Demographic History and the Site-Frequency Spectrum

What the Site-Frequency Spectrum Records

A population’s demographic history affects how often variants are observed at different frequencies. The site-frequency spectrum (SFS) summarizes this information across variable sites in a sample: instead of listing each site separately, it counts how many sites carry the derived allele in one, two, or more sampled chromosomes. Its shape can be informative about population history, but it does not identify that history on its own.

Consider \(n\) sampled chromosomes at a genomic site. If the ancestral state is known, the count of copies of the allele that differs from that ancestral state is the derived-allele count. Across many sites, the SFS records the number of sites at each possible derived-allele count. It describes sampled variants, not a direct census of all allele frequencies in the population.

The first class, \(\xi_1\), counts singletons: sites where the derived allele appears once. The class \(\xi_{n-1}\) contains sites where it appears in all but one sampled chromosome. If the ancestral allele cannot be identified reliably, one can instead count copies of the less frequent allele. This gives a folded SFS, which combines derived counts \(i\) and \(n-i\); it retains frequency information but loses the direction that distinguishes rare derived from rare ancestral alleles.

A Constant-Size Neutral Benchmark

A useful reference is the standard neutral model: a randomly sampled set of chromosomes from a large, randomly mating population of constant effective size \(N_e\); neutral mutations; and an infinite-sites model in which a site mutates at most once in the genealogy. Let \(\mu\) be the per-generation mutation rate per site for a diploid autosomal locus and define \(\theta=4N_e\mu\). Under the standard coalescent for this model, the expected number of sites in frequency class \(i\) is \(\theta/i\).

The reason for the \(1/i\) pattern is genealogical. Mutations are placed along branches of the sample’s coalescent tree. A branch subtending \(i\) of the \(n\) sampled chromosomes produces a mutation seen \(i\) times. Under the constant-size coalescent, the expected total branch length subtending \(i\) sampled chromosomes is \(4N_e/i\) generations. Mutations occur at rate \(\mu\) per generation per site on that branch length, so the expected number of mutations in the class is \(\mu(4N_e/i)=\theta/i\). Summing these expectations over the classes gives $$ E[S]=\sum_{i=1}^{n-1}E[\xi_i] =\sum_{i=1}^{n-1}\frac{\theta}{i} =\theta H_{n-1}. $$

This is a benchmark for expected counts, not a claim that every sample has this shape. It also gives a ratio of expected counts: $$ \frac{E[\xi_1]}{E[S]}=\frac{\theta}{\theta H_{n-1}}=\frac{1}{H_{n-1}}. $$ That quantity is not generally the expected fraction of realized segregating sites that are singletons. The realized fraction is \(\xi_1/S\), which is undefined when \(S=0\); its expectation, if a convention is chosen for \(S=0\), is a different calculation.

How Demographic History Changes the Shape

Demographic history changes the distribution of genealogical branch lengths, which in turn changes the SFS. A recent increase in population size, often called population growth or expansion, can produce relatively many recent mutations on external branches. Such mutations are often observed as singletons or other low-count variants. The usual qualitative expectation is therefore an excess of rare variants relative to the constant-size benchmark.

A bottleneck is a period when population size is small. It increases the role of drift: variants may be lost, while others may change substantially in frequency. Depending on the bottleneck’s timing, severity, and subsequent history, the observed spectrum may have fewer rare variants, more intermediate-frequency variants among those that remain segregating, or other distortions. It is not correct to expect every bottleneck to produce one fixed SFS shape. After a bottleneck, population growth can also generate many new rare variants, so the combined history matters.

Migration connects populations with different allele frequencies. It can introduce variants into a recipient population, and a variant that is rare in the recipient may be common in the source. The resulting SFS depends on the source population’s frequencies, the amount and timing of migration, and which populations and chromosomes were sampled. Migration does not necessarily increase the rare-variant class: a migrant allele can enter at low frequency, or it can be common in the recipient if migration is substantial or repeated.

Drift operates throughout these histories, especially strongly in small populations. It changes allele frequencies randomly and can remove variants or move them toward fixation. The SFS therefore reflects the joint effects of genealogy, mutation, drift, population size changes, and migration rather than a single demographic event in isolation.

Migration and the Meaning of a Pooled Spectrum

It matters whether a spectrum is calculated within each population or after pooling samples. Pooling can make a variant look rare overall even if it is common in one population, or can hide differences between populations. Moreover, migration should be described as gene copies entering the recipient population through reproduction, not automatically as a chromosome being removed from one fixed sample and transferred into another. In the latter bookkeeping setup, the allele count in the pooled sample would be conserved.

Reading an SFS With Appropriate Caution

The SFS is valuable because it compresses many site-level observations into a summary linked to genealogical patterns. It can help compare data with explicit demographic models and can reveal that a simple constant-size benchmark is inadequate. Its interpretation nevertheless depends on the data and model assumptions. In particular, the ancestral state must be reliable to interpret an unfolded spectrum, and the sampled chromosomes should represent the population or populations being described.

Also distinguish the observed spectrum from its normalized version. The counts \(\xi_i\) include information about the total number of variants, which depends on mutation rate, sample size, and history. A normalized spectrum such as \(\xi_i/S\) describes the distribution among observed segregating sites, but loses the total-count information and is undefined if \(S=0\). Neither summary should be treated as a direct readout of demographic history without accounting for how variants were sampled and selected for analysis.

Common Mistakes

• Calling a ratio of expectations an expected fraction. The expression \(E[\xi_1]/E[S]\) is not generally \(E[\xi_1/S]\). The latter needs its own derivation and a convention for samples with \(S=0\).
• Reading one pattern as a unique demographic diagnosis. An excess of rare variants can be consistent with population growth, but other processes and data-selection effects can also create or alter that pattern.
• Assuming a bottleneck always has one signature. Its effects depend on timing, severity, subsequent growth, migration, and which variants survive and are sampled.
• Confusing pooled sampling with chromosome transfer. A new descendant copy entering a recipient while source copies remain can change the pooled count. Moving a chromosome from one fixed sample to another conserves the pooled count.