Transcript Genetically Effective Population Size

Chapter 8
Evolution in Small Populations
The central problems are losses of genetic
diversity in small populations and changes in the
distribution of this diversity among populations.
Losses of genetic diversity can result in reduced
evolutionary flexibility and decline in fitness.
Changes in distribution of diversity can destroy
local adaptations and break up co-adapted gene
complexes (outbreeding depression).
Both of these problems can lead to a poorer
“match” of the organism to its environment, reducing
individual fitness and increasing the probability
of population or species extinction.
Therefore, conservation biologists should be
concerned with maintaining, as much as possible,
natural levels of genetic variation and natural
patterns of genetic diversity so that evolutionary
and ecological processes can continue!
Four factors that can reduce genetic variation
and are a function of population size:
Founder Effect
Population Bottleneck
Genetic Drift
Inbreeding
The severity of these factors is dependent upon
the Genetically Effective Population Size (Ne)
and not the absolute number of individuals or
census size (Nc).
Population size is defined in terms of the
equivalent size of a standardized population called
the “Idealized Population”.
We begin by assuming an infinitely large, randomly
mating base population, from which we take a
sample of N adults to form the “Ideal Population”.
The idealized population is maintained as a
randomly mating, closed population in succeeding
generations in which alleles may be lost by chance
and allele frequencies may vary due to sampling
variation.
Conditions of the Idealized Population:
No migration
Generations are distinct & do not overlap
Number of breeding individuals is the same in all
generations
All individuals are potential breeders
All individuals are hermaphroditic
Union of gametes is random, including the possibility
of selfing.
No selection at any stage of life-cycle
No mutation
Number of offspring per adult averages 1 and has
a variance of 1
The Effective Population Size (Ne) is the size of
an idealized population that would lose genetic
diversity, or become inbred, at the same rate as
the actual population.
In practice, the effective size of real populations
is usually much smaller than the number of breeding
individuals because real populations deviate in
structure from the idealized population by having
unequal sex-ratios, high variances in family size,
variable numbers of successive generations, and
in having overlapping generations.
Sex Ratio effective population size:
Ne = (4Nm X Nf)/(Nm + Nf)
Example: A census population of 500 individuals
would have an Ne of 500 (with respect to sex
ratio) if all individuals bred and there was a 1:1
sex ratio such as:
Ne = (4 X 250 X 250)/(250 + 250) = 500
However, if in this population of 500 with 250
females and 250 males, only 114 and 63 females
and males, respectively, bred, then Ne would be:
Ne = (4 X 114 X 63)/(114 + 63) = 28,728/117 = 162.31
This relationship can produce some surprising
results.
For example, one male breeding with four females
results in an Ne of 3.2, not much different than
one male breeding with 9 females (Ne = 3.6)
Effective population size is also strongly affected
by the distribution of progeny among females
(family size) and is estimated as:
Ne = (4N)/(2 + 2)
Where 2 is the variance in family size among
females.
Higher variances result in smaller effective
population sizes.
Effects of variance in number of progeny among
females on Ne.
2
0
1
2
5
10
Nc
100
100
100
100
100
Ne
200
133
100
57
33
Ne/Nc
2.00
1.33
1.00
0.57
0.33
If males can mate with more than one female, the
variation in family size (Vk) for males and females
must be incorporated since Vk is likely to be
different for males and females.
Ne = 8Nc/(Vkm + Vkf + 4)
Large population fluctuations also reduce Ne
because every time a population crashes to small
size, it experiences a demographic bottleneck.
The harmonic mean of population size in each
generation provides an estimate of Ne as follows:
1/Ne = 1/t(1/N1 + 1/N2 + 1/N3 + . . . . . + 1/Nt)
Where t is the time in generations.
The effects of a single population crash on
the effective population size can be seen as
follows.
Year
1
2
3
4
5
Nc
800
1000
25
500
1000
5-year mean of Nc = (800+1000+25+500+1000)/5
=
665
Ne = 1/5(1/800+1/1000+1/25+1/500+1/1000)
=
110.5
Two types of Genetically Effective Population Size
Inbreeding Effective Size (Nei)--measures the
rate of loss of heterozygous individuals from a
local population (a loss of variation within
individuals), or simply the increase in inbreeding.
Variance Effective Size (Nev)--measures the rate
of loss of total genetic variation from a population
whether the loss is experienced within individuals
or among individuals.
Typically, Nei and Nev differ substantially only
when population size is significantly increasing
or decreasing.
Unfortunately, no theories or equations exist that
simultaneously handle multiple deviations from
the ideal situation.
