# Towards More Realistic Leaf Shapes in Functional-Structural Plant Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}concentration, or focus on changes in controlled greenhouse conditions. While in the past two kinds of plant modeling approaches—architectural models and process-based models—were analyzed separately, they were combined to functional-structural plant models (FSPM, aka virtual plant models) more recently [1,2,3]. Using process-based models, whole-plant reactions on environmental conditions are studied, while architectural models focus on establishing algorithms to simulate plant morphological development often described by Lindenmayer-Systems (L-Systems) [4]. FSPMs allow studying both, the physiological and structural responses to global and local environmental conditions. Most prominent environmental factors include temperature, nutrient supply and light, e.g., photosynthetically active radiation (PAR) [1,2,5,6,7].

## 2. Materials and Methods

#### 2.1. Virtual Plant Model

_{L}(cm

^{2}) as follows:

#### 2.2. Experimental Data

#### 2.2.1. Digitized Leaves Dataset

_{L}(cm

^{2}) and corresponding equilateral triangle side w (cm) for 1125 digitized leaves used for shape estimation, with the majority being in the range of 150 cm

^{2}≤ A

_{L}≤ 2000 cm

^{2}or 9.3 cm ≤ w ≤ 34 cm.

#### 2.2.2. Plant Morphological Data

^{−2}s

^{−1}and 388 ± 134 µmol m

^{−2}s

^{−1}and for temperature of 22.0 ± 1.5 °C and 22.8 ± 1.4 °C, respectively.

#### 2.3. Morphometric Leaf Shape Analysis

#### 2.3.1. Preprocessing

_{L}= 100 cm

^{2}has a centroid size of 22.92 cm, estimated over all 17 landmarks, whereas a leaf, where both halves are folded downward to the maximum angle of 90° has a centroid size of approx. 16.44 cm. Hence, scaling to unit centroid size would lead to considerably different surfaces areas (sizes) of approx. 0.19 cm

^{2}and approx. 0.37 cm

^{2}, respectively. This would introduce variance in all three dimensions, although only two dimensions were altered. When scaling by equilateral triangle side w, estimated from the leaf area (Equation (3)), the surface area of both leaves will remain equal (here: w ≈ 7.60 cm). A general example of the differences introduced when scaling by centroid size or by w is given in the Supplementary Materials (Figure S1, Table S1).

#### 2.3.2. Morphometric Measures

#### 2.3.3. Symmetry Analysis

- calculate the left-right signed difference between the paired landmark shift for a single dimension (${s}_{x}$, ${s}_{y}$ or ${s}_{z}$)
- conduct robust Bayesian estimation on the differences ($n=1125$) to estimate posterior mean ($\widehat{s}$) and posterior standard deviation ($\widehat{\sigma}$) as measures for directional and fluctuating asymmetry
- (a)
- if 95% highest probability density interval (HDI) of $\widehat{s}$ includes zero⇒ no directional asymmetry
- (b)
- if 95% highest probability density interval (HDI) of $\widehat{s}$ excludes zero⇒ directional asymmetry

- conduct robust Bayesian estimation on the pooled left-right data ($n=2250$) to estimate overall posterior mean (absolute average displacement)
- if necessary: adjust average displacements using directional asymmetry information

#### 2.4. Virtual Plant Simulations

^{2}), daily stem red-to-far-red ratio from internodes weighted by internode length $\mathrm{R}:{\mathrm{FR}}_{\mathrm{Stem}}$ (-) and total plant length (${L}_{\mathrm{P}}$, cm).

## 3. Results and Discussion

#### 3.1. Average Leaf Shape

#### 3.2. Leaf Symmetry

^{2}to 2000 cm

^{2}(5 cm < w < 34 cm) (cf. Figure 3). Previous studies found measurement errors for similar magnetic field digitizers, calculated as the root-mean-square deviation (RMSD), to be in the range of 0.15 cm to 1.0 cm. Less deviation was found in laboratory trials compared to field experiments [23,48,49]. Hence, we cannot fully disregard this differences in symmetry on the basis of measurement error, when interpreted as the region of practical equivalence (ROPE) [50]. We therefore keep these results on directional asymmetry for the further study, although we do not expect significant effects on light interaction in the virtual plant simulations.

