Mutation Frequency Calculator
Mutation frequency = mutants ÷ total screened. With a Wilson score confidence interval, a frequency ↔ mutation-rate converter, and the Luria–Delbrück fluctuation test (p₀ method). Educational use — population genetics, 100% client-side, no data leaves your browser.
- As "1 in N"
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- Percent
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- Confidence interval
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Direction: under neutrality, f = 1 − e−μt, so μ = −ln(1−f) / t.
Valid only under neutrality (no selection, no back-mutation) and small μ·t.
p₀ method: m = −ln(p₀) mutations per culture, μ = m / N. The zero-mutant class is the only unambiguous one.
- p₀ (zero-class fraction)
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Worked examples (verified)
- 25 mutants / 1,000,000 screened
- f = 2.5 × ¹⁰⁻⁵ (25 per million ≈ 1 in 40,000)
95% CI: 1.69 × ¹⁰⁻⁵ – 3.69 × ¹⁰⁻⁵ - Luria–Delbrück exp. 16: p₀ = 11/20, N = 2×10⁷
- m = 0.5978 mutations/culture → μ = 2.99 × ¹⁰⁻⁸ /cell/generation
- rpoS exercise: p₀ = 10/50, N = 10⁷
- m = 1.6094 → μ = 1.61 × ¹⁰⁻⁷ /cell/generation
Order-of-magnitude guide
| Scale | Context | Note |
|---|---|---|
| 10⁻⁴ – 10⁻⁶ | per-locus mutation rate, varies by organism | upper end of spontaneous locus rates (WUSTL pop-gen notes) |
| ~10⁻⁸ per bp/gen | human de novo: ~38 new single-base mutations per genome per generation | ≈ 1 per 83 million bases (Univ. of Utah) |
| ~10⁻¹⁰ per bp/gen | per-site rate, fruit flies | why locus rates span 10⁻⁴–10⁻⁶ (WUSTL notes) |
| ≈ 3 × 10⁻⁸ | T1 resistance in E. coli (Luria–Delbrück exp. 16) | classic fluctuation-test anchor |
| ≈ 1.6 × 10⁻⁷ | rpoS fluctuation-test exercise | textbook-style worked example |
Methodology & assumptions
- Definition: mutation frequency f = k/n, mutants over total screened — a measured sample statistic.
- Confidence interval: Wilson score interval for a binomial proportion (honest for rare events).
- Frequency ↔ rate: f = 1 − e−μt under neutrality; shown approximation f ≈ μ·t for small μ·t.
- Fluctuation test: p₀ method, m = −ln(p₀), μ = m/N (Luria & Delbrück, 1943).
- Assumptions: neutral mutations (no selection), independent events, no back-mutation, full plating detection. Violations bias frequency away from the rate — frequency then measures the population snapshot, not μ.
Scope boundary — related but different tools
- Allele-frequency calculator (/tools/allele-frequency/, #1384): allele proportions p/q under Hardy–Weinberg from genotypes — standing variation, not new mutants.
- Frequency converter (/tools/frequency-converter/, #288): physical units — Hz, kHz, RPM. Unrelated to genetics.
- Mutation-rate tool (planned, #1426): μ per base per generation — the inverse direction of this tool's frequency ↔ rate tab.
Frequently asked questions
What is mutation frequency?
Mutation frequency is the fraction of mutants observed in a screened population: f = (number of mutants) / (total individuals or cells screened). It is a measured quantity — a sample statistic — not a theoretical constant. For example, finding 25 mutants among 1,000,000 screened cells gives a frequency of 2.5 × 10^-5 (25 per million, or about 1 in 40,000).
What is the mutation frequency formula?
f = k / n, where k is the number of individuals (or cells, colonies, offspring) carrying the mutation and n is the total number screened. With 3 mutants in 1,000,000 screened, f = 3 × 10^-6; a Wilson 95% confidence interval for that count runs from about 1.02 × 10^-6 to 8.82 × 10^-6.
