Evolution in structured populations: recent progress and new challenges
University of Oxford - March 18-20, 2024
Conference in honour of Alison Etheridge's contributions to population genetics
Programme
Monday 18th:
12:30-13:30: Registration
13:30-14:15: Steve Evans, "B cell phylodynamics"
14:15-15:00: John Wakeley, "Latent mutations in the ancestries of alleles under selection"
Coffee break
15:30-16:15: Anton Wakolbinger, "From clonal interference to Poissonian interacting trajectories"
16:15-17:00: Charline Smadi: "Quasi-equilibria and click times for Muller’s ratchet with (binary) tournament selection"
17:00-19:00: Drinks reception (Maths Institute)
Tuesday 19th:
9:15-10:00: Peter Koepernik, "The Brownian spatial coalescent"
10:00-10:45: Nick Barton, "Isolation by distance with a seed bank: dispersal in space and time"
Coffee break
11:15-12:00: Julie Tourniaire, "A branching particle system as a model of pushed fronts"
Lunch break
14:00-14:45: Julien Berestycki, "Branching selection particle systems and the selection principle"
14:45-15:30: Nina Gantert, "Survival and complete convergence for a branching annihilating random walk"
Coffee Break
16:00-16:45: Matthias Birkner, "Coalescence times of ancestral lineages in two-dimensional logistic branching random walks"
16:45-17:30: Special session
Conference dinner (Magdalen College)
Wednesday 20th:
9:15-10:00: Christina Goldschmidt: "Globally centred snakes"
10:00-10:45: Félix Foutel-Rodier, "A branching process with recombination"
Coffee Break
11:15-12:00: Maite Wilke Berenguer, "Weak and rare selection"
Lunch break
14:00-14:45: Jere Koskela, ""Bernoulli factories and duality in Wright-Fisher and Allen-Cahn models of population genetics"
14:45-15:30: Anita Winter, "The grapheme-valued Wright-Fisher diffusion with mutation"
Coffee Break
Coffee breaks and the conference dinner are provided by the conference to all registered participants. Suggestions of lunch options close to the Statistics department will also be provided. The drinks reception is kindly provided by the Mathematical Institute.
Talks
Nick Barton (IST Austria): "Isolation by distance with a seed bank: dispersal in space and time"
Sofia Backlund and Nick Barton, IST Austria
In our long-term study of snapdragons (Antirrhinum majus), we find that although pollen and seed mostly move only a few metres, some move much further; we also find that offspring flower several years after their parents, indicating a seed bank. We show how dormancy and leptokurtic dispersal influence genetic relatedness across space and time.
Julien Berestycki (University of Oxford): "Branching selection particle systems and the selection principle"
The N-branching Brownian motion with selection (N-BBM) is a particle system consisting of N independent particles that diffuse as Brownian motions in R, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. It is a very simple model for the evolution of a population under selection that has generated some fascinating research since its introduction by Brunet and Derrida in the early 2000s.
If one recentres the positions by the left most particle, this system has a stationary distribution. I will show that, as N converges to infinity, the stationary empirical measure of the N-particle system converges to the minimal travelling wave of an associated free boundary PDE. This resolves an open question going back at least to works of e.g. Maillard in 2012.
It follows a recent related result by Oliver Tough (with whom this is joint work) establishing a similar selection principle for the so-called Fleming-Viot particle system.
Matthias Birkner (Johannes-Gutenberg Universität, Mainz): "Coalescence times of ancestral lineages in two-dimensional logistic branching random walks"
Consider two ancestral lineages sampled from a system of two-dimensional branching random walks with logistic regulation in the stationary regime. We study asymptotics of their coalescence time for large initial separation and find that it agrees with well known results for a suitably scaled two-dimensional stepping stone model. Our main tool is a joint regeneration construction for the spatial embeddings of the two ancestral lineages.
Based on joint work in progress with Andrej Depperschmidt and Timo Schlüter.
Peter Koepernik (Oxford University): "The Brownian spatial coalescent"
In this talk, I introduce a Markov coalescent process on the continuous d-dimensional torus called the Brownian spatial Ξ-coalescent. It describes the genealogies of the Ξ-Fleming-Viot process, which is the large population size limit of a class of spatial Cannings models in which individuals follow independent Brownian motions.
The Brownian spatial Ξ-coalescent can also be axiomatically characterised by a spatial notion of sampling consistency among a certain class of Markov coalescent processes, similarly to how the well-known Ξ-coalescent can be characterised through sampling consistency within the set of all non-spatial coalescents. An interesting consequence of this characterisation is that all spatial population models in which individuals follow independent Brownian motions have non-Markovian genealogies if the branching mechanism depends non-trivially on the spatial distribution, for example through local regulation.