#### 3.3. Effects of Leaf Shape on Virtual Plant Simulations

^{®}Xeon

^{®}CPU E5-2660v3 with 64 GB Ram. This is mainly caused by an increased effort in Quasi-Monte Carlo light modeling, where, instead of the original four triangles, 20 triangles are evaluated for the new shape. Although computational times almost tripled, they are still on the scale of minutes and hence irrelevant in comparison to real life plant development cycles.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A_{L} | leaf area (cm^{2}) |

DW | produced dry weight per plant and day (g) |

$\mathrm{ER}\left(t\right)$ | internode elongation at day t (cm) |

${F}_{\mathrm{av}}\left(T\right)$ | temperature response function |

FIL | final internode length (cm) |

FSPM | functional-structural plant model |

GPA | Generalized Procrustes analysis (GPA) |

HDI | 95% highest probability density interval |

IL(t) | internode length at day t (cm) |

${L}_{\mathrm{P}}$ | plant length (cm) |

LAP | leaf area per plant (cm^{2}) |

L-System | Lindenmayer-System |

${P}_{1}$ | leaf base |

${P}_{2}$ | leaf tip |

PAR | photosynthetically active radiation |

R:FR | red-to-far-red ratio |

$\mathrm{R}:{\mathrm{FR}}_{\mathrm{Stem}}$ | daily steam red-to-far-red ratio from internodes weighted by their length (-) |

RMSD | root-mean-square deviation |

${s}_{x}$ | point wise displacement along the x-axis |

${s}_{y}$ | point wise displacement in distance to the x-axis |

${s}_{z}$ | point wise displacement in roll angle around the x-axis |

${\widehat{s}}_{x}$, ${\widehat{s}}_{y}$, ${\widehat{s}}_{z}$ | posterior mean of point wise displacement |

${\widehat{\sigma}}_{{s}_{x}}$, ${\widehat{\sigma}}_{{s}_{y}}$, ${\widehat{\sigma}}_{{s}_{z}}$ | posterior standard deviation of point wise displacement |

