What the Site-Frequency Spectrum Summarizes
A list of variants records which sites differ, but it does not by itself show whether most variants are observed in one chromosome or at intermediate frequency. The site-frequency spectrum (SFS) compresses that information: it counts segregating sites according to how many sampled chromosomes carry an allele. Its low-count bins are especially prominent in rare-variant data.
The SFS is a summary of a particular sample, not a direct inventory of every variant in a population. Its bins depend on how many chromosomes were sampled, which genomic sites were considered, and how variants were discovered or retained. Interpreting it therefore requires stating the sample and the way the spectrum was constructed.
“Derived” means the allele inferred to have arisen from the ancestral allele at that site. If ancestral states are unavailable or unreliable, the spectrum can instead be folded: each site is grouped by its minor-allele count \(m=\min(i,n-i)\). In a folded spectrum, a site with derived count \(n-1\) is in the same bin as one with derived count 1. Consequently, “singleton” in a folded spectrum means a minor allele observed once; it does not assert that the observed allele is derived.
An SFS can report counts of sites in each bin, or proportions after dividing bin counts by the total number of included segregating sites. Those summaries answer different questions. Counts depend on how many sites were examined, while proportions describe how the included variants are distributed across bins.
Worked Example: Reading Singleton and Doubleton Bins
Suppose a hypothetical study considers 100 sampled chromosomes and finds 10 biallelic segregating sites with derived-allele counts \(1,1,2,2,2,3,5,49,50,98\). Find the unfolded singleton and doubleton counts, and describe how folding changes the doubleton count.
In the unfolded spectrum, exactly two sites have derived count 1, so \(\xi_1=2\). Exactly three sites have derived count 2, so \(\xi_2=3\). Their sample frequencies are \(1/100=0.01\) and \(2/100=0.02\), respectively.
For the folded spectrum, replace each derived count \(i\) by \(m=\min(i,100-i)\). The site with derived count 98 has minor-allele count \(2\), so it joins the folded doubleton bin. The folded singleton count is still 2, and the folded doubleton count is \(3+1=4\). The distinction matters: folding loses the direction supplied by the ancestral-state assignment.
A Neutral Reference Pattern
A simple theoretical reference helps explain why low-frequency bins can be large even without a special process that creates rare variants. Consider a neutral, constant-size diploid population under the standard Wright–Fisher model, with random mating, no selection, no migration, and an infinite-sites mutation model. The infinite-sites assumption means that a site mutates at most once in the genealogy being considered, so recurrent mutation does not obscure the count. Let \(\mu\) be the mutation rate per site per generation and \(N_e\) the effective population size; define the population-scaled mutation rate \(\theta=4N_e\mu\).
The \(1/i\) pattern follows from the expected amount of genealogical branch length that subtends \(i\) of the sampled chromosomes. In coalescent time units of \(2N_e\) generations, the expected branch length in that class is \(2/i\). A lineage accumulates mutations at rate \(2N_e\mu=\theta/2\) per site per coalescent time unit. Multiplying the expected branch length by the mutation rate gives \( (2/i)(\theta/2)=\theta/i\) expected mutations per site. Across \(L\) sites, the expected count is \(L\theta/i\).
This result is a reference, not a universal prediction for real data. It assumes a particular population model and a consistent set of sites. It also concerns expected counts: it does not say that a realized singleton-to-doubleton ratio must equal 2, nor does it provide a test for sampling uncertainty.
Worked Example: Calculating Neutral Expected Bin Counts
Suppose a hypothetical analysis examines \(L=20{,}000\) comparable sites in a sample of 100 chromosomes. Under the neutral reference model, take \(\theta=0.001\) per site. Calculate the expected singleton, doubleton, and count-5 bins.
First, \(L\theta=20{,}000(0.001)=20\). Substituting into the expected-spectrum formula gives $$ E[\xi_1]=\frac{20}{1}=20,\qquad E[\xi_2]=\frac{20}{2}=10,\qquad E[\xi_5]=\frac{20}{5}=4. $$
Thus the model predicts 20 singleton sites, 10 doubleton sites, and 4 sites with derived count 5 on average. These are expected counts, not observed counts or guaranteed integer outcomes. The sample size identifies which count bins are possible; under this model, the expectation for a given count \(i\) has the stated form.
Why Mutation and Demography Change the Shape
Mutation supplies new variants. In the neutral reference formula, increasing a uniform mutation rate increases \(\theta\), scaling the expected count in every bin. If the number of examined sites also increases, expected counts scale with \(L\). Therefore, a larger number of singletons in one data set need not mean a larger proportion of singletons: the total number of observed variants and the distribution across bins should be considered separately.
Mutation rates are not necessarily uniform across genomic regions or mutation types. If one region has a higher mutation rate, it can contribute more variants across bins, and differences in mutation processes can complicate comparisons between regions. Comparisons of SFS counts are most interpretable when the sample size, genomic scope, and treatment of sites are made explicit.
