Going backward in time with the coalescent and other ancestral structures

Wednesday, June 16 at 11:30am (PDT)
Wednesday, June 16 at 07:30pm (BST)
Thursday, June 17 03:30am (KST)

SMB2021 SMB2021 Follow Wednesday (Thursday) during the "MS14" time block.
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Fernando Cordero (Bielefeld University, Germany), Sebastian Hummel (Bielefeld University, Germany)


Evolution shapes the currently observed patterns in biological populations and their future variation. Understanding the underlying process in its entirety requires to equally understand the effect of each force in isolation as well as their interplay. In many practical situations, evolutionary drivers have to be inferred in retrospect. In this context, the embedding of the corresponding genealogies into a rigorous mathematical framework has been extremely useful for the development of practical inference tools. The analysis of the corresponding ancestral processes – the most famous one being the coalescent – is mathematically challenging, and it has also incited a fair amount of mathematical research. The mini-symposium offers a selection of current theoretical studies of evolutionary effects when they are considered in isolation as well as in their interplay. The four talks have in common to use ancestral processes for the derivation of their results.

Cornelia Pokalyuk

(Goethe University Frankfurt, Institute for Mathematics, Germany)
"Haldane’s formula in Cannings models with moderate selection"
A rule of thumb known as Haldane’s formula states that the probability of fixation for a single beneficial individual with small selective advantage s >0 and offspring variance v in a large population of N individuals is approximately equal to 2s/v. In my presentation I will report on a proof of this asymptotics in the regime of moderate selection, i.e. s_N∼ N^{−b} and b∈(0,1), for a class of Cannings models which allow for a paintbox construction. A forwards as well as a backwards point of view of the paintbox construction turns out to be suitable for the analysis. Via the backwards view we arrive at a time-discrete analogue of the ancestral selection process which is in sampling duality to the wildtype frequency process. In the regime of moderately weak selection (i.e. 1/2< b <1) and under conditions on the paintbox which ensure convergence of the neutral genealogy to Kingman’s coalescent, this sampling duality leads to a proof of Haldane’s formula (EJP 26(4), 2021). In the case of moderately strong selection (0< b <1/2) we make use of the forward construction and approximate the frequency process by Galton-Watson processes (arxiv:2008.02225). The results are joint work with Florin Boenkost, Adrián González Casanova and Anton Wakolbinger.

Maite Wilke Berenguer

(Humboldt-Universität zu Berlin, Germany)
"Can dormancy induce skewed offspring distributions?"
Dormancy naturally occurs in several forms. A classic example is seasonal dormancy: populations that switch into a dormant form during 'winter', only to wake up in 'spring' to resume reproduction. If single individuals wake up significantly earlier than the main population, the additional time for reproduction might be reflected in the offspring numbers at the end of summer, with the early birds' offspring constituting a positive fraction of the population in the following year. We give a simple model for the evolution of such a population and show that for some choices of model parameters the genealogy of the population will be described by a Lambda-coalescent. In particular, the Beta-coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all Lambda-coalescents that can arise in this framework.

Airam Blancas

(Departamento de Estadística, ITAM, Mexico)
"A coalescent model with recombination and population structure"
We introduce a Markov model to describe the backwards evolution of l-linked loci from p-structured populations. More precisely, we define a continuous time Markov chain with jumps at recombination, coalescence, and migration events. We delineate the space states in a way a that it is possible to keep track the ancestral populations at every locus as well as the population location of the lineages. We prove that the state space cardinality of the process is polynomial for admixture populations and provide an analytic expression for 3-loci lineages from without admixture populations.

Dario Spanò

(University of Warwick, England)
"Asymptotic genealogies for interacting particle systems"
We study weighted particle systems in which new generations are resampled from current particles with probabilities proportional to their weights. This covers a broad class of sequential Monte Carlo (SMC) methods, widely-used in applied statistics and cognate disciplines. We consider the genealogical tree embedded into such particle systems, and identify conditions, as well as an appropriate time-scaling, under which they converge to the Kingman n-coalescent in the infinite system size limit in the sense of finite-dimensional distributions.Thus, the tractable n-coalescent can be used to predict the shape and size of SMC genealogies, as we illustrate by characterising the limiting mean and variance of the tree height. SMC genealogies are known to be connected to algorithm performance, so that our results are likely to have applications in the design of new methods as well.

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Virtual conference of the Society for Mathematical Biology, 2021.