
Research Article


Estimates of Additive and NonAdditive Genetic Variances with Varying Levels of Inbreeding


D. Norris,
W. Ngambi
and
C.A. Mbajiorgu


ABSTRACT

In this study, two populations with varying percentages
of animals in fullsib families were simulated. For each population, two
combinations of additive and dominance genetic variances of different
relative magnitudes were considered thereby creating 4 subpopulations.
For each subpopulation, a further 3 populations (I1, I2, I3) were created
with varying degree of inbreeding level (0, 0.02 and 0.05) resulting in
a total of 12 populations. Constant residual variance was used in all
populations. Variance components were estimated using the tildehat approximation
to REML based on siredam model. Both additive and dominance genetic variances
were estimated with high degree of accuracy and the level of inbreeding
did not seem to result in changes in the magnitudes of the genetic variances.
At low levels of inbreeding, accounting for inbreeding in genetic evaluations
may not be necessary.





INTRODUCTION
The partitioning of the total variance into its components allows the determination
of the relative importance of the various determinants of the phenotype, in
particular the role of heredity versus environment (Falconer,
1996). The role of heredity takes two forms: determination of individuals’
phenotype by their genotypes or determination of individuals’ phenotypes
by the genes transmitted from the parents. The assessment of both forms is increasingly
becoming important. However, livestock genetic evaluations in most livestock
species currently use the additive genetic model. This is the case even though
considerable research has shown the need for genetic evaluation accounting for
dominance effects for particular traits and populations (Norris
et al., 2006; Wei and Van der Werf, 1993;
Misztal et al., 1995, 1998; Varona et al.,
1997; Rye and Mao, 1998). The reason for not considering
dominance and epistatic genetic effects has largely been because of difficulties
associated with their estimation (Bridges and Knapp, 1987;
Fenster et al., 1997). A limiting factor in the
analysis of nonadditive genetic models has been the ability to compute the
inverses of nonadditive genetic covariance matrices for large populations.
Additionally, accounting for inbreeding in dominance analysis for medium to
large populations also poses a computational problem. However, new computing
programs have been developed to address this challenge. The wide use of artificial
insemination and embryo transfer coupled with the largescale application of
new reproductive technologies such as embryo splitting, cloning etc is increasing
the frequency of genetically identical, fullsib and threequarter sib groups
in livestock populations and thus necessitating the use of nonadditive genetic
models in livestock genetic evaluations (Chalh and El Gazzah,
2004).
Genetic covariance between individuals in random mating, noninbred population
for quantitative traits is a well defined linear function of the genetic variance
components assuming small contributions from many unlinked loci (Cockerham,
1954). In inbred populations, inbreeding my complicate the genetic covariance
structure of populations (De Boer and Hoeschele, 1993;
Kelly and Arathi, 2003). In the presence of inbreeding,
genetic variance is redistributed (Falconer, 1996). The
changes in gene frequency due to inbreeding cause some variance that was dominance
or epistatic in the ancestral population to become additive within the pure
breeds. Consequently, it is possible for additive and even total variance within
breeds to increase at low levels of inbreeding (Miller and
Goddard, 1998) and inability to account for inbreeding in genetic evaluations
may bias estimates of the genetic variances. Therefore, the objective of this
study was to investigate the changes in estimates of genetic variances under
varying levels of inbreeding.
MATERIALS AND METHODS
The study by Norris et al. (2002) showed that
in populations with a large number of animals having dominance genetic relationship
(20% or greater), estimates of dominance genetic variances can be obtained with
improved accuracy even when the dominance genetic effect in the population is
of small magnitude. The present study is an extension of the Norris
et al. (2002) study. This simulation study was carried out at the
University of Limpopo, South Africa in 2008.
Two populations with 50% (P1) and 100% (P2) of animals in fullsib families
were simulated. The number of animals in each population was 10000 and
each fullsib family had 25 animals. For each population, combinations
of additive variance, V_{A }and dominance variance, V_{D }were
considered: V1 (V_{A }= 800, V_{D }= 200), V2 (V_{A
}= 500, V_{D }= 500), thereby creating 4 populations, each
with 10000 animals. The residual variance was constant at 2000 in all
populations. From each population, a further 3 populations (I1, I2, I3)
were created with varying degree of inbreeding level (0, 0.02 and 0.05)
resulting in a total of 12 subpopulations. The populations are denoted
as follows in Table 1. Population P1V1I1 for instance
had 50% of animals in fullsib families with additive and dominance genetic
variances of 800 and 200, respectively. The inbreeding level was zero.
The descriptions of the other populations are given in the Table
1 under the properties column.
Each population was simulated for 20 replicates.
Records were simulated and analyzed according to the following sire and
dam model:
y_{ijk }= μ+s_{i}+m_{j}+sm_{ij}+e_{ijk} 
(1) 
where, μ is the population mean, s_{i }is the additive effect
of sire ~N (0, 1/4σ_{a}^{2}), m_{j } is the
additive effect of dam ~N (0, 1/4σ_{a}^{2}), sm_{ij
}is the dominance effect due to interaction of sire and dam ~ N (0,
1/4σ_{d}^{2}). ε_{ijk } is the residual
effect ~ N (0, σ_{e}^{2}+1/2σ_{a}^{2}+3/4σ_{d}^{2}).
Derivation of additive (a) and dominance (d) genetic values (Hoeschele
and VanRaden, 1991):

a = 0.5a_{s }+ 0.5a_{d }+ m_{a} 

d = fd_{s,d }+m_{d} 
where, a_{s} and a_{d }are the additive genetic effects
of sire and dam, respectively. fd_{s,d} is combination of sire
with dam due to interaction of genes from the sire with genes from the
dam. m_{a }and m_{d }are the respective additive and dominance
genetic effects due to mendelian sampling.
Var (m_{a}) = 0.5 σ_{a}^{2};
Var (m_{d}) = 0.75 σ_{d}^{2} 
The above siredam model can be written in matrix notation on an individual
animal basis as:
where, y is the data vector, a is the vector of random additive effects
for sire and dam, d is the vector of random dominance effects, e is the
vector of residuals and Z are known matrices corresponding to a and d,
respectively.
Variance components were estimated using the tildehat approximation to REML
(Van Raden and Jung, 1988). The inverse relationship
matrices, A^{1} and D^{1} were computed directly by algorithms
described by Henderson (1976) and Hoeschele
and VanRaden (1991). Computations were done using FORTRAN programs INVERS
and NONAD2 written by Hoeschele (1991).
Table 1: 
Populations description 

