Difficulties in parentage analysis: the probability that an offspring and parent have the same heterozygous genotype.

Parentage studies often estimate the number of parents contributing to half-sib progeny arrays by counting the number of alleles attributed to unshared parents. This approach is compromised when an offspring has the same heterozygous genotype as the shared parent, for then the contribution of the unshared parent cannot be unambiguously deduced. To determine how often such cases occur, formulae for co-dominant markers with n alleles are derived here for Ph, the probability that a given heterozygous parent has an offspring with the same heterozygous genotype, and Pa, the probability that a randomly chosen offspring has the same heterozygous genotype as the shared parent. These formulae have been derived assuming Mendelian segregation with either (1) an arbitrary mating system, (2) random mating or (3) mixed mating. The maximum value of Pa under random mating is 0.25 and occurs with any two alleles each at a frequency of 0.5. The behaviour with partial selfing (where reproduction is by selfing with probability s, and random mating otherwise) is more complex. For n < or = 3 alleles, the maximum value of Pa occurs with any two alleles each at a frequency of 0.5 if s < 0.25, and with three equally frequent alleles otherwise. Numerically, the maximum value of Pa for n > or = 4 alleles occurs with n* < or = n alleles at equal frequencies, where the maximizing number of alleles n* is an increasing function of the selfing rate. Analytically, the maximum occurs with all n alleles present and equally frequent if s > or = 2/3. In addition, the potential applicability of these formulae for evolutionary studies is briefly discussed.

Accuracy and precision of methods to estimate the number of parents contributing to a half-sib progeny array.

Molecular technologies have made feasible large-scale studies of genetic parentage in nature by permitting the genotypic examination of hundreds or thousands of progeny. One common goal of such studies is to estimate the true number of unshared parents who contributed to a large half-sib progeny array. Here we introduce computer programs designed to count the number of gametotypes contributed by unshared parents to each such progeny array, as well as assess the accuracy and precision of various estimators for the true number of unshared parents via computer simulation. These simulations indicate that under most biological conditions (1) a traditional approach (the multilocus MINIMUM METHOD) that merely counts the number of distinct haplotypes in offspring and divides by 2L, where L is the number of loci assayed, often vastly underestimates the true number of unshared parents who contributed to a half-sib progeny array; (2) a recently developed HAPLOTYPES estimator is a considerable improvement over the MINIMUM METHOD when parental numbers are high; and (3) the accuracy and precision of the HAPLOTYPES estimator increase as marker polymorphism and sample size increase, or as reproductive skew and the number of parents contributing to the progeny array decrease. Generally, HAPLOTYPES-based estimates of parental numbers in large half-sib cohorts should improve the characterization of organismal reproductive strategies and mating systems from genetic data.

On the number of reproductives contributing to a half-sib progeny array.

We address various statistical aspects of biological parentage in multi-offspring broods that arise via multiple paternity or multiple maternity and, hence, consist of mixtures of full- and half-sibs. Conditioned on population genetic parameters, computer simulations described herein permit estimation of: (1) the mean number of offspring needed to detect all parental gametes in a brood and (2) the relationship between the number of distinct parental gametes found in a brood and the number of parents. Results are relevant to the design of empirical studies employing molecular markers to assess genetic parentage in polygynous or polyandrous species with large broods, such as are found in many fishes, amphibians, insects, plants and other groups. The utility of this approach is illustrated using two empirical data sets.