# A Simple Method for Measuring the Bilateral Symmetry of Leaves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Methods

_{i}and B

_{i}represent the upper and lower intersecting areas, respectively, of the i-th strip with the leaf. The extent of bilateral symmetry can be measured using the following two equations:

## 3. Results and Discussion

^{β}, where the estimate of β usually ranges from 1 to 2 [25,26,27]. This mean–variance relationship was referred to as Taylor’s power law [26]. There are many biological and abiotic explanations about the empirical range of β’s estimate [26,28,29]. Our recent study shows that the exponent β actually reflects the variation degree in energy distribution or energy release among statistical units [29]. The present study is also valuable for demonstrating this hypothesis. Photosynthesis of plants is actually a process of transferring solar energy to bioenergy. The leaf arrangement of plants can significantly affect light capture, and light has vital influence on leaf shape, size, and symmetry [22,30]. The unsheltered part of a leaf usually has higher efficiency in utilizing light than the part that is sheltered by the upper leaves [31], which might lead to bilateral asymmetry of the leaf. It can be deemed as the difference in energy distribution (for maximizing the utilization of light resource) among leaves. The ‘bilat.measure’ function developed here can provide the upper-lower areal difference between the intersecting parts of a leaf and a strip. For a leaf, we can obtain the mean and variance of several areal differences (999 differences in the present study). There are many leaves for any species of plant, so a mean–variance dataset can be used to fit Taylor’s power law. Using the log-transformed data (natural logarithm) of mean and variance for four bamboo species from Indocalamus, a linear relationship was confirmed (Figure 5). The exponent’s estimate is 1.89 ± 0.03 (p < 0.01), which lies in the interval (1, 2). This result is in accord with previous reports on the range of β’s estimate for other study objects [25,26,27,28,29]. It is valuable to check whether there is a relationship between Taylor’s exponent and the extent of bilateral symmetry among different species. However, it is not the main topic of the present study. In addition, as only four species of bamboos were sampled, we did not check this. It merits further investigation using more species within the same taxon. In general, biomass is better than area in depicting Taylor’s power law because biomass is a good indicator for energy [29]. However, leaf weight can be expressed as a power function of leaf area for many plants [18,19]. Therefore, it is appropriate to check Taylor’s power law using the areal differences.

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Examples of four bamboo species: (

**a**) species 1; (

**b**) species 2; (

**c**) species 3; and (

**d**) species 4 (see Table 1 for details). The leaf base for every bamboo species lies on the coordinate origin.

**Figure 2.**Examples of six species of trees: (

**a**) species 5; (

**b**) species 6; (

**c**) species 7; (

**d**) species 8; (

**e**) species 9; and (

**f**) species 10 (see Table 1 for details). The leaf base for every tree species lies on the coordinate origin.

**Figure 3.**Illustration of how to divide a leaf into parts using 10 strips. In practice, n larger than 10 was used, which can better reflect the information regarding bilateral symmetry. In the present study, n is 999.

**Figure 4.**Comparison of leaf size and bilateral symmetry among 10 species of plants. The Latin names denoted by ‘species code’ on the x-axis can be found in Table 1. The labels on the y-axis represent leaf area (

**a**); leaf length (

**b**); root mean squared error (

**c**); standardized index for bilateral symmetry (

**d**); and areal ratio of the left side to the right side of a leaf (

**e**); respectively. The areal ratios of the left side to the right side for all the leaves from the 10 species of plants were plotted against the corresponding standardized indices for bilateral symmetry (

**f**). The letters A, B, C, D, E, F, and G are used to represent the significance of differences among different species; species sharing a common letter are not significantly different at the 0.05 significance level.

**Figure 5.**Taylor’s power law between mean and variance of areal differences between upper and lower parts of leaves. A leaf was divided into upper and lower sides. Then, there are two intersecting parts between a leaf and a strip. For a leaf, 999 strips were produced. Then, there are 999 intersecting parts for the upper side of a leaf with these strips, and the same number for the lower side with these strips. Thus, there are 999 areal differences between corresponding upper and lower intersecting parts. For a leaf, the mean and variance were obtained based on the absolute values of these differences. Each open circle represents a leaf. Different colors represent different bamboo species.

