L. P. Subedi

IAAS/Tribhuvan University, Rampur, Chitwan, Nepal

E-mail:<kiran@scn.mos.com.np>

**ABSTRACT**

Genotype-by-environment interaction (GEI) is a measure of differential response of genotypes to varied environmental conditions. Its effects are to limit the accuracy of yield estimation from multienvironmental evaluation of varietal performance. There are various models developed by different workers to predict the GEI. I have developed two models: variable active genic (VAG) and variable active parental genic proportions (VAPGeP) model. There are other models like AMMI (additive main effects and multiplicative interactions), that are extensively used these days. Non genetic parameters like eigen value and eigen vectors as well as principal components are computed in the latter model to predict GEI. In this presentation, I have equated different genetic parameters of the genetic model (my model) to eigen value, vectors and principal components and have provided their genetic meaning. For these purpose different populations: purelines and composites were computer generated. In general, the following relationship was established.

(GxE)_{ij}=(`A_{i}.-`A..)(a_{j }-`a.)+(D_{i}.-`D..)(d_{j }-`d.) +.. (In terms of genetic model = VAG model)

=±l_{1}v_{1i}u_{1j
}±l_{2}v_{2i}u_{2j }±.. (in terms of AMMI model) = Y_{ij} -`Y_{i}.-`Y._{j} +`Y..

Where A_{i}= difference between number of
homozygous dominant and homozygous recessive loci of i^{th} genotype D_{i}=
Number of heterozygous loci of i^{th} genotype; a_{j}=additive
genetic effect for j^{th} environment d_{j}=dominant
effect for j^{th} environment l_{m}=square
root of m^{th} eigen value v_{mi} and u_{mj}
are corresponding genotypic and environmental eigen vectors.Y_{ij}
refers to the yield of i^{th} genotype in j^{th} environment.
The variances of distribution generating a_{j}’s and d_{j}’s
were significantly correlated with eigen values, and A_{i} and D_{i}
effects were highly correlated with genotypic eigen vectors. Similarly, a_{j}
and d_{j} effects were found highly correlated with environmental eigen
vectors. However, the AMMI model was found mechanical as compared to the VAG
model. Moreover, active genetic constitution can also be computed by simulating
ANOVA and predicting genetic parameters.

**KEYWORD: purelines, composites, eigen
value, eigen vector, genetic constitution**

**INTRODUCTION**

Evaluation of genotypic performance in multi environment trials is the main activity in any plant-breeding program. Understanding and predicting of crop response to environment is very important. It is an outcome of the research work in physiology, agronomy, modeling and, genetics and plant breeding. Better understanding of adaptation leads to more efficient and targeted crop improvement strategy. Level of adaptation is a function of the genotype or the genetic constitution and its response to the environment it occupies. In other words, it is the function of the genes possessed by the plants and their regulation during growth and development. Therefore, the genotype-by-environment interaction (GEI), which is the differential response of genotypes to the varied environmental conditions, results from the genetic differences among varieties and their resulting influence on growth and development. This differential response is also a function of individual as well as population buffering. Whatever is the cause of GEI, a significant GEI complicates the selection of superior and widely adapted genotype thus encouraging biodiversity while a low level of GEI enhances the selection of widely adapted genotype resulting the reduced biodiversity.

**MATERIALS AND METHODS**

The materials and methods basically consist of description and computer application of two models: i) variable active genic (VAG) model and ii)additive main effects and multiplicative interactions (AMMI) model.

**VARIABLE
ACTIVE GENIC (VAG) MODEL**

The VAG model assumes the genetic dynamism
of active genetic constitution, where some basic loci and some other variable
loci control the character of interest and the species under consideration is
diploid with 2 alleles per locus. The additive, dominance and interaction
components of genetic mean are also assumed to vary from one level of
environment to other. Under this assumption, the phenotypic value (Y_{ijt})
of i ^{th} (I= 1,2,….,n) entry at j^{th} (j=1,2,…..,p)
environment in t^{th} (t= 1,2,…,.r) replication within environment, can
be represented as (Subedi, 1996):

Where:

m

the average of the phenotypic values of homozygous dominant and homozygous recessive genotypes of w

K_{ijt } = the number
of active loci i^{th} entry at j^{th} environment and t^{th
}replication within j^{th} environment. The numbers of active loci
vary among entries or genotypes because of hypostatic relation (interaction)
among some genes.

P_{iwuv}= the frequency of A_{v}A_{u}
genotype at w^{th} locus in the i^{th} entry.

Following relation holds good for different populations:

Purelines: P_{iwu }=0 or 1; P_{iwuv}= 0 if u¹v

Mixtures: 0£Piwu ³ 1; P_{iwuv}= 0 if u ¹ v

Hybrids: P_{iwu }= 0 or 0.5 or 1; P_{iwuv} = 0 or 1 if u ¹ v

Synthetic:P^{2}_{iwu} = P_{iwuu}; 2P_{iwu} P_{iwv}
= P_{iwuv} and P^{2}_{iwv} = P_{iwvv}

In all cases, P_{iwu} +P_{iwv} = 1.0

a_{wjt }= the mean of the
difference between the phenotypic values of homozygous dominant and homozygous
recessive genotype of w^{th} locus at j^{th} environment and t^{th}
replication within j^{th} environment . a_{wjt} is zero for all
inactive loci. A hypostatic gene (allele) is considered inactive.

d_{wjt }= the difference between
the phenotype values of a heterozygous genotype of w^{th} locus and its
corresponding m_{wjt} at the j^{th} environment and t^{th}
replication within the j^{th} environment.

q_{wi}
= 1, 0, -1 for homogygous dominant, heterozygote and homozygous recessive
genotype, respectively, of w^{th} locus in the i^{th } entry.

