Genetic drift is the change in the frequency of a gene variant in a
population due to random sampling of organisms. The alleles in the offspring
are a sample of those in the parents, and chance has a role in determining
whether a given individual survives and reproduces. A population's allele
frequency is the fraction of the copies of one gene that share a particular
form. Genetic drift may cause gene variants to disappear completely and
thereby reduce genetic variation. When there are few copies of an allele,
the effect of genetic drift is larger, and when there are many copies the
effect is smaller. In the early twentieth century vigorous debates
occurred over the relative importance of natural selection versus neutral
processes, including genetic drift. Ronald Fisher, who explained natural
selection using Mendelian genetics, held the view that genetic drift plays at the
most a minor role in evolution, and this remained the dominant view for several
decades. In 1968, Motoo Kimura rekindled the debate with his neutral theory of
molecular evolution, which claims that most instances where a genetic change
spreads across a population are caused by genetic drift. There is currently a
scientific debate about how much of evolution has been caused by natural
selection, and how much by genetic drift.
Analogy with marbles in a jar The process of genetic drift can be
illustrated using 20 marbles in a jar to represent 20 organisms in a population.
Consider this jar of marbles as the starting population. Half of the marbles
in the jar are red and half blue, and both colors correspond to two different
alleles of one gene in the population. In each new generation the organisms
reproduce at random. To represent this reproduction, randomly select a marble
from the original jar and deposit a new marble with the same color as its
"parent" into a new jar. Repeat this process until there are 20 new marbles
in the second jar. The second jar then contains a second generation of
"offspring," consisting of 20 marbles of various colors. Unless the second jar
contains exactly 10 red marbles and 10 blue marbles, a random shift occurred in
the allele frequencies. Repeat this process a number of times,
randomly reproducing each generation of marbles to form the next. The numbers of
red and blue marbles picked each generation fluctuates: sometimes more
red, sometimes more blue. This fluctuation is analogous to genetic
drift – a change in the population's allele frequency resulting from a random
variation in the distribution of alleles from one generation to the next.
It is even possible that in any one generation no marbles of a particular
color are chosen, meaning they have no offspring. In this example, if no red
marbles are selected the jar representing the new generation contains
only blue offspring. If this happens, the red allele has been lost permanently
in the population, while the remaining blue allele has become fixed: all future
generations are entirely blue. In small populations, fixation can occur in just
a few generations. Probability and allele frequency
The mechanisms of genetic drift can be illustrated with a simplified example.
Consider a very large colony of bacteria isolated in a drop of solution. The
bacteria are genetically identical except for a single gene with two
alleles labeled A and B. A and B are neutral alleles meaning that they do not
affect the bacteria's ability to survive and reproduce; all bacteria in this
colony are equally likely to survive and reproduce. Suppose that half the
bacteria have allele A and the other half have allele B. Thus A and B each
have allele frequency 1/2. The drop of solution then shrinks until
it has only enough food to sustain four bacteria. All other bacteria die without
reproducing. Among the four who survive, there are sixteen possible combinations
for the A and B alleles: (A-A-A-A),,,,
(A-A-B-A),,,, (A-A-A-B),,,,
(A-A-B-B),,,. Since all bacteria in the original
solution are equally likely to survive when the solution shrinks, the four
survivors are a random sample from the original colony. The probability that
each of the four survivors has a given allele is 1/2, and so the probability
that any particular allele combination occurs when the solution shrinks is
(The original population size is so large that the sampling effectively
happens without replacement). In other words, each of the sixteen possible
allele combinations is equally likely to occur, with probability 1/16.
Counting the combinations with the same number of A and B, we get the following
table. As shown in the table, the total number
of possible combinations to have an equal number of A and B alleles is six,
and its probability is 6/16. The total number of possible alternative
combinations is ten, and the probability of unequal number of A and B alleles is
10/16. Thus, although the original colony began with an equal number of A
and B alleles, chances are that the number of alleles in the remaining
population of four members will not be equal. In the latter case, genetic drift
has occurred because the population's allele frequencies have changed due to
random sampling. In this example the population contracted to just four
random survivors, a phenomenon known as population bottleneck.
