# ForGEM - genetic statistics

 FORGEM

The ForGEM model describes the genetic make up for a number of functional traits for every tree. Different trees in the same population may differ in genetic characteristics, i.e. alleles that determine the phenotypic value of the functional trait (model parameter). Based on this genetic information on individual trees, diversity measures within a population and differentiation measures between population can be calculated.

These population statistics, within and among populations, include diversity measures, differentiation measures between populations, heterozygozity and fixation, and F-statistics . The statistics described in this section are obtained from Hanssen (2000) and Hattemer (1991). Details on the use and interpretation can be found there and in Gregorius (1977, 1978, 1986, 1987, 1988).

## Diversity measures

### Genetic variety

The genetic variety can be measured as the number of different alleles or different genotypes in population (Gregorius, 1977; Gregorius et al., 1985).

### Genetic diversity

The genetic diversity characterizes the heterogeneity of the distribution of genetic variants in a population of a sample therefrom (Hattemer, 1991). It can thus be measured the allelic diversity of the k-th locus or genotype diversity of a deme.

${\displaystyle v_{k}={\frac {1}{\displaystyle \sum _{i=1}^{n_{k}}(p_{i}^{k})^{2}}}\qquad 1\leq v_{k}\leq n}$

where: n number different genetic types (alleles, genotypes); p frequency of i-th genetic type; v equals unity if there is only 1 genetic type, and equals n if every all genetic types are equally frequent (Gregorius, 1978).

### Mean effective number of alleles

In case of allele diversity, vk can be considered as the effective number of alleles for locus k if nk alleles occur with frequencies pik(i = 1, ... nk) (Hattemer, 1991). Thus, the mean effective number of alleles is the harmonic mean of vk at m loci.

${\displaystyle {\bar {v}}=m\cdot {\frac {1}{\displaystyle \sum _{k=1}^{m}{\frac {1}{v_{k}}}}}\qquad 1\leq v\leq {\frac {1}{n}}\sum _{k=1}^{n}n_{k}}$

### Hypothetical gametic multi-locus diversity

The diversity of the gametic output of populations is a special case of diversity and characterizes the adaptive potential of sexually reproducing populations (Gregorius 1978). It is hypothetical in the sense that the absence of fertility selection is assumed as well as the independence of the distributions of alleles at different loci (i.e. no linking) (Hattemer, 1991).

$gam} = \prod_{k=1}^m}v_k \qquad 1 \le v \le \prod_{k=1$

where: m the number of unlinked loci; vk the allelic diversity for the k-th locus; vgam is thus a measure for effective number of the multiloci gametes that can be produced in a population (Gregorius, 1978).

### Actual heterozygosity

${\displaystyle H_{a}=\sum _{i\neq j}P_{ij}}$

where: Pij the frequency of genotype with alleles i and j, with ij; Ha indicates the fraction of observed heterozygotes in the population.

### Fixation index

The fixation index indicates for the locus considered the surplus or deficit of heterozygotes compared to Hardy-Weinberg-equilibrium.

${\displaystyle F={\frac {H_{e}-H_{a}}{H_{e}}}}$

where: He the expected heterozygosity based on Hardy-Weinberg-equilibrium.

## Differentiation measures

### Genetic distance between demes

The differentiation between two demes is characterized by counting the number of genetic variants which the demes do not share. Thus, the allelic differentiation between demes X and Y represents the genetic distance between the demes (Gregorius, 1974; Gregorius and Roberts, 1986).

${\displaystyle d_{xy}={\tfrac {1}{2}}\cdot \sum _{i=1}^{n}\left|x_{i}-y_{i}\right|\qquad 0\leq d_{xy}\leq 1}$

where: xi and yi genetic frequencies (of alleles at a given locus or of a genotype) of deme X and Y.

### Genetic differentiation among demes

This statistic represents the genetic distance between a deme and its complement, i.e. the union of all other demes (Gregorius, 1985).

${\displaystyle D_{j}={\tfrac {1}{2}}\cdot \sum _{i=1}^{n}\left|p_{i}^{(j)}-{\bar {p}}_{i}^{(j)}\right|\qquad 0\leq D_{j}\leq 1}$

where: ${\displaystyle {p}_{i}^{(j)}}$ frequency of allele or genotype i in deme j; and ${\displaystyle {\bar {p}}_{i}^{(j)}}$ average allele or genotype frequency in the complement of deme j. The substructure of the complement has no influence of D, as different complement can yield the same ${\displaystyle {\bar {p}}_{i}^{(j)}}$. Thus, identical D’s do not necessarily indicate the demes with an identical genetic structure. However, vice versa demes with an identical genetic structure do possess an identical genetic structure (Hattemer, 1991).

### Average genetic differentiation

The average genetic differentiation among m demes is the weighted mean of Dj.

${\displaystyle \delta =\sum _{j=1}^{m}D_{j}\cdot c_{j}\qquad 0\leq \delta \leq 1}$

where: m number of populations; cj relative size of deme j (Gregorius 1984, 1988). δ attains zero if all demes have the same genetic structure, and reaches unity if all demes considers in pairs have no gene in common (Hattemer, 1991).

### Differentiation within a population

The concept of differentiation can also be applied within a population by considering each individual in that population a deme. The number of identical individuals can be counted and expressed relative to the number of other genetic types.

$N}{N-1} \cdot \left ( 1-\sum_{i=1}^n p_i^2 \right ) = \ \frac{N}{N-1} \cdot \left( 1-\frac{1}{v$

where: N the sample size; pi frequency of genetic type (allele or genotype). δT indicates the total genetic difference between all individuals of a population. δT equals zero if all individuals of the population are of the same genotype, and δT equals unity if all individuals are different (Gregorius 1987, 1988). δT represents the probability that two individuals samples from the sample population without replacement represent the same variant (Hattemer, 1991). Note that all differentiation measures range between zero and unity, whereas the genetic diversity measures range between unity and the number of genetic types, n (Gregorius 1987).

### F-statistics

F-statistics measure the degree of deviation of genotypic frequencies from those expected under random mating in structured populations (Falconer, 1996), [Weir and Cockerham, 1984; ref in (Larsen, 1996)].

FIS: Inbreeding coefficient of an individual relative to its on subpopulation. Measures inbreeding due to non-random mating in a sub-population. Within population fixation index.

FST: Average inbreeding of the subpopulation relative to the whole population, or correlation between two randomly chosen alleles in a sub-population relative to the alleles in the whole population. Measures inbreeding due to correlation among alleles cause by their occurrence in the same sub-population. Between populations fixation index.

FIT: Inbreeding coefficient of an individual relative to the whole population, or correlation between gametes for the total population. Measures the extend of inbreeding in the entire population (for neutral alleles). Total fixation index.

Note that:

• In a random mating population: FIS= 0 and FIT = FST.
• If all populations are genetically identical: FST = 0 and FIS = FIT.