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 w ork, it is found that there is a linear relation between the epidermis (tun ica) thickness and the wavelength between new leaves (primordia). This rela tion is near the buckling wavelength calculated from the geometry of the tu nica and interior (corpus) cells. The main focus is on the changes in patte rn 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 unif orm growth, represented by thermal expansion. It is found that the differen ce in the in-plane and out-of-plane buckling can be accounted for by a mode st change in one geometric parameter. (2) The second change is that in the unicellular alga Acetabularia. The geometry consists of a standard cylindri cal 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 fo rms. A mechanical model for the growth is proposed, in which the wall consi sts of a viscous material with a locally linear relation between mean stres s and creep (growth) rate. The result is that the elliptical shape for stab le growth can be regulated by one parameter of viscosity. The results reinf orce the suggestion that the stability of the surface is instrumental in th e generation of plant patterns, and that substantial change in pattern can be controlled by the modification of few mechanical parameters.