Supplementary File C: Extension of the genetic barrier 1 Alexandre Blanckaert *1,2 and Joachim Hermisson 1,3 2 1 Department of Mathematics, University of Vienna, 1090 Vienna, Austria 3 2 Instituto Gulbenkian de Ciˆ encia, 2780-156 Oeiras, Portugal 4 3 Mathematics and Biosciences Group, Max F. Perutz Laboratories, 1030 Vienna, Austria 5 February 28, 2018 6 C1 Extension of the genetic barrier 7 In this section, we investigate the impact of a new mutation, C, appearing at a locus in loose 8 linkage with any component of the previous genetic barrier. 9 C 1.1 Extension of the barrier from one to 2 loci 10 C 1.1.1 A single-locus barrier can be formed 11 We assume that C appears on the island. It interacts either with the island adaptation or 12 the continental one. In the first case, A is the polymorphic locus and in the second one, it is 13 B. This is in fact the same problem with just two different parametrizations. Indeed, with 2 14 loci and 2 alleles, the system is fully parametrized with 2 selection coefficients and one epistatic 15 one, as it generates a different fitness for each haplotype. We therefore focus on the interactions 16 between the B and C loci. Table C1 provides the two parametrizations and the link between 17 them. Equation (C1) gives the minimal condition on the selective advantage of C to strengthen 18 the genetic barrier. In addition, equation (C2) gives the necessary (γ nec ) and sufficient (γ suf ) 19 conditions for the mutation C to invade on the island, with locus B being polymorphic, ie 20 0 ≤ m ≤-β . These two conditions are obtained at m = 0 and m = m b max . If the fitness of 21 haplotype bC is smaller than the fitness of haplotype BC, then B impedes the invasion of C 22 * ablanckaert@igc.gulbenkian.pt 1