Multiple Mating But Not Recombination Causes Quantitative Increase in Offspring Genetic Diversity for Varying Genetic Architectures

Explaining the evolution of sex and recombination is particularly intriguing for some species of eusocial insects because they display exceptionally high mating frequencies and genomic recombination rates. Explanations for both phenomena are based on the notion that both increase colony genetic diversity, with demonstrated benefits for colony disease resistance and division of labor. However, the relative contributions of mating number and recombination rate to colony genetic diversity have never been simultaneously assessed. Our study simulates colonies, assuming different mating numbers, recombination rates, and genetic architectures, to assess their worker genotypic diversity. The number of loci has a strong negative effect on genotypic diversity when the allelic effects are inversely scaled to locus number. In contrast, dominance, epistasis, lethal effects, or limiting the allelic diversity at each locus does not significantly affect the model outcomes. Mating number increases colony genotypic variance and lowers variation among colonies with quickly diminishing returns. Genomic recombination rate does not affect intra- and inter-colonial genotypic variance, regardless of mating frequency and genetic architecture. Recombination slightly increases the genotypic range of colonies and more strongly the number of workers with unique allele combinations across all loci. Overall, our study contradicts the argument that the exceptionally high recombination rates cause a quantitative increase in offspring genotypic diversity across one generation. Alternative explanations for the evolution of high recombination rates in social insects are therefore needed. Short-term benefits are central to most explanations of the evolution of multiple mating and high recombination rates in social insects but our results also apply to other species.


Notation
To make the presentation for the derivation of the theoretical average value of S 2 G more compact, we will use the following notation: X i iid ∼ N(0, 1) i = 1, . . . , n The random variables X 1 , . . . , X n are independent of each other and all follow a standard normal distribution.
Y ∼ Bin(n, p) . . . . . . . . . . . . The random variable Y has a binomial distribution based on n trials and a probability of success p.

Modeling a Colony of Worker Bees
In order to assess the variability of a colony based on the number of paternal lines, we are considering a simple model, where the genotypic values at each locus are standard normal random variables. In particular, when considering L loci and M paternal lines, the Queen of the colony is modeled as where Q a (l) where D m (l) iid ∼ N(0, 1), m = 1, . . . , M, l = 1, . . . , L represents the genotypic value on locus l for mate m. With this, and disregarding recombination, any worker-offspring is modeled as follows: Each worker receives one allele from the Queen (without loss of generality we assume this to be the first allele) and one allele from one of the mates (we assume this to be the second allele). Since the mates are haploid, a mate will be selected as random to be the father of a specific worker and the worker inherits all genotypic values from that particular mate. As for the allele being inherited from the Queen, since she is diploid (as the workers) the worker could inherit either the genotypic value from the first or the second allele for any locus. Since we are also not considering recombination at the moment, all loci in the Queen can be seen as independent of each other, meaning that the inherited value on locus l has no influence on the inherited value at locus l + 1. Formally, a worker can represented where a(l) = 1, 2 and m = 1, . . . , M . Note that any of the a(l) = 1 with probability 0.5 and all values for m are equally likely with probability 1 M .

Assessing Genotypic Variability of One Colony
Now, having a colony of n workers generated this way, we can compute the genetic variability of the colony by determining a genotypic value for each worker, G, and then compute the sample variance of all the G's. Given the model from above, the genotypic value of a single worker is computed as For the following, note that although the Q a(l) (l)-and D m (l)-values are generated from a standard normal distribution, this does not necessarily imply that the G-values themselves are normally distributed. In particular, for a given colony with a Queen and m mates, there are only finitely many values for G possible (2 L M to be exact). Also, note that the The average of all the genotypic values of a colony is then given bȳ Now, since there are -as mentioned above -only finitely many genotypic values for the Q a i (l) (l)'s (2 at each locus) and the D m i 's (M total) the expression above can be written in their terms. Consider firstQ(l): In a colony of n workers, Y 1 (l) workers will have a(l) = 1 and n − Y 1 (l) workers will have a(l) = 2. Hence, Note that Y 1 (l) ∼ Bin(n, 0.5) since both alleles have the same likelihood to be inherited to the worker. In similar fashion, we can rewriteD. In a colony of n workers, Y 21 will have the first mate as father, Y 22 will have the second mate as father, etc. Clearly, M m=1 Y 2m = n. Since all mates have equal probability of being the father of a given worker, the random vector Y 2 = (Y 21 , . . . , Y 2M ) has a Multinomial distribution with parameters n and m = 1 M , . . . , 1 M . From this, we know that each Y 2m follows marginally a Binomial distribution, i.e. Y 2m ∼ Bin n, 1 M . Thus,D Having established the sample mean of the genotypic values of a colony, we can now assess the variability of the colony using the sample variance of the genotypic values: where For large n, S Q(l)Q(k) and S Q(l)D will be close to 0 due to the aforementioned independence between the all Q a(l) (l)-and D m -values. This fact will be illustrated and used when computing the expectation of S 2 G . Also, note that the alternate expressions for S 2 Q(l) and S 2 D are derived by a similar argument as the alternate expressions for the meansQ(l) andD and will also be useful in the computation of the expected value for S 2 G . Now, since the Y -variables are binomial random variables, we can take their expected values to compute the expected sample mean and expected sample variance for a colony of size n with fixed Queen and mates. Since Y 1 (l) ∼ Bin(n, 0.5), we know that E Y 1 (l) = n 2 . With this, we have Also, In a similar fashion, we can find E(D) and E(S 2 D ). We know that Y 2m ∼ Bin n, 1 M and thus E(Y 2m ) = n M . So, and We also could re-express S Q(l)Q(k) and S Q(l)D in similar manner, but we will forgo this here.
With all that the expected sample variance of a colony with a specific Queen and a specific set of M mates can be written as

The Expected Value of Colony Variability
To compute the expected value of the sample variance of genotypic values over all possible colonies, we will use the fact that if X i iid ∼ N(0, σ 2 ), i = 1, . . . , n, then n i=1 (X i −X) 2 σ 2 ∼ χ 2 (n−1) and thus E