APM 541: Population Genetics Models

41 Pages·2013·2.37 MB·English

by  Jay Taylor

APM 541: Stochastic Modelling in Biology
Population Genetics
Jay Taylor
Fall 2013
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;
Motivation
The central aim of population genetics is to understand the causes and
consequences of genetic variation ...
polymorphism within populations divergence between populations
(Gilbert, 2007)
(Durbin et al., 2010)
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Motivation
Why study genetic variation?
Genetic variation is the raw material of evolution. To understand how species arise
and adapt to their environments, we need to understand genetic variation and its
relationship to phenotypic differences.
Genetic variation can be used to infer demographic history and population
structure. Human genetic variation strongly supports a sub-Saharan African origin
for our species, followed by a colonization of the Near East some 100,000 years
ago and subsequent migrations into Europe and Asia (∼ 40-50 kya). However,
there is also growing evidence of admixture between modern and archaic humans,
including Neanderthals and Denisovans.
Genetic variation is increasingly relevant to medicine. Genome-wide association
studies can be used to identify mutations associated with disease phenotypes,
which in turn can be used to assess risk and to design individualized treatments.
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Motivation
Variation is influenced both by molecular and by demographic processes.
Mutation and recombination tend to increase variation by creating new
genotypes and sometimes re-creating old genotypes that have been lost.
Natural selection alters the genetic composition of populations and can either
reduce or increase variation depending on its mode of action.
Purifying selection tends to reduce variation.
In some cases, balancing selection and diversifying selection can act to
maintain variation.
Genetic drift (demographic stochasticity) tends to reduce genetic variation
through the random loss of rare alleles.
Migration can increase local levels of variation.
One of the challenges to understanding evolution and biodiversity is that these depend
on processes that occur on spatial and temporal scales spanning many orders of
magnitude.
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Motivation
Wing color variation in the scarlet tiger moth (Callimorpha dominula) is controlled by a
simple Mendelian polymorphism. However, the genetic composition of the Cothill Fen
population in the UK changed substantially between 1939 and 1979 (O’Hara, 2005).
Estimated frequency of the medionigra morph
in the Cothill scarlet tiger moth population
0.12
0.1
0.08
frequency0.06
0.04
0.02
0
1935 1940 1945 1950 1955 1960 1965 1970 1975 1980
year
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GeneticDrift:MoranModel
The Moran Model
Assumptions:
The population size is constant, with N haploid individuals.
Two alleles, A and A , are present in the population.
1 2
For now, we will ignore mutation.
Likewise, we will also assume that all individuals have the same fitness, i.e.,
reproduction and mortality do not depend on an individual’s genotype. When this
latter condition holds, we say that the two alleles are neutral.
Births and deaths occur continuously throughout the year. Specifically, at each
time step, one of the N individuals is chosen uniformly at random and gives birth
to a single offspring. A second individual is then chosen uniformly at random and
dies.
This model was introduced by P. A. P. Moran in 1954 and is one of the most
widely-studied processes in population genetics.
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GeneticDrift:MoranModel
Let Y denote the number of copies of the A allele contained in the population at time
t 1
t. Then Y takes values in the set E ={0,1,··· ,N} and Y and Y differ by at most
t t+1 t
one.
The number of copies of A will increase by 1 if an A individual reproduces and
1 1
an A individual dies.
2
Thenumberofcopieswilldecreaseby1ifanA individualreproduceswhileanA
2 1
individual dies.
The number of copies will stay the same if the individual that gives birth and the
individual that dies have the same genotype.
This shows that Y =(Y :t ≥0) is a birth-death process with the following transition
t
probabilities:
8
p(1−p) if j =k−1 or k+1
<
P(Y =j|Y =k)= 1−2p(1−p) if j =k
t+1 t
: 0 otherwise,
where p=k/n is the frequency of allele A in the population.
1
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GeneticDrift:MoranModel
Units
Frequencies: Since the population sizes of most species are unknown, the Moran model
is usually formulated in terms of the allele frequencies:
1
X = Y .
t N t
Here X =(X :t ≥0) is itself a birth-death process, but it takes values in the space
t
{0,1/N,··· ,1}. This is useful because we can often estimate allele frequencies (by
genotyping samples of individuals) even when we are unable to directly estimate N.
Time scale: We should also consider the time scale of this process. Here time is
measured in units such that there is one birth and one death event per time step. Since
it will be convenient to convert to a generational time scale for comparison with other
models, we note that the generation time for the Moran model is N time steps. To see
this, note that each individual has probability 1/N of dying per time step. Thus the life
span of each individual is geometrically distributed, with parameter 1/N, and the mean
of this distribution is N.
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GeneticDrift:MoranModel
Because 0 and N are the only absorbing states for this model, the ultimate fate of allele
A is either to be lost from the population (p=0) or to rise in frequency until it is the
1
only allele remaining at that locus (p=1). In the latter case, we say that the allele has
been fixed in the population. An important quantity in population genetics is the
fixation probability of an allele, which depends on its initial frequency and is defined as
u(p)≡P (X =1 for some t >0|X =p).
p t 0
Since this is the same as the probability of absorption at 1 given the initial frequency p,
we know that the quantities u(p) satisfy the following system of linear equations:
u(0) = 0
u(1) = 1
u(p) = p(1−p)u(p−1/N)+(1−2p(1−p))u(p)+p(1−p)u(p+1/N).
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GeneticDrift:MoranModel
By subtracting u(p) from both sides and then dividing by p(1−p), this last equation
can be rewritten as
0=u(p−1/N)−2u(p)+u(p+1/N)
which has the general solution
u(p)=Ap+C
where A and C are constants. In this case, we know that u(0)=0 and u(1)=1, which
shows that u(p)=p. Thus
The fixation probability of a neutral allele is equal to its initial frequency.
In particular, the fixation probability of a new neutral mutation is just 1/N.
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