Leaves as shell structures: Double curvature, auto-stresses, and minimal mechanical energy constraints on leaf rolling in grasses

Grass leaves are natural examples of shell structures because they are thin and display a double curvature. An important mechanical property of shells is that changes in longitudinal and transverse curvatures are not independent. The basis of this mechanical coupling is presented using simple diagrams. The relevance of the structural constraints for the processes of hydronastic rolling and developmental unrolling in grass leaves is then reviewed. I show that mechanical constraints can explain a large part of the genetic and developmental variability of hydronastic rolling in grasses, without reference to specific anatomic features such as bulliform cells. Mechanical analysis of a rolled maize mutant also revealed that developmental unrolling is not limited to a pure transverse expansion of hinge cells and involves both longitudinal and transverse dimensional changes in the upper epidermis. Interest in using mechanical models as a tool to reveal structural interactions at the tissue and organ level is discussed, and the importance of Paul Green's biophysical approach to the study of plant morphogenesis is emphasized.