Why we need a measure of uncertainty

• Last time’s tree is a point estimate from one finite alignment. A different sample of sites would give a (slightly) different tree.
• Ideal: repeat the sequencing experiment many times, re-estimate the tree each time, and see how often a clade recurs. We can’t — we only have one alignment.
• The bootstrap idea (Efron 1979): treat the observed data as a stand-in for the population and resample from it. Replicate data sets mimic what new data sets would look like.
• Applied to phylogenies by Felsenstein (1985); it is now the standard measure of clade support in ML and distance analyses.

Rate × time

A branch length in substitutions per site is

\[ b = r \times t \quad(\text{rate} \times \text{time}). \]

The molecular clock hypothesis: \(r\) is roughly constant across lineages, so branch lengths are proportional to time. This hypothesis is based in the neutral theory of molecular evolution. Recall the neutral substitution rate is the mutation rate (\(\rho=\mu\)), which is independent of population size. If \(\mu\) is similar across lineages, the molecular clock should be constant.

From distances to node ages

Building off of neighbor-joining, we can use the pairwise distances to estimate node ages using least-squares.

• Input: a matrix of pairwise distances \(d_{ij}\) (substitutions/site) and a rooted tree topology.
• Output: an age for every internal node.
• Idea: choose node ages so that the distances expected by the tree based on a linear regression are as close as possible to the distances we observe.

The least-squares criterion

Choose node heights \(h\) to minimise the weighted sum of squared residuals

\[ S(h) \;=\; \sum_{i<j} w_{ij}\,\bigl(d_{ij} - 2h_{m(i,j)}\bigr)^2 . \]

With equal weights (\(w_{ij}=1\)) each node can be solved on its own: set \(\partial S/\partial h_m=0\),

\[ \hat h_m \;=\; \frac{1}{2}\,\overline{d}_m, \qquad \overline{d}_m=\text{mean of } d_{ij} \text{ over all pairs whose MRCA is } m . \]

Each node age is half the average distance across it. Ages must also satisfy the condition that a child is younger than its parent

Rate–time non-identifiability

From sequence data we observe only the product \(b=rt\) — as the least-squares fit showed, the data alone give heights in substitutions, and the scale is set only by external time information. Doubling the rate and halving the time gives the identical likelihood:

\[ b = r\,t = (2r)\left(\tfrac{t}{2}\right). \]

• Molecular data alone cannot give absolute ages. We need external time information — as little as one calibration under a strict clock, but more are needed when there is rate variation.

Calibrations make rate and time identifiable

External evidence fixes ages (or age distributions) at nodes/tips:

• Fossils — a dated fossil places a minimum age on a clade (often a soft prior).
• Secondary calibrations — ages from a previous dated study.
• Biogeographic — island emergence, vicariance events (e.g. Hawaiian island ages).

Calibrations are priors on time, layered onto the rate–time product. Their choice and placement is data driven but requires justification from organismal experts.

Where do calibrations come from?

Crowns versus stems. Fig. 1 from Renner (2005)

Why multiple calibrations are needed

Faster rates of evolution in herbaceous plants compared to woody plants. Fig. 1 from Smith and Donoghue (2008)

Strong effect of calibration choice

Effect of different calibrations on estimated node ages. Fig. 3 from Barba-Montoya et al. (2018)

Strong effect of calibration choice

Effect of different calibrations on estimated node ages. Fig. 3 from Wu et al. (2026)

Strict vs. relaxed clocks

• Strict clock: one rate \(r\) for the whole tree. Simple, but real lineages vary (life history, generation time, mutation rate).
• Relaxed clock: each branch draws its own rate, e.g. from a lognormal:

\[ r_i \sim \text{Lognormal}(\mu_r, \sigma_r), \]

with \(\sigma_r\) measuring how clock-like the data are (Drummond et al. 2006).

• Uncorrelated (branch rates independent) vs. autocorrelated (parent→child) variants.
• Plants often violate a strict clock — herbs vs. trees differ in substitution rate.

Relaxed clocks with Bayesian divergence-time estimation

Put it together in one hierarchical model, sampled by MCMC:

\[ \underbrace{P(\text{data}\mid \tau, t, r)}_{\text{substitution + clock}} \;\times\; \underbrace{P(\tau, t)}_{\text{tree prior}} \;\times\; \underbrace{P(\text{calibrations})}_{\text{time priors}}. \]

• Output: a posterior distribution of time-calibrated trees.
• Every node age has a 95% HPD interval for time and every branch has a 95% HPD interval for its rate.

Answer 2: every node moves, not just the calibrated ones

• B vs A: the calibrated backbone nodes (red, old) sit on the 1:1 line; the uncalibrated angiosperm nodes shift younger (median 0.61 × the reference age).
• C vs A: no calibrated node is safe — a single rate has to explain both slow gymnosperms/ferns and fast angiosperms.

Lessons from the angiosperm case study

• Calibrations drive the timescale: removing 42 fossils moved the angiosperm crown ~75 My younger and widened intervals: more calibrations give older ages and tighter intervals.
• The clock model matters as much: a strict clock gave precise but conflicting ages; the relaxed clock (\(\sigma^2\approx\) 0.4–1.1) is needed.
• Many things to check for divergence times: posterior vs. prior, calibration effects, convergence, and taxon sampling.

Summary

• The bootstrap resamples alignment columns to measure sampling error: clade support is the fraction of replicates containing the clade — not the probability it is true, and blind to model bias.
• Branch length = rate × time.
• Least squares turns distances into node heights (half the mean distance across each node) in substitution units; one calibration rescales them to absolute time under a strict clock.
• Calibrations (fossils, secondary, biogeographic) are needed to obtain absolute time estimates.
• Relaxed clocks can accommodate many calibrations and thus rate variation across the tree.
• Bayesian relaxed clock models yield a posterior of chronograms. The posterior can be used to calculate the HPD to evaluate uncertainty.