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# Mathematical models of evolution

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Conferencia impartida por el Profesor José Cuesta en el marco de los Viernes Científicos en la Universidad de Almería el 5 de junio de 2009

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### Mathematical models of evolution

1. 1. Mathematical models of evolution Jose Cuesta Grupo Interdisciplinar de Sistemas Complejos Departamento de Matemáticas Universidad Carlos III de Madrid
2. 2. "Every body of discovery is mathematical in form, because there is no other guidance we can have." - Charles Darwin "Mathematics seems to endow one with something like a new sense." - Charles Darwin
3. 3. "A mathematician is a blind man in a dark room looking for a black cat which isn't there." - Charles Darwin
4. 4. 12 February 1809 – 12 February 2009
5. 5. Summary (1) Fundamentals of evolution (2) Genetic drift (3) Sequences (4) Fitness landscapes (5) Neutral evolution
6. 6. FUNDAMENTALS OF EVOLUTION
7. 7. Evolution: building blocks Replication Selection Mutation
8. 8. Evolution: building blocks Replication Selection Mutation
9. 9. Replication Bacterial reproduction: t x t 1=2 x t x t =2 x 0 3 divisions / hour = 144 divisions / 2 days 144 43 28 2 ≈2×10 ≈2×10 kg ≈3000 earths!
10. 10. Exponential growth d N t  =r N t dt Malthus law
11. 11. Saturation dN dt  =r N 1− N K  carrying capacity rt K N 0e N t= rt lim N t =K K  N 0  e −1 t ∞ saturation maintains a constant population
12. 12. Evolution: building blocks Replication Selection Mutation
13. 13. Fitness fitness: mean number of adult offspring in the next generation (separated generations) fitness: mean growth rate (mixed generations) dN =r N dt fitness
14. 14. Competition d NA dNB =r A N A =r B N B dt dt NA NB x= y= =1−x N A N B N A N B dx dy =x  r A − = y r B− dt dt average fitness: =x r A y r B
15. 15. Survival of the fittest dx =x 1−x  r A−r B  dt r Ar B 0 x 1 all-A r Ar B 0 x 1 all-B
16. 16. Survival of the fittest n d xk = x k  f k − = ∑ x k f k dt k =1 n ∑ x k =1 x k ≥0 ∀t k =1 n=2 n=3 n=4
17. 17. Survival of the fittest n d xk = xk ∑ x j  f k − f j  dt j =1 assume f k f j ∀ j≠k lim x k t =1 lim x j t =0  j≠ k  t ∞ t ∞
18. 18. Fundamental theorem of natural selection n d d xk n n =∑ fk =∑ f k x k  f k −=∑ x k  f k −2≡ 2f d t k=1 d t k=1 k=1 d 2 = f dt 2 Mean fitness never decreases   f 0  The speed of increase is determined by the variation within the population
19. 19. Composition-dependent fitness f k =f k  x 1 ,, x n ≡f k  x Example: two species r A  x , y =r  A y r B  x , y =r  B x symbiosis competition  A 0  A 0  B 0  B 0
20. 20. Symbiotic coexistence dx =x 1−x [ A− A  B  x ] dt  A 0 A x *=  B 0  A  B 0 x* 1 x
21. 21. Competitive exclusion dx =x 1−x [ A− A  B  x ] dt  A 0 A x *=  B 0  A  B 0 x* 1 x
22. 22. Evolution: building blocks Replication Selection Mutation
23. 23. Replication with error uA uB
24. 24. Replication with error dx = x r A 1−u A  y r B u B − x dt dy = x r A u A y r B 1−u B − y dt =x r A y r B dx =x 1−x  r A−r B 1−x  r B u B− x r A u A dt x =0 x=1 are not equilibria
