Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant

Comparison between computer recognition and manual measurement methods for the estimation of leaf area

BACKGROUND AND AIMS

Leaf area (A) is a crucial indicator of the photosynthetic capacity of plants. The Montgomery equation (ME), which hypothesizes that A is proportional to the product of leaf length (L) and width (W), is a valid tool for non-destructively measuring A for many broadleaved plants. At present, the methods used to compute L and W for the ME can be broadly divided into two kinds: using computer recognition and measuring manually. However, the potential difference in the prediction accuracy using either method has not been thoroughly examined in previous studies.

METHODS

In the present study, we measured 540 Alangium chinense leaves, 489 Liquidambar formosana leaves and 215 Liriodendron × sinoamericanum leaves, utilizing computer recognition and manual measurement methods to determine L and W. The ME was used to fit the data determined by the two methods, and the goodness of fits were compared. The prediction errors of A were analysed by examining the correlations with two leaf symmetry indices (areal ratio of the left side to the right side, and standardized index for bilateral asymmetry), as well as the leaf shape complexity index (the leaf dissection index).

KEY RESULTS

The results indicate that there is a neglectable difference in the estimation of A between the two methods. This further validates that the ME is an effective method for estimating A in broadleaved tree species, including those with lobes. Additionally, leaf shape complexity significantly influenced the estimation of A.

CONCLUSIONS

These results show that the use of computer recognition and manual measurement in the field are both effective and feasible, although the influence of leaf shape complexity should be considered when applying the ME to estimate A in the future.

Lead Tolerance and Remediation Potential of Four Indocalamus Species in Lead-Contaminated Soil

Indocalamus plants are low-growing shrubby bamboos with growth advantages, such as high biomass and strong resistance, and they are rich in germplasm resources in southern China. This study conducted soil lead (Pb) stress experiments on Indocalamus latifolius (Keng) McClure (LA), Indocalamus hunanensis B.M. Yang (HU), Indocalamus chishuiensis Y.L. Yang and Hsueh (CH) and Indocalamus lacunosus Wen (LC). Five Pb treatments (0, 500, 1000, 1500 mg·kg-1 Pb, and 1000 mg·kg-1 Pb + 1000 mg·kg-1 ethylenediamine tetraacetic acid (EDTA)) were established. EDTA was applied to explore the tolerance mechanism of different Indocalamus species after absorbing large amounts of heavy metals. The results were as follows: (1) under Pb treatment, the total relative biomass of LA, HU and LC was <100%, whereas the total relative biomass of CH was >100%; (2) after applying EDTA, the bioconcentration coefficient, translocation factor, and free proline content of the four Indocalamus species increased; and (3) the Pb mobility and distribution rates of the underground parts of the four Indocalamus species were consistently greater than those of the aboveground parts. The Pb mobility and distribution rates in the stems increased after applying EDTA, while those in the leaves decreased, as the plants tended to transfer Pb to their stems, which have lower physiological activity than their leaves.

Scaling relationships between the total number of leaves and the total leaf area per culm of two dwarf bamboo species

Total leaf area per plant is an important measure of the photosynthetic capacity of an individual plant that together with plant density drives the canopy leaf area index, that is, the total leaf area per unit ground area. Because the total number of leaves per plant (or per shoot) varies among conspecifics and among mixed species communities, this variation can affect the total leaf area per plant and per canopy but has been little studied. Previous studies have shown a strong linear relationship between the total leaf area per plant (or per shoot) (A T) and the total number of leaves per plant (or per shoot) (N T) on a log-log scale for several growth forms. However, little is known whether such a scaling relationship also holds true for bamboos, which are a group of Poaceae plants with great ecological and economic importance in tropical, subtropical, and warm temperate regions. To test whether the scaling relationship holds true in bamboos, two dwarf bamboo species (Shibataea chinensis Nakai and Sasaella kongosanensis 'Aureostriatus') with a limited but large number of leaves per culm were examined. For the two species, the leaves from 480 and 500 culms, respectively, were sampled and A T was calculated by summing the areas of individual leaves per culm. Linear regression and correlation analyses reconfirmed that there was a significant log-log linear relationship between A T and N T for each species. For S. chinensis, the exponent of the A T versus N T scaling relationship was greater than unity, whereas that of S. kongosanensis 'Aureostriatus' was smaller than unity. The coefficient of variation in individual leaf area increased with increasing N T for each species. The data reconfirm that there is a strong positive power-law relationship between A T and N T for each of the two species, which may reflect adaptations of plants in response to intra- and inter-specific competition for light.

Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution

The inequality in leaf and fruit size distribution per plant can be quantified using the Gini index, which is linked to the Lorenz curve depicting the cumulative proportion of leaf (or fruit) size against the cumulative proportion of the number of leaves (or fruits). Prior researches have predominantly employed empirical models-specifically the original performance equation (PE-1) and its generalized counterpart (GPE-1)-to fit rotated and right-shifted Lorenz curves. Notably, another potential performance equation (PE-2), capable of generating similar curves to PE-1, has been overlooked and not systematically compared with PE-1 and GPE-1. Furthermore, PE-2 has been extended into a generalized version (GPE-2). In the present study, we conducted a comparative analysis of these four performance equations, evaluating their applicability in describing Lorenz curves related to plant organ (leaf and fruit) size. Leaf area was measured on 240 culms of dwarf bamboo (Shibataea chinensis Nakai), and fruit volume was measured on 31 field muskmelon plants (Cucumis melo L. var. agrestis Naud.). Across both datasets, the root-mean-square errors of all four performance models were consistently smaller than 0.05. Paired t-tests indicated that GPE-1 exhibited the lowest root-mean-square error and Akaike information criterion value among the four performance equations. However, PE-2 gave the best close-to-linear behavior based on relative curvature measures. This study presents a valuable tool for assessing the inequality of plant organ size distribution.

Plant strategies for greatest height: tapering or hollowing

The tapered form and hollow cross-section of the stem and trunk of wild plants are rational mechanical approaches because they facilitate the plant simultaneously growing taller for photosynthesis and supporting its own weight. The purpose of this study is to clarify the advantages and disadvantages of tapering and hollowing from the perspective of the greatest probable height before self-buckling. We modelled woody plants using tapering or hollow cantilevers, formulated the greatest height before self-buckling, and derived a theoretical formula for the greatest probable height considering tapering and hollowing. This formula theoretically explains why almost all plants exhibit a tapered form: it allows for a greater height at an earlier growth stage than a hollow cross-section.

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 Two Simplified Versions of the Gielis Equation for Describing the Shape of Bamboo Leaves

Bamboo is an important component in subtropical and tropical forest communities. The plant has characteristic long lanceolate leaves with parallel venation. Prior studies have shown that the leaf shapes of this plant group can be well described by a simplified version (referred to as SGE-1) of the Gielis equation, a polar coordinate equation extended from the superellipse equation. SGE-1 with only two model parameters is less complex than the original Gielis equation with six parameters. Previous studies have seldom tested whether other simplified versions of the Gielis equation are superior to SGE-1 in fitting empirical leaf shape data. In the present study, we compared a three-parameter Gielis equation (referred to as SGE-2) with the two-parameter SGE-1 using the leaf boundary coordinate data of six bamboo species within the same genus that have representative long lanceolate leaves, with >300 leaves for each species. We sampled 2000 data points at approximately equidistant locations on the boundary of each leaf, and estimated the parameters for the two models. The root−mean−square error (RMSE) between the observed and predicted radii from the polar point to data points on the boundary of each leaf was used as a measure of the model goodness of fit, and the mean percent error between the RMSEs from fitting SGE-1 and SGE-2 was used to examine whether the introduction of an additional parameter in SGE-1 remarkably improves the model’s fitting. We found that the RMSE value of SGE-2 was always smaller than that of SGE-1. The mean percent errors among the two models ranged from 7.5% to 20% across the six species. These results indicate that SGE-2 is superior to SGE-1 and should be used in fitting leaf shapes. We argue that the results of the current study can be potentially extended to other lanceolate leaf shapes.