T | temperature (°C) |

t | time (day) |

w | equilateral triangle side (cm) |

## References

- Vos, J.; Evers, J.B.; Buck-Sorlin, G.H.; Andrieu, B.; Chelle, M.; de Visser, P.H.B. Functional-structural plant modelling: A new versatile tool in crop science. J. Exp. Bot.
**2010**, 61, 2101–2115. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sievänen, R.; Godin, C.; DeJong, T.M.; Nikinmaa, E. Functional-structural plant models: A growing paradigm for plant studies. Ann. Bot.
**2014**, 114, 599–603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vos, J.; Marcelis, L.; Evers, J. Functional-Structural plant modelling in crop production: Adding a dimension. In Functional-Structural Plant Modelling in Crop Production; Vos, J., Marcelis, L.F.M., Visser, P., Struik, P.C., Evers, J.B., Eds.; Springer: Dordrecht, The Netherlands, 2007; Volume 22, pp. 1–12. [Google Scholar]
- Prusinkiewicz, P.; Lindenmayer, A. The Algorithmic Beauty of Plants; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Kahlen, K.; Stützel, H. Modelling photo-modulated internode elongation in growing glasshouse cucumber canopies. New Phytol.
**2011**, 190, 697–708. [Google Scholar] [CrossRef] [PubMed] - Kahlen, K.; Chen, T.W. Predicting Plant Performance Under Simultaneously Changing Environmental Conditions—The Interplay Between Temperature, Light, and Internode Growth. Front. Plant Sci.
**2015**, 6, 1130. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, T.W.; Nguyen, T.M.N.; Kahlen, K.; Stützel, H. High temperature and vapor pressure deficit aggravate architectural effects but ameliorate non-architectural effects of salinity on dry mass production of tomato. Front. Plant Sci.
**2015**, 6, 887. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Viaud, G.; Loudet, O.; Cournède, P.H. Leaf Segmentation and Tracking in Arabidopsis thaliana Combined to an Organ-Scale Plant Model for Genotypic Differentiation. Front. Plant Sci.
**2017**, 7, 2057. [Google Scholar] [CrossRef] [PubMed] - Allen, M.T.; Prusinkiewicz, P.; DeJong, T.M. Using L-systems for modeling source–sink interactions, architecture and physiology of growing trees: The L-PEACH model. New Phytol.
**2005**, 166, 869–880. [Google Scholar] [CrossRef] [PubMed] - Hanan, J.; Hearn, A. Linking physiological and architectural models of cotton. Agric. Syst.
**2003**, 75, 47–77. [Google Scholar] [CrossRef] - Nikinmaa, E.; Sievänen, R.; Perttunen, J.; Hölttä, T. Simulated interaction between tree structure and xylem and phloem transport in 3D tree crowns using model LIGNUM. In Proceedings of the FSPM 2013 7th International Conference on Functional-Structural Plant Models, Saariselkä, Finland, 9–14 June 2013; Sievänen, R., Nikinmaa, E., Godin, C., Lintunen, A., Nygren, P.F.I., Eds.; [Google Scholar]
- Da Silva, D.; Han, L.; Faivre, R.; Costes, E. Influence of the variation of geometrical and topological traits on light interception efficiency of apple trees: Sensitivity analysis and metamodelling for ideotype definition. Ann. Bot.
**2014**, 114, 739–752. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Godin, C.; Sinoquet, H. Functional–structural plant modelling. New Phytol.
**2005**, 166, 705–708. [Google Scholar] [CrossRef] [PubMed] - Xu, J.; Li, J.; Cui, L.; Zhang, T.; Wu, Z.; Zhu, P.Y.; Meng, Y.J.; Zhang, K.J.; Yu, X.Q.; Lou, Q.F.; et al. New insights into the roles of cucumber TIR1 homologs and miR393 in regulating fruit/seed set development and leaf morphogenesis. BMC Plant Biol.
**2017**, 17, 130. [Google Scholar] [CrossRef] [PubMed] - Kahlen, K.; Wiechers, D.; Stützel, H. Modelling leaf phototropism in a cucumber canopy. Funct. Plant Biol.
**2008**, 35, 876–884. [Google Scholar] [CrossRef] - Kahlen, K. Towards functional-structural modelling of greenhouse cucumber. In Functional-Structural Plant Modelling in Crop Production; Vos, J., Marcelis, L.F.M., Visser, P., Struik, P.C., Evers, J.B., Eds.; Springer: Dordrecht, The Netherlands, 2007; Volume 22, pp. 209–217. [Google Scholar]
- Chambelland, J.C.; Dassot, M.; Adam, B.; Donès, N.; Balandier, P.; Marquier, A.; Saudreau, M.; Sonohat, G.; Sinoquet, H. A double-digitising method for building 3D virtual trees with non-planar leaves: Application to the morphology and light-capture properties of young beech trees (Fagus sylvatica). Funct. Plant Biol.
