Statistical Genomics Foundry
Paper
domain/molecular-evolutiontopic/dnds

Messer & Petrov 2013 — Frequent adaptation and the McDonald–Kreitman test

1. Citation

Messer PW, Petrov DA. 2013. “Frequent adaptation and the McDonald–Kreitman test.” PNAS 110(21):8615–8620. DOI 10.1073/pnas.1220835110. PMID 23650353; PMCID PMC3666677. Full text read via PMC: https://pmc.ncbi.nlm.nih.gov/articles/PMC3666677/ (access date 2026-07-05).

2. Access note

Full main text read (Abstract through Materials and Methods) from the PMC HTML rendering. Equations and inline math are served as images in that rendering; the three display equations ([1],[2],[3]) and the load-bearing inline expressions (exponential fit form, cutoff values, point estimates) were retrieved as image files and transcribed directly. The SI (SI Text / SI Materials and Methods, Table S1, Fig. S1) was not read — statements citing “(SI Text)” or “(SI Materials and Methods)” are referenced but their contents are outside this summary.

1.5 License mode

Own-words. The PMC record states the article is “Freely available online through the PNAS open access option,” but the specific license is not printed on the page [summarizer-inferred: the exact CC variant is not verifiable; Europe PMC reports no license]. Unverified → treated as restricted (LicenseRef-all-rights-reserved, own-words-only) under the default-deny rule. §7 is paraphrased; equation forms and functional strings are reproduced verbatim as facts regardless.

3. Thesis

Linked selection (genetic draft + background selection) systematically biases the MK test and its DFE/demography-correcting extensions — standard MK generally underestimates the adaptation rate α, but a simple asymptotic extension of MK (extrapolating a frequency-resolved α(x) to x→1) recovers accurate α in the authors’ forward simulations.

4. Problem & context

The MK test contrasts polymorphism vs. divergence at neutral (synonymous) and functional (nonsynonymous) sites to estimate α, the fraction of functional substitutions fixed by positive selection. It is robust to mutation-rate and coalescent-history variation across the genome, but is known to be confounded by (a) slightly deleterious mutations and (b) demography. Extensions correct for these by inferring the DFE and/or demography — but all assume site independence. Recent evidence that genetic draft (recurrent linked sweeps) and background selection pervasively violate independence motivates testing whether MK-type methods survive realistic linkage.

5. Method / approach

  • Forward population-genetic simulations of a 10-Mb chromosome with realistic gene structure (1 gene / 40 kb; each gene = 8 exons × 150 bp separated by 1.5-kb introns; 550-bp 5′ UTR, 250-bp 3′ UTR; three of four sites in exons/UTRs functional, every fourth exon/UTR site synonymous). ~4% of chromosome functional (deliberately conservative vs. denser genomes like Drosophila).
  • Mutations codominant, fitness effects additive across sites. Evolves under mutation, recombination, selection. Parameters chosen so products Nμ, Ns, and ratio s/r are what matter; tuned to human-like synonymous heterozygosity and to functional/synonymous substitution-rate ratio observed in humans.
  • Functional mutations of four types (neutral, beneficial, deleterious, strongly deleterious), each a specific selection coefficient; 40% of functional mutations always strongly deleterious. Free parameters: strength of purifying selection, strength of positive selection (s_b), and rate of adaptation α. α ranged 0 to 0.5; s_b from 0.001 to 0.05; deleterious Ns from −1 to −100 (Table S1).
  • Divergence estimated from fixations during a run; polymorphism/SFS from population samples of 100 randomly drawn chromosomes taken every N generations.
  • Studied linkage effects on: (i) neutral heterozygosity, (ii) SFS at functional & synonymous sites, (iii) fixation probabilities of deleterious mutations, then (iv) resulting MK α estimates.
  • Evaluated DFE-alpha (Eyre-Walker & Keightley) as representative DFE-based extension, under its two demographic models (constant size; single instantaneous size change), on simulated data whose nonadaptive DFE was drawn from a gamma distribution (shape β = 0.2, matching human estimates).
  • Proposed and demonstrated asymptotic MK (see §6/§7), applying it to simulations and to real D. melanogaster and human nonsynonymous polymorphism/divergence data.

MK α estimator — verbatim (eq. [1])

α ≈ 1 − (d₀/d)(p/p₀) where p, p₀ are polymorphism at functional and neutral sites; d, d₀ divergence at functional and neutral sites. (d₀ = neutral divergence, p₀ = neutral polymorphism.)

