Tajima's D

Tajima's D is a foundational statistical summary test in population genetics and molecular genomics, formulated by Japanese geneticist Fumio Tajima in 1989, that evaluates whether a sample of homologous DNA sequences is evolving neutrally at demographic equilibrium under the Wright-Fisher model. The statistic measures distortions in the Site Frequency Spectrum (SFS) by calculating the standardized difference between two mathematical estimators of the scaled population mutation parameter θ = 4*N*_eμ: the mean number of pairwise nucleotide differences (θ_π) and Watterson's segregating-sites estimator (θ_W).

Under the neutral theory of molecular evolution in a panmictic, constant-size population of effective population size N_e with infinite-sites mutation rate μ, both θ_π and θ_W have the same statistical expectation (E[θ_π] = E[θ_W] = θ), yielding Tajima's D ≈ 0. Deviations from zero reveal evolutionary processes that skew the genealogy of the sampled locus: negative values (D < 0) indicate an excess of rare single-nucleotide polymorphisms arising from a recent selective sweep, purifying selection, or demographic population expansion, whereas positive values (D > 0) indicate an excess of intermediate-frequency alleles produced by balancing natural selection, ancestral population structure, or a sudden demographic bottleneck.

🧬 Interactive Population Genetics Simulation: Observe how directional selection, balancing selection, and bottlenecks shift allele frequency spectra in Allele Drift.

Two Estimators of the Population Mutation Parameter (θ = 4*N*_eμ)

Consider a sample of n homologous DNA sequences of length L base pairs drawn randomly from a diploid population of effective size N_e evolving under Kimura's infinite-sites model (where every new point mutation occurs at a previously monomorphic nucleotide site). In coalescent theory, the expected number of neutral mutations separating two randomly chosen chromosomes is governed by the population-scaled mutation parameter θ = 4*N*_eμ (or 2*N*_eμ for haploid and mitochondrial DNA). Population geneticists estimate θ from sequence alignment data using two distinct summary statistics:

1. Tajima's Pairwise Estimator (θ_π or k̂): Defined as the average number of nucleotide differences (k_ij) across all n(n − 1) / 2 pairwise comparisons in the sample, or equivalently as the sum of single-site heterozygosities across all segregating sites:

θ_π = π = [ 2 / (n(n − 1)) ] · ∑_{i < j} k_ij = ∑_*i*=1ⁿ⁻¹ ξ_i · [ 2*i*(n − i) / (n(n − 1)) ] where ξ_i is the number of polymorphic sites at which the mutant allele appears i times in the sample of n chromosomes. Because the factor 2*i*(n − i) peaks when i ≈ n / 2, θ_π weights intermediate-frequency polymorphisms heavily and is insensitive to rare variants that occur in only one individual (singletons, i = 1).

1. Watterson's Estimator (θ_W): Formulated by Geoffrey Watterson in 1975, θ_W estimates θ from the total count of segregating (polymorphic) sites S = ∑_*i*=1ⁿ⁻¹ ξ_i in the alignment, divided by a harmonic sample-size correction factor a₁ that accounts for the expected total branch length of a neutral Kingman coalescent tree:

θ_W = S / a₁, where a₁ = ∑_*i*=1ⁿ⁻¹ (1 / i) Because S simply counts every variable nucleotide site regardless of whether the minor allele appears once (1/n) or in half the sample (n/2), θ_W weights rare and common alleles equally.

Normalization by Sampling and Coalescent Variance

To test whether the observed difference d = θ_π − θ_W departs significantly from zero, Tajima (1989) normalized d by its theoretical standard deviation under the neutral constant-size coalescent model without recombination:

D = ( θ_π − θ_W ) / √[ Var(θ_π − θ_W) ] = ( π − S / a₁ ) / √( e₁ S + e₂ S(S − 1) )

Here, the coefficients e₁ and e₂ depend exclusively on the sample size n through the harmonic sums a₁ = ∑_*i*=1ⁿ⁻¹ (1/i) and a₂ = ∑_*i*=1ⁿ⁻¹ (1/i²), capturing both stochastic Poisson mutation variance (e₁ S) and the genealogical variance of coalescent branch lengths (e₂ S(S − 1)). Under standard neutrality with no recombination, the null distribution of Tajima's D is approximately a beta distribution bounded between D_min and D_max with a mean near zero and unit variance; values outside the approximate interval −2.0 < D < +2.0 are statistically significant at the two-tailed p < 0.05 threshold. When intragenic genetic recombination occurs within the sequenced locus, the variance of θ_π − θ_W shrinks because adjacent sub-segments have partially independent coalescent trees, making the ±2.0 threshold conservative unless calibrated by coalescent simulations.

Negative Tajima's D (D < 0): Excess of Rare Alleles

A negative Tajima's D arises whenever θ_π < θ_W—meaning the alignment contains many segregating sites (S is large relative to pairwise differences), but the vast majority of those polymorphisms segregate at very low frequencies (singletons and doubletons, i = 1 or 2). In coalescent terms, D < 0 reflects a star-like genealogy dominated by long external (terminal) branches and compressed internal branches:

• Recent Hard Selective Sweep (Genetic Hitchhiking): When a beneficial mutation sweeps to fixation, it purges ancestral standing variation across the linked chromosomal neighborhood. New neutral mutations that accumulate after the sweep reside on single terminal lineages, creating a sharp spike of rare variants and driving D below −2.0.
• Purifying (Negative) Selection: In conserved protein-coding exons or ultraconserved non-coding regulatory elements, deleterious mutations (s < 0) are prevented by selection from drifting to intermediate frequencies, remaining trapped as rare alleles before being eliminated.
• Exponential Population Expansion: If a population undergoes rapid demographic growth (N(t) = N₀ exp(rt)), the rate of lineage coalescence per generation slows down in the recent past compared to the ancestral past, lengthening all terminal branches across the entire genome and generating a genome-wide negative Tajima's D.

