Zoology 510, Class Notes for Ridley, Chapter 6
Random Events in Population Genetics

Assignments.

Brief Outline

• Summary of Equations (listing significant equations from Chapter 6).
• Introductory comments on the strangeness of random processes.
• Overview
  • Context for this chapter.
  • Major points.

• Section-by-section comments
  • 6.1. Random sampling.
  • 6.2. Genetic drift.
  • 6.3. The founder effect.
  • 6.4. Allele substitution by drift.
  • 6.5. No "equilibrium" in small populations.
  • 6.6. The "march to homozygosity."
  • 6.7. Polymorphism resulting from neutral mutation and drift.
  • 6.8. Effective population size.

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Introductory comments on random processes.

The mathematics of probability yields some strange and often counter-intuitive expectations.  The general difficulty most people have in understanding probability is powerfully demonstrated by huge profits earned by casinos and other gambling businesses.  The following are some general points with some significance for understanding stochastic processes in evolution.

• Random events are unaffected by what has gone before.
  • There is never any "tendency" to restore an average.
  • There is no "balance" between alternative possibilities.
  • Events are not "directed" toward the most likely expectation.

• The "most likely" expectation can be extremely improbable.
  • If you flip a fair coin 100 times, the outcome of 50 heads and 50 tails is both:
    • the most likely expectation.
    • a very improbable result.
  • Explanation.  Of all the many possible outcomes, the 50:50 ratio is the one that is most likely.  However, the probabilities for 49:51 and 51:49 are both almost as high, while 48:52 and 52:48 are also close, etc.  Thus, although 50:50 has the highest probability of any particular outcome, the absolute probability for exactly 50:50 is actually quite low.

  • Here's an example which may be more intuitive.  If you throw darts at a target,
    • The most likely expectation for each dart is a bull's eye.
    • A bull's-eye is very unlikely (unless you are very skillful).
    • Explanation.  No matter how sloppy your aim is, all the throws will be scattered around the bull's-eye (unless there is systematic bias for one side or another).  Therefore, the most likely impact site must be the center, the average of all the scattered positions.  Certainly no other particular position is more likely.  But the sum of all the other, non-bull's-eye positions is so large, that the probability of a bull's-eye remains quite low (even though it remains the "most likely" expectation).

• How does this apply to population genetics?
  • For finite populations, the Hardy-Weinberg theorem gives the most likely expectation for gene frequencies after one round of random mating.
  • However, matching that expectation exactly is extremely improbable.
  • Therefore, at each generation, gene frequencies will almost certainly change rather than remain constant, although typically by only a small amount.
  • These many small changes lead to "drift", characterized as a "random walk".
    • At each generation, the current (changed) frequencies become the source for the next generation.  So the most likely expectation for the next generation is changed as well.
    • (This is unlike tosses of a fair coin, where current expectation remains 50:50.)
    • This is rather like a dart game where the target for each throw is not the bull's-eye but the site of the previous dart.
    • This process repeats at each generation, with a small, random change in gene frequency each time.
    • There is no "force" that would restore the original frequencies.

• The behavior of random walks can also appear paradoxical.
  • In a random walk, the "most likely" position expected in the future is the initial (or current) position.
  • However, the distance from the starting (or current) position is expected to increase over time.
  • Explanation.  There is no expectation for movement in one particular direction to be preferred over another.  Hence, the "most likely" position cannot be biased in any direction.  This is also the "average" position if many separate trials are averaged.
  • This applies not only at the beginning but indefinitely, so that from any point away from the starting position, there is no preference for moving back toward the start nor further away from the start.

• How does the "random walk" apply to population genetics?
  • Genetic drift is a random walk.
  • The most likely expectation, or the expected average over many trials, is given accurately by the Hardy-Weinberg theorem.
  • However, in any particular case, the more generations that pass, the farther are gene frequencies likely to be from the initial expectation.
  • In finite populations, one allele will be lost when its frequency drifts too close to zero.  From that point on, the other allele will be "fixed" in the population.
  • Drift to fixation is itself a stochastic expectation, a "most likely" result over enough generations.  But there is no "force" requiring drift to reach that result in any particular case.

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Overview.
Context for this chapter.

• To keep various topics in perspective, re-read the introduction to Evolutionary Genetics on p. 89 of the text.

