A New Information Theoretic Approach Shows that Mixture Models Outperform Partitioned Models for Phylogenetic Analyses of Amino Acid Data

By applying the newly introduced marginal Akaike information criterion (mAIC) to diverse empirical datasets, this study demonstrates that mixture models universally outperform partitioned models for phylogenetic analyses of amino acid data, highlighting the importance of further developing mixture models for accurate evolutionary inference.

The Gist

Imagine you are trying to reconstruct the family tree of a massive, ancient clan. You have a huge pile of old letters (DNA or protein sequences) from hundreds of different relatives. The goal is to figure out who is related to whom and how they evolved over millions of years.

To do this, scientists use mathematical "models" to guess how these letters changed over time. For a long time, there were two main ways to build these models: Partitioned Models and Mixture Models.

This paper is like a referee blowing the whistle to settle a decades-long debate: Which method actually works better? And the answer is a resounding victory for the Mixture Models.

Here is the breakdown in simple terms:

1. The Two Contenders

The Partitioned Model: The "Strict Teacher"
Imagine a classroom where the teacher divides the students into groups based on their height.

• Group A (Tall kids) gets a specific rule: "You can only wear blue shoes."
• Group B (Short kids) gets a different rule: "You can only wear red shoes."
• The Problem: In real life, a tall kid might sometimes wear red shoes, and a short kid might wear blue. The "Strict Teacher" forces everyone into a box. If you misclassify a student, the whole group's rules get messed up. In science, this is called "partitioning." You force different parts of your DNA into different buckets and apply one rule to the whole bucket.

The Mixture Model: The "Flexible Chef"
Now, imagine a chef making a giant stew. Instead of separating ingredients into bowls first, the chef throws everything into one pot.

• The chef knows that some ingredients (like carrots) behave one way, while others (like potatoes) behave differently.
• The chef doesn't force the carrots to stay in a "carrot zone." Instead, the chef calculates the flavor of every single ingredient based on how it actually behaves in the pot.
• The Advantage: It's flexible. It allows a specific spot in the DNA to act like a "carrot" even if its neighbors act like "potatoes." It doesn't need to force things into pre-defined boxes.

2. The Big Problem: The "Ruler" Was Broken

For years, scientists tried to compare these two methods using a standard ruler called AIC (Akaike Information Criterion). Think of AIC as a scorecard that tells you which model fits the data best. Lower scores are better.

The Catch: The old ruler was biased!

• It was designed to measure the "Strict Teacher" (Partitioned models).
• When they tried to measure the "Flexible Chef" (Mixture models) with this same ruler, the scores were unfair. It was like trying to measure a marathon runner's speed with a ruler meant for a snail.
• Because of this broken ruler, scientists often thought the "Strict Teacher" was better, even when the "Flexible Chef" was actually doing a better job.

3. The New Solution: A Fair Ruler (mAIC)

The authors of this paper (along with a colleague named Susko) invented a new, fair ruler called mAIC (marginal AIC).

• This new ruler knows how to measure both the "Strict Teacher" and the "Flexible Chef" on the same playing field.
• It levels the playing field so we can see who actually fits the data better.

4. The Results: The Chef Wins!

The researchers took nine massive datasets (representing insects, plants, fungi, bacteria, and ancient archaea) and ran them through both models using the new fair ruler.

The Outcome:

• The Flexible Chef (Mixture Models) won every single time.
• The "Strict Teacher" (Partitioned Models) was consistently outperformed.
• The difference wasn't small; it was massive. In some cases, the Mixture Model was thousands of points better on the scorecard.

Why did the Chef win?
Real evolution is messy. DNA sites don't always follow the neat rules of the "Strict Teacher." Sometimes a specific part of a protein behaves uniquely, regardless of which "bucket" you put it in. The Mixture Model captures this messy reality perfectly, while the Partitioned Model forces a rigid structure that doesn't exist in nature.

5. The "Robustness" Test: Does the Tree Hold Up?

To be sure, the scientists didn't just look at the scorecard. They also did a "stress test."

• They took the family tree and removed one relative at a time to see if the tree fell apart or stayed strong.
• Result: Both methods were pretty good at keeping the tree standing, but the Mixture Models were slightly more consistent.

The Takeaway for Everyone

For a long time, scientists were using a broken ruler that made them think the "Strict Teacher" (Partitioned Models) was the best way to study evolution.

This paper says: "Stop using the old ruler. Switch to the new one (mAIC), and you'll see that the 'Flexible Chef' (Mixture Models) is the superior method."

What does this mean for the future?

• Scientists should stop forcing their data into rigid boxes.
• They should embrace the flexible, "mix-and-match" approach of Mixture Models.
• This will lead to more accurate family trees of life, helping us understand how animals, plants, and bacteria actually evolved.

In short: Nature is too complex for rigid boxes. We need flexible models to understand it.

Technical Summary

1. Problem Statement

Phylogenetic analyses of large genomic datasets must account for evolutionary heterogeneity (sites evolving at different rates or under different processes). Two primary strategies exist:

• Partitioned Models: Sites are grouped a priori (e.g., by gene or codon position), and a single substitution model is assigned to each partition.
• Mixture Models: Sites are not pre-assigned; instead, the likelihood of each site is calculated as a marginal probability across a mixture of substitution profiles (e.g., C10–C60 models).

The Core Challenge: Historically, comparing these two model classes has been methodologically flawed. Standard information criteria (like AIC, AICc, BIC) rely on conditional likelihoods (cAIC) for partitioned models (where sites are assigned to specific partitions) but marginal likelihoods for mixture models. This discrepancy creates a systematic bias where cAIC artificially favors partitioned models, even when they are misspecified, leading to incorrect phylogenetic inferences. Until recently, there was no fair, computationally efficient metric to directly compare the fit of partitioned versus mixture models on empirical data.

