Population Genetics: Selection-Drift Balance

Kimura diffusion — allele frequency distribution, fixation, and the Ns criterion

Parameters

Kimura Diffusion

∂f/∂t = −∂/∂p[M(p)f] + ½∂²/∂p²[V(p)f]
M(p) = sp(1−p)[ph+(1−p)(1−h)]
V(p) = p(1−p)/(2N)
f*(p) ∝ p^(4Nu−1)(1−p)^(4Nu−1) e^(4Nsp)
Ns ≪ 1: drift dominates — nearly neutral evolution. Ns ≫ 1: selection dominates — beneficial alleles fix with probability ≈1−e^(−4Ns). Stationary distribution shows U-shape (drift) vs peak near 1 (selection).

Live

Ns
Regime
P_fix
T_fix
⟨p⟩