Download Lecture 7 Effective population size Linkage disequilibrium basics

Lecture 7
Effective population size
Bottlenecks
Coalescence theory basics
Sonja Kujala
[email protected]
Effective population size
• Ne
• the number of individuals in a Wright-Fisher model that would
produce the same amount of genetic drift as in the real
population
• reminder!
2N=10
N=5
•
2N=100
N=50
the actual number of individuals in a population
Effective population size
• population size has an effect on genetic drift
effective population size is a measure on the
amount of drift taking place in a population
• effective population size can be measured in
various ways - for instance by estimating
heterozygosity:
”population that harbours the amount of
heterozygosity as does a Wright-Fisher
population of 100 individuals, has effective
population size of 100 individuals, even if
the census size was much larger”
Effective population size
Factors that affect effective population size:
• Not all individuals are reproductive
not all leave offspring
preproductive stage
postproductive stage
• number of breeding individuals
effective population size
Effective population size
Factors that affect effective population size:
1)
2)
3)
4)
5)
unequal sex ratio
variance in offspring number
mode of inheritance - special cases in the genome
real change in population size
level of inbreeding
 all of these factors influence the genetic contribution to the next
generation
• Ne/N describes how much the population deviates from the ideal
Effective population size
3 approaches to estimate Ne
Inbreeding effective population size
• relates to increase in inbreeding in a given population to
that in the ideal population
Variance effective population size
• relates to the increase in variance in allele frequency in a
given population to that of the ideal population
Eigenvalue effective population size
• relates to the loss of heterozygosity in a given population
to that in the ideal population
Usually give very similar estimates
Effective population size
In populations of 16
Drosophila, the changes
were similar as in ideal
populations of size Ne=9
Effective population size
1) unequal sex ratio
• often the number of breeding males is smaller
than the number of breeding females
• sometimes opposite, e.g. honeybees
• Ne, Nf, Nm
4𝑁𝑓 𝑁𝑚
𝑁𝑒 =
𝑁𝑓 + 𝑁𝑚
cows/bull
Effective population size
• Biased sex-ratios in Finnish moose population
(Nygrén 2009)
• Less biased in the north (Ne > 200), more biased in the
south (Ne ≈ 100) (Kangas et al. 2013)
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Effective population size
generation t-1
1/N
1/N
generation t
• imagine monoecious population
• N individuals
• Pr(two alleles in generation t came from the same parent in
generation t-1) = (1/N)2
 Pr(any two individuals inheriting alleles from the same
parent) = N x (1/N)2 =1/N
Effective population size
• dioecious population
• half of the gametes must come from females and half from
males
• Pr(two alleles in generation t came from a female in t-1) =
(1/2)2
• Nf females
-> Pr(two alleles in generation t came from a same
female in t-1) = 1/Nf
• Nm males
-> Pr(two alleles in generation t came from a same male
in t-1) = 1/Nm
Effective population size
• Pr(two alleles in generation t came from a same parent
in generation t-1) =
in general
separate sexes
rearranges to
4𝑁𝑓 𝑁𝑚
𝑁𝑒 =
𝑁𝑓 + 𝑁𝑚
• if equal numbers of females and males, what is Ne?
• Nf=Nm=1/2N
• substitute in the equation
-> Ne=N
Effective population size
• the effective population size as a function of the proportions
of males, for three different total numbers of individuals:
Effective population size
• Southern elephant seal Mirounga
leonina
• polygynous mating structure
– behavioral observations suggest 40
females/male
– genetic data: 4-5 females/male
(Slade et al. 1998 Genetics 149, 1945-57)
– success of matings overestimated
– one male dominates only for a couple of
years
Effective population size
2) variance in offspring number
• imagine a Wright-Fisher population of constant size
-> average number of offspring must be 2
• the distribution of progeny size (family size, k) is often
described with the Poisson distribution
Effective population size
Poisson distribution
• “expresses the probability of a given number of
events (k) occurring in a fixed interval of time and/or
space if these events occur with a known average
rate and independently of the time since the last
event”
• known expected value λ
• mean = variance
Effective population size
• if the distribution of progeny size is nonPoisson, Ne is affected
• when ≠ 2, i.e. population size is changing
= average of family size
Vk = variance of family size
• when = 2, i.e. population size is constant
Effective population size
• large variation in number and survival
of salmon eggs and fry
• family size has become very uniform in Japan through birth
control
-> Ne got larger than N due to decreasing variation in k
Effective population size
• what if the variance in progeny number is different
between males and females?
