fixation time

Ewens WJ 2004 Mathematical population genetics I. Theoretical introduction. 2nd ed. Springer. ISBN:9780387201912

  • t*(x; p) is symmetric about x = 0.5
  • the mean time spent in any interval (x, x + δx) is the same as the mean time spent in (1 − x − δx, 1 − x)
  • even more surprisingly, t*(x; p) remains unchanged if α is replaced by −α
  • so that a selectively disadvantageous mutant, if destined for fixation, spends as much time, on the average, in any frequency range as a corresponding selectively advantageous mutant destined for fixation
  • this remarkable fact, noted first in effect by Maruyama (1974), will again be reconsidered later in the light of time-reversal properties
  • it is indeed easy to see that the entire behavior of the conditional process is independent of the sign of s
  • the diffusion coefficient b*(x), calculated from (4.56) and (5.5), is independent of s
  • the drift coefficient a*(x), calculated from (4.55), (5.43), and (5.47), is
  • a*(x) = ½αx(1 − x) / tanh(½αx)
  • clearly, a*(x) is independent of the sign of α
  • this more detailed conclusion was first noted by Watterson (1977b)
  • despite the symmetry of t*(x) around x = ½, it is not true that a*(x) = a*(1 − x)
  • (p. 170)
  • for general levels of dominance it is no longer true (as it was with no dominance) that t*(x, α, h, (2N)−1) = t*(x, −α, 1 − h, (2N)−1)
  • nor is it true that a*(x, α, h) = a*(x, −α, 1 − h)
  • there is, however, one relation, first noted by Maruyama and Kimura (1974), that does remain true
  • consider two cases
  • one where the initial frequency of A1 is (2N)−1 and the condition is made that A1 eventually fixes
  • the other where the initial frequency of A1 is 1 − (2N)−1 and the condition is made that A1 is eventually lost
  • by considering A2 rather than A1 it is clear that the equation
  • t*(1 − x, −α, 1 − h, (2N)−1) = t**(x, α, h, 1 − (2N)−1) ... (5.59)
  • must be true
  • this may be used with (5.58) to show that
  • t**(x, α, h, (2N)−1) = t**(x, α, h, 1 − (2N)−1) ... (5.60)
  • we noted above the special case of this equation when h = ½
  • thus the mean time spent in any frequency range is the same for both processes
  • it is not true that a*(x) = a**(x)
  • so that despite (5.60), the two processes have quite different properties
  • again, these perhaps paradoxical conclusions will be reconsidered, and to a large extent resolved, when we consider time-reversal properties of diffusion processes
  • (p. 171)