Mathematical models of
      evolution
               Jose Cuesta

Grupo Interdisciplinar de Sistemas Complejos
         D...
"Every body of discovery is mathematical in
form, because there is no other guidance we
can have."
                       ...
"A mathematician is a blind man in a dark
room looking for a black cat which isn't
there."
                            - C...
12 February 1809 – 12 February 2009
Summary

(1) Fundamentals of evolution
(2) Genetic drift
(3) Sequences
(4) Fitness landscapes
(5) Neutral evolution
FUNDAMENTALS
 OF EVOLUTION
Evolution: building blocks

        Replication

        Selection

        Mutation
Evolution: building blocks

        Replication

        Selection

        Mutation
Replication

Bacterial reproduction:

                                   t
        x t 1=2 x t         x t =2 x 0

3 divi...
Exponential growth

d N t 
         =r N t
  dt

  Malthus law
Saturation

dN
dt       
   =r N 1−
           N
           K      
  carrying capacity


                         rt
  ...
Evolution: building blocks

        Replication

        Selection

        Mutation
Fitness

fitness: mean number of adult offspring in the next
generation (separated generations)

fitness: mean growth rate...
Competition
  d NA                  dNB
       =r A N A             =r B N B
   dt                    dt
      NA         ...
Survival of the fittest
           dx
              =x 1−x  r A−r B 
           dt

                   r Ar B

 0    ...
Survival of the fittest
                                 n
 d xk
      = x k  f k −        = ∑ x k f k
  dt           ...
Survival of the fittest
                    n
        d xk
             = xk ∑ x j  f k − f j 
         dt       j =1


...
Fundamental theorem of natural
         selection
        n
  d           d xk n                    n
     =∑     fk     ...
Composition-dependent fitness
            f k =f k  x 1 ,, x n ≡f k  x

Example: two species

                   r A ...
Symbiotic coexistence
      dx
         =x 1−x [ A− A  B  x ]
      dt

       A 0                A
           ...
Competitive exclusion
      dx
         =x 1−x [ A− A  B  x ]
      dt

       A 0                A
           ...
Evolution: building blocks

        Replication

        Selection

        Mutation
Replication with error

       uA




       uB
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...
Replication with error
         rAuA
 x *≈1−                    r Ar B
        r A−r B
        r B uB
  x *≈             ...
Evolution with mutation
mutation matrix       Q=q ij          u=1, , 1
         n

        ∑ qij=1                 ⇔ ...
Evolution with mutation
                                            may be negative!
fundamental theorem

d d x     T    ...
GENETIC DRIFT
Genetic drift
neutral evolution (no selection, no mutation)
Small populations: bottlenecks
Fisher-Wright model
 A       A    A     A     A   a      a       a     generation t



              k                    ...
Fisher-Wright model
Fisher-Wright model
two absorbing states:



          k =N                         k =0

                 {
          i=...
Moran model
A   A   A      A   A   a    a     a


        k                  N −k

    A




A   A   A      A   A   A    a...
Moran model

             k  N −k 
P k , k ±1=        2
                 N
                    2   2
           N −k  ...
Birth-death processes
                                j                       i −1
                                       ...
Birth-death processes
two absorbing states (0 & N)
                                    Q N 1−i  j
                    ...
Moran model

                                                    i
                   q i =1          Qi =i        i=
   ...
Fixation of a mutant allele



                  1
              1=
                  N
Genetic drift under selection

   A       A      A        A   A   a    a     a


                  k                    N ...
Genetic drift under selection
               N −k        kfA
   P k , k 1=
                 N k f A  N −k  f a
       ...
Fixation of a mutant allele
          under selection

                  −1
             1−r                1−r
         ...
Diffusion approximation
                      w t 1=w t  P
master equation


         w i t 1−w i t =∑ w j t  ...
Diffusion approximation
        i                                                     1 2
     x≡             w i t ≡N f...
Diffusion approximation
absorption probability


                                   1
              0=−N a  x  '  x ...
SEQUENCES
Sequences: DNA
Chromosomes
Mitosis & meiosis
Genes
Transcription: genetic code
Proteins
Locus & alleles
Types of mutation
FITNESS LANDSCAPES
Fitness landscapes

• Metaphor introduced by Wright (1932)
• Representation of fitness of individuals or
  population
• Ke...
Fitness landscapes
Working example:
1 locus, 2 alleles, random mating
Working example:
    1 locus, 2 alleles, random mating

              x =[ A ]       y =[a ]
Hardy-Weinberg law:

        ...
Working example:
1 locus, 2 alleles, random mating




dx                              1    d  x 
   =x y [ f A  x − ...
Remarks
• First representation is a fundamental one:
  – Individuals sit at their genotypic positions
  – Population at ev...
Genotype space
L=2


                                             L=4

L=3




                                           ...
Genotype space

                            # different genotypes for L
                            loci and A alleles:

 ...
Hamming distance
a   a         a                  L
s =s s 2 ⋯s
                  d  s , s =∑  s , s 
    1         L...
The metaphor




rugged (Wright)                   Fujijama (Fisher)




                  flat (Kimura)
Speciation in rough landscapes
NEUTRAL EVOLUTION
Russian roulette model
                                   Ai
     A 1,  , A N                  B j1
     B1,  , B N
   ...
Russian roulette model
                2D site percolation
p=0.2                   p=0.51             p=0.594




        ...
Russian roulette for sequences
        D=L  A−1             L ≫1
            viable  p                     inviable
  ...
More general rugged lanscapes




       w2                1
    ∫w       f  w d w     ⇒   quasi-neutral network
      ...
New metaphor of fitness
     landscapes




most evolutionary steps are neutral (Kimura)
Neutral networks
WORD → WORE → GORE → GONE→ GENE




  RNA                proteins
RNA neutral networks


         mean number of sequences
         of length n folding into the
         same secondary str...
Aiming at a secondary structure
Steps to speciation
Steps to speciation
Speciation
speciation proceeds mostly trough random drift




      allopatric              peripatric




      parapatri...
Allopatric speciation
Paratric speciation
Some bibliography

• Nowak, Evolutionary Dynamics: Exploring the Equations
  of Life (Belknap Press, 2006)
• Ewens, Mathem...
Upcoming SlideShare
Loading in …5
×

Mathematical models of evolution

3,934 views

Published on

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

Published in: Education
  • Be the first to comment

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)

×