Mathematical models of molecular evolution

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Mathematical models of molecular evolution

• Agenda
• Motivation What implicit assumptions are often
  made about the
• the evolutionary process?
• Historical remarks the neutral theory of
  molecular evolution.
• The simplest mathematical models of evolution.
• random genetic drift.
• drift-mutation balance.
• drift-selection balance.
• Genotype space and fitness landscapes
• The Eigen quasispecies model.
• the error-threshold.
• More realistic fitness landscapes neutral
  networks.

Erik van Nimwegen
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Parsimony tree of D-loop mtDNA of several fish
species

• We are implicitly encouraged to believe that
• Each sequence is representative of its species.
• The relationships of the sequences in the tree
  reflect the evolutionary history of the species.
• The length of the branches correspond to the
  evolutionary distances in time.

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But why not?

• The variation in D-loop mtDNA within a species
  is as large as
• the variation between species.
• The tree just reflects the relationships
  between the single individuals
• from which the DNA for each of the species was
  extracted.
• The differences between the sequences dont
  reflect evolutionary history
• but rather selective pressures. Each sequence
  is optimized for the life-style
• and environment of its species.
• The tree reflects similarity in life-style and
  environment, not evolutionary
• history.

What do very simple models of evolution suggest?
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Historical remarksKimuras neutral theory of
molecular evolution (1968)

• Before Kimura, every locus in the genome was
  (implicitly) assumed to be selected.
• To maintain a population with this genome, each
  individual has to produce at least
• 1 offspring whose genome does not have any
  deleterious mutations.
• In the 1960s numbers started coming out
• the amount of DNA in mammalian genomes (109
  nucleotides).
• the number of amino acid substitutions in
  different proteins between different
• mammals (haemoglobin, cytochrome c). One amino
  acid change per 107 years.
• Those numbers did not make sense
• Human genome 109 nucleotides, mutation
  rate 10-8.
• The probability to produce an in tact
  offspring is 0.000045. Thus we should have
• 22,0000 offspring to produce one in tact
  offspring.
• 30 of Drosophila loci are polymorphic. All
  selected for?
• Kimuras suggestion The vast majority of
  single-nucleotide changes are
• selectively neutral.
• This is now well established, and neutral
  evolution is often used as the
• null model of molecular evolution.

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HistoryKimuras neutral theory of molecular
evolution (1968)

• Before Kimura, every locus in the genome was
  (implicitly) assumed to be selected.
• To maintain a population with this genome, each
  individual has to produce at least
• 1 offspring whose genome does not have any
  deleterious mutations.
• In the 1960s numbers started coming out
• the amount of DNA in mammalian genomes (109
  nucleotides).
• the number of amino acid substitutions in
  different proteins between different
• mammals (haemoglobin, cytochrome c). One amino
  acid change per 107 years.
• Those numbers did not make sense
• Human genome 109 nucleotides, mutation
  rate 10-8.
• The probability to produce an in tact
  offspring is 0.000045. Thus we should have
• 22,0000 offspring to produce one in tact
  offspring.
• 30 of Drosophila loci are polymorphic. All
  selected for?
• Kimuras suggestion The vast majority of
  single-nucleotide changes are
• selectively neutral.
• This is now well established, and neutral
  evolution is often used as the
• null model of molecular evolution.

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JBS Haldane (1892-1964)

• One of the founders of mathematical population
  genetics.
• Great popularizer of science.
• Influenced Aldous Huxleys Brave New World

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HistoryKimuras neutral theory of molecular
evolution (1968)

• Before Kimura, every locus in the genome was
  (implicitly) assumed to be selected.
• To maintain a population with this genome, each
  individual has to produce at least
• 1 offspring whose genome does not have any
  deleterious mutations.
• In the 1960s numbers started coming out
• the amount of DNA in mammalian genomes (109
  nucleotides).
• the number of amino acid substitutions in
  different proteins between different
• mammals (haemoglobin, cytochrome c). One amino
  acid change per 107 years.
• Those numbers did not make sense
• Human genome 109 nucleotides, mutation
  rate 10-8.
• The probability to produce an in tact
  offspring is 0.000045. Thus we should have
• 22,0000 offspring to produce one in tact
  offspring.
• 30 of Drosophila loci are polymorphic. All
  selected for?
• Kimuras suggestion The vast majority of
  single-nucleotide changes are
• selectively neutral.
• This is now well established, and neutral
  evolution is often used as the
• null model of molecular evolution.

8
Motoo Kimura (1924-1994)

• Introduced the neutral theory.
• Developed very important new mathematical tools
  in population genetics
• (the application of stochastic differential
  equations and diffusion models).

