A covarion model for phylogenetic estimation using discrete morphological datasets

This study introduces the "covariomorph" model in RevBayes to address lineage-specific rate variation in morphological evolution, demonstrating through simulations and empirical analysis that accounting for such heterogeneity significantly improves phylogenetic inference accuracy and impacts divergence time estimation.

The Gist

Imagine you are trying to reconstruct the family history of a group of animals based on their physical features—like the shape of their teeth, the number of toes, or the structure of their skulls. This is what scientists call morphological phylogenetics.

For a long time, scientists have used a standard "rulebook" (called the Mk model) to guess how these features changed over time. The rulebook had a simple, but flawed, assumption: every feature evolves at the same steady speed, everywhere, for everyone.

Think of it like a classroom where the teacher assumes every student learns math at the exact same speed, every single day, regardless of whether they are having a good day, a bad day, or if the subject suddenly becomes easier or harder. We know this isn't true in real life. Some students (or features) speed up when they get excited, and others slow down when they get bored.

The Problem: The "One-Size-Fits-All" Mistake

In nature, evolution is messy.

• The "Fast" Student: A bird's wing might evolve rapidly because it needs to adapt to flying.
• The "Slow" Student: A turtle's shell might stay exactly the same for millions of years because it's already perfect.
• The "Switcher": A feature might be slow for a long time, then suddenly speed up when the environment changes, and then slow down again.

Old models could handle the idea that some students are naturally fast and some are slow (this is called ACRV). But they couldn't handle the idea that a single student might change their speed halfway through the school year. They assumed if a feature was fast, it was always fast.

The Solution: The "Covariomorph" Model

The authors of this paper, Basanta Khakurel and Sebastian Höhn, created a new, smarter rulebook called the Covariomorph model.

Here is how it works, using a simple analogy:

Imagine a group of runners in a race.

• The Old Model: Assumes every runner has a fixed speed. Runner A is always a sprinter; Runner B is always a jogger. They never change.
• The New Model (Covariomorph): Realizes that runners can change gears. Runner A might start as a jogger, suddenly sprint when they see a finish line, and then slow down to a walk. Runner B might do the opposite.

The "Covariomorph" model allows each physical feature (like a tooth shape) to have its own internal speedometer that can switch between "Slow," "Medium," and "Fast" gears at any point in the family tree. It asks: "Did this feature speed up because of a specific event in this specific family line?"

How They Tested It

1. The Simulation (The Practice Run): They created fake family trees and fake features using a computer. They programmed some features to switch speeds randomly. When they fed this data into their new model, it successfully figured out the hidden speed switches. It proved the model could "see" the changes that the old models missed.
2. The Real World Test: They took 164 real datasets from nature (including ancient fossils and modern animals).
   • Result: About half of the datasets looked like the old "steady speed" model was fine.
   • Result: The other half showed clear signs of "speed switching." For these, the new model gave a much clearer picture of the family tree.

Why This Matters

When you get the family tree wrong, you get the history wrong.

• Branch Lengths: In these trees, the length of a branch represents how much change happened. If you assume a feature was always slow, but it actually sped up for a while, you might think the branch is short. The new model corrects this, making the branches the right length.
• Time Travel: If you want to know when two species split apart (divergence time), you need accurate branch lengths. If your speed assumptions are wrong, your timeline is wrong. The new model helps us get the dates right.

The Takeaway

The authors didn't just build a faster car; they built a car with a smart transmission.

They showed that evolution isn't a straight line at a constant speed. It's a journey with traffic jams, highway sprints, and pit stops. By allowing features to change their evolutionary "gear" as they move through history, the Covariomorph model gives us a much more accurate, nuanced, and realistic map of how life on Earth has evolved.

It's a small change in the math, but it could completely reshape how we understand the history of life.

Technical Summary

1. Problem Statement

Traditional phylogenetic inference using discrete morphological data typically relies on the Markov k-state (Mk) model (Lewis, 2001). This model assumes:

• Homogeneity: All characters evolve at the same rate across the entire phylogeny.
• Stationarity: The rate of evolution for a specific character does not change over time or across lineages.

While extensions like Among-Character Rate Variation (ACRV) (e.g., Mk+$\Gamma$) allow different characters to evolve at different average rates, they fail to account for lineage-specific rate variation (heterotachy). In biological reality, selective pressures often cause specific characters to evolve rapidly in certain lineages (e.g., during adaptive radiations) and slowly in others, or even become invariable for periods. Existing models either assume a character is always "fast" or always "slow" (ACRV) or apply a single branch-length multiplier to all characters (relaxed clocks), neither of which captures the dynamic switching of evolutionary regimes for individual characters.

2. Methodology: The Covariomorph Model

The authors introduce the covariomorph model, a novel extension of the covarion model (originally developed for molecular data) adapted for discrete morphological characters within the Bayesian framework of RevBayes.

Core Mechanism

The model treats morphological evolution as a continuous-time Markov process involving two simultaneous stochastic processes for each character:

1. State Transitions: Changes between observable character states (e.g., 0 to 1) governed by a base substitution rate matrix ($Q$, typically Mk).
2. Rate Switching: Transitions between different rate categories (regimes) governed by a switching rate parameter ($\delta$).