Steve Evans (UC Berkeley, Departments of Statistics and Mathematics, and Center for Computational Biology): "B cell phylodynamics"
Germinal centers (GC) are micro-anatomical structures that transiently form in lymph nodes during an adaptive immune response. In a GC, B cells—the cells that make antibodies—diversify and compete based on the ability of the antibodies they express to recognize a foreign antigen molecule. As GC B cells proliferate, they undergo targeted mutations in the genomic locus encoding the antibody protein that can modify its antigen binding affinity. Via signaling from other GC cell types, the GC can monitor the binding phenotype of the B cell population it contains and provide survival signals to B cells with the highest-affinity antibodies (i.e., birth and death rates depend on type). Motivated by this mechanism, we develop a mean-field model that couples the birth and death rates in a focal multi-type birth and death process (MTBDP) with d types to the empirical distribution of states—i.e., the mean-field over an exchangeable system of N replica MTBDPs. The empirical distribution process of the N replicas converges to a deterministic probability measure-valued flow as N goes to infinity. In the limit, the focal process evolves as a multi-type birth and death process with rates governed by the probability measure-valued flow which is in turn the flow of one-dimensional marginal distributions of the focal process. Individual focal processes become independent in the limit and this holds out the hope of inference being feasible for this model. This is joint work with William S. DeWitt, Ella Hiesmayr, and Sebastian Hummel.
Félix Foutel-Rodier (University of Oxford): "A branching process with recombination"
Consider a biparental Wright-Fisher model in which all individuals carry a unique haploid chromosome represented by an interval. At each generation, each individual samples two parents from the previous generation and a crossover occurs between their chromosomes with a small probability. As time goes, the chromosome of any focal individual gets fragmented into smaller pieces due to recombination that spread in the population. Under a branching approximation, I will describe the distribution of these fragments at a large time. In particular, I will provide the limit of their lengths, genealogy, and locations on the chromosome. This relies on a method of moments for random trees and on a many-to-few formula to compute these moments which are of general interest.
This is joint work with Emmanuel Schertzer.
Nina Gantert (Technische Universität München): "Survival and complete convergence for a branching annihilating random walk"
We discuss extinction, survival and complete convergence for discrete-time branching annihilating random walk (BARW) on the d-dimensional lattice. The BARW model is given as follows: Each particle produces a Poissonian number of offspring with mean μ which independently move to a uniformly chosen site within a fixed distance R from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated.
Based on joint work with Matthias Birkner, Alice Callegaro, Jiří Černý and Pascal Oswald
Christina Goldschmidt (University of Oxford): "Globally centred snakes"
We prove that any globally centred discrete snake on a Bienaymé tree with critical, finite third moment offspring distribution which satisfies a finite global variance condition plus a natural tail condition on the displacements has the Brownian snake driven by a Brownian excursion as its scaling limit. This improves on earlier results of various authors including Janson and Marckert for the case where the displacements are independent of the offspring numbers, and Marckert for the globally centred, global finite variance case restricted to bounded offspring distributions. Our results imply, in particular, the convergence of a special discrete snake which gives the difference between the height process of such a Bienaymé tree and a constant multiple of its Łukasiewicz path, when rescaled by n^{-1/4}, under optimal tail conditions. Our proof of the convergence of the finite dimensional distributions makes essential use of a discrete line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin; the tightness proof adapts a method deployed by Haas and Miermont in the context of Markov branching trees.
This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.
Jere Koskela (University of Warwick): "Bernoulli factories and duality in Wright-Fisher and Allen-Cahn models of population genetics"
Mathematical models of genetic evolution often come in pairs connected by a so-called duality relation. Seminal examples include the neutral Wright-Fisher diffusion and the Kingman coalescent, as well as the Wright-Fisher diffusion with selection and the ancestral selection graph. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. I will show how the so-called Bernoulli factory, a celebrated tool in simulation-based computing, can be used to derive duality relations for broad classes of genetics models. Concrete examples will include a class of Wright--Fisher diffusions with general drift functions, and Allen-Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To my knowledge, this is the first time a connection has been made between Bernoulli factories and duality in models of population genetics. This is joint work with Dario Spanò and Krzysztof Łatuszyński.
Charline Smadi (INRAE): "Quasi-equilibria and click times for Muller’s ratchet with (binary) tournament selection"
Consider a population of N individuals, each of them carrying a type in N0. 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. I will describe how, 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 click times of the ratchet, 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. 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. In a work of Etheridge et al. [EPW09], the authors obtained approximations of the classical ratchet, allowing to identify a crucial parameter for the speed of accumulation of deleterious mutations. One of their approximations for the dynamics of the best class, after a parameter change, correspond exactly to the dynamics of the best class in the tournament racthet. We will discuss how this could help us studying the classical ratchet.
