# Initialisation of allelelic effects

 FORGEM

For the ForGEM model, it is necessary to assign allelic effects to each of the alleles that compose the genotype of the individual tree. Allelic effects are kept constant during the entire simulation. If information is lacking on the actual number of loci, the number of alleles and the allelic effects that determine quantitative phenotypic traits, a statistical approach is taken. This is done by designing for each trait a genotype distribution over the population such that the observed mean and variance of the phenotypic trait of the population are attained, under the constraint that the allele frequencies follow the U-shaped initial distribution. If information becomes available on the QTLs or candidate genes of the phenotypic traits considered, this statisticaly procedure can be replaced by actual data on the genetic make-up of these traits for a particular population.

The approach followed in ForGEM to obtain the observed mean phenotypic value is:

1. assign initially arbitrary allelic effecs of +1 and -1 to each of the alleles
2. calculate mean and variance under the constraint of the the U-shaped distribution of allele frequencies
3. scale allelic effects such that the distribution of phenotypic values over over all possible genotypes is normalised (mean equals zero, variance equals unity)
4. add the mean and multiply with the variance of the functional trait in question

The mean and variance of a genotype are:

${\displaystyle m=\sum _{l=1}^{Nloci}pi+qj}$

${\displaystyle var=\sum _{l=1}^{Nloci}p(i-m)^{2}+q(j-m)^{2}}$

This assignment of +1 and -1 values can be done for all alleles in a multi-locus 2 allele system.

The following steps are made to arrive at a mean of zero, and a variance of unity for the whole population.

First, make expectations zero by offset and sum of individual effect.

${\displaystyle \sum _{l=1}^{Nloci}(p(i-c)+q(j+c)=0}$

${\displaystyle =>\sum _{l=1}^{Nloci}pi-\sum _{l=1}^{Nloci}pc+\sum _{l=1}^{Nloci}qj+\sum _{l=1}^{Nloci}qc=0}$

${\displaystyle =>\sum _{l=1}^{Nloci}(pi+qj)-c\sum _{l=1}^{Nloci}(p-q)=0}$

${\displaystyle =>m-c\sum _{l=1}^{Nloci}(p-q)=0}$

${\displaystyle =>m=c\sum _{l=1}^{Nloci}(p-q)}$

${\displaystyle =>c=m/\sum _{l=1}^{Nloci}(p-q)}$

${\displaystyle var=\sum _{l=1}^{Nloci}p(i-m-c)^{2}+q(j-m+c)^{2}}$

${\displaystyle =>var=\sum _{l=1}^{Nloci}pi^{2}+qj^{2}}$

This leads to a large number of possible allelic values. Arbitrarily, the first combination of allelic effects that yield the lowest expectancy (m) is selelected. in the example above this is:

q					p
a	b	c	d	e	A	B	C	D	E
0.006	0.044	0.141	0.299	0.499	0.994	0.956	0.859	0.701	0.501	m	var	c
-1	-1	-1	-1	-1	1	1	1	1	1	3.0306	2.5001	-1.0028

${\displaystyle i-c}$:

a	b	c	d	e	A	B	C	D	E
0.006	0.044	0.141	0.299	0.499	0.994	0.956	0.859	0.701	0.501	m	var
0.002848	0.002848	0.002848	0.002848	0.002848	-0.002848	-0.002848	-0.002848	-0.002848	-0.002848	-0.008606	0.000041

${\displaystyle m=0=>}$ ${\displaystyle var=\sum _{l=1}^{Nloci}pi^{2}=4.87}$

${\displaystyle (i-c)/sqrt(var):}$

a	b	c	d	e	A	B	C	D	E
0.006	0.044	0.141	0.299	0.499	0.994	0.956	0.859	0.701	0.501	m	var
0.447213595	0.447213595	0.447213595	0.447213595	0.447213595	-0.447213595	-0.447213595	-0.447213595	-0.447213595	-0.447213595	0	1

The population values are then be obtained by adding the observed mean and multiplying by the observed standard deviation.

The allelic effect thus depends on the number of loci.