polygenic adaptation

Weissman DB, Desai MM, Fisher DS & Feldman MW 2009 The rate at which asexual populations cross fitness valleys. Theor Popul Biol 75:286-300.

  • once a beneficial mutation is established (upon reaching a size of order 1 / s), its frequency will increase roughly deterministically until the population is dominated by beneficial mutants
  • we refer to any mutant (beneficial or not) whose descendants will include a beneficial multiple-mutant that will establish as successfu
  • we begin by considering the simplest case, where a double mutation increases fitness by s (i.e. K = 2), but each of the single-mutants is neutral (i.e. δ1 = 0)
  • the essential property of the single-mutant lineage that determines its probability of producing a double-mutant is its time-integrated population size, ∫n(t)dt
  • this is the number of mutational opportunities this lineage presents
  • we call the value of this integral at time t the "weight" at time t of the single-mutant lineage
  • the total number of mutational opportunities before the lineage goes extinct is W ≡ limt → ∞ W(t) = ∫0∞n(t)dt, the total weight of the lineage
  • to calculate W, we must understand the dynamics of the single-mutant lineages
  • with probability of order 1 / T, the lineage will survive for more than T generations
  • if it does, its population size n(T) will be of order T
  • this will produce a weight of order T2
  • (a population size of order T for a time of order T)
  • the probability that such a lineage gives rise to at least one double-mutant that establishes is thus 1 − e−Cμ1sT2, where C is an unknown constant of order 1
  • we have used the fact that the occurrence of successful mutations is a Poisson process
  • this means that lineages that survive longer than T ~ 1 / √(μ1s) generations (and hence reach size 1 / √(μ1s) individuals) are very likely to produce established double-mutants
  • thus with probability 1 / √(μ1s) a single-mutant lives long enough that it is extremely likely to produce a double-mutant that establishes
  • since the probability of a single-mutant lineage having weight at least T2 falls off only as 1 / T, while the expected number of double-mutants produced by the single-mutant lineage increases as T2 for T < 1 / √(μ1s), the rate at which double-mutants are produced is dominated by these rare lucky single-mutant lineages that reach this 1 / √(μ1s)
  • thus the overall probability that a single-mutant gives rise to a double-mutant that establishes is simply
  • p1 ~ √(μ1s) ... (3)
  • if the single-mutant intermediates are deleterious, things are only slightly more complicated
  • if δ1 < √(μ1s), ... we have as before p1 ~ √(μ1s)
  • the single-mutant is effectively neutral for the purposes of producing double-mutants
  • note this can be true even if Nδ1 ≫ 1
  • (where the single-mutant is not effectively neutral by conventional definitions)
  • if on the other hand δ1 > √(μ1s), then the fact that the single-mutant is deleterious matters
  • the single-mutant lineage will reach a size of at most order 1 / δ1, have a weight of order 1 / δ12, and give rise to a double-mutant that establishes with probability of order μ1s / δ12
  • p1 ~ μ1s / δ1 ... (4)
  • when δ1 ~ √(μ1s), this reduces to the neutral result
  • all of our discussion to this point has implicitly assumed that the population size N is large enough that the intermediates can drift to the sizes described above
  • for the neutral case this means
  • N ≫ 1 / √(μ1s) ... (5)
  • when this is true, lineages that typically produce double-mutants that establish can do so while staying small compared to N
  • when Eq. (5) fails, double-mutants establish primarily after a lucky single-mutant has first drifted to fixation