Thus, influences of bottlenecks, skewed sex ratios,
and family size cannot at present be simultaneously
estimated with these equations.
Real Meaning of Effective Population Size
There is no such thing as the “effective population
size” in the sense that many people think about
this.
Effective size is defined with respect to a genetic
parameter of interest, and as the parameter of
interest changes, the effective size can change.
Hence, a population can be characterized by
several different “effective sizes” simultaneously.
Thus, the phrase “the effective population size”
is meaningless unless the genetic parameter of
interest is also supplied.
The important point is that, due to properties
associated with sex ratio, family size, and
population fluctuations, Ne is nearly always
significantly smaller than the census size.
Importance of Small Populations in Conservation
Biology
Small or declining populations are more prone to
extinction than large stable populations.
Population size is the most influential of the five
criteria for listing species as endangered under the
IUCN.
Species whose adult population sizes are less than
50, 250, or 1,000 are listed as critically endangered,
endangered, and vulnerable, respectively.
There are special evolutionary problems confronted
by small populations.
In small populations, the role of chance
predominates and the effects of selection are
typically reduced or even eliminated.
Chance introduces a random, or stochastic, element
into the evolution of populations.
Small populations become inbred at a faster rate
than do larger populations, as inbreeding is
unavoidable.
When a small population reproduces, the subsequent
generation is derived from a sample of parental
gametes.
Each offspring receives 1 allele, selected at random,
from each parent.
Just by chance, some alleles, especially rare ones,
may not be passed on to the offspring and may be
lost.
The frequencies of alleles that are transmitted
to the next generation are likely to differ from
those in the parental generation.
Over multiple generations, allele frequencies
change, or “drift”, from one generation to the
next, a process termed “genetic drift”.
Chance Effects -- Genetic Drift
Basic concept of genetic drift -- Evolution can be
thought of as a change in allele frequency and
finite population size alone ensures that evolution
will occur through sampling error.
Broadly speaking, there are four consequences of
random genetic drift.
These are not really consequences, but rather
different ways in which the consequences may
be seen.
Random Drift -- Random changes of gene frequency
which change in an erratic manner from generation
to generation, with no tendency to revert to its
original value.
Differentiation between subpopulations -- Random
genetic drift occurring independently in different
subpopulations leads to genetic differentiation
between subpopulations.
Uniformity within subpopulations -- Genetic
variation within each subpopulation becomes
gradually reduced, and the individuals become more
alike genotypically.
Increased homozygosity -- Homozygosity increases
in frequency at the expense of heterozygosity.
This coupled with the general tendency for
deleterious alleles to be recessive, is the genetic
basis for the loss of fertility and viability that
almost always results from inbreeding.
Genetic Drift -- Random sampling of gametes within
small populations has three consequences of major
importance in evolution and conservation:
Random changes in allele frequencies from one
generation to the next.
Loss of genetic diversity and fixation of alleles
within populations.
Diversification among replicate populations from
the same original source (e.g. fragmented pops.)
Fixation -- Genetic drift will ultimately cause all
except one allele to be lost and the remaining allele
is said to be “fixed”.
The probability of losing an allele is dependent on
its frequency and the population size.
In the case of two alleles, the probability of losing
one allele is the probability of fixing the other
allele.
The probability that a gamete does not contain
allele A1 is (1 - p).
Consequently, the probability that a randomly
mating population losses allele A1 (all individuals
become A2A2) is:
Pr(losing A1) = (1 - p)2N but,
= (1 - p)2Ne
The rarer the allele, the far greater the probability
of being lost.
The gene frequencies in these samples will have
an average value equal to that in the base
population and will be distributed about this mean
with a variance of:
variance = 2 = p0q0/2Ne
Although genetic drift is random, we can calculate
a 95% confidence interval for the magnitude of
change as follows:
q ± 2q
Example: You have a population of 200 lions
consisting of 100 males and 100 females,
with an allele frequency of the slow allele at the
ADH locus of 0.33. The breeding structure of
this group is that each male controls a pride of
5 adult, breeding females.
Calculate the 95% confidence interval for the
change in gene frequency due to genetic drift.
Step I: Calculate effective population size.
Ne = (4NfNm)/(Nf + Nm)
= (4 X 100 X 20)/(100 + 20)
= 8000/120 = 66.67
Step II: Calculate Variance in q
2 = (p0q0)/2Ne
= (0.67 X 0.33)/(2 X 66.67)
= 0.221 / 133.34
= 0.0017
Step III: Calculate 95% CI
q ± 2q
2q = 2 X (0.0017)0.5
= 2 X 0.04
= 0.08
Thus, 0.33 ± 0.08
Therefore, in the next generation
0.25 < q < 0.41
Founder Effect -- This occurs when a few
individuals establish a new population, the genetic
constitution of which depends upon the genes of
the founders.
