With a little help from my friends: Cooperation can accelerate crossing of adaptive valleys

Natural selection favors changes that lead to genotypes possessing high fitness. A conflict arises when several mutations are required for adaptation, but each mutation is separately deleterious. The process of a population evolving from a genotype encoding for a local fitness maximum to a higher fitness genotype is termed an adaptive peak shift. Here we suggest cooperative behavior as a factor that can facilitate adaptive peak shifts. We model cooperation in a public goods scenario, wherein each individual contributes resources that are later equally redistributed among all cooperating individuals. We use mathematical modeling and stochastic simulations to study the effect of cooperation on peak shifts in well-mixed populations and structured ones. Our results show that cooperation can accelerate the rate of complex adaptation. Furthermore, we show that cooperation increases the population diversity throughout the peak shift process, thus increasing the robustness of the population to drastic environmental changes. Our work could help explain adaptive valley crossing in natural populations and suggest that the long term evolution of a species depends on its social behavior.

Here we consider an additional factor that can affect the process of crossing 95 adaptive valleys: cooperative behavior. We show that for a population subdivided 96 into demes and connected by migration, cooperation between individuals within the 97 same deme can considerably increase the rate of adaptive peak shifts. 98 We focus on a public goods form of cooperation (Kagel and Roth 1995): all 99 individuals within a deme contribute some resources (thus contributing fitness) to 100 other deme members, and all deme members receive an equal amount of the 101 redistributed resources. This cooperative behavior reduces the fitness difference 102 between different genotypes and therefore effectively "smooths" the landscape to 103 some degree. As a result, less fit mutants are more likely to survive with 104 cooperation, increasing the rate of appearance of multiple mutants. Cooperation has 105 an opposite effect on the fixation of the fittest genotype. Precisely because the population diversity. This increase in diversity is beneficial in evolutionary terms, 113 as it can help populations to overcome environmental changes, parasites etc. (Clarke 114 1979). 115 Overall, our results show that cooperation affects adaptive peak shifts substantially 116 and might be an important and overlooked component of complex adaptation. 117 118 Model 119 We model a population of sexually reproducing haploid individuals, containing two 120 bi-allelic loci. The ab genotype is the wild type with fitness 1; the fitness of the single 121 mutant genotypes, and , is 1 − ; the fitness of the double mutant is 122 1 + . and are the selection coefficients of the single mutant and the relative 123 advantage of the double mutant, respectively ( > 0, ≥ 1). We assume equal 124 forward and backward mutation rates for both loci (defined in units of mutations per 125 generation per locus), and denote them by . Recombination occurs with rate per 126 generation per loci pair. 127 We model a population composed of demes connected by migration, each of size 128 . Thus the total size of the population is = ⋅ . Each generation, individuals of with probability . Setting = 1 − 1 ⁄ defines a population in which offspring 137 are uniformly distributed among all demes in every generation; setting = 0 138 determines each deme to be an effectively isolated population. For simplicity, we 139 assume that migration to each deme is equiprobable. Finally, mating occurs between 140 individuals of the newly formed demes. The offspring generation replaces the parent 141 generation, so that population size remains constant and generations do not 142 overlap. Within this framework, we first analyzed the expected waiting time to a peak shift 155 when each deme contains two individuals ( = 2) and the population is fully mixed 156 ( = 1 − 1 ⁄ ) using Branching processes analysis (Eshel 1981

179
A fully mixed population 180 We start with the special case where demes only have two individuals ( = 2) and 181 the population is fully mixed by migration in every generation ( = 1 − 1 ⁄ ). We 182 estimate the relative difference in adaptation time due to cooperation, ρ(c). peak shift, as recombination tends to break beneficial gene combinations (Eshel and 187 Feldman 1970). This is consistent for cooperative behavior as well (Fig. 1, 198 Note that some conditions allow for a peak shift only in a non-cooperative 199 population, whereas a cooperative population is not expected to cross the adaptive

238
, and the y-axis represents the double mutant advantage, . White areas represent parameters 239 where both cooperatives and non-cooperatives are not expected to achieve a peak shift; black areas 240 are parameters for which only non-cooperatives are expected to shift a peak; blue areas represent 241 parameters for which both cooperatives and non-cooperatives achieve a peak shift, but non-242 cooperatives do so faster. Teal to red areas are parameters for which cooperatives achieve a peak 243 shift faster than non-cooperatives, and the color represents the average relative difference in the 244 expected time for a peak shift due to cooperation, ρ, averaged over ≥ 1200 simulations per 245 parameter set. Panels A and B represent results for low and high recombination rates: = 0.01, 0.1,  Using Branching processes (Harris 1948) we analyze the probability of a peak shift for 522 the case of = 2 and = 1 − 1 . We define _ to be the fitness of genotype x, 523 when genotype y is its partner.

524
Since the fitness of and is the same, and so is their contribution to their 525 partners we will denote (single mutant) as either or . Since is the wild type and is the most common genotype, and since the 527 population is fully mixed, most cooperating couples are of two individuals.

528
Therefore the fitness of most individuals is _ . 529 We normalize the fitness of all individuals relative to the most common fitness Hence, the probability that the first individual would appear in the population at 538 a certain generation, given that it had not appeared earlier is: which can be approximated by ⋅ , where = ⋅ is the size of the population. 540 The expected time for appearance of the first individual ( ), which is 541 geometrically distributed, is: The approximation of the average fitness, marked by ̅, is: The progeny of an individual, marked by , is: Therefore, if we assume that the number of offspring is Poisson distributed and its 545 mean is slightly above one (Eshel 1981;Hadany 2003), the probability that an genotype would fixate in the population, rather than go extinct, denoted by , is 547 approximated by: Thus we can approximate the expected time for appearance of an individual that 549 will go to fixation by: In order for to fixate, we require that > 1. Since < 1 and ≤ 1, we have that (2 − 2 + )(1 − ) > 0 and therefore 570 (eq.s7) > 2 (1− + ) (1− )(2−2 + ) Differentiating the right hand side of (eq.s7) with respect to yields: 572 (eq.s8) In order to verify our approximations we used stochastic simulations. Fig. S1 shows 580 the results of the simulation (blue line) and the approximation (red line), for various 581 cooperation levels. This is shown for the time of first appearance of the double 582 mutant ( Fig. 2A) and the double mutant's fixation probability (Fig. 2B), as a function 583 of the cooperation level ( ). When = 0, we are reduced to the results presented, 584 and verified against simulations, in (Hadany 2003). We can see that our 585 approximation is accurate when increases, but it does remain slightly biased. 586 Note that both the waiting time for appearance of the double mutant and its fixation 587 probability decrease with . The parameters we use are , , , , , ℎ, , , , as defined in Table 1  According to eq.s9, we can define a migration matrix that would simulate the 643 change in the population composition due to migration. For instance, if there are 644 3 demes in the population, the matrix would be defined as follows: The frequency of each genotype after mutation takes place can be written as: