📄 evolutionary biology

The shape of evolution: persistent homology of genetic-distancedata as an observable of reticulate processes in pathogen andplant genomes

This paper demonstrates that persistent homology applied to genetic-distance data serves as a robust topological observable for detecting and characterizing reticulate evolutionary processes, such as recombination and selective sweeps, across diverse pathogen and plant genomes.

Original authors: Feged-Rivadeneira, A., Gonzalez-Casabianca, F., Cardenas Ramirez, P., Angel, A., Corredor, V.

Published 2026-06-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Feged-Rivadeneira, A., Gonzalez-Casabianca, F., Cardenas Ramirez, P., Angel, A., Corredor, V.

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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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

Imagine you are trying to understand the history of a group of people by looking at how similar their family recipes are. Usually, scientists think of evolution like a family tree: one ancestor splits into two branches, those split again, and so on. It's a straight line of descent.

But in the real world, evolution isn't always a tree. Sometimes, different branches cross-pollinate (like two families swapping recipes), or a specific recipe becomes so popular that everyone suddenly starts using it, wiping out the old variations. These messy, tangled histories are hard to spot with standard tools.

This paper introduces a new way to look at genetic data using a mathematical concept called "persistent homology." Think of this not as a microscope, but as a shape-shifter. Instead of just counting differences, the authors turn genetic data into a 3D sculpture.

Here is how they break it down:

The Shape of History

The researchers take a map of genetic distances (how different organisms are from each other) and build a geometric shape out of it. They look for specific features in this shape:

Loops (The "Donut" Shape): If the data forms a loop, it suggests reticulation. In everyday terms, this is like a web where different lines cross each other. This happens when organisms swap genetic material (recombination) or when different lineages mix.
Solid Blocks or Cavities: These represent more complex, higher-order mixing events.

They created a simple "four-letter alphabet" to describe the shape of these histories:

K1 (Clonal): A straight line or a single point. No mixing.
K2 (Divergence): A simple split, like a tree branch.
K3 (Reticulation): A loop or a web. This is the "messy" mixing we are looking for.
K4 (Higher-order): Even more complex webs.

What the Simulations Showed

Before looking at real bugs and plants, the authors ran computer simulations (like a flight simulator for evolution). They found two main things:

The Loop Count Rises with Mixing: As the rate of genetic mixing (recombination) goes up, the number of "loops" in the shape increases steadily. It's a reliable thermometer for mixing.
Spotting the Difference: This method can tell the difference between a "clean" tree history and a "tangled" web history with near-perfect accuracy (98–100%). If the shape has loops, it's a web; if not, it's a tree.

Real-World Examples

The team applied this "shape detector" to four real-world scenarios:

1. Malaria in Africa and Asia
They looked at 20,000 samples of the malaria parasite from 33 countries.

The Surprise: You might expect places with high mixing (like Africa) to have the most loops. Instead, the "loop count" was highest in Southeast Asia and Papua.
The Explanation: At the population level, the shape is dominated by demographics (how populations are structured and separated) rather than just the speed of mixing. The "clonal" or "swept" populations in Asia created a shape that looked more like a complex web than the freely mixing African populations.

2. Drug Resistance in Colombia
They compared malaria samples that were resistant to a drug (Sulfadoxine-Pyrimethamine) against those that were sensitive.

The Result: The resistant samples had a much higher "loop count" (12 vs. 5).
The Meaning: This suggests the drug resistance didn't come from one single ancestor. Instead, it appeared on several different genetic backgrounds, creating a tangled web of resistance.

3. The Artemisinin Sweep in Cambodia
They tracked a specific drug-resistance event over a decade (2008–2018).

The Journey: The shape started as a web (K3), peaked in complexity when the resistance was spreading, and then collapsed into a simple line (K1) once the resistance took over completely.
The Significance: This is the first time scientists have directly watched a "selective sweep" (where one trait takes over) change the shape of evolution in real-time.

4. Wild Plants in Europe
They looked at Arabidopsis thaliana (a common weed).

The Finding: Ancient populations in Spain (relicts) showed more complex shapes (loops) than the newer populations that expanded into Sweden and the UK after the ice age. This proves the method works for plants, not just disease.

The "Control" Check

To make sure their tool wasn't just seeing shapes where there were none, they tested it on the mitochondria of the malaria parasite. Since mitochondria don't mix (they are passed down in a straight line), the tool correctly found zero loops. This confirmed the tool is specific and accurate.

A Glimpse at the Future

The authors also looked at 3D "cavities" (holes inside the shape, not just loops on the surface). They found these appear in mixed systems but not in simple ones. They suggest that the ratio of these 3D holes to the 2D loops might help distinguish between small-scale evolution (micro) and large-scale evolution (macro), though they admit this needs more testing.

