Download Evolution and speciation on holey adaptive landscapes

Fitness landscapes
Sergey Gavrilets
Departments of Ecology and
Evolutionary Biology and
Mathematics, University of
Tennessee, Knoxville
Table of contents
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General notion of fitness landscapes
Fitness landscapes in simple population
genetic models
Rugged landscapes
Single-peak landscapes
Flat landscapes
Holey landscapes
Sewall Wright (1889-1988)
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A founder of theoretical population genetics
(with Fisher and Haldane)
Introduced the notion of “fitness landscapes”
(a.k.a. adaptive landscapes, adaptive
topographies, surfaces of selective values) in
1931
His last publication on fitness landscapes was
published in 1988
Papers on fitness landscapes
Title only
1980-1989
1990-1999
2000-
8
59
34
Title, keywords, abstract
no data
212
181
Some of the journals that publish these papers: JOURNAL OF THEORETICAL BIOLOGY,
PROTEIN ENGINEERING, PHYSICAL REVIEW E, CANCER RESEARCH, EVOLUTION ,
JOURNAL OF MATHEMATICAL BIOLOGY, LECTURE NOTES IN COMPUTER SCIENCE,
CURRENT OPINION IN BIOTECHNOLOGY, MARINE ECOLOGY-PROGRESS SERIES,
INTEGRATED COMPUTER-AIDED ENGINEERING, PHYSICA A-STATISTICAL MECHANICS
AND ITS APPLICATIONS, BIOLOGY & PHILOSOPHY, INTERNATIONAL JOURNAL OF
TECHNOLOGY MANAGEMENT, BIOSYSTEMS , JOURNAL OF GENERAL VIROLOGY ,
ECOLOGY LETTERS, RESEARCH POLICY , SYSTEMS RESEARCH AND BEHAVIORAL
SCIENCE, ANNALS OF APPLIED PROBABILITY, BIOPOLYMERS
Working example: one-locus twoallele model of viability selection
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Two allele at a single locus: A and a
Allele frequencies: p and 1-p
Three diploid genotypes: AA, Aa and aa
2
2
p
,
2
p
(
1

p
),
(
1

p
)
Genotype frequencies:
Viabilities: waa , waA , wAA
Average fitness of the population:

w  wAA p 2  wAa 2 p(1  p )  waa (1  p )2
Fitness landscape as fitness of
gene combinations
Fitness landscape as the
average fitness of populations

p(1  p ) d w
p 

2 w dp
Genotype space
L=2,A=3 case
Dimensionality: D=L(A-1) for haploids and D=2L(A-1) for diploids
One-locus multi-allele model of
stepwise mutation
Fitness landscape in a twolocus two-allele model
Dimensionality of the population
state space
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General case:
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Randomly mating population under constant
viability selection:
D  A 1
2L
D  A 1
L
Average fitness of the population in
a 2-locus 2-allele model with additive
fitnesses
D=2 (because of linkage equilibrium)
Fitness landscapes for mating
pairs: fertility
Fitness landscapes for mating
pairs: mating preference
Drosophila silvestris, D.heteroneura and hybrids
Fitness landscapes for
quantitative characters
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Relationship between a set of Q
quantitative characters that an individual
has and its fitness; dimensionality of
phenotype space is Q
Relationship between the average
fitness of the population and its genetic
structure; dimensionality is equal to the
number of phenotypic moments affecting
the average fitness
Fitness landscape with two
quantitatie characters
Mating preference function as
fitness landscape
( x  y )2
( x, y )  exp( 
)
2V
Average fitness of the population
under stabilizing selection
z2
w  exp( 
)
2Vs

