Supplementary Material for: The efficacy of natural selection in producing optimal sex ratio adjustments in a fig wasp species Jaco M. Greeff1*, Karina Pentz1 and Marié Warren1 1 Section Genetics, Department of Biochemistry, Genetics and Microbiology, University of Pretoria, Pretoria 0002, South Africa * Corresponding author. jaco.greeff@up.ac.za Proceedings of the Royal Society B DOI: 10.1098/rspb.2020.1377 1. Calculating the probability of sibmating The inbreeding coefficient, F, determines the relatedness of daughters to a mother and in turn F = s/(4-3s), with s being the fraction of females that are sibmated [1]. When sex ratios vary conditionally upon foundress number (n), then sex ratio affects s and hence F, giving 𝑠𝑠 = ∑�λ𝑛𝑛 ⋅𝑛𝑛⋅𝑐𝑐𝑛𝑛 ⋅(1−𝑟𝑟𝑛𝑛 )⋅1�𝑛𝑛� ∑(λ𝑛𝑛 ⋅𝑛𝑛⋅𝑐𝑐𝑛𝑛 ⋅(1−𝑟𝑟𝑛𝑛 )) , (1) with λn = the probability of an n-foundress fig, rn = proportion of sons laid by a female in an n-foundress fig and cn = the clutch size in an n-foundress fig [2]. Herre [3] argued that since rn increases as n increases while cn decreases as n increases, it will be approximately correct to assume that all ncn(1 - rn) are equal to each other so that (1) simplifies to 𝑠𝑠 = = ∑�λ𝑛𝑛 ⋅1�𝑛𝑛� ∑ λ𝑛𝑛 ∑�λ𝑛𝑛 ×1�𝑛𝑛� 1 = 𝑛𝑛 , 1 ℎ (2) with nh = the harmonic mean foundress number. Frank [2] assumed that cn are identical to obtain his equation (6). If we in addition assume that all rn are identical, we obtain 𝑠𝑠 = ∑�λ𝑛𝑛 ⋅𝑛𝑛⋅1�𝑛𝑛� ∑(λ𝑛𝑛 ⋅𝑛𝑛) ∑ λ𝑛𝑛 = ∑(λ 𝑛𝑛 ⋅𝑛𝑛) 1 = ∑(λ 1 , 𝑛𝑛 ⋅𝑛𝑛) = 𝑛𝑛 (3) 𝑎𝑎 with na = the arithmetic mean foundress number. The true value of s must fall between equations (2) and (3), with (2) being too high and (3) being too low. In C. galili nh = 1.624 and na = 2.909. Given that clutches are only constrained for three and more females, the truth should be between these two points. When females have different clutch sizes and compositions the degree of inbreeding increases [2,4]. 2. Derivation of the optimal sex ratio for indiscriminate females We consider figs with two foundresses of which at least one is a C. galili female. Let p be the fraction of figs containing two C. galili females and 1 - p be the fraction of figs containing one C. galili and one C. arabicus female, then a fraction 2p/(1 + p) of C. galili females are in a two-foundress fig and (1 - p)/(1 + p) of C. galili females are actually in a one-foundress fig. The fitness effect of each female is equivalent as they all produce approximately the same number of daughters (figure 2c). If the kin value, K, of each sex is the product of the sex's reproductive value and relatedness to the foundress, then for males Km = ½ and for females Kf 1+3𝐹𝐹 = 2(1+𝐹𝐹) = 1 2−𝑠𝑠 , where F is the inbreeding coefficient and s is the fraction of females that are sibmated [5,6]. Consider a mutant foundress that produce a sex ratio (= fraction of sons) of r2 in a population where females produce sex ratios of 𝑟𝑟̂2 . If all females lay the same clutch size, then the fitness per egg can be written as 2𝑝𝑝 2 − 𝑟𝑟̂2 − 𝑟𝑟2 1 − 𝑟𝑟2 1 − 𝑝𝑝 �𝐾𝐾𝑓𝑓 (1 − 𝑟𝑟2 ) + 𝐾𝐾𝑚𝑚 𝑟𝑟2 � �� + �𝐾𝐾𝑓𝑓 (1 − 𝑟𝑟2 ) + 𝐾𝐾𝑚𝑚 𝑟𝑟2 � �� 1 + 𝑝𝑝 𝑟𝑟̂2 + 𝑟𝑟2 𝑟𝑟2 1 + 𝑝𝑝 𝑤𝑤 = and the ESS sex ratio, r2*, can be found by setting the partial derivative of w with respect to r2 equal to 0 𝜕𝜕𝑤𝑤 � 𝜕𝜕𝑟𝑟2 𝑟𝑟̂ =0 2 =𝑟𝑟2 =𝑟𝑟2 ∗ and solving for r2* gives, 1 2𝑝𝑝 r2*= 2 𝐾𝐾𝑚𝑚 1+𝑝𝑝 𝐾𝐾𝑓𝑓 +𝐾𝐾𝑚𝑚 1 2𝑝𝑝 =2 1+𝐹𝐹 1+𝑝𝑝 2(1+2𝐹𝐹) 1 2𝑝𝑝 2−𝑠𝑠 =2 1+𝑝𝑝 4−𝑠𝑠 