The probabilities for the number of copies of allele A that survive can be
calculated directly from the Binomial distribution where the "success"
probability is 1/2 i.e. the probability that there are k copies of A alleles in
the combination is given by where n=4 is the number of surviving
bacteria. Mathematical models of genetic drift
Mathematical models of genetic drift can be designed using either branching
processes or a diffusion equation describing changes in allele frequency
in an idealised population. = Wright–Fisher model=
Think of a gene with two alleles, A or B. In diploid populations consisting of
N individuals there are 2N copies of each gene. An individual can have two
copies of the same allele or two different alleles. We can call the
frequency of one allele p and the frequency of the other q. The
Wright–Fisher model assumes that generations do not overlap and that each
copy of the gene found in the new generation is drawn independently at
random from all copies of the gene in the old generation. The formula to
calculate the probability of obtaining k copies of an allele that had frequency p
in the last generation is then where the symbol "!" signifies the
factorial function. This expression can also be formulated using the binomial
coefficient, = Moran model=
The Moran model assumes overlapping generations. At each time step, one
individual is chosen to reproduce and one individual is chosen to die. So in
each timestep, the number of copies of a given allele can go up by one, go down
by one, or can stay the same. This means that the transition matrix is
tridiagonal, which means that mathematical solutions are easier for
the Moran model than for the Wright–Fisher model. On the other hand,
computer simulations are usually easier to perform using the Wright–Fisher
model, because fewer time steps need to be calculated. In the Moran model, it
takes N timesteps to get through one generation, where N is the effective
population size. In the Wright–Fisher model, it takes just one.
In practice, the Moran model and Wright–Fisher model give qualitatively
similar results, but genetic drift runs twice as fast in the Moran model.
= Other models of drift= If the variance in the number of
offspring is much greater than that given by the binomial distribution
assumed by the Wright–Fisher model, then given the same overall speed of genetic
drift, genetic drift is a less powerful force compared to selection. Even for
the same variance, if higher moments of the offspring number distribution exceed
those of the binomial distribution then again the force of genetic drift is
substantially weakened. = Random effects other than sampling
error= Random changes in allele frequencies can
also be caused by effects other than sampling error, for example random
changes in selection pressure. One important alternative source of
stochasticity, perhaps more important than genetic drift, is genetic draft.
Genetic draft is the effect on a locus by selection on linked loci. The
mathematical properties of genetic draft are different from those of genetic
drift. The direction of the random change in allele frequency is
autocorrelated across generations. Drift and fixation
The Hardy–Weinberg principle states that within sufficiently large populations,
the allele frequencies remain constant from one generation to the next unless
the equilibrium is disturbed by migration, genetic mutations, or
selection. However, in finite populations, no new
alleles are gained from the random sampling of alleles passed to the next
generation, but the sampling can cause an existing allele to disappear. Because
random sampling can remove, but not replace, an allele, and because random
declines or increases in allele frequency influence expected allele
distributions for the next generation, genetic drift drives a population
towards genetic uniformity over time. When an allele reaches a frequency of 1
it is said to be "fixed" in the population and when an allele reaches a
frequency of 0 it is lost. Smaller populations achieve fixation faster,
whereas in the limit of an infinite population, fixation is not achieved.
Once an allele becomes fixed, genetic drift comes to a halt, and the allele
frequency cannot change unless a new allele is introduced in the population
via mutation or gene flow. Thus even while genetic drift is a random,
directionless process, it acts to eliminate genetic variation over time.
= Rate of allele frequency change due to drift=
Assuming genetic drift is the only evolutionary force acting on an allele,
after t generations in many replicated populations, starting with allele
frequencies of p and q, the variance in allele frequency across those
populations is = Time to fixation or loss=
Assuming genetic drift is the only evolutionary force acting on an allele,
at any given time the probability that an allele will eventually become fixed
in the population is simply its frequency in the population at that
time. For example, if the frequency p for allele A is 75% and the frequency q
for allele B is 25%, then given unlimited time the probability A will
ultimately become fixed in the population is 75% and the probability
that B will become fixed is 25%. The expected number of generations for
fixation to occur is proportional to the population size, such that fixation is
predicted to occur much more rapidly in smaller populations. Normally the
effective population size, which is smaller than the total population, is
used to determine these probabilities. The effective population takes into
account factors such as the level of inbreeding, the stage of the lifecycle
in which the population is the smallest, and the fact that some neutral genes are
genetically linked to others that are under selection. The effective
population size may not be the same for every gene in the same population.
One forward-looking formula used for approximating the expected time before a
neutral allele becomes fixed through genetic drift, according to the
Wright–Fisher model, is where T is the number of generations, Ne
is the effective population size, and p is the initial frequency for the given
allele. The result is the number of generations expected to pass before
fixation occurs for a given allele in a population with given size and allele
frequency. The expected time for the neutral allele
to be lost through genetic drift can be calculated as
When a mutation appears only once in a population large enough for the initial
frequency to be negligible, the formulas can be simplified to
for average number of generations expected before fixation of a neutral
mutation, and for the average number of generations
expected before the loss of a neutral mutation.
= Time to loss with both drift and mutation=
The formulae above apply to an allele that is already present in a population,
and which is subject to neither mutation nor selection. If an allele is lost by
mutation much more often than it is gained by mutation, then mutation, as
well as drift, may influence the time to loss. If the allele prone to mutational
loss begins as fixed in the population, and is lost by mutation at rate m per
replication, then the expected time in generations until its loss in a haploid
population is given by where is equal to Euler's constant. The
first approximation represents the waiting time until the first mutant
destined for loss, with loss then occurring relatively rapidly by genetic
drift, taking time Ne