Common ancestor type distribution: A Moran model and its deterministic limit

Cordero, Fernando

doi:10.1016/j.spa.2016.06.019

Cited by 16 publications

(34 citation statements)

References 19 publications

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“…For the case 1/2 < b < 1, Theorem 3. 5 gives an affirmative answer for subclasses of Cannings models admitting a paintbox representation, and in particular also for the Wright-Fisher model with selection. In Theorem 3.5a) we prove Haldane's formula under the condition…”

Section: Introductionmentioning

confidence: 92%

Haldane’s formula in Cannings models: the case of moderately weak selection

Boenkost¹,

Casanova²,

Pokalyuk³

et al. 2021

Electron. J. Probab.

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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 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 probability of fixation is asymptotically (as N → ∞) equal to the selective advantage of haploids sN divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences sN obeying N −1 sN N −1/2 , which is a regime of "moderately weak selection". It turns out that for sN 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.

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Section: Introductionmentioning

confidence: 92%

Haldane’s formula in Cannings models: the case of moderately weak selection

Boenkost¹,

Casanova²,

Pokalyuk³

et al. 2021

Electron. J. Probab.

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“…Backward in time potential ancestors of a sample of the population are traced back with the help of the ASG. An appropriate dynamical ordering and pruning of its lines leads to the pruned lookdown ASG, which in turn permits to characterise the common ancestor type distribution (see [10]). The block counting process L N := (L N t ) t≥0 of the pruned lookdown ASG describes the number of potential ancestors of a given sample of individuals.…”

Section: Preliminaries: Fearnhead-type Recursionsmentioning

confidence: 99%

“…Let L N ∞ be a random variable distributed according to (p N n ) n∈ [N ] . The stationary tail probabilities a N n := P (L N ∞ > n), n ∈ [N − 1] 0 , are characterised by the recurrence relation (see [10,Prop. 4.7])…”

Section: Preliminaries: Fearnhead-type Recursionsmentioning

confidence: 99%

On the stationary distribution of the block counting process for population models with mutation and selection

Cordero

Möhle

2019

Journal of Mathematical Analysis and Applications

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We consider two population models subject to the evolutionary forces of selection and mutation, the Moran model and the Λ-Wright-Fisher model. In such models the block counting process traces back the number of potential ancestors of a sample of the population at present. Under some conditions the block counting process is positive recurrent and its stationary distribution is described via a linear system of equations. In this work, we first characterise the measures Λ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. Next, we solve the linear system of equations corresponding to the Moran model. For the Λ-Wright-Fisher model we show that the probability generating function associated to the stationary distribution of the block counting process satisfies an integro differential equation. We solve the latter for the Kingman model and the star-shaped model. 1 coalescence mechanism is given by the Λ-coalescent. Additionally, selection introduces binary branching at constant rate per ancestral line. This approach differs from the one in [22] in that the ASG describes the ancestries of an untyped sample of the population. In absence of mutations, the block counting process of the Λ-ASG is in moment duality with the type-frequency process in the Λ-Wright-Fisher model (see, e.g., [27]). In presence of mutations, two relatives of the Λ-ASG permit to resolve mutation events on the spot and encode relevant information of the model: the Λ-killed ASG and the Λ-pruned lookdown ASG (the three ancestral processes coincide in absence of mutations). The killed ASG is reminiscent to the coalescent with killing [18, Chap. 1.3.1] and it was introduced in [4] for the Wright-Fisher diffusion model with selection and mutation. Its construction generalises in a natural way to Λ-Wright-Fisher models. The killed ASG helps to determine weather or not all the individuals in a sample of the population at present are unfit and is related to the type-frequency process via a moment duality. The pruned lookdown ASG in turn helps to determine the type of the common ancestor of a given sample of the population. It was introduced in [41] for the Wright-Fisher diffusion model and extended to the Λ-Wright-Fisher model in [3] and to the Moran model and its deterministic limit in [10] (see also [2]). Moreover, the block counting process of the pruned lookdown ASG is Siegmund dual to the fixation line process (see [26,30] for the neutral case and [3] for the general case). In what follows, if not explicitly mentioned, whenever we talk of the block counting process we refer to the block counting process of the Λ-pruned lookdown ASG. In the Wright-Fisher diffusion and the Moran model the block counting process is positive recurrent for any strictly positive selection parameter. For the Λ-Wright-Fisher model, there is a critical value σ Λ such that the block counting process is positive recurrent for any selection parameter σ ∈ (0, σ Λ ) (see [25] and the discussion in [3, p....

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“…In the diffusion limit [36], one is left with branching events (at rate σ per line), coalescence events (at rate 1 per pair of lines), and mutation events (at rate ϑ ν 0 and ϑ ν 1 per line, respectively). In the deterministic limit [12], one loses the coalescence events, so that only branching (rate s per line) and mutation (rate uν 0 and uν 1 per line) survive. In the diffusion limit, the number of lines in the graph always remains finite (with probability 1), whereas it diverges in the deterministic limit as time tends to infinity (for any s > 0).…”

Section: The Ancestral Selection Graphmentioning

confidence: 99%

“…We are particularly interested in the limit t → ∞, that is, in the type of the ancestor of a random individual sampled from the equilibrium distribution, given that the initial frequency of the beneficial type was x. The probability h(x) that this ancestor is of type 0 is then given by the probability of at least one success in a random number of L ∞ coin tosses, each with success probability x, as summarised in the following theorem, once more from [5,12]. x(1 − x) n a n , where a n := P(L ∞ > n) = p n with p from Proposition 3 (including the limiting cases).…”

Section: Propositionmentioning

confidence: 99%

Lines of Descent Under Selection

Baake

Wakolbinger

2017

J Stat Phys

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We review recent progress on ancestral processes related to mutationselection 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 individual's potential ancestry back until the mutations that define the individual's 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.

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Common ancestor type distribution: A Moran model and its deterministic limit

Cited by 16 publications

References 19 publications

Haldane’s formula in Cannings models: the case of moderately weak selection

Haldane’s formula in Cannings models: the case of moderately weak selection

On the stationary distribution of the block counting process for population models with mutation and selection

Lines of Descent Under Selection

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