A method for comparing inferred evolutionary accumulation dynamics across covariates and model structures

Technical Summary: A Method for Comparing Inferred Evolutionary Accumulation Dynamics

Problem Statement
Evolutionary Accumulation Models (EvAMs) describe systems acquiring binary features over time, such as mutations in cancer or drug resistance in pathogens. While diverse methods exist to infer these dynamics—ranging from deterministic graphs and partially ordered sets (posets) to stochastic transition matrices—there is a lack of a unified framework for comparing inferred dynamics across different algorithms, datasets, and covariates. Existing comparison methods often struggle with specific complexities inherent to EvAMs:

1. "Frameshift" Differences: Two pathways may share similar relative orderings (e.g., feature A before B) but differ entirely in absolute timing or state transitions (e.g., 0000→0001 vs. 0000→1000). Transition-based or absolute-ordering methods may flag these as dramatic differences, while relative-ordering methods might miss the shift in absolute state.
2. Model Flexibility: Many existing comparison techniques rely on deterministic relationships or fixed-length sequences, making them unsuitable for reversible dynamics, stochastic dependencies, or cases where sets of features (rather than single features) influence acquisition.
3. Statistical vs. Scientific Significance: Distinguishing between statistically significant differences in parameter estimates (due to uncertainty) and scientifically meaningful differences in the underlying dynamic behavior is challenging.

Methodology
The authors propose a method based on ordering matrices to summarize and compare inferred EvAM dynamics. This approach focuses on probabilistic summaries of the conditions under which features are acquired, accommodating stochasticity, reversibility, and multi-step acquisitions.

1. Ordering Matrices:

   • Relative Ordering Matrix ($P^{rel}_{ij}$): Estimates the probability that feature $i$ is acquired after feature $j$ is already present. This captures dependencies between specific features.
   • Absolute Ordering Matrix ($P^{abs}_{in}$): Estimates the probability that feature $i$ is acquired when at most $n-1$ other features are already present. This captures the timing of acquisition relative to the total number of accumulated features.
   • These matrices are derived from inferred transition probabilities, whether from exact transition matrices or sampling-based approaches (e.g., random walkers in state space).

2. Statistical Comparison Criterion:
   To ensure differences are both statistically robust and scientifically meaningful, the authors propose a dual-threshold criterion comparing distributions of matrix elements derived from resampling (e.g., bootstrap or posterior samples).

   • Let $P^{rel}_{ij}$ be the probability that $i$ precedes $j$.
   • A difference is reported if, for a proportion $1-p$ of samples in Instance 1, $P^{rel}_{ij} < q$, and for a proportion $1-p$ of samples in Instance 2, $P^{rel}_{ij} > 1-q$.
   • Default Parameters: $p=0.2$ and $q=0.25$. This requires that >80% of samples in Instance 1 show a <25% chance of $i$ preceding $j$, while >80% of samples in Instance 2 show a >75% chance. This ensures non-overlapping distributions with a substantial magnitude gap, filtering out trivial statistical noise.

3. Implementation:
   The method is implemented in the hyperinf R package. It utilizes ape for tree manipulation and ggplot2/ggarrange for visualization.

Key Results
The method was validated using synthetic data and applied to two real-world datasets:

• Synthetic Data:

  • The method successfully distinguished between simple linear pathways and more complex scenarios involving "frameshift" differences.
  • In small datasets ($n=4$), the method detected differences in relative ordering despite high uncertainty.
  • In larger datasets ($n=12$), uncertainty decreased, allowing the detection of differences in both relative and absolute ordering for multiple features.
  • The approach worked for both irreversible (HyperHMM) and reversible (HyperMk) models.

• Drug Resistance in Klebsiella pneumoniae (KpAMR):

  • Applied to genomic data from Gambia and South Korea using HyperTraPS.
  • The method confirmed previously identified differences in absolute ordering for features like Bla_Carb_acquired, Flq_mutations, and Omp_mutations.
  • It additionally detected robust differences in Sul_acquired and Tmt_acquired features, demonstrating higher sensitivity than previous non-targeted approaches.

• Chromosomal Aberrations in Cancer:

  • Applied to 2,658 cancer samples across 38 types to compare chromosomal aberration accumulation.
  • The method identified distinct dynamics between tumor types, even within the same tissue (e.g., CNS-PiloAstro vs. CNS-GBM) and across different tissues (e.g., Kidney-RCC variants).
  • Adenocarcinomas showed particularly high variability in inferred dynamics across tissue types.

Significance and Claims
The authors position this work not as a replacement for specific EvAM inference tools, but as a general quantitative framework for comparing the outputs of these tools.

• Flexibility: The primary contribution is the ability to compare dynamics across models that support stochasticity, reversibility, higher-order interactions, and multi-step acquisitions, which many existing comparison metrics cannot handle.
• Robustness: By focusing on probabilistic summaries and enforcing a dual-threshold criterion, the method distinguishes between statistical uncertainty and genuine biological divergence in evolutionary pathways.
• Interpretability: The use of ordering matrices provides a standardized, interpretable summary of complex dynamics, facilitating comparisons across different datasets and covariates (e.g., geography, tumor type).
• Modesty: The authors acknowledge that while the method addresses specific limitations (frameshifts, reversibility), it is one of several approaches. They note that the intersection of stochastic modeling and ranking statistics (e.g., Plackett-Luce models) offers further avenues for exploration. The method is presented as a tool to formalize comparisons in diverse domains, including cancer progression, drug resistance, and evolutionary biology.