# The Pitfalls of Heterosis Coefficients

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## Abstract

**:**

## 1. Introduction

## 2. The Dominance and Heterosis Coefficients

- (i)
- Wright [1] defined:$${D}_{\mathrm{W}}=\frac{{z}_{2}-{z}_{12}}{{z}_{2}-{z}_{1}}$$
- (ii)
- Falconer [5] proposed the following coefficient:$${D}_{\mathrm{F}}=\frac{{z}_{12}-\overline{z}}{\frac{{z}_{2}-{z}_{1}}{2}}$$

## 3. Relationship between the Potence Ratio and the other Heterosis Coefficients

## 4. The Pitfalls of the Most Commonly Used Heterosis Coefficients

## 5. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationship between the potence ratio ${H}_{\mathrm{PR}}$ and the two heterosis coefficients ${H}_{\mathrm{MP}}$ and ${H}_{\mathrm{BP}}$. (

**A**,

**B**) Six traits were measured in maize (FLO: flowering time [days between 50% flowering and 12 August], PH: plant height, EH: ear height, GY: grain yield, TKW: thousand-kernel weight, KM: kernel moisture) in four crosses (H1: B73 × F252, H2: F2 × EP1, H3: F252 × EP1, H4: F2 × F252) grown in three environments in France (E1: Saint-Martin-de-Hinx, E2: Jargeau, E3: Rhodon). (

**A**) Relationship between ${H}_{\mathrm{PR}}$ and ${H}_{\mathrm{MP}}$. (

**B**) Relationship between ${H}_{\mathrm{PR}}$ and ${H}_{\mathrm{BP}}$. (For clarity, four of the 72 trait-cross-environment combinations are not represented because they have high ${H}_{\mathrm{PR}}$ values.) (

**C**,

**D**) Five traits were measured in cotton (SY: seed yield [grams per plant], LY: lint yield [grams per plant], BNP: bolls per plant, BW: boll weight [grams], LP: lint percentage) in two crosses (H1: X1135 × GX100-2 and H2: GX1135 × VGX100-2) grown in three environments in China (E1: Handan, E2: Cangzhou, E3: Xiangyang) (Data from [10]). (

**C**) Relationship between ${H}_{\mathrm{PR}}$ and ${H}_{\mathrm{MP}}$. (

**D**) Relationship between ${H}_{\mathrm{PR}}$ and ${H}_{\mathrm{BP}}$.

**Figure 2.**Heterosis values obtained with different coefficients. (

**A**) Heterosis coefficients for six traits measured in the F252 × EP1 cross grown in Saint-Martin-de-Hinx (France) in 2014. (

**B**) Heterosis coefficients for ear height in four crosses grown in Saint-Martin-de-Hinx (France) in 2014. (

**C**) Heterosis coefficients for plant height in the F2 × F252 cross grown in the three environments. The six traits and the three environments are the same as in Figure 1A. The scales of the heterosis coefficients are normalized by the maximum value in each dataset (figures at the top right of the vertical lines).

**Figure 3.**Heterosis for flowering in two maize hybrids. (

**A**) Percentage of flowering over time (number of days since 1 January) for parents W117 and F192 and their hybrid, adjusted with a Hill function. (

**B**) Percentage of flowering over time for parents W117 and F252 and their hybrid. (

**C**) Heterosis coefficient profiles over time for the W117 × F192 cross. (

**D**) Heterosis coefficient profiles over time for the W117 × F252 cross.

**Table 1.**Dominance and heterosis coefficients. ${D}_{\mathrm{W}}$: Wright’s dominance coefficient [1]. ${D}_{\mathrm{F}}$: Falconer’s dominance coefficient [5]. ${H}_{\mathrm{mp}}$, ${H}_{\mathrm{MP}}$, ${H}_{\mathrm{PR}}$, ${H}_{\mathrm{bp}}$ and ${H}_{\mathrm{BP}}$: heterosis coefficients. Subscripts: mp or MP, mid-parent; PR, potence ratio; bp or BP, best-parent. ${z}_{1}$ (resp. ${z}_{2}$): the phenotypic value of parental homozygote 1 or of parent 1 (resp. 2). ${z}_{12}$: the heterozygote or hybrid value. $\overline{z}$: the mean parental value. By convention, ${z}_{2}>{z}_{1}$.

Reference | Coefficient | Coefficient Scales with Their Characteristic Values |
---|---|---|

High homozygote | ${D}_{\mathrm{W}}={\displaystyle \frac{{z}_{2}-{z}_{12}}{{z}_{2}-{z}_{1}}}$ | |

Mean homozygote | ${D}_{\mathrm{F}}={\displaystyle \frac{{z}_{12}-\overline{z}}{({z}_{2}-{z}_{1})/2}}$ | |

Mid-parent | ${H}_{\mathrm{mp}}={z}_{12}-\overline{z}$ | |

${H}_{\mathrm{MP}}={\displaystyle \frac{{z}_{12}-\overline{z}}{\overline{z}}}$ | ||

${H}_{\mathrm{PR}}={\displaystyle \frac{{z}_{12}-\overline{z}}{({z}_{2}-{z}_{1})/2}}$ | ||

Best-parent | ${H}_{\mathrm{bp}}={z}_{12}-{z}_{2}$ | |

${H}_{\mathrm{BP}}={\displaystyle \frac{{z}_{12}-{z}_{2}}{{z}_{2}}}$ |

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**MDPI and ACS Style**

de Vienne, D.; Fiévet, J.B.
The Pitfalls of Heterosis Coefficients. *Plants* **2020**, *9*, 875.
https://doi.org/10.3390/plants9070875

**AMA Style**

de Vienne D, Fiévet JB.
The Pitfalls of Heterosis Coefficients. *Plants*. 2020; 9(7):875.
https://doi.org/10.3390/plants9070875

**Chicago/Turabian Style**

de Vienne, Dominique, and Julie B. Fiévet.
2020. "The Pitfalls of Heterosis Coefficients" *Plants* 9, no. 7: 875.
https://doi.org/10.3390/plants9070875