**TODAY**,
we explore the effects of finite population size and inbreeding on genetic
variation, and show that this can lead to random
evolutionary change (or "drift"). Mutation is, of course, a sort of random
genetic change, but genetic drift can work much faster.

First
we must study the theory of inbreeding, which can be "regular", for instance
in sib-sib mating such as the Pharaohs of Ancient Egypt, or as a simple
effect of random mating in small populations. We first study **regular
systems of inbreeding**, then go on to how small population sizes
can cause both **genetic drift** and inbreeding.

**MEASURING INBREEDING**

If an individual mates with a relative
(or with itself! as in some plants or snails), the offspring may be homozygous
for a copy of an allele which is** identical by
descent** from one of the ancestors:

... in the diagram, a male is homozygous
for two copies of an allele `--`
inherited from a single copy in an ancestor. This is partly because his
mum was also his dad's niece (a type of inbreeding that is common in many
human societies).

The **INBREEDING
COEFFICIENT, F, **is used to gauge the strength
of inbreeding.

Note: two alleles that are **identical
by descent** must be **identical in state**.
However, a homozygote for an identifiable allele can often be produced
without inbreeding in its recent ancestry. Thus *identity
in state does not necessarily imply identity by descent*.

*Is inbreeding
always bad?*

Inbreeding is not
generally recommended because of the existence of deleterious recessive
alleles in most populations. Although these should be rare per gene
(usually much less than 10^{-3}, see mutation-selection
balance), there will be many deleterious alleles per genome.
According to some estimates, you and I each carry about 1 strongly deleterious
hidden mutation. When homozygous, these mutations reduce fitness; inbreeding
will therefore lead to **inbreeding depression**
as the homozygous mutations become expressed.

However, inbreeding isn't all bad, and many organisms habitually inbreed. Animals such as fig wasps and certain parasites regularly mate with their siblings, and selfing is common in many of the most aggressive weeds of agriculture. The advantage is presumably ecological, since a single female can then colonize an empty resource or host. There may also be a genetic advantage by preventing recombination between adaptive loci. One assumes deleterious recessives in habitually inbreeding species have mostly been purged by selection.

In human societies where some families have a lot of wealth, or where a bridal dowry is paid, inbreeding is common. Examples are European royal families, and on the Indian subcontinent. Perhaps here the idea is to prevent the "recombination" of wealth with other families!

In any case, mild
inbreeding, such as mating between first cousins, or uncle-niece isn't
so dangerous. Charles Darwin married his first cousin, Emma Wedgewood,
and had an astonishing 10 children. Some were sickly or died young,
but this was common in the days before penicillin.

We can measure** F** easily in
regular systems of inbreeding, using Sewall Wright's method of "path analysis":

1) Find each path that alleles may take to becomeCalculations like these are used in genetic counselling, and in animal breeding and in zoos to avoid inbreeding depression. Some examples:IBD.

2) Find the number of path segments () between gametes (eggs or sperm) through a single ancestor in common in each path.x

3) Calculate the probability ofIBDfor each path. The probability that an allele isIBDbetween two gametes connected through an individual is1/2. Thus, the probability ofIBDfor each path is(1/2).^{x}

4) Add up the probabilities of each path to get the total probability ofIBD.

Consider two alleles, **A**, and **a**
with frequencies *p,q* with inbreeding (**IBD**) at rate ** F**:

Frequency of homozygotes:

**AA**` = (1- F)p^{2}`
[outbred]

Similarly the frequency of the other homozygotes,
**aa**`=
q^{2
}+
Fpq`

All genotype frequencies must add to 1,
so the extra `2 Fpq AA`
and

So,genotypeAA Aa aafrequencyp^{2}+Fpq2pq(1-F)q^{2}+FpqSum = 1

Therefore, as well as measuring a probability (of=HetHet(1 -_{HW}F)

*Deterministic
vs. stochastic evolution*

The Hardy-Weinberg
law is the basis of all population genetics theory, but it assumes that
in the absence of selection or other evolutionary forces, absolutely ** no**
gene frequency change occurs during reproduction. This would be true
in an infinitely large population; under these conditions, selection would
be completely predictable and

However, this is
only approximately true in real populations of finite size. Assume
a diploid population of constant size ** N**. Each of

Below is an example
of drift. Imagine a rare species kept in a zoo with a population of only
six diploid individuals. There are a total of 12 alleles (numbered 1-12
in generation 0). All alleles are assumed equally fit, so that evolution
is neutral. The alleles may also be genetically distinguishable, or "different
in state" (represented by colours).

If the wild source population were large,
all the alleles in generation 0 would have come from different ancestors;
none would be ** identical by descent (IBD)**.
However, by chance some alleles are lost in each generation. After a moderate
number of generations, every allele will ultimately become a copy of just
one of the original alleles, or

Alleles that are
** IBD**
must also be

This kind of evolution
is not predictable; it is random or ** stochastic**.
Stochastic evolution occurs in any finite population, whether or not selection
is operating

Drift is slower in larger populations. Why? If I tossed a coin twice, and get 2 heads, you would not be surprised. If I tossed 20 times, and got 20 heads you would be very surprised. If I scored 200 heads in as many tosses, you would rightly suspect me of cheating. Similarly, if we have two alleles in a population (equivalent to heads and tails), we get a larger variance of allele frequency if we have a small population. This is equivalent to getting a more variable fraction of heads when tossing a coin a small number of times.

