🧬 biology

Radial Interaction Tomography: Recognizing Non-Transitive Evolutionary Games from One Range-Expansion Image

This paper introduces Radial Interaction Tomography, a computer-vision framework that reconstructs pairwise evolutionary interaction flows and detects non-transitive cyclic dynamics from a single endpoint image of a microbial range expansion, enabling robust mechanistic inference and active control design without requiring time-series data.

Original authors: Faruk Alpay, Baris Basaran

Published 2026-07-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Faruk Alpay, Baris Basaran

Original paper licensed under CC BY 4.0 (http://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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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 a petri dish where different types of bacteria are growing outward from a central point, like ripples in a pond or slices of a pie. As they grow, they push against each other, creating colorful, swirling boundaries. Usually, scientists look at these patterns to guess which bacteria are "stronger" (more fit) than others. If Bacteria A beats B, and B beats C, we expect A to beat C. This is a simple, linear hierarchy, like a rock-paper-scissors game where there is always a winner.

But what if the game isn't simple? What if A beats B, B beats C, but C beats A? This is a "non-transitive" or cyclic relationship, like a real Rock-Paper-Scissors game.

This paper introduces a new method called Radial Interaction Tomography. Think of it as a "time machine" that works in reverse. Instead of watching the bacteria grow over time, the authors ask: "Can we look at just one final photo of the colony and figure out the entire history of who fought whom, and who won?"

Here is a breakdown of their approach using simple analogies:

1. The Frozen Movie

When bacteria grow in a colony, the cells in the middle stop moving and get "frozen" in place as new layers grow on top of them. The distance from the center acts like a timestamp. The outer edge is "today," and the center is "yesterday."

The Analogy: Imagine a tree ring. By looking at the rings, you can tell how the tree grew. Here, the "rings" are the colored sectors of bacteria. The authors treat the final image not just as a picture, but as a recorded history of battles.

2. Unwrapping the Spiral

To analyze the image, the computer "unwraps" the circular colony into a flat strip (like unrolling a carpet).

The Math: They measure the angle of the boundary lines between different bacteria types. If the line between Type A and Type B tilts one way, it means A is pushing B back. If it tilts the other way, B is pushing A.
The Goal: They calculate a "flow" for every pair of bacteria. Does A gain territory from B? Does B gain from A?

3. The "Lie Detector" Test

The core innovation is a mathematical test to see if the pattern makes sense as a simple hierarchy (A > B > C) or if it reveals a cycle (A > B > C > A).

The Analogy: Imagine a detective trying to solve a crime. If the clues fit a simple story (the butler did it), the detective is happy. But if the clues show a loop (the butler did it, but the gardener did it, but the butler did it again), the simple story fails.
The Result: The authors proved that their method can mathematically separate the "simple story" (transitive) from the "cyclic loop" (non-transitive). They can even tell you how much of the pattern is a cycle and how much is just noise.

4. The "Minimum Ingredients" Rule

The paper also figured out a rule for how many different bacteria you need to see a cycle.

The Finding: If you have 4 types of bacteria, you need at least 8 distinct boundary lines in the image to be sure you are seeing a true cycle and not just a random mess. If you have fewer lines, the math says, "I can't tell the difference, so I won't guess." This prevents false alarms.

5. From "What Happened" to "What If"

Once the computer figures out the history of the battles (the "cyclic residuals"), it doesn't just stop there. It uses that information to run simulations.

The Analogy: Think of it like a flight simulator. First, the computer analyzes a real flight path to see where the pilot made mistakes (the cyclic residuals). Then, it uses that data to design a new flight plan that avoids those mistakes.
The Application in the Paper: They used the detected "cycles" to design a new set of rules (control parameters) for a simulation. The goal was to create a scenario where the "bad" bacteria are suppressed while the "good" bacteria thrive. They tested millions of scenarios and found a specific recipe that worked very well in their computer model.

6. What They Did NOT Do

It is important to note what this paper is not claiming:

No New Biology: They didn't discover a new species of bacteria.
No Lab Experiments: They didn't grow bacteria in a real lab for this specific test; they used computer-generated images and public photos to prove their math works.
No Medical Cures: They are not claiming this will immediately cure diseases. They are building a mathematical tool to understand how these patterns form.

Summary

The paper presents a computer vision tool that looks at a single snapshot of a bacterial colony and reconstructs the "story" of their competition. It can tell you if the competition was a simple ladder (one winner) or a complex cycle (rock-paper-scissors). Furthermore, it uses that story to simulate how we might engineer these colonies to behave in a desired way, all without needing to watch them grow in real-time.

The authors are essentially saying: "We can read the history of a fight from the scars left on the battlefield, and then use that knowledge to design a better battlefield."

Technical Summary: Radial Interaction Tomography

Problem Formulation
The paper addresses a computer-vision inverse problem: recovering the history of pairwise evolutionary interactions from a single endpoint image of a microbial range expansion. While prior work utilized the "frozen front" property of expanding colonies (where radius correlates with deposition time) to estimate constant relative fitness or model stochastic wandering, these approaches assume a transitive scalar fitness hierarchy. This paper investigates whether a single endpoint image can reveal non-transitive interactions (e.g., cyclic dominance like rock-paper-scissors or bacteriocin-mediated antagonism) that cannot be explained by a scalar fitness ordering. The core challenge is distinguishing geometric signals of cyclic competition from artifacts of imaging noise, boundary wandering, or scalar fitness variations.