25. 25. Replication with error rAuA x *≈1− r Ar B r A−r B r B uB x *≈ r A r B r B −r A uA x *= r A =r B u A u B mutation is the source of variability
26. 26. Evolution with mutation mutation matrix Q=q ij  u=1, , 1 n ∑ qij=1 ⇔ Q u =u T T j=1 W ≡RQ   r1 0 R= ⋱ 0 rn mutation-selection matrix n dx =x W − x =x W u =∑ x j r j T dt j=1 quasispecies equation
27. 27. Evolution with mutation may be negative! fundamental theorem d d x T 2 T 2 2 T = W u =x W u − =x W − I  u dt dt equilibria: x * W = * x * if     is irreducible,  * is the largest eigenvalue of Q W as x * normally corresponds to a mixed population, * need not be the absolute maximum
28. 28. GENETIC DRIFT
29. 29. Genetic drift neutral evolution (no selection, no mutation)
30. 30. Small populations: bottlenecks
31. 31. Fisher-Wright model A A A A A a a a generation t k N −k j N− j sample with replacement     P kj = N j k N N −k N  A A a a a a a a generation t+1 j N−j
32. 32. Fisher-Wright model
33. 33. Fisher-Wright model two absorbing states: k =N k =0 { i=P lim X t =N X 0=i t ∞ ∣ } N i= ∑ P ij  j 0 =0  N =1 j=0 i E { X t 1∣ X t=k }=k ⇒ i= N
34. 34. Moran model A A A A A a a a k N −k A A A A A A A a a k 1 N −k −1
35. 35. Moran model k  N −k  P k , k ±1= 2 N 2 2  N −k  k Pk ,k = 2 N P k , j =0 if ∣k − j∣1 birth-death process with two absorbing states
36. 36. Birth-death processes j i −1 P k , k−1 q 0=1 q j =∏ Q i =∑ q j k =1 P k , k1 j =0 if absorption { i=P lim X t =N X 0=i ∞ t ∞ ∣ } t ij =∑  P ij n T = P T I t 0j=t Nj =0 n=0 if ergodic w =w P
37. 37. Birth-death processes two absorbing states (0 & N) Q N 1−i  j t ij = if ji Qi q j P j , j1 i= QN Q N i 1− j  t ij = if ji q j P j , j1 one absorbing state (N) Q N −Q i Q N −Q j t ij = if ji t ij = if j i q j P j , j1 q j P j , j 1 no absorbing state  q i P i , i1 −1 wi= N ∑  q j P j , j 1−1 j=0
38. 38. Moran model i q i =1 Qi =i i= N t ij = N  N −i N−j  if j i t ij =N  i j if ji mean absorption time N −1 t i= ∑ t ij =O  N  j =1
39. 39. Fixation of a mutant allele 1  1= N
40. 40. Genetic drift under selection A A A A A a a a k N −k probability proportional to fitness A
41. 41. Genetic drift under selection N −k kfA P k , k 1= N k f A  N −k  f a k  N −k  f a P k , k−1= N k f A N −k  f a P k , k−1 f a −1 = ≡r P k , k1 f A 1−r −i i= −N 1−r
42. 42. Fixation of a mutant allele under selection −1 1−r 1−r A= −N a = N 1−r 1−r • If r 1 selection favors A over neutral case • If r 1 selection favors a over neutral case
43. 43. Diffusion approximation w t 1=w t  P master equation w i t 1−w i t =∑ w j t  P ji −w i t  ∑ P ij j j birth-death process  wi t=∑ [ wi ±1 t  P i ±1,i −wi t  P i ,i ±1 ] ±
44. 44. Diffusion approximation i 1 2 x≡ w i t ≡N f  x , t   t≡  x  = 2 N N a  x ≡ P i ,i 1−P i , i−1 b  x ≡ P i , i1 P i ,i −1 forward Kolmogorov (Fokker-Planck) equation ∂ f  x ,t  ∂ 1 ∂2 =−N [a  x  f  x , t ] 2 [ b x f  x , t] ∂t ∂x 2 ∂x
45. 45. Diffusion approximation absorption probability 1 0=−N a  x  '  x b x ' '  x 2 mean absorption time 1 −1=−N a  x   '  x  b  x  ' '  x  2
46. 46. SEQUENCES
47. 47. Sequences: DNA
48. 48. Chromosomes
