bioRxiv preprint doi: https://doi.org/10.1101/190249; this version posted September 18, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. bioRxiv preprint Coalescent theory of migration network motifs NICOLAS ALCALA∗, 1, AMY GOLDBERG∗, UMA RAMAKRISHNAN† and NOAH A. ROSENBERG∗ ∗Department of Biology, Stanford University, Stanford, CA 94305-5020, USA, †National Centre for Biological Sciences, Bangalore, Karnataka 560065, India ABSTRACT Natural populations display a variety of spatial arrangements, each potentially with a distinctive impact on genetic diversity and genetic differentiation among subpopulations. Although the spatial arrangement of populations can lead to intricate migration networks, theoretical developments have focused mainly on a small subset of such networks, emphasizing the island-migration and stepping-stone models. In this study, we investigate all small network motifs: the set of all possible migration networks among populations subdivided into at most four subpopulations. For each motif, we use coalescent theory to derive expectations for three quantities that describe genetic variation: nucleotide diversity, FST, and half-time to equilibrium diversity. We describe the impact of network properties on these quantities, finding that motifs with a large mean node degree have the largest nucleotide diversity and the longest time to equilibrium, whereas motifs with small density have the largest FST. In addition, we show that the motifs whose pattern of variation is most strongly influenced by loss of a connection or a subpopulation are those that can be split easily into several disconnected components. We illustrate our results using two example datasets—sky island birds of genus Brachypteryx and Indian tigers—identifying disturbance scenarios that produce the greatest reduction in genetic diversity; for tigers, we also compare the benefits of two assisted gene flow scenarios. Our results have consequences for understanding the effect of geography on genetic diversity and for designing strategies to alter population migration networks to maximize genetic variation in the context of conservation of endangered species. KEYWORDS coalescent theory; genetic differentiation; network; population structure 1 such predictions enable descriptions of the impact of migration 11 2 OALESCENT theory is a powerful tool to predict patterns of as one of the main evolutionary forces influencing allele frequen- 12 3 C genetic variation in models of population structure, and cies. In molecular ecology, they help evaluate the consequences 13 4 many studies have investigated the predictions of coalescent of abiotic factors such as geographic barriers, and biotic factors 14 5 models about genetic variation under a variety of different as- such as assortative mating, on levels of genetic diversity and 15 6 sumptions about the genetic structure of populations (Donnelly genetic differentiation. In conservation genetics, they can be 16 7 and Tavaré 1995; Fu and Li 1999; Rosenberg and Nordborg 2002). used to quantify the impact of past and future disturbance, as 17 8 Correctly predicting the effect of connectivity patterns on the well as to predict the outcome of management initiatives. 18 9 expected amount of nucleotide diversity and genetic differentia- 10 tion is important in a range of settings. In population genetics, The two most frequently examined models of population 19 Copyright © 2017 by the Authors structure are the island-migration and stepping-stone models. 20 In the island model, individuals can migrate from any subpopu- 21 Manuscript compiled: Monday 18th September, 2017 1 Affiliation correspondence address and email for the corresponding author. lation to any other subpopulation, all with the same rate (Wright 22 bioRxiv September 2017 1 bioRxiv preprint doi: https://doi.org/10.1101/190249; this version posted September 18, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 1 x s 44 e f arbitrary connectivity patterns. The number of patterns grows t i r t e o v Common models - 45 m rapidly with the number of subpopulations, however, and the 1 island model x 2 3 circular stepping-stone model 46 s comprehensive description of networks of arbitrary size is a e f t i linear stepping-stone model r t e o single population model v 47 - combinatorial challenge. Because small network motifs are the m 2 x 4 5 6 7 “building blocks” of large networks (Milo et al. 2002), the deriva- 48 s e f t i r t e o 49 v tion of their features can be a step in predicting properties of - m 3 12 complex connectivity networks. We thus characterize coales- 