SPP 1590 - Workshop
Ancestral lines under recombination and selection
The deterministic selection-recombination equation is a system of ordinary differential equations describing the evolution of the genotype distribution in a population of fixed size under single-crossover recombination and natural selection in the deterministic limit.
Despite its apparent nonlinearity, we obtained an explicit integral representation of the solution by exploiting the underlying genealogical structure. In addition, this genealogical way of thinking allows us to relate the solution of the differential equation (forward in time) via duality to a weighted fragmentation process (backward in time), which can in turn be expressed in terms of a collection of independent Yule processes with restarting.
This is joint work with Ellen Baake and Carolin Herrmann.
Scaling limits of discrete population models with random environment and interactions
We are interested in the large population approximations of discrete time individual based models. In each generation, individuals behave independently conditionally on the environment and the size of the population. In particular, we are focusing on an extension of the approximation of Wright Fisher model with selection to random environment. We take into account both small fluctuations of the selection coefficient and accumulation of dramatic events. The scaled process then converges to Wright Fisher diffusions in a Lévy environment. Our analysis relies on the study of characteristic triplet of semi-martingale and we prove that this latter can be reduced to a rich enough functional space, adapted to the structure of the process. We may also consider another model, extending the classical convergence of Galton Watson processes to sexual reproduction. This latter is a work in progress.
This work is I collaboration with Maria Emilia Cabaellero and Sylvie Méléard.
Local limit of Galton-Watson trees
We shall first recall some results on local limits of critical and sub-critical Galton-Watson trees conditioned to be large. Then we will present some new results on local limit of super-critical Galton-Watson trees conditioned to have at a very large population at generation n. We will exhibit different regime for the possible limits. In the low regime we get the (super-critical) Kesten tree; in the moderate regime we get a continuous family of random trees; in the high regime we get results only for the geometric offspring distribution and for offspring distributions with bounded support (Harris case). The proofs rely on strong ratio theorems for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes.
On some aspects of tree-valued Feller diffusion
Feller's branching diffusion is a well studied stochastic process on [0,∞) which models the evolution of the total mass of a branching population and arises as a limit of appropriately rescaled sequences of nearly critical Galton-Watson processes.
We construct a process for the joint evolution of the mass and the genealogy of the population currently alive in a Feller diffusion model. The state space of this "tree-valued Feller diffusion" is modeled by equivalence classes of extended ultrametric measure spaces. The space is Polish and has a rich semigroup structure for the genealogy valued process. We briefly discuss existence, uniqueness and Feller property of the solutions of the corresponding martingale problem. For the uniqueness we use Feynman-Kac duality with the distance matrix augmented Kingman coalescent (the Log-Laplace duality with a deterministic evolution, cannot be formulated on the ultrametric measure spaces). As time permits, we discuss how some results and representations known for the [0,∞)-valued Feller diffusion can be lifted to the case of tree-valued Feller diffusions. In particular we obtain a representation as a Cox point process of genealogies of single ancestor subfamilies, as well as various representations of the longtime limits arising under different conditioning regimes.
This is joint work in progress with Andreas Greven.
The evolution of ant slavery and parasite host-coevolution in Temnothorax ants
The transition to parasitism is a drastic shift in lifestyle, involving rapid changes in gene function and expression. Slavery evolved several times independently within the ant genus Temnothorax. To investigate the genomic basis of slavemaker evolution, we made use of transcriptome sequence information of three slavemaker and three host species a) to characterize gene expression patterns of slavemaker raiding and host defensive phenotypes b) to construct a phylogeny and phylogenetic network based on over 5000 loci and c) to identify genes with signatures of positive selection. More, but species-specific genes altered their expression with slave-raiding, whereas fewer, but more commonly expressed genes expression characterized the non-raiding phenotype. As expected, recognition genes (e.g. olfactory receptors and cuticular hydrocarbon synthesis) were under positive selection. The lineage-specific evolutionary patterns among both slavemakers and hosts points to convergent rather than parallel trajectories in the evolution of the slavemaker lifestyle.
The geographic mosaic theory of coevolution predicts that species interactions vary between locales. Depending on who leads the coevolutionary arms-race, the effectivity of parasite attack or host defence strategies will explain parasite prevalence. We compared behaviour and brain transcriptomes of Temnothorax longispinosus ant workers when defending their nest against an invading social parasite, the slavemaking ant T. americanus. A full-factorial design allowed us to test whether behaviour and gene expression are linked to parasite pressure on host populations or to the ecological success of parasite populations. Albeit host defences were shown before to co-vary with local parasite pressure, we found parasite success to be much more important. Our chemical and behavioural analyses revealed that parasites from high prevalence sites carry less recognition substances and are less often attacked by hosts and this link was further supported by gene expression analysis. Host-parasite interactions were strongly influenced by social parasite strategies, so that variation in parasite prevalence is determined by parasite traits rather than the efficacy of host defence. Gene functions associated with parasite success indicated strong neuronal responses in hosts, including long-term changes in gene regulation indicating an enduring impact of parasites on host behaviour.
