How does the cytoskeleton read the laws of geometry in aligning the division plane of plant cells?

Since Robert Hooke observed the froth-like texture of sectioned plant tissue, there have been numerous attempts to describe the geometrical properties of cells and to account for the patterns they form. Some aspects of biological patterning can be mimicked by compressed spheres and by liquid foams, implying that compression or surface tension are physical bases of patterning. The 14-sided semi-regular tetrakaidecahedron encloses a given volume most efficiently and packs to fill space. However, observations of real plant tissue (and of soap bubbles) in the first half of this century established that plant cells only rarely form this mathematically ideal figure composed predominantly of 6-sided polygons. Instead, they tend to form a topologically transformed variant having mainly pentagonal faces although there is variability in the number of sides and the angles formed. But the one irreducible component of normal cell and tissue geometry is that only three edges meet at a point in a plane. In solid space, this gives rise to tetrahedral junctions and it is from this that certain limitations on sidedness flow. For three edges to meet at a point means that there must be an avoidance mechanism which prevents a new cell plate from aligning with an existing 3-way junction. Sinnott and Bloch (1940) saw that the cytoplasmic strands which precede the cell plate, predicted its alignment and also avoided 3-way junctions in unwounded tissues. Recently, F-actin and microtubules have been detected in these pre-mitotic, transvacuolar strands. The question considered here is why those cytoskeletal elements avoid aligning with the vertex where a neighbouring cross wall has already joined the mother wall. An hypothesis is discussed in which tensile strands – against a background of cortical re-organization during pre-mitosis – tend to seek the minimal path between nucleus and cortex. In this way, it is suggested that unstable strands are gradually drawn into a transvacuolar baffle (the phragmosome) within which cell division occurs. Vertices are avoided by the strands because they constitute unfavoured longer paths. The demonstrable tendency of tensile strands to contact mother walls perpendicularly would seem to account for Hofmeister's and Sachs' rules involving right-angled junctions. As others have discussed, such right-angled junctions give way to co-equal 120° angles between the three walls during subsequent cell growth. It is this asynchrony of cell division – where attachment of a cell plate causes the neighbouring wall to buckle – that forms a vertex to be avoided by subsequent pre-mitotic strands in that neighbouring cell. In this way, successive division planes would not co-align. It is therefore suggested that the exceptional formation of 4-way junctions in wounded tissue results from the fact that adjacent cells divide simultaneously; the lack of prebuckling of a common wall under these circumstances means that there is no vertex to be avoided by the minimal path mechanism.