Positive Tajima's D (D > 0): Excess of Intermediate-Frequency Alleles

Conversely, a positive Tajima's D occurs when θ_π > θ_W—meaning most polymorphic sites segregate at intermediate frequencies (25%–50%), inflating pairwise nucleotide divergence (π) without adding rare segregating sites (S). Coalescently, D > 0 reflects a genealogy with deep internal branches separating two or more ancient clade lineages:

• Balancing Selection: Long-term heterozygote advantage (overdominance) or negative frequency-dependent selection maintains two or more functional allele classes at intermediate frequencies for millions of generations (T_MRCA ≫ 4*N*_e), allowing neutral linked variants to accumulate along the deep internal branches separating the balanced alleles.
• Recent Population Bottleneck or Contraction: During a sudden population crash, rare alleles (singletons) are lost first by stochastic sampling, while surviving ancestral lineages coalesce rapidly on deep internal branches, transiently elevating θ_π above θ_W across the genome.
• Cryptic Population Subdivision (Admixture): Pooling sequences from two partially isolated subpopulations with fixed or high-F_ST differences creates artificial intermediate-frequency variants across the pooled sample.

Because Tajima's D does not use an outgroup species to distinguish ancestral alleles from derived mutant alleles, a rare derived allele at frequency p = 0.05 and a nearly fixed derived allele at frequency p = 0.95 contribute identically to both θ_π and θ_W (since 2*p*(1 − p) is symmetric around p = 0.5). To capture the asymmetric information in the unfolded Site Frequency Spectrum, subsequent theorists formulated complementary statistics:

1. Fu and Li's D and F (1993): Formulated by Yun-Xin Fu and Wen-Hsiung Li, these statistics explicitly compare the number of mutations sitting on external terminal branches (singletons, η_e) against the total number of mutations (η) or pairwise differences (θ_π), using either an outgroup species (D, F) or unpolarized singletons (*D*\*, *F*\*). Because purifying selection restricts deleterious variants to terminal branches, Fu and Li's tests exhibit higher statistical power than Tajima's D for detecting background purifying selection.
2. Fay and Wu's H (2000): Polarizing polymorphic sites with a closely related outgroup (such as chimpanzee for human alignments), Fay and Wu defined θ_H = ∑_*i*=1ⁿ⁻¹ ξ_i · [ 2*i*² / (n(n − 1)) ], which weights high-frequency derived alleles (i → n − 1) quadratically. Whereas a demographic expansion makes Tajima's D negative without creating high-frequency derived alleles, a selective sweep with flanking recombination drives both Tajima's D and Fay and Wu's H (H = θ_π − θ_H) negative.
3. Zeng's E (2006): Kai Zeng combined the high-frequency derived estimator θ_L = ∑ i ξ_i / (n − 1) with Watterson's θ_W to track post-sweep recovery after intermediate-frequency polymorphisms have been wiped out.

How can researchers tell whether a negative Tajima's D is caused by a selective sweep or by population expansion?

Both a recent selective sweep and an exponential growth phase (N(t) = N₀ exp(rt)) produce an excess of rare singletons and a negative Tajima's D. However, demographic history acts genome-wide, altering the coalescent branch lengths of all autosomes simultaneously, whereas natural selection acts locus-specifically on a target gene and its linked recombinational neighborhood. Researchers therefore calculate the empirical genome-wide distribution of Tajima's D across thousands of sliding windows (e.g., 10 kb windows) and identify candidate selective sweeps as localized statistical outliers in the bottom 1% tail that also exhibit negative Fay and Wu's H and extended haplotype homozygosity.

Why do non-African human populations have a more positive average Tajima's D than African populations?

Sub-Saharan African populations maintained a larger long-term effective population size (N_e ≈ 15,000–20,000) followed by recent demographic growth, resulting in a slightly negative genome-wide average Tajima's D (D ≈ −0.5) due to abundant rare variants. By contrast, ancestral Eurasian and American populations experienced a severe Out-of-Africa founder bottleneck roughly 50,000 to 70,000 years ago (N_e dropping to ~1,500–3,000), which purged rare singletons while retaining common ancestral polymorphisms at intermediate frequencies, shifting the genome-wide background Tajima's D toward positive values (D ≈ +0.2 to +0.8).

What biological loci consistently show strongly positive Tajima's D values?

The classic examples of strongly positive Tajima's D (D > +2.0 to +3.5) are host-pathogen coevolutionary loci maintained by balancing selection: the peptide-binding groove exons of vertebrate Major Histocompatibility Complex (MHC) Class I and II genes, the ABO blood group glycosyltransferase locus, plant self-incompatibility (S-RNase) loci, and surface antigen genes in the malaria parasite Plasmodium falciparum (such as AMA1 and MSP1), where host immune memory selects for rare or diverse antigenic morphs.

1. Tajima, F. (1989). "Statistical method for testing the neutral mutation hypothesis by DNA polymorphism." Genetics, 123(3), 585–595. doi:10.1093/genetics/123.3.585
2. Watterson, G. A. (1975). "On the number of segregating sites in genetical models without recombination." Theoretical Population Biology, 7(2), 256–276. doi:10.1016/0040-5809(75)90020-990020-9)
3. Fu, Y. X., & Li, W. H. (1993). "Statistical tests of neutrality of mutations." Genetics, 133(3), 693–709. doi:10.1093/genetics/133.3.693
4. Fay, J. C., & Wu, C. I. (2000). "Hitchhiking under positive Darwinian selection." Genetics, 155(3), 1405–1413. doi:10.1093/genetics/155.3.1405