Different populations:
purelines and composites were generated as of Subedi (1996). The yield of a
genotype is generated at two levels: a) first of all, the genotype is generated
and then genetic constitution parameters (K_{i}, A_{i} , D_{i})
are computed based on its genetic constitution, and b) these parameters are
then multiplied by m_{j}, a_{j} and d_{j},
respectively and added them together.

Different magnitudes and types
of GEI are basically generated by changing the variance of distributions
generating a_{j}’s and d_{j}’s.

**AMMI MODEL**

The basic linear model used in the analysis of GEI is of the form:

Y_{ij}= m +g_{i} +Î_{j}
+ d_{ij :} Where, (GXE)_{ij }=d_{ij}
g_{i} = genotypic effect of i^{th} genotype, and Î_{j}= environmental effect of j^{th} environment
and m=grand mean = `Y.. In Additive Main Effects and Multiplicative
Interactions (AMMI) model (Shafii et al.,1992), the yield is partitioned as:

Y_{ij}= m + g_{i} + Î_{j} + (l_{1} v_{1i}u_{1j} + l_{2} v_{2i}u_{2j} +…)

l_{m}
is the singular value of the matrix E^{T}E (square root of eigen value
of PCA axis m), where E is the n x p matrix of residuals (d_{ij}) . The v_{mi} and u_{mj} are
corresponding normalized eigen vectors of matrix EE^{T} and E^{T}E
, respectively. Genotype PCA score and environment PCA score are expressed as
unit vector times the square root of l_{m
}. The GEI sum of squares
is divided into PCA axes where axis m is regarded as having n+p-1-2m degrees of
freedom, where n and p are the number of genotypes and environments,
respectively.

Different criteria are used to compare
these two models. First of all, variance of distribution generating a_{j}’s
of VAG model and l_{1} ^{2} (sum of squares explained) by
first component of AMMI model were regressed to find any relationship that
exits between the models. Secondly, the GEI sum of squares explained by
different components of the two models was compared.

Actual analysis of variance (ANOVA) of
multi-environment tested performance of a set of genotypes is simulated and
compared. If there exits a significant relationship among the different
components of the two models between actual and simulated ANOVA, then these two
ANOVA are declared similar. The A_{i} and D_{i} effects if
found similar to either first or second PCA scores warrant a situation where an
approximate genetic constitution or architecture of the tested genotypes can be
computed. For the sake of example, the rapeseed data of Shafii et al. (1992)
was used in this study, to illustrate this point.

**RESULTS AND DISCUSSION**

The diagrams generated through varying magnitude and types of GEI are presented in Figure 1. Diagrams indicate basically two main points: a) the number of cross-over interactions are equal to or less than minimum of (n,p):number of genotypes or number of environments , and b) most of the variations in GEI is predicted by first two PCA scores (PCA1 and PCA2).

The sum of squares of GEI predicted by different components of the VAG and the AMMI models are presented in Table 1. In some situations (first three cases), each component of both models predict almost equal magnitude of GEI, while rest of the cases, the same is not true. In the latter case also, the magnitude of GEI predicted by first two components (VAG2 vs AMMI2) are almost equal to themselves and to the magnitude of actual GEI. This indicates the existence of following relationship:

VAG2 = ±l_{1}^{ }v_{1i }u_{1j}
± l_{2}^{ }v_{2i }u_{2j }±…= Y_{ij }- `Y_{i}. - `Y._{j }+ `Y..

It has been found (data not presented here) that the components of AMMI model may have similar or differing signs, throughout.

The relationship between the distribution generating a_{j}’s
and weighted eigen values (l_{1}^{2}/np) is presented in Figure 2. The figure compares
this relationship for two types of the populations: purelines and composites.
This relationship should be viewed in the light of the last 3 sets of data in
table 1. It indicates that there exists a significant relationship, although
may not be 1:1, between the parameters of the two models.

**Table 1.
Comparison between different components of AMMI and **

**VAG models for
their predictability of sum of squares due to GEI **

AMMI1 |
VAG 1 |
AMMI 2 |
VAG 2 |
Actual |
Df (n=p) |

12.33 |
12.77 |
19.27 |
19.76 |
20.38 |
49 |

19.41 |
19.47 |
31.16 |
30.03 |
37.18 |
49 |

42.61 |
51.68 |
63.1 |
68.1 |
67.0 |
49 |

11.06 |
1.38 |
13.51 |
12.13 |
12.93 |
4 |

19.89 |
7.75 |
19.76 |
17.86 |
19.91 |
4 |

11.44 |
0.61 |
12.81 |
11.51 |
12.26 |
4 |

AMMI 1 = l_{1}v_{1i }u_{1j} AMMI
2 = l_{1}v_{1i }u_{1j} + l_{2}^{ }v_{2i
}u_{2j }

VAG 1 = (`A_{i}.
- `A.) (a_{j}-`a.)

VAG 2 = (`A_{i}. - `A.) (a_{j}-`a.) + (`D_{i}._{
}-`D.) (d_{j} -`d.)

n = no. of genotypes p = no. of environments