Species Code | Latin Name | Family | Leaves | Sampling Time |
---|---|---|---|---|

1 | Indocalamus pedalis (Keng) P. C. Keng | Poaceae | 100 | early July 2014 |

2 | Indocalamus barbatus McClure | Poaceae | 100 | early July 2014 |

3 | Indocalamus victorialis P. C. Keng | Poaceae | 100 | early July 2014 |

4 | Indocalamus pumilus Q. H. Dai et C. F. Keng | Poaceae | 100 | early July 2014 |

5 | Chimonanthus praecox (Linn.) Link | Calycanthaceae | 72 | 20 October 2017 |

6 | Ginkgo biloba L. | Ginkgoaceae | 84 | 20 October 2017 |

7 | Aucuba japonica Thunb. var. variegata D’ombr. | Cornaceae | 100 | 20 October 2017 |

8 | Liriodendron tulipifera L. | Magnoliaceae | 100 | 11 October 2017 |

9 | Phoebe sheareri (Hemsl.) Gamble | Lauraceae | 100 | 26 October 2017 |

10 | Pittosporum tobira (Thunb.) Ait. | Pittosporaceae | 100 | 27 October 2017 |

Species | r (RMSE ^{1}, LA ^{2}) | r (SI ^{3}, LA) | r (AR ^{4}, LA) | CV_{RMSE} | CV_{SI} | CV_{AR} |
---|---|---|---|---|---|---|

1 | 0.686 (p < 0.01) | 0.013 (p > 0.05) | −0.182 (p > 0.05) | 52.3% | 37.2% | 12.1% |

2 | 0.786 (p < 0.01) | 0.192 (p > 0.05) | −0.025 (p > 0.05) | 40.2% | 21.8% | 9.9% |

3 | 0.500 (p < 0.01) | −0.21 (p = 0.036) | −0.139 (p > 0.05) | 50.4% | 35.8% | 10.7% |

4 | 0.753 (p < 0.01) | 0.132 (p > 0.05) | 0.172 (p > 0.05) | 43.4% | 24.1% | 12.0% |

5 | 0.263 (p < 0.05) | −0.071 (p > 0.05) | −0.053 (p > 0.05) | 65.2% | 54.2% | 20.9% |

6 | 0.467 (p < 0.01) | 0.119 (p > 0.05) | −0.160 (p > 0.05) | 47.7% | 42.4% | 17.2% |

7 | 0.345 (p < 0.01) | −0.048 (p > 0.05) | 0.041 (p > 0.05) | 59.8% | 51.4% | 21.3% |

8 | 0.545 (p < 0.01) | −0.101 (p > 0.05) | −0.008 (p > 0.05) | 48.2% | 42.4% | 10.9% |

9 | 0.472 (p < 0.01) | −0.143 (p > 0.05) | −0.090 (p > 0.05) | 69.5% | 54.3% | 24.7% |

10 | 0.290 (p < 0.01) | −0.027 (p > 0.05) | −0.008 (p > 0.05) | 55.6% | 46.5% | 12.6% |

^{1}RMSE: root-mean-squared error;

^{2}LA: leaf area;

^{3}SI: standardized index for bilateral symmetry;

^{4}AR: areal ratio of the left side to the right side of a leaf.

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**MDPI and ACS Style**

Shi, P.; Zheng, X.; Ratkowsky, D.A.; Li, Y.; Wang, P.; Cheng, L.
A Simple Method for Measuring the Bilateral Symmetry of Leaves. *Symmetry* **2018**, *10*, 118.
https://doi.org/10.3390/sym10040118

**AMA Style**

Shi P, Zheng X, Ratkowsky DA, Li Y, Wang P, Cheng L.
A Simple Method for Measuring the Bilateral Symmetry of Leaves. *Symmetry*. 2018; 10(4):118.
https://doi.org/10.3390/sym10040118

**Chicago/Turabian Style**

Shi, Peijian, Xiao Zheng, David A. Ratkowsky, Yang Li, Ping Wang, and Liang Cheng.
2018. "A Simple Method for Measuring the Bilateral Symmetry of Leaves" *Symmetry* 10, no. 4: 118.
https://doi.org/10.3390/sym10040118