Omega Decomposition (ω, ω_a, ω_na)

MKado reports the rate of nonsynonymous substitution per site (ω), and where appropriate, its decomposition into adaptive (ω_a) and non-adaptive (ω_na) components. The decomposition was introduced by Gossmann, Keightley & Eyre-Walker (2012) and applied in the McDonald-Kreitman context — the same form MKado uses — by Coronado-Zamora et al. (2019) in their analysis of selection across the Drosophila life cycle.

Background

The classical McDonald-Kreitman summary statistics (α, NI, DoS) are proportions — they describe what fraction of substitutions were adaptive, but not the rate at which adaptation occurs. As Gossmann et al. (2012) argue, comparing α between species or genomic regions is misleading when the underlying neutral substitution rate varies. The authors recommend reporting ω_a — the rate of adaptive substitution per site, scaled by the synonymous substitution rate — as a more appropriate cross-comparison statistic. Coronado-Zamora et al. (2019) later applied the same decomposition directly to MK-style fixed/polymorphic counts (the form MKado computes), showing how ω_a and ω_na partition the substitution rate into adaptive and slightly-deleterious components across functional categories.

The decomposition

Following Coronado-Zamora et al. (2019):

\[\begin{split}\omega &= \frac{D_n}{D_s} \cdot \frac{L_s}{L_n} \\ \omega_a &= \alpha \cdot \omega \\ \omega_{na} &= (1 - \alpha) \cdot \omega = \omega - \omega_a\end{split}\]

Where:

• \(D_n, D_s\) — nonsynonymous and synonymous fixed differences

• \(L_n, L_s\) — nonsynonymous and synonymous site totals (Nei-Gojobori averaged across analyzed codons in both groups)

• \(\alpha\) — proportion of adaptive substitutions

ω = (Dn × Ls) / (Ds × Ln) is the standard dN/dS ratio with Nei-Gojobori site weighting; ω_a is the adaptive-substitution rate per synonymous site; ω_na is its complement.

Site counting

MKado computes \(L_n\) and \(L_s\) with the Nei-Gojobori (1986) method: for each clean codon, count the fraction of single-nucleotide substitutions at each site that would be synonymous, sum across the three codon positions, and average across all clean codons in both ingroup and outgroup at each analyzed position.

Note

Gossmann et al. (2012) used the F3×4 codon-frequency model implemented in PAML, not Nei-Gojobori. The two methods give numerically different site counts, so MKado’s ω, ω_a, and ω_na are not strictly comparable to numbers reported under the F3×4 convention. The decomposition itself (α × ω) is identical; the difference is in how \(L_n\) and \(L_s\) are estimated.

When ω_a / ω_na are reported (and when they are not)

ω is well-defined and well-behaved per gene: it is the standard dN/dS ratio, routinely reported per-gene by tools like PAML’s codeml. MKado therefore reports ω on every result type.

ω_a and ω_na, by contrast, inherit the variance of α. Per-gene Smith & Eyre-Walker α is known to be noisy on the small Dn/Ds/Pn/Ps counts typical of a single gene; multiplying by ω propagates that noise into ω_a. Following the convention of Gossmann et al. (2012), Galtier (2016), and Coronado-Zamora et al. (2019), MKado therefore restricts ω_a / ω_na to result types whose α estimator is appropriate for the decomposition:

Result type	ω	ω_a / ω_na	α estimator feeding ω_a
MKResult (per-gene MK)	✓	✗	per-gene Smith-Eyre-Walker α
PolarizedMKResult (per-gene)	✓	✗	per-gene polarized α
AsymptoticMKResult (per-gene)	✓	✓	asymptotic α (Messer & Petrov 2013)
AsymptoticMKResult (aggregated)	✓	✓	asymptotic α — canonical Gossmann-style ω_a
ImputedMKResult	✓	✓	imputed α (Murga-Moreno et al. 2022)
AlphaTGResult	✓	✓	Tarone-Greenland weighted α

Rationale: per-gene Smith-Eyre-Walker α is noisy because Dn, Ds, Pn, Ps counts on a single gene are typically small. Multiplying that noisy α by ω produces a ω_a whose variance is dominated by α-variance, not by any biological signal. The asymptotic, Tarone-Greenland, and imputed estimators were designed for use with sparse counts and are explicitly intended for this kind of decomposition.

If you need a per-gene ω_a estimate with proper uncertainty quantification, consider the SnIPRE Bayesian hierarchical estimator (Eilertson et al. 2012) — not currently implemented in MKado.

Confidence intervals

For result types that already perform a sampling procedure to estimate α, MKado also reports 95% CIs on ω_a and ω_na (and on ω itself where it has a sampling distribution):

Result type	ω CI	ω_a CI	ω_na CI
AsymptoticMKResult	n/a (constant)	α_CI × ω	(1 − α_CI) × ω (flipped)
AlphaTGResult	bootstrap	bootstrap	bootstrap

For AsymptoticMKResult the asymptotic-fit Monte Carlo (or exponential-fit bootstrap) only resamples polymorphisms — Dn, Ds, Ln, Ls are constants of the alignment. ω therefore has no sampling distribution and is reported as a point estimate. The CIs on ω_a and ω_na are derived analytically by scaling the α CI (ω_a_ci_low = α_ci_low × ω, etc.); the ω_na percentiles flip because (1 − α) is monotonically decreasing in α.

For AlphaTGResult the bootstrap resamples genes, so Dn, Ds, Ln, Ls all vary per replicate and ω has its own bootstrap distribution. ω, ω_a, and ω_na are recomputed per replicate and 2.5%/97.5% percentiles taken.

ImputedMKResult does not currently report CIs on ω_a / ω_na — the imputed test produces a single point estimate. If you need CIs there, consider running the imputed test on bootstrap replicates of the input alignment.

Edge cases

ω, ω_a, and ω_na return None (rendered as NA in TSV, null in JSON) when:

• \(D_s = 0\) — denominator undefined

• \(L_s = 0\) or \(L_n = 0\) — site counts undefined

• \(L_n\) / \(L_s\) were not computed (e.g. legacy mk_test_from_counts() calls without the ln / ls arguments)

Additionally, ω_a and ω_na are None when α is None — typically when \(D_n = 0\) or \(P_s = 0\).

References

Gossmann TI, Keightley PD, Eyre-Walker A (2012). The effect of variation in the effective population size on the rate of adaptive molecular evolution in eukaryotes. Genome Biology and Evolution 4(5):658-667. https://doi.org/10.1093/gbe/evs027

Coronado-Zamora M, Salvador-Martínez I, Castellano D, Barbadilla A, Salazar-Ciudad I (2019). Adaptation and conservation throughout the Drosophila melanogaster life-cycle. Genome Biology and Evolution 11(5):1463-1482. https://doi.org/10.1093/gbe/evz046

Galtier N (2016). Adaptive protein evolution in animals and the effective population size hypothesis. PLoS Genetics 12(1):e1005774. https://doi.org/10.1371/journal.pgen.1005774

Nei M, Gojobori T (1986). Simple methods for estimating the numbers of synonymous and nonsynonymous nucleotide substitutions. Molecular Biology and Evolution 3(5):418-426.