Internal stress drives ferromagnetic-like ordering in networks of proliferating bacteria

Abstract

Proliferation is a defining feature of life. Through growth, division, and death, living systems consume energy and inject mass, breaking conservation laws and driving collective phenomena from biofilm formation to embryonic development. Yet, while active matter physics has advanced our understanding of self-propelled agents, quantitative frameworks for proliferating systems are still emerging, and most work focuses on simplified settings. Here, we study \textit{E.coli} bacteria growing inside a network of single-file microchannels as a minimal model of structured environments. Competition for free volume drives the spontaneous emergence of coherent growth patterns that persist across generations but vanish when the channel links exceed the typical cell size at birth. Despite the strongly out-of-equilibrium character of the dynamics, the observed phenomenology can be quantitatively captured by an effective equilibrium description in which the flow state at each node is represented by a spin variable with ferromagnetic interactions. Simulations of growing elastic cells show that this coupling arises from internal stress accumulated at network nodes due to dynamical constraints. Our results reveal a surprising correspondence between proliferating active matter and equilibrium statistical mechanics, highlighting open fundamental questions and offering a first step toward describing growth phenomena in real-world complex environments.

Figures (4)

• Figure 1: Transition from ordered growth in networks of microchannels. (A) Bright-field image of the SU-8 microchambers containing the square network features. (B) Fluorescence images of GFP-expressing bacteria growing in microchambers of different sizes. Scalebar 5$\mu$m.(C) Timelapse images showing ordered bacterial growth within a network of size L=3 $\mu$m, with arrows indicating the output flux of bacteria. The absence of an arrow indicates that the node is temporarily unoccupied.
• Figure 2: Bacterial flux dynamics in microchannel networks: experiments and model.(A) Left: Possible flows in a corner node. Right: Schematic of the spin model applied to the output flow of bacteria in the network. (B) Mean magnetization over time in networks of different sizes (single experiment). Black markers indicate the measured values, while the gray line represents an interpolation applied when no bacteria are passing through the network node. (C) Probability distribution of the mean magnetization from experimental data (orange) and from the model (black lines). The cyan dotted line represent the binomial distribution for four statistically independent spin variables. Data were collected from more than ten independent experiments for each network size $L$. (D) Coupling parameter $K$ derived from the model as a function of the network length $L$. The dashed line is an exponential fit to guide the eye. (E) Time correlation function of the mean magnetization for the four network sizes, represented by round markers. The shaded area shows the standard error of the mean. Each data point is averaged over more than ten independent experiments. Dashed lines represent exponential fits, while solid lines show Monte Carlo dynamics extracted using the corresponding $K$ values from the statics. The inset highlights the exponential growth of the characteristic time $\tau_c$ with the parameter $K$.
• Figure 3: Energy associated with spin configurations. The table summarizes the interaction energy associated with each configuration of adjacent spins. Illustrative examples of network configurations are shown in the bottom panel. The value shown above each schematic denotes the associated total energy, while the value below corresponds to the mean magnetization $M$. For symmetry reasons, we show only representative configurations with non-negative magnetization, as configurations with reversed spins are energetically equivalent.
• Figure 4: Bacterial flow and spin-model analysis in simulations.(A) Schematic of the simulation setup. Top panel: Each bacterium is modeled as two spheres of diameter $a$ separated by a distance $\ell$ and connected by a spring with rest length $\ell_0$. Bottom panel: Snapshot of a simulation configuration with two zooms highlighting elastic interactions between neighboring bacteria. (B) Probability distribution of the mean magnetization for different network lengths normalized to the mean length of bacteria at birth $L/\langle \ell_b \rangle$. Orange indicates data from simulations, while the black line represents the probability distribution extracted from the model. (C) Coupling parameter $K$ extracted from simulations using the average energy of the three states (blue triangles; $\beta = 4$ in simulation units) and from the square of the mean magnetization (orange squares). Data show excellent agreement between the two methods and with experimental data when the length is normalized. (D) Median energy of the three states (blocked, partially blocked, and free) as a function of normalized network size. (E) Time correlation function of the mean magnetization for the four network sizes, represented by round markers. Dashed lines represent exponential fits, while solid lines show Monte Carlo dynamics extracted using the corresponding $K$ values from the statics. The inset highlights the exponential growth of the characteristic time $\tau_c$ with the parameter $K$, as obtained from simulations and experiments. Both timescales are normalized by the doubling time $\tau_d$. For panels (B,C,D,E) data points referring to simulations are averaged over more than 50 independent simulations. Standard error bars are smaller than the marker size.