The Bottom Line

This paper argues that the topology (the geometric shape) of genetic data is a direct, observable fact of evolution. By turning genetic distances into shapes, we can instantly see if a population is a clean tree or a tangled web, helping us understand how drug resistance spreads and how species evolve.

Technical Summary: The Shape of Evolution

Problem Statement
Evolutionary biology has long recognized that processes such as recombination, lineage divergence, drug-resistance sweeps, introgression, and refugial persistence generate genomic signatures incompatible with strict tree-like structures. Traditional phylogenetic models often struggle to capture these reticulate (network-like) histories. The paper posits that the "shape" of genetic-distance data serves as a direct observable for these complex processes, offering a topological alternative to tree-based inference.

Methodology
The authors formalize the shape of genetic-distance data using persistent homology, a tool from topological data analysis. The core methodology involves:

Simplicial Complex Construction: Genetic-distance matrices are converted into Vietoris-Rips filtrations.
Topological Quantification: The filtration yields Betti numbers:
- $\beta_0$ : Connected components.
- $\beta_1$ : Loops (1-dimensional holes).
- $\beta_2$ : Cavities (2-dimensional holes).
Interpretation Framework: Rather than treating $\beta_1$ as a literal count of recombination events, the authors define it as a quantity monotone with effective recombination above a sampling-dependent geometric baseline.
Topological Alphabet: The resulting shapes are categorized into four primitives:
- K1: Clonal.
- K2: Divergence.
- K3: Reticulation.
- K4: Higher-order reticulation.

Key Contributions and Validation
The paper establishes two primary claims through Coalescent and Wright-Fisher simulations:

Monotonicity: $\beta_1$ rises monotonically with the recombination rate across six orders of magnitude.
Discriminatory Power: Persistent homology features successfully separate reticulate from non-reticulate histories with 98–100% recall. The authors note that residual confusion is confined entirely to the non-reticulate K1/K2 pair, which $\beta_1$ does not distinguish.

Empirical Results
The pipeline was applied to four distinct empirical systems:

Plasmodium falciparum (MalariaGEN Pf7):
- Analyzed 20,864 samples across 33 countries.
- Per-population $\beta_1$ values spanned two orders of magnitude (median 20.5, range 8–32), significantly diverging from a label-shuffled null.
- Counter-intuitive Finding: The ordering of $\beta_1$ was opposite to the recombination rate. Freely recombining African populations exhibited the lowest $\beta_1$ , while clonal/swept Southeast Asian and Papuan populations showed the highest. The authors reconcile this by arguing that at the population scale, $\beta_1$ is dominated by demographic structure rather than recombination rate alone.
Colombian Cauca SP-resistant Samples:
- Sulfadoxine-pyrimethamine (SP) resistant samples showed $\beta_1 = 12$ , compared to a near-clonal SP-sensitive baseline of $\beta_1 = 5$ .
- The resistant group exhibited two orders of magnitude more total persistence and a high- $\beta_1$ , multi-origin band consistent with K3 (reticulation), suggesting resistance was carried on several genomic backgrounds.
P. falciparum Artemisinin Sweep (Cambodia, 2008–2018):
- Traced a K3 $\to$ K1 trajectory.
- $\beta_1$ rose to a mid-sweep peak of 45 and collapsed to 13 at fixation.
- The authors claim this is the first direct observation of a selective-sweep transient in topological coordinates, noting that per-bin values represent medians of three subsamples with wide confidence intervals.
Arabidopsis thaliana (1001 Genomes):
- Applied the framework to plant genomes.
- Iberian relict populations (Spain) showed higher normalized $\beta_1$ ( $\beta_1/n = 0.64$ ) compared to post-glacial expansion populations (Sweden: 0.54; United Kingdom: 0), demonstrating the framework's applicability beyond pathogens.

Specificity and Higher-Order Topology

Negative Control: A P. falciparum mitochondrial dataset (non-recombining) recovered $\beta_1 = 0$ across all subsamples, establishing pipeline specificity.
$\beta_2$ Analysis: Moving beyond the 1-skeleton, $\beta_2$ was zero at clonal/expansion limits but positive across reticulate systems.
Admixture Simulation: A controlled two-vs-three-way admixture simulation confirmed that $\beta_2$ can separate regimes that share a similar $\beta_1$ profile.
Hypothesis: The authors present the ratio $\beta_2/\beta_1$ as a potential separator of microevolutionary from macroevolutionary timescales, though they explicitly frame this as a hypothesis for future testing due to the small number of systems and the absence of a $\beta_2$ null.

Significance
The paper concludes that the topology of genetic-distance data is a valid evolutionary observable. The immediate implication highlighted is the utility of this approach for drug-resistance surveillance in P. falciparum, offering a method to detect and characterize reticulate evolutionary processes that are otherwise obscured in tree-based models.