# QTL×QTL×QTL Interaction Effects for Total Phenolic Content of Wheat Mapping Population of CSDH Lines under Drought Stress by Weighted Multiple Linear Regression

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## Abstract

**:**

_{gw}parameter compared to aaa

_{p}in five cases, with the exception of severe drought in 2012. The results show that by using weighted regression on marker observations, the obtained estimates are closer to the ones obtained by the phenotypic method. The coefficients of determination for the weighted regression model were significantly higher than for the unweighted regression and ranged from 46.2% (control in 2010) to 95.0% (control in 2011). Considering this, it is clear that a three-way interaction had a significant effect on the expression of quantitative traits.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Plant Material

#### 2.2. Estimation of the QTL×QTL×QTL Interaction Effects

#### 2.2.1. Estimation Based on the Phenotype

#### 2.2.2. Estimation Based on the Genotype

_{g}was based on the assumption that genes responsible for the total phenolic content were completely linked to observed molecular markers. After the selection of p markers (among all observed), determining the total phenolic content, the phenotypic observation model for DH lines is given as

**1**is the n-vector of ones, $\mu $ is the general mean,

**A**is the (n×p)-matrix of the form $\mathit{A}=\left[\begin{array}{cc}\begin{array}{cc}{\mathit{m}}_{{l}_{1}}& {\mathit{m}}_{{l}_{2}}\end{array}& \begin{array}{cc}\cdots & {\mathit{m}}_{{l}_{p}}\end{array}\end{array}\right]$, l

_{1}, l

_{2}, …, l

_{p}∈ {1, 2, …, q}, $\mathit{\beta}$ is the p-vector of unknown parameters of the form ${\mathit{\beta}}^{\prime}=\left[\begin{array}{cc}\begin{array}{cc}{a}_{{l}_{1}}& {a}_{{l}_{2}}\end{array}& \begin{array}{cc}\cdots & {a}_{{l}_{p}}\end{array}\end{array}\right]$,

**E**is the matrix for which columns are products of some columns of matrix

**A**, $\mathit{\gamma}$ is the vector of unknown parameters of the form ${\mathit{\gamma}}^{\prime}=\left[\begin{array}{cc}\begin{array}{cc}a{a}_{{l}_{1}{l}_{2}}& a{a}_{{l}_{1}{l}_{3}}\end{array}& \begin{array}{cc}\cdots & a{a}_{{l}_{p-1}{l}_{p}}\end{array}\end{array}\right]$,

**T**is the matrix for which columns are three-way products of some columns of matrix

**A**, $\mathit{\delta}$ is the vector of unknown parameters of the form ${\mathit{\delta}}^{\prime}=\left[\begin{array}{cc}\begin{array}{cc}aa{a}_{{l}_{1}{l}_{2}{l}_{3}}& aa{a}_{{l}_{1}{l}_{2}{l}_{4}}\end{array}& \begin{array}{cc}\cdots & aa{a}_{{l}_{p-2}{l}_{p-1}{l}_{p}}\end{array}\end{array}\right]$, and

**e**is the n-vector of random variables such that E(e

_{i}) = 0, Cov(e

_{i}, e

_{j}) = 0 for i≠j, i, j = 1, 2, …, n. The parameters ${a}_{{l}_{1}}$, ${a}_{{l}_{2}}$, …, ${a}_{{l}_{p}}$ are the additive effects of the genes controlling the total phenolic content, parameters $a{a}_{{l}_{1}{l}_{2}}$, $a{a}_{{l}_{1}{l}_{3}}$, …, $a{a}_{{l}_{p-1}{l}_{p}}$ are the additive×additive interaction effects and parameters $aa{a}_{{l}_{1}{l}_{2}{l}_{3}}$, $aa{a}_{{l}_{1}{l}_{2}{l}_{4}}$, …, $aa{a}_{{l}_{p-2}{l}_{p-1}{l}_{p}}$ are the additive×additive×additive interaction effects. It was assumed that epistatic and three-way epistatic interaction effects show only loci with significant additive gene action effects. Consequently, this assumption results in a decrease in the number of possible significant effects and makes the regression model more useful. Loci with significant additive effects of genes were located and estimated previously by Czyczyło-Mysza et al. [42].

