Mutation-Selection Balance Calculator

A mutation-selection balance calculator solves one of population genetics’ most useful questions: given how often a harmful mutation arises and how strongly natural selection removes it, what is the equilibrium frequency of the deleterious allele in a population?

This calculator takes a mutation rate, a selection coefficient, and a dominance coefficient, and returns the predicted allele frequency along with the affected population fraction, carrier frequency, and mutational load, which measures the reduction in mean fitness associated with deleterious mutations.

What Is Mutation-Selection Balance?

Every generation, natural selection removes some of those copies by reducing the reproductive success of individuals whose genotypes carry a fitness cost. The mutation-selection balance is the point where these two forces cancel: the rate at which selection removes alleles exactly equals the rate at which mutation introduces them. The allele frequency stops changing and the population sits at a stable equilibrium.

The classic result, worked out by J. B. S. Haldane in 1927, gives a simple approximation for the equilibrium allele frequency q̂ of a rare deleterious allele, based on the mutation rate μ, selection coefficient s, and dominance coefficient h.

This makes mutation-selection balance one of the cleanest applications of population genetics theory: predict a gene’s disease frequency from first principles, then check the prediction against observed data.

Overview of Chromosomal Mutations, Types & Examples

Chromosomal mutations are any alterations or errors that occur on a chromosome. In living organisms, mutations occur at one in every ten million cell replications. Explore what happens when a chromosome encounters such changes in its structure, number, and type. Learn the pros and cons of chromosomal mutations.

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The Two Equilibrium Formulas

The formula splits into two cases based on dominance. For a deleterious allele that is partially or fully dominant (h > 0), the equilibrium frequency is given by the Haldane formula:

q̂ = μ / (h × s)

This form applies to additive, partially dominant, and fully dominant alleles. Heterozygotes also pay a fitness cost, so selection acts on both heterozygotes and homozygotes. The higher the mutation rate or the weaker the selection, the higher the equilibrium frequency.

For a fully recessive deleterious allele (h = 0), the formula changes. Heterozygotes are unaffected, so selection acts only on the rare aa homozygotes. In that case:

q̂ = √(μ / s)

This is the celebrated square-root formula. The intuition: new aa homozygotes appear at rate q², selection removes them at rate s, and the equilibrium balances the two. Recessive deleterious alleles can drift to much higher frequencies than dominant ones because heterozygotes are invisible to selection.

The fully recessive case is a boundary case. As h becomes very small but remains nonzero, the simple dominant approximation q-hat = μ / (hs) can behave very differently from the recessive square-root formula, so very small dominance values should be interpreted with care.

The calculator above uses the sqrt formula for h = 0 and the Haldane formula for h > 0. It also returns the population-level “cost of selection” Haldane derived: the mutational load L, which is 2μ for any allele with nonzero dominance and μ for fully recessive alleles.

This is the reduction in mean fitness associated with maintaining the equilibrium, and in the simple Haldane model it is independent of the selection coefficient s.

How to Use This Calculator

The Mutation Rate (μ)

This is the per-gene, per-generation rate at which the wild-type allele mutates into the deleterious form. Mutation rates vary widely by locus and mutation type. For textbook single-gene problems, values around 10^-6 per gene per generation are often used, while specific loci can differ by orders of magnitude. The text input accepts scientific notation: 1e-6, 1.4e-5, and so on.

The Selection Coefficient (s)

This is the fitness reduction for aa homozygotes, ranging from 0 (no selection) to 1 (lethal). A coefficient of 0.1 means aa homozygotes have 90% the reproductive success of AA homozygotes.

Real values for disease alleles vary widely. In classroom models, values near 1 represent very strong selection, values around 0.5 represent substantial fitness reduction, and values near 0.01 represent weak selection. The slider updates the displayed value as you drag.

The Dominance Coefficient (h)

This describes how much of the fitness cost applies to heterozygotes. At h = 0, only aa homozygotes are affected (full recessivity). At h = 0.5, heterozygotes pay half the cost (additive inheritance). At h = 1, heterozygotes and homozygotes have the same reduced fitness (full dominance). The boundary h = 0 is the only mathematically distinct case; everything else uses the Haldane formula.

All three parameters update the output cards live as you change them. The “Load a preset” dropdown fills in three textbook example values: cystic fibrosis, achondroplasia, and a generic partial-dominance case. The calculator also displays a “regime” label that tells you which formula is being used.

Worked Examples

Example 1: Why Does Cystic Fibrosis Persist?

