Lecture 9
Molecular clocks and dated phylogenies
The question this lecture answers
A phylogeny with a calendar axis tells you when lineages last shared an ancestor. That is what turns a tree into a statement about history: when a virus entered a country, when two species diverged, how old a fossil lineage is.
Figure 4A of Douglas et al. (2021), Virus Evolution 7(2):veab052 (CC BY).
A maximum-clade-credibility tree of SARS-CoV-2 genomes from New Zealand's first wave.
Rings are calendar months; red is the New Zealand deme and grey the rest of the world,
so each red cluster is a separate introduction. $t_1$ marks the first reported case and
$t_2$ the decrease in mobility.
The sample dates at the tips are known. Every internal date is estimated, and so has a posterior distribution and a credible interval, exactly like any other parameter from L8. The squares are posterior clade probabilities.
The problem to solve: sequences record genetic distance, which is
rate multiplied by time. Before a tree can carry a calendar axis, something has to
separate those two.
1. The Molecular Clock Hypothesis
Genetic Distance = Rate × Time
The fundamental equation of molecular evolution:
The Molecular Clock Equation
$$d = \mu \times t$$
Where:
- $d$ = genetic distance (expected substitutions per site)
- $\mu$ = substitution rate (substitutions per site per unit time)
- $t$ = time
The Strict Molecular Clock
Under a strict molecular clock, all lineages evolve at the same rate:
The strict clock: a single rate $\mu$ converts the time tree to a substitution tree
Strict Molecular Clock
- Single evolutionary rate $\mu$ for all branches
- Ultrametric trees (all tips equidistant from root)
- Proposed by Zuckerkandl and Pauling (1962)
- Enables dating without fossils if rate is known
2. The Identifiability Problem
Non-identifiability of Rate and Time
Without calibration, infinitely many combinations of rate and time produce the same genetic distances
Fundamental problem: From genetic distances alone, we cannot separate rate from time. If we double all times and halve the rate, we get the same likelihood!
Solutions to the Identifiability Problem
To separate rate and time, we need additional information:
1. Node Calibrations
Use fossil or biogeographic evidence to constrain node ages:
Fossil evidence constraining the age of the ABC ancestor to 25-35 Mya
Node Calibration in Practice
- Fossil provides minimum age (fossil must be younger than clade)
- Biogeography can provide maximum age (e.g., island age)
- Often specified as probability distributions (e.g., lognormal)
- Multiple calibrations improve precision
2. Tip Calibrations
Use samples from different time points:
Ancient DNA from sample C (20,000 years old) provides temporal information
Applications of Tip Calibration
- Ancient DNA: Subfossil remains, museum specimens
- Rapidly evolving pathogens: Virus samples from different years
- Laboratory evolution: Samples from known time points
Key principle: With either node or tip calibration, knowing even one absolute time allows us to estimate all other times under a strict clock.
3. Relaxed Molecular Clocks
The strict molecular clock is often too restrictive. Relaxed clocks allow rate variation across branches.
Relaxed Clock Parameterization
Each branch has its own rate, converting time tree to substitution tree
Relaxed Molecular Clock
Instead of a single rate $\mu$, we have a vector of rates $\vec{\mu} = (\mu_1, \mu_2, ..., \mu_{2n-2})$
The substitution tree is computed as: $T = \vec{\mu} \star g$
Where $\star$ denotes element-wise multiplication of rates and branch durations
Identifiability Under Relaxed Clocks
With relaxed clocks, the identifiability problem is even more severe
Critical issue: Relaxed clocks are only identifiable through their priors! The likelihood alone cannot distinguish between fast rates with short times vs. slow rates with long times.
