Below are explanations of the assumptions and theory that underlie the simulations in each of the tabs.
This simulation tracks the frequency of the \(A\) allele over generations in one or more populations of randomly mating, diploid individuals. The simulation initiates with a starting allele frequency of \(p_A\). Realizations of the number of \(A\) alleles in each individual are drawn from a bionomial distribution of size \(N\) and success probability \(p_A\), or \(x_{0i} \sim \textrm{binom}(N, p_{A0})\), where \(g = 0\) indicates the generation number (\(0\) for the base generation) and \(i\) indicates the individual number.
The mean of this realization is the average number of \(A\) alleles in an individual. This estimate, divided by the number of alleles in an individual (2 for a diploid) gives the estimated frequency of \(A\) alleles in the population, or \(\hat{p}_{A0} = \frac{1}{2} \sum{x_{0i}}\).
We assume that each individual is equally likely to pass alleles on to the next generation. Therefore, the realization of the number of \(A\) alleles in the next generation is drawn from a similar binomial distribution, except the success probability is changed to the estimated allele frequency (\(\hat{p}_A\)) in the previous generation, or \(x_{gi} \sim \textrm{binom}(N, \hat{p}_A)\). This process is repeated for \(g = 1, 2, ..., G\) generations.
The whole simulation can be summarized as follows:
The genetic variance simulation relies on the theoretical equation for the additive genetic variance and dominance genetic variance at a single gene (Bernardo 2010):
\(V_A = 2pq[a+d(q-p)]^2 \\ V_D = 4p^2q^2d^2\),
where \(p\) is the frequency of the \(A\) allele, \(q\) is the frequency of the \(B\) allele (\(q = 1 - p\)), \(a\) is the additive effect of the gene, and \(d\) is the dominance effect at the gene.
Since we only consider a single gene, epistatic variance (i.e. \(V_I\)) is absent. Therefore, the total genetic variance at the single gene is the sum of the additive variance and the dominance variance, or \(V_G = V_A + V_D\).
By using these equations, we assume that:
This simulations combines the basic elements of the first (namely, a randomly mating population) and the second (namely, additive effects at genes). This simulation is expanded to included many genes and the dimension of selection.
As before, the number of \(A\) alleles at each gene is simulated from a binomial distribution given a starting allele frequency that is identitical for all genes. The number of genes influencing the quantitative trait (i.e. QTL) is assumed to be \(L = 25\). The genes are assumed independent (i.e. no linkage) and dominance is assumed absent.
The additive effects of genes are drawn from a geometric series. The effect of the kth QTL (where \(k = 1, 2, ..., L\)) is equal to \(b^k\), where \(b = \frac{L - 1}{L + 1}\). Greater additive effects are more favorable, and the \(A\) allele is always assumed to be the favorable allele.
The genotypic value of an individual is the sum of the additive allele effects carried by that individual. The genetic variance (\(V_G\)) is the the variance among these genotypic values. To generate phenotypic values, residuals are sampled from a normal distribution with mean \(0\) and standard deviation \(\sqrt{V_R}\) and added to the genotypic values. \(V_R\) is the residual (i.e. error) variance and is calculated using the user-defined heritability (\(h^2\)), where \(V_R = \frac{V_G}{h^2} - V_G\).
Selection is simulated by taking the top (\(i \times 100\))th percent of the individuals based on phenotypic values. The estimated allele frequencies in the selected population are used to generate the allele numbers in the next generation.
Bernardo, Rex. 2010. Breeding for Quantitative Traits in Plants. 2nd ed. Woodbury, Minnesota: Stemma Press.