**2008**, 35, 1059–1069. [Google Scholar] [CrossRef] - Chéné, Y.; Rousseau, D.; Lucidarme, P.; Bertheloot, J.; Caffier, V.; Morel, P.; Belin, É.; Chapeau-Blondeau, F. On the use of depth camera for 3D phenotyping of entire plants. Comput. Electron. Agric.
**2012**, 82, 122–127. [Google Scholar] [CrossRef] [Green Version] - An, N.; Palmer, C.M.; Baker, R.L.; Markelz, R.C.; Ta, J.; Covington, M.F.; Maloof, J.N.; Welch, S.M.; Weinig, C. Plant high-throughput phenotyping using photogrammetry and imaging techniques to measure leaf length and rosette area. Comput. Electron. Agric.
**2016**, 127, 376–394. [Google Scholar] [CrossRef] [Green Version] - Loch, B.I. Surface Fitting for the Modelling of Plant Leaves. Ph.D. Thesis, University of Queensland, Queensland, Australia, 2004. [Google Scholar]
- Zhang, L.; Weckler, P.; Wang, N.; Xiao, D.; Chai, X. Individual leaf identification from horticultural crop images based on the leaf skeleton. Comput. Electron. Agric.
**2016**, 127, 184–196. [Google Scholar] [CrossRef] - Kahlen, K. 3D Architectural Modelling of Greenhouse Cucumber (Cucumis sativus L.) Using L-Systems; International Society for Horticultural Science (ISHS): Leuven, Belgium, 2006; pp. 51–58. [Google Scholar]
- Kahlen, K.; Stützel, H. Estimation of Geometric Attributes and Masses of Individual Cucumber Organs Using Three-dimensional Digitizing and Allometric Relationships. J. Am. Soc. Hort. Sci.
**2007**, 132, 439–446. [Google Scholar] - Whitlock, M. The heritability of fluctuating asymmetry and the genetic control of developmental stability. Proc. R. Soc. Lond. Ser. B
**1996**, 263, 849–853. [Google Scholar] [CrossRef] [PubMed] - DONGEN, S.V. Fluctuating asymmetry and developmental instability in evolutionary biology: Past, present and future. J. Evol. Biol.
**2006**, 19, 1727–1743. [Google Scholar] [CrossRef] [PubMed] - Klingenberg, C.P. Analyzing Fluctuating Asymmetry with Geometric Morphometrics: Concepts, Methods, and Applications. Symmetry
**2015**, 7, 843–934. [Google Scholar] [CrossRef] [Green Version] - Valen, L.V. A study of fluctuating asymmetry. Evolution
**1962**, 16, 125–142. [Google Scholar] [CrossRef] - Klein, L.L.; Caito, M.; Chapnick, C.; Kitchen, C.; O’Hanlon, R.; Chitwood, D.H.; Miller, A.J. Digital Morphometrics of Two North American Grapevines (Vitis: Vitaceae) Quantifies Leaf Variation between Species, within Species, and among Individuals. Front. Plant Sci.
**2017**, 8, 373. [Google Scholar] [CrossRef] [PubMed] - Chitwood, D.H.; Rundell, S.M.; Li, D.Y.; Woodford, Q.L.; Yu, T.T.; Lopez, J.R.; Greenblatt, D.; Kang, J.; Londo, J.P. Climate And Developmental Plasticity: Interannual Variability In Grapevine Leaf Morphology. Plant Physiol.
**2016**. Available online: http://www.plantphysiol.org/content/170/3/1480 (accessed on 9 July 2018). - Chitwood, D.H.; Ranjan, A.; Martinez, C.C.; Headland, L.R.; Thiem, T.; Kumar, R.; Covington, M.F.; Hatcher, T.; Naylor, D.T.; Zimmerman, S.; et al. A Modern Ampelography: A Genetic Basis for Leaf Shape and Venation Patterning in Grape. Plant Physiol.
**2014**, 164, 259–272. [Google Scholar] [CrossRef] [PubMed] - Martinez, C.C.; Chitwood, D.H.; Smith, R.S.; Sinha, N.R. Left–right leaf asymmetry in decussate and distichous phyllotactic systems. Philos. Trans. R. Soc. Lond. Ser. B
**2016**, 371. [Google Scholar] [CrossRef] [PubMed] - Graham, J.H.; Whitesell, M.J., II; Hel-Or, H.; Nevo, E.; Raz, S. Fluctuating Asymmetry of Plant Leaves: Batch Processing with LAMINA and Continuous Symmetry Measures. Symmetry
**2015**, 7, 255–268. [Google Scholar] [CrossRef] [Green Version] - Chen, T.W.; Henke, M.; de Visser, P.H.B.; Buck-Sorlin, G.; Wiechers, D.; Kahlen, K.; Stützel, H. What is the most prominent factor limiting photosynthesis in different layers of a greenhouse cucumber canopy? Ann. Bot.
