8We study the establishment probabilities of locally adapted mutations using a multi-9 type branching process framework. We find a surprisingly simple and intuitive analytical 10 approximation for the establishment probabilities in a symmetric two-deme model under 11 the assumption of weak (positive) selection. This is the first analytical closed-form ap-12 proximation for arbitrary migration rate to appear in the literature. We find that the 13 establishment probability lies between the weak and the strong migration limits if we 14 condition the origin of the mutation to the deme where it is advantageous. This is not 15 the case when we condition the mutation to first occur in a deme where it is disadvanta-16 geous. In this case we find that an intermediate migration rate maximizes the probability 17 of establishment. We extend our results to the cases of multiple demes, two demes with 18 asymmetric rates of gene flow, and asymmetric carrying capacities. The latter case al-19 lows us to illustrate how density regulation can affect establishment probabilities. Finally 20 we use our results to investigate the role of gene flow on the rate of local adaptation 21 and identify cases in which intermediate amounts of gene flow facilitate the rate of local 22 adaptation as compared to two populations without gene flow. 23 1 Keywords: 24 local adaptation; branching process; establishment probabilities 25 Introduction 26 Studying the maintenance of genetic variation under migration-selection balance has a long 27 tradition in population genetics. While most theoretical research on the establishment and 28 maintenance of local adaptation and population divergence has focused on deterministic models 29 considerably less work has been done on the probability of establishment of locally adapted 32 mutations. Even in infinitely large populations, new beneficial mutations experience genetic 33 drift while they are rare, and hence can get lost from the population despite their selective 34 advantage. The probability that a new beneficial mutation evades extinction due to stochas-35 tic fluctuations has been called the invasion probability, establishment probability or fixation 36 probability, depending on the context. In the simplest case of a single panmictic popula-37 tion of infinite size, Haldane's classical result states that the establishment probability of a 38 mutation with time-and frequency-independent selection coefficient s is approximately 2s 39 [ Haldane, 1927]. Since then, Haldane's result has been generalized and extended to several 40 scenarios (see [Patwa and Wahl, 2008] for a review about fixation probabilities of beneficial 41 mutations). 42Traditionally, there are two main approaches to study establishment probabilities: branch-43 ing processes and diffusion approximations. Branching processes often allow for the derivation 44 of simple and intuitive results [Harris, 2002], but are restricted to beneficial mutations in (in-45 finitely large) populations. The diffusion approximation, first used by [Kimura...

Experimental and theoretical studies have highlighted the impact of gene flow on the probability of evolutionary rescue in structured habitats. Mathematical modeling and simulations of evolutionary rescue in spatially or otherwise structured populations showed that intermediate migration rates can often maximize the probability of rescue in gradually or abruptly deteriorating habitats. These theoretical results corroborate the positive effect of gene flow on evolutionary rescue that has been identified in experimental yeast populations. The observations that gene flow can facilitate adaptation are in seeming conflict with traditional population genetics results that show that gene flow usually hampers (local) adaptation. Identifying conditions for when gene flow facilitates survival chances of populations rather than reducing them remains a key unresolved theoretical question. We here present a simple analytically tractable model for evolutionary rescue in a two‐deme model with gene flow. Our main result is a simple condition for when migration facilitates evolutionary rescue, as opposed as no migration. We further investigate the roles of asymmetries in gene flow and/or carrying capacities, and the effects of density regulation and local growth rates on evolutionary rescue.

The SIB Swiss Institute of Bioinformatics (www.isb-sib.ch) provides world-class bioinformatics databases, software tools, services and training to the international life science community in academia and industry. These solutions allow life scientists to turn the exponentially growing amount of data into knowledge. Here, we provide an overview of SIB's resources and competence areas, with a strong focus on curated databases and SIB's most popular and widely used resources. In particular, SIB's Bioinformatics resource portal ExPASy features over 150 resources, including UniProtKB/Swiss-Prot, ENZYME, PROSITE, neXtProt, STRING, UniCarbKB, SugarBindDB, SwissRegulon, EPD, arrayMap, Bgee, SWISS-MODEL Repository, OMA, OrthoDB and other databases, which are briefly described in this article.