Methodology
The authors propose Radial Interaction Tomography, a deterministic pipeline that treats the colony image as a pattern-recognition object rather than relying on learned embeddings or neural networks.

Geometric Observable: The method extracts a radius-indexed pairwise boundary-flow field, $q_{ij}(\rho)$ , defined as the derivative of the unwrapped angle of a sector boundary with respect to a deposition coordinate $\rho$ (typically $\log(r/r_0)$ ).
Hodge Decomposition: The observed edge-flow vector is decomposed into a scalar-compatible component (gradients of a node potential, representing transitive fitness) and a residual component (non-potential flows, representing cyclic interactions). This decomposition is weighted by a precision matrix to handle noise.
Identifiability Analysis: The paper establishes strict boundaries on what is recoverable:
- Radial Clock Non-identifiability: Physical time cannot be recovered without independent calibration; only the deposition coordinate is identifiable.
- Mechanism Non-identifiability: The image reveals pairwise boundary flow but not isolated front speeds or microscopic mechanisms without additional data.
- Contact-Complete Design: A theorem proves that to observe every pair in an unrestricted antisymmetric game of $n$ genotypes on a single frontier, a minimum of $\binom{n}{2}$ (for odd $n$ ) or $\binom{n}{2} + n/2$ (for even $n$ ) boundaries is required. For $n=4$ , at least eight boundaries are necessary, not six.
Statistical Testing: The pipeline employs an exact Gaussian generalized likelihood-ratio test (GLRT) to detect cyclic residuals, coupled with Bonferroni corrections for interval scanning along the radial path.

Key Contributions

Formulation of Radial Interaction Tomography: Defining the inverse problem of recovering radius-indexed transitive and cyclic patterns from a single image.
Separation of Observable and Interpretation: Explicitly decoupling the geometric edge-flow field from biological interpretations (like isolated front speed), preventing invalid inferences.
Contact-Design Theorem: Establishing the minimum number of sector boundaries required to fully resolve pairwise interactions in a circular design.
Deterministic Benchmarking: Creating a suite of tests including analytic endpoint images, pixel-level round trips, scalar-null stress tests, and public-image tracing, all without neural networks.
Active Design Control: Extending the inverse result to a control layer where recovered cyclic residuals serve as forcing signals for dimensionless reaction-diffusion control and strain-design optimization.

Experimental Results
The paper validates the method through several deterministic and stochastic benchmarks:

Pixel-Level Recovery: In a "pixel round trip" (rendering a blurred/noisy image and recovering flows), the method achieved an edge-flow RMSE of $\approx 2.86 \times 10^{-3}$ and cyclic-norm RMSE of $\approx 2.95 \times 10^{-3}$ .
Scalar-Null Stress Tests: When the ground truth is purely transitive (scalar), the method correctly identifies false positives only when nuisance factors (noise, extinction, distortion) are severe. Under high nuisance, the system is designed to abstain rather than infer non-transitive ecology, with abstention rates rising appropriately (e.g., 0.564 false detection rate under worst-case nuisance, prompting abstention).
Mechanistic Validation: Using a non-learning frozen-front simulator with four genotypes, the method successfully recovered a stable annulus and detected a cyclic residual with a peak norm of 0.358, matching the programmed cyclic interaction.
Dose Response: A sweep of programmed cyclic-antagonism scales showed a strong correlation ( $r=0.776$ ) between the programmed scale and the recovered cyclic residual.
Active Control: The recovered residuals were used to drive a reaction-diffusion control sweep. An adjoint-state refined design reduced the undesired population state by 99.35% while retaining 161.39% of the beneficial density.
Robustness: A Monte Carlo robustness test with 1,000,000 trajectories confirmed that the selected designs maintained at least 40% adverse-state reduction and 85% beneficial-strain retention under parameter perturbations.
Public Image Audit: The pipeline was applied to a public figure from Weinstein et al., passing as a "computer-vision sanity check" but explicitly noting that publication annotations prevent its use as raw quantitative ground truth.

Significance and Claims
The paper claims that a single endpoint image contains sufficient geometric information to reconstruct pairwise interaction histories and detect non-transitive signatures, provided the experimental design meets specific contact-completeness criteria. The primary contribution is the falsifiable visual recognition claim: the ability to separate geometric observables from downstream biological interpretations.

The authors emphasize that while the method successfully recovers "frozen-front" interaction histories and drives active design simulations, it does not claim to replace physical experiments. The "active design-control" layer is presented as a dimensionless simulation target, and the paper explicitly states that biological translation requires independent calibration, measured phenotype tensors, and physical replicates. The work serves as a rigorous framework for interpreting range-expansion images, defining the limits of what can be inferred from geometry alone, and providing a deterministic tool for hypothesis generation in evolutionary ecology.