Despite the common use of the blooming metaphor, its floral inspiration remains poorly understood. Here we study the physical process of blooming in the asiatic lily Lilium casablanca. Our observations show that the edges of the petals wrinkle as the flower opens, suggesting that differential growth drives the deployment of these laminar shell-like structures. We use a combination of surgical manipulations and quantitative measurements to confirm this hypothesis and provide a simple theory for this change in the shape of a doubly curved thin elastic shell subject to differential growth across its planform. Our experiments and theory overturn previous hypotheses that suggest that blooming is driven by differential growth of the inner layer of the petals and in the midrib by providing a qualitatively different paradigm that highlights the role of edge growth. This functional morphology suggests new biomimetic designs for deployable structures using boundary or edge actuation rather than the usual bulk or surface actuation. mechanics | curvature reversal P lant and fungal movements, although limited by the absence of dedicated motile elements such as muscles, are nevertheless varied. They span a range of length and time scales, from the rapid movements in microscopic nematophagous fungi (1), to the slow opening and closing of leaf stomata. A striking example of rapid movements is seen in the Venus flytrap, which snaps in a fraction of a second as a result of an actively controlled release of turgor across the thickness of the leaf, switching its curvature (2). This and other examples of rapid movements in plants and fungi usually involve slender geometries and may be quantified in terms of the motion of water flow in a soft porous solid (3). Morphological transitions that guide cell,tissue, leaf, branch, root, and stem shape, on the other hand, are much slower and are driven by differential growth. Here we consider a ubiquitous example of plant movement driven by differential growth, the blooming of a flower, the inspiration for much art and poetry but surprisingly little science. When a flower blossoms, its petals change curvature on a time scale of a few hours to days, consistent with the idea that these movements are growth driven. In flowers that bloom just once, differential cell proliferation is the dominant mode of growth, whereas in those that open and close repeatedly, cell elongation plays an important role (4). Frequently proposed explanations for petal movements posit a difference in growth rate between the abaxial and adaxial sides (surfaces) of a petal or an active role for the midribs (4, 5). However many flower petals have rippled edges, and this raises another qualitatively different possibility that we will explore here, namely that it is possible to change the shape of a lamina via excess growth of the margins relative to the center. This mechanism of differential planar growth has recently been shown to both explain the morphology of saddle and ripple shaped leaves and algal blades (6-10) and to engineer the shape soft thin sheets using differential swelling (11). Indeed, our approach quantifies and synthesizes the observations of von Goethe (12) who argued that flower petals are analogous to leaves physiologically. We study the process of blooming in the common lily Lilium casablanca as its popularity and large size makes it easily available and amenable to manipulation. Anatomically, a lily bud consists of 3 inner petals embraced by 3 outer sepals shown in Fig. 1 A and B. All petals/sepals are convex doubly curved surfaces (i.e., they are curved simultaneously in two orthogonal directions at all points along the mid surface), with their longitudinal curvature less than their lateral curvature. Both petal and sepals have midribs that are bilayer composite structures ( Fig. 2A) with a soft leafy part (gray) that is contiguous with the petal/sepal and a stiff woody part (green) that is distinct from the lamina. Whereas the midrib of the outer sepals are featureless, those of the inner petals have grooves into which the margin of the outer sepals are tucked, as shown in Fig. 1A . This locking mechanism maintains the structural integrity of the bud while allowing for the rapid deployment of the petals once blooming starts. To follow the bud of the Lily Lilium casablanca, we place it with its stem immersed in water in an environment of uniform humidity and temperature, under continuous fluorescent lighting and then film it with time lapse video to record the blooming process at intervals of 1 min. The blooming process takes four and a half days until the lily opens fully over which time we ensure that the ambient conditions are kept constant (Movie S1). Fig. 1C shows a typical opening sequence of a lily. During the first 4 days, the young green bud (10 cm long and 2.5 cm wide) absorbs about 0.2 L of water, increasing in length by 10% and in diameter by 20% and turns white. Although the bud is closed at the end of this stage, it is ready to burst forth quite literally; evidence for this can be seen by peeling back the outer sepals that show that the inner petals have wrinkled edges, as shown in Fig. 2C . It is worth pointing out that the inner petals wrinkle before blooming whereas the outer sepals wrinkle at the end of blooming. At the end of the fourth day ( Fig. 1C and Movie S1) these growthinduced stresses in the bud reach a critical value large enough to overcome the petal-sepal lock. This causes the flower to bloom rapidly as the petals/sepals reverse curvature and bend outward. Simultaneously, wrinkles of typical wavelength (approximately 1.5 cm) develop along the edges of these laminae. To quantify the growth of these tissues, we paint equidistant black dots (1-cm apart) along both the periphery, the center, and along the midribs of both the petal and sepal in their closed state and track them over time. This allows us to measure the total in-plane relative growth strain of the tissue along the long axis of the petal over the period associated with blooming. We find that there is a growth strain 10% of the midrib in both petals and sepals is relatively uniform. However, the marginal growth strain in the sepal in the longitudinal direction is more than 20% basally and increases to nearly 50% distally (Fig. 3A) . This lateral strain gradient in a lamina of finite width can give rise to global saddleshaped structures as well as rippled edges (10) and is consistent with our observations of wrinkling of the petal edges in the prestressed bud and in the fully bloomed lily. In addition to
Haiyi et al. REF use a combination of surgical manipulations and quantitative measurements to confirm this hypothesis and provide a simple theory for changes in the shape of a doubly curved thin elastic shell subject to differential growth across its plan-form.
Growth, geometry, and mechanics of a blooming lily
Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America
Medicine Biology