#### Unweighted Regression

**G**is of full rank, the estimate of ${\mathit{\alpha}}_{\mathit{u}}$ from traditional (unweighted) multiple linear regression model is given by [46]

_{gu}effect of genes influencing the total phenolic content from a traditional (unweighted) multiple linear regression model can be found as

#### Weighted Regression

**W**of unknown variances of observations, which, however, may be empirically found by estimation. The selection of markers for the weighted regression is made by a similar method described for the unweighted case but concerns the weights for lines. In this model, the estimate of ${\mathit{\alpha}}_{\mathit{w}}$ is

_{gw}effect of genes influencing the total phenolic content from the weighted multiple linear regression model can be found as

## 3. Results

#### 3.1. Estimation Based on the Phenotype

_{p}) are presented in Table 1. In four of the six cases, the total aaa

_{p}effect was positive. A negative effect was observed for the 2011 and 2012 control groups. The highest total additive×additive×additive effect was observed for the 2012 severe drought group and the lowest for the 2011 control group. Triples interaction effects were statistically significant for all groups except the control group in 2012.

#### 3.2. Estimation Based on the Genotype

_{gu}) and weighted (aaa

_{gw}) multiple linear regression are presented in Table 2.

#### 3.2.1. Unweighted Regression

_{gu}effect ranged from –0.043 (control, 2010) to 0.688 (severe drought, 2012) (Table 2). The phenotypic variation of total phenolic content explained by the total triples interaction was similar in all five cases where the number of triples interactions was greater than zero and ranged from 32.4% (control, 2010) to 44.5% (severe drought, 2012) (Table 2).

#### 3.2.2. Weighted Regression

_{gu}), and weighted (aaa

_{gw}) multiple linear regression are presented in Table 3. Sixty-seven statistically significant triples interactions were observed. None of the QTL×QTL×QTL threes were significant in at least two cases: threes significant using weighted regression were not significant using unweighted regression and vice versa (Table 3). The QTLs most frequently found in triples interaction were gwm165.3 (22 times), gwm269.2 (11 times) and wmc429 (11 times). Using unweighted regression, the genes most frequently found in triples interaction were gwm165.3 (19 times), gwm269.2 (11 times) and wPt-0391 (9 times) (Table 3). Using weighted regression, the genes most frequently observed in triples interaction were wmc429 (eleven times), wPt-3738 (five times), wPt-6239 (five times) and wPt-6316 (also five times) (Table 3).

## 4. Discussion

_{gu}compared to assessment based on phenotypes alone in three cases (severe drought in 2010, control in 2012 and severe drought in 2012). In contrast, weighted regression yielded an improvement (in absolute value) in the evaluation of the aaa

_{gw}parameter compared to aaa

_{p}in five cases, with the exception of severe drought in 2012 (Table 1 and Table 2).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Minimal and maximal values of average for lines, means for all doubled haploid lines for total phenolic content, coefficient of variation, phenotypic estimates of the total additive×additive×additive effect (aaa

_{p}) and the number of genes (the number of effective factors).

Year | 2010 | 2011 | 2012 | |||
---|---|---|---|---|---|---|

Stress | Control | Severe Drought | Control | Severe Drought | Control | Severe Drought |

Minimal | 7.104 | 6.499 | 7.164 | 6.231 | 7.645 | 4.626 |

Maximal | 14.414 | 12.682 | 16.321 | 15.070 | 13.781 | 13.023 |

Mean | 10.450 | 9.435 | 11.993 | 10.214 | 10.736 | 8.204 |

Coefficient of variation | 18.06 | 18.72 | 15.39 | 15.51 | 14.34 | 23.63 |

aaa_{p} | 0.309 * | 0.156 * | −0.250 * | 0.437 ** | −0.023 | 0.620 *** |

The number of genes (the number of effective factors) | 4.617 | 3.824 | 5.050 | 6.021 | 4.164 | 4.846 |

**Table 2.**Genotypic estimates of the total additive×additive×additive effects estimated based on unweighted (aaa

_{gu}) and weighted (aaa

_{gw}) multiple linear regression.

