Whenever a new point mutation arises in a gene, a single individual in the population comes to carry a sequence that differs from all other members of that population. The subsequent fate of this new variant is a central question in population genetics. Most new mutations are lost from the population within a few generations, while a small minority spread widely enough to replace all other variants at that site a process termed fixation. Understanding the probability and dynamics of fixation, for both neutral and selected mutations, forms the basis of the neutral theory of molecular evolution and of population genetic models such as the Wright–Fisher model.
1. Fixation of Neutral Mutations
1.1 Concept of Fixation
When a mutation first arises, it is present in only one copy of the gene within the population. Using the framework of genealogical trees, a mutation can be traced forward through successive generations. Some mutations are transmitted to several descendants and persist for a number of generations, but ultimately disappear because they do not lie on the line of descent leading to the present generation. Other mutations happen to arise precisely on the lineage that leads to the present generation, allowing them to rise to high frequency and stand a strong chance of eventually spreading through the entire population — that is, of becoming fixed. The overwhelming majority of new mutations are lost within a few generations, and only a small fraction ever become fixed.
1.2 Probability of Fixation for a Neutral Mutation
For a neutral mutation, the probability of eventual fixation can be derived using a simple genealogical argument. Consider an ancestral population containing N copies of a gene, in which a mutation has just occurred in one copy. Since every copy of the gene is equally likely to become the ancestor of the entire present-day population (through the process of coalescence), the probability that the mutated copy happens to be the one that becomes this ancestor is 1/N.
The same result can be obtained by reasoning forward in time: if the population is followed sufficiently far into the future, only one individual from the present generation will have descendants that survive indefinitely. A mutation will only become fixed if it happens to occur in the gene copy belonging to this one fortunate lineage. Hence, the fixation probability for a neutral mutation is:
P(fix) = 1/N
1.3 Rate of Fixation of Neutral Mutations
Let u denote the probability that a new mutation arises at a given site, per generation, per copy of the gene. Since there are N copies of the gene in the population, the mean number of new mutations arising at that site, across the whole population, in a single generation is Nu.
The overall rate of fixation of neutral mutations is obtained by multiplying the rate at which new mutations arise by the probability that each one is eventually fixed:
u(fix) = Nu × (1/N) = u
This result shows that the rate of fixation of neutral mutations is exactly equal to the underlying mutation rate, and is completely independent of population size. This is one of the most fundamental and widely cited results of the neutral theory of molecular evolution (Kimura, 1983). It is important to note that this simple equality (fixation rate = mutation rate) holds only for neutral mutations; for mutations under selection, the fixation probability is not equal to 1/N, and the relationship is more complex.
2. Simulating Random Drift and Fixation
2.1 The Wright–Fisher Model
Neutral mutations spread through a population purely by chance, through a process called random genetic drift — meaning that the number of copies of a mutation changes from generation to generation due to random sampling effects alone, not due to any fitness advantage or disadvantage. The Wright–Fisher model is the standard population genetics model used to study this process.
Under this model, each gene copy in the new generation is assumed to be descended from a single, randomly chosen gene copy in the previous generation. If there are m copies of a neutral mutant sequence among the N gene copies in one generation, then the probability that any single gene copy in the next generation carries the mutation is:
a = m/N
and the probability that it does not carry the mutation is (1 − a). Because each of the N gene copies in the new generation is chosen independently in this way, the number of copies, n, of the mutation in the next generation follows a binomial distribution with parameters N and a. On average, the expected number of copies in the next generation equals m (i.e., the mean value of n is Na = m), but the actual number fluctuates around this mean purely due to chance — this chance fluctuation is what constitutes random drift.
2.2 Computer Simulation of Fixation
This binomial model provides the basis for computer simulations of the fixation process. In a representative simulation, a population of N = 200 gene copies is considered. At time zero, a single copy of a new neutral mutation is introduced. In each subsequent generation, the number of copies is updated by drawing a random value from the binomial distribution described above. The simulation is continued until the mutation either becomes extinct (n = 0) or reaches fixation (n = N).
When such a simulation is repeated many times (for example, 2,000 independent runs), the results show that:
- The vast majority of runs result in the mutation becoming extinct after only a few generations.
- A small number of runs show the mutation rising to a substantial frequency before eventually disappearing.
- An even smaller number of runs result in the mutation reaching fixation.
In one such set of 2,000 simulation runs (with N = 200), fixation was observed to occur 11 times. This closely matches the theoretical expectation, since the predicted number of fixation events is:
2,000 × (1/N) = 2,000 × (1/200) = 10
The close agreement between the simulated and theoretically expected number of fixations confirms the validity of the P(fix) = 1/N result for neutral mutations.
3. Introducing Natural Selection
3.1 Fitness and the Selection Coefficient
The simulation framework can be extended to describe mutations subject to natural selection rather than pure drift. An advantageous mutation is assigned a fitness of (1 + s) relative to the original (non-mutant) sequence, which has a fitness of 1. The parameter s is called the selection coefficient, and represents the proportional increase in expected number of offspring conferred by the mutation; an individual carrying the advantageous mutation is expected, on average, to produce (1 + s) times as many offspring as an individual without it.