Demographic Bottleneck -- Occurs when a
population experiences a severe, temporary
reduction in size.
The magnitude of loss of genetic variation depends
on the size of the bottleneck and the growth
rate of the population afterwards.
The proportion of genetic diversity remaining after
one generation, t, the next generation is:
1 - (1/2Ne)
This proportion can range from 0.5 (50% variation)
with an Ne of 1 (the gametes of 1 individual carry
on average 50% of the genetic diversity of the
population) to near 1.0 (100%) with a large Ne,
(e.g., 1,000)
Generally, a bottleneck rarely has severe genetic
or fitness consequences if population size
quickly recovers in a generation or two.
Population genetics theory tells us that perhaps
more important than depletion of quantitative
genetic variation by founder effects, bottlenecks,
or genetic drift, is loss of rare alleles from the
population.
However, little empirical evidence supports this
idea.
We know that rare alleles contribute little to
overall genetic diversity but they may be
important to a population during infrequent
or periodic events such as unusual temperatures
or exposure to new pathogens, and may offer
unique responses to future evolutionary change.
Impact of a bottleneck on heterozygosity:
H = -(1/2Ne)H0
Impact of a bottleneck on allelic diversity:
# alleles
A = n -

(1
i=1
- pi)2Ne
where n is the number of alleles before the
bottleneck and pi is the frequency of the ith
allele.
Inbreeding -- mating of individuals by common
ancestry.
Probability of occurrence increases in smaller
populations if mating occurs at random.
In a population of bisexual organisms, every
individual has 2 parents, 4 grandparents,
8 great grandparents, etc.
t generations back, an individual has 2t ancestors.
Not very many generations back, the number of
individuals required to provide separate ancestors
for all individuals in the population becomes
larger than any real population could attain.
For example, 50 generations back, would mean that
an individual would have 250 = 1.2 X 1015 ancestors.
Therefore, any pair of individuals must be related
to each other through one or more common
ancestors in the more or less remote past.
The smaller the size of the population in
previous generations, the less remote the common
ancestor.
Thus, pairs mating at random are more closely
related to each other in a small population than a
larger population.
For this reason, the properties of small populations
can be treated as a consequence of inbreeding.
Inbred individuals -- offspring produced by
inbreeding -- may carry two genes at a locus that
are replicates of the same gene in a previous
generation.
There are two types of identity among allelic
states -- two types of homozygotes.
Identical by Descent or Autozygous -- two genes
that originate from the replication of a single
gene pair in a previous generation.
Independent by Descent or Allozygous -homozygous individuals not known to be autozygous.
It is the production of autozygosity that gives
rise to increase of homozygotes as a consequence
of inbreeding.
The inbreeding coefficient (F) is the probability
that two genes at a locus in an individual are
identical by descent.
F refers to an individual and expresses the degree
of relationship between the individuals parents.
If the parents of any generation have mated
randomly, then F of the progeny is the probability
that 2 gametes taken at random from the parent
generation carry autozygous genes at a locus.
This is the average coefficient of inbreeding of
all progeny.
Individuals of different families will have different
inbreeding coefficients because with random
mating, some pairs of parents will be more closely
related than others.
The degree of relationship expressed in the
inbreeding coefficient is a comparison between the
population in question and some specified or
implied base populations.
Without this reference point, it is meaningless
because all genes now present at a given locus
would be found to be identical by descent if traced
back far enough.
Therefore, F only becomes meaningful if we
specify some time in the past beyond which
ancestries will not be considered and at which
time all genes in the populations are to be
considered allozygous (= independent).
This point of reference is the base population
and by definition, it has an inbreeding coefficient
of zero (F = 0).
Inbreeding in the “Idealized Population”
We will deduce F in successive generations,
beginning with the base population.
Examine a hermaphroditic marine organism capable
of self-fertilization, shedding eggs and sperm into
the sea.
There are N individuals, each shedding equal
numbers of gametes at random.
Because it is the base population, all genes at a
locus have to be considered as non-identical.
Therefore, considering only one locus, among the
gametes shed there are 2N different sorts,
bearing genes A1, A2, A3, . . . . , AN at the “A” locus.
The gametes of any one sort carry identical genes,
those of different sort carry genes of independent
origin.
Question: What is the probability that a pair of
gametes taken at random carry identical genes?
This is the inbreeding coefficient of generation 1.
Answer: Any gamete has a (1/2N)th chance of
uniting with another of the same sort.