Year | 2010 | 2011 | 2012 | |||||
---|---|---|---|---|---|---|---|---|

Stress | Control | Severe Drought | Control | Severe Drought | Control | Severe Drought | ||

Unweighted | QTLs number | 6 | 5 | 4 | 6 | 7 | 6 | |

Number of aaa_{gu} | 8 | 1 | 0 | 2 | 14 | 8 | ||

aaa_{gu} effects | Min. | −0.288 | 0.350 | −0.489 | −0.410 | −0.138 | ||

Max. | 0.533 | 0.350 | 0.569 | 0.538 | 0.409 | |||

Total | −0.043 | 0.350 | 0.110 | 0.170 | 0.688 | |||

R^{2} [in %] | 32.4 | 42.0 | 36.3 | 39.1 | 44.5 | |||

Weighted | QTLs number | 16 | 15 | 20 | 14 | 26 | 14 | |

Number of aaa_{gw} | 3 | 5 | 9 | 5 | 6 | 6 | ||

aaa_{gw} effects | Min. | −0.659 | −0.494 | −0.607 | −0.499 | −0.526 | −1.124 | |

Max. | 0.598 | 0.612 | 0.548 | 0.368 | 0.557 | 1.182 | ||

Total | 0.443 | −0.265 | 1.178 | −1.302 | −0.184 | −0.228 | ||

R^{2} [in %] | 46.2 | 61.4 | 95.0 | 58.8 | 78.9 | 58.5 |

**Table 3.**Genotypic estimates of the additive×additive×additive interaction effects for the particular QTL×QTL×QTL threes based on unweighted (aaa

_{gu}) and weighted (aaa

_{gw}) multiple linear regression.

Year | 2010 | 2011 | 2012 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Stress | Control | Severe Drought | Control | Severe Drought | Control | Severe Drought | ||||||||

QTL1 (LG) | QTL2 (LG) | QTL3 (LG) | aaa_{gu} | aaa_{gw} | aaa_{gu} | aaa_{gw} | aaa_{gu} | aaa_{gw} | aaa_{gu} | aaa_{gw} | aaa_{gu} | aaa_{gw} | aaa_{gu} | aaa_{gw} |

