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Anton Wakolbinger

Publications and source records attributed to Anton Wakolbinger.

At least 19 recordsLinked to original sources
arXiv2026

A small noise approximation for Muller's Ratchet

We consider an infinite system of SDEs with Fleming-Viot noise indexed by $k=0,1,2,\dots$, whose parameters $α,λ$, and $ν$ are the (deleterious) selection coefficient, the (uni-directional) mutation rate, and a quantity which determines the size of the system's fluctuations. The SDE's unique weak solution $X(t) = (X_k(t))_{k=0,1,2,...}$ models what is known in population genetics as Muller's ratchet. Here, $X_k(t)$ stands for the frequency of individuals carrying $k$ deleterious mutations. Since the mutation process is uni-directional, $t\mapsto \inf\{k: X_k(t)> 0\}$ is non-decreasing for almost every path of $X$, and we refer to an increase as a click of Muller's ratchet. A long standing question concerns the clicking rate of Muller's ratchet. Using Duhamel's principle for semigroups, we give a partial answer by approximating $E(\sum_{k=1}^\infty kX_k(t) )$ and $E\big(X_0(t)\big)$ up to $O(1/ν^2)$ for fixed $α$, $λ$ and $t>0$. Our results suggest that $ψ:=ναe^{-λ/α}$ is a crucial quantity also when the mutation/selection ratio $θ= λ/α$ is moderately large: for large $να$, clicking of the ratchet on the time scale $\frac 1α\log θ$ becomes rare as soon as $ψ$ becomes large.

math.PR↗
arXiv2026

Logarithmic scaling of selective sweep curves: from tents to houses

One of the classical results of mathematical population genetics states that the frequency of a beneficial mutant's offspring, on its way to fixation in a large population, looks like a logistic curve. A logarithmic scaling (both in height and time) of these selective sweep curves leads (in the case of strong selection) to a tent-like shape in the large population limit: First the logarithmic frequency of the mutant increases linearly from 0 to 1, then that of the former resident decreases from 1 to 0. For moderate selection the logarithmic frequencies develop (in the large population limit) a jump at the beginning/the end of the sweep, which takes the shape of the tent into that of a house. Our main result (proved for the Moran model) assesses the regularity of this convergence in the large population limit: It is uniform in the house's roof (phases of linear growth and decline) and "Skorokhod $M_1$" in the house's walls (closely around the jumps). Apart from interest in its own right, we anticipate that this result and the proof techniques will be instrumental for extending the description of clonal interference by Poissonian interacting trajectories (as it was done in Hermann et al. (2024) for strong selection) also to moderate selection.

math.PR↗
arXiv2026

Killing versus catastrophes in birth-death processes and an application to population genetics

We establish connections between the absorption probabilities of a class of birth-death processes with killing, and the stationary tail of a related class of birth-death processes with catastrophes. The major ingredients of the proofs are a decomposition of the dynamics of these processes, a Feynman--Kac type relationship for Markov chains with reset and rebirth, and the concept of Siegmund duality, which allows us to invert the relationship between the processes. We apply our results to a pair of ancestral processes in population genetics, namely the killed ancestral selection graph and the pruned lookdown ancestral selection graph, in a finite population setting and its diffusion limit.

math.PR↗
arXiv2025

The tournament ratchet's clicktime process, and metastability in a Moran model

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~$N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. In the classical variant, an individual's selective advantage is proportional to the difference between the population average and the individual's mutation load, whereas in the ratchet with {\em tournament selection} only the signs of the differences of the individual mutation loads matter. In a parameter regime which leads to slow clicking (i.e. to a loss of the currently fittest class at a rate $\ll 1/N$) we prove that the rescaled process of click times of the tournament ratchet converges as $N\to \infty$ to a Poisson process. Central ingredients in the proof are a thorough analysis of the metastable behaviour of a two-type Moran model with selection and deleterious mutation (which describes the size of the fittest class up to its extinction time) and a lower estimate on the size of the new fittest class at a clicktime.

math.PR↗
arXiv2025

Muller's ratchet in a near-critical regime: tournament versus fitness proportional selection

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~$N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as $N\to \infty$). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as $N\to \infty$.

