A generalization of the informational view of non-random mating: Models with variable population frequencies

Mate choice may generate non-random mating patterns. It has been recently shown that the mating distribution caused by mate choice can be expressed as a gain in information with respect to random mating. In that model, the population phenotypic frequencies were assumed as constant during the breeding season. In the present work such restriction was relaxed to consider different encounter-mating processes in which the population frequencies of available individuals change over mating rounds. As with the constant case, here we describe the change in the mating phenotypes by the flow of information with respect to random mating. This information can be partitioned into sexual selection, sexual isolation and a mixed effect. Likewise, the pairwise statistics for total change, sexual selection and sexual isolation are generalized for variable population frequencies. The new tests had more power for the detection of the effects of non-random mating when the population frequencies vary during the breeding season. The differences in power were high for sexual selection but slight for sexual isolation scenarios. However, the application of the new formulas require the estimation of frequencies at each mating round. Therefore, choosing one or another type of statistics would depend on the biological scenario as well as in the availability and easiness to split the sampling in more than one mating round.


Introduction 21
Mate choice can be defined in general, as any aspect of an organism phenotype that 22 leads to an individual to engage in sexual activity with some partners more likely than 23 with others (Rosenthal, 2017). 24 From a population genetics point of view, mate choice is defined by its effects: it is the 25 observed mating frequency deviation with respect to random mating. So defined, the 26 effects of mate choice can be partitioned into sexual selection and sexual or behavioural 27 isolation (intersexual selection). Sexual selection is defined as the observed change in 28 gene or phenotype frequencies in mated individuals with respect to population 29 frequencies. Sexual isolation is defined as the deviation from random mating in mated 30 individuals (Rolán-Alvarez and Caballero, 2000). 31 In a previous work (Carvajal-Rodríguez, 2018), the mating distribution caused by mate 32 choice was expressed as the gain in information with respect to random mating. In that 33 model, the population phenotypic frequencies were assumed as constant over the 34 breeding season. In the present work, the previous results were extended to include the 35 case where population frequencies vary over different mating rounds during the 36 breeding season. 37 The extension to include variable population frequencies is appropriate for a better 38 description of monogamous species in which pair formation occurs by different mating 39 rounds. In this case, the frequencies of available individuals can change during the same 40 breeding season (Gimelfarb, 1988). 41 We show that, by relaxing the condition of constant population frequencies, a new 42 generalized information partition can be obtained. As before, this information can be 43 different mating rounds. Similarly, for males with phenotype j we have p2rj = n2rj / n2r. 93 During the breeding season, the absolute frequencies of adults will be updated 94 depending on the type of the encounter-mating scheme. Under the individual encounter-95 mating model (Gimelfarb, 1988) there is only one mate by round. In this case, if the 96 new mating includes a female of i-type then in the next round we have n1(r+1)i = n1ri -1 97 and n1r+1 = n1r -1, and similarly for males. In general, if each mating round involves a 98 number Ur of matings then in the next round, we will have n1r+1 = n1r -Ur available 99 females and n2r+1 = n2r -Ur available males, and the corresponding phenotypic classes 100 updated depending on the specific matings that occurred. 101 Now, consider the encounter of a female of type i with a male of type j. The mutual 102 mating propensity mij is the number of matings, after encounter of i with j, in a given 103 environment (Carvajal-Rodríguez, 2018). 104 In mating round r there is a probability qrij = p1ri × p2rj of encounter between i and j. 105 Therefore, the number of matings i × j at this round is qrijmij. The sum over the mating 106 rounds gives the total number of i × j matings and so, the observed relative frequencies 107 of these matings after R rounds can be expressed as 108 . 110 Note that MR defines the total number of matings at the end of the season and can be 111 expressed as the sum of matings at each round so that

113
Also note that the maximum possible number of matings occurring in round r is 114 max(Mr) = max{mij, …} for every i and j. 115 When mating is at random, the mating propensity is constant i.e. mij = m for every pair i, 116 j and so 117 Then, by noting the expected frequencies under random mating as qRîj and by 119 substituting the constant propensity in (1) we obtain 120 Note that if the frequencies are constant over the different mating rounds then we 122 recover the formulas as given in (Carvajal-Rodríguez, 2018). So that with constant 123 frequencies 124 When the population frequencies are variable over the breeding season, the correct 126 pairwise statistic for measuring the deviation from random mating is given by RPTIij = 127 q'Rij/qRij, which is the ratio of the frequency of the observed pair types divided by the 128 expected pair types under random mating, calculated from the population frequencies 129 when these are variable over the same breeding season. If the population frequencies 130 can be considered constant, the RPTIij statistic becomes the pair total index (PTIij) as 131 originally defined in (Rolán-Alvarez and Caballero, 2000). 132 As with the constant frequency case we can measure the change with respect to random 133 mating 134 Then, after R rounds we get the probability of random mating i × j computed from 167 matings as 168 Now, we may express the components of JRPTI in (2) as Δ(qRij) = (q'Rij -P'Rij) + (P'Rij -qRij) = Δ(P'Rij) + Δ(PRij) By substitution of (4) and (5) in (2) we obtain the information partition 178 where 180 We have generalized the information partition of total change due to non-random 184 mating, for models without replacement in which the population frequencies are 185 updated after each mating round. The original partition developed in (Carvajal-186 Rodríguez, 2018), emerges as a particular case from (6) when the population 187 frequencies are constant or the number of mating rounds are reduced to one (R = 1). 188 Concerning the sexual selection information, in the previous work we showed that 190 under the constant frequency assumption, sexual selection can be divided into its female 191 and male components. However, the generalized index (JRPSS), is no longer the exact 192 sum of the information in both sexes. There is a new error term that increases with the 193 number of mating rounds. To appreciate this, let consider the female and male 194 population frequencies averaged over R mating rounds 195 And the averages for the frequencies calculated from mated individuals, i.e. where Δp1Ri = p'1Ri -p1Ri and Δp2Rj = p'2Rj -p2Rj.
Then, the sexual selection information is 206 the error term arises because the logarithm for P'Rij and qRij correspond to a sum of 208 products over R, which makes sense because encounters only happens within the same 209 round i.e., for r ≠ s, the quantities p1ri × p2sj or p1si × p2rj have no biological sense here. 210 For the partition in (7) to be exact (εR = 0) we need to equate the sum of products to the 211 product of the sums, i.e. Σr (p1ri × p2rj) = (Σr p1ri ) × (Σr p2rj) which is not true in general. 212 The same problem occurs with p'1ri and p'2rj.

