A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo

A New Program to Estimate the Parameters of Preston’s Equation, A General Formula for Describing the Egg Shape of Birds

Preston’s equation is a general model describing the egg shape of birds. The parameters of Preston’s equation are usually estimated after re-expressing it as the Todd-Smart equation and scaling the egg’s actual length to two. This method assumes that the straight line through the two points on an egg’s profile separated by the maximum distance (i.e., the longest axis of an egg’s profile) is the mid-line. It hypothesizes that the photographed egg’s profile is perfectly bilaterally symmetrical, which seldom holds true because of photographic errors and placement errors. The existing parameter estimation method for Preston’s equation considers an angle of deviation for the longest axis of an egg’s profile from the mid-line, which decreases prediction errors to a certain degree. Nevertheless, this method cannot provide an accurate estimate of the coordinates of the egg’s center, and it leads to sub-optimal parameter estimation. Thus, it is better to account for the possible asymmetry between the two sides of an egg’s profile along its mid-line when fitting egg-shape data. In this paper, we propose a method based on the optimization algorithm (optimPE) to fit egg-shape data and better estimate the parameters of Preston’s equation by automatically searching for the optimal mid-line of an egg’s profile and testing its validity using profiles of 59 bird eggs spanning a wide range of existing egg shapes. We further compared this method with the existing one based on multiple linear regression (lmPE). This study demonstrated the ability of the optimPE method to estimate numerical values of the parameters of Preston’s equation and provide the theoretical egg length (i.e., the distance between two ends of the mid-line of an egg’s profile) and the egg’s maximum breadth. This provides a valuable approach for comparing egg shapes among conspecifics or across different species, or even different classes (e.g., birds and reptiles), in future investigations.

Comparison of Leaf Shape between a Photinia Hybrid and One of Its Parents

Leaf shape and size can vary between hybrids and their parents. However, this has seldom been quantitatively tested. Photinia × fraseri is an important landscaping plant in East Asia as a hybrid between evergreen shrubs P. glabra and P. serratifolia. Its leaf shape looks like that of P. serratifolia. To investigate leaf shape, we used a general equation for calculating the leaf area (A) of broad-leaved plants, which assumes a proportional relationship between A and product of lamina length (L) and width (W). The proportionality coefficient (which is referred to as the Montgomery parameter) serves as a quantitative indicator of leaf shape, because it reflects the proportion of leaf area A to the area of a rectangle with L and W as its side lengths. The ratio of L to W, and the ellipticalness index were also used to quantify the complexity of leaf shape for elliptical leaves. A total of >4000 leaves from P. × fraseri and P. serratifolia (with >2000 leaves for each taxon) collected on a monthly basis was used to examine: (i) whether there is a significant difference in leaf shape between the two taxa, and (ii) whether there is a monotonic or parabolic trend in leaf shape across leaf ages. There was a significant difference in leaf shape between the two taxa (p < 0.05). Although there were significant differences in leaf shape on a monthly basis, the variation in leaf shape over time was not large, i.e., leaf shape was relatively stable over time for both taxa. However, the leaf shape of the hybrid was significantly different from its parent P. serratifolia, which has wider and more elliptical leaves than the hybrid. This work demonstrates that variations in leaf shape resulting from hybridization can be rigorously quantified and compared among species and their hybrids. In addition, this work shows that leaf shape does not changes as a function of age either before or after the full expansion of the lamina.

'biogeom': An R package for simulating and fitting natural shapes

Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called 'biogeom' that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree-rings, and three growth-rate equations that generate the profiles of ovate leaves and the ontogenetic growth curves of animals and plants. 'biogeom' includes several empirical datasets comprising the boundary coordinates of bird eggs, fruits, lanceolate and ovate leaves, tree rings, seeds, and sea stars. The package can also be applied to other kinds of natural shapes similar to those in the datasets. In addition, the package includes sigmoid curves derived from the three growth-rate equations, which can be used to model animal and plant growth trajectories and predict the times associated with maximum growth rate. 'biogeom' can quantify the intra- or interspecific similarity of natural outlines, and it provides quantitative information of shape and ontogenetic modification of shape with important ecological and evolutionary implications for the growth and form of the living world.

A nondestructive method of calculating the wing area of insects

Most insects engage in winged flight. Wing loading, that is, the ratio of body mass to total wing area, has been demonstrated to reflect flight maneuverability. High maneuverability is an important survival trait, allowing insects to escape natural enemies and to compete for mates. In some ecological field experiments, there is a need to calculate the wing area of insects without killing them. However, fast, nondestructive estimation of wing area for insects is not available based on past work. The Montgomery equation (ME), which assumes a proportional relationship between leaf area and the product of leaf length and width, is frequently used to calculate leaf area of plants, in crops with entire linear, lanceolate leaves. Recently, the ME was proved to apply to leaves with more complex shapes from plants that do not have any needle leaves. Given that the wings of insects are similar in shape to broad leaves, we tested the validity of the ME approach in calculating the wing area of insects using three species of cicadas common in eastern China. We compared the actual area of the cicadas’ wings with the estimates provided by six potential models used for wing area calculation, and we found that the ME performed best, based on the trade‐off between model structure and goodness of fit. At the species level, the estimates for the proportionality coefficients of ME for three cicada species were 0.686, 0.693, and 0.715, respectively. There was a significant difference in the proportionality coefficients between any two species. Our method provides a simple and powerful approach for the nondestructive estimation of insect wing area, which is also valuable in quantifying wing morphological features of insects. The present study provides a nondestructive approach to estimating the wing area of insects, allowing them to be used in mark and recapture experiments.