49. 49. Mitosis & meiosis
50. 50. Genes
51. 51. Transcription: genetic code
52. 52. Proteins
53. 53. Locus & alleles
54. 54. Types of mutation
55. 55. FITNESS LANDSCAPES
56. 56. Fitness landscapes • Metaphor introduced by Wright (1932) • Representation of fitness of individuals or population • Key points: – Fitness is affected by environment (external factors) – Fitness depends on phenotype, which is determined by genotype (permanent factor)
57. 57. Fitness landscapes
58. 58. Working example: 1 locus, 2 alleles, random mating
59. 59. Working example: 1 locus, 2 alleles, random mating x =[ A ] y =[a ] Hardy-Weinberg law: 2 2 [AA ]= x [ Aa]=2 x y [aa]= y f A  x = f AA x f Aa y f a  x = f Aa x f aa y  x = f AA x 2 f Aa 2 x y f aa y 2
60. 60. Working example: 1 locus, 2 alleles, random mating dx 1 d  x  =x y [ f A  x − f a  x ]= x y dt 2 dx
61. 61. Remarks • First representation is a fundamental one: – Individuals sit at their genotypic positions – Population at every position changes in time • Second representation is not because: – Depends on evolution law – Assumes maximization of mean fitness (not always true; e.g. mutations) – Can't be generalized to multilocus models
62. 62. Genotype space L=2 L=4 L=3 L=5 vertices of L-dimensional hypercubes
63. 63. Genotype space # different genotypes for L loci and A alleles: G= AL dimensionality (number of nearest neighbors): D=L  A−1 e.g. L=100, A=2: D=100 G≈10 30 L=2 (loci), A=3 (alleles)
64. 64. Hamming distance a a a L s =s s 2 ⋯s d  s , s =∑  s , s  1 L a b a b b b b b i i s =s s ⋯s 1 2 L i=1 d  abc , aBC=2 d  Abc ,aBC =3
65. 65. The metaphor rugged (Wright) Fujijama (Fisher) flat (Kimura)
66. 66. Speciation in rough landscapes
67. 67. NEUTRAL EVOLUTION
68. 68. Russian roulette model Ai A 1,  , A N B j1 B1,  , B N  Ai −1  Ai  Ai 1 Bj Bj Bj  p viable Ai 1− p Ai Bj inviable B j−1
69. 69. Russian roulette model 2D site percolation p=0.2 p=0.51 p=0.594 size=65x65 p =0.592746 c neutral network
70. 70. Russian roulette for sequences D=L  A−1 L ≫1 viable  p  inviable 1− p  branching process p k = k   D−1 p k 1− p D −1−k 1 1 E {k }= D−1 p ⇔ pc= ≈ D−1 D
71. 71. More general rugged lanscapes w2 1 ∫w f  w d w ⇒ quasi-neutral network 1 D fitness distribution
72. 72. New metaphor of fitness landscapes most evolutionary steps are neutral (Kimura)
73. 73. Neutral networks WORD → WORE → GORE → GONE→ GENE RNA proteins
74. 74. RNA neutral networks mean number of sequences of length n folding into the same secondary structure: ~0.6735 n3 / 2 2.1635n RNA
75. 75. Aiming at a secondary structure
76. 76. Steps to speciation
77. 77. Steps to speciation
78. 78. Speciation speciation proceeds mostly trough random drift allopatric peripatric parapatric sympatric
79. 79. Allopatric speciation
80. 80. Paratric speciation
81. 81. Some bibliography • Nowak, Evolutionary Dynamics: Exploring the Equations of Life (Belknap Press, 2006) • Ewens, Mathematical Population Genetics (Springer, 2004) • Durret, Probability Models of DNA Sequence Evolution (Springer, 2008) • Gavrilets, Fitness Landscapes and the Origin of Species (Princeton, 2004)