Diminishing returns: A comparison between fresh mass vs. area and dry mass vs. area in deciduous species

"Diminishing returns" in leaf economics occurs when increases in lamina mass (M), which can either be represented by lamina dry mass (DM) or fresh mass (FM), fail to produce proportional increases in leaf surface area (A), such that the scaling exponent (α) for the M vs. A scaling relationship exceeds unity (i.e., α > 1.0). Prior studies have shown that FM vs. A is better than DM vs A in assessing diminishing returns in evergreen species. However, the superiority of FM vs. A over DM vs. A has been less well examined for deciduous species. Here, we applied reduced major axis protocols to test whether FM vs. A is better than DM vs. A to describe the M vs. A scaling relationship, using a total of 4271 leaves from ten deciduous and two evergreen tree species in the Fagaceae and Ulmaceae for comparison. The significance of the difference between the scaling exponents of FM vs. A and DM vs. A was tested using the bootstrap percentile method. Further, we tested the non-linearity of the FM (DM) vs. A data on a log-log scale using ordinary least squares. We found that (i) the majority of scaling exponents of FM vs. A and DM vs. A were >1 thereby confirming diminishing returns for all 12 species, (ii) FM vs. A was more robust than DM vs. A to identify the M vs. A scaling relationship, (iii) the non-linearity of the allometric model was significant for both DM vs. A and FM vs. A., and (iv) the evergreen species of Fagaceae had significantly higher DM and FM per unit area than other deciduous species. In summary, FM vs. A is a more reliable measure than DM vs. A when dealing with diminishing returns, and deciduous species tend to invest less biomass in unit leaf light harvesting area than evergreen species.

Quantifying the Variation in the Geometries of the Outer Rims of Corolla Tubes of Vinca major 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, r is the polar radius corresponding to the polar angle φ; m 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 Vinca major 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 k = 1 and m = 5: r=acos54φn2+sin54φn3−1/n1 (Model 1), and r=acos54φn2+sin54φn2−1/n1 (Model 2). The adjusted root mean square error (RMSE_adj) 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 n₂ and n₃ in Model 1 on a log-log scale. The results validated the two simplified Gielis equations. The RMSE_adj values for all corolla tubes were smaller than 0.05 for both models. The numerical values of n₂ and n₃ were demonstrated to be statistically equal based on the regression analysis, which suggested that the ORCTs of V. major 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.

'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.

Competitive performance of Pinus massoniana is related to scaling relationships at the individual plant and branch levels

PREMISE

Competition is an important driver of tree mortality and thus affects forest structure and dynamics. Tree architectural traits, such as height-to-diameter (H-D) and branch length-to-diameter (L-d) relationships are thought to influence species competitiveness by affecting light capture. Unfortunately, little is known about how the H vs. D and L vs. d scaling exponents are related to tree performance (defined in the context of growth vigor) in competition.

METHODS

Using data from field surveys of 1547 individuals and destructive sampling of 51 trees with 1086 first-order branches from a high-density Pinus massoniana forest, we explored whether the H vs. D and the L vs. d scaling exponents respectively differed numerically across tree performance and branch vertical position in crowns.

RESULTS

The results indicated that (1) the H vs. D scaling exponent decreased as tree performance declined; (2) the L vs. d scaling exponent differed across tree performance classes (i.e., the scaling exponent of "inferior" trees was significantly larger than that of "moderate" and "superior" trees); (3) the L vs. d scaling exponent decreased as branch position approached ground level; and (4) overall, the branch scaling exponent decreased as tree performance improved in each crown layer, but decreased significantly in the intermediate layer.

CONCLUSIONS

This study highlights the variation within (and linkage among) length-to-diameter scaling relationships across tree performance at the individual and branch levels. This linkage provides new insights into potential mechanisms of tree growth variation (and even further mortality) under competition in subtropical forests.

Comparison of a universal (but complex) model for avian egg shape with a simpler model

Recently, a universal equation by Narushin, Romanov, and Griffin (hereafter, the NRGE) was proposed to describe the shape of avian eggs. While NRGE can simulate the shape of spherical, ellipsoidal, ovoidal, and pyriform eggs, its predictions were not tested against actual data. Here, we tested the validity of the NRGE by fitting actual data of egg shapes and compared this with the predictions of our simpler model for egg shape (hereafter, the SGE). The eggs of nine bird species were sampled for this purpose. NRGE was found to fit the empirical data of egg shape well, but it did not define the egg length axis (i.e., the rotational symmetric axis), which significantly affected the prediction accuracy. The egg length axis under the NRGE is defined as the maximum distance between two points on the scanned perimeter of the egg's shape. In contrast, the SGE fitted the empirical data better, and had a smaller root-mean-square error than the NRGE for each of the nine eggs. Based on its mathematical simplicity and goodness-of-fit, the SGE appears to be a reliable and useful model for describing egg shape.

