Questions: Fixation Probability and Diffusion Models

Question 1

A beneficial mutation with selection coefficient s = 0.02 appears as a single copy in a population with effective size Ne = 10,000. What is the approximate probability that this mutation will eventually reach fixation?

• A. Nearly 100% — selection is strong enough to guarantee fixation in a large population
• B. About 50% — beneficial alleles have roughly even chances of fixing or being lost
• C. About 4% — even beneficial mutations are usually lost to drift while they are rare
• D. Exactly 1/20,000 — the same as a neutral mutation, because drift dominates in large populations

Answer: (C)

Using the approximation 2s = 2 × 0.02 = 0.04, the fixation probability is about 4%. This means the mutation will be lost ~96% of the time. This counterintuitive result arises because when the mutation first appears as a single copy among 20,000 alleles, it is highly vulnerable to chance elimination — the carrier might die before reproducing or happen to pass on the other allele. Selection becomes reliably effective only once the allele has drifted to a high enough frequency to avoid stochastic loss. Option A (the most tempting wrong answer) incorrectly assumes selection guarantees fixation regardless of frequency.

Question 2

A weakly deleterious mutation with s = -0.0001 appears in a bacterial pathogen with effective population size Ne = 100 during a bottleneck. What is the likely fate of this mutation?

• A. It will be rapidly purged — natural selection efficiently removes all deleterious mutations regardless of population size
• B. It will behave approximately as a neutral mutation and may fix by drift, because Ne × s is much less than 1
• C. It will be maintained at intermediate frequency indefinitely by balancing selection
• D. It cannot fix because a negative selection coefficient means fixation probability is exactly zero

Answer: (B)

Ne × s = 100 × 0.0001 = 0.01, which is much less than 1. When this product is << 1, drift overwhelms selection and the allele behaves essentially as neutral — it may fix, be lost, or drift to intermediate frequencies by chance, with selection having minimal effect. This is the key practical implication of the Ne × s threshold: in small populations (bottlenecks, endangered species, pathogens), mildly deleterious mutations accumulate as if they were neutral, with important consequences for fitness and disease evolution.

Question 3

A new neutral mutation arising as a single copy in a diploid population of effective size Ne has a fixation probability of 1/(2Ne).

• T. True
• F. False

Answer: True

In a diploid population with Ne individuals, there are 2Ne allele copies. Each copy has an equal probability of being ancestral to all future alleles, so a new neutral mutation (present once) has probability 1/(2Ne) of fixing. This is a foundational result of neutral theory. It also leads to the elegant conclusion that neutral mutations fix at a rate equal to the mutation rate itself, independent of population size — because although larger populations have lower per-allele fixation probability, they also produce proportionally more new mutations.

Question 4

A strongly beneficial mutation with a large selection coefficient is virtually very likely to fix once it appears in a population, because strong selection overcomes drift.

• T. True
• F. False

Answer: False

The fixation probability is approximately 2s regardless of how large s is within biologically realistic ranges. A mutation with s = 0.10 has only ~20% fixation probability; a mutation with s = 0.20 has ~40%. Even very strongly beneficial mutations are usually lost — because when a mutation first appears as a single copy, it is at the mercy of drift before selection has a chance to act. 'Virtually guaranteed' would require a fixation probability near 100%, which never occurs for a single new mutation. Most beneficial mutations are simply unlucky in their early generations.

Question 5

What does the product Ne × s reveal about the fate of a mutation, and why does this product matter more than the selection coefficient s alone?

Model answer: Ne × s marks the boundary between drift-dominated and selection-dominated evolution. When Ne × s >> 1, selection reliably determines allele fate — beneficial alleles tend to fix and deleterious alleles tend to be purged. When Ne × s << 1, drift dominates and even deleterious alleles can fix as if they were neutral. The selection coefficient s alone is insufficient because what matters is whether selection is strong relative to drift, and drift strength is inversely proportional to Ne. A mutation with s = 0.001 is strongly favored in a population of Ne = 10,000 (Ne × s = 10) but effectively invisible to selection in a population of Ne = 100 (Ne × s = 0.1).

This threshold is why population size matters so much for evolutionary outcomes. Large populations purge mildly deleterious mutations efficiently and can selectively fix weakly beneficial ones. Small populations (endangered species, populations through bottlenecks, some pathogens) accumulate deleterious mutations and lose beneficial ones to drift — with real consequences for adaptation, fitness, and disease evolution.