Cystic fibrosis is caused by loss-of-function mutations in the CFTR gene, most commonly ΔF508. The disease is autosomal recessive: only aa homozygotes show the disease phenotype, while Aa carriers have normal fitness. Using a per-generation mutation rate of about 6.7 × 10^-7 for the ΔF508 allele, and a selection coefficient of 0.9 (severe childhood mortality in the pre-treatment era), the calculator predicts:

• Equilibrium allele frequency q̂: 8.6 × 10^-4 (about 1 in 1160 alleles)
• Disease frequency (q²): 7.4 × 10^-7 (about 1 in 1.3 million births)
• Carrier frequency (2q): 1.7 × 10^-3 (about 1 in 580 people)
• Mutational load: 6.7 × 10^-7

That carrier frequency is the result predicted by this simple one-allele mutation-selection model. It depends strongly on the assumed mutation rate and selection coefficient. In real populations the observed disease frequency is often higher than the simple model predicts, especially in European-derived populations where CF reaches about 1 in 2,500 to 3,500 white newborns.

The discrepancy is often discussed in terms of higher historical mutation rates, possible heterozygote advantage, founder effects in isolated populations, or the fact that real CFTR disease variation involves many alleles rather than a single mutation.

The Hardy-Weinberg Calculator is the natural tool for converting between observed carrier frequencies and the underlying allele frequency, which is the input this calculator needs.

Example 2: Achondroplasia and the De Novo Paradox

Achondroplasia is the most common form of dwarfism, caused by gain-of-function mutations in FGFR3 (Fibroblast Growth Factor Receptor 3). The disease is autosomal dominant with nearly complete penetrance: heterozygotes have the condition, and aa homozygotes are typically stillborn or die shortly after birth. About 80% of cases are de novo mutations, meaning the parents are unaffected.

As a simplified dominant-allele classroom model, using μ = 1.4 × 10^-5, s = 0.99, and h = 1 gives:

• Equilibrium allele frequency q̂: 1.4 × 10^-5
• Disease frequency (2q for dominant): 2.8 × 10^-5 (about 1 in 35,000 births)
• Mutational load: 2.8 × 10^-5 (twice the mutation rate, the classic Haldane result)

Best estimates place the frequency of achondroplasia at about 1 in 15, 000 to 40,000 live births. The high mutation rate is exactly what makes this work as a mutation-selection balance.

Most affected individuals have a newly mutated allele rather than an inherited one; GeneReviews reports that about 80% of individuals with achondroplasia have unaffected parents and a de novo FGFR3 pathogenic variant.

Advanced paternal age is associated with de novo FGFR3 pathogenic variants, so paternal age can affect observed incidence. If you want to check whether an observed achondroplasia count in a small sample is consistent with the model, the Chi-Square Test Calculator can compare expected and observed counts directly.

Example 3: Solving for Selection from Observed Data

The calculator also works in reverse. If you have observed disease frequency data and know the mutation rate, you can solve for the implied selection coefficient. For example, suppose a population has a recessive disease with a known mutation rate of 10^-6 and an observed disease frequency of 1 in 10,000.

The observed q is √(1/10000) = 0.01. Using the recessive formula q̂ = √(μ/s) and solving for s, you get s = μ/q̂² = 10^-6 / (0.01)² = 10^-2. The implied selection coefficient against homozygotes is 0.01, meaning affected individuals have 99% the reproductive success of unaffected individuals.

This kind of reverse calculation is a simple textbook way to estimate the implied selection coefficient under the assumptions of the model.

Limitations of the Simple Model

The mutation-selection equilibrium above assumes an idealized population: infinite size, random mating, no migration, no back mutation, and a single deleterious allele at a single locus. Real populations violate all of these assumptions to varying degrees. The most important caveats for genetics students and researchers:

• Genetic drift in small populations: The deterministic equilibrium only applies when the population is large enough that random sampling does not dominate. In small populations, drift can push allele frequencies away from the predicted equilibrium, especially when selection is weak relative to effective population size.
• Heterozygote advantage: If carriers have higher fitness than wild-type homozygotes (h < 0), the simple model breaks down. The classic example is sickle cell anemia, where heterozygotes are resistant to malaria. The result is a balanced polymorphism with much higher allele frequency than mutation-selection balance alone would predict.
• Back mutation: The model assumes mutation is one-directional. In reality, deleterious alleles can mutate back to wild type at some rate. The effect is usually small because the back-mutation rate is much lower than the forward rate, but it becomes important when the allele is already at high frequency.
• Multiple deleterious alleles: Real genes carry many different loss-of-function mutations, not just one. The total mutation rate to any deleterious allele is what matters, and each allele can have its own dominance and selection profile. Aggregating across many alleles requires a different framework.
• Non-random mating: Assortative mating, inbreeding, and population structure all shift the equilibrium. The Pedigree Analyzer handles the inbreeding case for individual families, which is a related but separate problem.

For a worked example of the most common related calculation (predicting offspring genotype ratios given parental genotypes), the Punnett Square Calculator covers that ground.

This calculator sits within the genetics tools sub-hub alongside other population genetics tools including Hardy-Weinberg, Chi-Square, and Epistasis.

It is the natural next step after Hardy-Weinberg: where Hardy-Weinberg gives a no-evolution baseline under ideal conditions, mutation-selection balance models what happens when mutation and selection act at the same time.

Frequently Asked Questions