Heterostyly

• Basic tenet of population genetics: bigger populations mean more genetic diversity
• When genetic diversity is high, selection can act more efficiently
• Thus, more diversity should mean lower extinction rates

A great way to increase genetic diversity is to have mechnical barriers to inbreeding!

Heterostyly in Primulaceae

Fig. 1 from de Vos et al. (2014). Only the different stigma and anther morphs are compatible in Primula. Morphs are controlled by a single bi-allelic locus.

Heterostyly in Primulaceae

Fig. 2 from de Vos et al. (2014). Variable diversification rates and extinction fractions associated with clades of Primulaceae

Heterostyly

Heterosytly by the numbers

Multiple indepdent origins of mechanisms to promote outcrossing. Does that mean outcrossing always confers a fitness benefit?

Photosynthesis in Poaceae

Fig. 1 from Spriggs, Christin, and Edwards (2014). Distribution of C4 photosynthesis in Poaceae

Photosynthesis in Poaceae

Was diversification of C4 lineages a response to changing climate and ecosystems?

Fig. 2a from Spriggs, Christin, and Edwards (2014). Evidence for higher net diversification rates in C4 lineages compared to C3 lineages in Poaceae

Polyploidy across angiosperms

There is an apparent cost to polyploidy. A reason for rediploidization?

Fig. 1 from Mayrose et al. (2011). Diploids more frequently have higher diversification than polyploids across genera.

Sex determination systems in plants

• Hermaphroditism - male and female reproductive organs on the same individual, often in the same flower. Most common in angiosperms and assumed the ancestral state.

• Monoecy - male and female reproductive organs on the same individual, but separate flowers. Corn is a classic example.

• Dioecy - separate male and female individuals. Plants can have sex chromosomes and sex determination systems too. Aparagus is a classic XY example.

Sex determination systems in plants

Fig. 1 from Harkess et al. (2020). Sex determination by two Y-linked genes in garden asparagus

Sex determination systems in plants

Sex determination systems in plants

Not all traits lead to differences in diversification rates among lineages. Although Dioecy is rare, there is no consistent pattern across angiosperms that it has less lineages per unit time. Even when accounting for other factors.

Fig. 2 from Sabath et al. (2016). Dioecy is rare across angiosperms, but does not consistently lead to lower diversification rates.

The Mk model

• A trait with discrete states (e.g. self-compatible or self-incompatible) changes along the branches at rates in a matrix \(Q\) (Lewis 2001):

\[ Q = \begin{pmatrix} -q_{01} & q_{01} \\ q_{10} & -q_{10} \end{pmatrix} \]

• It’s the same continuous-time Markov model as a DNA substitution model, with two states instead of four.
• Apply the same principles of maximum likelihood to estimate the rates.

From counting events to a likelihood

• Lineages split at rate \(\lambda\) and die at rate \(\mu\).
• Net diversification is \(r = \lambda - \mu\)
• The extinction fraction is \(\epsilon = \mu/\lambda\).
• Given a dated tree, which rates surprise us the least?

From counting events to a likelihood

Motivation for a model - many events go unobserved!

The Yule likelihood

Adding extinction: two probabilities instead of one

Need to keep track of lineages that might go extinct or be unsampled too . - \(D(t)\): probability a lineage alive at time \(t\) gives rise to the observed children. - \(E(t)\): probability a lineage alive at time \(t\) leaves no sampled descendants.

• With constant rates and every species sampled, \(r = \lambda - \mu\) and \[ D(t) = \frac{r^2 e^{-rt}}{\left(\lambda - \mu e^{-rt}\right)^2}, \qquad E(t) = \frac{\mu\left(1 - e^{-rt}\right)}{\lambda - \mu e^{-rt}}. \]

Statistical non-identifiability of speciation and extinction rates

Not every species is in the tree

• Trees rarely have every species. This Thursday’s tree has 356 of about 2,300.
• Missing species are unsampled taxa and the model does not known those are not extinct.
• Overcome with the sampling fraction \(f\), the chance that a species made it into the tree (FitzJohn, Maddison, and Otto 2009). Now \(E\) starts at the tips from \(E(0) = 1 - f\), not from 0.
• BiSSE can take a different \(f\) for each state.
• \(f\) is something we assume, not something the tree tells us. It treats sampling as random.

When trait changes affect diversification

From Mk to BiSSE

• BiSSE lets two states have their own speciation and extinction rates (Maddison, Midford, and Otto 2007).
• Six parameters: \(\lambda_0, \lambda_1, \mu_0, \mu_1, q_{01}, q_{10}\).
• Trait change and diversification are fit together, on the same tree.
• We can test the hypothesis: is net diversification in state 1 different from state 0?