q-bio.PE↗
arXiv2025

From clonal interference to Poissonian interacting trajectories

We consider a population whose size $N$ is fixed over the generations, and in which random beneficial mutations arrive at a rate of order $1/\log N$ per generation. In this so-called Gerrish--Lenski regime, typically a finite number of contending mutations are present together with one resident type. These mutations compete for fixation, a phenomenon addressed as clonal interference. We introduce and study a Poissonian system of interacting trajectories (PIT), and prove that it arises as a large population scaling limit of the logarithmic sizes of the contending clonal subpopulations in a continuous-time Moran model with strong selection. We show that the PIT exhibits an almost surely positive asymptotic rate of fitness increase (speed of adaptation), which turns out to be finite if and only if fitness increments have a finite expectation. We relate this speed to heuristic predictions from the literature. Furthermore, we derive a functional central limit theorem for the fitness of the resident population in the PIT.

math.PR↗
arXiv2023

Quasi-equilibria and click times for a variant of Muller's ratchet

Consider a population of $N$ individuals, each of them carrying a type in $\mathbb N_0$. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type $k$ has the same selective advantage over all individuals with type $k' > k$, and type $k$ mutates to type $k+1$ at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to $k'-k$. For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the {\em click times} of the ratchet (i.e. the times at which the hitherto minimal (`best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a ``dual'' hierarchical multitype logistic system, and also via the distribution of the final minimal displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.

math.PR↗
arXiv2023

A diploid population model for copy number variation of genetic elements

We study the following model for a diploid population of constant size $N$: Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability $\tfrac 12$ on to the offspring. We study the process $X^N = (X^N(1), X^N(2),...)$, where $X_t^N(k)$ is the frequency of individuals at time $t$ that carry $k$ elements, and prove convergence (in some weak sense) of $X^N$ jointly with its empirical first moment $Z^N$ to the ``slow-fast'' system $(Z,X)$, where $X_t = \text{Poi}(Z_t)$ and $Z$ evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.

math.PR↗
arXiv2023

Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn

For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law Pólya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three and four individuals that are randomly sampled from $[n]$. This allows us to obtain a new, conceptual proof of the asymptotic Gaussianity (including the functional convergence) of the renormalised sums of more general colourings, which can be seen as an invariance principle beyond the main result of Hammond and Sheffield. In this proof, a key ingredient of independent interest is a sufficient criterion for the asymptotic Gaussianity of the renormalised sums in randomly coloured random partitions of $[n]$, based on Stein's method. Along the way we also prove a statement on the asymptotics of the coalescence probabilities in the long-range seedbank model of Blath, González Casanova, Kurt, and Spanò.

math.PR↗
arXiv2022

Evolving genealogies for branching populations under selection and competition

For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by Donnelly and Kurtz (1999), namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of ``potential'' selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the ``active'' selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz' Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation.

math.PR↗
arXiv2022

Haldane's formula in Cannings models: The case of moderately strong selection

For a class of Cannings models we prove Haldane's formula, $π(s_N) \sim \frac{2s_N}{ρ^2}$, for the fixation probability of a single beneficial mutant in the limit of large population size $N$ and in the regime of moderately strong selection, i.e. for $s_N \sim N^{-b}$ and $0< b<1/2$. Here, $s_N$ is the selective advantage of an individual carrying the beneficial type, and $ρ^2$ is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele's frequency process with slightly supercritical Galton-Watson processes in the early phase of fixation.

math.PR↗
arXiv2020

Haldane's formula in Cannings models: The case of moderately weak selection

We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new \emph{Cannings ancestral selection graph} in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane's formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the prob\-ability of fixation is asymptotically (as $N\to \infty$) equal to the selective advantage of haploids $s_N$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_N$ obeying $N^{-1} \ll s_N \ll N^{-1/2} $, which is a regime of "moderately weak selection". It turns out that for $ s_N \ll N^{-2/3} $ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.

math.PR↗
arXiv2020

The height process of a continuous state branching process with interaction

For a generalized continuous state branching process with non-vanishing diffusion part, finite expectation and a directed ("left-to-right") interaction, we construct the height process of its forest of genealogical trees. The connection between this height process and the population size process is given by an extension of the second Ray--Knight theorem. This paper generalizes earlier work of the two last authors which was restricted to the case of continuous branching mechanisms. Our approach is different from that of Berestycki et al. There the diffusion part of the population process was allowed to vanish, but the class of interactions was more restricted.

math.PR↗
arXiv2020

Microbial populations under selection

This chapter gives a synopsis of recent approaches to model and analyse the evolution of microbial populations under selection. The first part reviews two population genetic models of Lenski's long-term evolution experiment with Escherichia coli, where models aim at explaining the observed curve of the evolution of the mean fitness. The second part describes a model of a host-pathogen system where the population of pathogenes experiences balancing selection, migration, and mutation, as motivated by observations of the genetic diversity of HCMV (the human cytomegalovirus) across hosts.

