Topological flowscape reveals state transitions in nonreciprocal living matter

TL;DR

A transition between nonreciprocal states using starfish embryos at different stages of development is demonstrated, revealing that weak nonreciprocity promotes structural order while stronger asymmetry disrupts it.

Abstract

Nonreciprocal interactions-- where forces between entities are asymmetric-- govern a wide range of nonequilibrium phenomena, yet their role in structural transitions in living and active systems remains elusive. Here, we demonstrate a transition between nonreciprocal states using starfish embryos at different stages of development, where interactions are inherently asymmetric and tunable. Experiments, interaction inference, and topological analysis yield a nonreciprocal state diagram spanning crystalline, flocking, and fragmented states, revealing that weak nonreciprocity promotes structural order while stronger asymmetry disrupts it. To capture these transitions, we introduce topological landscapes, mapping the distribution of structural motifs across state space. We further develop topological flowscapes, a dynamic framework that quantifies transitions between collective states and detects an informational rate shift from the experimental state transition. Together, these results establish a general approach for decoding nonequilibrium transitions and uncover how asymmetric interactions sculpt the dynamical and structural architecture of active and living matter.

Figures (4)

• Figure 1: A nonreciprocal mixture of starfish embryos transitions from a traveling state to a fluctuating state. (a) Nonreciprocity arises from a developmental gap between embryos. An E1 embryo (24 hours post-fertilization) and an E2 embryo (48 hours post-fertilization) form a bound pair at the air-water interface. Scale bar: $\mathrm{100\ \mu m}$. (b) The velocity of the embryo pair, $\mathbf{v}_{\mathrm{pair}}$, tends to align with the displacement vector $\mathbf{r}_{12}$ from E1 to E2. The average drift speed toward E2, $\mathbf{v}_{\mathrm{pair}} \cdot \hat{\mathbf{r}}_{12}$, is $\mathrm{12 \pm 3\ \mu m/s}$ (SEM, n = 14). Scale bar: $\mathrm{100\ \mu m}$. (c) Snapshot of an E1--E2 mixture at 1 hour, overlaid with 2-minute trajectories (blue: E1, red: E2). Scale bar: $\mathrm{1\ mm}$. (d) Time series of velocity polarization, $\mathbf{P}(t) \equiv \langle \hat{\mathbf{v}}_i(t) \rangle_i$, reveals a transition from a traveling state to a fluctuating state. Oscillations and a phase shift between $\mathbf{P}_x$ and $\mathbf{P}_y$ highlight the chirality of the velocity polarization, which rotates clockwise during the traveling state. (e) Snapshot of the same mixture at 6 hours, overlaid with 2-minute trajectories (blue: E1, red: E2). Scale bar: $\mathrm{1\ mm}$.
• Figure 2: Topological order parameter distinguishes emergent states in an inference-based model of nonreciprocal mixtures. (a) In the inference-based model, we use experimentally inferred pairwise interactions $f_{L/T}^{ij}$, except for longitudinal inter-type interactions whose nonreciprocity is scaled by $\mathcal{N}$. The model exhibits four distinct states as the nonreciprocity $\mathcal{N}$ increases: crystalline, self-propelled crystalline, flocking, and fragmented states. (b) Velocity polarization magnitude $|\mathbf{P}|$ as a function of $\mathcal{N}$, averaged over 20 initial configurations (error bars: SEM). $|\mathbf{P}|$ decreases for $\mathcal{N}>1$ . (c) Left: The structural metric $d_{\mathrm{hex}}$ quantifies the number of local topological (T1) transitions relative to a perfect hexagonal crystal; red edges highlight topology-changing transitions. Right: The structural order parameter $\langle d_{\mathrm{hex}} \rangle$ reaches a minimum at $\mathcal{N} = 0.5$ and rises sharply at $\mathcal{N} = 1$ (error bars: SEM, n=20). (d) State diagram of nonreciprocal mixtures, summarizing emergent states across the pairwise nonreciprocity $\mathcal{N}$.
• Figure 3: Topological landscape reveals a symmetry-breaking transition in self-organized structures as nonreciprocity increases. (a) We construct a low-dimensional manifold of topological motifs based on their pairwise topological distances. The resulting topological landscape visualizes the probability distribution of local topologies (from each level of nonreciprocity $\mathcal{N}$) on this manifold. (b) Contour plots of topological landscapes display a transition from single-peaked to multi-peaked distributions as $\mathcal{N}$ increases. (c) At $\mathcal{N} = 1$ (experimentally inferred nonreciprocity), 14 topological motifs exhibit a frequency greater than 1%. In particular, the top two dominant motifs correspond to the two distinct peaks of the topological landscape. Top: Motifs from $\mathrm{M_1}$ to $\mathrm{M_7}$. $\mathrm{M_1}$ is a perfect crystal, while $\mathrm{M_2}$ contains a 5-defect. Bottom: Motifs from $\mathrm{M_8}$ to $\mathrm{M_{14}}$. See Supplementary Section V.C for an enlarged motif atlas with full annotations. (d) Negative log-likelihood ratio relative to $\mathrm{M_1}$, measured along the $\mathrm{M_1}$--$\mathrm{M_2}$ axis. At $\mathcal{N} = 0$, the system shows a single minimum at $\mathrm{M_1}$. Near $\mathcal{N} = 1$, the structures undergo a first-order-like transition from $\mathrm{M_1}$-dominated to $\mathrm{M_2}$-dominated states, with coexistence at $\mathcal{N} = 1$.
• Figure 4: Topological flowscape characterizes the time evolution of topological landscapes in the experiment. (a) Contour plots of experimental topological landscapes reveal a shifting balance between two dominant peaks over time. These peaks correspond to motifs $\mathrm{M_1}$ and $\mathrm{M_2}$, previously identified in the model at the experimentally inferred nonreciprocity $\mathcal{N} = 1$. (b) Negative log-likelihood relative to $\mathrm{M_1}$, measured along the $\mathrm{M_1}$--$\mathrm{M_2}$ axis. The self-organized structures transition from an $\mathrm{M_2}$-dominated to an $\mathrm{M_1}$-dominated state as the embryo mixture shifts from a traveling to a fluctuating state (Fig. \ref{['fig:fig1']}d). (c) Topological flowscape of the experiment. Each landscape is mapped to a point using KL divergence from two synthetic reference distributions centered at $\mathrm{M_1}$ and $\mathrm{M_2}$, with widths $\sigma_M$ matched to the kernel of experimental landscape (Supplementary Section VI.B). The trajectory flows from a region near $\mathcal{N} = 1$ toward one resembling $\mathcal{N} = 0$. (d) Flowscape constructed using experimental states at different times as dynamic references. Using 0.5 h and 6.5 h as references captures a long-timescale transition. Inset: Using 2 h and 2.5 h as references reveals short-timescale cyclic transitions associated with cluster merging and breaking (Supplementary Section VI.E). (e) Diagonal displacement in the flowscape quantifies the log-likelihood difference between reference states. The progress is rapid during the traveling state and slows significantly in the fluctuating state. (f) Entropy production rate (EPR), estimated from changes in topological motif frequencies using thermodynamic speed limits, exhibits a similar transition as in (e), with higher rates during the traveling state and reduced rates in the fluctuating state.