Figure 1.
The Rl indicatrices representing various growth at a point: isotropic (A) and anisotropic (B-D): (B) symmetry with respect to y, i.e. the Rl along each direction in xz plane is the same, (C) pure elongation along z, i.e. there is no growth in xy plane, (D) elongation along z with contraction (green) along x.
The scheme on the left shows deformation of the exemplary cell resulting from each growth. In every case Rl in a considered direction is proportional to the distance from the calculation point to the indicatrix surface along this direction; the growth rate along z axis is always the same.
Figure 2.
The Arabidopsis root apex (longitudinal section adopted from Van der Berg et al. 1998) with the root-natural coordinate system, R-NC (u,v,φ) for φ =const., applied to it.
The exemplary initial cells (red) and two of three principal growth directions (Ga, Gp) are indicated; the insert shows all three PDGs in 3D. The u and v lines (thin blue) represent PDG trajectories, two of them u0 and v0 turning into -v0 (thick blue), divide the apex into four zones corresponding to: 1, 2 -the root proper without epidermis, 3, 4- the root cap with epidermis, the zone 1 represents QC. Bar = 20 µm .
Figure 3.
Displacement velocity field assumed for the Arabidopsis root apex (after Nakielski and Lipowczan 2012); in the background the outermost cell row and lines defining the root zones (see Fig. 2) are shown.
The V vectors are represented by line segments, the segment indicated by circle corresponds to 0.11 µm min-1.
Figure 4.
Anisotropy of growth rates in Arabidopsis root apex obtained for V field from Figure 3.
The 3D plots show Rl indicatrices; those drawn in red are for initial cells (see Fig.2). In the indicatrix labelled by circle maximal Rl is about 8.2% h-1. The green plots represent negative values of the rate.
Figure 5.
Anisotropy of growth rates in the planes defined by pairs of PDGs, visualized by the degree of growth anisotropy (DGA); the inserts show orientation of the considered planes.
(A) DGA given by the ratio of Rl in Gp to Rl in Ga; (B) DGA given by the ratio of Rl in Gp to Rl in Gl, (C) DGA given by the ratio of Rl in Ga to Rl in Gl. The DGA values are attributed to cells from Figure 2 using color-coding, the negative ones (dark green) result from compression in Ga. For the cells localized in QC (white), the DGA has not been computed.
Figure 6.
Anisotropy of growth rate in exemplary cell walls (red in the cell pattern), which are oblique with respect to PDGs.
In every case, Rl indicatrices and their intersections by two planes: one representing the wall (red) and the other, defined by Ga and Gl (yellow), are shown. Both intersections are symmetrical with respect to these directions. The values of DGA calculated as the ratio of Rl in Ga to Rl in Gl (i.e. as in Fig. 5C) for yellow and red plots, respectively, are the following: (A) 0.92 and 2.06, (B) 1.56 and 1.62, (C) 0.44 and 0.52.
Figure 7.
Maps of Rl indicatrices obtained for V field with modified values of the parameters c (B, C) and d (D, E); the map (A) corresponding to Figure 4 is a reference.
The change in c leads to decrease (B) and increase (C) of Rl in zones 2 and 4 whereas the rates in zone 3 remain unchanged. In turn, the change in d leads to decrease (D) and increase (E) of Rl in zones 3 and 4 whereas the rates in zone 2 remain unchanged. The indicatrices indicated by circles are considered in the text.
Figure 8.
The maps of Rl indicatrices as in Figure 7 but obtained assuming modification of the proximal border of the quiescent centre (zone 1) and the central part of the root cap (zone 3); the indicatrices drawn in red are for initials from Figure 2: (A) the reference map corresponding to Figure 4 where the border is represented by the line u0=0.35; (B, C) as in (A) but for u0=0.30 and u0=0.40, respectively; (D) the border drawn on the basis of cell pattern in Figure 2; (E) as in (D) but the velocity V=0.01 m min-1 at the whole QC border is assumed.