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.

"Diminishing returns" for leaves of five age-groups of Phyllostachys edulis culms

PREMISE

Leaf mass (M) and lamina surface area (A) are important functional traits reported to obey a scaling relationship called "diminishing returns" (i.e., M ∝ A^α>1 ). Previous studies have focused primarily on eudicots and ignored whether the age of leaves affects the numerical value of the scaling exponent (i.e., α).

METHODS

The effect of age was examined using 1623 Phyllostachys edulis leaves from culms differing in age collected in Nanjing, China. The scaling relationships among leaf A, fresh mass (FM), and dry mass (DM) were evaluated using reduced major axis protocols. The bootstrap percentile method was used to test the significance of differences in α-values.

RESULTS

Overall, the numerical values of α exceeded 1.0. The scaling relationship between FM and A was statistically more robust than that between DM and A. The scaling exponents of FM vs. A exhibited a "high-low-high-low-high" numerical trend from the oldest to the youngest age-group. FM increased linearly as culm age decreased; the leaf DM per unit area (LMA) exhibited a parabolic trend across the age-groups.

CONCLUSIONS

"Diminishing returns" is confirmed for all but one age-group of an important monocot species. The relationship between FM and A was statistically more robust than that between DM and A for each age-group. The FM per unit A decreased with increasing age-groups, whereas the middle age-groups had a greater LMA than the oldest and youngest age-groups. These data are the first to show that the age of shoots affects the scaling relationship between leaf mass and area.

A General Model for Describing the Ovate Leaf Shape

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.

Plant Age Has a Minor Effect on Non-Destructive Leaf Area Calculations in Moso Bamboo (Phyllostachys edulis)

Leaf area is among the most important leaf functional traits, and it determines leaf temperature and alters light harvesting. The calculation of individual leaf area is the basis of calculating the leaf area index (i.e., the total leaf area per unit ground area) that is directly associated with the ability of plants to intercept light for photosynthesis. It is valuable to provide a fast and reliable approach to measuring leaf area. Here, we examined the validity and calculation accuracy of the Montgomery equation (ME), which describes the area of a leaf as a product of leaf length, width and a specific coefficient referred to as the Montgomery parameter, MP. Using ME, we calculated leaf areas of different age groups of bamboo culms. For most broad-leaved plants, leaf area is proportional to the product of leaf length and width, and MP falls within a range of 1/2 to π/4, depending on leaf shape. However, it is unknown whether there is an intra-specific variation in MP resulting from age structure and whether such a variation can significantly reduce the predictability of ME in calculating leaf area. This is relevant as a population of perennial plants usually composes of different age groups. We used Moso bamboos as model as this species is of ecological and economic importance in southern China, and pure stands can cover six to seven plant age groups. We used five age groups of moso bamboo and sampled 260–380 leaves for each group to test whether ME holds true for each group and all groups combined, whether there are significant differences in MP among different age groups, and whether the differences in MP can lead to large prediction errors for leaf area. We observed that for each age group and all groups combined, there were significant proportional relationships between leaf area and the product of leaf length and width. There were small but significant differences in MP among the five age groups (MP values ranged from 0.6738 to 0.7116 for individual plant ages; MP = 0.6936 for all age groups combined), which can be accounted for by the minor intergroup variation of leaf shape (reflected by the leaf width/length ratio). For all age classes, MP estimated for the pooled data resulted in <4% mean absolute percentage error, indicating that the effect of variation in MP among different age groups was small. We conclude that ME can serve as a useful tool for accurate calculations of leaf area in moso bamboo independent of culm age, which is valuable for estimation of leaf area index as well as evaluating the productivity and carbon sequestration capacity of bamboo forests.

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.