Random mating 282
As a trait not related to choice, we considered the belly which in this species is yellow, 283 ranging from pale to dark, and it has not known effect on mate choice. Thus, we may 284 classify the matings by the belly color phenotype as pale yellow (y) or dark (d). The 285 initial frequency of the phenotype was assumed to be equally distributed with 286 intermediate frequency in both sexes so p1y = p2y = 0.5 and p1d = p2d = 0.5. 287 Because mating is at random (mutual propensities are equal) we simply performed the 288 Mont Carlo sampling based on the phenotypic frequencies. That is, to incorporate a 289 female into the mating pair, we checked if a uniform U(0,1) random number was lower 290 than p1y and then a yellow female entered to the pair, if not, a dark female did it. For 291 incorporating the male partner, we checked if a new random number was lower than p2y 292 then a yellow male also entered to the pair, if not, a dark male did it. After 1 (individual 293 model) or 10 (mass model) pairs were formed, the frequencies were updated, and the 294 process repeated until a total of 100 pairs was reached. 295 In Table 1, we can see the result of 10 mating rounds under the mass encounter model. 296 During the mating rounds, the population frequencies were recorded jointly with the 297 number of matings for the different classes. At the end of the breeding season (after 298 round 10), the sum of the observed matings over rounds corresponded to the finally 299 observed matings that we would have recorded if we do not distinguish among the 300 different mating rounds. 301 With the recorded data, we can compute JRPTI and JPTI which for the specific case in the 302

Non-random mating: Sexual isolation 316
As a putative assortative mating trait, we considered the length polymorphism of 317 microsatellite DNA related with Oxtr. Thus, we may classify the matings by long (L) 318 versus short (S) length polymorphism. The population size, mating rounds and initial 319 frequencies were defined as in the previous example, so, p1L = p2L = 0.5 and p1S = p2S = 320 0.5. As before, there would be 100 mating rounds under the individual encounter model 321 and at least 10 mating rounds under the mass encounter model. In the latter, there are 10 322 encounters per round but some of them may not succeed, so more than 10 mating 323 rounds would be necessary for completing 100 matings. 324 To model assortative mating, the mutual propensities were different and we defined the 325 homotypic matings having three times more propensity than the heterotypic, so mLL = mSS = 3mLS and mSL = mLS. The specific value of the maximum propensity depended on 327 the encounter model i.e. mLL = 1 for the individual encounter and mLL = 10 for the mass 328 encounter model. 329

Mass encounter 330
In each mating round, 10 random encounters occurred based on the population 331 frequencies. Once the encounter happened, the mating succeeds if a uniform U(0,1) 332 random number was lower than the pair propensity normalized by Mr, if so, the mating 333 was recorded. The frequencies were updated after the 10 encounters. As in the random 334 mating case, the simulation ended when 100 matings pairs were formed. 335 With the recorded frequencies we computed JRPTI and JPTI, and performed the 336 corresponding tests, which for the specific mass encounter example shown in Table 2, 337 were significant for JRPTI (JRPTI =0.13, p-value = 0.004) but non-significant for JPTI (JPTI 338 = 0.04, p-value = 0.256). 339 In general, the power for detection of non-random mating was about 90% for JRPTI and 345 73% for JPTI (over 10,000 runs). In the case of detection of sexual isolation effects, the 346 JRPSI test had a power of 87% while a remaining 3% corresponded to the miss-detection 347 of sexual selection (JRPSS) instead of isolation. However, in the case of JPSI all the non-348 random mating detected cases were identified as sexual isolation (73%). 349 350

Individual encounter 351
In the case of the individual encounter, the differences between the indices for detection 352 of non-random mating were lower, with 81% of detection by JRPTI against 74% by JPTI.
However, in the case of the detection of sexual isolation, the JRPSI test had 0% power 354 while JPSI had 74% (all the non-random mating detected cases). 355 As mentioned before, when we want to distinguish sexual isolation versus sexual 356 selection effects, we need to identify the expectation P'Rij of random mating from the 357 frequencies in mated individuals. This expectation is formalized by equation (3)  Here we considered the same length polymorphism trait so that females with long 385 length polymorphism had more mating vigor. The population size, mating rounds and 386 initial frequencies were defined as before. 387 To model female sexual selection, the mutual propensities were different so that L 388 females had three times more propensity than the S ones, so mLL = mLS = 3mSL and mSS = 389 mSL. The specific value of the maximum propensity depended on the encounter model 390 i.e. mLL = 1 for the individual encounter and mLL = 10 for the mass encounter model. 391 In Table 3 we see a specific mass encounter example, with significant JRPTI (JRPTI 392 =0.123, p-value = 0.007) but non-significant JPTI (JPTI = 0.006, p-value = 0.887). Sexual 393 selection was also detected.