📄 evolutionary biology

On the mapping between evolutionary scenarios and governing rules in state-dependent speciation-extinction models

This study establishes a mathematical mapping between evolutionary scenarios and governing rules in state-dependent speciation-extinction models, revealing that distinct scenarios can produce identical tip state distributions and that inference on extant-only trees is biased toward certain scenarios due to the interplay of mixing rates, time, and likelihood conditioning.

Original authors: Soewongsono, A. C., Landis, M. J.

Published 2026-09-29

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Soewongsono, A. C., Landis, M. J.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

In the grand study of life on Earth, biologists often ask a simple but profound question: why is one trait common while another remains rare? Is a specific color, a particular habitat preference, or a unique behavior found in most species simply because it is better suited for survival, or is it a historical accident? To answer this, scientists look at the "family trees" of life, known as phylogenies. These trees map out how species are related, showing when they split from a common ancestor and how long ago. By combining these trees with data on which species have which traits, researchers can try to reconstruct the history of evolution. They use mathematical models to estimate the rates at which new species form, how often they go extinct, and how frequently they switch from one trait to another. This field, known as macroevolution, relies heavily on the assumption that the patterns we see in living species today are a direct reflection of the processes that shaped them over millions of years. However, a fundamental challenge remains: we only see the survivors. The vast majority of species that ever lived are extinct, and the family trees we build are based only on the branches that reached the present day. This creates a blind spot, making it difficult to know if the current balance of traits is the result of a steady, long-term process or a fleeting moment in time.

A team of researchers at Washington University in St. Louis set out to explore this blind spot by asking a bold question: if we strip away the family tree entirely and look only at the final count of species with different traits, can we still figure out what happened? They focused on a specific type of evolutionary model where the rate of speciation and extinction depends on the trait a species possesses. In these models, a species might be more likely to survive or reproduce if it has one trait rather than another. The researchers wanted to know if different evolutionary histories could lead to the exact same final outcome. Imagine two different stories: in one, a trait helps species multiply rapidly but also causes them to switch to a different trait quickly; in another, the trait is neutral but the species rarely switch. Could both stories end with the same number of species having each trait? The team developed a new mathematical framework to map out every possible "story" or scenario that could produce a specific pattern of trait frequencies. They found that the answer is yes: multiple, completely different evolutionary scenarios can produce identical long-term patterns of trait distribution.

The researchers discovered that for any given final mix of species, there are several distinct sets of rules that could have created it. They called these sets of rules "rulesets." Each ruleset represents a unique combination of birth rates, death rates, and switching rates between traits. For instance, a scenario where a trait is a "source" of new species while the alternative is a "sink" where species die out can produce the same final numbers as a scenario where both traits are sources, provided the rates of switching between them are adjusted in a specific way. This means that simply counting the species at the end of the day does not tell you which of these underlying stories is the true one. The study showed that while these different stories are mathematically distinct, they are all capable of arriving at the same destination.

To understand which of these stories is most likely to be true, the team turned to computer simulations. They generated thousands of imaginary evolutionary histories, letting them run forward in time to see which ones actually produced surviving families of species. They found that not all stories are created equal. Scenarios where both traits act as "sources"—meaning species with either trait tend to multiply rather than die out—were far more likely to produce large, surviving trees than scenarios where one or both traits acted as "sinks." This makes intuitive sense: if a trait causes species to go extinct, that branch of the tree is likely to disappear before it can be observed today. Consequently, the evolutionary histories that survive to the present are biased toward those where species are generally thriving.

The researchers then tested whether standard statistical methods, which work backward from the surviving trees to guess the original rules, could distinguish between these different scenarios. They found that these methods are not neutral. When analyzing trees that contain only living species, the statistical tools strongly favored the "source-source" scenarios, even when the true history was something else. This bias arises because the mathematical likelihoods used in these analyses are conditioned on the fact that the tree survived. A tree that survived is more likely to have come from a history where species were multiplying, so the analysis naturally leans toward that conclusion. The study also revealed that the speed at which a system settles into its final pattern matters. Some evolutionary scenarios reach a stable balance of traits very quickly, while others take a long time. If a tree is young, it might not have had enough time to reach that balance, making it harder to infer the correct history. The simulations showed that the ability to correctly identify the true evolutionary story depends heavily on how much time has passed and how quickly the system was converging toward its final state.

