1 / 151

Lecture 9: Population genetics, first-passage problems

Lecture 9: Population genetics, first-passage problems. Outline: population genetics Moran model fluctuations ~ 1/ N but not ignorable effect of mutations effect of selection neurons: integrate-and-fire models interspike interval distribution no leak with leaky cell membrane

prudence-bender
Download Presentation

Lecture 9: Population genetics, first-passage problems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 9: Population genetics, first-passage problems • Outline: • population genetics • Moran model • fluctuations ~ 1/N but not ignorable • effect of mutations • effect of selection • neurons: integrate-and-fire models • interspike interval distribution • no leak • with leaky cell membrane • evolution • traffic

  2. Population genetics: Moran model 2 alleles, N haploid organisms

  3. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces

  4. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are

  5. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N

  6. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N n1 – 1 with probability x(1 – x)

  7. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N n1 – 1 with probability x(1 – x) n1 with probability x2 + (1 – x)2.

  8. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N n1 – 1 with probability x(1 – x) n1 with probability x2 + (1 – x)2. So

  9. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N n1 – 1 with probability x(1 – x) n1 with probability x2 + (1 – x)2. So

  10. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N n1 – 1 with probability x(1 – x) n1 with probability x2 + (1 – x)2. So or

  11. Population genetics: Moran model 2 alleles, N haploid organisms choose 2 individual at random: 1 dies, the other reproduces If there are n1 organisms of type 1 before this step, then afterwards there are n1 + 1 with probability x(1 – x) x = n1/N n1 – 1 with probability x(1 – x) n1 with probability x2 + (1 – x)2. So or

  12. continuum limit: FP equation (N steps/generation)

  13. continuum limit: FP equation (N steps/generation)

  14. continuum limit: FP equation (N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0

  15. continuum limit: FP equation (N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0 (once an allele dies out, it can not come back)

  16. continuum limit: FP equation (N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0 (once an allele dies out, it can not come back) stochastic differential equation:

  17. continuum limit: FP equation (N steps/generation) boundary conditions: P(0,t) = P(1,t) = 0 (once an allele dies out, it can not come back) stochastic differential equation: notice

  18. heterozygocity Eventually P(x,t) gets concentrated at one boundary,

  19. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other.

  20. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one.

  21. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity

  22. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:

  23. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:

  24. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:

  25. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma:

  26. heterozygocity Eventually P(x,t) gets concentrated at one boundary, i.e., one allele completely dominates the other. But it is random which one. Measure this by the heterozygocity use Ito’s lemma: i.e., diversity dies out in about N generations

  27. fluctuations of x

  28. fluctuations of x

  29. fluctuations of x

  30. fluctuations of x

  31. fluctuations of x

  32. fluctuations of x So mean-square fluctuations of x grow initially linearly in t and then saturate

  33. with mutation: Mutation induces a drift term in the FP and sd equation

  34. with mutation: Mutation induces a drift term in the FP and sd equation

  35. with mutation: Mutation induces a drift term in the FP and sd equation

  36. with mutation: Mutation induces a drift term in the FP and sd equation stationary solution:

  37. with mutation: Mutation induces a drift term in the FP and sd equation stationary solution:

  38. fluctuations Use Ito’s lemma on F(x) = x2:

  39. fluctuations Use Ito’s lemma on F(x) = x2:

  40. fluctuations Use Ito’s lemma on F(x) = x2: at steady state:

  41. fluctuations Use Ito’s lemma on F(x) = x2: at steady state:

  42. fluctuations Use Ito’s lemma on F(x) = x2: at steady state:

  43. fluctuations Use Ito’s lemma on F(x) = x2: at steady state: mean square fluctuations:

  44. heterozygocity:

  45. heterozygocity:

  46. heterozygocity: small noise (large population):

  47. heterozygocity: small noise (large population):

  48. heterozygocity: small noise (large population): large noise (small population):

  49. heterozygocity: small noise (large population): large noise (small population):

  50. heterozygocity: small noise (large population): large noise (small population): usually one allele dominates, rare transitions

More Related