A General Model for Describing the Ovate Leaf Shape

Abstract

Many plant species produce ovate leaves, but there is no general parametric model for describing this shape. Here, we used two empirical nonlinear equations, the beta and Lobry–Rosso–Flandrois (LRF) equations, and their modified forms (referred to as the Mbeta and MLRF equations for convenience), to generate bilaterally symmetrical curves along the x-axis to form ovate leaf shapes. In order to evaluate which of these four equations best describes the ovate leaf shape, we used 14 leaves from 7 Neocinnamomum species (Lauraceae) and 72 leaves from Chimonanthus praecox (Calycanthaceae). Using the AIC and adjusted root mean square error to compare the fitted results, the modified equations fitted the leaf shapes better than the unmodified equations. However, the MLRF equation provided the best overall fit. As the parameters of the MLRF equation represent leaf length, maximum leaf width, and the distance from leaf apex to the point associated with the maximum leaf width along the leaf length axis, these findings are potentially valuable for studying the influence of environmental factors on leaf shape, differences in leaf shape among closely related plant species with ovate leaf shapes, and the extent to which leaves are bilaterally symmetrical. This is the first work in which temperature-dependent developmental equations to describe the ovate leaf shape have been employed, as previous studies lacked similar leaf shape models. In addition, prior work seldom attempted to describe real ovate leaf shapes. Our work bridges the gap between theoretical leaf shape models and empirical leaf shape indices that cannot predict leaf shape profiles.

‘biogeom’: An R package for simulating and fitting natural shapes

“…When two axially symmetrical rate equations along the x-axis are combined into one, we find that it can describe and fit a variety of ovate leaf shapes especially well. 19 Therefore, we added three rate equations to 'biogeom' . The three rate equations can be used to produce sigmoid curves by integration.…”

“…In addition, because of the high degree of leaf-shape variation for ovate leaves (i.e., variations in the curvature of leaf boundaries across different plant species), this class of leaf shapes cannot be accurately represented by the Gielis equation, and additional equations are needed to describe such variations. 19 In thermal biology, many temperature-dependent development models are used to reflect the effect of temperature on organismal developmental rates (i.e., the reciprocal of the developmental time required to complete a specific developmental stage). Prior studies have shown that the integral forms of temperature-dependent developmental (or growth) rate models can produce sigmoid growth curves, which can be used to describe the growth trajectories of many animals and plants.…”

“…However, this method has not been included in any software packages so far. In addition, because of the high degree of leaf‐shape variation for ovate leaves (i.e., variations in the curvature of leaf boundaries across different plant species), this class of leaf shapes cannot be accurately represented by the Gielis equation, and additional equations are needed to describe such variations 19 …”

The Modified Brière Equation and Its Applications

“…Given that the MBE is flexible in fitting a skewed bell-shaped curve, it is potentially useful to model the profiles of ovate leaves [32]. Shi et al [32] formed two axially symmetrical curves using the modified beta equation (also the modified LRF equation) to fit the ovate leaf shapes of Neocinnamomum plants (Lauraceae). Given that the MBE can generate similar skewed curves, the MBE should be similarly useful in describing ovate leaf shape.…”

Tree Size Influences Leaf Shape but Does Not Affect the Proportional Relationship Between Leaf Area and the Product of Length and Width

“…It cannot be used to accurately evaluate the degree of leaf bilateral asymmetry, or predict the leaf centroid from the base of an oval or oboval leaf shape. In order to cope with this limitation, Shi et al (2021b) developed an ovate and obovate leaf shape model using leaf length and width and a third parameter representing the distance from the leaf base to the point on the leaf length axis associated with maximum leaf width. Consequently, Li et al (2021c) defined the “centroid ratio” (as the ratio of this third parameter to leaf length) to quantify the extent of the deviation of the leaf centroid from the midpoint of leaf length.…”