Bursts of coalescence within population pedigrees whenever big families occur

We consider a simple diploid population-genetic model with potentially high variability of offspring numbers among individuals. Specifically, against a backdrop of Wright-Fisher reproduction and no selection, there is an additional probability that a big family occurs, meaning that a pair of individuals has a number of offspring on the order of the population size. We study how the pedigree of the population generated under this model affects the ancestral genetic process of a sample of size two at a single autosomal locus without recombination. Our population model is of the type for which multiple-merger coalescent processes have been described. We prove that the conditional distribution of the pairwise coalescence time given the random pedigree converges to a limit law as the population size tends to infinity. This limit law may or may not be the usual exponential distribution of the Kingman coalescent, depending on the frequency of big families. But because it includes the number and times of big families, it differs from the usual multiple-merger coalescent models. The usual multiple-merger coalescent models are seen as describing the ancestral process marginal to, or averaging over, the pedigree. In the limiting ancestral process conditional on the pedigree, the intervals between big families can be modeled using the Kingman coalescent but each big family causes a discrete jump in the probability of coalescence. Analogous results should hold for larger samples and other population models. We illustrate these results with simulations and additional analysis, highlighting their implications for inference and understanding of multilocus data.

Ancestries of a recombining diploid population

We derive the exact one-step transition probabilities of the number of lineages that are ancestral to a random sample from the current generation of a bi-parental population that is evolving under the discrete Wright-Fisher model with n diploid individuals. Our model allows for a per-generation recombination probability of r . When r = 1, our model is equivalent to Chang's (Adv Appl Probab 31:1002-1038, 1999) model for the karyotic pedigree. When r = 0, our model is equivalent to Kingman's (Stoch Process Appl 13:235-248, 1982) discrete coalescent model for the cytoplasmic tree or sub-karyotic tree containing a DNA locus that is free of intra-locus recombination. When 0 < r < 1 our model can be thought to track a sub-karyotic ancestral graph containing a DNA sequence from an autosomal chromosome that has an intra-locus recombination probability r . Thus, our family of models indexed by r ∈ [0, 1] connects Kingman's discrete coalescent to Chang's pedigree in a continuous way as r goes from 0 to 1. For large populations, we also study three properties of the ancestral process corresponding to a given r ∈ (0, 1): the time Tn to a most recent common ancestor (MRCA) of the population, the time Un at which all individuals are either common ancestors of all present day individuals or ancestral to none of them, and the fraction of individuals that are common ancestors at time Un. These results generalize the three main results of Chang's (Adv Appl Probab 31:1002-1038, 1999). When we appropriately rescale time and recombination probability by the population size, our model leads to the continuous time Markov chain called the ancestral recombination graph of Hudson (Theor Popul Biol 23:183-201, 1983) and Griffiths (The two-locus ancestral graph, Institute of Mathematical Statistics 100-117, 1991).

The largest strongly connected component in the cyclical pedigree model of Wakeley et al

We establish a link between Wakeley et al.'s (2012) cyclical pedigree model from population genetics and a randomized directed configuration model (DCM) considered by Cooper and Frieze (2004). We then exploit this link in combination with asymptotic results for the in-degree distribution of the corresponding DCM to compute the asymptotic size of the largest strongly connected component S(N) (where N is the population size) of the DCM resp. the pedigree. The size of the giant component can be characterized explicitly (amounting to approximately 80% of the total populations size) and thus contributes to a reduced 'pedigree effective population size'. In addition, the second largest strongly connected component is only of size O(logN). Moreover, we describe the size and structure of the 'domain of attraction' of S(N). In particular, we show that with high probability for any individual the shortest ancestral line reaches S(N) after O(loglogN) generations, while almost all other ancestral lines take at most O(logN) generations.

Inbreeding, pedigree size, and the most recent common ancestor of humanity

How many generations ago did the common ancestor of all present-day individuals live, and how does inbreeding affect this estimate? The number of ancestors within family trees determines the timing of the most recent common ancestor of humanity. However, mating is often non-random and inbreeding is ubiquitous in natural populations. Rates of pedigree growth are found for multiple types of inbreeding. This data is then combined with models of global population structure to estimate biparental coalescence times. When pedigrees for regular systems of mating are constructed, the growth rates of inbred populations contain Fibonacci n-step constants. The timing of the most recent common ancestor depends on global population structure, the mean rate of pedigree growth, mean fitness, and current population size. Inbreeding reduces the number of ancestors in a pedigree, pushing back global common ancestry times. These results are consistent with the remarkable findings of previous studies: all humanity shares common ancestry in the recent past.