Taylor’s Power Law for Leaf Bilateral Symmetry

Inequality Measure of Leaf Area Distribution for a Drought-Tolerant Landscape Plant

Measuring the inequality of leaf area distribution per plant (ILAD) can provide a useful tool for quantifying the influences of intra- and interspecific competition, foraging behavior of herbivores, and environmental stress on plants' above-ground architectural structures and survival strategies. Despite its importance, there has been limited research on this issue. This paper aims to fill this gap by comparing four inequality indices to measure ILAD, using indices for quantifying household income that are commonly used in economics, including the Gini index (which is based on the Lorenz curve), the coefficient of variation, the Theil index, and the mean log deviation index. We measured the area of all leaves for 240 individual plants of the species Shibataea chinensis Nakai, a drought-tolerant landscape plant found in southern China. A three-parameter performance equation was fitted to observations of the cumulative proportion of leaf area vs. the cumulative proportion of leaves per plant to calculate the Gini index for each individual specimen of S. chinensis. The performance equation was demonstrated to be valid in describing the rotated and right shifted Lorenz curve, given that >96% of root-mean-square error values were smaller than 0.004 for 240 individual plants. By examining the correlation between any of the six possible pairs of indices among the Gini index, the coefficient of variation, the Theil index, and the mean log deviation index, the data show that these indices are closely related and can be used interchangeably to quantify ILAD.

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.

Application of an Ovate Leaf Shape Model to Evaluate Leaf Bilateral Asymmetry and Calculate Lamina Centroid Location

Leaf shape is an important leaf trait, with ovate leaves common in many floras. Recently, a new leaf shape model (referred to as the MLRF equation) derived from temperature-dependent bacterial growth was proposed and demonstrated to be valid in describing leaf boundaries of many species with ovate leaf shape. The MLRF model's parameters can provide valuable information of leaf shape, including the ratio of lamina width to length and the lamina centroid location on the lamina length axis. However, the model wasn't tested on a large sample of a single species, thereby limiting its overall evaluation for describing leaf boundaries, for evaluating lamina bilateral asymmetry and for calculating lamina centroid location. In this study, we further test the model using data from two Lauraceae species, Cinnamomum camphora and Machilus leptophylla, with >290 leaves for each species. The equation was found to be credible for describing those shapes, with all adjusted root-mean-square errors (RMSE) smaller than 0.05, indicating that the mean absolute deviation is smaller than 5% of the radius of an assumed circle whose area equals lamina area. It was also found that the larger the extent of lamina asymmetry, the larger the adjusted RMSE, with approximately 50% of unexplained variation by the model accounted for by the lamina asymmetry, implying that this model can help to quantify the leaf bilateral asymmetry in future studies. In addition, there was a significant difference between the two species in their centroid ratio, i.e., the distance from leaf petiole to the point on the lamina length axis associated with leaf maximum width to the leaf maximum length. It was found that a higher centroid ratio does not necessarily lead to a greater investment of mass to leaf petiole relative to lamina, which might depend on the petiole pattern.

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

Many cross-sectional shapes of plants have been found to approximate a superellipse rather than an ellipse. Square bamboos, belonging to the genus Chimonobambusa (Poaceae), are a group of plants with round-edged square-like culm cross sections. The initial application of superellipses to model these culm cross sections has focused on Chimonobambusa quadrangularis (Franceschi) Makino. However, there is a need for large scale empirical data to confirm this hypothesis. In this study, approximately 750 cross sections from 30 culms of C. utilis were scanned to obtain cross-sectional boundary coordinates. A superellipse exhibits a centrosymmetry, but in nature the cross sections of culms usually deviate from a standard circle, ellipse, or superellipse because of the influences of the environment and terrain, resulting in different bending and torsion forces during growth. Thus, more natural cross-sectional shapes appear to have the form of a deformed superellipse. The superellipse equation with a deformation parameter (SEDP) was used to fit boundary data. We find that the cross-sectional shapes (including outer and inner rings) of C. utilis can be well described by SEDP. The adjusted root-mean-square error of SEDP is smaller than that of the superellipse equation without a deformation parameter. A major finding is that the cross-sectional shapes can be divided into two types of superellipse curves: hyperellipses and hypoellipses, even for cross sections from the same culm. There are two proportional relationships between ring area and the product of ring length and width for both the outer and inner rings. The proportionality coefficients are significantly different, as a consequence of the two different superellipse types (i.e., hyperellipses and hypoellipses). The difference in the proportionality coefficients between hyperellipses and hypoellipses for outer rings is greater than that for inner rings. This work informs our understanding and quantifying of the longitudinal deformation of plant stems for future studies to assess the influences of the environment on stem development. This work is also informative for understanding the deviation of natural shapes from a strict rotational symmetry.

