Persistent symmetry frustration in pollen tubes

Abstract

Pollen tubes are extremely rapidly growing plant cells whose morphogenesis is determined by spatial gradients in the biochemical composition of the cell wall. We investigate the hypothesis (MP) that the distribution of the local mechanical properties of the wall, corresponding to the change of the radial symmetry along the axial direction, may lead to growth oscillations in pollen tubes. We claim that the experimentally observed oscillations originate from the symmetry change at the transition zone, where both intervening symmetries (cylindrical and spherical) meet. The characteristic oscillations between resonating symmetries at a given (constant) turgor pressure and a gradient of wall material constants may be identified with the observed growth-cycles in pollen tubes.

Figure 1. Schematic diagram of the apical region of pollen tube – the two considered axisymmetric zones: cylindrical for the distal part and semispherical for apex.

(A) A narrow, symbolically denoted by ‘a’, ring of cylindrical symmetry. (B) Description as for (A) but for the spherical symmetry. (A’), (B’) – corresponding ‘breathing modes’ resulting from changing symmetries at the ‘a‘ limit (see the text).

Figure 2. Spatial distribution of mechanical properties in the apex and apical shank of the pollen tube.

(A) Poisson coefficient and (B) Young’s modulus [Pa].

Figure 3. Tensile strain (A and B) due to the turgor pressure P acting on the cell wall as a function of continuously changing wall mechanical parameters (indicated in the Chart) - cylindrical symmetry at the transition zone.

Radial part of the tensile strain (color bar) visualized on the pollen tube surface (C). Symmetry change induces frustration at the transition ‘a’ zone (A) – a red (dash-dotted) line immersed in the blue background.

Figure 4. Tensile strain due to the turgor pressure P acting on the cell wall as a function of changing wall mechanical parameters (indicated) - spherical symmetry at the transition region.

Symmetry change induces frustration at the transition ‘a’ zone (A) visible as blue (solid) line immersed in the red background.

Figure 5. Tensile strain due to the turgor pressure P acting on the cell wall as a function of changing wall mechanical parameters (indicated) of a given symmetry (B).

Different symmetry stripes show intermingling of tensile strain lines (mixed red and blue area) causing frustration (A).

Figure 6. Radial stress tensor σ_rr [Pa] as a function of radial distance r from the symmetry axis (pointing in z direction) and meridional distance z from the tip.

A clearly visible step in stress manifold induced solely by the symmetry brake from the spherical to the cylindrical one. Calculation parameters are the same for cylindrical and spherical symmetries. Turgor pressure P = 0.3 MPa. Radial distance 1–1.4 µm is taken in this case to better visualize the step in the stress tensor.

Figure 7. Tensile stress σ_rr [Pa] calculated for cylindrical (solid line) and spherical (dashed line) symmetries parameterized by turgor pressure: red –0.3 MPa, green –0.2 MPa, blue –0.1 MPa.

Radial distance 1–1.4 µm is taken to better visualize the differences (solid and dash-dotted lines) in the stress tensor.