Designing topological edge states in bacterial active matter

TL;DR

The paper demonstrates geometry-driven topological edge states in a prototypical active matter system by engineering directional kagome networks of ratchet channels in dense bacterial suspensions. Through measured velocity fields and a lattice transport model, the authors show edge-localized bacterial density arising from rectified flows, and they establish a topological origin via a non-Hermitian band structure with Chern numbers $N^{\text{top}}_{Ch}=-1$, $N^{\text{mid}}_{Ch}=0$, and $N^{\text{bottom}}_{Ch}=1$, along with gap-crossing edge modes in half-periodic geometries. A reduced nonlinear model and direct simulations reveal edge transport consistent with these edge modes, and network chirality is identified as a key design parameter for enabling edge localization. Overall, the work introduces a geometry-based route to robust topological transport in active matter, offering a principled design framework for microfluidic and bioengineering applications that leverage environmental geometry rather than particle-level chirality.

Abstract

Topology provides a unifying framework for understanding robust transport through protected edge states arising from nontrivial wavenumber topology. Extending these concepts to active matter, however, remains largely unexplored experimentally, with realizations limited to systems composed of chiral active particles. Here, we realize topological edge states in dense bacterial suspension, which represents a prototypical active matter system, using microfabricated geometrical structures with nontrivial wavenumber topology. Inspired by previous theoretical studies, we constructed a directional kagome network composed of ratchet-shaped channels that induce unidirectional bacterial flow. In this network, we found clear edge localization of bacterial density. A steady-state analysis based on the bacterial transport model and experimentally measured velocity field reveals how the characteristic collective flow generates edge localization. The model also uncovers the topological origin of the observed edge states. By tuning the geometry of the microfabricated networks, we identified directional channel design and network chirality as the key design features essential for the emergence of the edge state. Our results pave the way for establishing a control and design principle of topological transport in such active matter systems.

Figures (4)

• Figure 1: Edge localization in a directional kagome network.a, Experimental setup. Bacterial collective motion is observed near the liquid-air interface. b, Schematics of directional ratchet-shaped and non-directional straight channels. Wide-field images of the channels filled with dyed bacterial suspension are also shown. c, d, Confocal images of dyed bacterial suspension confined in triangular structures ( c) and a directional kagome network ( d). In the panel c, time-averaged velocity field obtained by PIV analysis is overlaid. The length of the arrow in the box corresponds to a speed of $20µ m/s$. e, Wide-field image of a large directional kagome network filled with dyed bacterial suspension, captured using epifluorescence microscopy with a tiling scan. The network boundary is indicated by orange lines. f, Bacterial density profiles in the directional and non-directional kagome networks. Background-subtracted, spatially-averaged fluorescence intensity at wells along diagonal lines of the kagome network is plotted (see also Supplementary Fig. 1). The horizontal axis indicates the distance from the edge: $d=1$ corresponds to wells closest to the vertex of the orange hexagon in e, while $d=21$ corresponds to wells in the central triangular motif of the network. The error bars represent the standard error over six data obtained from different diagonal lines.
• Figure 2: Bacterial transport driven by characteristic collective flow.a, Time-averaged velocity field of bacterial collective flow in a small kagome network. Velocity vectors are overlaid on a confocal image of fluorescently labeled, dense bacterial suspension. b, Map of channel flow $U$ calculated from the velocity field in each channel. The length of the arrow in the box corresponds to a speed of $10µ m/s$. c, Comparison of channel flow $U$ in the bulk and along the edge. We focused on two types of channels in the small kagome network: those connecting wells within the triangular motif at the center of the network (blue, labeled bulk) and those connecting wells along the edge (red, labeled edge). The mean channel flow for each type is shown, with error bars indicating the standard error. d, Schematic of the bacterial transport driven by the channel flow. Bacteria enter and leave a well through inlet and outlet channels, respectively. The time evolution of the density at the well labeled “C” in the figure is given by $\partial_t\rho_{\text{C}}=\rho_{\text{A}} U_{\text{A}\to\text{C}}+\rho_{\text{E}} U_{\text{E}\to\text{C}}-\rho_{\text{C}}U_{\text{C}\to\text{B}}-\rho_{\text{C}} U_{\text{C}\to\text{D}}$ [Eq. (\ref{['main:eq1']})]. e, f, Steady-state bacterial density distribution, evaluated from the channel flow pattern (see Methods). In f, the area of each green circle represents the bacterial density at that site. Two types of wells were analyzed: one type at the center of the network (blue, labeled bulk) and the other along the edge (pink, labeled edge). In e, the mean steady-state bacterial density $\rho_{\text{ss}}$ for each type is shown, with error bars indicating the standard error.
• Figure 3: Theoretical characterization of bacterial transport on the directional kagome network.a, Schematic of the directional kagome network. Each unit triangular motif, composed of three sites $\nu=\alpha,\beta,\gamma$, is labeled by $(i,j)$. b, Bulk band structure under the fully periodic boundary condition. For $c>0$, the bands are separated by gaps, and the Chern numbers of the top, middle, and bottom bands are $-1$, $0$, and $1$, respectively. c, Dispersion relation in the half-periodic geometry for $c\rho_0/(2U_0)=10$. d, $y$-coordinate of the center of mass $\langle j\rangle$ for each eigenmode in the half-periodic geometry for $c\rho_0/(2U_0)=10$. Gap-crossing modes have their centers of mass concentrated near the bottom ($y\approx 0$) and top ($y\approx 20$) rows of the directional kagome network. e, Snapshot at time $t=3.0$ from the direct simulation of the nonlinear model in the half-periodic geometry for $c\rho_0/(2U_0)=10$. The color shows the density deviation $\delta\rho_I=\rho_I-\rho_{0,I}$. At $t=0$, the density deviation $\delta\rho_I$ is set to 0.01 at the top-right corner, marked in green, and $\delta\rho_I=0$ at all other sites. The red arrow indicates the direction of the observed edge transport. See also Supplementary Movie 3. f, Total travel distance $D(t)$ of the density peak along the top triangular motifs. The travel distance is defined as the distance from the triangular motif where the initial density deviation was applied (marked in green in e). The orange line shows a linear fit, $D(t)=v_{\text{fit}} t$. The fit is performed over the time interval $t<3$ (shaded in gray), during which the density peak propagates at an approximately constant speed. From this fit, we obtain $v_{\text{fit}}=8.4\pm 1.2$. g, Expansion coefficients $|c^{(m)}_{k^x}|$ at $t=0$ obtained by projecting density deviation in the simulation onto the eigenmodes of the half-periodic Hamiltonian for $c\rho_0/(2U_0)=10$.
• Figure 4: Geometrical origin of edge states.a, Schematics of the chiral and achiral square networks (top), along with corresponding epifluorescence images of microfabricated structures filled with fluorescently labeled bacterial suspension (bottom). In the chiral square network, the local chirality of each square motif is preserved. In contrast, in the achiral square network, for which yellow arrows are added, negative and positive chiralities both exist and cancel each other out between neighboring square motifs. b, Epifluorescence microscopy image of a large-scale chiral square network (top) and an achiral square network (bottom) filled with fluorescently labeled bacteria, acquired by a tiling scan. The chiral (achiral) square network is outlined in orange (blue). c, Bacterial density profiles for chiral and achiral networks. Background-subtracted spatially-averaged brightness is plotted for the wells along the diagonal lines of the networks (green in panel b). Error bars represent the standard error measured over four diagonal lines.