Increase in Absolute Leaf Water Content Tends to Keep Pace with That of Leaf Dry Mass—Evidence from Bamboo Plants

Leaves, as the most important photosynthetic organ of plants, are intimately associated with plant function and adaptation to environmental changes. The scaling relationship of the leaf dry mass (or the fresh mass) vs. leaf surface area has been referred to as “diminishing returns”, suggesting that the leaf area fails to increase in proportion to leaf dry mass (or fresh mass). However, previous studies used materials across different families, and there is lack of studies testing whether leaf fresh mass is proportional to the leaf dry mass for the species in the same family, and examining the influence of the scaling of leaf dry mass vs. fresh mass on two kinds of diminishing returns based on leaf dry mass and fresh mass. Bamboo plants (Poaceae: Bambusoideae) are good materials for doing such a study, which have astonishingly similar leaf shapes across species. Bamboo leaves have a typical parallel venation pattern. In general, a parallel venation pattern tends to produce a more stable symmetrical leaf shape than the pinnate and palmate venation patterns. The symmetrical parallel veins enable leaves to more regularly hold water, which is more likely to result in a proportional relationship between the leaf dry mass and absolute water content, which consequently determines whether the scaling exponent of the leaf dry mass vs. area is significantly different from (or the same as) that of the leaf fresh mass vs. area. In the present study, we used the data of 101 bamboo species, cultivars, forms and varieties (referred to as 101 (bamboo) taxa below for convenience) to analyze the scaling relationships between the leaf dry mass and area, and between leaf fresh mass and area. We found that the confidence intervals of the scaling exponents of the leaf fresh mass vs. dry mass of 68 out of the 101 taxa included unity, which indicates that for most bamboo species (67.3%), the increase in leaf water mass keeps pace with that of leaf dry mass. There was a significant scaling relationship between either leaf dry mass or fresh mass, and the leaf surface area for each studied species. We found that there was no significant difference between the scaling exponent of the leaf dry mass vs. leaf area and that of the leaf fresh mass vs. leaf area when the leaf dry mass was proportional to the leaf fresh mass. The goodness of fit to the linearized scaling relationship of the leaf fresh mass vs. area was better than that of the leaf dry mass vs. area for each of the 101 bamboo taxa. In addition, there were significant differences in the normalized constants of the leaf dry mass vs. fresh mass among the taxa (i.e., the differences in leaf water content), which implies the difference in the adaptabilities to different environments across the taxa.

Leaf Bilateral Symmetry and the Scaling of the Perimeter vs. the Surface Area in 15 Vine Species

The leaves of vines exhibit a high degree of variability in shape, from simple oval to highly dissected palmatifid leaves. However, little is known about the extent of leaf bilateral symmetry in vines, how leaf perimeter scales with leaf surface area, and how this relationship depends on leaf shape. We studied 15 species of vines and calculated (i) the areal ratio (AR) of both sides of the lamina per leaf, (ii) the standardized symmetry index (SI) to estimate the deviation from leaf bilateral symmetry, and (iii) the dissection index (DI) to measure leaf-shape complexity. In addition, we examined whether there is a scaling relationship between leaf perimeter and area for each species. A total of 14 out of 15 species had no significant differences in average ln(AR), and mean ln(AR) approximated zero, indicating that the areas of the two lamina sides tended to be equal. Nevertheless, SI values among the 15 species had significant differences. A statistically strong scaling relationship between leaf perimeter and area was observed for each species, and the scaling exponents of 12 out of 15 species fell in the range of 0.49−0.55. These data show that vines tend to generate a similar number of left- and right-skewed leaves, which might contribute to optimizing light interception. Weaker scaling relationships between leaf perimeter and area were associated with a greater DI and a greater variation in DI. Thus, DI provides a useful measure of the degree of the complexity of leaf outline.