When the researchers applied their framework to real-world data, looking at the evolutionary history of lizards and snakes (squamates) and flowering plants (angiosperms), they found that the patterns from their simulations held up. In both groups, the most commonly inferred evolutionary scenarios were those where both traits acted as sources, consistent with the idea that surviving trees are biased toward thriving lineages. However, there were differences between the simulated data and the real world. In the real data, the methods inferred certain complex scenarios more often than the simulations predicted, suggesting that the real evolutionary processes might be more nuanced or that other factors, such as hidden traits not accounted for in the models, are influencing the results. The study highlights that while we can learn a great deal from the traits of living species, we must be cautious about assuming we know the exact path evolution took to get there. The same final picture can be painted with many different brushes, and without the full history of the tree, including the extinct branches, we may only see a fraction of the true story.

This work provides a crucial theoretical foundation for understanding the limits of what we can learn from the living world. It shows that the relationship between the data we observe and the processes that generated it is not one-to-one. By mapping out all the possible rules that could lead to a specific outcome, the researchers have given biologists a way to see the full range of possibilities. This helps in choosing better starting points for future studies and in understanding why certain evolutionary stories are more frequently told than others. It serves as a reminder that the history of life is complex, and that the patterns we see today are the result of a delicate interplay between survival, time, and the specific rules of evolution that governed each lineage. The study does not claim to solve the mystery of trait evolution, but it clarifies the landscape of possibilities, ensuring that future inquiries are grounded in a realistic understanding of what the data can and cannot tell us.

Technical Summary: Mapping Evolutionary Scenarios to Governing Rules in State-Dependent Speciation-Extinction Models

Problem Statement
Biologists frequently seek to explain trait frequency distributions among extant species by inferring underlying evolutionary processes, such as asymmetries in diversification (speciation and extinction) or character evolution. State-dependent speciation-extinction (SSE) models, such as BiSSE and GeoSSE, provide a framework for jointly modeling these processes. However, a fundamental challenge remains: multiple distinct evolutionary scenarios can produce identical long-term stationary distributions of tip state frequencies. Furthermore, when phylogenetic trees are reconstructed solely from extant species (conditioned on survival), the inference process may be biased toward specific scenarios. This study addresses the question: In the absence of a resolved phylogenetic tree, what information about the underlying data-generating process is retained in present-day state frequencies alone? Specifically, the authors investigate whether identical tip state frequencies can arise from distinct evolutionary scenarios and how the conditioning on extant-only trees affects the preferential inference of these scenarios.

Methodology
The authors develop a mathematical framework to define and map "rulesets" to "evolutionary scenarios" within discrete-state SSE models.

Definitions and Mathematical Derivation:
- Evolutionary Scenario: Defined qualitatively by the signs of net diversification rates ( $\lambda_i - \mu_i$ ) in each state and the relative magnitudes of transition rates ( $q_{ij}$ ).
- Ruleset: Defined as a precise system of equalities and inequalities among model parameters that are compatible with a specific set of asymptotic stationary state frequencies ( $\hat{\Pi}$ ).
- The authors derive general conditions (Theorem 1) and specific corollaries for BiSSE (Corollaries 1 & 2) and GeoSSE models. These derivations establish a bijection between distinct rulesets and distinct evolutionary scenarios for a given set of stationary frequencies.
- They distinguish between "trivial" scenarios (where all states share the same dynamical phase, e.g., all supercritical or all subcritical) and "non-trivial" scenarios (where states occupy opposing phases, e.g., source-sink dynamics).
Convergence Analysis:
- The study introduces the concept of relaxation time ( $\hat{t} = \gamma^{-1}$ ), where $\gamma$ is the spectral gap of the mean growth matrix. This metric characterizes the rate at which a system converges to its stationary distribution.
- The authors analyze how the product of the spectral gap and elapsed time ( $\gamma t$ ) influences the distance between observed tip frequencies and the true stationary frequencies.
Simulation Experiments:
- The authors simulate 100 valid extant-only trees for various tree sizes (25 to 800 tips) under different rate regimes (slow and fast) and experimental settings (e.g., with/without missing taxa, with/without rejection of imbalanced tip states).
- They calculate the probability of each true scenario generating a valid tree ( $P(S_j | \text{valid})$ ) and the probability of inferring a specific scenario given a valid tree ( $P(\hat{S}_i | \text{valid})$ ).
- Confusion matrices are constructed to assess type-I (false positive) and type-II (false negative) errors across scenarios.
Empirical Application:
- The framework is applied to a reconstructed timetree of 6,885 squamate species (Title et al. 2024) and a compilation of 49 previous BiSSE studies on angiosperm clades (Helmstetter et al. 2023).
- Estimated parameters from these empirical datasets are mapped to the defined evolutionary scenarios to assess their prevalence.