Demography changes genealogies, and mutations occurring on different parts of a genealogy tend to appear at different sample counts. For example, rapid recent population growth can produce many recent branches leading to individual sampled chromosomes. Mutations on those terminal branches are often singletons, so an excess of low-count variants can be compatible with recent growth. A bottleneck can instead change frequencies through drift, remove some rare variants, and move others to higher frequencies. The resulting spectrum depends on the timing and severity of demographic events; there is no single direction that describes every bottleneck history.
These patterns are not unique demographic fingerprints. Mutation-rate differences, ancestry composition, selection, and the way sites entered the data can also affect the spectrum. A low-frequency excess may motivate demographic investigation, but the SFS alone does not establish one particular history.
Worked Example: Comparing Two Hypothetical Spectra
Two hypothetical samples each contain 100 chromosomes and have the same stated number of comparable sites. Sample A has 100 singletons, 50 doubletons, and 50 variants with derived counts from 3 through 10. Sample B has 180 singletons, 55 doubletons, and 45 variants with counts from 3 through 10. Compare the low-frequency composition and explain what it can and cannot show.
Sample A has \(100+50+50=200\) listed variants, of which \(100/200=0.50\), or 50%, are singletons. Sample B has \(180+55+45=280\) listed variants, of which \(180/280\approx0.643\), or 64.3%, are singletons. In these hypothetical data, Sample B has both more singleton sites in absolute count and a larger singleton share among the listed variants.
The comparison is consistent with a stronger contribution of recent mutations or genealogical branches leading to individual sampled chromosomes in Sample B. It does not identify population growth as the cause: mutation processes and ascertainment could also differ. Nor does this comparison alone establish that the difference is beyond sampling variation. Interpreting the contrast requires comparable sampling and site inclusion, and an appropriate analysis of uncertainty.
Ascertainment Can Reshape the Observed Spectrum
Ascertainment is the process by which variants are discovered and selected for inclusion. If a variant is more likely to enter a data set when it is observed repeatedly in a discovery sample, singleton and doubleton variants will be underrepresented in the resulting SFS. The observed spectrum then describes variants that passed the discovery process, not all variants at the considered sites.
For example, a common-variant array might include only sites meeting a minimum frequency threshold in an initial discovery cohort. A later sample can still contain rare alleles at those sites, but the set of sites was selected because of their discovery-cohort frequencies. A frequency-based filter applied directly to the analyzed sample has a related effect: bins below the filter threshold are removed by design. Neither procedure reveals the unfiltered spectrum.
Worked Example: Applying a Discovery-Frequency Threshold
In a hypothetical discovery sample of 100 chromosomes, 200 candidate variants have minor-allele counts as follows: 80 singletons, 50 doubletons, 30 with counts 3–4, 20 with counts 5–9, and 20 with counts at least 10. Suppose a variant is retained only if its minor-allele frequency in this sample is at least 0.05. How many variants remain, and what happens to the low-count bins?
With 100 chromosomes, a minor-allele frequency of at least 0.05 requires a minor-allele count of at least \(0.05(100)=5\). The first three groups—counts 1, 2, and 3–4—are excluded: \(80+50+30=160\) variants. The retained groups have counts 5–9 and at least 10, so \(20+20=40\) variants remain.
The retained data contain no discovery singletons or doubletons, not because those variants were absent from the underlying population, but because the inclusion rule excluded them. Any SFS computed from these 40 variants is conditional on that ascertainment rule. Treating it as an unfiltered population spectrum would misrepresent what was measured.
Reading and Reporting an SFS Carefully
A useful SFS description states the sample size in chromosomes, whether counts are unfolded or folded, the genomic sites included, and whether the entries are counts or proportions. If bins are grouped—for example, counts 3–5—report the exact range. These details prevent a “rare-variant excess” from being ambiguous across studies with different sample sizes or filters.
A common mistake is to conclude that a large singleton count proves recent population growth. The reasoning is incomplete: recent growth can increase the relative contribution of very recent branches, but a higher mutation rate or a discovery process that treats frequency differently can also alter observed counts. The correction is to describe the observed spectrum first, check whether samples and site inclusion are comparable, and treat demographic explanations as hypotheses requiring evidence beyond the pattern alone.
Check Your Understanding
Answer these questions using the definitions and assumptions given above.
- In a sample of 80 chromosomes, what sample frequency corresponds to a singleton? What is the derived-allele count of a doubleton?
- A site has derived-allele count 79 in a sample of 80 chromosomes. Which bin does it enter in the folded spectrum?
- Under the stated neutral reference model, how does the expected count in bin 4 compare with the expected count in bin 2?
- Why can a discovery rule requiring minor-allele count at least 5 reduce the observed singleton and doubleton counts?
- Name one reason an excess of singletons might be compatible with recent population growth, and one reason that pattern alone cannot establish growth.