Quantifying the Variation in the Geometries of the Outer Rims of Corolla Tubes of L.

Many geometries of plant organs can be described by the Gielis equation, a polar coordinate equation extended from the superellipse equation, r=a|cosm4φ|n2+|1ksinm4φ|n3-1/n1. Here, is the polar radius corresponding to the polar angle φ; is a positive integer that determines the number of angles of the Gielis curve when φ ∈ [0 to 2π); and the rest of the symbols are parameters to be estimated. The pentagonal radial symmetry of calyxes and corolla tubes in top view is a common feature in the flowers of many eudicots. However, prior studies have not tested whether the Gielis equation can depict the shapes of corolla tubes. We sampled randomly 366 flowers of L., among which 360 had five petals and pentagonal corolla tubes, and six had four petals and quadrangular corolla tubes. We extracted the planar coordinates of the outer rims of corolla tubes (in top view) (ORCTs), and then fitted the data with two simplified versions of the Gielis equation with = 1 and = 5: r=acos54φn2+sin54φn3-1/n1 (Model 1), and r=acos54φn2+sin54φn2-1/n1 (Model 2). The adjusted root mean square error (RMSE) was used to evaluate the goodness of fit of each model. In addition, to test whether ORCTs are radially symmetrical, we correlated the estimates of and in Model 1 on a log-log scale. The results validated the two simplified Gielis equations. The RMSE values for all corolla tubes were smaller than 0.05 for both models. The numerical values of and were demonstrated to be statistically equal based on the regression analysis, which suggested that the ORCTs of are radially symmetrical. It suggests that Model 1 can be replaced by the simpler Model 2 for fitting the ORCT in this species. This work indicates that the pentagonal or quadrangular corolla tubes (in top view) can both be modeled by the Gielis equation and demonstrates that the pentagonal or quadrangular corolla tubes of plants tend to form radial symmetrical geometries during their development and growth.