evolutionary rescue - senecio7の日記

Kirkpatrick M & Peischl S 2013 Evolutionary rescue by beneficial mutations in space and time. Phil Trans R Soc Lond B 368:20120082.

φ_t(x) = ∑_{k = 0}^∞f_k_,tx^k ... (2.1)
P_t = 1 − φ₀(φ₁(. . . φ_{t − 1}(0))) ... (2.2)
assume that changes in the environment for the first few generations after the mutation appears are small
the PGFs for each generation can be written as the sum of a reference (or average) PGF and a small deviation
by neglecting terms in equation (2.2) that include products of two or more of those deviations, we can derive a remarkably simple approximation for the probability that the mutation survives indefinitely
p ≈ 2s_e / V ... (2.3)
V is the average variance in the number of offspring copies that the mutation leaves
s_e is the effective selection coefficient
s_e = s ∑_{k = 0}^∞(1 − s)^ks_k ... (2.4)
s is the mean selection coefficient over time
s_k is the selection coefficient in generation k
the fate of a mutation is usually determined during the first couple of generations after which it appears
the approximation is most accurate if |s − s_k|≪1 during this initial period

an important conclusion implied by equation (2.3) is that a mutation's fate is largely determined by its fitnesses in the first few generations after it appears
selection coefficient for the kth generation after the mutation appears, s_k, is weighted by (1 − s)^k, which becomes successively smaller with larger k

(a) environments that change systematically in time
a mutation has fitness s₀ when it first appears
its fitness increases or decreases linearly over the next T generations to a final fitness s₁
calculating the summation in equation (2.4) gives the fixation probability
p = 2[s₀ − (s₀ − s₁) (1 − s₁) (1 − (1 − s₁)^T) / (s₁T)] ... (2.5)
when the environmental change is slow (so T is large), the fixation probability is close to what is predicted by the mutation's initial fitness: p ≈ 2s₀
when the change is fast, the probability is close to what is predicted by its final fitness: p ≈ 2s₁
the transition between these two regimes occurs when T is roughly equal to 1 / s₁
that behavior is sensible, because whether a mutation is established in a constant environment is largely decided in the first 1 / s generations of its life