Beta-n-coalescents with exponential growth
In Matuszewski, S., Hildebrandt, M. E., Achaz, G. and Jensen, J. D. (2018). Coalescent processes with skewed o spring distributions and nonequilibrium demography. Genetics, 208(1), 323-338., the genealogy of a sample of n individuals taken from a population described by a modified Moran model (where the maximum of o spring per individual can be much bigger than 2) with an exponentially growing population size was analysed. For population size → ∞, the (rescaled) genealogical trees converge to a Dirac n-coalescent, whose branch lengths are changed by a deterministic function. Dirac n-coalescents are multiple merger coalescents: random trees with n leaves, a Markovian structure and which allow for multifurcation. We expand this approach to a broader class of modified Moran models and other Cannings-type models, e.g. leading to (multiple merger) Beta n-coalescent limit genealogies, whose branch lengths are transformed by a deterministic function. The class B of time-changed Beta coalescents include the standard genealogy model Kingman's n-coalescent with (and without) exponential growth. For several samples of DNA sequence data, we infer the best fitting genealogy model from B using an approximate maximum likelihood approach based on the site frequency spectrum, accounting for misspecification of the ancestral allele.
This is a joint work with S. Matuszewski, J. Jensen (U. Arizona), M. Lapierre, A. Lambert (SMILE Paris), E. Kerdoncu (SMILE Paris) and G. Achaz (SMILE Paris).
Evolving genealogies for branching populations under selection and competition
We give a pathwise construction for the evolution of mass, type distribution and genealogical trees for a branching population under weak fecundity selection and weak type-dependent competition in the infinite population size limit. To this aim, we use the lookdown approach of Donnelly and Kurtz and we build on results from the neutral setting. This is joint work (in progress) with Airam Blancas, Sandra Kliem, Viet Chi Tran and Anton Wakolbinger.
Tree-valued Markov processes and tree lengths under additive selection
When studying evolutionary models of a constant-size population, individuals are related through joint ancestry, which lead to the genealogy relating all individuals of the population. While neutral evolution leads to Kingman's coalescent, we are interested in a scenario of two allelic types under weak selection. It is generally believed that genealogical distances under additive selection are shorter than under neutrality. For a sample of size two, this is shown by Depperschmidt, Greven and Pfaffelhuber (2012), who investigate the Laplace-transform of pairwise genealogical distances in the selective case for small selection coefficients. We extend this result and give an explicit formula for the Laplace-transform of the total tree length of n sampled points. We do so by investigating the equilibrium of the tree-valued Fleming-Viot process with mutation and selection.
The Kingman dynamic on algebraic measure trees
The Kingman dynamic corresponds to the resampling of leaves on binary trees. That is, we pick at random a leaf that we remove and insert to a random external edge. It is first defined as a Markov chain on the space of binary algebraic measure trees with a fixed number of leaves. We show that this chain converges in distribution as the number of leaves goes to infinity, and we give a construction of the dynamic on the space of all binary algebraic trees as a solution of a well-posed martingale problem. The Kingman algebraic measure tree, which can be seen as the Kingman metric measure tree where we forgot about the metric and focused on the tree structure, appears as an invariant distribution for the Kingman dynamic.
This is joint work with Anita Winter.
Altruistic defense traits in structured populations: Many-demes limit in the sparse regime
We discuss spatially structured Wright-Fisher type diffusions modelling the frequency of an altruistic defense trait. These diffusions arise as the limit of spatial Lotka-Volterra type models with a host population and a parasite population, where one type of host individuals (the altruistic type) is more effective in defending against the parasite but has a weak reproductive disadvantage. For the many-demes limit (mean-field approximation) hereof, we obtain a propagation of chaos result in the case where only a few diffusions start outside of an accessible trap. In this "sparse regime", the system converges in distribution to a forest of trees of excursions from the trap.
The genealogical process in the finite Moran model
The genealogy of a finite Moran population at a single point in time can be represented by a tree generated by the Yule Process (a Yule Tree). As the population evolves in time, the tree can be changed accordingly, such that the Moran model is emulated by a Markov Chain on the set of Yule Trees of a given size. I will give a detailed description of this chain and point out some possible implications and applications. Importantly, time can be reversed in this process, which leads to a process similar to the Aldous Chain on cladograms and appears useful when studying topological properties of the genealogy over time. I will also discuss how the genealogical process can be modified to incorporate selection, and how it can be conditioned on the fixation of a given individual. Combining both, we will find a backward-intime formulation significantly different from the neutral one.