50 cent quantities under all possible motifs describing the spatial 51 52 8 9 10 13 15 17 18 arrangements of up to four subpopulations. We first derive x s e f t i r t the expected coalescence times between pairs of lineages sam- 53 e o v - m 4 pled in each of the subpopulations and pairs sampled from 54 11 14 16 different subpopulations. For each subpopulation, we compute 55 three population-genetic quantities: expected nucleotide diver- 56 Figure 1 All possible network motifs for sets of at most 4 ver- sity, expected FST values between pairs of subpopulations, and 57 tices. Purple motif backgrounds highlight motifs that follow half-time to equilibrium after a perturbation. For each motif, 58 standard models, island or stepping-stone or both. Note that we compute four network statistics—number of vertices, num- 59 we take the term “motif” to indicate a specific small undi- ber of edges, mean degree, and density—correlating them with 60 rected graph (rather than a small directed or undirected sub- the population-genetic quantities. Finally, we investigate the 61 graph statistically overrepresented in large empirical net- nucleotide diversity lost after a connectivity loss or a subpopula- 62 works, as in many applications). tion loss—a transition between motifs. We interpret the results 63 in relation to problems in conservation genetics, considering two 64 case studies, birds of genus Brachypteryx and Indian tigers. For 65 23 1951). In the stepping-stone model, individuals can only migrate both examples, we (i) consider genetic data in a network motif 66 24 to neighboring subpopulations (Kimura 1953; Maruyama 1970). framework, and (ii) evaluate the potential impacts of connectiv- 67 25 Stepping-stone models can represent multiple spatial arrange- ity change on population-genetic variation. 68 26 ments. Under the circular stepping-stone model, subpopulations 27 are arranged in a circle, so that all individuals can migrate to Model 69 28 exactly two subpopulations. 29 Although the island and stepping-stone models can accom- Population connectivity 70 30 modate a variety of patterns of connectivity among subpopu- We consider K haploid or diploid subpopulations of equal size 71 31 lations, they represent only some of the possible patterns, or N individuals. We denote by Mij the scaled backward migration 72 32 network “motifs.” Indeed, these models account for only 7 of rate, representing twice the number of lineages per generation 73 33 18 motifs possible for sets of one to four subpopulations (Fig- from subpopulation i that originate from subpopulation j. Thus, 74 34 ure 1). Numbering motifs by the classification from Read and Mij = 2Nmij for haploids and 4Nmij for diploids, where mij is 75 35 Wilson (2005, p. 8), motif 1 corresponds to the panmictic pop- the probability for a lineage of subpopulation i to originate from 76 36 ulation model, motif 18 to the island model, motifs 6, 14, and subpopulation j in the previous generation. The total scaled 77 37 16 to stepping-stone models, and motifs 3 and 7 to both island migration rate of subpopulation i, or twice the scaled number 78 K 38 and stepping-stone models. Although tools of coalescent theory of lineages that originate elsewhere, is Mi = ∑j=1,j6=i Mij. We 79 39 to study arbitrary migration models are available (Wilkinson- further assume that the numbers of migrants from each non- 80 40 Herbots 1998), to our knowledge, patterns of variation expected isolated subpopulation are all equal to M, so that for two non- 81 41 from the remaining 11 motifs have not been described. isolated subpopulations i and j, Mi = Mj = M. Time is a 82 42 An objective in the study of spatial arrangements of popu- continuous variable t, scaled in units of the size of a single 83 43 lations is to examine the properties of networks representing subpopulation (N for haploids, 2N for diploids). We focus on 84 2 Alcala et al. bioRxiv preprint doi: https://doi.org/10.1101/190249; this version posted September 18, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 85 cases with 1 ≤ K ≤ 4, and we consider all possible connectivity State (11) State (12) State (13) 2 2 2 86 patterns between subpopulations, where each pattern represents ● ● ● 1 3 1 3 1 3 87 a distinct graph on at most four vertices (Figure 1). ● ● ● ● ● ● State (22) State (23) State (33) State (00) 2 2 2 88 Coalescence ● ● ● ● 1 3 1 3 1 3 89 We consider the fate of two gene lineages drawn from a specific ● ● ● ● ● ● ● ● 90 pair of subpopulations, either the same or different subpopula- Figure 2 Schematic representation of all states for two lineage 91 tions.