Mathematical models of
evolution
Jose Cuesta
Grupo Interdisciplinar de Sistemas Complejos
Departamento de Matemáticas
Universidad Carlos III de Madrid

"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

"A mathematician is a blind man in a dark
room looking for a black cat which isn't
there."
- Charles Darwin

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 divisions / hour = 144 divisions / 2 days
144 43 28
2 ≈2×10 ≈2×10 kg ≈3000 earths!

Exponential growth
d N t 
=r N t
dt
Malthus law

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

Evolution: building blocks
Replication
Selection
Mutation

Fitness
fitness: mean number of adult offspring in the next
generation (separated generations)
fitness: mean growth rate (mixed generations)
dN
=r N
dt
fitness

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

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

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

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 ∞

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

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

Symbiotic coexistence
dx
=x 1−x [ A− A  B  x ]
dt
 A 0 A
x *=
 B 0  A  B
0 x* 1
x

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

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−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

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

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

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

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 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

Fisher-Wright model

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

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

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

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

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

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

Fixation of a mutant allele
1
 1=
N

Genetic drift under selection
A A A A A a a a
k N −k
probability proportional
to fitness
A

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

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

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 ]
±

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

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

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
• Key points:
– Fitness is affected by environment (external
factors)
– Fitness depends on phenotype, which is
determined by genotype (permanent factor)

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:
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

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

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

Genotype space
L=2
L=4
L=3
L=5
vertices of L-dimensional hypercubes

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)

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

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

Ai −1  Ai  Ai 1
Bj Bj Bj

p viable
Ai
1− p Ai
Bj
inviable B j−1

Russian roulette model
2D site percolation
p=0.2 p=0.51 p=0.594
size=65x65 p =0.592746
c
neutral network

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

More general rugged lanscapes
w2 1
∫w f  w d w ⇒ quasi-neutral network
1 D
fitness distribution

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 structure:
~0.6735 n3 / 2 2.1635n
RNA

Aiming at a secondary structure

Steps to speciation

Steps to speciation

Speciation
speciation proceeds mostly trough random drift
allopatric peripatric
parapatric sympatric

Allopatric speciation

Paratric speciation

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)