# Catch me if you can: Adaptation from standing genetic variation to a moving phenotypic optimum

Catch me if you can: Adaptation from standing genetic variation to a moving phenotypic optimum

Sebastian Matuszewski , Joachim Hermisson , Michael Kopp
doi: http://dx.doi.org/10.1101/015685
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Abstract

# Partitioning, duality, and linkage disequilibria in the Moran model with recombination

Partitioning, duality, and linkage disequilibria in the Moran model with recombination
Mareike Esser, Sebastian Probst, Ellen Baake
Subjects: Probability (math.PR); Populations and Evolution (q-bio.PE)

The Moran model with recombination is considered, which describes the evolution of the genetic composition of a population under recombination and resampling. There are $n$ sites (or loci), a finite number of letters (or alleles) at every site, and we do not make any scaling assumptions. In particular, we do not assume a diffusion limit. We consider the following marginal ancestral recombination process. Let $S = \{1,…c,n\}$ and $\mathcal A=\{A_1, …c, A_m\}$ be a partition of $S$. We concentrate on the joint probability of the letters at the sites in $A_1$ in individual $1$, $…c$, at the sites in $A_m$ in individual $m$, where the individuals are sampled from the current population without replacement. Following the ancestry of these sites backwards in time yields a process on the set of partitions of $S$, which, in the diffusion limit, turns into a marginalised version of the $n$-locus ancestral recombination graph. With the help of an inclusion-exclusion principle, we show that the type distribution corresponding to a given partition may be represented in a systematic way, with the help of so-called recombinators and sampling functions. The same is true of correlation functions (known as linkage disequilibria in genetics) of all orders.
We prove that the partitioning process (backward in time) is dual to the Moran population process (forward in time), where the sampling function plays the role of the duality function. This sheds new light on the work of Bobrowski, Wojdyla, and Kimmel (2010). The result also leads to a closed system of ordinary differential equations for the expectations of the sampling functions, which can be translated into expected type distributions and expected linkage disequilibria.

# Transition densities and sample frequency spectra of diffusion processes with selection and variable population size

Transition densities and sample frequency spectra of diffusion processes with selection and variable population size
Daniel Zivkovic, Matthias Steinrücken, Yun S. Song, Wolfgang Stephan
doi: http://dx.doi.org/10.1101/014639

Advances in empirical population genetics have made apparent the need for models that simultaneously account for selection and demography. To address this need, we here study the Wright-Fisher diffusion under selection and variable effective population size. In the case of genic selection and piecewise-constant effective population sizes, we obtain the transition density function by extending a recently developed method for computing an accurate spectral representation for a constant population size. Utilizing this extension, we show how to compute the sample frequency spectrum (SFS) in the presence of genic selection and an arbitrary number of instantaneous changes in the effective population size. We also develop an alternate, efficient algorithm for computing the SFS using a method of moments. We apply these methods to answer the following questions: If neutrality is incorrectly assumed when there is selection, what effects does it have on demographic parameter estimation? Can the impact of negative selection be observed in populations that undergo strong exponential growth?

# Feller’s Contributions to Mathematical Biology

Feller’s Contributions to Mathematical Biology

Ellen Baake, Anton Wakolbinger
(Submitted on 21 Jan 2015)

This is a review of William Feller’s important contributions to mathematical biology. The seminal paper [Feller1951] “Diffusion processes in genetics” was particularly influential on the development of stochastic processes at the interface to evolutionary biology, and interesting ideas in this direction (including a first characterization of what is nowadays known as “Feller’s branching diffusion”) already shaped up in the paper [Feller 1939] (written in German) “The foundations of a probabistic treatment of Volterra’s theory of the struggle for life”. Feller’s article “On fitness and the cost of natural selection” [Feller 1967] contains a critical analysis of the concept of “genetic load”.

# Rise and fall of asexual mutators in adapted populations

Rise and fall of asexual mutators in adapted populations
Ananthu James, Kavita Jain
Subjects: Populations and Evolution (q-bio.PE)

In an adapted population in which most mutations are deleterious, the mutation rates are expected to be low. Indeed, in recent experiments on adapted populations of asexual mutators, beneficial mutations that lower the mutation rates have been observed to get fixed. Using a multitype branching process and a deterministic argument, we calculate the time to fix the wildtype mutation rate in an asexual population of mutators, and find it to be a ${\tt U}$-shaped function of the population size. In contrast, the fixation time for mutators is known to increase with population size. On comparing these two time scales, we find that a critical population size exists below which the mutators prevail, while the mutation rate remains low in larger populations. We also discuss how our analytical results compare with the experiments.

# The SMC’ is a highly accurate approximation to the ancestral recombination graph

The SMC’ is a highly accurate approximation to the ancestral recombination graph

Peter R. Wilton, Shai Carmi, Asger Hobolth
(Submitted on 12 Jan 2015)

Two sequentially Markov coalescent models (SMC and SMC’) are available as tractable approximations to the ancestral recombination graph (ARG). We present a model of coalescence at two fixed points along a pair of sequences evolving under the SMC’. Using our model, we derive a number of new quantities related to the pairwise SMC’, thereby analytically quantifying for the first time the similarity between the SMC’ and ARG. We use our model to show that the joint distribution of pairwise coalescence times at recombination sites under the SMC’ is the same as it is marginally under the ARG, demonstrating that the SMC’ is the canonical first-order sequentially Markov approximation to the pairwise ARG. Finally, we use these results to show that population size estimates under the pairwise SMC are asymptotically biased, while under the pairwise SMC’ they are approximately asymptotically unbiased.