**2014**, 114, 677–688. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sarlikioti, V.; de Visser, P.H.B.; Buck-Sorlin, G.H.; Marcelis, L.F.M. How plant architecture affects light absorption and photosynthesis in tomato: Towards an ideotype for plant architecture using a functional-structural plant model. Ann. Bot.
**2011**, 108, 1065–1073. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lewis, P. 3D canopy modelling as a tool in remote-sensing research. In Functional-Structural Plant Modelling in Crop Production; Vos, J., Marcelis, L.F.M., Visser, P., Struik, P.C., Evers, J.B., Eds.; Springer: Dordrecht, The Netherlands, 2007; Volume 22, pp. 219–229. [Google Scholar]
- Cieslak, M.; Lemieux, C.; Hanan, J.; Prusinkiewicz, P. Quasi-Monte Carlo simulation of the light environment of plants. Funct. Plant Biol.
**2008**, 35, 837–849. [Google Scholar] [CrossRef] - Bookstein, F.L. Morphometric Tools for Landmark Data: Geometry and Biology; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]
- Walker, J.A. Ability of Geometric Morphometric Methods to Estimate a Known Covariance Matrix. Syst. Biol.
**2000**, 49, 686–696. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zelditch, M.L.; Swiderski, D.L.; Sheets, H.D.; Fink, W.L. Geometric Morphometrics for Biologists; Academic Press: San Diego, CA, USA, 2004. [Google Scholar]
- Claes, P.; Daniels, K.; Walters, M.; Clement, J.; Vandermeulen, D.; Suetens, P. Dysmorphometrics: The modelling of morphological abnormalities. Theor. Biol. Med. Model.
**2012**, 9, 5. [Google Scholar] [CrossRef] [PubMed] - Kim, K.; Sheets, H.D.; Haney, R.A.; Mitchell, C.E. Morphometric Analysis of Ontogeny and Allometry of the Middle Ordovician Trilobite Triarthrus becki. Paleobiology
**2002**, 28, 364–377. [Google Scholar] [CrossRef] - Webster, M.; Sheets, H.D.; Hughes, N.C. Allometric patterning in trilobite ontogeny: Testing for heterochrony in Nephrolenellus. In Beyond Heterochrony: The Evolution of Development; Zelditch, M.L., Ed.; Wiley-Liss: Hoboken, NJ, USA, 2001; pp. 105–144. [Google Scholar]
- Bookstein, F.L. Size and Shape Spaces for Landmark Data in Two Dimensions. Stat. Sci.
**1986**, 1, 181–222. [Google Scholar] [CrossRef] - Mitteroecker, P.; Gunz, P. Advances in Geometric Morphometrics. Evol. Biol.
**2009**, 36, 235–247. [Google Scholar] [CrossRef] [Green Version] - Kruschke, J.K. Bayesian estimation supersedes the t test. J. Exp. Psychol. Gen.
**2013**, 142, 573. [Google Scholar] [CrossRef] [PubMed] - Kruschke, J.K.; Meredith, M. BEST: Bayesian Estimation Supersedes the t-Test, R Package Version 0.5.0; 2017. Available online: https://cran.r-project.org/package=BEST (accessed on 9 July 2018).
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2017. [Google Scholar]
- Rakocevic, M.; Sinoquet, H.; Christophe, A.; Varlet-Grancher, C. Assessing the geometric structure of a white clover (Trifolium repens L.) canopy using3-D digitising. Ann. Bot.
**2000**, 86, 519–526. [Google Scholar] [CrossRef] - Thanisawanyangkura, S.; Sinoquet, H.; Rivet, P.; Cretenet, M.; Jallas, E. Leaf orientation and sunlit leaf area distribution in cotton. Agric. For. Meteorol.
**1997**, 86, 1–15. [Google Scholar] [CrossRef] - Kruschke, J.K.; Liddell, T.M. The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective. Psychon. Bull. Rev.
**2017**, 25, 178–206. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Merila, J.; Biorklund, M. Fluctuating Asymmetry and Measurement Error. Syst. Biol.
**1995**, 44, 97–101. [Google Scholar] [CrossRef] - Schultheis, J.R.; Wehner, T.C.; Walters, S.A. Optimum planting density and harvest stage for little-leaf and normal-leaf cucumbers for once-over harvest. Can. J. Plant Sci.
**1998**, 78, 333–340. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Top and front view of the original prototype leaf shape used in L-Cucumber. Four equilateral triangles. Midrib defines symmetry axis. Leaf halves are rotated (rolled left/right) around the midrib to form a 15° downward angle.

