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Wang L, He K, Hui C, Ratkowsky DA, Yao W, Lian M, Wang J, Shi P. Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution. Ecol Evol 2024; 14:e11072. [PMID: 38435001 PMCID: PMC10905244 DOI: 10.1002/ece3.11072] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2023] [Revised: 01/28/2024] [Accepted: 02/09/2024] [Indexed: 03/05/2024] Open
Abstract
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.
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Affiliation(s)
- Lin Wang
- Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingChina
| | - Ke He
- Architectural Design and Research InstituteShenzhen UniversityShenzhenChina
| | - Cang Hui
- Department of Mathematical Sciences, Centre for Invasion BiologyStellenbosch UniversityStellenboschSouth Africa
- Mathematical and Physical Biosciences, African Institute for Mathematical SciencesCape TownSouth Africa
| | - David A. Ratkowsky
- Tasmanian Institute of AgricultureUniversity of TasmaniaHobartTasmaniaAustralia
| | - Weihao Yao
- Bamboo Research Institute, College of Ecology and EnvironmentNanjing Forestry UniversityNanjingChina
| | - Meng Lian
- Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingChina
| | - Jinfeng Wang
- Bamboo Research Institute, College of Ecology and EnvironmentNanjing Forestry UniversityNanjingChina
| | - Peijian Shi
- Department of Applied Mathematics, College of ScienceNanjing Forestry UniversityNanjingChina
- Bamboo Research Institute, College of Ecology and EnvironmentNanjing Forestry UniversityNanjingChina
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Shi P, Gielis J, Quinn BK, Niklas KJ, Ratkowsky DA, Schrader J, Ruan H, Wang L, Niinemets Ü. 'biogeom': An R package for simulating and fitting natural shapes. Ann N Y Acad Sci 2022; 1516:123-134. [PMID: 35879250 DOI: 10.1111/nyas.14862] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
Abstract
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.
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Affiliation(s)
- Peijian Shi
- Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing, China
| | - Johan Gielis
- Department of Biosciences Engineering, University of Antwerp, Antwerp, Belgium
| | - Brady K Quinn
- St. Andrews Biological Station, Fisheries and Oceans Canada, St. Andrews, New Brunswick, Canada
| | - Karl J Niklas
- School of Integrative Plant Science, Cornell University, Ithaca, New York, USA
| | - David A Ratkowsky
- Tasmanian Institute of Agriculture, University of Tasmania, Hobart, Tasmania, Australia
| | - Julian Schrader
- School of Natural Sciences, Macquarie University, Sydney, New South Wales, Australia.,Biodiversity, Macroecology & Biogeography, University of Göttingen, Göttingen, Germany
| | - Honghua Ruan
- Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing, China
| | - Lin Wang
- Co-Innovation Centre for Sustainable Forestry in Southern China, Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing, China
| | - Ülo Niinemets
- Institute of Agricultural and Environmental Sciences, Estonian University of Life Sciences, Tartu, Estonia.,Estonian Academy of Sciences, Tallinn, Estonia
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Jin J, Quinn BK, Shi P. The Modified Brière Equation and Its Applications. PLANTS 2022; 11:plants11131769. [PMID: 35807720 PMCID: PMC9269267 DOI: 10.3390/plants11131769] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/15/2022] [Revised: 06/30/2022] [Accepted: 07/01/2022] [Indexed: 11/16/2022]
Abstract
The Brière equation (BE) is widely used to describe the effect of temperature on the development rate of insects, and it can produce both symmetrical and asymmetrical bell-shaped curves. Because of its elasticity in curve fitting, the integrated form of BE has been recommended for use as a sigmoid growth equation to describe the increase in plant biomass with time. However, the start time of growth predicted by the sigmoid growth equation based on the BE is not completely comparable to empirical crop growth data. In the present study, we modified the BE by adding an additional parameter to further increase its elasticity for data fitting. We termed this new equation the modified Brière equation (MBE). Data for the actual height and biomass of 15 species of plants (with two cultivars for one species) were fit with the sigmoid growth equations based on both the BE and MBE assuming that the growth start time was zero for both. The goodness of fit of the BE and MBE sigmoid growth equations were compared based on their root-mean-square errors and the corresponding absolute percentage error between them when fit to these data. For most species, we found that the MBE sigmoid growth equation achieved a better goodness of fit than the BE sigmoid growth equation. This work provides a useful tool for quantifying the ontogenetic or population growth of plants.
