The ancestral selection graph for a Λ-asymmetric Moran model.

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ-asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.

Segregational instability of multicopy plasmids: A population genetics approach.

Plasmids are extra-chromosomal genetic elements that encode a wide variety of phenotypes and can be maintained in bacterial populations through vertical and horizontal transmission, thus increasing bacterial adaptation to hostile environmental conditions like those imposed by antimicrobial substances. To circumvent the segregational instability resulting from randomly distributing plasmids between daughter cells upon division, nontransmissible plasmids tend to be carried in multiple copies per cell, with the added benefit of exhibiting increased gene dosage and resistance levels. But carrying multiple copies also results in a high metabolic burden to the bacterial host, therefore reducing the overall fitness of the population. This trade-off poses an existential question for plasmids: What is the optimal plasmid copy number? In this manuscript, we address this question by postulating and analyzing a population genetics model to evaluate the interaction between selective pressure, the number of plasmid copies carried by each cell, and the metabolic burden associated with plasmid bearing in the absence of selection for plasmid-encoded traits. Parameter values of the model were estimated experimentally using Escherichia coli K12 carrying a multicopy plasmid encoding for a fluorescent protein and bla TEM-1, a gene conferring resistance to ß-lactam antibiotics. By numerically determining the optimal plasmid copy number for constant and fluctuating selection regimes, we show that plasmid copy number is a highly optimized evolutionary trait that depends on the rate of environmental fluctuation and balances the benefit between increased stability in the absence of selection with the burden associated with carrying multiple copies of the plasmid.

The role of connectivity on COVID-19 preventive approaches.

Preventive and modeling approaches to address the COVID-19 pandemic have been primarily based on the age or occupation, and often disregard the importance of heterogeneity in population contact structure and individual connectivity. To address this gap, we developed models based on Erdos-Rényi and a power law degree distribution that first incorporate the role of heterogeneity and connectivity and then can be expanded to make assumptions about demographic characteristics. Results demonstrate that variations in the number of connections of individuals within a population modify the impact of public health interventions such as lockdown or vaccination approaches. We conclude that the most effective strategy will vary depending on the underlying contact structure of individuals within a population and on timing of the interventions.

Haldane's formula in Cannings models: the case of moderately strong selection.

For a class of Cannings models we prove Haldane's formula, [Formula: see text], for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for [Formula: see text] and [Formula: see text]. Here, [Formula: see text] is the selective advantage of an individual carrying the beneficial type, and [Formula: see text] is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele's frequency process with slightly supercritical Galton-Watson processes in the early phase of fixation.

Separation of timescales for the seed bank diffusion and its jump-diffusion limit.

We investigate scaling limits of the seed bank model when migration (to and from the seed bank) is 'slow' compared to reproduction. This is motivated by models for bacterial dormancy, where periods of dormancy can be orders of magnitude larger than reproductive times. Speeding up time, we encounter a separation of timescales phenomenon which leads to mathematically interesting observations, in particular providing a prototypical example where the scaling limit of a continuous diffusion will be a jump diffusion. For this situation, standard convergence results typically fail. While such a situation could in principle be attacked by the sophisticated analytical scheme of Kurtz (J Funct Anal 12:55-67, 1973), this will require significant technical efforts. Instead, in our situation, we are able to identify and explicitly characterise a well-defined limit via duality in a surprisingly non-technical way. Indeed, we show that moment duality is in a suitable sense stable under passage to the limit and allows a direct and intuitive identification of the limiting semi-group while at the same time providing a probabilistic interpretation of the model. We also obtain a general convergence strategy for continuous-time Markov chains in a separation of timescales regime, which is of independent interest.

The Wright-Fisher model with efficiency.

In this article, we propose a Wright-Fisher model with two types of individuals: the inefficient individuals, those who need more resources to reproduce and can have a higher growth rate, and the efficient individuals. In this model, the total amount of resource N is fixed, and the population size varies randomly depending on the number of efficient individuals. We show that, as N increases, the frequency process of efficient individuals converges to a diffusion which is a generalization of the Wright-Fisher diffusion with selection. The genealogy of this model is given by a branching-coalescing process that we call the Ancestral Selection/Efficiency Graph, and that is an extension of the Ancestral Selection Graph (Krone and Neuhauser, 1997a,b). The main contribution of this paper is that, in evolving populations, inefficiency can arise as a promoter of selective advantage and not necessarily as a trade-off.