RESULTS AND DISCUSSION
The estimates of genetic variances under varying levels of inbreeding in subpopulations
with 50% of animals in fullsib families are shown in Table 2.
The estimated genetic variances are shown in Table 2 and 3
while on the table footnotes the simulated genetic variances are indicated.
The estimated genetic variances (additive and dominance) are similar (p>0.05)
to the true simulated variances in all populations. Additionally, these variances
are similar to the simulated variances irrespective of the level of inbreeding.
The results on the ability to accurately estimate nonadditive genetic variances
in populations with a high number of animals with nonadditive relationships
support an earlier study by Norris et al. (2002)
which indicated that in populations with at least 20% of animals with nonadditive
relationships, accurate estimations of dominance variance is achievable. There
is conflicting information on the effect of inbreeding or inbreeding depression
on the estimation of genetic variances. De Boer and Arendonk
(1992) used a genetic model with either 64 or 1,600 unlinked biallelic loci
and complete dominance to study prediction of additive and dominance effects
in selected or unselected populations with inbreeding. When changes in mean
and genetic covariances associated with dominance due to inbreeding were ignored,
significantly biased predictions of additive and dominance effects in generations
with inbreeding resulted. Bias, assessed as the average difference between predicted
and simulated genetic effects in each generation, increased almost linearly
with the inbreeding coefficient. When the genetic variation underlying a quantitative
trait is controlled by genes that act additively within and between loci, the
additive genetic variance within a population following a bottleneck event or
inbreeding is expected to decrease by a proportion F, the inbreeding coefficient
of the population (Wang et al., 1998). However,
there indication that additive genetic variance and heritability of some quantitative
traits within populations can actually increase following population bottlenecks
(Wang et al., 1998; Whitlock
and Fowler, 1999). The study by Russell et al.
(1984) investigated changes in variances with increased inbreeding in beef
cattle showed that the changes do not generally follow the theoretical expectations
for the redistribution of genetic variances. Whitlock and
Fowler (1999) performed a largescale experiment on the effects of inbreeding
and population bottlenecks on the additive genetic and environmental variance
for morphological traits in Drosophila melanogaster lines and found that
the mean change in additive genetic variance was in very good agreement with
classical additive theory, decreasing proportionally to the inbreeding coefficient
of the lines.
In a study by Takayoshi et al. (2007) on the
relationships among estimates of additive genetic variance, dominance genetic
variance and inbreeding depression for type traits in Holstein population of
Japan, the estimates of inbreeding depressions had significantly negative relationship
with the estimates of dominance genetic standard deviations. De
Boer and Hoeschele (1993) showed that for a biallelic locus with complete
dominance and for favorable gene frequencies that are about 0.20 or about 0.80,
large effects of inbreeding depression were linked to high dominance variance.
The estimates of genetic variances under varying levels of inbreeding
in subpopulations with 100% of animals in fullsib families are shown
in Table 3. Similar results are observed in these subpopulations
as those with 50% of animals in fullsib families. The estimated genetic
variances (additive and dominance) are similar (p>0.05) to the true
simulated variances and these variances are similar to the simulated variances
irrespective of the level of inbreeding.
Table 2: 
Genetic variances in populations with 50% fullsibs under
different levels of inbreeding 

P150% fullsib families, V1(V_{A }= 800 and
V_{D }= 200): Simulated (true) genetic variances, V2 (V_{A
}= 500 and V_{D }= 500): Simulated (true) genetic variances,
I1: No. inbreeding, I2: 0.02 inbreeding, I3: 0.05 inbreeding, SE:
Standard error 
Table 3: 
Genetic variances in populations with 100% fullsibs
under different levels of inbreeding 

P2100% fullsib families, V1(V_{A }= 800 and
V_{D }= 200): Simulated (true) genetic variances, V2(V_{A
}= 500 and V_{D }= 500): Simulated (true) genetic variances,
I1: No. inbreeding, I2: 0.02 inbreeding, I3: 0.05 inbreeding, SE:
Standard error 
It is worth noting that the inbreeding levels in this study were very low
compared to other studies and therefore the stability in the estimation of genetic
variances may be due to these low levels of inbreeding. However, a study by
Miller and Goddard (1998) in which they modeled additive
and nonadditive genetic effects within and between breeds in multi breed evaluation,
showed that at low levels of inbreeding within a breed, most nonadditive genetic
variation exists withinbreed, indicating that models which do not account for
withinbreed nonadditive genetic variation could be missing the largest source
of variation.
CONCLUSION
In the presence of inbreeding, the common practice is account for it
by including inbreeding depression as a covariate. This study seems to
indicate that at low levels of inbreeding, genetic variances can still
be estimated with accuracy. Therefore, it may not necessary to account
for inbreeding in such cases thus removing some of the computational challenges
associated with incorporating dominance relationships and inbreeding coefficients
in genetic analysis.

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