q-bio.PE↗
arXiv2020

Doob--Martin boundary of Rémy's tree growth chain

Rémy's algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the $n^{\mathrm{th}}$ tree is uniformly distributed over the set of rooted, planar, binary trees with $2n+1$ vertices. We obtain a concrete characterization of the Doob--Martin boundary of this transient Markov chain and thereby delineate all the ways in which, loosely speaking, this process can be conditioned to "go to infinity" at large times. A (deterministic) sequence of finite rooted, planar, binary trees converges to a point in the boundary if for each $m$ the random rooted, planar, binary tree spanned by $m+1$ leaves chosen uniformly at random from the $n^{\mathrm{th}}$ tree in the sequence converges in distribution as $n$ tends to infinity -- a notion of convergence that is analogous to one that appears in the recently developed theory of graph limits. We show that a point in the Doob--Martin boundary may be identified with the following ensemble of objects: a complete separable $\mathbb{R}$-tree that is rooted and binary in a suitable sense, a diffuse probability measure on the $\mathbb{R}$-tree that allows us to make sense of sampling points from it, and a kernel on the $\mathbb{R}$-tree that describes the probability that the first of a given pair of points is below and to the left of their most recent common ancestor while the second is below and to the right. The Doob--Martin boundary corresponds bijectively to the set of extreme points of the closed convex set of normalized nonnegative harmonic functions, in other words, the minimal and full Doob--Martin boundaries coincide. These results are in the spirit of the identification of graphons as limit objects in the theory of graph limits.

math.PR↗
arXiv2020

Probabilistic aspects of $Λ$-coalescents in equilibrium and in evolution

We present approximation methods which lead to law of large numbers and fluctuation results for functionals of $Λ$-coalescents, both in the dust-free case and in the case with a dust component. Our focus is on the tree length (or total branch length) and the total external branch length, as well as the time to the most recent common ancestor and the size of the last merger. In the second part we discuss evolving coalescents. For certain Beta-coalescents we analyse fluctuations of a class of functionals in appropriate time scales. Finally we review results of Gufler on the representation of evolving $Λ$-coalescents in terms of the lookdown space.

math.PR↗
arXiv2019

PATRICIA bridges

A radix sort tree arises when storing distinct infinite binary words in the leaves of a binary tree such that for any two words their common prefixes coincide with the common prefixes of the corresponding two leaves. If one deletes the out-degree $1$ vertices in the radix sort tree and "closes up the gaps", then the resulting PATRICIA tree maintains all the information that is necessary for sorting the infinite words into lexicographic order. We investigate the PATRICIA chains -- the tree-valued Markov chains that arise when successively building the PATRICIA trees for the collection of infinite binary words $Z_1,\ldots, Z_n$, $n=1,2,\ldots$, where the source words $Z_1, Z_2,\ldots$ are independent and have a common diffuse distribution on $\{0,1\}^\infty$. It turns out that the PATRICIA chains share a common collection of backward transition probabilities and that these are the same as those of a chain introduced by Rémy for successively generating uniform random binary trees with larger and larger numbers of leaves. This means that the infinite bridges of any PATRICIA chain (that is, the chains obtained by conditioning a PATRICIA chain on its remote future) coincide with the infinite bridges of the Rémy chain. The infinite bridges of the Rémy chain are characterized concretely in Evans, Grübel, and Wakolbinger 2017 and we recall that characterization here while adding some details and clarifications.

math.PR↗
arXiv2019

Modelling and simulating Lenski's long-term evolution experiment

We revisit the model by Wiser, Ribeck, and Lenski (Science \textbf{342} (2013), 1364--1367), which describes how the mean fitness increases over time due to beneficial mutations in Lenski's long-term evolution experiment. We develop the model further both conceptually and mathematically. Conceptually, we describe the experiment with the help of a Cannings model with mutation and selection, where the latter includes diminishing returns epistasis. The analysis sheds light on the growth dynamics within every single day and reveals a runtime effect, that is, the shortening of the daily growth period with increasing fitness; and it allows to clarify the contribution of epistasis to the mean fitness curve. Mathematically, we explain rigorous results in terms of a law of large numbers (in the limit of infinite population size and for a certain asymptotic parameter regime), and present approximations based on heuristics and supported by simulations for finite populations.

q-bio.PE↗