Fixation probabilities and hitting times for low levels of frequency-dependent selection.

In population genetics, diffusions on the unit interval are often used to model the frequency path of an allele. In this setting we derive approximations for fixation probabilities, expected hitting times and the expected frequency spectrum for low levels of frequency-dependent selection. Specifically, we rederive and extend the one-third rule of evolutionary game theory (Nowak et al., 2004) and effects of stochastic slowdown (Altrock and Traulsen, 2009). Since similar effects are of interest in other application areas, we formulate our results for general one-dimensional diffusions.

The independent loss model with ordered insertions for the evolution of CRISPR spacers.

Today, the CRISPR (clustered regularly interspaced short palindromic repeats) region within bacterial and archaeal genomes is known to encode an adaptive immune system. We rely on previous results on the evolution of the CRISPR arrays, which led to the ordered independent loss model, introduced by Kupczok and Bollback (2013). When focusing on the spacers (between the repeats), new elements enter a CRISPR array at rate θ at the leader end of the array, while all spacers present are lost at rate ρ along the phylogeny relating the sample. Within this model, we compute the distribution of distances of spacers which are present in all arrays in a sample of size n. We use these results to estimate the loss rate ρ from spacer array data for n=2 and n=3.

A Brownian ratchet for protein translocation including dissociation of ratcheting sites.

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate γ. Extending our previous approach (Depperschmidt and Pfaffelhuber in Stoch Processes Appl 120:901-925, 2010) we allow the ratcheting molecules to dissociate (at rate δ) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speeds of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation. This speed gives an approximation for the time it takes to translocate a protein of given length.

Selective sweeps for recessive alleles and for other modes of dominance.

A selective sweep describes the reduction of linked genetic variation due to strong positive selection. If s is the fitness advantage of a homozygote for the beneficial allele and h its dominance coefficient, it is usually assumed that h=1/2, i.e. the beneficial allele is co-dominant. We complement existing theory for selective sweeps by assuming that h is any value in [0, 1]. We show that genetic diversity patterns under selective sweeps with strength s and dominance 0 < h < 1 are similar to co-dominant sweeps with selection strength 2hs. Moreover, we focus on the case h=0 of a completely recessive beneficial allele. We find that the length of the sweep, i.e. the time from occurrence until fixation of the beneficial allele, is of the order of √(N/s) generations, if N is the population size. Simulations as well as our results show that genetic diversity patterns in the recessive case h=0 greatly differ from all other cases.

Linkage disequilibrium under genetic hitchhiking in finite populations.

The model of genetic hitchhiking predicts a reduction in sequence diversity at a neutral locus closely linked to a beneficial allele. In addition, it has been shown that the same process results in a specific pattern of correlations (linkage disequilibrium) between neutral polymorphisms along the chromosome at the time of fixation of the beneficial allele. During the hitchhiking event, linkage disequilibrium on either side of the beneficial allele is built up whereas it is destroyed across the selected site. We derive explicit formulas for the expectation of the covariance measure D and standardized linkage disequilibrium sigma 2D between a pair of polymorphic sites. For our analysis we use the approximation of a star-like genealogy at the selected site. The resulting expressions are approximately correct in the limit of large selection coefficients. Using simulations we show that the resulting pattern of linkage disequilibrium is quickly-i.e., in <0.1N generations-destroyed after the fixation of the beneficial allele for moderately distant neutral loci, where N is the diploid population size.

Approximating genealogies for partially linked neutral loci under a selective sweep.

Consider a genetic locus carrying a strongly beneficial allele which has recently fixed in a large population. As strongly beneficial alleles fix quickly, sequence diversity at partially linked neutral loci is reduced. This phenomenon is known as a selective sweep. The fixation of the beneficial allele not only affects sequence diversity at single neutral loci but also the joint allele distribution of several partially linked neutral loci. This distribution can be studied using the ancestral recombination graph for samples of partially linked neutral loci during the selective sweep. To approximate this graph, we extend recent work by Etheridge et al. (Ann Appl Probab 16:685-729, 2006) and Schweinsberg and Durrett (Ann Appl Probab 15:1591-1651, 2005) using a marked Yule tree for the genealogy at a single neutral locus linked to a strongly beneficial one. We focus on joint genealogies at two partially linked neutral loci in the case of large selection coefficients alpha and recombination rates rho = theta(alpha/log alpha) between loci. Our approach leads to a full description of the genealogy with accuracy of theta((log alpha)(-2)) in probability. As an application, we derive the expectation of Lewontin's D as a measure for non-random association of alleles.

Approximate genealogies under genetic hitchhiking.

The rapid fixation of an advantageous allele leads to a reduction in linked neutral variation around the target of selection. The genealogy at a neutral locus in such a selective sweep can be simulated by first generating a random path of the advantageous allele's frequency and then a structured coalescent in this background. Usually the frequency path is approximated by a logistic growth curve. We discuss an alternative method that approximates the genealogy by a random binary splitting tree, a so-called Yule tree that does not require first constructing a frequency path. Compared to the coalescent in a logistic background, this method gives a slightly better approximation for identity by descent during the selective phase and a much better approximation for the number of lineages that stem from the founder of the selective sweep. In applications such as the approximation of the distribution of Tajima's D, the two approximation methods perform equally well. For relevant parameter ranges, the Yule approximation is faster.