Brownian motion is the null. It says a trait wanders with no destination, at one constant rate, everywhere on the tree. Real traits often do something more interesting:

• Constraint. A trait is held near an optimum and stops wandering: the Ornstein–Uhlenbeck model.
• Bursts. A trait diversifies fast early then slows: the early-burst model.
• Lost signal. Relatives resemble each other less than the tree predicts: Pagel's transformations.

Each is BM plus one extra parameter. Setting that parameter to its default value recovers BM exactly, so each is a testable departure from the null.

Three families of model

1. Reshape the tree. Pagel's $\lambda$, $\delta$, $\kappa$; multi-rate models.
2. Change the process. Ornstein–Uhlenbeck; early burst.
3. Discrete traits. The Mk model and stochastic mapping.

Choosing between them is a model-selection problem: fit each, then compare with AIC, exactly the aicw() step from the lab.

• Pagel proposed three ways to stretch and squeeze the variance–covariance matrix $\mathbf{C}$ before fitting BM.
• Each has one parameter that, set to 1, recovers the original BM tree:
  • $\lambda$ (lambda): phylogenetic signal,
  • $\delta$ (delta): the tempo of evolution through time,
  • $\kappa$ (kappa): gradual versus speciational change.
• Fitting the parameter and comparing to $1$ tests whether the trait obeys BM on the true tree.

$\lambda$ multiplies the internal (shared) branches, leaving the tips at the present.

$\lambda$: 1.00

Raises node heights to the power $\delta$, rescaling when change happens.

• $\delta=1$: unchanged BM.
• $\delta<1$: change concentrated early (deep branches do more work).
• $\delta>1$: change concentrated late (recent branches do more).

Raises each branch length to the power $\kappa$.

• $\kappa=1$: standard BM (change $\propto$ time).
• $\kappa=0$: all branches length 1; change depends only on the number of speciation events, not on time (punctuated evolution).

$\kappa$ raises each branch length to the power $\kappa$. The horizontal axis is now expected change, not time.

$\kappa$: 1.00

• BM assumes a single rate $\sigma^2$ across the whole tree. What if one clade evolves faster?
• Three ways to test:
  1. Contrast-based rate test: compare the spread of squared contrasts between clades.
  2. ML + AIC: fit a model with a separate $\sigma^2$ per clade and compare to single-rate BM.
  3. Bayesian MCMC: let every branch have its own rate; the posterior flags rate shifts.
• Caveat: methods 1 and 2 need the clades to be specified in advance. Scanning every clade and keeping the most significant inflates false positives.

• The Ornstein–Uhlenbeck (OU) model is BM plus a pull toward an optimum $\theta$: $$dX = \underbrace{\alpha(\theta - X)\,dt}_{\text{pull to optimum}} + \underbrace{\sigma\,dW}_{\text{random walk}}$$
• $\alpha$ is the strength of the pull. The further $X$ strays from $\theta$, the harder it is pulled back.
• This is the model of stabilizing selection: a trait kept near an adaptive peak (right).
• Special case: $\alpha=0$ removes the pull and OU collapses to plain BM. So OU nests BM.

Lineages start away from the optimum $\theta$ (green line) and are pulled toward it as they wander.

Pull strength $\alpha$: 1.0 Optimum $\theta$: 2.0 Rate $\sigma^2$: 1.0 Re-simulate

Unlike BM, the variance does not grow forever; it approaches an equilibrium $\sigma^2/2\alpha$. Once divergence exceeds a few phylogenetic half-lives the trait reaches this equilibrium and further time adds little difference; on short branches OU still behaves like BM.

The phylogenetic half-life $\ln 2/\alpha$ says how fast ancestry is forgotten. Large $\alpha$ means the tree barely matters.

OU is easy to over-fit on small trees, and measurement error mimics a high $\alpha$. A favourable AIC is not proof of an optimum.

• Let different parts of the tree be pulled toward different optima $\theta_1,\theta_2,\dots$ This is the Hansen model.
• Paint the branches with regimes; each regime has its own peak in trait space. Lineages sharing a regime are drawn to the same point.
• This formalises Simpson's "adaptive zones": shifts between peaks are shifts between adaptive strategies.
• When independent lineages are pulled to the same peak, the model captures convergence, exactly the test behind the Caribbean Anolis result from last lecture (Mahler et al. 2013).

• An opportunity opens (a new island, a key innovation, a mass extinction). Lineages diversify fast to fill empty niches, then slow as niches fill.
• The rate decays through time: $$\sigma^2(t) = \sigma^2_0\, e^{rt}, \quad r<0$$
• Most disparity accumulates early; the tips are relatively similar.

Decay $r$: -2.0

Four models fit to one trait. Set each log-likelihood; weights update live.

Akaike weights for BM (yellow), SSP (= OU, blue), and EB (black) across 49 datasets: body size (top), body shape (bottom).
Harmon et al. (2010) Evolution 64: 2385–2396, Fig 2.

• Many traits are discrete: present/absent, 2/3/4 colour morphs, diet category, limbed/limbless.
• These have no meaningful average, so BM and contrasts do not apply.
• Instead, model evolution as jumps between states along the branches, using a continuous-time Markov chain.
• This is the same modelling approach used for DNA substitution earlier in the course, now applied to phenotypes.

• The Mk model (Lewis 2001) describes a character with $k$ states changing at constant rates, a direct generalisation of the Jukes–Cantor model for DNA.
• Rates live in a transition matrix $\mathbf{Q}$. For $k=2$: $$\mathbf{Q} = \begin{pmatrix} -q_{01} & q_{01} \\ q_{10} & -q_{10} \end{pmatrix}$$
• The two flavours you fit in the lab:
  • ER (equal rates): $q_{01}=q_{10}$, one rate.
  • ARD (all rates different): each transition its own rate.
• More states or asymmetric rates simply add parameters to $\mathbf{Q}$; compare them with AIC, as before.

A binary trait (state 0 / state 1) evolves down the branches. Each branch is coloured by its current state.

Rate $q$: 1.0 ARD (asymmetric): bias toward state 0 Re-simulate

• A single simulation is easy; the likelihood sums over all possible histories, which is astronomically many.
• Felsenstein's pruning algorithm makes it tractable: work from the tips to the root, storing at each node the probability of the data below it, for each possible state.
• At each node, combine the two daughter probabilities using $\mathbf{Q}$ and the branch lengths; recurse down to the root.
• This is the same algorithm used for DNA substitution models earlier in the course; only the states and $\mathbf{Q}$ differ.

• Squamate reptiles (lizards and snakes) have lost their limbs many times independently: snakes, but also numerous lizard lineages.
• Fitting an Mk model and reconstructing ancestral states shows limblessness (red) scattered across the tree, not confined to one clade.
• Each red patch on a black background is an independent transition: a striking case of convergent evolution in a discrete trait.
• The same fit also estimates whether the limbed→limbless transition is effectively irreversible: regaining limbs is rare or impossible.