Key Contributions

Formal Mapping: The paper establishes a rigorous bijection between mathematical rulesets (parameter constraints) and qualitative evolutionary scenarios for SSE models. It demonstrates that identical stationary frequencies do not imply identical evolutionary processes.
Identification of Non-Uniqueness: The study proves that multiple distinct evolutionary scenarios (specifically, different combinations of net diversification signs and transition rate directions) can yield the same long-term tip state frequencies.
Characterization of Inference Bias: Through simulations, the authors show that not all scenarios are equally likely to be inferred from extant-only data. There is a systematic bias toward scenarios involving supercritical processes (source-source dynamics) and specific source-sink configurations.
Relaxation Time Framework: The introduction of the spectral gap and relaxation time provides a theoretical tool to understand how the "age" of a tree and the speed of convergence interact to determine observed state frequencies.

Results

Forward-in-Time (Data Generation): Scenarios where both states are sources (supercritical, $\rho > 0$ ), specifically Scenarios 3a/b and 4a/b, are the most frequent generators of valid, non-extinct trees. Scenarios where both states are sinks (subcritical) rarely produce valid trees. Among valid trees, Scenarios 3a/b are more prevalent than 4a/b because 4a/b requires transition rates that oppose the stationary frequency ordering, restricting the admissible parameter space.
Backward-in-Time (Inference): When fitting models to simulated extant-only trees, Scenarios 3a/b are the most frequently inferred, followed by 2a/b. Scenarios 1a/b and 4a/b are inferred less frequently.
- The frequent inference of 3a/b aligns with their high prevalence in valid trees and the conditioning of likelihoods on survival.
- The frequent inference of 2a/b (source-sink) is partly attributed to their faster convergence rates (larger spectral gaps), which allow observed frequencies to approach stationarity more quickly, making them easier to recover.
- Scenarios 1a/b and 4a/b exhibit high type-II error rates (low sensitivity), meaning they are often misclassified as other scenarios (particularly 2a/b or 3a/b).
Empirical Findings: Analyses of squamate and angiosperm datasets mirror simulation results, with Scenarios 3a/b being the most commonly inferred. However, empirical data shows a higher relative frequency of Scenario 4a/b compared to simulations, potentially due to model misspecification (e.g., unmodeled hidden states) or specific biological constraints in these clades.

Significance and Claims
The authors claim that this study provides a theoretical foundation for directly linking data-generating processes to observed tip state frequencies without relying solely on the topology of a reconstructed tree.

Complementary Approach: The framework is presented not as a replacement for tree-based inference, but as a complementary tool to characterize the range of plausible evolutionary scenarios given minimal information (species counts and traits).
Prior Specification: The findings suggest that priors in Bayesian phylogenetics should be informed by the likelihood of a scenario generating a valid, non-extinct tree. Specifically, supercritical processes are naturally favored in extant-only datasets.
Interpretation of Bias: The paper clarifies that the preference for certain scenarios in empirical studies may arise from the interplay between mixing rates, elapsed time, and the conditioning inherent in tree likelihoods (survival bias), rather than solely from the biological reality of the clades.
Extensibility: While demonstrated on BiSSE and GeoSSE, the authors assert that the concepts of rulesets and the bijection to evolutionary scenarios are extensible to other discrete-state SSE models, though the complexity increases with the number of states.

The authors maintain a modest tone, noting that while their framework clarifies theoretical identifiability issues, empirical results may still be influenced by model misspecification (e.g., hidden states) and that the ranking of scenarios is not always one-to-one when considering forward versus backward time perspectives.

Technical Summary: Mapping Evolutionary Scenarios to Governing Rules in State-Dependent Speciation-Extinction Models

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