Wright–Fisher fixation probability under free recombination — verbatim (eq. [2])

π(s) = (1 − e^(−2s)) / (1 − e^(−4Ns))

6. Key claims / findings

  • Standard MK underestimates α, even with frequency cutoffs. Considering only polymorphisms with derived allele frequency x ≥ 0.1 (Fig. 2 Left) or x ≥ 0.5 (Fig. 2 Right), MK estimates still underestimate α, often substantially. Example: true α = 0.4 → MK estimate −0.2 at cutoff x ≥ 0.1; raising the cutoff to x ≥ 0.5 reduces but does not remove the discrepancy.
  • The bias is downward (toward negative α) and is worse for weakly deleterious than strongly deleterious mutations: weakly deleterious variants contribute to polymorphism but have low fixation probability; strongly deleterious contribute to neither polymorphism nor divergence and don’t bias α. Stronger positive selection mitigates the bias (weakly deleterious become effectively neutral).
  • No single effective population size (N_e) works. Fixation probabilities of deleterious mutations under linkage map to different inferred N_e for different selection coefficients; no scalar N_e reproduces fixation across all four deleterious classes. Because N enters eq. [2] exponentially, small N differences yield large fixation-probability differences.
  • Draft distorts the SFS like a population expansion — skews toward low-frequency variants — at both functional and synonymous sites, and these distortions do not fit any constant-N_e mutation–selection–drift model once adaptation is even moderate.
  • DFE-alpha (constant-size model) systematically overestimates α, underestimates strength of purifying selection, and overestimates gamma shape β by up to ~2×.
  • DFE-alpha (step-change model) gives more accurate α and β but overestimates mean deleterious selection strength, and always infers a population expansion that never occurred — even with zero adaptation it infers a ~5-fold expansion (background selection alone biases demographic inference); at α ≈ 0.1 it already infers ~10-fold expansion (a built-in DFE-alpha limit).
  • Asymptotic MK recovers α. Proof-of-principle run (s_b = 0.01, s_d = −0.0002, true α = 0.42): standard MK at cutoff x ≥ 0.1 gives α = 0.08; asymptotic MK gives α = 0.38. Across the gamma-DFE runs it is much more accurate than cutoff-MK and than DFE-alpha without demographic correction, and comparable to DFE-alpha with correction.
  • Real-data application. Human nonsynonymous: asymptotic α = 0.13 (0.09, 0.19), consistent with α = 0.1–0.2 from ref. 22; note standard MK excluding all polymorphisms below 10% frequency gives a negative value, α = −0.05. D. melanogaster: asymptotic α = 0.57 (0.54, 0.60), similar to (slightly higher than) prior estimates. CIs from 1,000 bootstrap replicates.

7. Load-bearing statements (own-words — all-rights-reserved; equations/functional strings kept verbatim as facts)

  • Downward-bias headline (Abstract): when slightly deleterious mutations are present, standard MK estimates of α severely underestimate the true adaptation rate — even after excluding all polymorphisms below 50% population frequency.
  • Asymptotic α(x) definition — verbatim (eq. [3]): α(x) = 1 − (d₀/d)(p(x)/p₀(x)) where p(x), p₀(x) are levels of polymorphism at functional and synonymous sites for the specific derived allele frequency x.
  • Why it cancels demography/draft (Discussion): because α(x) depends only on the ratio p(x)/p₀(x), any bias that shifts the SFS at functional and synonymous sites equally — whether from demography or genetic draft — cancels; extrapolating α(x) to x → 1 should then converge near the true α, provided adaptive mutations contribute little to polymorphism and purifying selection has been roughly stable over time.
  • Fitted functional form — verbatim: an exponential of the form α(x) = a + b·exp(−cx) fitted for all x ≥ 0.1 by nonlinear least-squares and extrapolated to x → 1 (Fig. 3A). The authors motivate the exponential for the case of a single shared deleterious selection coefficient (functional polymorphism then decays roughly exponentially relative to neutral with increasing frequency), while noting the correct form is unclear when selection coefficients follow a broad distribution unless the DFE is known.
  • Mechanism cutoffs don’t fix it (Discussion): under recurrent sweeps a slightly deleterious mutation can hitchhike to substantial frequency, become unlinked in the late phase, and persist as a common polymorphism while slowly declining — at every stage its frequency overstates its fixation probability, and excluding low-frequency polymorphisms does not remove such mutations from the sample.

8. Stated scope, assumptions, limitations (the source’s own caveats)

  • Asymptotic MK assumes (i) adaptive mutations do not significantly contribute to polymorphism and (ii) purifying selection has been sufficiently stable over time.
  • The exponential fit is justified for a single deleterious selection coefficient; the correct functional form for a broad DFE is not known unless the DFE is known. The authors show the exponential still “works reasonably well” empirically for gamma-distributed coefficients but flag it as unproven generally.
  • future work must check whether the asymptotic approach also holds for a wider class of DFEs and complex demographic scenarios — and it is equally unclear whether DFE-alpha-with-correction would.
  • Simulation scenario is one gene structure / recombination regime at ~4% functional density; presented as conservative, not exhaustive.
  • Conclusions rest on forward simulations plus two real datasets; not a general analytic proof.

