🔬 condensed matter

Topological flowscape reveals state transitions in nonreciprocal living matter

This paper utilizes starfish embryos to demonstrate how nonreciprocal interactions drive structural transitions in living matter, introducing a novel framework of "topological flowscapes" to map and quantify the dynamic shifts between ordered and disordered collective states.

Original authors: Hyunseok Lee, EliseAnne Koskelo, Shreyas Gokhale, Junang Li, Chenyi Fei, Chih-Wei Joshua Liu, Lisa Lin, Jorn Dunkel, Dominic J. Skinner, Nikta Fakhri

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hyunseok Lee, EliseAnne Koskelo, Shreyas Gokhale, Junang Li, Chenyi Fei, Chih-Wei Joshua Liu, Lisa Lin, Jorn Dunkel, Dominic J. Skinner, Nikta Fakhri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a ballroom dance floor, but instead of humans, it's filled with thousands of tiny, swimming starfish embryos. These aren't just any dancers; they are from two different generations. Some are "teenagers" (24 hours old), and some are "adults" (48 hours old). In a normal, fair world, if two dancers pushed each other, they'd push back with equal force. But in this living ballroom, the rules are broken: the interactions are nonreciprocal. The teenagers and adults don't just push; they play a game of "run and chase." The younger ones actively chase the older ones, while the older ones try to swim away, creating a one-way street of force.

The big question the scientists asked was: What happens when you mix these two generations together? Does the chaos of the chase destroy order, or does it create something new?

The Great Dance Shift
The researchers watched this microscopic ballroom for about 7 hours. At the start, the mix of young and old embryos didn't just swim randomly; they spontaneously formed a traveling state. It was like a massive, swirling flock of birds moving in perfect unison. They rotated clockwise and drifted together at a speed of about 12 ± 3 µm/s (that's roughly the width of a human hair every second). This was a surprise: in a fair, reciprocal world, they wouldn't have moved together like this. The "chase" dynamic was the engine driving the whole group forward.

But then, something magical happened. After about 3 hours, the music changed. The flock didn't just stop; it transformed. The embryos settled down into a fluctuating state. Instead of zooming around, they arranged themselves into a giant, ordered crystal lattice, like a honeycomb. They still wiggled and vibrated, but the massive, collective drift stopped. The group had gone from a chaotic, high-speed chase to a stable, structured community.

The "Topological" Map
How did the scientists understand this shift? They invented a new way to look at the dance floor, which they called a "topological landscape."

Imagine you are trying to describe a messy room. You could say, "It's messy," or you could count exactly how many times you'd have to move a chair to make the room perfectly tidy. That's what the scientists did. They counted the "topological steps" (called T1 transitions) needed to turn the messy, wiggling group of embryos into a perfect hexagonal crystal.

They found a counter-intuitive secret: Weak nonreciprocity actually helps order. When the "chase" was just a little bit uneven, it acted like a magical annealing process. It helped the embryos fix their mistakes, smoothing out defects and making the crystal more perfect than it would have been if everyone were fair and equal. But if the "chase" became too strong (too much nonreciprocity), the crystal shattered, and the group broke into tiny, fragmented pieces.

The "Flow" of Information
To understand how the group changed from a traveling flock to a crystal, they built a "topological flowscape." Think of this as a GPS for the dance floor. It didn't just show where the embryos were; it showed the path they took to get there and how much energy they burned to do it.

They discovered that the transition wasn't a slow, gradual slide. It was a sharp turn. Around the 3-hour mark, the system underwent a rapid shift. The "energy" (or dissipation) of the system dropped as the embryos settled down. Interestingly, the scientists found that when the system was burning more energy (during the traveling state), it was better at "proofreading" itself. It was more likely to fix errors and move toward the perfect crystal shape. When the energy dropped, this proofreading slowed down.

What They Ruled Out
It's important to note what this dance floor is not. The scientists explicitly ruled out the idea that this order comes from the embryos just bumping into each other randomly or that the order exists because of some external coordination. The order emerged purely from the internal, asymmetric "chase" between the two generations. They also showed that if you mix embryos of the same age, this traveling flock never happens; the nonreciprocity is the key ingredient.

How Sure Are They?
The team didn't just guess; they measured it. They filmed thousands of embryos, used computer vision to track every single one, and even built a computer simulation that mimicked the real physics. In these simulations, they could dial the "nonreciprocity" up and down like a volume knob. When they turned it up to match the real experiment, the simulation perfectly recreated the shift from a traveling flock to a crystal. They even found a "hysteresis" effect in the simulations: if you slowly turn the knob back and forth, the system doesn't switch states at the exact same point, suggesting a complex, memory-like behavior in how these living crystals form.

In short, this paper reveals that in the world of living matter, being a little bit unfair (nonreciprocal) isn't a bug; it's a feature. It's the secret sauce that allows a chaotic group of swimmers to self-organize into a beautiful, structured crystal, proving that sometimes, a little bit of chasing is exactly what you need to find your place in the crowd.