Structural properties of the seed bank and the two island diffusion.

We investigate various aspects of the (biallelic) Wright-Fisher diffusion with seed bank in conjunction with and contrast to the two-island model analysed e.g. in Kermany et al. (Theor Popul Biol 74(3):226-232, 2008) and Nath and Griffiths (J Math Biol 31(8):841-851, 1993), including moments, stationary distribution and reversibility, for which our main tool is duality. Further, we show that the Wright-Fisher diffusion with seed bank can be reformulated as a one-dimensional stochastic delay differential equation, providing an elegant interpretation of the age structure in the seed bank also forward in time in the spirit of Kaj et al. (J Appl Probab 38(2):285-300, 2001). We also provide a complete boundary classification for this two-dimensional SDE using martingale-based reasoning known as McKean's argument.

Modelling and simulating Lenski's long-term evolution experiment.

We revisit the model by Wiser et al. (2013), which describes how the mean fitness increases over time due to beneficial mutations in Lenski's long-term evolution experiment. We develop the model further both conceptually and mathematically. Conceptually, we describe the experiment with the help of a Cannings model with mutation and selection, where the latter includes diminishing returns epistasis. The analysis sheds light on the growth dynamics within every single day and reveals a runtime effect, that is, the shortening of the daily growth period with increasing fitness; and it allows to clarify the contribution of epistasis to the mean fitness curve. Mathematically, we explain rigorous results in terms of a law of large numbers (in the limit of infinite population size and for a certain asymptotic parameter regime), and present approximations based on heuristics and supported by simulations for finite populations.

Genetic Variability Under the Seedbank Coalescent.

We analyze patterns of genetic variability of populations in the presence of a large seedbank with the help of a new coalescent structure called the seedbank coalescent. This ancestral process appears naturally as a scaling limit of the genealogy of large populations that sustain seedbanks, if the seedbank size and individual dormancy times are of the same order as those of the active population. Mutations appear as Poisson processes on the active lineages and potentially at reduced rate also on the dormant lineages. The presence of "dormant" lineages leads to qualitatively altered times to the most recent common ancestor and nonclassical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with a seedbank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seedbank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Estimates (obtained by simulations) of the distributions of commonly employed distance statistics, in the presence and absence of a seedbank, are compared. The effect of a seedbank on the expected site-frequency spectrum is also investigated using simulations. Our results indicate that the presence of a large seedbank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect from genetic data the presence of a large seedbank in natural populations.

Strong seed-bank effects in bacterial evolution.

Bacterial genomes are mosaics with fragments showing distinct phylogenetic origins or even being unrelated to any other genetic information (ORFan genes). Thus the analysis of bacterial population genetics is in large part a collection of explanations for anomalies in relation to classical population genetic models such as the Wright-Fisher model and the Kingman coalescent that do not adequately describe bacterial population genetics, genomics or evolution. The concept of "species" as an evolutionary coherent biological group that is genetically isolated and shares genetic information through recombination among its members cannot be applied to any bacterial group. Recently, a simple probabilistic model considering the role of strong seed-bank effects in population genetics has been proposed by Blath et al. This model suggests the existence of a genetic pool with high diversity that is not subject to classical selection and extinction. We reason that certain bacterial population genetics anomalies could be explained by the prevalence of strong seed-bank effects among bacteria. To address this possibility we analyzed the genome of the bacterium Azotobacter vinelandii and show that genes that code for functions that are essential for the bacterium biology do not have a relation of ancestry with closely related bacteria, or are ORFan genes. The existence of essential genes that are not inherited from the most recent ancestor cannot be explained by classical population genetics models and is irreconcilable with the current view of genes acquired by horizontal transfer as being accessory or adaptive.