normality

Kimura M 1954 Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics 39:280-295.

  • the gene A is selectively neutral on the average
  • Mδx = 0
  • Vδx = Vsx2(1 − x)2
  • ∂φ/∂t = (Vs/2) ∂2/∂x2 {x2 (1 − x)2 φ} ... (2)
  • as was demonstrated in the previous report (KIMURA 1952a), if the gene frequency x is transformed into a variate ξ = log(x / (1 − x)), ξ changes continuously from −∞ to +∞ as x changes from 0 to 1
  • the distribution of ξ becomes approximately normal
  • the process of change of ξ is approximately represented by a Gaussian process
  • to solve the equation (2), the same transformation turns out to be very useful
  • u = (1/2) e(Vs/8) t x(3/2) (1 − x)(3/2) φ
  • ∂u/∂t = (Vs/2) ∂2u/∂ξ2 ... (3)