Four possibilities for a single lineage

\(D_i(t)\) and \(E_i(t)\) are now estimated for each state \(i\). Here is what can happen to it in a short time \(dt\):

• A split counts twice, because either daughter could be the one that’s lost.
• Extinction can’t happen on a branch that reached the present, so \(\mu\) is only accounted for in \(E_i(t)\).

BiSSE is Mk AND birth–death

Allow differnet speciation and extition rates for different traits!

The allows us to test hypotheses by contstraining or relaxing parameters.

Asking a question with BiSSE

lik <- make.bisse(tree, states)
same_rates <- constrain(lik, lambda1 ~ lambda0, mu1 ~ mu0)
speciation_differs <- constrain(lik, mu1 ~ mu0)
extinction_differs <- constrain(lik, lambda1 ~ lambda0)

Plant results from SSE models

The false-positive problem

Adding a hidden trait

Add a trait we never observed, A or B, with its own rates of change. It can also change \(\lambda\) and \(\mu\), so rate shifts have somewhere else to go (Beaulieu and O’Meara 2016). Boxes of one color share rates:

CID is character-independent diversification: the CID-2 and CID-4 rows, where both observed states share a color. Details next.

In hisse, rates go in the order 0A, 1A, 0B, 1B: CID-2 is c(1, 1, 2, 2), HiSSE is c(1, 2, 3, 4).

Character-Independent Diversification (CID)

• Character: the observed trait we care about, state 0 or 1.
• Character-independent: \(\lambda\) and \(\mu\) do not depend on that trait. State 0 and state 1 share rates within a hidden class.
• Rates still vary among lineages. They depend on a hidden trait, A or B, and a lineage can switch between hidden classes along the tree.
• The observed trait still evolves between 0 and 1. It has no effect on speciation or extinction.

HiSSE and the CID null

The observed trait only helps explain diversification if BiSSE beats CID-2 or if HiSSE beats CID-4. Otherwise background rate variation explains the data just as well.

Plant examples with hidden states

• Onagraceae: self-compatible lineages have higher extinction and lower net diversification, even with hidden states in the model (Freyman and Höhna 2019).
• Nightshades: diversification is explained better by breeding system plus an unmeasured factor than by ploidy. Polyploidy mostly arose straight from self-incompatible lineages, when whole-genome duplication broke self-incompatibility (Zenil-Ferguson et al. 2019).

Plant examples with hidden states

A helpful way to express models is to draw them. The \(q_{ij}\) parameters can be constrained to test hypotheses.

Fig. 5 from Freyman and Höhna (2019). A model of self incompatibility with hidden states.

Plant examples with hidden states

Fig. 6 from Freyman and Höhna (2019). The self-incompatible vs self-compatible comparison is still consequential and consistent with previous findings.

Plant examples with hidden states

Increasing levels of complexity can be incorporated, such as a third polyploid self-compatible state.

Fig. 2 from Zenil-Ferguson et al. (2019). A model of self incompatibility and polyploidy with hidden states.

What drives angiosperm diversification?

• A review of 152 SSE studies on angiosperms found that reproductive and morphological traits were often linked to diversification, but no trait had a consistent effect across clades (Helmstetter et al. 2023).
• Results also tracked the data: larger, older or less completely sampled trees more often found a trait effect.
• So treat SSE results as hypotheses, compare against hidden-state nulls, and report tree size and sampling.

SSE models and their extensions

Many traits remain to be explored extensively!

Fig. 2b from Helmstetter et al. (2023). Distribution of SSE studies across angiosperms.

The SSE family

• MuSSE handles more than two states; QuaSSE handles continuous traits (FitzJohn 2010).
• ClaSSE lets traits change at speciation. In nightshades, self-incompatibility was lost mostly at speciation events (Goldberg and Igić 2012).
• ChromoSSE models chromosome number, including polyploidy (Freyman and Höhna 2018).
• Software: diversitree (FitzJohn 2012) and RevBayes (Höhna et al. 2016).

Summary

• A dated tree gives a likelihood for birth–death rates. Extinction leaves a trace, but \(\lambda\) and \(\mu\) suffer from non-identifiability, so \(r\) is usually discussed.
• BiSSE is Mk and birth–death joined by state-dependent rates.
• BiSSE is the simplest SSE model, handling binary traits. This can be extended and used to test sophisticated hypotheses.
• SSE tests give false positives easily, so compare them against hidden-state (CID) nulls.

For Thursday

• We will use the Goldberg et al. (2010) data.
• We will use the diversitree R package. Consider installing before class: install.packages("diversitree").
• We’ll fit BiSSE to the nightshades, then see how far we get with HiSSE.
• Is selfing an evolutionary dead end?