Abstract:
We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \to \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \to \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$.

Abstract:
In the area of evolutionary theory, a key question is which portions of the genome of a species are targets of natural selection. Genetic hitchhiking is a theoretical concept that has helped to identify various such targets in natural populations. In the presence of recombination, a severe reduction in sequence diversity is expected around a strongly beneficial allele. The site frequency spectrum is an important tool in genome scans for selection and is composed of the numbers , where is the number of single nucleotide polymorphisms (SNPs) present in from individuals. Previous work has shown that both the number of low- and high-frequency variants are elevated relative to neutral evolution when a strongly beneficial allele fixes. Here, we follow a recent investigation of genetic hitchhiking using a marked Yule process to obtain an analytical prediction of the site frequency spectrum in a panmictic population at the time of fixation of a highly beneficial mutation. We combine standard results from the neutral case with the effects of a selective sweep. As simulations show, the resulting formula produces predictions that are more accurate than previous approaches for the whole frequency spectrum. In particular, the formula correctly predicts the elevation of low- and high-frequency variants and is significantly more accurate than previously derived formulas for intermediate frequency variants.

Abstract:
We study the effects of fast spatial movement of molecules on the dynamics of chemical species in a spatially heterogeneous chemical reaction network using a compartment model. The reaction networks we consider are either single- or multi-scale. When reaction dynamics is on a single-scale, fast spatial movement has a simple effect of averaging reactions over the distribution of all the species. When reaction dynamics is on multiple scales, we show that spatial movement of molecules has different effects depending on whether the movement of each type of species is faster or slower than the effective reaction dynamics on this molecular type. We obtain results for both when the system is without and with conserved quantities, which are linear combinations of species evolving only on the slower time scale.

Abstract:
We study the following model for an evolving random graph $\mathcal G = (G_n)_{n=n_0, n_0+1,...}$, where $G_n = (V_n, E_n)$ is a graph with $|V_n|=n$ vertices, $n=n_0,n_0+1,...$ In state $G_n = (V_n, E_n)$, a vertex $v\in V_n$ is chosen from $V_n$ uniformly at random and is $p$-copied. Upon such an event, a new vertex $v'\notin V_n$ is created and every edge $\{v,w\} \in E_n$ is copied with probability~$p$, i.e.\ $E_{n+1}$ has an edge $\{v',w\}$ with probability $p$, independently of all other edges. Within this graph, we study several aspects for large $n$. (i) The frequency of isolated vertices converges to~1 if $p\leq p^* \approx 0.567143$, the unique solution of $pe^p=1$. (ii) The expected frequency of $k$-cliques converges to $0$ (or $\infty$) if $pp_k$), if the starting graph contains at least one $k$-clique. In particular, the expected degree of a vertex converges to $0$ (or $\infty$) for $p<0.5$ (or $p>0.5$) and we obtain that the transitivity ratio of the random graph is of the order $n^{-2p(1-p)}$. (iii) The evolution of the degrees of the vertices in the initial graph can be described explicitly. Here, we obtain the full distribution as well as convergence results.

Abstract:
Protein translocation in cells has been modelled by \emph{Brownian ratchets}. In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion $(X_t)_{t\geq 0}$ with a changing reflection point $(R_t)_{t\geq 0}$. The rate of change of $R_t$ is $\gamma(X_t-R_t)$ and the new reflection boundary is distributed uniformly between $R_{t-}$ and $X_t$. The asymptotic speed of the ratchet scales with $\gamma^{1/3}$ and the asymptotic variance is independent of $\gamma$.