Mathematical Formulation

• Expanded State Space: The model expands the state space from $k$ (character states) to $k \times m$, where $m$ is the number of discrete rate categories.
• Rate Matrix ($\tilde{Q}$): The total rate matrix is constructed as a block matrix:
  • Diagonal Blocks: Contain the scaled rate matrices ($r_i Q$) for each rate category $i$, where $r_i$ is a rate scalar derived from a discretized probability distribution (e.g., lognormal).
  • Off-Diagonal Blocks: Contain the switching rates ($\frac{\delta}{m-1} I$), representing the probability of a character moving from one rate category to another without changing its observable state.
• Parameters:
  • $\sigma$: Standard deviation of the lognormal distribution determining the magnitude of rate variation among categories.
  • $\delta$: The switching rate between categories.
  • $m$: The number of rate categories (treated as a fixed hyperparameter in the study, though tested across a range).

Implementation

• Implemented in RevBayes using the expandCharacters() and fnCovarion() functions.
• Likelihood Calculation: Uses Felsenstein's pruning algorithm on the expanded state space. The likelihood is marginalized over all possible hidden rate categories at internal nodes.
• Ascertainment Bias: The current implementation does not include the Mkv correction for ascertainment bias (conditioning on variable sites), as standard corrections do not yet apply to the expanded virtual state space of the covariomorph model.

3. Key Contributions

1. Model Development: First implementation of a covarion-like model specifically for discrete morphological data, allowing characters to switch between evolutionary rate regimes (heterotachy) along a phylogeny.
2. Theoretical Generalization: Demonstrates that the covariomorph model nests simpler models:
   • If $\delta \to 0$, it collapses to Mk+ACRV (static rate variation).
   • If $\sigma \to 0$ or $\delta \to \infty$, it collapses to the standard Mk model (homogeneous rates).
3. Empirical Validation: A comprehensive analysis of 164 empirical morphological datasets to test for the presence of covarion-like dynamics.
4. Case Studies: Detailed analysis of two specific datasets (Rays and Sharks) to demonstrate the impact of the model on topology and branch lengths.

4. Results

Simulation Study

• Parameter Recovery: The model successfully recovers true switching rates ($\delta$) and rate variation ($\sigma$) when the data is generated under covariomorph processes.
• Tree Length Dependency: Recovery accuracy is highly dependent on total tree length. Short trees struggle to distinguish between high switching rates and static models.
• Model Collapse: When data is generated under a simple Mk model, the covariomorph model correctly infers high switching rates and low rate variation, effectively collapsing to the simpler Mk behavior.

Empirical Analysis (164 Datasets)

• Bimodal Distribution: The 164 datasets fell into two distinct clusters:
  1. ~50% (77 datasets): Exhibited parameters consistent with the standard Mk model (low $\sigma$, extremely high $\delta$), suggesting no significant heterotachy.
  2. ~50%: Showed evidence of rate heterogeneity ($\sigma > 0$). Some fit ACRV, while others showed patterns compatible with covarion-like switching.
• Model Selection: For datasets showing heterogeneity, the covariomorph model was often preferred over Mk and Mk+ACRV based on Bayes Factors.

Case Studies: Rays and Sharks

• Model Fit: Both datasets showed decisive evidence (high Log Bayes Factors) favoring the covariomorph model over Mk and Mk+ACRV. The Sharks dataset required a higher number of rate categories ($m \ge 8$) for optimal fit compared to Rays.
• Topological Impact:
  • Accounting for heterotachy (covariomorph) resulted in significantly different tree topologies compared to Mk and ACRV models.
  • Clades supported by the covariomorph model often had different posterior probabilities than those from static models.
• Branch Lengths: Covariomorph models inferred substantially longer branch lengths (1.5x to 1.7x longer) than the Mk model. This suggests that ignoring heterotachy leads to underestimating evolutionary time or rates.

5. Significance and Implications

• Improved Phylogenetic Accuracy: By modeling lineage-specific rate shifts, the covariomorph model provides a more biologically realistic framework, potentially resolving topological conflicts caused by model misspecification.
• Divergence Time Estimation: The significant increase in inferred branch lengths implies that traditional models may systematically underestimate divergence times or evolutionary rates in datasets with heterotachy.
• Future Directions:
  • Ascertainment Bias: Integrating the Mkv correction is a critical next step.
  • Decoupling Rates: Future models should decouple the rate of state transitions from the rate of regime switching to avoid conflating fast evolution with high heterotachy.
  • Mixture Models: The authors propose a future extension where individual characters within a single dataset are assigned to either a simple Mk process or a complex covariomorph process, acknowledging that not all traits evolve heterotachously.

In conclusion, the covariomorph model offers a robust, flexible framework for morphological phylogenetics that addresses the limitation of assuming uniform evolutionary rates across lineages, significantly impacting tree topology and branch length estimation in datasets exhibiting dynamic evolutionary patterns.