.. _genetic_structure:

Genetic structure
======================================

The genetic structure is described by a polygenic multi-allelic model with 
finite numbers of loci and alleles for both the functional and neutral parts of the 
genome. The value of trait :math:`Z` thus results from the expression of :math:`l_Z`  functional loci, each 
of which has a pool of :math:`n_{Z, l}` (with :math:`l \in [1, 2, ..., l_Z]`) possible alleles in the initial population characterized by their 
allelic value :math:`A_{Z, l, k}`  (with  :math:`k \in [1, 2, ... n_{Z, l}]`).

Following classical quantitative genetics :cite:`lynch1998genetics`, we assume that 
the genotypic values :math:`G_Z(i)`  of trait :math:`Z`  in the population follow initially a normal distribution:

.. math:: 

    N(\overline{G_Z}(0), \sigma^2_{A, Z}(0))

with :math:`\overline{G_Z}(0)`  the initial genotypic mean and :math:`\sigma^2_{A, Z}(0)` the initial additive genetic variance. 
It follows (see justification in the next section) that the allelic 
values :math:`A_{Z, l, k}`  of the :math:`n_{Z, l}`  alleles of locus :math:`l`  initially present in the population are randomly drawn from a normal 
distribution :math:`N\left(0, \dfrac{\sigma^2_{A, Z}(0)}{2 l_Z}\right)`
:cite:`soularue2012assortative`. This allelic model defines allelic values as deviations 
around the initial genotypvic mean :math:`\overline{G_Z}(0)` of the population and allows for heterogeneous allelic values 
across loci coding for the same trait, many of them with minor effects and a few ones with major effects.

Similarly, the neutral part of the genome is described by :math:`l_b`  neutral loci, each of which has a pool 
of :math:`n_{b, l}`  (with :math:`l \in [1, 2, ..., l_b]`) 
possible alleles in the initial population characterized by their allelic 
identity :math:`b_{l,k}`  (with :math:`k \in [1, 2, ... n_{b, l}]`) with no effect of 
evolving trait values. The allelic identities :math:`b_{l,k}`  of the  alleles of 
locus :math:`l`  initially present in the population are randomly drawn from a discrete uniform distribution with probability 
mass function :math:`\dfrac{1}{n_{b,l}}`.