Abstract:
Genetic hitchhiking describes evolution at a neutral locus that is linked to a selected locus. If a beneficial allele rises to fixation at the selected locus, a characteristic polymorphism pattern (so-called selective sweep) emerges at the neutral locus. The classical model assumes that fixation of the beneficial allele occurs from a single copy of this allele that arises by mutation. However, recent theory (Pennings and Hermisson, 2006a; Pennings and Hermisson, 2006b) has shown that recurrent beneficial mutation at biologically realistic rates can lead to markedly different polymorphism patterns, so called soft selective sweeps. We extend an approach that has recently been developed for the classical hitchhiking model (Schweinsbergand Durrett, 2005; Etheridge, Pfaffelhuber, Wakolbinger, 2006) to study the recurrent mutation scenario. We show that the genealogy at the neutral locus can be approximated (to leading orders in the selection strength) by a marked Yule process with immigration. Using this formalism, we derive an improved analytical approximation for the expected heterozygosity at the neutral locus at the time of fixation of the beneficial allele.

Abstract:
We consider the sensitivity, with respect to a parameter \theta, of parametric families of operators A_{\theta}, vectors \pi_{\theta} corresponding to the adjoints A_{\theta}^{*} of A_{\theta} via A_{\theta}^{*}\pi_{\theta}=0 and one parameter semigroups t\mapsto e^{tA_{\theta}}. We display formulas relating weak differentiability of \theta\mapsto \pi_{\theta} (at \theta=0) to weak differentiability of \theta\mapsto A_{\theta}^{*}\pi_{0} and [e^{A_{\theta}t}]^{*}\pi_{0}. We give two applications: The first one concerns the sensitivity of the Ornstein--Uhlenbeck process with respect to its location parameter. The second one provides new insights regarding the Wright--Fisher diffusion for small mutation parameter.

Abstract:
Let $Z = (Z_t)_{t\in[0,\infty)}$ be an ergodic Markov process and, for $n\in\mathbb{N}$, let $Z^n = (Z_{n^2 t})_{t\in[0,\infty)}$ drive a process $X^n$. Classical results show under suitable conditions that the sequence of non-Markovian processes $(X^n)_{n\in\mathbb{N}}$ converges to a Markov process and give its infinitesimal characteristics. Here, we consider a general sequence $(Z^n)_{n\in\mathbb{N}}$. Using a general result on stochastic averaging from [Kur92], we derive conditions which ensure that the sequence $(X^n)_{n\in\mathbb{N}}$ converges as in the classical case. As an application, we consider the diffusion limit of branching random walk in quickly evolving random environment.

Abstract:
The genome of bacterial species is much more flexible than that of eukaryotes. Moreover, the distributed genome hypothesis for bacteria states that the total number of genes present in a bacterial population is greater than the genome of every single individual. The pangenome, i.e. the set of all genes of a bacterial species (or a sample), comprises the core genes which are present in all living individuals, and accessory genes, which are carried only by some individuals. In order to use accessory genes for adaptation to environmental forces, genes can be transferred horizontally between individuals. Here, we extend the infinitely many genes model from Baumdicker, Hess and Pfaffelhuber (2010) for horizontal gene transfer. We take a genealogical view and give a construction -- called the Ancestral Gene Transfer Graph -- of the joint genealogy of all genes in the pangenome. As application, we compute moments of several statistics (e.g. the number of differences between two individuals and the gene frequency spectrum) under the infinitely many genes model with horizontal gene transfer.

Abstract:
Evolutionary models for populations of constant size are frequently studied using the Moran model, the Wright-Fisher model, or their diffusion limits. When evolution is neutral, a random genealogy given through Kingman's coalescent is used in order to understand basic properties of such models. Here, we address the use of a genealogical perspective for models with weak frequency-dependent selection, i.e. N s =: {\alpha} is small, and s is the fitness advantage of a fit individual and N is the population size. When computing fixation probabilities, this leads either to the approach proposed by Rousset (2003), who argues how to use the Kingman's coalescent for weak selection, or to extensions of the ancestral selection graph of Neuhauser and Krone (1997) and Neuhauser (1999). As an application, we re-derive the one-third law of evolutionary game theory (Nowak et al., 2004). In addition, we provide the approximate distribution of the genealogical distance of two randomly sampled individuals under linear frequency-dependence.