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Affiliation(s)
- Jun Jin
- Research Institute of Architecture, Southeast University, Nanjing 210096, China;
| | - Brady K. Quinn
- Biological Effects Section, St. Andrews Biological Station, Fisheries and Oceans Canada, St. Andrews, NB E5B 0E4, Canada;
| | - Peijian Shi
- Bamboo Research Institute, College of Science, Nanjing Forestry University, Nanjing 210037, China
- Correspondence: ; Tel.: +86-25-8542-7231
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Ma J, Niklas KJ, Liu L, Fang Z, Li Y, Shi P. Tree Size Influences Leaf Shape but Does Not Affect the Proportional Relationship Between Leaf Area and the Product of Length and Width. FRONTIERS IN PLANT SCIENCE 2022; 13:850203. [PMID: 35755713 PMCID: PMC9221507 DOI: 10.3389/fpls.2022.850203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2022] [Accepted: 05/23/2022] [Indexed: 05/12/2023]
Abstract
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.
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Affiliation(s)
- Jianzhong Ma
- Yunnan Academy of Forestry and Grassland, Kunming, China
- Bamboo Research Institute, Nanjing Forestry University, Nanjing, China
| | - Karl J. Niklas
- Plant Biology Section, School of Integrative Plant Science, Cornell University, Ithaca, NY, United States
| | - Leyi Liu
- College of Landscape Architecture and Horticulture Science, Southwest Forestry University, Kunming, China
| | - Zhendong Fang
- Shangri-la Alpine Botanical Garden, Shangri-la, China
| | - Yirong Li
- Bamboo Research Institute, Nanjing Forestry University, Nanjing, China
| | - Peijian Shi
- Bamboo Research Institute, Nanjing Forestry University, Nanjing, China
- *Correspondence: Peijian Shi,
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Hang H, Bauer M, Mio W, Mander L. Geometric and topological approaches to shape variation in Ginkgo leaves. ROYAL SOCIETY OPEN SCIENCE 2021; 8:210978. [PMID: 34849242 PMCID: PMC8611351 DOI: 10.1098/rsos.210978] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2021] [Accepted: 10/25/2021] [Indexed: 05/09/2023]
Abstract
Leaf shape is a key plant trait that varies enormously. The range of applications for data on this trait requires frequent methodological development so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 468 leaves produced by Ginkgo biloba, and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo. We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth and the angle of the two lobes at the leaf base that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. We have applied both methods to modern and fossil material: the methods are complementary, identifying similar primary patterns of variation, but also revealing different aspects of morphological variation. Our topological approach distinguishes long-shoot leaves from short-shoot leaves, both methods indicate that leaf shape influences or is at least related to leaf area, and both could be applied in palaeoclimatic and evolutionary studies of leaf shape.
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Affiliation(s)
- Haibin Hang
- Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
| | - Martin Bauer
- Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
| | - Washington Mio
- Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
| | - Luke Mander
- School of Environment, Earth and Ecosystem Sciences, The Open University, Milton Keynes, MK7 6AA, UK
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Li Y, Zheng Y, Ratkowsky DA, Wei H, Shi P. Application of an Ovate Leaf Shape Model to Evaluate Leaf Bilateral Asymmetry and Calculate Lamina Centroid Location. FRONTIERS IN PLANT SCIENCE 2021; 12:822907. [PMID: 35111188 PMCID: PMC8801803 DOI: 10.3389/fpls.2021.822907] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2021] [Accepted: 12/23/2021] [Indexed: 05/21/2023]
Abstract
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.
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Affiliation(s)
- Yirong Li
- Bamboo Research Institution, College of Science, Nanjing Forestry University, Nanjing, China
| | - Yiwen Zheng
- Bamboo Research Institution, College of Science, Nanjing Forestry University, Nanjing, China
| | - David A. Ratkowsky
- Tasmanian Institute of Agriculture, University of Tasmania, Hobart, TAS, Australia
| | - Hailin Wei
- Hunan Academy of Forestry, Changsha, China
- *Correspondence: Hailin Wei,
| | - Peijian Shi
- Bamboo Research Institution, College of Science, Nanjing Forestry University, Nanjing, China
- *Correspondence: Hailin Wei,
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