2. Methodology

The authors employed a multi-faceted approach to compare model performance across nine diverse empirical amino acid datasets (covering animals, plants, fungi, bacteria, and archaea).

A. Data and Experimental Design

• Datasets: Nine large-scale alignments (ranging from ~15k to ~9.5M sites) were selected.
• Subsampling: To manage computational costs, datasets were subsampled to 400 loci (or all available for smaller datasets) and varying taxon counts (20, 10, and 5 taxa) to test robustness across dataset sizes.
• Models Tested:
  • Partitioned Models: Optimized using PartitionFinder in IQ-TREE under three branch-length linkage settings: edge-equal, edge-proportional, and edge-unlinked.
  • Mixture Models: Specifically the C60 profile mixture model (the most complex standard profile model) with three variants:
    1. C60-wfix: Fixed profile weights.
    2. C60-wopt: Optimized profile weights.
    3. C60+F: C60 with an additional empirical frequency profile estimated from the data.
• Control: All models were evaluated on the same fixed tree topology (inferred from the best partitioned model) to ensure a conservative comparison favoring partitioned models.

B. Evaluation Metrics

The study utilized three complementary criteria:

1. Marginal Akaike Information Criterion (mAIC):
   • Based on the recent work of Susko et al. (2026), this metric calculates AIC scores using marginal probabilities for both partitioned and mixture models. This removes the conditional/marginal bias, allowing for a direct, fair comparison.
2. Parametric Bootstrap Tests (Model Adequacy):
   • Extending the framework of Giacomelli et al. (2025), the authors simulated 100 datasets under each fitted model.
   • Statistics: They compared the distribution of site-wise Shannon entropy (and amino acid diversity, div) between simulated and empirical data.
   • Metric: Used the Cramér–von Mises (CvM) test to compare the full distribution of entropy values, rather than just the mean, providing a more rigorous test of whether the model captures the generative process.
3. Tree Robustness Test:
   • A "leave-one-taxon-out" jackknife approach was used. Trees were inferred on 19-taxon subsets and compared to the full 20-taxon tree.
   • Metric: The Lin-Rajan-Moret (LRM) distance was used (instead of Robinson-Foulds) to quantify topological differences, as LRM is less sensitive to minor branch shifts.

3. Key Contributions

• Methodological Correction: The study validates and applies the mAIC as the first computationally tractable, information-theoretic method capable of directly comparing partitioned and mixture models without bias.
• Empirical Benchmarking: It provides the first large-scale empirical comparison showing that mixture models consistently outperform partitioned models across diverse taxonomic groups.
• Refinement of Bootstrap Methods: The authors improved upon existing parametric bootstrap methods by utilizing Shannon entropy (accounting for amino acid frequencies) and the CvM test (comparing full distributions) rather than simple mean diversity.
• Clarification of Branch-Length Linkage: The study investigates the performance of different branch-length linkage schemes (edge-equal, proportional, unlinked) under the new mAIC framework.

4. Key Results

• Superiority of Mixture Models (mAIC):
  • Across all nine datasets and taxon subsamples, C60 mixture models (particularly C60+F and C60-wopt) achieved significantly lower mAIC scores than the best-fitting partitioned models.
  • The difference in scores was often in the thousands of units, indicating overwhelming evidence in favor of mixture models.
  • As dataset size increased, the advantage of mixture models became more pronounced.
• Model Adequacy (Parametric Bootstrap):
  • In 7 of 9 datasets, C60 models (especially C60-wfix) best reproduced the empirical distribution of site-wise Shannon entropy.
  • Partitioned models tended to overestimate amino acid diversity because they force a single average profile on all sites within a partition, failing to capture the reality that many protein sites accept only a subset of amino acids.
  • Exception: In Sac Fungi and Budding Yeast datasets, edge-unlinked partitioned models performed better in bootstrap tests, likely due to specific compositional heterogeneity not captured by the fixed C60 profiles.
• Tree Robustness:
  • Both C60+F mixture models and edge-proportional partitioned models showed similar robustness to taxon removal.
  • No single model type was uniformly superior in tree topology stability, though mixture models generally performed slightly better or equally well.
• Branch-Length Linkage:
  • mAIC favored edge-proportional partitioned models (consistent with BIC), whereas cAIC and AICc favored edge-unlinked models.
  • The authors argue that mAIC is the correct metric for selecting models based on global parameters (like tree topology), while cAIC may be more appropriate for local parameters (like partition-specific rates).

5. Significance and Implications

• Paradigm Shift: The study challenges the long-standing reliance on partitioned models in phylogenomics. It suggests that for amino acid data, mixture models are universally superior in terms of model fit and ability to capture evolutionary heterogeneity.
• Future Research Direction: The authors advocate for a shift in the field toward developing and utilizing more complex mixture models rather than refining partitioning schemes.
• Practical Guidance:
  • Researchers should use mAIC (not cAIC) when comparing partitioned and mixture models.
  • For global phylogenetic inference, edge-proportional branch linking is preferred over edge-unlinked when using partitioned models, but mixture models remain the superior choice overall.
  • The C60+F model is recommended as a robust, high-performance standard for amino acid phylogenomics.
• Tooling: The study demonstrates that efficient, user-friendly tools (like IQ-TREE) can now implement these rigorous comparisons, making advanced model selection accessible for routine empirical analyses.

In conclusion, this paper provides definitive evidence that mixture models offer a more accurate representation of amino acid evolution than partitioned models, resolving a decades-old debate through the application of the newly developed marginal Akaike Information Criterion.