Nondestructive estimation of leaf area for 15 species of vines with different leaf shapes

PREMISE

The nondestructive measurement of leaf area is important for expediting data acquisition in the field. The Montgomery equation (ME) assumes that leaf area (A) is a proportional function of the product of leaf length (L) and width (W), i.e., A = cLW, where c is called the Montgomery parameter. The ME has been successfully applied to calculate the surface area of many broad-leaved species with simple leaf shapes. However, whether this equation is valid for more complex leaf shapes has not been verified.

METHODS

Leaf A, L, and W were measured directly for each of 5601 leaves of 15 vine species, and ME and three other models were used to fit the data. All four models were compared based on their root mean square errors (RMSEs) to determine whether ME provided the best fit.

RESULTS

The ME was a reliable method for estimating the A of all 15 species. In addition, the numerical values of 13 of the 15 values of c fell within a previously predicted numerical range (i.e., between 1/2 and π/4). The data show that the numerical values of c are largely affected by the value of W/L, the concavity of the leaf base, and the number of lobes on the lamina.

CONCLUSIONS

The Montgomery parameter can reflect the influence of leaf shape on leaf-area calculations and can serve as an important tool for nondestructive measurements of leaf area for many broad-leaved species and for the investigation of leaf morphology.

Comparison of the Scaling Relationships of Leaf Biomass versus Surface Area between Spring and Summer for Two Deciduous Tree Species

The scaling relationship between either leaf dry or fresh mass (M) and surface area (A) can reflect the photosynthetic potential and efficiency of light harvesting in different broad-leaved plants. In growing leaves, lamina area expansion is typically finished before the completion of leaf biomass accumulation, thereby affecting the M vs. A scaling relationship at different developmental stages of leaves (e.g., young vs. adult leaves). In addition, growing plants can have different-sized leaves at different plant ages, potentially also changing M vs. A scaling. Furthermore, leaf shape can also change during the course of ontogeny and modify the M vs. A scaling relationship. Indeed, the effect of seasonal changes in leaf shape on M vs. A scaling has not been examined in any previous studies known to us. The study presented here was conducted using two deciduous tree species: Alangium chinense (saplings forming leaves through the growing season) and Liquidambar formosana (adult trees producing only one leaf flush in spring) that both have complex but nearly bilaterally symmetrical leaf shapes. We determined (i) whether leaf shapes differed in spring versus summer; (ii) whether the M vs. A scaling relationship varied over time; and (iii) whether there is a link between leaf shape and the scaling exponent governing the M vs. A scaling relationship. The data indicated that (i) the leaf dissection index in spring was higher than that in summer for both species (i.e., leaf-shape complexity decreased from young to adult leaves); (ii) there was a significant difference in the numerical value of the scaling exponent of leaf perimeter vs. area between leaves sampled at the two dates; (iii) spring leaves had a higher water content than summer leaves, and the scaling exponents of dry mass vs. area and fresh mass vs. area were all greater than unity; (iv) the scaling relationship between fresh mass and area was statistically more robust than that between leaf dry mass and area; (v) the scaling exponents of leaf dry and fresh mass vs. area of A. chinense leaves in spring were greater than those in summer (i.e., leaves in younger plants tend to be larger than leaves in older plants), whereas, for the adult trees of L. formosana, the scaling exponent in spring was smaller than that in summer, indicating increases in leaf dry mass per unit area with increasing leaf age; and (vi) leaf shape appears not to be related to the scaling relationship between either leaf dry or fresh mass and area, but is correlated with the scaling exponent of leaf perimeter vs. area (which tends to be a ½ power function). These trends indicate that studies of leaf morphometrics and scaling relationships must consider the influence of seasonality and plant age in sampling of leaves and the interpretation of data.

The Generalized Gielis Geometric Equation and Its Application

Many natural shapes exhibit surprising symmetry and can be described by the Gielis equation, which has several classical geometric equations (for example, the circle, ellipse and superellipse) as special cases. However, the original Gielis equation cannot reflect some diverse shapes due to limitations of its power-law hypothesis. In the present study, we propose a generalized version by introducing a link function. Thus, the original Gielis equation can be deemed to be a special case of the generalized Gielis equation (GGE) with a power-law link function. The link function can be based on the morphological features of different objects so that the GGE is more flexible in fitting the data of the shape than its original version. The GGE is shown to be valid in depicting the shapes of some starfish and plant leaves.