• We are building up a list of general processes that act to affect gene frequencies.
  • Processes that can lead to fixation of alleles (maximal homozygosity).
    • Selection (Chapter 5)
    • Genetic drift (Chapter 6)
    • Founder effect (Chapter 6)
    • Inbreeding (Chapter 6)

  • Processes that can create or maintain allele polymorphism (heterozygosity).
    • Heterozygote advantage (Chapter 5)
    • Mutation-selection balance (Chapter 5)
    • Multiple-niche selection (Chapter 5)
    • Frequency-dependent selection (Chapter 5)
    • Migration (Chapter 5)
    • Mutation together with drift (Chapter 6)

  • Note the following key points:
    • In Chapter 5 we assume large populations in which drift is unimportant.
    • In Chapter 6, we assume that alleles are neutral so that selection is unimportant.
    • In real life, we must remember that drift and selection are both important evolutionary processes.  Even when alleles appear neutral, small differences in selection coefficient may in fact exist, so selection must never be completely disregarded.  But drift can override weak selection if a populations is sufficiently small, so drift can never be ignored either.

• Allele frequencies fluctuate at random, but eventually become fixed.
• Neutral genetic variation in a population is eventually lost to drift.
• Initially similar sub-populations will diverge in allele frequencies, and may eventually become fixed for different alleles (depending on gene flow).
• The probability, at any given time, that an allele will eventually become fixed equals the frequency of the allele at that time.
• The rate at which these events occur is greater for small populations.
  • For an allele which eventually becomes fixed, the average time to reach fixation is 4N.
  • Heterozygosity decreases by 1 / 2N per generation.
• The effective population size is the size for which expected heterozygosity matches observed heterozygosity.
• Effective population size will be smaller than census size if:
  • Some individuals do not breed.
  • Individuals vary greatly in number of progeny.
  • Numbers of males and females are not equal.
  • Generations overlap.
  • Population size fluctuates.
  • Mating is not panmictic, which may result from:
    • Subdivided population structure.
    • Geographic distance.
    • Mating among relatives (in the extreme case, self-fertilization).
• All of these factors will decrease the expected heterozygosity (or, equivalently, increase the homozygosity, or inbreeding) of the actual population.

Chapter 6, Section-by-Section Comments

6.1. "Successive generations are a random sample from the parental gene pool."

• Make sure you understand what is meant by the term random sample in this context.
• Note that for small populations, the Hardy-Weinberg theorem gives an expectation (i.e., a most probable outcome) rather than a deterministic prediction.
• In Ridley's initial example (20 gametes sampled at random from a gene pool in which A and a alleles have frequency 0.5), he asserts that the most likely outcome is 10 of each, for frequencies of 0.5 A and 0.5 a in the next generation.  That is, indeed, the Hardy-Weinberg expectation.
• But do note that the most-likely Hardy-Weinberg expectation can be quite improbable.  The probability of the most likely outcome in this case of 20 gametes is actually less than 1 in 5, for this particular case of 20 gametes.
• The probability that an actual random sample will exactly match the most likely expected outcome actually becomes lower, the bigger sample (although the size of probable discrepency grows smaller.)
• Understand that random samples are NOT selection.

• We will consider three processes - drift, founder effect, and inbreeding -- that all operate most strongly in small populations and that all reduce heterozygosity.
  • Drift occurs gradually over many generations.
  • The founder effect results from a single small sample when a population is first founded.
  • Inbreeding can reduce the effective size of a population.
• We will also consider one process -- drift in conjunction with neutral mutation -- which creates and maintains allele polymorphism.

6.2. "The frequency of alleles with the same fitness will change at random through time in a process called genetic drift."

• We give this name, genetic drift, to the effect of random changes in allele frequency, because even though it is itself a random process it leads to definite expectations (6.4 - 6.7) when many loci or many populations are considered.
• Note that Figure 6.1 does NOT show "pure" drift (since there is a average change in gene frequencies suggestive of selection), but does show the greater variability in the small populations that is expected to result from random sampling / drift.

6.3. "A small founder population may have a nonrepresentative sample of the ancestral population's genes."

• This founder effect is the result of a single small random sample.
• The founder effect is not the same as genetic drift, since drift is cumulative change over many generations.
• Both drift and founder effect result from random sampling.
• A sizable population with a high frequency of a deleterious allele can be explained by the founder effect, by considering that founder allele frequencies can be maintained during population growth (by standard Hardy-Weinberg expectation), while selection may take many generations to reduce the frequency of the deleterious allele (especially if it is recessive with respect to fitness).

6.4. "One gene can be substituted for another by random drift."

