form of wiring cost which applies even when the exact form of the cost function is not known. We conclude that nature selects for developmental mechanisms which produce planar bifurcations.