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Genetic drift Evolution without selection or
mutation
A population of fixed size. Each individual has
its own type (color). Individuals reproduce
clonally.
Parent generation
Offspring generation
Each individual has the same reproductive success
on average Each offspring individual in the new
generation has a parent chosen at random from
the parent generation.
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Genetic drift Evolution without selection or
mutation
A population of fixed size. Each individual has
its own type (color). Individuals reproduce
clonally.
Parent generation
Offspring generation
Each individual has the same reproductive success
on average Each offspring individual in the new
generation has a parent chosen at random from
the parent generation.
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Genetic drift Evolution without selection or
mutation
A population of fixed size. Each individual has
its own type (color). Individuals reproduce
clonally.
Parent generation
Offspring generation
Each individual has the same reproductive success
on average Each offspring individual in the new
generation has a parent chosen at random from
the parent generation.
12
Genetic drift Evolution without selection or
mutation
A population of fixed size. Each individual has
its own type (color). Individuals reproduce
clonally.
Parent generation
Offspring generation
Each individual has the same reproductive success
on average Each offspring individual in the new
generation has a parent chosen at random from the
parent generation.
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Genetic drift Evolution without selection or
mutation
A population of fixed size. Each individual has
its own type (color). Individuals reproduce
clonally.
Parent generation
Offspring generation
Each individual has the same reproductive success
on average Each offspring individual in the new
generation has a parent chosen at random from the
parent generation.
Some have no offspring!
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Genetic drift Eventually one color will take
over
2 N generations.

• For a clonally reproducing population of size N.
  After on average 2 N
• generations all but 1 lineage will go extinct.
• More complex for sexually reproducing entities
  but qualitative the same
• idea Almost all genetic material stems from a
  very small fraction of the
• ancestral population more than 2 N generations
  ago.

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Genetic drift with mutation

• Each time an individual reproduces there is a
  probability µ that it
• mutates and introduces a new color.

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Genetic drift with mutation
(1-µ)

• Each time an individual reproduces there is a
  probability µ that it
• mutates and introduces a new color.

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Genetic drift with mutation
µ
(1-µ)

• Each time an individual reproduces there is a
  probability µ that it
• mutates and introduces a new color.

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Genetic drift with mutation
2 N generations.

• Each time an individual reproduces there is a
  probability µ that it
• mutates and introduces a new color.
• In the limit of large time the number of
  different colors will on
• average equal C 1 2 N µ.
• In the example above µ 1/8.

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Genetic drift with mutation
The product Nµ determines the amount of genetic
diversity in a population. When Nµ 1 almost
each individual is unique. When Nµ 1 a single
type will dominate the population. Example HIV
virus. µ 10-5 per nucleotide and N 107-108
infected cells in a host. This means almost
every nucleotide is variable in the
population. Example Human µ 10-8 per
nucleotide and N 103-105 (?) A typical
nucleotide shows almost no variation in the
population. µ 10-5 per gene. A typical gene
will have few variants in a population. µ 1
per genome. Every genome is essentially unique.
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Drift and SelectionDoes a beneficial mutant
always take over?
? Fitness of mutant relative to the rest of the
population. i.e 2 means reproducing twice as
fast. p probability that the mutant will take
over the population. Thus, even with a 25
increase in fitness, the probability of the
mutant spreading is about 40 Mutants with small
fitness effects likely need to be discovered
several times before they spread.
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Ronald Aylmer Fisher (1890-1962)

• The theory of natural selection.
• Major contributions to the theory of statistics.

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Can a deleterious mutant take over?
Yes
Threshold s 1/N
When s0, the mutant is neutral and the take over
probability is 1/N. The larger the decrease in
fitness s, the smaller the chance of take over.
Selection can not see fitness differences less
than 1/N !
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SummaryDrift-mutation and Drift-selection
balance
Ns
1
1
Nµ
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Genotype space and fitness landscapes

• Genetic information is carried by the DNA. A
  genotype is thus a
• long string over a 4 letter alphabet A,C,G,T.
• Genetic variations
• single point mutations (replacing a single
  nucleotide).
• small insertions and deletions.
• recombination (gene conversion).
• excision and integration of mobile genetic
  elements.

Focus on point mutations only.

• Genotypes DNA sequence of length L.
• Genotype space has 4L points.
• Each point has 3L single point mutant
  neighbors.

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Example genotype spaces Two nucleotide alphabet
A,T, sequences of length L1,2,3,4.
and so on
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Fitness landscapes
fAA
Intuitive picture
fitness
fAT
fTA
fitness
fTT
genotypes
In evolution populations move uphill
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Sewall Wright (1889-1988)

• Inventor of the adaptive landscape metaphor
  (1932).
• One of the founders of mathematical population
  genetics.
• Here together with Kimura in 1968 (the year of
  the neutral
• theory!)

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The Eigen Quasispecies model

• Genotypes g are strings of length L over
  alphabet A,C,G,T or A,T.
• Each genotype g has an associated fitness fg
  which denotes
• the reproduction rate of g.
• The population rate is kept constant by randomly
  killing individuals
• at the same rate as the overall reproduction
  rate.
• At each replication, each letter mutates with
  probability µ.
• Pg is the fraction of the population with
  genotype g.

