Figure 1.
Venation pattern of a Gloeospermum sphaerocarpum leaf.
This leaf was subjected to a chemical treatment to remove all the soft tissues, leaving only the veins. The network-like structure as well as many open ends of the thinnest segments can be observed.
Figure 2.
Snapshots of the development process.
The values of the growing parameter, from top left to bottom right, are η = 1.2, 2.4, 3.6, and 4.8. The seed we use as the initial condition is shown in the first panel with a different color. The numerical lattice has 1024×1024 nodes.
Figure 3.
Snapshots of the development process with a different starting configuration.
The values of η and the system size are the same as in the previous figure. In both figures the hierarchical process can be clearly observed. Note also the open ends of some of the thinnest segments.
Figure 4.
Histograms of the average length of the vein segments of width w.
(A) Actual leaves. Each curve is the histogram of a given dycotiledon leaf: Gloeospermum sphaerocarpum (square symbols), Amphirrhox longifolia (full triangles), and Rinorca amapensis (open circles). Inset: the same quantity, but averaged over more than 1,200,000 segments of eight different leaves. Note that thicker veins tend to be slightly larger than thinner ones. (B) Numerical leaves. Histograms for three different realizations (size: 1024×1024). Inset: Histogram of 30,000 segments obtained in twelve realizations for η = 3.6 and three different sizes (512×512, 768×768, and 1024×1024).
Figure 5.
Histograms of the number of vein segments of width w.
(A) Actual leaves. Histograms for the same three leaves showed in the previous figure. For all the leaves analyzed, a power decay with an exponent close to 3 is observed. Inset: Average over four leaves. A shoulder for thick veins can be observed in both figures. (B) Numerical leaves. Histograms for three different realizations. In the region of intermediate values of thickness, a power decay with an exponent close to 2 is obtained. Inset: Average for the same realizations as in the previous figure, showing a shoulder for the region of thick veins.
Figure 6.
Evolution of the histograms of widths.
Each curve corresponds to one of the four stages of growth shown in Figure 2 of this paper. Note that the distribution of thick veins is quite constant during the evolution.
Figure 7.
Angles between veins as a function of the ratio between the radius of the thinnest (RS) and thickest (RL) segments. The angle between thin and intermediate radius is labeled αIS. The angle between thin and thick segments is αLS, whereas the angle between thick and intermediate segments is αLI. Isolated symbols are data obtained from actual leaves, and were taken from Figure 14 of [30]. Colored lines with small symbols are our numerical results.
Figure 8.
(A) Mechanical analogy. Elastic stresses are accounted for by the springs indicated. Horizontal springs represent the cells of the mesophyll, and its deviation from its equilibrium length is a measure of the deformation energy of the cell. Vertical interlayer springs account for the interaction between mesophyll and epidermis. We suppose that the epidermis grows at a lower rate than the mesophyll, and thus the mismatch between layers will increase with time. A collapsed cell in this schema is represented by a horizontal spring suffering a stress higher than its elastic limit. Once this threshold is reached, the spring has a permanent deformation. (B) Representation of the mesophyll layer with a group of cells in the collapsed state. Note that the initial three-dimensional problem was reduced to two dimensions, as we only describe the intermediate plane where horizontal springs lie.
Figure 9.
Values of Φ vs. y for a fixed value of x, in two stages of growth. The values of the growing parameter are η = 1.2 (A) and η = 2.4 (B). Positive values of Φ correspond to veins, whereas negatives values are associated to intact tissue.
Figure 10.
Patterns without irreversibility.
Result of a simulation with the same parameters as in Figures 2A and 3A, but without the irreversibility condition. Note the lateral wandering and thinning of the veins with respect to the initial condition (shown in a different color). The numerical lattice has 512×512 nodes.