Theoretical development of effective number of parents and gene diversity estimation
Parental fertility is defined as the proportional gametic contribution of female and male parents to their progeny (Griffin 1982; Reynolds and El-Kassaby 1990). Assuming that female and male strobili production count is a good representative of their gametic contribution (Gregorius 1989; Kang and El-Kassaby 2002; Funda et al. 2011), then this count can be used to estimate potential gametic contribution and hence parental fertility.
Fertility variation is described by the sibling coefficient (Ψ), which is the probability that two alleles randomly chosen from the gamete gene pool originated from the same parent (Kang 2000). Furthermore, the sibling coefficient is related to the coefficient of variation (CV) of female and male reproductive outputs (Kang 2000; Nebi et al. 2006). Female and male parents are defined as those parents contributing female and male gametes, respectively. Thus, the sibling coefficient of parental fertility (Ψ), which is based on zygotes (i.e., seeds), can be further described separately as female (Ψf) and male (Ψm) sibling coefficients as:
where N is the population census number, fi and mi are the proportional contributions of female and male of the i-th individual, and CVf and CVm are the coefficients of variation of female and male reproductive outputs in the population.
The effective number of female (Np(f)) and male (Np(m)) parents can be calculated separately from the female (Ψf) and male (Ψm) sibling coefficients, and are connected with their respective coefficient of variation (female CVf and male CVm) (Kang 2000; Bilir et al. 2006) as follows:
where N is the population census number, Ψf and Ψm are the female and the male fertility variation (i.e., sibling coefficients), and CVf and CVm are the female and male reproductive output’s coefficients of variation in the population under study.
- Scenario A (dioecious species): no covariation between female and male fertility
When there is no covariation between female and male reproductive outputs, the sibling coefficient (Ψ) is calculated from equations (1.1) and (1.2) components as:
where N is the population census number, pi is the total contribution (fertility) of the i-th individual, fi and mi are the proportional contributions of the i-th individual as female and male parents, and Ψf and Ψm are the female and male parents’ sibling coefficients, respectively.
The parental effective number of parents (Np) can be calculated from the sibling coefficient (Ψ) (see also formula 2.1 and 2.2). The Np is equivalent to the status number (Ns) when the parents are non-inbred and unrelated (Lindgren and Mullin 1998; Kang and El-Kassaby 2002).
where N is the population census number, Ψf and Ψm are the female and male parent’s sibling coefficients, and CVf and CVm are the coefficients of variation for female and male reproductive outputs in the population under study, respectively.
- Scenario B (monoecious or hermaphrodite species): positive or negative correlation between female and male fertility
Under covariation between female and male fertility (i.e., between female and male reproductive outputs), the sibling coefficient (Ψ) can be developed with the Person’s correlation coefficient (r) as follows:
where Ψf and Ψm are the female and male parent’s sibling coefficients, and r is the Person’s product-moment correlation coefficient between female and male reproductive outputs in the population.
With the covariation (i.e., correlation) between female and male reproductive outputs, the formulae (4) for the parental effective number of parents (Np) can further be developed with the correlation coefficient (r) as:
where N is the population census number, Ψ is the parental sibling coefficient, Ψf and Ψm are the sibling coefficients of female and male parents, CVf and CVm are the female and male reproductive outputs coefficients of variation, and r is the Person’s correlation coefficient between female and male reproductive outputs.
Animal breeders and geneticists use the number of fathers (Nf) and mothers (Nm) to estimate the effective population size as Ne(v) = 4NfNm / (Nf + Nm) when the sex ratio of a population departs from Fisherian sex ratio (1:1), dealing with dioecies species (Lynch 2007; Allendorf et al. 2013). In woody plant breeding, however, most gymnosperms are monoecious species so that the correlated fertility between gender should be considered for estimation the effective population size.
In this study, we provided different formula for dioecious species (Scenario A) and monoecious or hermaphrodite species (Scenario B); however, the formulae (4) has the same function when r is equal to zero as the formulae (6), so we propose to use the formulae (6) as a general equation of genetic indicator.
Relative effective number of parents and loss of gene diversity
The relative effective number of parents (Nr) is calculated as the relative proportion of the effective number of parents (Np) divided by census number (N) and it is a description of the percentage of the real population functioning as the idealized population. It is estimated for female, male and combined parents as:
The loss of gene diversity (GD) between generations (from parents to offspring) is estimated following Nei (1973), Lacy (1995) and Lindgren and Mullin (1998) as:
In small populations such as tree seed orchards, the effective population size and the genetic diversity of progeny can be calculated from equations 4, 6 and 8. In seed orchards setting, determining the effective population size and the genetic diversity of progeny can be estimated easily using both coefficient of variation (CV) and coefficient of correlation (r) for parental reproductive outputs (e.g., either strobili, seed-cone or seed production).
Pinus koraiensis seed orchard as a case population
Based on the above-theoretical representation, we estimated the effective number of parents (genetic diversity of the seed crops) and the factors influencing its pattern in the 1.5-generation Pinus koraiensis clonal seed orchard. The seed orchard was established by the National Institute of Forest Science, Republic of Korea in 1995 and located in the Gangwon province, South Korea (N37°23′; E127°38′) with 52 clones (total of 713 ramets; average of 37 ramets/clone). Clones/ramets were randomly allocated to the orchard’s grid at 5×5m spacing. The seed orchard is now owned and managed by the National Seed Variety Center of the Korea Forest Service.
Over a consecutive four-year period (2014-2017), the numbers of female and male strobili were assessed for all ramets (100% sampling). The female strobili were individually counted over the entire crown while the numbers of male strobili were estimated by multiplying the average number of strobili per branch by the total number of strobili-bearing branches.
Parental reproductive output balance was assessed using a cumulative gamete contribution curve (Griffin 1982; El-Kassaby and Reynolds 1990) after sorting the number of female and male strobili produced per clone in descending order and the cumulative contribution percentages were plotted against the proportion of clones.
Equal-cone harvest, collecting equal proportions of cones from each clone, was proposed to mitigate the female fertility variation among clones. The equal-cone harvest among clones was imposed in the seed orchard of P. koraiensis, thus the female parents’ fertility variation was negated. It should be noted that equal-cone harvest should be principally given to the most productive clones and thus accepting some loss of cone production is considered.