The largest strongly connected component in Wakeley et al’s cyclical pedigree model
Jochen Blath, Stephan Kadow, Marcel Ortgiese
Comments: 21 pages, 2 figures
Subjects: Probability (math.PR); Populations and Evolution (q-bio.PE)

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 $O(\log N)$. 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 $O(\log \log N)$ generations, while almost all other ancestral lines take at most $O(\log N)$ generations.

Tractable stochastic models of evolution for loosely linked loci
Paul A. Jenkins, Paul Fearnhead, Yun S. Song
Comments: 32 pages, 1 figure
Subjects: Probability (math.PR); Populations and Evolution (q-bio.PE)

Of fundamental importance in statistical genetics is to compute the sampling distribution, or likelihood, for a sample of genetic data from some stochastic evolutionary model. For DNA sequence data with inter-locus recombination, standard models include the Wright-Fisher diffusion with recombination and its dual genealogical process, the ancestral recombination graph. However, under neither of these models is the sampling distribution available in closed-form, and their computation is extremely difficult. In this paper we derive two new stochastic population genetic models, one a diffusion and the other a coalescent process, which are much simpler than the standard models, but which capture their key properties for large recombination rates. In the former case, we show that the sampling distribution is available in closed form. We further demonstrate that when we consider the sampling distribution as an asymptotic expansion in inverse powers of the recombination parameter, the sampling distributions of the two models agree with the standard ones up to the first two orders.

Dynamics of a combined medea-underdominant population transformation system
Chaitanya Gokhale, Richard Guy Reeves, Floyd A Reed

Transgenic constructs intended to be stably established at high frequencies in wild popu- lations have been demonstrated to “drive” from low frequencies in experimental insect populations. Link- ing such population transformation constructs to genes which render them unable to transmit pathogens could eventually be used to stop the spread of vector-borne diseases like malaria and dengue. Generally, population transformation constructs with only a single transgenic drive mechanism have been envisioned. Using a theoretical modelling approach we describe the predicted properties of a construct combining autosomal Medea and underdominant population transformation systems. We show that when combined they can exhibit synergistic properties which in broad circumstances surpass those of the single systems. With combined systems, intentional population transformation and its reversal can be achieved readily. Combined constructs also enhance the capacity to geographically restrict transgenic constructs to targeted populations. It is anticipated that these properties are likely to be of particular value in attracting regulatory approval and public acceptance of this novel technology.