Millar et al. (2008) sequenced a segment of 344 base pairs in a fast evolving region of first hypervariable region (a region they argued was not under strong selective pressure) from 1931 Adélie penguins (508 families) at Antarctic nesting sites. This was a follow-up to the study of Lambert et al. (2002) of ancient and contemporary samples of the same region of mitochondrial DNA. In the data reported in Millar et al. (2008), a low proportion of the sites in individual birds exhibited heteroplasmies, where two bases were called on the electropherogram at that site, each with a signal greater than θ=0.23 of the total signal. Signals below the detection threshold θ, could not be reliably distinguished from background noise and were not reported there. Fifty-five mothers (out of 508) exhibited site heteroplasmies (above θ), with seven having two heteroplasmic sites. Hence, the proportion of sites with an observed heteroplasmy across all mothers wasIn all but three cases, the observed site heteroplasmies persisted above the threshold in both mother and chicks. Site heteroplasmies in a chick were only checked at sites where a heteroplasmy had been observed in its mother.

Following Lambert et al. (2002) and Millar et al. (2008), we define mutation rate as the rate at which a base substitution is incorporated into all mitochondrial genomes of an individual. Previous studies linking mutation rate with observed site heteroplasmies include two studies (Brandstätter et al. 2004; Santos et al. 2005) of human populations of up to three generations, where the mutation rate was estimated from assuming that each observed site heteroplasmy represented a transitional substitution. We find most of the substitutions giving rise to an observed site heteroplasmy are subsequently lost, and those that eventually become fixed may persist as an observed site heteroplasmy for many generations and be oversampled.

For our study, we follow the maternal ancestry of an individual, tracking the trajectory of a site substitution introduced into the germ line by some ancestor. At each generation, the population of mitochondrial genomes will pass through a bottleneck when the oocyte segregates a small number of its mother's genomes. Lambert et al. (2002) argued that this region of the mt-genome is not under strong selective pressure, so our model assumes random drift among the N_x segregating genomes.

Consider the maternal ancestry of an individual A₀, with A₁ its mother and A₂ its maternal grandmother, etc. Suppose a nucleotide substitution X→Y has occurred at a site in one genome of a maternal line ancestor A_g (g≥1), which is inherited by A_g−1. Our model shows that the probability of an additional substitution inherited at that site among A_g−1,…,A₀ is less than 10⁻², so we will neglect this possibility. Suppose this site heteroplasmy persists for (exactly) k≥1 generations, inherited by A_g−1,…,A_g−k, but not by A_g−k−1. If k≥g, then A₀ will be heteroplasmic at that site, although not necessarily observable. If k<g, A_g−k−1,…,A₀ are not heteroplasmic at that site, with their genomes either containing all X or all Y.

We propose a model where each oocyte recruits N_x mitochondrial genomes independently from the population of its mother's genomes. (The number N_x is sometimes referred to as the number of segregating units. For the Adélie penguins, we estimated N_x to have a 95% confidence interval (CI) of 25<N_x<69.)

The observed levels of most site heteroplasmies from the blood samples of a mother and her chicks closely agree, which reflects a close agreement in the germ line. We will assume that the variation, after accounting for measurement uncertainty, is due to sampling at the recruiting bottleneck and that the proportions in the blood sample estimate the inherited proportions.

If A₁ (the mother of A₀) contains a site heteroplasmy with the nucleotides Y and X appearing in proportions ϕ and 1−ϕ, then the probability (using a binomial selection with replacement) that A₀ inherits i genomes with allele Y and N_x−i genomes of allele X isIf A₁ had inherited j copies of allele Y and N_x−j of allele X from her mother A₂, we assume she exhibits the proportions ϕ=j/N_x and 1−ϕ=1−j/N_x of the alleles in the genomes available for inheritance. Hence,is the probability that A₀ inherits i copies of allele Y, and N_x−i copies of allele X, given her mother had inherited j and N_x−j corresponding copies. Letbe the matrix of these probabilities, where the N_x−1 rows and columns are indexed by i,j{1,2,…,N_x−1}.

Given that A_g has a somatic mutation in a descendant of one of her N_x founding genomes, the probability that A_g−1 inherits more than one mutated genome is very small. Hence, we will assume A_g−1 is heteroplasmic at that site, with proportion 1/N_x of its genomes containing Y. If the heteroplasmy is lost g generations later, then A₀ has either all X or all Y at that site. Let h_X,_Y be the proportion of cases where the heteroplasmy persists and h_X and h_Y be the proportions where the mutation is lost or fixed. The neutral model predicts that as g increases

In a simulation study of 10⁷ heteroplasmic site histories, we followed the introduction of one mutation until the site heteroplasmy was lost, for N_x=20 and for N_x=40. We found (table 1) for θ=0.23 that h_X≈((N_x−1)/N_x) and h_Y≈1/N_x. Table 1 also gives the average numbers of generations that the site heteroplasmies persist, and are observable. We note that over all histories, the average numbers of generations a site heteroplasmy was observable were almost identical for N_x=20 and 40, but the average variation in the levels of a heteroplasmy at a site between a mother and her chick differed significantly.

For each A_k in the ancestry, let n_k be the number of its founding genomes with nucleotide Y at that site. We have assumed that n_g−1=1. If for k<g−1, n_k=0 or N_x, the heteroplasmy is lost. Suppose 1≤n_k+1=j<N_x, then the probability that n_k=i, (1≤i<N_x) isIn lemma 1 of the electronic supplementary material, we show thatwhich is the ith entry of the leading column of . Assuming that the probability that A_g introduces a somatic mutation into the germ line at a selected site is α, then summing over all generations g≥1, the probability that A₀ has a site heteroplasmy with n₀=i (1≤i<N_x) iswhere is the first entry in the ith row ofAs is the expected number of generations that a new site heteroplasmy persists with i copies (figure 1), the expected number of generations a heteroplasmy is observable at that site isMost site heteroplasmies never reach the detection threshold, those that do, usually persist for many more than generations (table 1).

In figure 2, we plot the values of (Q₂₀)_i,1 and (Q₄₀)_i,1, noting that these two distributions are almost identical, and that is closely approximated by 2/i within the observed region. (The limit as N_x→∞ was noted by Fisher and Wright (Ewens 2004, eqn (1.56)).)

Accounting for the effects modelled here may illuminate the apparent discrepancies reported between molecular substitution rates and those estimated from pedigree studies, such as for human studies, e.g. Parsons et al. (1997), Howell et al. (2003) and Santos et al. (2005). As these studies do not report observational thresholds, our model cannot be applied directly to their data.