** Predictable
unpredictability** (remember, science
= accurate prediction!)

We can't predict
*exactly*
what is going to happen in genetic drift, but the *distribution* of
results is known, and useful. We can quantify the following:

1) The **mean
gene frequency**. The probabilities for
two alleles in a single generation are given by the binomial distribution,
with binomial probability ** p** and numbers of trials

2) The **variance
of gene frequency** after one generation.
The binomial variance is:

Knowing the variance for a single generation, we can predict the long-term consequences of drift, including the probability distribution for allele frequency after a given number of generations. (The maths is, unfortunately, beyond this course!).

3) The **probability
that a particular allele will eventually be fixed**.
We know that one of the alleles will eventually take over; the probability
that it will be any particular allele is simply the fraction that the allele
has in the population initially, or .

4) Eventually, any
population will become fixed for one of the original alleles, and we can
also predict approximately how long this will take. Looking backwards,
this is the **coalescence time**
of a given population. The coalescence time is given by (rate of fixation)^{-1}
(see below) and will therefore be about **2 N** generations.

Genetic drift is important in nature.
Here is a recent example from an Asian bramble (*Rubus alceifolius*)
which is an introduced weed on some Pacific islands. Genetic variation
was studied by means of a DNA fingerprint technique called "Amplified Fragment
Length Polymorphisms" - AFLP for short. Each vertical "lane" on the
gel represents DNA from a single individual; each AFLP band is thought
to represent an independent DNA fragment, and polymorphisms are revealed
by presence or absence of bands. In its native range (Vietnam, right),
this species is highly polymorphic, while in an introduced population (the
island of Réunion, left), no polymorphisms are observed. This suggests
that the founder population was very small, and that all variation has
been lost. (see Amsellem L et al. 2000.Mol. Ecol.
9: 443-455, reproduced by permission).

BUT the **2 N** alleles in the
previous generation may be

By definition, the heterozygosity after a single generation of inbreeding,

(a) This is true only

(b) ** F** can also measure inbreeding
as a result of subdivision into two or more finite populations. Remember
that when we assumed Hardy-Weinberg, we also assumed a lack of migration
(i.e. mixing of populations).

When we sample from a number of sub-populations with different gene frequencies which do not mate randomly with each other, the heterozygote deficit gives us a measure of identity by descent produced by the population subdivision.

This between-population inbreeding is usually
written ** F_{ST}**, meaning
inbreeding (

For example, assume many populations of
finite size * N *start from from the same gene frequency and
drift apart for

You can try some simulations of drift yourself;
go to natural
selection and drift simulations. You can use some of these (DRIFT.EXE,
and PDRIFT.EXE) to get an estimate of the level of inbreeding and heterozygote
deficit (** F** or

** F_{ST}**
is widely used to study gene frequency variation over a geographic range
as a measure of population subdivision. This topic, which we can't
cover here (shame!), is often referred to as

Even with no deterministic
bias, or natural selection, alleles usually do not have identical probability
of being passed on, as required in these simple models. Population
geneticists get around this by calculating an idealized, or ** effective
population size** that produces approximately
the same rate of genetic drift in their simple models as does the actual
population with all its complexity. The effective population size
may be rather different from the actual population size. Two examples:

1) Separate sexes. The simple theory above assumes that a single individual may have two alleles IBD for a single allele in the previous generation. In fact, they can only do this if there is selfing. In dioecious organisms like us, this is not (yet!) possible. Separate sexes therefore enforce some outbreeding, and slow the buildup of identity by descent: the effective size is marginally larger than the actual population size.2) Unequal sex ratio. In species which maintain harems, like the elephant seal (see later in

SEX AND SEXUAL SELECTION), a single male may commandeer almost all the matings by fighting off other males. Similarly, in modern cow herds almost all females are fertilized artificially; a single bull provides enough sperm for thousands of offspring. Although there are millions of cows in Britain, calves are mostly progeny of very few bulls. The effective population size may therefore be in the hundreds rather than millions, because genes in the population are funnelled through these few bulls in every generation.

During this lecture, we measured inbreeding
using the **inbreeding coefficient**,
** F**.
We applied this method to

The Hardy-Weinberg law is very useful,
and simple models of natural selection work well most of the time. However,
these models have the ever-so-slight drawback that they depend on an assumption
of infinite population sizes. Before today, we modeled evolution in terms
of infinitely divisible gene frequencies. In fact this is simply doesn't
work: some of the most interesting evolution happens when we mix **random**
genetic drift -- due to finite population sizes -- with **deterministic**
forces -- selection. Drift may or, may not be important in evolution, but
it always happens, because populations are always finite.

For now, it is worth knowing that the equation
characterizes perhaps the most important genetic problem in conservation.
The equation will be important in any species with low overall ** N**;
for instance in many endangered large mammals, such as tigers in the Gir
forest in India, Florida panthers, and Sumatran rhinos.

Well! That's probably enough for today!

**FURTHER READING**

FUTUYMA, DJ 1998.
Evolutionary Biology. Chapter 11 (pp. 297-314).
**Population
Structure** lecture notes (optional!).

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