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.

Proportional Relationship between Leaf Area and the Product of Leaf Length and Width of Four Types of Special Leaf Shapes

The leaf area, as an important leaf functional trait, is thought to be related to leaf length and width. Our recent study showed that the Montgomery equation, which assumes that leaf area is proportional to the product of leaf length and width, applied to different leaf shapes, and the coefficient of proportionality (namely the Montgomery parameter) range from 1/2 to π/4. However, no relevant geometrical evidence has previously been provided to support the above findings. Here, four types of representative leaf shapes (the elliptical, sectorial, linear, and triangular shapes) were studied. We derived the range of the estimate of the Montgomery parameter for every type. For the elliptical and triangular leaf shapes, the estimates are π/4 and 1/2, respectively; for the linear leaf shape, especially for the plants of Poaceae that can be described by the simplified Gielis equation, the estimate ranges from 0.6795 to π/4; for the sectorial leaf shape, the estimate ranges from 1/2 to π/4. The estimates based on the observations of actual leaves support the above theoretical results. The results obtained here show that the coefficient of proportionality of leaf area versus the product of leaf length and width only varies in a small range, maintaining the allometric relationship for leaf area and thereby suggesting that the proportional relationship between leaf area and the product of leaf length and width broadly remains stable during leaf evolution.

A General Leaf Area Geometric Formula Exists for Plants—Evidence from the Simplified Gielis Equation

Plant leaves exhibit diverse shapes that enable them to utilize a light resource maximally. If there were a general parametric model that could be used to calculate leaf area for different leaf shapes, it would help to elucidate the adaptive evolutional link among plants with the same or similar leaf shapes. We propose a simplified version of the original Gielis equation (SGE), which was developed to describe a variety of object shapes ranging from a droplet to an arbitrary polygon. We used this equation to fit the leaf profiles of 53 species (among which, 48 bamboo plants, 5 woody plants, and 10 geographical populations of a woody plant), totaling 3310 leaves. A third parameter (namely, the floating ratio c in leaf length) was introduced to account for the case when the theoretical leaf length deviates from the observed leaf length. For most datasets, the estimates of c were greater than zero but less than 10%, indicating that the leaf length predicted by the SGE was usually smaller than the actual length. However, the predicted leaf areas approximated their actual values after considering the floating ratios in leaf length. For most datasets, the mean percent errors of leaf areas were lower than 6%, except for a pooled dataset with 42 bamboo species. For the elliptical, lanceolate, linear, obovate, and ovate shapes, although the SGE did not fit the leaf edge perfectly, after adjusting the parameter c, there were small deviations of the predicted leaf areas from the actual values. This illustrates that leaves with different shapes might have similar functional features for photosynthesis, since the leaf areas can be described by the same equation. The anisotropy expressed as a difference in leaf shape for some plants might be an adaptive response to enable them to adapt to different habitats.

Taylor’s Power Law for Leaf Bilateral Symmetry

Leaf shape and symmetry is of interest because of the importance of leaves in photosynthesis. Recently, a novel method was proposed to measure the extent of bilateral symmetry in leaves in which a leaf was divided into left and right sides by a straight line through the leaf apex and base, and a number of equidistant strips were drawn perpendicular to the straight line to generate an equivalent number of differences in area between the left and right parts. These areal differences are the basis for a measure of leaf bilateral symmetry, which was then examined to see how well it follows Taylor’s power law (TPL) using three classes of plants, namely, 10 geographical populations of Parrotia subaequalis (H.T. Chang) R.M. Hao et H.T. Wei, 10 species of Bambusoideae, and 10 species of Rosaceae. The measure of bilateral symmetry followed TPL for a single species or for a class of closely related species. The estimate of the exponent of TPL for bamboo plants was significantly larger than for the dicotyledonous trees, but its goodness of fit was the best among the three classes of plants. The heterogeneity of light falling on branches and leaves due to above-ground architectural patterns is an important contributor to leaf asymmetry.

Why Does Not the Leaf Weight-Area Allometry of Bamboos Follow the 3/2-Power Law?

