Shell stability related to pattern formation in plants

In the last few years we have studied the possible relation of instability of a shell surface to the patterns that develop in plants. In the present work, it is found that there is a linear relation between the epidermis (tunica) thickness and the wavelength between new leaves (primordia). This relation is near the buckling wavelength calculated from the geometry of the tunica and interior (corpus) cells. The main focus is on the changes in pattern that occur. (1) The wild variety of snapdragon has primordia that bulge out of plane, while a mutant has in-plane folding. A crude mechanical model is an elastic ring constrained at the outer diameter and subjected to uniform growth, represented by thermal expansion. It is found that the difference in the in-plane and out-of-plane buckling can be accounted for by a modest change in one geometric parameter. (2) The second change is that in the unicellular alga Acetabularia. The geometry consists of a standard cylindrical pressure vessel with a nearly hemispherical end cap. At a point in time, the end cap flattens and a uniform circumferential array of new shoots forms. A mechanical model for the growth is proposed, in which the wall consists of a viscous material with a locally linear relation between mean stress and creep (growth) rate. The result is that the elliptical shape for stable growth can be regulated by one parameter of viscosity. The results reinforce the suggestion that the stability of the surface is instrumental in the generation of plant patterns, and that substantial change in pattern can be controlled by the modification of few mechanical parameters.