[EPW09] A. M. Etheridge, P. Pfaffelhuber, and A. Wakolbinger. How often does the ratchet click? Facts, heuristics, asymptotics, page 365–390. London Mathematical Society Lecture Note Series. Cambridge University Press, 2009.
This is a joint and ungoing work with with A. Gonzalez Casanova, J. Lukas Igelbrink and A. Wakolbinger
Julie Tourniaire (IST Austria): "A branching particle system as a model of pushed fronts"
We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate r(x), negative drift −μ, and killed upon reaching 0. More precisely, the particles branch at rate ρ/2 in [0, 1], for some ρ ≥ 1, and at rate 1/2 in (1,+∞). The drift μ = μ(ρ) is chosen in such a way that the system is critical.
This system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies by Birzu, Hallatschek and Korolev on the noisy FKPP equation with Allee effect (or cooperation) suggest the existence of three classes of fluctuating fronts: pulled, semi-pushed and fully-pushed fronts. They also observe a rapid diversity loss in pulled and semi-pushed fronts.
In this talk, I will focus on the pushed regime. I will show that the BBM exhibits the same phase transitions as the noisy FKPP equation. I will then use this particle system to explain how the internal mechanisms driving the invasion shape the genealogy of an expanding population.
John Wakeley (Harvard University): "Latent mutations in the ancestries of alleles under selection"
Authors: Wai-Tong (Louis) Fan and John Wakeley
We consider a single genetic locus with two alleles A1 and A2 in a large haploid population. The locus is subject to selection and two-way, or recurrent, mutation. Assuming the allele frequencies follow a Wright-Fisher diffusion and have reached stationarity, we describe the asymptotic behaviors of the conditional gene genealogy and the latent mutations of a sample with known allele counts, when the count n1 of allele A1 is fixed, and when either or both the sample size n and the selection strength |α| tend to infinity. The model of Barton, Etheridge and Sturm (2004) is central to our analysis. Our study extends previous work under neutrality to the case of non-neutral rare alleles, asserting that when selection is not too strong relative to the sample size, even if it is strongly positive or strongly negative in the usual sense (α → −∞ or α → +∞), the number of latent mutations of the n1 copies of allele A1 follows the same distribution as the number of alleles in the Ewens sampling formula. On the other hand, very strong positive selection relative to the sample size leads to neutral gene genealogies with a single ancient latent mutation. We also present results on large-sample approximations for the sample frequency of A1 from the diffusion.
Anton Wakolbinger (Goethe Universität Frankfurt): "From clonal interference to Poissonian interacting trajectories"
We consider a Moran model with recurrent beneficial mutations of random fitness. These mutations compete for fixation, a phenomenon addressed as clonal interference. The rate of mutations is assumed to be in the so-called Gerrish-Lenski regime, with mutations along individual lineages arriving slightly faster than on the evolutionary timescale. We show that for large population size the population at generic times consiss of one currently resident type together with a random (finite) number of contending mutant families of mesoscopic sizes that compete for fixation. In the large population limit the rescaled logarithmic sizes of the contending mutant families constitute a system of interacting piecewise linear trajectories driven by a Poisson point process. We relate the speed of adaptation in this system to heuristic predictions from the literature.
The talk is based on joint work in progress with Felix Hermann, Adrián González Casanova, Renato Soares dos Santos, and András Tobiás.
Maite Wilke Berenguer (Humboldt Universität, Berlin): "Weak and rare selection"
We analyse a two-type Lambda-Fleming-Viot process with selection: in addition to classic weak selection we model rare (but strong) selective events that lead to additional jumps. We provide a criterion to quantify the impact of different shapes of rare selection on the fate of the weakest allele.
As is common for these models we start from an individual based model where selection is dictated by a random environment. With the ususal scaling, we actually obtain said two-type Lambda-Fleming-Viot process with weak and rare selection and a moment-dual branching-coalescing process describing the genealogy of the population. The jumps describing the rare selection events in the frequency process correspond to correlated behaviour in the dual particle system.
The generator of the rare selective jumps has a representation in terms of the generator of weak selection similar to the representation of the generator of the two-type Lambda-Fleming-Viot process in terms of the generator of the Wright-Fisher diffusion given by Griffiths. This is a key ingredient in the proof of the criterion for extinction, because it allows for a Lypaunov-function-type argument.
This is joint work with A. Gonzalez Casanova, N. Hansen, J.L. Perez, and D. Spano
Anita Winter (Universität Duisburg Essen): "The grapheme-valued Wright-Fisher diffusion with mutation"
Recently models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of sequences of dense graphs. In this talk we consider a Markov chain in the space of finite graphs that comes from the Moran model with mutation - also known as the infinite alleles model. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex space, an adjacency matrix and a sampling measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity.
We show that the grapheme-valued diffusion has a stationary distribution that is linked to the GEM distribution.