• This is a fundamental concept.  When averaged over many loci and many populations, random genetic drift turns out to be a very effective agency for evolutionary change (but not for adaptation).
• In a population of N individuals, there are 2N genes at a given locus.  Consider what can happen to a new, neutral mutation.  ("Neutral" means that it has no selective advantage or disadvantage.)
  • A new mutation, present as only single copy in the population, has a frequency of 1 / (2N).
  • The chance that a new neutral mutation will be lost from the population by chance alone is (2N - 1) / (2N), or (equivalent expression) 1 - 1 / 2N.
  • The chance that a new neutral mutation will eventually be fixed in the population is 1 / 2N.
• That chance that any particular new neutral mutation is fixed is therefore low.  But what happens if new neutral mutations are continually introduced into the populations at rate u per gene per generation?
  • With 2N genes at any given locus in the population, in every generation there will be (on average) 2Nu new mutations.
  • Each of those alleles will be fixed with probability 1 / (2N).
  • Multiplying, we get 2Nu times 1 / (2N) = u per generation as the rate at which new alleles at each locus are fixed in a population.
  • Random drift thus leads to an expectation of a very definite average rate of neutral evolution by mutation and allele replacement.
• However, remember.  Drift is evolution, but drift is NOT selection.  Allele replacement by drift is not adaptive.

6.5. "The Hardy-Weinberg 'equilibrium' is not an equilibrium in a small population.."

• The Hardy-Weinberg theorem does give the most likely expectation for gene frequencies at each generation.
• But, as noted above in 6.1, in real populations this most likely expectation is actually quite improbable (like the dart hitting the bull's-eye").
• Each bit of "noise", departing from the Hardy-Weinberg expectation, resets the gene frequencies for the next generation.
• So there is no "equilibrium", there is random drift with no stability until one or another (neutral) allele is fixed.
• Of course, selection, which is a deterministic process, can override drift if alleles do not all have equivalent fitness.

6.6. "Neutral drift over time produces a march to homozygosity."

• The "march to homozygosity" described by Ridley is itself a probabilistic expectation, not a deterministic process such as the word "march" would imply.
• The "march to homozygosity" is predictable on average, when many alleles at many loci are considered.  But any particular allele will "drift" rather than "march" to homozygosity (or loss).
• To make the concept a bit more intuitive, consider a locus at which there are as many different alleles as there were genes in the population.  Then the odds that each and every one of these alleles would be passed along so that there would be exactly one of each in the next generation, would be fantastically low.  In such a case, some loss of heterozygosity would indeed be expected at every generation (and, of course, with any loss of heterozygosity, there is a gain in homozygosity).
• In his explanation, Ridley introduces a population of "hermaphrodites" capable of self-fertilizing.  This is done solely to make the math technically correct for a simple model.  Remember the Hardy-Weinberg conditions? The only reason hermaphroditism is not also required for the Hardy-Weinberg theorem is that when a population is assumed to be indefinitely large, one need not worry about self-fertilization in a "randomly mating" population.  But as soon as a finite population is considered, an assumption of hermaphroditism is needed to assure at every generation that every individual can mate with every other individual.  Of course, without that assumption (as Ridley points), the basic concept still applies but the math gets messier.
• Note that there is an error in the second-to-last paragraph on p. 144 of Ridley, which explains the definition of heterozygosity.  Where Ridley says "...p² is the sum of all chances", the p² should be preceded by a summation sign (a capital sigma).  Likewise in the subsequent line, in the expression 1 - p², a summation sign should precede the p².

6.7. "A calculable amount of polymorphism will exist in a population because of neutral mutation."

• This is an essential concept for understanding the Neutral Theory of molecular evolution in Chapter 7.
• The process of genetic drift leads to eventual homozygosity, IF no new alleles are introduced (and, of course, only in the absence of a deterministic process like selection that maintains polymorphism).
• But IF there is a continual supply of new neutral alleles (i.e., mutation), then drift assures the likelihood that some fraction of those alleles will increase in frequency and thereby increase the overall heterozygosity.
• Because there are so many genetic loci, so many ways to produce neutral alleles, and so much time for drift to operate, the statistical predictions of drift lead to very reliable conclusions.
• Equation 6.7 is important for appreciating the extent to which heterozygosity is expected from the drift of neutral mutations.  This equation is presented graphically in Fig. 6.6 on p. 147.

6.8. "Population size and effective population size."

• Actual population count is NOT the same as effective population size for evolutionary theory.  Reasons include:
  • Some countable individuals may not breed, for a wide variety of biological reasons, such as polygyny (in which a few males mate with "harems" of many females while many other males fail to mate altogether).
  • Individuals may vary significantly in fertility, even if all do reproduce.
  • Population size may vary from generation to generation, with significant "bottlenecks" of low population size.
  • Subdivided population or non-random mating may accelerate inbreeding over the expectation of panmixus.
• The effective population size is usually smaller than actual population count.
• Effective population size significantly affects the expectations of drift, founder effect, and march to homozygosity.
• Any factor which decreases the effective population size will increase inbreeding (or reduce expected heterozygosity).

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