Therefore, 1/2N is the probability that uniting
gametes carry identical genes, and is thus the
coefficient of inbreeding of the progeny in
generation 1.
Question: What is F in the second generation?
Answer: There are now two ways in which
identical homozygotes can be produced.
New replication of genes
Replication of genes from previous replication.
The probability of newly replication genes coming
together is again (1/2N) with the remaining portion
(1 - 1/2N) of zygotes carrying genes that are
independent in their origin from generation 1 but
may have been identical in origin in generation 0.
Thus, the total probability of identical homozygotes
in generation 2 is:
F2 = [(1/2N) + (1 - 1/2N)F1]
An increment
attributable to
new inbreeding
“remainder”, attributable
to the previous inbreeding
and having the inbreeding
coefficient of the previous
generation.
We designate the “increment” or new inbreeding
as: F = 1/2N then,
Ft = 1 - (1 - F)t where F0 = 0
We can also calculate the probability of pulling
out two genes that are NOT identical by
descent, or heterozygous as: Ht = 1 -Ft
Then by subtracting Ft, we can obtain the rate of
change in heterozygosity from random genetic
drift as: Ht = (1 - F)t ≈ H0e-(t/2N)
Note: IF EFFECTIVE POPULATION SIZE IS
KNOWN OR CAN BE CALCULATED, IT WOULD
BE MORE APPROPRIATE TO USE Ne IN THE
PREVIOUS EQUATIONS.
Example 1: In a population of 200 randomly mating
white rhinos, p0 = 0.73, what will heterozygosity
be in 50 generations?
Step I: Calculate q0.
q0 = 1 - 0.73 = 0.27
Step II: Calculate H0,
H0 = 2p0q0 = 2 X 0.73 X 0.27 = 0.394
Step III: Use equation Ht = H0e-(t/2N)
H50 = (0.394)e-(50/400)
H50 = (0.394)e-0.125
H50 = (0.394)(0.88)
H50 = 0.3477
Example 2: In a fish hatchery, all eggs (an
infinitely large number) are fertilized by the sperm
of a single male in each generation. Calculate rate
of inbreeding.
Step I: Calculate Ne
Ne = (4NmNf)/(Nm + Nf)
Ne = (4 X 1 X 10,000)/(1 + 10,000)
Ne = 3.9996
Step II: Calculate rate of Inbreeding
F = 1/2Ne
F = 1/(2 X 3.9996) = 1/7.9992 = 0.125
In small populations, matings among relatives is
inevitable not as a result of deliberate inbreeding
but simply as a consequence of small numbers of
founders and the small population size.
Inbreeding also becomes inevitable in larger
populations, but it takes longer.
For example, a population of size 100 over 57
generations becomes, on average, as inbred as
the progeny of brother-sister matings.
Inbreeding is of profound importance in
conservation biology as it leads to:
reductions in heterozygosity
reduced reproduction & survival
increased risk of extinction
Inbreeding results in predictable increase in
homozygosity and may be manifested as
Inbreeding Depression such as:
reductions in fecundity
reductions in offspring size
reductions in growth
reductions in survivorship
changes in age at maturity
physical deformities.
Two competing hypotheses for the mechanism
leading to inbreeding depression.
Dominance Hypothesis -- Inbreeding results in
more instances of deleterious recessive alleles
appearing in homozygous form, where they are
clearly expressed, rather than being masked by
dominance in the heterozygous state.
This hypothesis suggests that inbred populations
have already experienced exposure of deleterious
recessive alleles, and most have probably been
purged from the population.
Further inbreeding should not then have large
effects on fitness.
Overdominance Hypothesis -- focuses on the loss
of genome-wide heterozygosity and its presumed
fitness advantages.
This hypothesis predicts that further inbreeding
should result in continued loss of fitness
through further heterozygosity loss.
Data on inbreeding depression in the wild
are difficult to compile, as the level of inbreeding
is not easily determined.
Data from domesticated animals indicate F of 10%
will result in a 5% -- 10% decline in individual
reproductive traits such as clutch size or survival
rates; in aggregate, total reproductive attributes
may decline by 25%.
Many species are known to avoid inbreeding in the
wild, further evidence that inbreeding depression
is real and important.
However, not all inbreeding is cause for alarm.
Some natural populations apparently have
experienced low levels of inbreeding for many
generations with no ill effects.
Fundamental points of this section:
Small isolated populations will loose some fraction
of their original genetic diversity over time,
approximately at a rate of 1/2Ne per generation.
Small population numbers over prolonged periods
of time are to be avoided in conservation
programs whenever possible.
There should be concern about low genetic
variation in small populations, but it is by no means
a universal pattern.