Scaling Relationships between Leaf Shape and Area of 12 Rosaceae Species

Leaf surface area (A) and leaf shape have been demonstrated to be closely correlated with photosynthetic rates. The scaling relationship between leaf biomass (both dry weight and fresh weight) and A has been widely studied. However, few studies have focused on the scaling relationship between leaf shape and A. Here, using more than 3600 leaves from 12 Rosaceae species, we examined the relationships of the leaf-shape indices including the left to right side leaf surface area ratio (AR), the ratio of leaf perimeter to leaf surface area (RPA), and the ratio of leaf width to length (RWL) versus A. We also tested whether there is a scaling relationship between leaf dry weight and A, and between PRA and A. There was no significant correlation between AR and A for each of the 12 species. Leaf area was also found to be independent of RWL because leaf width remained proportional to leaf length across the 12 species. However, there was a negative correlation between RPA and A. The scaling relationship between RPA and A held for each species, and the estimated scaling exponent of RPA versus A approached −1/2; the scaling relationship between leaf dry weight and A also held for each species, and 11 out of the 12 estimated scaling exponents of leaf dry weight versus A were greater than unity. Our results indicated that leaf surface area has a strong scaling relationship with leaf perimeter and also with leaf dry weight but has no relationship with leaf symmetry or RWL. Additionally, our results showed that leaf dry weight per unit area, which is usually associated with the photosynthetic capacity of plants, increases with an increasing A because the scaling exponent of leaf dry weight versus A is greater than unity. This suggests that a large leaf surface area requires more dry mass input to support the physical structure of the leaf.

Effects of Salt Stress on the Leaf Shape and Scaling of Pyrus betulifolia Bunge

Leaf shape can reflect the survival and development of plants in different environments. In particular, leaf area, showing a scaling relationship with other leaf-shape indices, has been used to evaluate the extent of salt stress on plants. Based on the scaling relationships between leaf area and other leaf-shape indices in experiments at different levels of salt stress, we could examine which leaf-shape indices are also related to salt stress. In the present study, we explored the effects of different salt concentration treatments on leaf dry mass per unit area (LMA), the quotient of leaf perimeter and leaf area (QPA), the quotient of leaf width and length (QWL), the areal quotient (AQ) of left and right sides of a leaf and the standardized index (SI) for bilateral symmetry. We treated Pyrus betulifolia Bunge under NaCl salt solution of 2‰, 4‰ and 6‰, respectively, with fresh water with no salt as the control. The reduced major axis (RMA) was used to fit a linear relationship of the log-transformed data between any leaf trait measures and leaf area. We found that leaf fresh weight and dry weight decrease with salt concentration increasing, whereas the exponents of leaf dry weight versus leaf area exhibit an increasing trend, which implies that the leaves expanding in higher salt environments are prone to have a higher cost of dry mass investment to increase per unit leaf area than those in lower salt environments. Salt concentration has a significant influence on leaf shape especially QWL, and QWL under 6‰ concentration treatment is significantly greater than the other treatments. However, there is no a single increasing or decreasing trend for the extent of leaf bilateral symmetry with salt concentration increasing. In addition, we found that the scaling exponents of QPA versus leaf area for four treatments have no significant difference. It indicates that the scaling relationship of leaf perimeter versus leaf area did not change with salt concentration increasing. The present study suggests that salt stress can change leaf functional traits especially the scaling relationship of leaf dry weight versus leaf area and QWL, however, it does not significantly affect the scaling relationships between leaf morphological measures (including QPA and the extent of leaf bilateral symmetry) and leaf area.