How is mutation frequency different from mutation rate?
Mutation rate (mu) is the probability of a mutation per unit — e.g. per base pair per generation. Mutation frequency is the proportion of mutants actually observed in a population. Rate is a per-event constant; frequency is a population snapshot that depends on how many generations have passed, selection, and sampling. This tool computes frequency; the companion tool mutation-rate (planned, #1426) will compute rates.
How do I convert between mutation frequency and mutation rate?
Under neutrality (no selection), frequency and rate are linked by f = 1 - exp(-mu * t), so mu = -ln(1 - f) / t for t generations. When mu * t is small, mu ≈ f / t is an excellent approximation: a frequency of 10^-6 after 1 generation implies a rate of about 1.0000005 × 10^-6 per generation. With selection, back-mutation, or drift this conversion breaks down.
What is the fluctuation test (Luria-Delbruck) p0 method?
The fluctuation test distinguishes spontaneous from induced mutations. In the p0 method you count the fraction p0 of parallel cultures with zero mutants; the mean number of mutations per culture is m = -ln(p0), and the mutation rate is mu = m / N where N is the number of cells per culture. The zero class is used because it is the only unambiguous one — a plate with colonies cannot tell 100 mutations from 1 mutation with 100 daughter cells.
What did the Luria-Delbruck experiment actually find?
In experiment 16, 11 of 20 cultures had no T1-resistant mutants (p0 = 0.55), with about 2 × 10^7 bacteria per culture. Then m = -ln(0.55) ≈ 0.5978 mutations per culture, giving mu ≈ 2.99 × 10^-8 mutations per cell per generation — roughly 3 × 10^-8. This tool reproduces that calculation exactly.
What order of magnitude should a mutation frequency be?
Per-locus spontaneous rates typically fall between 10^-4 and 10^-6 depending on organism (per WUSTL population-genetics lecture notes); per-base rates are much smaller — about 10^-10 in fruit flies, and humans carry roughly 38 new single-base mutations per genome per generation (about 1 per 83 million bases, per the University of Utah). The classic bacterial anchor is ~3 × 10^-8 mutations per cell per generation for T1 resistance in E. coli.
Why does this calculator use a Wilson score confidence interval?
Mutation counts are usually tiny relative to the sample, so the naive plus/minus normal interval can dip below zero or under-cover. The Wilson score interval is the standard honest interval for a binomial proportion with rare events. For 3 mutants in 1,000,000, the 95% interval is roughly 1.0 × 10^-6 to 8.8 × 10^-6 — wide, which is the truth when events are this rare.
What assumptions does this tool make?
Honest assumptions: (1) mutations are neutral — if mutants grow faster or slower than wild type, frequency no longer tracks the rate; (2) each scored mutant is an independent event (no jackpot clones inflating counts); (3) no back-mutation; (4) for rate conversion, mu * t must be small and selection absent. The p0 fluctuation method additionally assumes full plating (every mutant is detected).
Is this the same as the allele-frequency calculator?
No. The allele-frequency tool (#1384) computes allele proportions p and q in a population from genotype counts under Hardy-Weinberg equilibrium — a statement about standing variation. Mutation frequency counts new mutants among screened individuals — a statement about the mutation process. Different denominators, different questions; neither substitutes for the other.
Is this related to the frequency converter tool?
No. The frequency-converter tool (#288) converts physical signal units — hertz, kilohertz, RPM, radians per second. 'Frequency' there means cycles per second. Here 'mutation frequency' means mutants per individual screened. Same word, unrelated domains.
Can mutation frequency exceed the mutation rate?
Yes, easily. Rate is per generation per unit; frequency accumulates over generations and can be amplified by selection. Under neutrality with small mu * t, frequency ≈ mu * t, so after 1,000 generations at mu = 10^-8 the expected frequency is about 9.99995 × 10^-6 — roughly a thousand times the rate. They only coincide when t = 1.
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