Shape Matters: Hofmeister’s Rule, Primordium Shape, and Flower Orientation

Abstract

Hofmeister's rule is an empirical heuristic derived from the observation that new leaf primordia are formed in the largest space between the existing flanks of the older primordia. These observations have been repeatedly validated in studies of leaf arrangement, but there has been little attempt to extend them to inflorescence and floral organs. This investigation demonstrates the validity of Hofmeister's observations to cincinnus and early flower development in Phenakospermum guyannense (Strelitziaceae) and Heliconia latispatha (Heliconiaceae) and relates these results to Paul Green's work on the biophysics of organ formation. The cincinni of Phenakospermum and Heliconia arise in the axils of primary bracts and produce a prophyll, continuation apex, and flower in regular succession. The shapes and orientations of the apical regions of the cincinni are correlated with the placement of these organs, which in turn effect the positions of the sepals and their sequence of formation. The result is two rows of mirror-image flowers. The mirror-image symmetry of the flowers is a direct result of Hofmeister's rule in connection with the shape of the apical region. These two factors create a self-sustaining developmental system that produces prophylls, continuation apices, flowers, and sepals in regular succession. Although the cincinni and flowers of these species are built on a common plan, slight differences in apical shape and orientation produce differences in mature floral orientation. Understanding these orientations, and identifying the sequence of sepal formation, allows a proper identification of organ homologies. A study of Green's results and theories shows that Hofmeister's rule can be considered as an empirical condensation of the biophysical factors that influence organ position. These biophysical factors are widely applicable to organ formation in many species.

Emergence of complex patterns in a higher-dimensional phyllotactic system

“…2d). Thus, we observe the quantitatively identical planar spiral structures as in the two-dimensional case, and, without loss of generality, δ n (1) = δ (1) < π in the equilibrium, while unchangedly δ (2) = π by definition (Interval II in Fig. 2a).…”

“…The two generalized divergence angles δn (1) and δn (2) in three dimensions. Like in the two-dimensional case, δn (1) is the angle between the locations of the two most recent primordia, with respect to (0,0,0).…”

“…Like in the two-dimensional case, δn (1) is the angle between the locations of the two most recent primordia, with respect to (0,0,0). δn (2) is given by the angle between the two planes that are spanned by xn, xn−1 and (0,0,0), and xn−1, xn−2 and (0,0,0), respectively. As such it is equal to the spherical angle between the arcs (on S 2 ) from xn to xn−1, and from xn−1 to xn−2, as shown in the figure. unwanted artifacts as demonstrated by simulation results proving robust against small random perturbations of the grid.…”

“…The hypothesis considers the scenario of a regular time interval between the formation of two successive primordia, the plastochrone, and states that the formation of the newest primordium on the apical ring occurs in the largest available space left by the existing primordia. Kirchoff [2] pointed out that this common presentation of Hofmeister's rule in the contemporary literature, albeit biologically reasonable, does not exactly correspond to Hofmeister's original work. Here, we use the term "modified Hofmeister rule" in order to connect to related works while accommodating historical fact.…”

The <I>Winteraceae</I> of the Old World. VII. <I>Zygogynum</I> in the Solomon Islands (incl. Bougainville)

“…Within a flower the length of the stamens can vary very little to rather considerably, even within orthostichies. As the floral appendages are not arranged in a fixed pattern, but according to the principle of first available space (Hofmeister 1868: 508, Vink 1970, Leins & Erbar 1997, Kirchoff 2003, for history), the form and size of a location on the floral apex available for the development of a stamen varies and with it the size of a stamen.…”