Environmental-biomechanical reciprocity and the evolution of plant material properties

Abiotic-biotic interactions have shaped organic evolution since life first began. Abiotic factors influence growth, survival, and reproductive success, whereas biotic responses to abiotic factors have changed the physical environment (and indeed created new environments). This reciprocity is well illustrated by land plants who begin and end their existence in the same location while growing in size over the course of years or even millennia, during which environment factors change over many orders of magnitude. A biomechanical, ecological, and evolutionary perspective reveals that plants are (i) composed of materials (cells and tissues) that function as cellular solids (i.e. materials composed of one or more solid and fluid phases); (ii) that have evolved greater rigidity (as a consequence of chemical and structural changes in their solid phases); (iii) allowing for increases in body size and (iv) permitting acclimation to more physiologically and ecologically diverse and challenging habitats; which (v) have profoundly altered biotic as well as abiotic environmental factors (e.g. the creation of soils, carbon sequestration, and water cycles). A critical component of this evolutionary innovation is the extent to which mechanical perturbations have shaped plant form and function and how form and function have shaped ecological dynamics over the course of evolution.

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

The Montgomery equation predicts leaf area as the product of leaf length and width multiplied by a correction factor. It has been demonstrated to apply to a variety of leaf shapes. However, it is unknown whether tree size (measured as the diameter at breast height) affects leaf shape and size, or whether such variations in leaf shape can invalidate the Montgomery equation in calculating leaf area. Here, we examined 60 individual trees of the alpine oak (Quercus pannosa) in two growth patterns (trees growing from seeds vs. growing from roots), with 30 individuals for each site. Between 100 and 110 leaves from each tree were used to measure leaf dry mass, leaf area, length, and width, and to calculate the ellipticalness index, ratio of area between the two sides of the lamina, and the lamina centroid ratio. We tested whether tree size affects leaf shape, size, and leaf dry mass per unit area, and tested whether the Montgomery equation is valid for calculating leaf area of the leaves from different tree sizes. The diameters at breast height of the trees ranged from 8.6 to 96.4 cm (tree height ranged from 3 to 32 m). The diameter at breast height significantly affected leaf shape, size, and leaf dry mass per unit area. Larger trees had larger and broader leaves with lower leaf dry mass per unit area, and the lamina centroid was closer to the leaf apex than the leaf base. However, the variation in leaf size and shape did not negate the validity of the Montgomery equation. Thus, regardless of tree size, the proportional relationship between leaf area and the product of leaf length and width can be used to calculate the area of the leaves.

Evidence That Supertriangles Exist in Nature from the Vertical Projections of Koelreuteria paniculata Fruit

Many natural radial symmetrical shapes (e.g., sea stars) follow the Gielis equation (GE) or its twin equation (TGE). A supertriangle (three triangles arranged around a central polygon) represents such a shape, but no study has tested whether natural shapes can be represented as/are supertriangles or whether the GE or TGE can describe their shape. We collected 100 pieces of Koelreuteria paniculata fruit, which have a supertriangular shape, extracted the boundary coordinates for their vertical projections, and then fitted them with the GE and TGE. The adjusted root mean square errors (RMSEadj) of the two equations were always less than 0.08, and >70% were less than 0.05. For 57/100 fruit projections, the GE had a lower RMSEadj than the TGE, although overall differences in the goodness of fit were non-significant. However, the TGE produces more symmetrical shapes than the GE as the two parameters controlling the extent of symmetry in it are approximately equal. This work demonstrates that natural supertriangles exist, validates the use of the GE and TGE to model their shapes, and suggests that different complex radially symmetrical shapes can be generated by the same equation, implying that different types of biological symmetry may result from the same biophysical mechanisms.

Half-leaf width symmetric distribution reveals buffering strategy of Cunninghamia lanceolata

BACKGROUND

Leaf length and width could be a functioning relationship naturally as plant designs. Single-vein leaves have the simplest symmetrical distribution and structural design, which means that fast-growing single-vein species could interpret the scheme more efficiently. The distribution of leaf length and width can be modulated for better adaptation, providing an informative perspective on the various operational strategies in an emergency, while this mechanism is less clear. Here we selected six age groups of Cunninghamia lanceolata pure forests, including saplings, juveniles, mature, and old-growth trees. We pioneered a tapering model to describe half-leaf symmetric distribution with mathematical approximation based on every measured leaf along developmental sequence, and evaluated the ratio of leaf basal part length to total length (called tipping leaf length ratio).

RESULTS

The tipping leaf length ratio varied among different tree ages. That means the changes of tipping leaf length ratio and leaf shape are a significant but less-noticed reflection of trees tradeoff strategies at different growth stages. For instance, there exhibited relatively low ratio during sapling and juvenile, then increased with increasing age, showing the highest value in their maturity, and finally decreased on mature to old-growth transition. The tipping leaf length ratio serves as a cost-benefit ratio, thus the subtle changes in the leaf symmetrical distribution within individuals reveal buffering strategy, indicating the selection for efficient design of growth and hydraulic in their developmental sequences.

CONCLUSIONS

Our model provides a physical explanation of varied signatures for tree operations in hydraulic buffering through growth stages, and the buffering strategy revealed from leaf distribution morphologically provides evidence on the regulation mechanism of leaf biomechanics, hydraulics and physiologies. Our insight contributes greatly to plant trait modeling, policy and management, and will be of interest to some scientists and policy makers who are involved in climate change, ecology and environment protection, as well as forest ecology and management.