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Gene Substitution
Dan Graur
1
Gene substitution is the process
whereby a mutant allele completely
replaces the predominant or wild
type allele in a population.
Gene substitution occurs when a mutant
allele arises in a population as a single
copy in a single individual, increases its
frequency to 1 (i.e., becomes fixed) after
a certain number of generations.
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Frequency of 1
Very low frequency
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Not all mutants, however, reach
fixation. In fact, the majority of them
are lost after a few generations.
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Very low frequency
Frequency of 0
5
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Fixation probability
The probability that a particular allele will
become fixed in a population depends on
(1) its frequency
(2) its selective advantage or disadvantage
(3) the effective population size
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The case of genic selection
1. three genotypes A1A1, A1A2, A2A2
2. fitness values:
1, 1 + s, 1 + 2s,
The probability of fixation of A2 is:
P
1 e
4Ne sq
4N s
e
1 e
where q is the frequency of allele A2.
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P
1 e
4Ne sq
4N s
e
1 e
As s approaches 0 (neutral mutation),
the equation reduces to
P q
The fixation probability for a neutral allele
equals its frequency in the population.
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A new mutant arising in a diploid population
of size N has an initial frequency of 1/(2N).
If the mutation is neutral the probability of
fixation is P = 1/(2N).
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For a neutral mutation, i.e., s = 0
1
P
2N
For positive values of s and large values
of N
P  2s
Less than
100%
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Thus, if an advantageous mutation arises
in a large population and its selective
advantage over the rest of the alleles is
small (up to ~5%), then the fixation
probability is approximately twice its
selective advantage.
For example, the probability of fixation of
a new codominant mutation with s = 0.01
is 2%.
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Population
size
Neutral
mutation
Advantageous
mutation
(s = 0.01)
Deleterious
mutation
(s = –0.001)
1,000
0.05%
2%
0.004%
10,000
0.005%
2%
–
20
~10
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Mutation accumulation assay
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Fixation Time
The time required for the fixation (or the
loss) of an allele depends on:
(1) its frequency
(2) its selective advantage or disadvantage
(3) the effective population size
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Conditional Fixation Time
The time of fixation of mutants which
do not undergo fixation is ∞.
Thus, we only deal with the mean fixation
time of those mutants that will eventually
become fixed in the population. This
variable is called the conditional fixation
time.
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Conditional Fixation Time
In the case of a new neutral mutation whose initial frequency
in a diploid population is by definition q = 1/(2N), the mean
conditional fixation time is approximated by
t  4N generations
For a mutation with a selective advantage of s, the mean
conditional fixation time is approximated by
t  (2 /s)ln( 2N) generations
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Population Generation
size
time
Neutral
mutation
1,000,000
8 million
years
2 years
Advantageous Deleterious
mutation
mutation
(s = 0.01)
(s = –0.01)
5,800
years
?
a. Less than 5,800 years
b. More than 8 million years
c. More than 5,800 but less than 8 million years
d. 8 million years
e. 5,800 years
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Population Generation
size
time
Neutral
mutation
1,000,000
8 million
years
2 years
Advantageous Deleterious
mutation
mutation
(s = 0.01)
(s = –0.01)
5,800
years
5,800
years
a. Less than 5,800 years
b. More than 8 million years
c. More than 5,800 but less than 8 million years
d. 8 million years
e. 5,800 years ✔
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Rate of Gene (or Allele)
Substitution
= number of mutants reaching
fixation per unit time
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Rate of Gene Substitution
Neutral mutations:
If neutral mutations occur at a rate of u per
gene per generation, then the number of
mutants arising at a locus in a diploid
population of size N is 2Nu per generation.
The probability of fixation for each neutral
mutation is 1/(2N).
The rate of substitution of neutral alleles is
obtained by multiplying the total number of
mutations by the probability of their
fixation.