 z  VG

 ln w

z
Metaphor of fitness landscapes
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Two or three dimensional visualization of
certain features of multidimensional
fitness landscapes [Wright 1932]
Rugged fitness landscape
Hill climbing on a rugged fitness
landscape (Kauffman and Levin
1987)
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L diallelic haploid loci
Fitnesses are assigned randomly
The walk starts on a randomly chosen
genotype
At each time step, the walk samples one of the
L one-step neighbors. If the neighbor has
higher fitness, the walk moves there.
Otherwise, no change happens. The walk
stops when it reaches a local fitness peak, so
that all L neighbors have smaller fitness
Sample of Kauffman and Levin’s
results
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Expected number of local peaks is 2 /( L  1)
Expected fraction of fitter neighbors dwindles by ½ on
each improvement step
Average number of steps till a local peak is log 2 ( L  1)
Ratio of accepted to tried mutations scales as ln k / k
From most starting points, a walk can climb only to an
extremely small fraction of the local peaks. Any one
local peak can be reached only from an extremely
small fraction of starting points.
“Complexity catastrophe”: as L increases, the heights
of accessible peaks fall towards the average fitness
L
Single-peak fitness landscape
Ronald Fisher (1890-1962)
Fisher’s geometric model of
adaptation
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Each organism is characterized by Q continuous
variables
There is a single optimum phenotype O and fitness
decreases monotonically with increasing (Euclidean)
distance from the optimum
Let d/2 be the current distance to the optimum O
Each mutation is advantageous if it moves the organism
closer to O.
Let r be the mutation size (i.e. distance between the
current state and the mutant)
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For large Q, the probability that a mutation is
advantageous is P(r)=1-F(r) where F is the
cumulative distribution function of a standard
normal distribution, and x  r Q / d
Mutations of small size
are the most important in
evolution
Corrections to the Fisher model
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Kimura (1983): the probability that an
advantageous mutation with effect s is fixed is
2s. Therefore, the rate of adaptive substitutions
is 2x(1-F(x)). Thus, mutation of intermediate
size are most important.
Orr (1998): distance to the optimum
continuously decreases. The distribution of
factors fixed during adaptation is exponential.
“Error threshold” (Manfred
Eigen)
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Assume that there is a single optimum
genotype (“master sequence”) that has fitness
1; all other genotypes have fitness 1-s. Let n be
the mutation rate per sequence per generation
Then, if n<s, then the equilibrium frequency of
the master sequence is 1-n/s.
If n>s, the master sequence is not maintained
in the population
Flat fitness landscape (of the
neutral theory of molecular
evolution)
Motoo Kimura (1924-1994)
Evolution of flat landscapes
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Random walk on a hypercube
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Equilibrium distribution: equal probability to be at
any vertex; time to reach the equilibrium
distribution is order  L log L steps
Transient dynamics of the distance to the initial
state
L
d t  [1  exp( 2 t )]
2
The index of dispersion (i.e. var(x)/E(x), where x is
the number of steps per unit of time) is equal to 1.
Evolution of flat landscapes
(cont.)
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In a population of N alleles, any two alleles can
be traced back to a common ancestor about N
generations ago (under the Fisher-Wright
binomial scheme for random genetic drift)
The average number of mutations fixed per
generation is equal to the mutation rate n
The average genetic distance between two
organisms is 2Nn
Population can be clustered into 2(2Nn)/d
clusters such that the average distance within
the same cluster is d.
How many dimensions do real
fitness landscapes have?
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The world as we perceive it is three dimensional
Superstring theory: 10 to 12 dimensions are
required to explain physical world
Biological evolution takes place in a space with
millions dimensions
(3/27/03)
SuperKingdom # of species
# of sequences range (in million
base pairs)
Archae
16
16
1.5-5.8
Bacteria
101
130
0.4-9.1
Eukaryotes
11
11
0.2-282
Extremely high dimensionality of
the genotype space results in:
redundancy in the genotype-fitness map
a possibility that high-fitness genotypes form
networks that extend throughout the genotype
space (=> substantial genetic divergence without
going through adaptive valleys)
increased importance of chance and contingency in
evolutionary dynamics (=>mutational order as a
major source of stochasticity)
Russian roulette model
Genotype is viable with probability p and is inviable otherwise:
There exists a giant cluster of viable genotypes if p>0.5973
(percolation in two dimensions)
Percolation on a hypercube
Each genotype has L “neighbors.”
In the L-dimensional hypercube (e.g. if there are L
diallelic loci), viable genotypes form a percolating neutral
network if p>1/L (assuming that L is very large).
Uniformly rugged landscape
Fitness w is drawn from a distribution on (0,1):
The nearly neutral network of genotypes with fitnesses
between w1 and w2 percolates if w2-w1>1/L.
Metaphor of holey fitness landscapes disregards fitness differences between
different genotypes belonging to the network of high-fitness genotypes and
treats all other genotypes as holes
Microevolution and local adaptation ~ climbing from a “hole”
macroevolution ~ movement along the holey landscape
speciation takes place when populations come to be on opposite sides of a
"hole" in the landscape
The origin of the idea
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Verbal arguments
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Bateson (1909)
Dobzhansky (1937)
Muller (1940, 1942)
Maynard Smith
(1970, 1983)
• Nei (1976)
• Barton and
Charleswoth (1984)
• Kondrashov and
Mina (1986)
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Formal models
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Nei (1976)
Wills (1977)
Nei et al (1983)
Bengtsson and
Christiansen (1983)
• Bengtsson (1985)
• Barton and Bengtsson
(1986)
Dobzhansky model (1937)
(1900-1975)
“This scheme may appear fanciful, but it is worth considering
further since it is supported by some well-established facts and
contradicted by none.” (Dobzhansky, 1937, p.282)
Maynard Smith (1970):
“It follows that if evolution by natural selection is to occur,
functional proteins must form a continuous network which
can be traversed by unit mutational steps without passing
through nonfunctional intermediates” (p.564)
Terminology
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A neutral network is a contiguous set of genotypes
(sequences) possessing the same fitness.
A nearly neutral network is a contiguous set of
genotypes possessing approximately the same
fitness.
A holey fitness landscape is a fitness landscape in which
relatively infrequent high-fitness genotypes form a
contiguous set that expands throughout the genotype
space.
Conclusions from models
The existence of percolating nearly-neutral networks of highfitness combinations of genes which allow for “nearly-neutral”
divergence is a general property of fitness landscapes with a
very large number of dimensions.
Experimental evidence
 Direct analyses of relationships between genotype
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and fitness in plants, Drosophila, mammals and
moths
Ring species and hybrid zones
Artificial selection experiments
Natural hybridization in plants and animals
Intermediate forms in the fossil record
Properties of RNA and proteins
Patterns of molecular evolution
Artificial life
Applications
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Speciation
Hybrid zones
Morphological macroevolution
RNA and proteins
Adaptation
Molecular evolution
Gene and genome duplication
Canalization of development