Timepiece XV: CLUES
Coalescent Likelihood Under Effects of Selection
The Mechanism at a Glance
CLUES is a full-likelihood method for estimating the selection coefficient \(s\) acting on a biallelic SNP, using modern and ancient DNA. It answers the most direct question in molecular evolution: is this allele being favored or disfavored by natural selection, and by how much?
The key insight: an allele’s frequency trajectory through time is shaped by selection. Under neutrality (\(s = 0\)), allele frequencies drift randomly – the Wright-Fisher diffusion. Under selection (\(s \neq 0\)), the trajectory is biased: a beneficial allele drifts upward on average, a deleterious allele drifts downward. By modeling the frequency trajectory as a Hidden Markov Model and conditioning on the observed genealogy, CLUES computes the likelihood of the data for each value of \(s\) and finds the maximum.
Strong positive selection (\(s > 0\)) = the allele frequency rises faster than drift alone can explain. Coalescence times among derived-allele carriers are shorter (a selective sweep compresses the genealogy).
Neutrality (\(s = 0\)) = the frequency trajectory is consistent with random drift. Coalescence times follow the standard coalescent.
Negative selection (\(s < 0\)) = the allele frequency is pushed downward. Rare alleles stay rare; common alleles become rarer.
CLUES reads these patterns with a Hidden Markov Model whose hidden states are discretized allele frequencies and whose emissions come from the coalescent structure of the gene tree.
If PSMC is a two-hand watch that reads population size from a single genome, then CLUES is a chronometer with a compass – it reads both the tempo (coalescence times) and the direction (frequency change) to detect the invisible hand of natural selection.
Primary Reference
The four gears of CLUES:
The Wright-Fisher HMM (the escapement) – A discretized Wright-Fisher diffusion with selection, modeling how the allele frequency changes from one generation to the next. The transition matrix encodes the strength and direction of selection. This is where population genetics meets probability: the mean frequency shift depends on \(s\), and the variance depends on \(N\).
Emission Probabilities (the gear train) – Two types of evidence constrain the allele frequency at each time point: (a) coalescent events in the gene tree (which lineages merge, and when) and (b) ancient genotype likelihoods (direct observations of the allele in the past). These are the data that the HMM “observes.”
Importance Sampling (the mainspring) – The genealogy at a locus is not known exactly – it is estimated from an ARG sampler like Relate or SINGER. CLUES averages over multiple sampled genealogies using importance weights, properly accounting for genealogical uncertainty. This is where the output of SINGER feeds directly into CLUES.
Inference and Testing (the case and dial) – Maximum likelihood estimation of \(s\) via Brent’s method, a likelihood ratio test calibrated against \(\chi^2\), multi-epoch selection estimation, and posterior trajectory reconstruction. The final step: reading the watch.
These gears mesh together into a complete selection inference machine:
Gene tree samples from Relate/SINGER + optional ancient genotypes
|
v
+-----------------------+
| BUILD FREQUENCY BINS |
| Beta(1/2,1/2) |
| quantile spacing |
+-----------------------+
|
v
+--------> BUILD TRANSITION MATRIX |
| (Wright-Fisher + selection) |
| | |
| v |
| COMPUTE EMISSIONS |
| (coalescent + ancient GL) |
| | |
| v |
| BACKWARD ALGORITHM |
| (for each gene tree sample)|
| | |
| v |
| IMPORTANCE-WEIGHTED |
| LIKELIHOOD |
| | |
| v |
| OPTIMIZE s (Brent/NM) |
| | |
+-------- update s, recompute? |
| |
DONE |
| |
v |
+-------------------------+
| TEST & RECONSTRUCT |
| LRT vs chi-squared |
| Posterior trajectory |
+-------------------------+
|
v
Selection coefficient + allele frequency history
Why Detect Selection?
Population genetics has two great engines: drift and selection. Most of the Timepieces in this collection focus on drift – using the coalescent to infer demographic history, date ancestors, or reconstruct genealogies. CLUES flips the lens: given a genealogy (from SINGER or Relate) and a demographic model (from PSMC or similar), it asks whether the pattern of coalescence times at a specific locus departs from what neutral drift would predict.
This question matters because:
Medical genetics: Identifying loci under selection reveals genes that affect fitness – often the same genes that affect disease susceptibility. The alleles that natural selection has targeted (lactase persistence, malaria resistance, skin pigmentation) are often medically relevant.
Evolutionary biology: Measuring the strength of selection across the genome tells us how evolution actually works in practice – how strong are selective pressures? How often do they change? Do they differ between populations?
Ancient DNA integration: With the explosion of ancient DNA data, we can now directly observe allele frequencies in the past. CLUES integrates this direct evidence with the genealogical signal, providing a uniquely powerful framework for studying selection through time.
Prerequisites for this Timepiece
Before starting CLUES, you should have worked through:
Coalescent Theory – coalescence times, exponential waiting times, and how population size affects the coalescent
Hidden Markov Models – the forward-backward algorithm and numerical stability in log space
The SMC – the sequential Markov coalescent approximation
Familiarity with SINGER or Relate is helpful, since CLUES uses their gene tree samples as input
PSMC – to understand coalescent time distributions under variable population size (CLUES reuses the concept of a piecewise-constant \(N(t)\))
If you’ve built those earlier mechanisms, you have every tool you need. CLUES adds one new ingredient – selection – and shows how a single parameter \(s\) reshapes the entire frequency trajectory.