Population Size and Genetic Drift
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© 1997
David H.A. Fitch
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Lecture notes

Deviations from the null hypotheses:  Finite populations sizes and genetic drift, mutation and gene flow

I.  Genetic drift:  the effect of population size alone on allele and genotype frequencies

A.  Finite population size alone results in a change in allele frequency (which results in a decline in heterozygosity)

1.  The probability of an individual being autozygous in a population of finite size N (by random sampling alone) is the probability of getting any gene copy as one of the alleles in a diploid genotype (=1) AND the probability of getting the same gene copy by random sampling (= 1/(2N)).  That is, for any particular individual in a randomly mating population of finite size N,
     F = 1/(2N).

2.  The probability of an individual in this population being allozygous for these gene copies is thus:
     1-F = 1 - (1/(2N))...
UNLESS these gene copies came from a population already inbred to an extent Ft-1, (where t is the present generation), in which case,
     Ft = 1/(2N) + [1 - 1/(2N)]Ft-1
(i.e., the probability of either being autozygous in the present generation OR the probability that an individual comes from that fraction of the population that is already autozygous due to inbreeding in the previous generation).

3.  Because Ft = (H0-Ht)/H0 and Ft-1 = (H0-Ht-1)/H0,
     Ht = Ht-1(1 - 1/(2N))...
That is, in each successive generation (from t-1 to t), the heterozygosity (H) declines by 1/(2N).

4.  Also, the smaller the population size (N), the faster the decline in heterozygosity.

5.  This decline in heterozygosity is due to the increase in frequency of one of the alleles, which approaches fixation.

6.  But only chance (i.e., random sampling) dictates which allele leaves more descendents and becomes fixed.

7.  Note that this kind of loss in heterozygosity differs from that due to nonrandom mating, since a HW equilibrium is still approximately maintained in this finite population.

8.  This stochastic change in allele frequency resulting simply from the finite size of a population is called "genetic drift".

II.  Genetic drift can result in evolutionary divergence

1.  Because of genetic drift, the variance between demes (small subpopulations) increases over time (i.e., demes will diverge, esp. if they become isolated).

2.  Given enough time, allele A or a will become fixed (p = 1 or p = 0).

3.  The allele that is already more frequent will have a higher probability of being fixed:
The probability of a ("neutral") allele is its frequency (= 1/(2N) for a single gene copy).

4.  Thus, the main features of genetic drift are:

  1. A loss of genetic variation results within populations
  2. Genetic divergence results between populations
  3. Evolution results (i.e., allele frequencies change, until H = 0)

III.  Mutations, their rate of fixation and Kimura's molecular "clock"

A.  Mutations form new alleles which thus arise at an initial frequency of 1/(2N).  Because the probability that an allele will be fixed is equal to its frequency, there is a small probability (1/(2N)) that it will become fixed (by chance alone, if it is neutral to selection).

B.  Rate of fixation of new mutations

  1. In any generation, the number of possible copies at which a new mutation could originate is 2N (i.e., new mutants could exist at any of the gene copies in the population).
  2. The probability of getting a new mutant is thus 2Nu (i.e., the rate at which mutation occurs at any one gene copy, applied to all gene copies)
  3. Thus, the total number of mutations that will be fixed per generation is:
    (2Nu) (1/(2N)) = u
    (i.e., the probability of getting a new mutant in the population and then the probability that this mutant will be fixed)
  4. That is, the rate at which new (neutral) mutants are fixed is 1/u (i.e., the number of generations for a mutant to be fixed, which is the inverse of the number of mutants fixed per generation); this rate is INDEPENDENT of population size, N.
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  1. However, the time it takes a particular mutant to achieve fixation from the time it arises is dependent on population size (this time is 4Ne generations, where Ne is the effective breeding population (N if everybody contributes progeny).  (This result is obviously not derived here, but see Kimura (1983) if you want more.)

C.  The "molecular clock" (M. Kimura, 1983:  The Neutral Theory)

  1. That is, if u is constant, populations (and species) should evolve (i.e., new alleles will replace old alleles by drift) at a constant per-generation rate.
  2. Because different populations that become separated will accumulate different mutations by chance, such populations will diverge (doesn't this sound like "descent with modification"!).

IV.  Mutation balances the loss of heterozygosity that results from drift

A.  Although it is a weak force for evolution (changing allele frequencies) in the short term (the probability of fixation is very low in a large population

B.  Prediction of heterozygosity (H) in a small population

  1. In the absence of mutation, the probability of being autozygous is:
    Ft = (1/(2N)) + (1 - (1/(2N)))Ft-1
  2. But either of the two autozygous allles could mutate, so the probability that BOTH will NOT mutate is (1-u)2
  3. Hence, Ft = (1-u)2 [(1/(2N)) + (1 - (1/(2N)))Ft-1]
  4. At equilibrium, Ft = Ft-1; this reduces to Feq = 1/(1+4Nu)
  5. Because H = 1-Feq, then H = (4Nu) / (1+4Nu)

C.  Thus, H decreases as u or N decrease.

  1. That is, H (heterozygosity, a measure of variability) becomes lower as N gets smaller or if u becomes lower (i.e., H depends on both mutation rate and population size)
  2. Also, drift causes allele frequencies to fluctuate randomly, and these fluctuations occur more rapidly and drastically in smaller populations
  3. This also means that, if a population becomes separated into 2 populations, the variance between the populations increases due to drift (they diverge)

V.  The "founder effect":  the result of the small size of founder population

A.  N can get temporarily small (a "bottleneck")
B.  If so, H will decrease more the smaller the population and the lower the population growth rate

VI.  Gene flow:  another way to introduce new alleles into the population

A.  There are several models for gene flow:

  1. Continent-island model (one-way migration to a separated population)
  2. Island model (random migration between separate populations)
  3. Stepping-stone model (migrants only come from neighboring populations)
  4. Isolation-by-distance model (gene flow from local neighborhoods in a continuously distributed population)

B.  For example, in the island model, migrant gene copies can be treated as if they were new mutations:
     F = 1 / (1+4Nm), where m is the probability that a migrant contributes to the next generation, and m is small

VII.  Conclusions integrating nonrandom mating, genetic drift, mutation and gene flow

A.  Inbreeding:
1.  Inbreeding distributes genes from heterozygous to homozygous states
2.  Inbreeding thereby increases genetic variance between different demes

B.  Genetic drift:
1.  Because of drift, allele frequencies fluctuate randomly in each deme
2.  This fluctuation is more rapid and more severe in smaller demes
3.  Because of finite sizes of populations, genetic drift results in driving loci to homozygosity

C.  Mutation and gene flow:
1.  The trend toward homozygosity from inbreeding and drift is offset ("balanced") by mutation and gene flow
2.  Also, populations (and thus species) will diverge genetically (i.e., evolve:  allele frequencies will change and different populations will thus accumulate differences)
3.  This divergence is expected to occur at a constant clocklike rate (if mutation rate and generation time are constant, which they aren't)

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Exercises

  1. A mutation (A*) arises (repeatedly) at a slow rate (u) from an allele (A) that already exists in the population at a frequency f(A) = p.  But there is also a rate of reversion (v) back to A.  The change in q is up-vq = u(1-q) - vq.  At equilibrium, this change is 0.  What is the predicted equilibrium frequency of A* (q) if u = 10-5 and v = 10-7?
     
  2.  

 

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Simulations

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 Nonrandom
Mating  Genetic
Drift  Selection:
One Locus  Selection:
Many Loci
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[Nonrandom Mating] [Genetic Drift] [Selection: One Locus] [Selection: Many Loci]

[Natural Deviations from the Null Hypotheses]

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