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2Nu
K
2N
K u
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Intuitive explanation:
In a large population the number of mutations
arising in every generation is high, but the
fixation probability of each mutation is low.
In a small population the number of mutations
arising in every generation is low, but the fixation
probability of each mutation is high.
The rate of substitution for
neutral mutations is independent
of population size.
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Rate of Gene Substitution
Advantageous mutations:
If advantageous mutations occur at a rate
of u per gene per generation, then the
number of mutants arising at a locus in a
diploid population of size N is 2Nu per
generation.
The probability of fixation for each
mutation is 2s.
The rate of substitution of advantageous
alleles is 4Nsu.
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Deleterious mutations
Neutral mutations
Advantageous mutations
Overdominant mutations
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Mutational Meltdown: The double jeopardy of small
populations
It is possible for deleterious mutations to become
fixed via genetic drift.
Deleterious mutations occur more frequently than
advantageous mutations.
In small populations, random genetic drift is more
important than selection.
Small populations may be driven to extinction due to
(1) accumulation of deleterious alleles, and (2) the
fact that selection is too week to allow for
advantageous mutations to accumulate.
Michael Lynch
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Multilocus models
Previously, we assumed that the genetic transmission of an
allele at one locus was independent of the transmission of
another allele at a different locus. Under this assumption,
we could treat each locus separately.
In practice, however, the transmission of an allele at a
locus may be dependent on the transmission of alleles at
other loci. The most common cause for this lack of
independence is linkage, i.e., the close physical proximity
of two loci on the same chromosome and the finite rate of
meiotic recombination in the sequence separating the two
loci from each other.
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Linkage equilibrium and disequilibrium
A diploid organism.
Two autosomal loci, A and B.
Each locus with two alleles, A1 and A2 at locus A, and
B1 and B2 at locus B.
Linkage equilibrium occurs if the association between
the alleles at the two loci is random.
Linkage disequilibrium occurs if some combinations of
alleles occur significantly more or significantly less
frequently in a population than would be expected from a
random association between the alleles at the two loci.
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Hitchhiking and genetic draft
 A population withtwo neutral haplotypes, A2B1 and A2B2,
coexist with frequencies of p2 and q2, respectively.
 An advantageous mutation, A1, arises on the haplotype
carrying the B1 allele. (Completely arbitrary, it could have arisen
on on the haplotype carrying the B2 allele.)
Without the advantageous allele arising at locus A, the
probability of fixation for alleles B1 and B2 would have been p2
and q2, respectively.
The linkage to the advantageous allele A1, however, alters
these expectations. On its way to fixation, the advantageous
mutation A1 will carry along the linked B1 allele, and will
ultimately render the population monomorphic at locus B.
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Hitchhiking and genetic draft
 Advantageous mutations reduce or eliminate genetic variation at
genetically linked sites (selective sweep).
 A neutral or even deleterious allele that is sufficiently tightly
linked to a positively selected allele increases its frequency and may
be swept to fixation (genetic hitchhiking).
In genetic hitchhiking, only the initial conditions are stochastic, the
rest of the process is deterministic (genetic draft).
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Selective sweeps leave several characteristic molecular
signatures in the population:
1.Eliminate nucleotide variation in the region of the
genome close to the beneficial allele.
2.Cause an excess of high-frequency derived (new)
alleles.
3.Create long-range associations with neighboring loci—
the “long-range haplotype,” That is, a selective sweep will
lead to creation of linkage disequilibrium over large
swaths of DNA around the positively selected variant.
4.The positive selection in one population causes large
frequency differences between populations—larger than
for neutrally evolving alleles.
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A selective sweep takes approximately
2N e 
2ln 
generations.
 s 
In addition, the signature of positive selection
may be identifiable for an additional amount of
time, depending on the rates of mutation and
recombination in the relevant region.
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For how long after the fact can an
evolutionary detective identify a
selective sweep in the human
population?
35
The estimated human effective population size is
~10,000. The mean generation time is 25 years.