dupw004b (1A) | wPt-3094 (3A) | wmc1 (3B) | 0.471 | |||||||||||

dupw004b (1A) | wPt-668160 (1A) | gwm635b (7A) | 0.612 | |||||||||||

wPt-3094 (3A) | gwm165.3 (4A) | wPt-0391 (4B) | 0.350 | |||||||||||

wPt-3094 (3A) | gwm635b (7A) | wPt-671748 (7D) | −0.494 | |||||||||||

cfa2262 (3A) | dupw004a (4A) | blt101.t7 (4D) | −0.499 | |||||||||||

blt4.1 (3B) | wPt-4048 (3B) | gwm269.2 (4A) | −0.138 | |||||||||||

blt4.1 (3B) | gwm269.2 (4A) | gwm165.3 (4A) | 0.049 | |||||||||||

wPt-6239 (3B) | blt101.t7 (4D) | wPt-2697 (5A) | −0.337 | |||||||||||

wPt-6239 (3B) | blt101.t7 (4D) | wmc83 (7A) | −0.348 | |||||||||||

rPt-8896 (3B) | wmc1 (3B) | gwm52 (3D) | −0.376 | |||||||||||

wPt-0021 (3B) | gwm52 (3D) | barc60 (4B) | −0.659 | |||||||||||

gwm269.2 (4A) | gwm165.3 (4A) | gwm205 (5D) | 0.008 | |||||||||||

gwm269.2 (4A) | gwm165.3 (4A) | barc44 (5D) | −0.152 | |||||||||||

gwm269.2 (4A) | gwm165.3 (4A) | m69p78.1 (7A) | −0.100 | |||||||||||

gwm269.2 (4A) | gwm205 (5D) | barc44 (5D) | 0.063 | |||||||||||

gwm269.2 (4A) | gwm205 (5D) | m69p78.1 (7A) | −0.263 | |||||||||||

gwm269.2 (4A) | barc44 (5D) | wPt-9834 (5A) | 0.533 | |||||||||||

gwm165.3 (4A) | gwm165.3 (4A) | wPt-667091 (7D) | 0.158 | |||||||||||

gwm165.3 (4A) | wPt-0391 (4B) | wPt-3883 (7A) | 0.026 | |||||||||||

gwm165.3 (4A) | wPt-0391 (4B) | wPt-667091 (7D) | −0.179 | |||||||||||

gwm165.3 (4A) | wPt-0391 (4B) | wmc243b (7D) | 0.247 | |||||||||||

gwm165.3 (4A) | gwm205 (5D) | m69p78.1 (7A) | 0.156 | |||||||||||

gwm165.3 (4A) | barc44 (5D) | m69p78.1 (7A) | −0.288 | |||||||||||

gwm165.3 (4A) | gwm174 (5D) | wPt-3883 (7A) | 0.026 | |||||||||||

gwm165.3 (4A) | gwm174 (5D) | wPt-667091 (7D) | 0.031 | |||||||||||

gwm165.3 (4A) | gwm174 (5D) | wmc243b (7D) | −0.161 | |||||||||||

gwm165.3 (4A) | wPt-3883 (7A) | wmc243b (7D) | −0.008 | |||||||||||

wPt-0391 (4B) | wPt-0391 (4B) | wPt-667091 (7D) | −0.410 | |||||||||||

wPt-0391 (4B) | gwm174 (5D) | wPt-3883 (7A) | 0.128 | |||||||||||

wPt-0391 (4B) | gwm174 (5D) | wmc243b (7D) | 0.538 | |||||||||||

wPt-0391 (4B) | wPt-3883 (7A) | wPt-667091 (7D) | −0.151 | |||||||||||

barc152 (1B) | wPt-6239 (3B) | gwm191b (3D) | −0.400 | |||||||||||

barc152 (1B) | m65p64.8_4B (4B) | wPt-8149 (7A) | −0.526 | |||||||||||

barc152 (1B) | m92p78.10 (2A) | m60p64.13_3B (3B) | 0.557 | |||||||||||

barc152 (1B) | wPt-8072 (2B) | gwm165.3 (4A) | −0.493 | |||||||||||

m65p64.8a (5B) | gwm271 (5B) | m69p78.1 (7A) | 0.504 | |||||||||||

gwm174 (5D) | wPt-3883 (7A) | wPt-667091 (7D) | −0.130 | |||||||||||

gwm174 (5D) | wPt-3883 (7A) | wmc243b (7D) | 0.055 | |||||||||||

psr648_1B (1B) | wmc181 (2A) | gwm285 (3B) | 0.598 | |||||||||||

m17p65.1 (1B) | cfd73 (2D) | wmc468 (4A) | −0.478 | |||||||||||

wmc432 (1D) | wPt-3738 (1D) | psp2151.3 (2A) | −0.640 | |||||||||||