After some time Pg will take on a limit
distribution. That is, the Pg will not change
anymore. This distribution is called the
quasispecies.
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Two Quasispecies examples
Fitness
Quasispecies
Quasispecies
Fitness
Distance from all A string
Distance from all T string.

• Length 40 strings A,T. Mutation rate µ0.02
• The right peak is higher and steeper than the
  left peak.
• Most of the population is concentrated below
  the summit.
• Mutation-selection balance determines this
  distance from the summit.
• The average fitness in both populations is the
  same, i.e. the population
• at a broad peak may outcompete a population at a
  higher, steeper peak.

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The Error threshold
d0 (summit)
error threshold
Fraction of the population
d3
d22
d25
d2
d1
mutation rate µ

• Length L50, A,T strings.
• Fitness all A string (d0) is 3, fitness of all
  other strings is 1.
• Error threshold occurs (roughly) when f0 (1-µ)L
  1. Here at µ 0.0217.
• This threshold generally determines the balance
  between selection,
• mutation, and the amount of genetic
  information necessary to maintain the
• fitness.

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Error threshold numerical examples

• The minimal increase in relative fitness
  necessary to sustain dL nucleotides of genetic
• information.
• Bands show the selection/drift balance.
• For human mutational meltdown only becomes an
  issue for dL 10000. Below that
• drift is the main issue.
• For HIV drift is never the issue. Adding of a
  whole gene to the HIV genome needs
• fitness increase of 1-10 or more.

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More realistic fitness landscapesRNA secondary
structure
Each RNA molecule will (in solution) fold into
some secondary structure.

• Assume that the fitness/reproduction rate of a
  RNA molecule depends
• only on its secondary structure.
• Examples, fitness decreases with distance from
  structure to a particular
• target structure, e.g. tRNA.

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More realistic fitness landscapesRNA secondary
structure

• Population evolving from random starting RNAs.
• Long periods in which the same best structure
  dominates the population. The genetic make-up of
  the
• population keeps changing during these periods.
• Every transition corresponds to a single point
  mutation.

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More realistic fitness landscapesNeutral
networks

• Every color corresponds to a phenotype.
• Sets of genotypes with the same phenotype form
  neutral networks that are intertwined with one
• another.
• During evolution populations drift over these
  neutral networks without observable changes in
• phenotypes.
• Populations are constrained by to go where
  neutral evolution can take them.

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So what about this case?

• Each sequence is representative of its species.
• Only when Nµ Laverage number of neutral mutations D-loop mtDNA
  can undergo, N is the species population size
  and µ is the per nucleotide mutation rate.
• The relationships of the sequences in the tree
  reflect the evolutionary history of the species.
• Generally yes, if the selective pressures
  on the D-loop mtDNA in these species is
  comparable.
• That is, if these pieces of DNA evolved on a
  common (or very similar neutral network).
• The length of the branches correspond to the
  evolutionary distances in time.
• Assumes (in addition the the previous
  assumptions) that the parts of the neutral
  network that the species evolved on have similar
  structure.

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Neutral evolution of mutational robustness

• Some genotypes have more neutral neighbors than
  others.
• Evolution will automatically concentrate the
  population on these genotypes that have most
• neutral neighbors, i.e. those that are robust
  against mutations.
• The magnitude of this effect is a structural
  feature of the neutral network.

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References and further reading

• General population genetics theory
• Hartl and Clark, Principles of population
  genetics. Sinauer Associates.
• The neutral theory of molecular evolution
• M. Kimura, The neutral theory of molecular
  evolution, Cambridge University press.
• M. Kimura, Population genetics, molecular
  evolution, and the neutral theory
• (selected papers), edited by Naoyuki Takahata.
  The University of Chichage press.
• The Eigen Quasispecies model
• M. Volkenstein, Physical approaches to
  biological evolution, Springer-Verlag
• M. Eigen and P. Schuster, The Hypercycle a
  principle of natural self-organization,
• Springer 1979.
• M. Eigen, J. McCaskill, P. Schuster, The
  Molecular Quasi-species, Adv. in Chem. Phys., 75
  (1989)149-263.
• Neutral networks (only specialized literature
  unfortunately)
• W. Huynen, P. Stadler, and W. Fontana,
  Smoothness within ruggedness, the role of
  neutrality in adaptation. PNAS 93 (1996) 397-401.
• W. Fontana and P. Schuster. Continuity in
  evolution On the nature of transitions,
• Science 280 (1998) 1451-1455.
• E. van Nimwegen, M. Huynen and J. Crutchfield,
  Neutral evolution of mutational robustness, PNAS
  96 (1999) 9716-9720.