4. Bayesian Framework for Molecular Clocks
Posterior with Strict Clock
Strict Clock Posterior
Substitution tree: $T = \color{darkgreen}{\mu} \times \color{red}{g}$
$$P(\color{red}{g}, \color{darkgreen}{\mu}, \color{blue}{\Theta}|D) \propto \Pr(D|T)P(\color{red}{g}|\color{blue}{\Theta})P(\color{darkgreen}{\mu})P(\color{blue}{\Theta})$$
Where:
- $\color{red}{g}$ = time tree
- $\color{darkgreen}{\mu}$ = clock rate
- $\color{blue}{\Theta}$ = tree prior parameters (as in L7 and L8)
Posterior with Relaxed Clock
Relaxed Clock Posterior
Substitution tree: $T = \color{darkgreen}{\vec{\mu}} \star \color{red}{g}$
$$P(\color{red}{g}, \color{darkgreen}{\vec{\mu}},\color{blue}{\Theta}|D) \propto \Pr(D|T)P(\color{red}{g}|\color{blue}{\Theta})P(\color{darkgreen}{\vec{\mu}})P(\color{blue}{\Theta})$$
Where $P(\color{darkgreen}{\vec{\mu}})$ is the prior for rate variation
Key observations:
- The phylogenetic likelihood only depends on the substitution tree $T$: $\Pr(D|T)$
- The tree prior only depends on the time tree $\color{red}{g}$: $P(\color{red}{g}|\color{blue}{\Theta})$
- By fixing $\color{darkgreen}{\vec{\mu}} = 1$, we get a time tree in units of substitutions
5. Models of Rate Variation
Autocorrelated Models
Rates evolve along the tree, with child rates similar to parent rates:
Autocorrelated Rate Model
$$P(\color{darkgreen}{\vec{\mu}}) = \prod_i P(\mu_i | \mu_{\text{parent}(i)})$$
Common implementation: Lognormal autocorrelated model
$$\log(\mu_i) \sim \text{Normal}(\log(\mu_{\text{parent}(i)}), \sigma \sqrt{t_i})$$
Where:
- $t_i$ = time duration of branch $i$
- $\sigma$ = rate of evolution of rates (volatility parameter)
Uncorrelated Models
Rates are drawn independently for each branch:
Uncorrelated Rate Model
$$P(\color{darkgreen}{\vec{\mu}}) = \prod_i P(\mu_i | \nu)$$
Common implementations:
- Lognormal: $\log(\mu_i) \sim \text{Normal}(M, S)$
- Exponential: $\mu_i \sim \text{Exponential}(\lambda)$
- Gamma: $\mu_i \sim \text{Gamma}(\alpha, \beta)$
Autocorrelated
- Biologically motivated
- Smooth rate changes
- Good for closely related species
- Can extrapolate rates
Uncorrelated
- More flexible
- Allows sudden rate changes
- Good for diverse datasets
- Simpler to implement
6. Advantages of Molecular Clock Models
Relaxed molecular clocks offer several benefits over unconstrained models:
- Improved phylogenetic accuracy:
- Rate smoothing helps identify correct topology
- Reduces long-branch attraction artifacts
- Better performance on empirical datasets
- Automatic rooting:
- No need for outgroup
- Root position estimated from rate variation
- Particularly useful when outgroup is distant
- Temporal information:
- Relative divergence times always available
- Absolute times with calibration
- Useful for studying evolutionary rates
- Integration with other models:
- Natural combination with coalescent priors
- Enables epidemiological inference
- Links to fossil data
Best practice: Use relaxed clocks unless you have strong evidence for a strict clock. The added flexibility usually outweighs the increased parameter count.
Summary
This lecture answered one question: how do we put a date, with an interval, on a common ancestor?
- The obstacle:
- Sequences record genetic distance, and distance = rate × time
- Rate and time are non-identifiable: a fast rate over a short time and a slow rate over a long time give identical likelihoods
- The solution, calibration:
- Node calibration from a dated fossil
- Tip calibration from samples of different known ages: ancient DNA, or a fast-evolving virus sequenced over months
- Either one puts the whole tree on a scale of years
- Rate variation:
- Strict clocks assume one rate; relaxed clocks give every branch its own
- Relaxed clocks are usually more realistic, but add non-identifiability that only the priors resolve, so priors must be reported
- Uncorrelated draws each rate independently; autocorrelated lets rate evolve along the tree
- What you get out:
- A posterior distribution of dates for every ancestor, hence credible intervals
- A rooted tree without needing an outgroup
- Dates that are conditional on the clock model, substitution model and tree prior
Key takeaway: Molecular clocks transform phylogenetics from estimating relationships to understanding the tempo and mode of evolution.
Recommended Reading:
- Stadler et al. (2024) Decoding Genomes - Ch 6.4 (strict clock, time trees, calibrations)
- Drummond & Bouckaert (2015) "Bayesian Evolutionary Analysis with BEAST" - Chapters 6-7 (relaxed clocks in detail)
- Yang (2014) "Molecular Evolution: A Statistical Approach" - Chapter 7
- Drummond et al. (2006) "Relaxed phylogenetics and dating with confidence" - PLOS Biology
- Douglas et al. (2021) "Phylodynamics reveals the role of human travel and contact tracing in controlling the first wave of COVID-19 in four island nations" - Virus Evolution 7(2):veab052. Source of the opening figure.
Check Your Understanding
- What makes rate and time non-identifiable in molecular evolution?
- How do node and tip calibrations solve the identifiability problem?
- The SARS-CoV-2 tree in the opening figure has a calendar axis even though there are no fossils. What calibrates it?
- What's the key difference between autocorrelated and uncorrelated relaxed clocks?
- Why do relaxed clocks often produce better phylogenetic estimates than no-clock models, and what do they make you more dependent on?
- A paper reports a divergence date of 12.4 Mya. What else must it report before you can judge that number?