separation of timescales

Martin G & Lambert A 2015 A simple, semi-deterministic approximation to the distribution of selective sweeps in large populations. Theor Popul Biol 101:40-46.

• the main tool to obtain the distribution of allelic frequencies at at given time is by numerical solutions of the diffusion equation by perturbation analysis for small N_es (Kimura, 1957), or more recently, via spectral analysis of the diffusion operator (Song and Steinrücken, 2012)

• we start from the Wright-Fisher diffusion approximation of the exact stochastic process of allelic frequency under selection and genetic drift
• we approximate this diffusion itself, via a well-known separation of timescales
• this separation of timescales is explicitly connected to the simpler Feller diffusion process

• the dynamics of p_t follow a stochastic differential equation (SDE) corresponding to the Wright-Fisher diffusion approximation
• dp_t = p_t q_t (s_a q_t + s_b p_t) dt + √(p_t q_t / (2N_e)) dB_t ... (1)
• s_a = s (h + (1 − h) F)
• s_b = s (1 − (1 − F) h)
• p₀ = k / 2N
• the haploid model is characterized by s_a = s_b = s_*
• conditional on fixation, p_t is in fact only 'stochastic' while p_t → 0 (early phase A: t → 0) or q_t → 0 (late phase C: t → τ), the intermediate phase B being approximately deterministic
• the resulting early phase A is characterized by a simpler (linear) diffusion (Feller, 1951), akin to a branching process with independently growing types, as first noted by Haldane (1927)
• the same goes for the late phase C
• phase A : p_t = o(1) : dp_t ≈ s_a p_t dt + √(p_t / (2N_e)) dB_t + O(p_t²dt) ... (2)
• phase B : p_t = O(1) : dp_t ≈ p_t q_t (s_a q_t + s_b p_t) dt + O(1 / √(N_e) dt)
• phase C : q_t = o(1) : dp_t = −dq_t ≈ s_b q_t dt + √(q_t / (2N_e)) dB_t + O(q_t²dt)
• phase A in Eq. (2) is the SDE of a supercritical Feller diffusion (Feller, 1951) with 'drift' term s_a > 0 and 'diffusion' term 1/2N_e
• p_t ~ Feller(s_a, 1/2N_e)
• q_t ~ Feller(−s_b, 1/2N_e)

• conditional on non-extinction, a Feller diffusion starting at some p₀ > 0, when properly rescaled by its expectation E(p_t) = p₀ e^s_at, converges to some fixed distribution as time grows to infinity
• more precisely, define the positive random variable lim_t→∞ ω_t = p_t / E(p_t) (called a 'terminal value')
• for a trajectory destined to fixation, the distribution of this variable converges, at large times, to an exponentially distributed random variable ω_t = ω^~ ~ Exp(P₁)
• P₁ is the non-extinction probability of the Feller diffusion
• any density independent branching process ends up being roughly deterministic and exponentially growing at a fixed rate, if avoiding initial extinction
• the long term behavior of non-extinct processes thus appears as a distribution of exponentially growing processes, all with same rates (s_a here) and variable 'starting points' (ω^~p₀ here)
• applying this property to the phase A of our selective sweep model, starting from a single copy (p₀ = 1/2N), yields the following long term behavior of phase A
• lim_t→∞ p_t e^−s_at = p^~₀ ~ Exp(2NP₁)
• the allele will have reached its asymptotic behavior by the time it enters phase B
• the trajectories entering this phase are approximately distributed as p^~₀ e^s_at, namely as if they had been deterministic from the start, but with variable initial frequency p^~₀
• if an arbitrary number 1 ≤ k ≤ n of the k initial copies reach establishment, they will do so independently, yielding several independent Feller diffusion, all conditioned by fixation, during phase A
• the allele then rises into phase B approximately as the sum of n processes with i.i.d. exponentially distributed initial states, yielding a gamma distribution for p^~₀
• lim_t→∞ p_t e^−s_at = p^~₀ ~ Γ(n, 1/(2NP₁))