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We consider the mutation--selection differential equation with pairwise interaction (or, equivalently, the diploid mutation--selection equation) and establish the corresponding ancestral process, which is a random tree and a variant of the ancestral selection graph. The formal relation to the forward model is given via duality. To make the tree tractable, we prune branches upon mutations, thus reducing it to its informative parts. The hierarchies inherent in the tree are encoded systematically via tripod trees with weighted leaves; this leads to the stratified ancestral selection graph. The latter also satisfies a duality relation with the mutation--selection equation. Each of the dualities provides a stochastic representation of the solution of the differential equation. This allows us to connect the equilibria and their bifurcations to the long-term behaviour of the ancestral process. Furthermore, with the help of the stratified ancestral selection graph, we obtain explicit results about the ancestral type distribution in the case of unidirectional mutation.
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We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equations solution (and, in particular, of its equilibrium) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.
We review recent progress on ancestral processes related to mutation-selection models, both in the deterministic and the stochastic setting. We mainly rely on two concepts, namely, the killed ancestral selection graph and the pruned lookdown ancestral selection graph. The killed ancestral selection graph gives a representation of the type of a random individual from a stationary population, based upon the individuals potential ancestry back until the mutations that define the individuals type. The pruned lookdown ancestral selection graph allows one to trace the ancestry of individuals from a stationary distribution back into the distant past, thus leading to the stationary distribution of ancestral types. We illustrate the results by applying them to a prototype model for the error threshold phenomenon.
We study ancestral structures for the two-type Moran model with two-way mutation and frequency-dependent selection that follows the nonlinear dominance or fittest-type-wins scheme. Both schemes lead, in distribution, to the same type-frequency process. Reasoning through the mutation structure on the ancestral selection graph (ASG), we derive processes suitable to determine the type distribution of the present and ancestral population, leading to, respectively, the killed and pruned lookdown ASG. To this end, we establish factorial moment dualities to the Moran model and a relative thereof, respectively. Finally, we extend the results to the diffusion limit.
We consider a spatial model of cancer in which cells are points on the $d$-dimensional torus $mathcal{T}=[0,L]^d$, and each cell with $k-1$ mutations acquires a $k$th mutation at rate $mu_k$. We will assume that the mutation rates $mu_k$ are increasing, and we find the asymptotic waiting time for the first cell to acquire $k$ mutations as the torus volume tends to infinity. This paper generalizes results on waiting for $kgeq 3$ mutations by Foo, Leder, and Schweinsberg, who considered the case in which all of the mutation rates $mu_k$ were the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates.
Using graphical methods based on a `lookdown and pruned version of the {em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.
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