**Figure 2.**(

**a**) Seventeen unique landmarks in the digitalization process of cucumber leaves; (

**b**) Triangulation scheme for leaf area calculation; (

**c**) 17 similar landmarks assigned to the original L-Cucumber prototype leaf. Dashed arrows connect symmetrical landmark pairs. (Own representation based on Kahlen and Stützel [23].)

**Figure 3.**Leaf area A

_{L}(cm

^{2}) and corresponding equilateral triangle side w (cm) of the digitized leaves used for shape estimation (n = 1125, bin width: 50 cm

^{2}).

**Figure 4.**Exemplary preprocessing transformation of a measurement leaf to align with the corresponding prototype leaf shape of the original L-Cucumber model. Alignment of $\overline{{P}_{1}{P}_{2}}$ by translation to origin, and yaw- and pitch rotation.

**Figure 5.**Example of the point-wise displacements in three dimensions between a measurement point 5′ (blue) and its prototype equivalent 5 (red) (

**a**) ${s}_{x}$—displacement along the x-axis; (

**b**) ${s}_{y}$—displacement in distance to the x-axis; (

**c**) ${s}_{z}$—displacement in roll angle around the x-axis.

**Figure 6.**Size-free comparison of new leaf shape, original L-Cucumber shape and all 1125 measurements by their outlines. Additionally, closed and open circles indicate the landmark positions in the new and original shape. Axis limits represent maximum spread of measurement data.

**Figure 7.**Three-dimensional impression of equal-sized original (

**left**) versus new (

**right**) leaf shape. Color representation is based on pairs of corresponding triangles from the left and right halve.

**Figure 8.**Representation of new leaf shape’s symmetry by mirroring the right (R) leaf half onto the left (L) half across the $xz$-plane. Non-overlapping paired landmarks (cf. Table 1) indicate directional asymmetry. The shaded area represents the outline of the leaf.

**Figure 9.**Measure of directional asymmetry: Posterior means (${\widehat{s}}_{x}$, ${\widehat{s}}_{y}$, ${\widehat{s}}_{z}$) of point-wise differences ($n=1125$) in three-dimensional displacement of landmark pairs. Mean value (○) and 95% HDI (

**–**).

**Figure 10.**Extent of fluctuating asymmetry: Posterior standard deviations (${\widehat{\sigma}}_{{s}_{x}}$, ${\widehat{\sigma}}_{{s}_{y}}$, ${\widehat{\sigma}}_{{s}_{z}}$) of point-wise differences ($n=1125$) in three-dimensional displacement of landmark pairs. Mean value (○) and 95% HDI (

**–**).

**Figure 11.**Internode length from experiment E1 for rank 5 to maximal rank 20 against L-Cucumber simulation results at four timepoints for original and new leaf shape. (

**A**) Rank-wise average and standard deviation of 5 experimental measurements or 100 simulations. (

**B**) Root-mean-square deviation per day.

**Figure 12.**Internode length from experiment E2 for rank 5 to maximal rank 20 against L-Cucumber simulation results at four timepoints for original and new leaf shape. (

**A**) Rank-wise average and standard deviation of 5 experimental measurements or 100 simulations. (

**B**) Root-mean-square deviation per day.

**Figure 13.**Relative deviation of summary parameters from new leaf shape simulations at the final simulation step in reference to the L-Cucumber simulations using the original shape for both experimental conditions (E1, E2). Posterior means and 95% HDI estimated by robust Bayesian estimation of relative differences from 100 simulations varying in initial plant orientation.

**Table 1.**Overall posterior means ($n=2250$) of point-wise three-dimensional displacement (${s}_{x}$, ${s}_{y}$, ${s}_{z}$) for each landmark with consideration of differences in pairs from robust Bayesian estimation (no zero-overlap of the 95% HDI) (see Figure 9). A single value for a landmark pair indicates no directional asymmetry.

Landmark | 3 | 8 | 4 | 9 | 5 | 10 | 6 | 11 | 7 | 12 | 14 | 16 | 15 | 17 | 2 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Pair | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||||||||

${s}_{x}$ ($w$) | −0.12 | 0.01 | 0.02 | 0.29 | 0.31 | 0.02 | −0.12 | 0.009 | 0.001 | 0.29 | −0.12 | |||||

${s}_{y}$ ($w$) | 0.05 | −0.11 | −0.17 | −0.40 | −0.39 | −0.17 | −0.16 | −0.23 | −0.19 | – | 0.16 | |||||

${s}_{z}$ (°) | 34.03 | 16.86 | −4.52 | −16.63 | −19.85 | 33.00 | 12.65 | – | 0 |

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**MDPI and ACS Style**

Schmidt, D.; Kahlen, K.
Towards More Realistic Leaf Shapes in Functional-Structural Plant Models. *Symmetry* **2018**, *10*, 278.
https://doi.org/10.3390/sym10070278

**AMA Style**

Schmidt D, Kahlen K.
Towards More Realistic Leaf Shapes in Functional-Structural Plant Models. *Symmetry*. 2018; 10(7):278.
https://doi.org/10.3390/sym10070278

**Chicago/Turabian Style**

Schmidt, Dominik, and Katrin Kahlen.
2018. "Towards More Realistic Leaf Shapes in Functional-Structural Plant Models" *Symmetry* 10, no. 7: 278.
https://doi.org/10.3390/sym10070278