9. Failure modes / invalidity patterns

  • Standard MK (with or without a frequency cutoff): downward-biased / can go negative whenever slightly deleterious mutations + linked selection are present. Symptom/detector named: a negative α estimate (human data yielded α = −0.05 at a 10% cutoff), and estimates that stay below the true value even after excluding sub-50% polymorphisms.
  • Frequency-cutoff remedy fails because draft lets slightly deleterious variants persist as frequent polymorphisms whose frequency overstates fixation probability — they survive the cutoff.
  • DFE-based methods (e.g., DFE-alpha) on mutation–selection–drift assumptions: misestimate DFE mean and shape → tend to overestimate α (constant-size model). Named symptom: gamma shape β overestimated up to ~2×.
  • Demography-correcting methods: recover α but infer spurious demography — DFE-alpha’s step-change model always infers a population-size expansion even though none occurred. Named detector: an inferred large past expansion (≥5-fold even at α = 0, ~10-fold by α ≈ 0.1) from synonymous-site SFS is a red flag that draft/background selection, not demography, is being fit. General warning: estimating demography from neutral sites that lie close to functional ones (such as synonymous sites) will in general produce erroneous inferences of population expansions.
  • No scalar N_e can simultaneously reproduce fixation probabilities across selection-coefficient classes — so plugging a single effective N into diffusion formulas is invalid under moderate draft.

10. What the source does NOT address

  • No closed-form/analytic theory correcting MK for linkage (authors call for “new analytics”).
  • No general derivation of the correct α(x) functional form for arbitrary DFEs.
  • No treatment of complex/realistic demographies combined with draft for the asymptotic method (flagged as future work).
  • No explicit minimum-data prescription (see §12) — sample size in sims is 100 chromosomes, but no stated per-bin site-count threshold for the asymptotic fit.
  • SI-resident details (exact simulation code, background-selection/draft analytic approximations, Table S1 full parameter grid) not covered here (SI not read).

11. Open questions / ambiguities left unresolved

  • Which functional form to fit α(x) when selection coefficients come from a broad DFE.
  • Whether asymptotic MK holds under complex demography + high draft.
  • Whether any current method (asymptotic MK or DFE-alpha-with-correction) is reliable across the full realistic parameter space.
  • How to obtain trustworthy demographic inference when synonymous sites are draft-distorted.

12. Guidance answers

  • Mechanism of slightly-deleterious downward bias, and why toward negative α: Answered (§6, §7, §9). Slightly deleterious variants contribute to polymorphism (p up) but have low fixation probability (little contribution to divergence d), so eq. [1] α ≈ 1 − (d₀/d)(p/p₀) inflates the subtracted term (d₀/d)(p/p₀) and drives α down, even negative. Frequency cutoffs don’t rescue it because under recurrent sweeps a slightly deleterious mutation hitchhikes to high frequency, unlinks late in the sweep, and lingers as a frequent (yet ultimately non-fixing) polymorphism — at every stage the mutation’s frequency overstates its fixation probability. Bias is worse for weakly than strongly deleterious mutations.
  • Asymptotic-MK procedure + functional form + estimator equation verbatim: Answered (§7). Compute frequency-resolved α(x) = 1 − (d₀/d)(p(x)/p₀(x)) [eq. 3] across derived-allele-frequency bins, fit α(x) = a + b·exp(−cx) (exponential) for x ≥ 0.1 by nonlinear least-squares, and extrapolate to x → 1; that limit ≈ true α. CIs via 1,000 bootstrap resamples of the observed α(x) values.
  • Data requirements (min sites/bin, SFS density, sample size): The paper states no explicit minimum sites-per-frequency-bin and no “≥50 sites/bin” threshold. What it does state: simulation population samples of 100 chromosomes; the exponential fit is over derived-allele-frequency bins for x ≥ 0.1; goodness-of-fit and CIs come from 1,000 bootstrap replicates. A skill claiming “≥50 sites/bin” is not supported by this source — report as convention, not from this paper.
  • Robustness to demography (and limits): Answered (§7, §8). By construction α(x) depends only on the ratio p(x)/p₀(x), so any biases that shift the SFS at functional and synonymous sites in the same way, whether from demography or genetic draft, effectively cancel out — i.e., robust to demography to the extent it perturbs functional and synonymous SFS equally. Limits: assumes adaptive mutations don’t contribute much to polymorphism and purifying selection is stable over time; the exponential form is only justified for a single deleterious coefficient; authors explicitly flag that robustness under complex demographic scenarios and broad DFEs is unverified.