Technical Summary: Topological Flowscape Reveals State Transitions in Nonreciprocal Living Matter

Problem Statement
Nonreciprocal interactions, where forces between entities are asymmetric, are fundamental to a wide range of nonequilibrium phenomena, driving complex dynamics such as traveling waves, spontaneous oscillations, and chiral flows. However, the specific role of nonreciprocity in governing structural transitions and the emergence of stable configurations in living and active systems remains elusive. Furthermore, the extent to which nonreciprocity dictates information transfer and state evolution in biological systems is unclear. This study addresses the gap in understanding how nonreciprocity orchestrates collective self-organization and structural transitions in a biologically tunable system.

Methodology
The authors employ a multi-faceted approach combining biological experiments, data-driven interaction inference, and novel topological analysis frameworks.

Experimental System: The study utilizes Patiria miniata starfish embryos at two distinct developmental stages: E1 (24 hours post-fertilization) and E2 (48 hours post-fertilization). These embryos interact via asymmetric, fluid-mediated forces at an air-water interface. The developmental age gap creates a tunable nonreciprocal interaction where E1 embryos actively pursue E2 embryos, while E2 embryos move away, establishing a "run-and-chase" dynamic.
Interaction Inference: Using Bayesian inference on many-body experimental trajectories, the authors reconstruct the most probable longitudinal and transverse forces between embryo pairs as functions of separation and type. This allows for the quantification of nonreciprocity and its evolution over time.
Inference-Based Modeling: A pairwise interaction model is constructed using the inferred forces. A nonreciprocity parameter $N$ is introduced to systematically tune the asymmetry of inter-type longitudinal interactions ( $N=0$ for reciprocal, $N=1$ for experimentally inferred values, and $N>1$ for stronger nonreciprocity).
Topological Metrics and Landscapes:
- Topological Order Parameter: A metric $\langle d_{hex} \rangle$ is defined based on topological packing statistics. It quantifies the number of local T1 topological transitions (neighbor exchanges) required to transform a given local motif into a perfect hexagonal crystal.
- Topological Landscapes: A low-dimensional manifold is constructed using classical Multidimensional Scaling (MDS) on the pairwise topological distances between motifs. Motif frequencies are mapped onto this manifold to visualize the distribution of local structural states.
- Topological Flowscapes: A dynamic framework is introduced where the temporal evolution of topological landscapes is represented as trajectories in a reduced space. Coordinates are defined by Kullback-Leibler (KL) divergences from reference distributions (representing specific structural motifs), quantifying the informational "cost" of describing the system state.

Key Contributions and Results

State Transitions in Nonreciprocal Mixtures: The experiments reveal a transition from a "traveling state" (characterized by collective translation and flock-like dynamics) to a "fluctuating state" (characterized by a crystal-like structure with reduced collective motion). This transition is driven by the time-dependent decrease in nonreciprocity as the embryos develop.
Nonreciprocity-Dependent State Diagram: The inference-based model maps a rich state space as a function of nonreciprocity $N$ $N$ :
- $N=0$ (Reciprocal): A rotating crystalline state.
- $0 < N < 1$ (Weak Nonreciprocity): A self-propelled crystalline state. Interestingly, weak nonreciprocity enhances structural order, a phenomenon termed "nonreciprocal error correction," where defects are annealed into stable crystalline motifs.
- $N \approx 1$ (Experimental): A flock-like state with dynamic merging and fragmentation of clusters.
- $N > 1$ (Strong Nonreciprocity): A fragmented state with independently moving clusters.
Symmetry-Breaking Transition: Topological landscapes reveal that the transition near $N=1$ involves a shift from single-peaked to multi-peaked distributions of motifs. The system exhibits a first-order-like transition with a coexistence of a perfect crystal motif (M1) and a motif with a five-fold defect (M2). This transition is accompanied by nonreciprocal hysteresis, where the structural state depends on the history of the nonreciprocity parameter.
Information Flow and Proofreading: Topological flowscapes quantify the trajectory of state transitions. The analysis reveals:
- A distinct rate shift in information transfer coinciding with the macroscopic transition from traveling to fluctuating states.
- A correlation between entropy production rate (dissipation) and the speed of information flow.
- An emergent "proofreading" mechanism: in the high-dissipation traveling state, the system exhibits a directional bias toward low-defect configurations (M1), effectively suppressing structural defects. This bias diminishes as dissipation decreases in the fluctuating state.

Significance
The paper establishes a general, physics-informed approach for decoding nonequilibrium transitions in active and living matter. By introducing topological landscapes and flowscapes, the authors provide a versatile framework that connects microscopic interactions to macroscopic organization, structure, dynamics, and information flow. The work demonstrates that self-organization in nonequilibrium systems depends not only on the states they occupy but also on the directed flows between these states. The findings uncover how asymmetric interactions sculpt the dynamical and structural architecture of active matter, offering a new perspective on how biological systems utilize nonreciprocity to achieve order and perform functions like error correction and proofreading.

More like this