Recent advances in sequencing technologies have made available an ever-increasing amount of ancient genomic data. In particular, it is now possible to target specific single nucleotide polymorphisms in several samples at different time points. Such timeseries data are also available in the context of experimental or viral evolution. Time-series data should allow for a more precise inference of population genetic parameters and to test hypotheses about the recent action of natural selection. In this manuscript, we develop a likelihood method to jointly estimate the selection coefficient and the age of an allele from time-serial data. Our method can be used for allele frequencies sampled from a single diallelic locus. The transition probabilities are calculated by approximating the standard diffusion equation of the Wright-Fisher model with a one-step process. We show that our method produces unbiased estimates. The accuracy of the method is tested via simulations. Finally, the utility of the method is illustrated with an application to several loci encoding coat color in horses, a pattern that has previously been linked with domestication. Importantly, given our ability to estimate the age of the allele, it is possible to gain traction on the important problem of distinguishing selection on new mutations from selection on standing variation. In this coat color example for instance, we estimate the age of this allele, which is found to predate domestication.T IME-series analysis is widespread in several fields, such as meteorology, economics, and physics (Hamilton 1994) with the relation being statistical models designed to deal with a time-ordered sequence of observations. Such observations are also prevalent in several areas of biology. Until recently, however, time-series molecular data were only available for time spanning a few generations in higher organisms. Therefore, in the context of population genetics, time-serial data were mostly limited to viral or experimental evolution (Wichman et al. 2005;Bollback and Huelsenbeck 2007;Nelson and Holmes 2007;Gresham et al. 2008).However, with recent advances in DNA sequencing and DNA preparation techniques, the study of extinct and long dead organisms is now entering a new era, an era in which time-sampled measurements spanning hundreds or thousands of generations for even mammalian species may be obtained. For example, while previous studies were limited to short segments of mitochondrial DNA, whole nuclear genomes are now available from several ancient samples (Rasmussen et al. 2010;Reich et al. 2010), and it is now additionally possible to target specific DNA regions in ancient organisms (Lalueza-Fox et al. 2007;Ludwig et al. 2009;Rusk 2009). Therefore, time-serial data will become increasingly available for a whole range of organisms allowing one to test evolutionary questions using not only present day samples, but also samples from extinct populations.The relevant theory to describe such temporal changes in allele frequency has existed since the adve...
Various ways of implementing boundary conditions for the numerical solution of the Navier-Stokes equations by a lattice Boltzmann method are discussed. Five commonly adopted approaches are reviewed, analyzed, and compared, including local and nonlocal methods. The discussion is restricted to velocity Dirichlet boundary conditions, and to straight on-lattice boundaries which are aligned with the horizontal and vertical lattice directions. The boundary conditions are first inspected analytically by applying systematically the results of a multiscale analysis to boundary nodes. This procedure makes it possible to compare boundary conditions on an equal footing, although they were originally derived from very different principles. It is concluded that all five boundary conditions exhibit second-order accuracy, consistent with the accuracy of the lattice Boltzmann method. The five methods are then compared numerically for accuracy and stability through benchmarks of two-dimensional and three-dimensional flows. None of the methods is found to be throughout superior to the others. Instead, the choice of a best boundary condition depends on the flow geometry, and on the desired trade-off between accuracy and stability. From the findings of the benchmarks, the boundary conditions can be classified into two major groups. The first group comprehends boundary conditions that preserve the information streaming from the bulk into boundary nodes and complete the missing information through closure relations. Boundary conditions in this group are found to be exceptionally accurate at low Reynolds number. Boundary conditions of the second group replace all variables on boundary nodes by new values. They exhibit generally much better numerical stability and are therefore dedicated for use in high Reynolds number flows.
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