Mutation and selection

Mutation changes one genetic state into another, while selection changes frequencies because genetic types differ in reproductive success. When both processes act, their effects can oppose one another and produce an equilibrium.

Core idea. Mutation can continually introduce an allele that selection tends to remove. Mutation–selection balance is the frequency at which these opposing changes cancel under a specified model.

Mutation alone

Let \(p(t)\) be the frequency of allele \(A\). Suppose \(A\) mutates to \(a\) at rate \(\mu\), while \(a\) mutates to \(A\) at rate \(\nu\). A continuous-time approximation is

\[\boxed{\frac{dp}{dt}=\nu(1-p)-\mu p}.\]

The first term adds \(A\) alleles through reverse mutation. The second removes them through forward mutation.

Mutation equilibrium

At equilibrium, \(dp/dt=0\):

\[0=\nu(1-p^*)-\mu p^*.\]

Therefore

\[\nu=(\mu+\nu)p^*,\]

and

\[\boxed{p^*=\frac{\nu}{\mu+\nu}}.\]

The corresponding frequency of \(a\) is

\[\boxed{1-p^*=\frac{\mu}{\mu+\nu}}.\]

Thus mutation alone need not drive one allele to fixation when mutation occurs in both directions.

Discrete-generation mutation

If \(p_n\) is the frequency of \(A\) before mutation in generation \(n\), then after mutation

\[\boxed{p_{n+1}=(1-\mu)p_n+\nu(1-p_n)}.\]

This has the same equilibrium \(\nu/(\mu+\nu)\). The discrete and continuous formulations describe different time conventions but express the same balance of gains and losses.

Selection alone

For a simple haploid model, let allele \(A\) have fitness \(w_A\) and allele \(a\) have fitness \(w_a\). After selection,

\[\boxed{p'=\frac{pw_A}{pw_A+(1-p)w_a}}.\]

If \(w_A>w_a\), selection tends to increase \(A\); if \(w_A

Combining mutation and selection

A model must specify the life-cycle order. For example, if selection acts first and mutation acts afterwards, define

\[p_s=\frac{pw_A}{pw_A+(1-p)w_a}.\]

Then mutation gives

\[\boxed{p_{n+1}=(1-\mu)p_s+\nu(1-p_s)}.\]

This recurrence can be iterated numerically or solved for equilibrium under particular parameter choices.

Order matters in exact discrete models. Selection followed by mutation and mutation followed by selection need not give exactly the same one-generation recurrence. Under weak evolutionary forces the difference may be small, but the model should state the assumed life cycle.

A deleterious allele

It is common to track the frequency \(q\) of a deleterious allele \(a\). Mutation from the normal allele \(A\) introduces \(a\) at rate \(\mu\), while selection removes it.

The equilibrium depends strongly on whether the deleterious effect is expressed in heterozygotes.

Haploid mutation–selection balance

Suppose the deleterious haploid allele has relative fitness \(1-s\), with \(0\[\frac{dq}{dt}\approx \mu(1-q)-sq(1-q).\]

For an interior equilibrium with \(q\ne1\),

\[\mu-sq^*=0,\]

\[\boxed{q^*\approx\frac{\mu}{s}}.\]

When \(\mu\ll s\), this equilibrium is small.

Diploid selection and dominance

For a diploid locus, assign relative fitnesses

Genotype	Relative fitness
\(AA\)	\(1\)
\(Aa\)	\(1-hs\)
\(aa\)	\(1-s\)

Here \(s\) measures the selective disadvantage of the deleterious homozygote and \(h\) describes how much of that disadvantage is expressed in the heterozygote.

When the deleterious allele is partly dominant

If \(h>0\) and the deleterious allele is rare, most copies occur in heterozygotes and selection can act on them. Under the usual assumptions of weak mutation and a rare deleterious allele,

\[\boxed{q^*\approx\frac{\mu}{hs}}.\]

This approximation requires \(hs\) to be large enough relative to \(\mu\) for \(q^*\) to remain small.

Completely recessive deleterious allele

If \(h=0\), selection does not act against heterozygotes in this simple model. When \(q\) is rare, deleterious homozygotes occur with frequency approximately \(q^2\). Balancing mutation against selection then gives the classical approximation

\[\boxed{q^*\approx\sqrt{\frac{\mu}{s}}}.\]

This is generally larger than \(\mu/s\) because a recessive deleterious allele can be hidden from selection in heterozygotes.

Do not use one mutation–selection formula universally. The equilibrium depends on ploidy, dominance, the fitness model, mutation direction and the approximations being made.

Back mutation

If mutation also converts \(a\) back to \(A\), an additional loss term for \(a\) appears. The equilibrium then reflects forward mutation, reverse mutation and selection together.

Strong selection versus weak selection

When selection is strong relative to mutation, a deleterious allele may remain rare. When selection is weak, mutation can maintain a higher frequency. The ratio of the relevant rates is therefore often more informative than either parameter considered alone.

Deterministic versus finite populations

The equations above describe expected systematic changes. In a finite population, genetic drift also changes allele frequencies randomly. A deterministic mutation–selection equilibrium is therefore not a guarantee that a realised finite population remains exactly at that frequency.

Why mutation is evolutionarily important

Mutation supplies new heritable variants, but mutation by itself does not determine whether a variant spreads. Its future frequency depends on selection, drift, migration and the genetic context.

Transition to genetic drift

So far the allele-frequency equations have been deterministic: the same starting state gives the same future trajectory. The next lesson removes that assumption and studies random allele-frequency change caused by finite-population sampling.

Key idea. Mutation introduces or converts alleles; selection changes their representation according to fitness. Their balance can maintain non-zero allele frequencies, but the equilibrium formula depends on the biological assumptions. Finite populations add genetic drift, which is considered next.