3. Supplementary Tables Table S1. A comparison of the frequencies of wasp abundance as numbers of figs and number of wasps for figs with and without Ceratosolen galili. C. galili C. galili present absent Presence of C. % % arabicus and S. number % of excluding number % of number of sycomori of figs figs neither of wasps wasps of figs figs Neither 158 48 492 51 C. arabicus 105 32 62 267 28 594 62 S. sycomori 46 14 26 129 13 231 24 Both 22 6 12 76 8 132 14 Total 331 963 957 Table S2. The number of 263 figs that did not contain S. sycomori that contain a specific number of foundresses of each Ceratosolen species. Percentages rounded to nearest 1 below numbers in italics; left open in case it equals 0. Bold type indicates figs with less than three foundresses. The total percentage of figs containing a certain number of C. galili females are given in the final row. 2 22 8 14 5 3 1 1 3 25 10 6 2 1 0 0 1 0 0 0 0 0 0 5 1 65 25 30 11 13 5 7 3 2 1 1 Number of foundresses C. galili 4 5 6 7 8 9 10 16 8 5 4 1 4 5 6 3 2 2 2 2 3 3 4 1 1 0 0 1 1 2 5 0 1 0 0 0 1 2 0 0 1 0 1 0 0 1 0 1 0 0 0 0 6 1 0 0 0 0 0 0 Total % 46 16 12 10 4 5 2 C. arabicus 0 1 2 3 4 11 1 12 0 17 1 19 0 23 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 Table S3. The number of 758 C. galili females found in 263 figs without S. sycomori that contain a specific number of foundresses of each Ceratosolen species. Percentages rounded to nearest 1 below numbers in italics; left open in case it equals 0. Bold type indicates figs with less than three foundresses. The total percentage of females in figs containing a certain number of C. galili females are given in the final row. Number of foundresses C. galili C. arabicus 1 2 3 4 5 6 7 8 9 10 11 12 17 19 23 0 65 44 75 64 40 30 28 8 36 50 11 0 17 0 23 5 4 4 1 5 7 1 2 3 9 6 10 8 1 8 0 0 0 0 0 0 0 30 28 18 12 15 24 7 4 2 2 2 3 1 1 4 2 13 6 3 20 0 6 0 0 0 10 0 12 0 0 0 2 1 3 1 1 2 3 7 2 0 0 0 6 0 8 0 0 0 0 0 0 0 1 1 1 4 2 0 3 0 0 0 0 0 0 0 0 0 0 19 0 3 5 1 2 0 4 0 0 0 0 0 0 0 0 0 0 0 1 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total % 16 11 13 13 7 9 5 3 5 8 1 2 2 3 Table S4. Three figs with unusually high sex ratios that were discarded from analyses. foundress sons daughters sex ratio number 2 Foundresses 2 254 148 0.63 treatment Mixed species 1 52 45 0.54 Mixed species 1 150 18 0.89 3 Table S5. Summary of the mean C. galili clutch composition for each treatment. Treatment 1 Foundress 2 Foundresses 3 Foundresses Mixed species Sex ratio 0.14 0.18 0.26 0.23 Clutch size 202.7 232.6 153.3 184.4 Number of Daughters 175.4 190.3 113.6 142.1 Number of Sons 27.3 42.3 39.7 42.3 References 1. Li CC. 1976 A First Course in Population Genetics. Pacific Grove, CA: Boxwood. 2. Frank SA. 1985 Hierarchical selection theory and sex ratios. II on applying the theory, and a test with fig wasps. Evolution 39, 949–964. 3. Herre EA. 1985 Sex ratio adjustment in fig wasps. Science 228, 896–898. 4. Molbo D, Machado CA, Herre EA, Keller L. 2004 Inbreeding and population structure in two pairs of cryptic fig wasp species. Mol. Ecol. 13, 1613–1623. (doi:10.1111/j.1365-294X.2004.02158.x) 5. Hamilton WD. 1972 Altruism and related phenomena, mainly in social insects. Annu. Rev. Ecol. Syst. 3, 193–232. 6. Suzuki Y, Iwasa Y. 1980 A sex ratio theory of gregarious parasitoids. Res. Popul. Ecol. 22, 366–382. (doi:10.1007/BF02530857)