If a lucky mutation has a selective advantage of
5%, the sweep will be complete in ∼10,000 years.
If a lucky mutation has a selective advantage of
1%, the sweep will be complete in ∼50,000 years.
SELECTIVE SWEEPS CAN ONLY BE
DETECTED FOR VERY SHORT PERIODS OF
TIME
36
Detecting recent selective sweeps due to selection
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Why are we (adult UH students) able to drink milk?
38
The digestion of the disaccharide lactose, the
primary sugar present in milk, into its
monosaccharide constituents, glucose and
galactose, is catalyzed by a small-intestine
enzyme called lactase-phlorizin hydrolase (LPH
or lactase).
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Lactase persistence
In mammals, levels of lactase decline rapidly after
weaning, and adults are not able to digest lactose. In
humans, most individuals are unable to digest
lactose as adults (lactose intolerant), i.e., they carry
the trait lactase nonpersistence. Digestion of fresh
milk in individuals who are lactose intolerant can
result in diarrhea, which for most of human history
was lethal.
In populations in which the only source of milk is the
mother, lactase nonpersistence is a selectively
advantageous trait, since breastfeeding is a potent,
albeit imperfect, contraceptive, which inhibits
menstruation and delays resumption of ovulation.
However, in some populations, a derived genetic trait
has appeared, in which the ability to digest lactase is
maintained in adults. Such individuals are lactose
tolerant due to lactase persistence. This trait is
particularly common in populations that have
traditionally practiced dairying, i.e., in populations
which can obtain milk extramaternally.
Lactase persistence
Lactase persistence arose at least
twice in human populations
The lactase-persistence haplotypes
West Africa
North Europe
Bersaglieri et al. 2004
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Background selection
In the case of strong negative selection on a locus,
genetically linked (neutral & advantageous) variants will
also be removed, producing a decrease in the level of
variation surrounding the locus under purifying
selection. This process of purging non-deleterious alleles
from the population due to spatial proximity to
deleterious alleles is called background selection.
Background selection is the opposite of Selective sweep.
Because the deleterious mutations driving background
selection are removed from the population, they are
extremely difficult to detect.
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Epistasis
Previously, we assumed that each locus contributes
independently to the fitness of the individual (i.e.,
different loci do not interact with one another in any
manner that affects the fitness). Thus, each locus can
be dealt with separately.
This is not, however, always the case!
Epistasis refers to interactions among alleles at
different loci resulting in “non-independent effects.” In
other words, epistasis occurs when the effects of an
allele at one locus are modified by one or several alleles
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at other loci.
Epistasis
Epistasis may be defined at the fitness level or at the
level of the phenotype. We distinguish between
functional epistasis, in which alleles at different loci
produce non-independent phenotypic effects, and
fitness epistasis, in which alleles at different loci nonindependently determine the fitness of their carrier,
whether or not epistasis is detectable at the level of the
phenotype.
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Epistasis
The genetic-background effect, according to which a
mutation may have different effects on fitness
depending on the genome in which it occurs, may be
regarded as a generalized kind of fitness epistasis.
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Epistasis
Positive epistasis means that the phenotype (or the
fitness) is higher than expected.
Negative epistasis means that the phenotype (or the
fitness) is lower than expected.
In the literature, one may find different terms, such as, synergistic,
diminishing, antagonistic, aggravating, ameliorating, buffering,
compensatory, and reinforcing… Confusing!
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Epistasis
Positive epistasis means that the phenotype (or the
fitness) is higher than expected.
Negative epistasis means that the phenotype (or the
fitness) is lower than expected.
 Mutation a at locus 1 increases IQ by 1 point.
 Mutation b at locus 2 increase IQ by 2 points.
 The two mutations together (say, following recombination)
increase IQ by 12 points.
 Is the epistasis positive or negative?
 Is the epistasis functional or fitness epistasis?
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