Fig 1.
Patterns on a growing tissue: Affine or non-affine behavior?
(A) A square-shaped tissue grows anisotropically into a rectangle. Is the pattern passively stretched like a drawing on piece of rubber, or does the pattern geometry influence growth distribution? (B) Typical experiment. A leaf is stretched by applying external forces using a U-shaped steel wire (drawn schematically in black, forces in red). Note the difference in the venation patterns between the stretched and non-stretched sides. (C) Using the texture tensor to quantify vasculature. A part of the vascular network of a leaf, after digitization. Black lines represent veins; the texture tensor of an areole is represented by an ellipse (red) computed from all the vectors connecting its center to the center of neighboring areoles (blue); every ellipse is a representation of the local geometry of the network.
Fig 2.
Numerical simulations of a growing network: Anistropy depends on the applied stress.
(A) The effect of external stress. Each vein is represented by a black line. The red box is the unit cell with periodic boundary conditions. For visualization, the texture tensor of each areole is shown in the middle of the areole (blue, not to scale), and the averaged texture tensor in the middle of each figure (green, not to scale). The three tissues grew under turgor pressure; in addition, the stress σxx along the x direction or σyy along the y direction were set equal to the magnitude of turgor pressure Ptur. (B) The distribution of the main orientation (the orientation of the eigenvector associated with the highest eigenvalue) of the texture tensor in the simulation. The distribution is seen to be relatively widely scattered around 90° when no external stress is present (green). When stress is applied, the distribution becomes sharply concentrated around the direction of the external stress (orange and blue). (C) The anisotropy of the texture tensor field averaged over 40 simulations, as a function of the external stress. The stress is measured in units of the turgor pressure, and the anisotropy is measured when the network has doubled its area from the end of the vein creation stage (see Methods).
Fig 3.
Veins in numerical simulations undergo non-affine deformations.
(A) Colormap of the non-affinity index q of each areole, for one realization of the numerical simulation with stretching in the y−direction (σyy = Ptur as in Fig 2); in the case of an affine dilation q would be equal to one for all areoles; q > 1 (resp. q < 1) means that the areole grew more (resp. less) than its neighbourhood. (B) Histogram of q over all realizations with stretching in the x−direction and for a stress σxx ranging from 0 to 2Ptur. As in Fig 2, the simulations were stopped when the leaves doubled their area.
Fig 4.
Stretching leaves: Tracing of the venation pattern throughout growth.
The transparent drops visible in panel (A) are epoxy glue. The tensile force was applied between these points. The distance between the stretching points is 3.5mm on day 1 and 9.3mm on day 15. In panel (a) we plot the locations of the stretched (right) and freely-grown (left) regions in the leaf on day 1. A close up shows that the freely-grown region grew by isotropic expansion, while the stretched region grew anisotropically (note the different scales for day 1 and day 15). This is confirmed by superimposing the rescaled venation patterns from day 1 (red) and day 15 (blue). (B) Quantification of the large scale anisotropy of growth: The elongation of two perpendicular segments in each region. The endpoints of each segments are marked by colored points in panel (A), and the same color code is used in panel (B). The length of each segment is normalized by its initial value.
Fig 5.
Quantifying the effect of stretching on a leaf.
Various scalar quantities defined from the averaged texture tensor M, for bay leaves that grew free of external stress (top row), or stretched (bottom row, arrows indicate the force direction). (B,C) The determinant detM, which quantifies the size of an areole, and the maximum eigenvalue λ of M, which quantifies the greater dimension of an areole. These two quantities are normalized by their average value for each leaf. Note the larger areoles at the base of the non-stretched leaf, and in the stretched region of the stretched leaf. λ attains its maximum on the edge of the leaf, while the determinant is maximal in the interior. (D,E) The anisotropy and orientation of the texture tensor (or of the corresponding ellipse); larger values of anisotropy imply elongated areoles. The orientation is noisy in regions where the texture tensor is isotropic.
Fig 6.
Veins in experiments undergo non-affine deformations.
(A) Colormap of the non-affinity index q of each areole, after 15 days of external stretching; in the case of an affine deformation q would be equal to one for all areoles; q > 1 (resp. q < 1) means that the areole grew more (resp. less) than its neighbors. (B) Histogram of the orientation of veins before and after stretching showing that veins reorient in the stretching direction. (C) Histogram of q in the stretched and unstretched regions.
Fig 7.
Model with randomness in vein thickness and threshold in the growth law.
All panels show the stress-dependent distribution of non-affinity index q in all the simulations for a noise amplitude of r = 40%. (A) With no growth threshold (η = 0) the distribution in the presence of anisotropic stress (red) is wider than without it (blue), in contrast to experimental finding (Fig 6C). (B) When a threshold, η = η0, is introduced, the trend is reversed, and the stressed tissue features a narrower distribution, in agreement with experiments. (C,D) Sensitivity of the results to the threshold; the normalized threshold η⋆ = η/η0 ranges from 0.1 to 1.3. Panels (C) and (D) correspond, respectively, to σx = 0 and σx = 2Ptur.