wPt-3738 (1D) | tPt-0202 (3A) | gwm269.2 (4A) | −1.124 | |||||||||||

wPt-3738 (1D) | tPt-0202 (3A) | dupw004a (4A) | 1.182 | |||||||||||

wPt-3738 (1D) | gwm513 (4B) | wmc157 (7D) | 0.387 | |||||||||||

wPt-3738 (1D) | wmc429 (1D) | tPt-0202 (3A) | 0.354 | |||||||||||

wmc429 (1D) | cfd11 (2D) | wPt-10291 (3D) | −0.505 | |||||||||||

wmc429 (1D) | gwm30.1 (2D) | tPt-0202 (3A) | 0.481 | |||||||||||

wmc429 (1D) | wPt-7466 (2D) | wPt-9749 (2D) | −0.607 | |||||||||||

wmc429 (1D) | gwm269.2 (4A) | dupw004a (4A) | −0.387 | |||||||||||

wmc429 (1D) | wPt-6316 (1D) | wPt-7466 (2D) | 0.548 | |||||||||||

wmc429 (1D) | wPt-6316 (1D) | wPt-10291 (3D) | 0.436 | |||||||||||

wmc429 (1D) | wPt-6316 (1D) | gwm161 (3D) | 0.493 | |||||||||||

wmc429 (1D) | wPt-6316 (1D) | gwm165.3 (4A) | −0.511 | |||||||||||

wmc429 (1D) | wPt-6316 (1D) | gwm639.2 (5B) | 0.419 | |||||||||||

wmc429 (1D) | wPt-732556 (1D) | gwm30.1 (2D) | 0.424 | |||||||||||

m92p78.10 (2A) | wPt-6239 (3B) | gwm161 (3D) | 0.360 | |||||||||||

m92p78.10 (2A) | wPt-6239 (3B) | wPt-8149 (7A) | 0.317 | |||||||||||

wmc453a (2A) | wPt-3883 (7A) | wPt-8919 (7B) | 0.569 | |||||||||||

wmc453a (2A) | wmc283.1 (7A) | wPt-8919 (7B) | −0.459 | |||||||||||

psp2151.3 (2A) | blt4.1 (3B) | wPt-4048 (3B) | 0.201 | |||||||||||

psp2151.3 (2A) | blt4.1 (3B) | gwm269.2 (4A) | 0.136 | |||||||||||

psp2151.3 (2A) | blt4.1 (3B) | gwm165.3 (4A) | −0.095 | |||||||||||

psp2151.3 (2A) | wPt-4048 (3B) | gwm269.2 (4A) | 0.409 | |||||||||||

psp2151.3 (2A) | wPt-4048 (3B) | gwm165.3 (4A) | 0.091 | |||||||||||

psp2151.3 (2A) | gwm269.2 (4A) | gwm165.3 (4A) | 0.035 | |||||||||||

wPt-3949 (2B) | wmc257 (2B) | gwm165.3 (4A) | 0.368 | |||||||||||

wmc257 (2B) | wPt-2697 (5A) | gwm292_5D (5D) | −0.485 |

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## Share and Cite

**MDPI and ACS Style**

Cyplik, A.; Czyczyło-Mysza, I.M.; Jankowicz-Cieslak, J.; Bocianowski, J.
QTL×QTL×QTL Interaction Effects for Total Phenolic Content of Wheat Mapping Population of CSDH Lines under Drought Stress by Weighted Multiple Linear Regression. *Agriculture* **2023**, *13*, 850.
https://doi.org/10.3390/agriculture13040850

**AMA Style**

Cyplik A, Czyczyło-Mysza IM, Jankowicz-Cieslak J, Bocianowski J.
QTL×QTL×QTL Interaction Effects for Total Phenolic Content of Wheat Mapping Population of CSDH Lines under Drought Stress by Weighted Multiple Linear Regression. *Agriculture*. 2023; 13(4):850.
https://doi.org/10.3390/agriculture13040850

**Chicago/Turabian Style**

Cyplik, Adrian, Ilona Mieczysława Czyczyło-Mysza, Joanna Jankowicz-Cieslak, and Jan Bocianowski.
2023. "QTL×QTL×QTL Interaction Effects for Total Phenolic Content of Wheat Mapping Population of CSDH Lines under Drought Stress by Weighted Multiple Linear Regression" *Agriculture* 13, no. 4: 850.
https://doi.org/10.3390/agriculture13040850