Conference Agenda

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation. To depict repeating changes of the environment, all of the model parameters vary over time as piecewise constant and periodic functions, on an intermediate time scale between those of stabilization of the resident population (fast) and exponential growth of mutants (slow). This can biologically interpreted as the influence of seasons or the variation of drug concentration during medical treatment. The typical evolutionary behaviour can be studied by looking at limits of large populations and rare mutations. Analysing the crossing of fitness valleys in a changing environment leads to various interesting phenomena on different time scales which depend on the length of the valley. By carefully examining the influences of the changing environment carefully on each time scale, we are able to determine the effective growth rates of emergent mutants and their ability to invade the resident population. Eventually, we investigate the special situation of pit stops, where single intermediate mutants within the valley have phases of positive fitness and can thus grow to a diverging size before dying out again. This significantly accellerates the traversal of the valley and leads to a interesting new time scale. This is joint work with Anna Kraut.

Consider a population with two types of (one-dimensional) individuals, where type is interpreted as size (length): large individuals are of size $1$ and small individuals are of fixed size $\vartheta$, $\vartheta \in (0,1)$. Each generation has an available space of length $R$. To form a new generation, individuals from the current generation are sampled one by one, and if there is at least some available space, they reproduce and their offspring are added to the new generation. The probability of sampling an individual whose offspring is small is given by $\mu^R(x)$, where $x$ is the proportion of small individuals in the current generation. We call this stochastic model in discrete time the two-size Wright--Fisher model. The function $\mu^R$ can be used to model mutation and/or various forms of frequency-dependent selection. Denoting by $(X_t^R)_{t \geq 0}$ the frequency process of small individuals, we show convergence on the evolutionary time scale $Rt$ to the solution of the SDE $$\mathrm{d} X_t = \big(-(1-\vartheta) X_t(1-X_t)+\mu(X_t)\big)\mathrm{d} t + \sqrt{X_t(1-X_t)(1-(1-\vartheta) X_t)}\, \mathrm{d} B_t,$$ where $\mu(x)=\lim_{R \to \infty} R(\mu^R(x)-x)$, and $B$ is a standard Brownian motion.

To prove this statement, the dynamics inside one generation of the model are considered as a renewal process, with the population size as the first-passage time $\tau(R)$ above level $R$. Methods from (uniform) renewal theory are applied and in particular a uniform version of Blackwell's renewal theorem (for binary, non-arithmetic random variables) is established.

In order to understand the underlying genealogical picture of the model, different concepts of duality are used.

Multi-type models have recently experienced renewed interest in the stochastic modeling of evolution. This is partially due to their mathematical analysis often being more challenging than their single-type counterparts; an example of this is the site-frequency spectrum of a colony-based population with moderate migration.

In this talk, we model the genealogy of such a population via a multi-type coalescent starting with $N(K)$ colored singletons with $d \geq 2$ possible colors (colonies). The process is described by a continuous-time Markov chain with values on the colored partitions of the colored integers in $\{1, \ldots, N(K)\}$; blocks of the same color coalesce at rate $1$, while they are also allowed to change color at a rate proportional to $K$ (migration).

Given this setting, we study the asymptotic behavior, as $K\to\infty$ at small times, of the vector of empirical measures, whose $i$-th component keeps track of the blocks of color $i$ at time $t$ and of the initial coloring of the integers composing each of these blocks. We show that, in the proper time-space scaling, it converges to a multi-type branching process (case $N(K) \sim K$) or a multi-type Feller diffusion (case $N(K) \gg K$). Using this result, we derive an applicable representation of the site-frequency spectrum.

This is joint work with Fernando Cordero and Emmanuel Schertzer.

The Admixture Model describes the probability that an individual $i$ possesses zero, one, or two copies of allele $j \in \{1, \ldots, J\}$ at marker $m$, based on the frequencies $p_{k,j,m}$ of allele $j$ in population $k$ at marker $m$ and the ancestry proportions $q_{i,k}$​, which represent the fraction of individual $i$'s genome inherited from population $k$. A key extension of this model is the Recent Admixture Model, which generalizes the framework by accounting for the ancestry of the individual’s parents, rather than just the individual. This allows for a more refined representation of genetic inheritance in recently admixed populations. We denote the data of individual $i$ at marker $m$ and allele $j$ by $X_{i,j,m}$ and assume (in the Admixture Model)

$$X_{i, \cdot,m} \sim Multi(2, (\langle q_{i, \cdot}, p_{\cdot, j,m}\rangle)_{j = 1, \ldots, J}).$$

In the Recent Admixture Model, we have the same assumption about the distribution of the data, but for the parents and not for the individual.

In both the Admixture and Recent Admixture Models, two settings are typically considered: the supervised setting, where the allele frequencies $p_{k,j,m}$ are known a priori, and the unsupervised setting, where these frequencies must be estimated from the data. This study focuses on the theoretical properties of the maximum likelihood estimators (MLEs) for both models, in both contexts. Specifically, it examines the consistency and central limit behavior of these estimators, which are important for understanding the reliability and accuracy of ancestry inference. The MLEs in these models can be efficiently computed using popular algorithms such as STRUCTURE or ADMIXTURE, which are widely used in population genetics.

Since the standard theory from Hoadley concerning the consistency of MLEs for not identically distributed random variables is not directly applicable to our case, we changed his proof to show consistency in the supervised setting under weak constraints. In the unsupervised setting, the MLEs are usually non-unique (see Heinzel, Baumdicker, Pfaffelhuber "Revealing the range of maximum likelihood estimates in the admixture model.", bioRxiv). Hence, we name constraints that consequences, even in this setting, the consistency of the MLEs. In addition, we have proven that the constraints imposed in all cases are indeed necessary. Our results on the consistency of the estimators form the basis for establishing central limit theorems in both the supervised and unsupervised settings. A key aspect of our analysis is the comparison of the asymptotic behavior of the estimators when the true parameter lies on the boundary of the parameter space versus when it is located in the interior. From a mathematical standpoint, the boundary case is particularly intriguing. In this case, we demonstrate that the asymptotic variance of the estimators is much smaller compared to the case where the parameter space is open, highlighting the impact of boundary constraints on the efficiency of the estimators.

We apply our theoretical results to simulated data and to data from the 1000 Genomes Project, i.e. we see that the central limit results in the (Recent) Admixture Model can be used to estimate the variance of the estimator even for a small number of markers and individuals well. Finally, we demonstrate the usefulness of our results in an application settings, e.g. in the forensic genetics to select genetic markers and to find an optimal marker set.