Morphometric methods.

<p>(A) A skeleton graph with the increased level of occlusion of the volumetric data in the background, (B–C) Visualization of spheres used for calculating morphometric traits - diameter of a sphere at the terminal branch is defined as terminal branch thickness –<i>dc</i>, (D) A visualization of the volumetric data of TS_002 coral (See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002849#pcbi-1002849-t001" target="_blank">Table 1</a> for label), (E) Visualization of symmetry angles <i>h_angle</i> and <i>v_angle</i> , (F) Visualization of the associated vectors used for calculation of symmetry vector <i>sm_mag</i>.</p

The simulated growth forms.

<p>(A) Simulated coral in a no-flow condition. (B–F) Simulated corals from various flow simulations (B) <i>Pe_branch</i> = 0.00113, (C) <i>Pe_branch</i> = 0.0105, (D) <i>Pe_branch</i> = 0.0970, (E) <i>Pe_branch</i> = 1.13, (F) <i>Pe_branch</i>∼11.3, Arrow indicates flow direction. The labels of the simulated corals are located on the bottom of each figure (See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002849#pcbi-1002849-t001" target="_blank">Table 1</a> for labels).</p

An example of three consecutive accretive growth steps.

<p>(A–C) Accretive growth steps; vertex v_i represents a simulated corallite. The new layer is constructed along the direction of normal vector n_i of the vertex v_i. A, B and C are three consecutive growth steps where triangles are inserted once the surface of the object increases.</p

Schematic diagram of the simulation.

<p>(A) A spherical object represents an initial growth state of the simulation (first growth step) (B) A simulation phase involves solving the Navier-Stokes equations (i) and the advection-diffusion equation (ii). (C) Accretion phase translocates absorbed nutrients from previous simulation phase to a new growth layer hence, after a few consecutive growth steps, spontaneous branching occurs.</p

Random bifurcations have a tendency to be planar.

<p><b>A.</b> Diagram showing how random bifurcations can be mapped onto a unit sphere. The bifurcation point is fixed at the center of the sphere and non-bifurcation points are projected onto its surface. Intuitively, the probability of finding a cone with cone angle can be thought of in terms of choosing three non-bifurcating points that fall onto the base circle of this cone, i.e. the intersection of the cone with the sphere. <b>B.</b> The circumference of the base circle gets larger as the cone angle increases. <b>C.</b> Overlay of the random bifurcations' distribution of cone angles with the biological distributions shows that the distribution of all biological bifurcations deviates significantly from the random one (KS test, p-value = 10⁻⁵). <b>D.</b> The probability distribution for random bifurcations is (where is in radians). The probability of cone angles >160° is 26%.</p

Mean values of symmetry magnitude <i>sm_mag</i> versus <i>Pe_branch</i> for simulated corals (A) and CT-scanned (B) corals.

<p>Error bars indicate 95% confidence interval. (C-D) surface/volume ratio of simulated and CT-scanned corals versus <i>Pe_branch</i>.</p

Visualization of the volume rendering of the CT-scanned corals with their associated histograms of the local morphometric traits.

<p>Red lines indicate projected branches vector on the substratum plane (visualized from the bottom up perspective). For the <i>in situ</i> flow-controlled corals, flow direction is from right to left. The morphometric traits measured here are as follow: symmetry angles <i>h_angle</i> and <i>v_angle</i>, and the symmetry attitude <i>sm_mag</i> (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002849#pcbi-1002849-t001" target="_blank">table 1</a> for labels). (A) controlled coral CT_456, <i>Pe_branch</i> = 0.136, (B) reduced flow coral (TS_002), <i>Pe_branch</i> = 0.0163 (C), enhanced-flow coral TS_001, <i>Pe_branch</i> = 0.188, (D) enhanced-flow coral TS_003, <i>Pe_branch</i> = 0.227, and (E) enhanced-flow coral CT455, <i>Pe_branch</i> = 0.288.</p

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

<div><p>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 <em>Madracis</em> 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.</p> </div

Finite element meshes representation of the environment around the simulated coral.

<p>The arrow indicates the direction of the flow.</p