The principle of similarity (Thompson, 1917) states that the weight of an organism follows the 3/2-power law of its surface area and is proportional to its volume on the condition that the density is constant. However, the allometric relationship between leaf weight and leaf area has been reported to greatly deviate from the 3/2-power law, with the irregularity of leaf density largely ignored for explaining this deviation. Here, we choose 11 bamboo species to explore the allometric relationships among leaf area (A), density (ρ), length (L), thickness (T), and weight (W). Because the edge of a bamboo leaf follows a simplified two-parameter Gielis equation, we could show that A ∝ L² and that A ∝ T². This then allowed us to derive the density-thickness allometry ρ ∝ T^b and the weight-area allometry W ∝ A^(b+3)/2 ≈ A^9/8, where b approximates -3/4. Leaf density is strikingly negatively associated with leaf thickness, and it is this inverse relationship that results in the weight-area allometry to deviate from the 3/2-power law. In conclusion, although plants are prone to invest less dry mass and thus produce thinner leaves when the leaf area is sufficient for photosynthesis, such leaf thinning needs to be accompanied with elevated density to ensure structural stability. The findings provide the insights on the evolutionary clue about the biomass investment and output of photosynthetic organs of plants. Because of the importance of leaves, plants could have enhanced the ratio of dry material per unit area of leaf in order to increase the efficiency of photosynthesis, relative the other parts of plants. Although the conclusion is drawn only based on 11 bamboo species, it should also be applicable to the other plants, especially considering previous works on the exponent of the weight-area relationship being less than 3/2 in plants.

A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation

The size and shape of plant leaves change with growth, and an accurate description of leaf shape is crucial for describing plant morphogenesis and development. Bilateral symmetry, which has been widely observed but poorly examined, occurs in both dicot and monocot leaves, including all nominated bamboo species (approximately 1,300 species), of which at least 500 are found in China. Although there are apparent differences in leaf size among bamboo species due to genetic and environmental profiles, bamboo leaves have bilateral symmetry with parallel venation and appear similar across species. Here, we investigate whether the shape of bamboo leaves can be accurately described by a simplified Gielis equation, which consists of only two parameters (leaf length and shape) and produces a perfect bilateral shape. To test the applicability of this equation and the occurrence of bilateral symmetry, we first measured the leaf length of 42 bamboo species, examining >500 leaves per species. We then scanned 30 leaves per species that had approximately the same length as the median leaf length for that species. The leaf‐shape data from scanned profiles were fitted to the simplified Gielis equation. Results confirmed that the equation fits the leaf‐shape data extremely well, with the coefficients of determination being 0.995 on average. We further demonstrated the bilateral symmetry of bamboo leaves, with a clearly defined leaf‐shape parameter of all 42 bamboo species investigated ranging from 0.02 to 0.1. This results in a simple and reliable tool for precise determination of bamboo species, with applications in forestry, ecology, and taxonomy.

Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant

The relationship between spatial density and size of plants is an important topic in plant ecology. The self-thinning rule suggests a -3/2 power between average biomass and density or a -1/2 power between stand yield and density. However, the self-thinning rule based on total leaf area per plant and density of plants has been neglected presumably because of the lack of a method that can accurately estimate the total leaf area per plant. We aimed to find the relationship between spatial density of plants and total leaf area per plant. We also attempted to provide a novel model for accurately describing the leaf shape of bamboos. We proposed a simplified Gielis equation with only two parameters to describe the leaf shape of bamboos one model parameter represented the overall ratio of leaf width to leaf length. Using this method, we compared some leaf parameters (leaf shape, number of leaves per plant, ratio of total leaf weight to aboveground weight per plant, and total leaf area per plant) of four bamboo species of genus Indocalamus Nakai (I. pedalis (Keng) P.C. Keng, I. pumilus Q.H. Dai and C.F. Keng, I. barbatus McClure, and I. victorialis P.C. Keng). We also explored the possible correlation between spatial density and total leaf area per plant using log-linear regression. We found that the simplified Gielis equation fit the leaf shape of four bamboo species very well. Although all these four species belonged to the same genus, there were still significant differences in leaf shape. Significant differences also existed in leaf area per plant, ratio of leaf weight to aboveground weight per plant, and leaf length. In addition, we found that the total leaf area per plant decreased with increased spatial density. Therefore, we directly demonstrated the self-thinning rule to improve light interception.