Effective population size
Pumas (Puma concolor) in Yellowstone
• Murphy 1998, Culver et al. 2004
• number of lifetime offspring
produced by individual females and
males in a population
• 1987-1995
• microsatellite markers
• Nef / Nf = 9.14 / 14
• Nem / Nm = 4.45 / 24
-> Ne = 11.97
Effective population size
3) mode of inheritance
- special cases in the genome
- X-linked genes
XX female
• if Nf = Nm -> Ne of X-linked genes is 3/4 of the Ne of autosomes
• if Nf ≠ Nm :
- genes inherited only through one sex
• if Nf = Nm -> Ne of these genes is 1/4 of the Ne of autosomes
• (transmitted through one sex only AND haploid in this sex)
• if Nf ≠ Nm :
• Y chromosome (if paternally inherited)
• mtDNA (if maternally inherited)
XY male
Effective population size
4) change in real population size
• if population size varies in time, across generations
• e.g. N0=1000, N1=10, N2=1000 (a bottleneck),
what is Ne across these three generations?
= the harmonic mean
• Ne = 29.4
• (the arithmetic mean would be 670)
Effective population size
• the lowest population numbers determine, to a large
extent, the overall Ne
• because after the size reduction, all individuals are
decendants of the bottleneck survivors
Effective population size
Estimating Ne from genetic data
• methods based on the expectation that genetic drift
increases as effective population size decreases
• data over generations
- temporal changes of allele frequency
- greater change in allele frequencies between
generations is expected in small populations
- heterozygote excess
- small population size will increase allele
frequency differences between males and females
-> results in excess of heterozygotes in their
progeny
- loss of heterozygosity
- after a bottleneck event
- due to founder effect
• short term estimates
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Reminder
Bottlenecks
• the size of a population decreases at least for one
generation
• founder effect
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Bottlenecks
• strength of the bottleneck =  / f (Gil McVean 2002)
27
Bottlenecks
Lion (Panthera leo )
• Driscoll et al. 2002, Genome Res
• Gir-population in eastern India
• A previously large and continuous
population decreased to 20
individuals due to hunting and
reduction of habitats
• Presently about 250 individuals left
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Bottlenecks & Coalescence theory
Coalescence theory basics
• very sequence data oriented way of thinking about
population genetic processes
• what kind of processes generated the current data
• genealogical trees describe the history of a piece of DNA
(a gene)
• genealogical trees
phylogenetic trees!)
• based on models such as Wright-Fisher model
• this is about drift – probability that two sequences share
a common ancestor - coalesce – in the previous
generation
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Bottlenecks & Coalescence theory
• modelling backwards
in time
• coalescence event ( )
• most recent common
ancestor (MRCA)
• time in 2N generations
MRCA
past
present
Bottlenecks & Coalescence theory
• stochastic variation among gene genealogies
Bottlenecks & Coalescence theory
• adding mutations
• the expected number of
mutations on a branch is
proportional to the
branch length
Bottlenecks & Coalescence theory
• coalescence time is proportional to the population size
• lineages in small populations coalesce quickly (short
branches, fitting fewer mutations)
• in large populations lineages coalesce slowly (longer
branches, fitting more mutations)
• if pop size changes, so will the rate of coalescence
population size growth -> long external branches
population size decrease -> short external branches
• ”average” shape of the tree:
past
present
constant
expanding
decreasing
Bottlenecks & Coalescence theory
• what do the gene genealogies look like after
bottleneck?
past
present
Bottlenecks & Coalescence theory
• age of bottleneck
- t2 represent sampling shortly after bottleneck
- t1 sampling after a longer time
• severity of bottleneck
• throw in mutations, where will they land?
• shape of the tree determines the relative abundance of common vs.
rare variants
Bottlenecks & Coalescence theory
• bottlenecks reduce diversity
• long term effective population size
𝜃 = 4𝑁𝑒 𝜇
Wall et al. 2008
Genome Research
•
•
•
•
parameters estimated from sequence data
Ne of the African populations is larger
DOES NOT mean that there are more individuals
DOES mean that African populations have been able to maintain
more genetic diversity than other populations
• bottleneck effects in other populations, ”Out of Africa”