3.2 Modified Wright–Fisher Model Under Selection
If there are m copies of the advantageous mutation in the population at a given time, the mean fitness of the whole population is:
w̄ = [m(1 + s) + (N − m)] / N
Under the Wright–Fisher model with selection, each gene copy is chosen to be a parent of the next generation with a probability proportional to its fitness. The probability that a gene copy in the next generation carries the advantageous mutation is therefore:
a = m(1 + s) / (N w̄)
The number of copies of the mutation in the next generation still follows a binomial distribution, as in the neutral case, but now using this modified value of a. Because a is increased relative to the neutral case, the expected (mean) number of copies of the mutation in the next generation is greater than m, reflecting the systematic advantage conferred by selection.
3.3 Simulation Results Under Selection
Simulations using a selection coefficient of s = 0.05 show that, even though the mutation is advantageous, a substantial number of simulation runs still result in extinction of the mutation, due to the continuing influence of random drift. Nevertheless, some runs do proceed to fixation.
3.4 The Concept of “Nearly Neutral” Mutations
The relationship between the fixation probability and the selection coefficient is central to understanding weak selection. When the selection coefficient is very small, such that s ≪ 1/N, the fate of the mutation is governed predominantly by random drift rather than by selection, and its fixation probability is approximately equal to 1/N, just as for a strictly neutral mutation. Mutations in this regime are termed “nearly neutral”, since they behave, for practical purposes, in the same way as neutral mutations.
An important consequence of this result is that whether a mutation of a given selection coefficient behaves as effectively neutral or as genuinely advantageous depends on population size:
- The same mutation, with the same value of s, may behave as a nearly neutral mutation in a small population(where 1/N is relatively large).
- The same mutation may behave as a genuinely advantageous mutation in a large population (where 1/N is much smaller than s).
3.5 Stronger Selection and Selective Sweeps
When the selection coefficient is increased further — for example, to s = 0.2 — a substantially greater proportion of simulation runs proceed to fixation, although extinction still occurs in some runs. Additionally, in runs where fixation does occur, the time taken to reach fixation becomes progressively shorter as the selection coefficient increases.
When the selective advantage of a mutation is very large, the mutation is able to spread through the population rapidly, in a phenomenon known as a selective sweep. In such cases, the dynamics of spread can be approximated using deterministic theory, which ignores the effects of random sampling and treats allele frequency change as a smooth, predictable function of time — an approximation that becomes increasingly accurate as the selection coefficient increases.
3.6 Deleterious Mutations
For a deleterious mutation, fitness is defined as (1 − s) rather than (1 + s), with all other aspects of the model remaining the same. As expected, the probability of fixation for a deleterious mutation is lower than that for a neutral mutation. However, the fixation probability for a deleterious mutation is not exactly zero — due to the influence of random drift, even a disadvantageous mutation retains a small but non-zero chance of eventually becoming fixed, particularly in small populations.
4. Interpreting Simulation Parameters
Simulation studies of this kind typically use relatively small population sizes (such as N = 200) for practical reasons of computational convenience. Because fixation probabilities scale with 1/N, using a small population size means that comparatively large selection coefficients must be used in the simulation in order for the effect of selection to be clearly visible against the background of random drift. For example, a selection coefficient of s = 0.05 behaves as only mildly advantageous when N = 200, with substantial random drift still evident.
In real biological populations, a selection coefficient of 5% would generally be regarded as a very large selective effect, since:
- Mutations conferring such a strong selective advantage are likely to be rare in nature.
- Real population sizes are typically far larger than 200 individuals.
This distinction highlights the complementary value of simulation and analytical approaches in population genetics: simulations provide an intuitive, visual demonstration of stochastic processes such as drift and selection, while analytical (mathematical) results allow calculations to be extended to realistic parameter values — such as very large population sizes — for which direct simulation would be computationally impractical. Simulations are also useful for verifying that the approximations made in deriving analytical formulas remain valid.
Conclusion
The fate of a new mutation in a population is governed by an interplay between random genetic drift and natural selection. For neutral mutations, the probability of eventual fixation is simply 1/N, leading to the fundamental neutral-theory result that the rate of fixation equals the underlying mutation rate, independent of population size.
The Wright–Fisher model, based on a binomial sampling process, provides the mathematical and simulation framework for studying this process, and can be extended to incorporate a selection coefficient s for advantageous or deleterious mutations. Weak selection (s ≪ 1/N) produces “nearly neutral” behavior indistinguishable from drift, while strong selection can drive a mutation to fixation rapidly through a selective sweep, describable by deterministic theory. The dependence of these outcomes on population size underscores that the classification of a mutation as neutral, nearly neutral, or strongly selected is not absolute, but relative to the size of the population in which it occurs.
