Figure 1.
Image series illustrating some of the steps involved in going from an image of a cleared leaf (left panel), to a binary representation (middle panel), to one in which vein orders have been assigned (right panel).
The central inset shows a vein bifurcation point; The radius of the parent branch (r0) and daughter branches (r1 and r2) at all branching points are determined, and used to determine cross sectional area (e.g. A = πr02), and to solve for α in the equation r0α=r1α + r2α (see Materials and Methods).
Figure 2.
Frequency distribution for the value of α approximated by solving r0α=r1α + r2α, for α (see Methods) for 1,514,771 individual vein junctions across all 349 leaves.
Note that while the distribution fails a normality test, it is well approximated by a normal curve (hashed line) and strongly overlaps the expectation from Murray’s law, log10 (3) = 0.49 (red vertical line), with a mean value of 0.52. Thus, the distribution of estimated α values is strongly consistent with the expectation for Murray’s law.
Figure 3.
Frequency distribution of the area ratio for 1,514,771 individual vein junctions across all leaves.
The mean and median are equal to the expectation for Murray’s law when daughter branches are symmetric. The distribution spans the range of expected values from 0 to 2, which includes area-preserving branching as well.
Figure 4.
Box and whisker plot of the distribution of area ratios as a function of node order.
Within each box, the central red mark is the median value, the box edges represent the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers, and outliers (red plus symbols) are plotted individually. Outliers are considered those values which are larger than P75 + 1.5(P75-P25) or smaller than P25 - 1.5(P75-P25), where P75 and P25 are the 75th and 25th percentiles, respectively [53]. We classify data points as outliers only for the purposes of visualization; no data points were removed from our analyses. The red and black dashed lines are the expectations from Murray’s law and elastic similarity, respectively. Note that for vein orders 1-4 agreement with the expectation for Murray’s law is strong, but begins to depart as branch order increases, moving closer to the expectation for area-preserving branching. Note that in contrast to the convention used by leaf anatomists, here first order veins are the smallest, “terminal” veins in the network (see Methods).
Figure 5.
Frequency distributions for area ratios for each branch order 1-8.
Note that as in Figure 4, the mode of distribution shifts from overlapping Murray’s law in the lowest order branches, to overlapping area preserving branching in the highest order branches. In addition, the shape of the distribution changes with increasing branch order, decreasing in variance and becoming increasingly leptokurtic (see Results).
Figure 6.
The influence of parent diameter and order on vein symmetry.
Panel A: Heat map showing daughter branch symmetry, the ratio of the smaller daughter branch to the larger daughter branch, as a function of parent branch size. As parent branch diameter increases, the ratio of parent to daughter branch diameters becomes more asymmetric. Note that the abundance values for the 2D histogram are log transformed for image clarity, but the values on the key are not transformed. Panel B: Box and whisker plot of the decrease in symmetry with increasing vein orders. Within each box, the central red mark is the median value, the box edges represent the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers, and outliers (red plus symbols) are plotted individually (see Fig. 4 caption for the definition of outliers). First order veins are largely symmetric, but symmetry decreases as vein order increases.