Abstract
Despite their inherently nonequilibrium nature^{1}, living systems can selforganize in highly ordered collective states^{2,3} that share striking similarities with the thermodynamic equilibrium phases^{4,5} of conventional condensedmatter and fluid systems. Examples range from the liquidcrystallike arrangements of bacterial colonies^{6,7}, microbial suspensions^{8,9} and tissues^{10} to the coherent macroscale dynamics in schools of fish^{11} and flocks of birds^{12}. Yet, the generic mathematical principles that govern the emergence of structure in such artificial^{13} and biological^{6,7,8,9,14} systems are elusive. It is not clear when, or even whether, wellestablished theoretical concepts describing universal thermostatistics of equilibrium systems can capture and classify ordered states of living matter. Here, we connect these two previously disparate regimes: through microfluidic experiments and mathematical modelling, we demonstrate that lattices of hydrodynamically coupled bacterial vortices can spontaneously organize into distinct patterns characterized by ferro and antiferromagnetic order. The coupling between adjacent vortices can be controlled by tuning the intercavity gap widths. The emergence of opposing order regimes is tightly linked to the existence of geometryinduced edge currents^{15,16}, reminiscent of those in quantum systems^{17,18,19}. Our experimental observations can be rationalized in terms of a generic lattice field theory, suggesting that bacterial spin networks belong to the same universality class as a wide range of equilibrium systems.
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Main
Lattice field theories (LFTs) have been instrumental in uncovering a wide range of fundamental physical phenomena, from quark confinement in atomic nuclei^{20} and neutron stars^{21} to topologically protected states of matter^{22} and transport in novel magnetic^{23} and electronic^{24,25} materials. LFTs can be constructed either by discretizing the spacetime continuum underlying classical and quantum field theories^{20}, or by approximating discrete physical quantities, such as the electron spins in a crystal lattice, through continuous variables. In equilibrium thermodynamics, LFT approaches have proved invaluable both computationally and analytically, for a single LFT often represents a broad class of microscopically distinct physical systems that exhibit the same universal scaling behaviours in the vicinity of a phase transition^{4,26}. However, until now there has been little evidence as to whether the emergence of order in living matter can be understood within this universality framework. Our combined experimental and theoretical analysis reveals a number of striking analogies between the collective cell dynamics in bacterial fluids and known phases of condensedmatter systems, thereby implying that universality concepts may be more broadly applicable than previously thought.
To realize a microbial nonequilibrium LFT, we injected dense suspensions of the rodlike swimming bacterium Bacillus subtilis into shallow polydimethyl siloxane (PDMS) chambers in which identical circular cavities are connected to form one and twodimensional (2D) lattice networks (Fig. 1, Supplementary Fig. 6 and Methods). Each cavity is 50 μm in diameter and 18 μm deep, a geometry known to induce a stably circulating vortex when a dense bacterial suspension is confined within an isolated flattened droplet^{15}. For each cavity i, we define the continuous vortex spin variable V_{i}(t) at time t as the total angular momentum of the local bacterial flow within this cavity, determined by particle imaging velocimetry (PIV) analysis (Fig. 1b, f; Supplementary Movies 1 and 2 and Methods). To account for the effect of oxygenation variability on suspension motility^{9}, flow velocities are normalized by the overall rootmeansquare (r.m.s.) speed measured in the corresponding experiment. Bacterial vortices in neighbouring cavities interact through a gap of predetermined width w (Fig. 1f). To explore different interaction strengths, we performed experiments over a range of gap parameters w (Methods). For square lattices, we varied w from 4 to 25 μm and found that for all but the largest gaps, w ≤ w_{∗} ≈ 20 μm, the suspensions generally selforganize into coherent vortex lattices, exhibiting domains of correlated spins whose characteristics depend on coupling strength (Fig. 1a, e). If the gap size exceeds w_{∗}, bacteria can move freely between cavities and individual vortices cease to exist. Here, we focus exclusively on the vortex regime w < w_{∗} and quantify preferred magnetic order through the normalized mean spin–spin correlation χ = 〈Σ_{i∼j}V_{i}(t)V_{j}(t)/Σ_{i∼j}V_{i}(t)V_{j}(t)〉, where Σ_{i∼j} denotes a sum over pairs {i, j} of adjacent cavities and 〈⋅〉 denotes time average.
Square lattices reveal two distinct states of preferred magnetic order (Fig. 1a, e, i), one with χ < 0 and the other with χ > 0, transitioning between them at a critical gap width w_{crit} ≈ 8 μm (Fig. 1j). For subcritical values w < w_{crit}, we observe an antiferromagnetic phase with anticorrelated (χ < 0) spin orientations between neighbouring chambers on average (Fig. 1a and Supplementary Movie 1). By contrast, for w > w_{crit}, spins are positively correlated (χ > 0) in a ferromagnetic phase (Fig. 1e and Supplementary Movie 2). Noting that the r.m.s. spin 〈V_{i}(t)^{2}〉^{1/2} decays only slowly with increasing gap width w → w_{∗} (Fig. 1k), and that the chambers do not impose any preferred handedness on the vortex spins (Supplementary Fig. 1 and Supplementary Section 1), we conclude that the observed phase behaviour is caused by spin–spin interactions. However, although both phases possess a welldefined average vortex–vortex correlation, the individual spins fluctuate randomly over time as ordered domains split, merge and flip (Fig. 1i and Supplementary Figs 1 and 3) while the system explores configuration space inside a statistical steady state (Supplementary Sections 1 and 3). Thus, although the bacterial vortex spins {V_{i}(t)} define a realvalued lattice field, the phenomenology of these continuous bacterial spin lattices is qualitatively similar to that of the classical 2D Ising model^{4} with discrete binary spin variables s_{i} ∈ {±1}, whose configurational probability at finite temperature T = (k_{B}β)^{−1} is described by a thermal Boltzmann distribution ∝exp(− βJΣ_{i∼j}s_{i}s_{j}), where J > 0 corresponds to ferromagnetic and J < 0 to antiferromagnetic order. The detailed theoretical analysis below shows that the observed phases in the bacterial spin system can be understood quantitatively in terms of a generic quartic LFT comprising two dual interacting lattices. The introduction of a double lattice is necessitated by the microscopic structure of the underlying bacterial flows. By analogy with a lattice of interlocking cogs, one might have intuitively expected that the antiferromagnetic phase would generally be favoured, because only in this configuration does the bacterial flow along the cavity boundaries conform across the intercavity gap, avoiding the potentially destabilizing headtohead collisions that would occur with opposing flows (Fig. 1b, c). However, the extent of the observed ferromagnetic phase highlights a competing biofluidmechanical effect.
Just as the quantum Hall effect^{17} and the transport properties of graphene^{18,19} arise from electric edge currents, the opposing order regimes observed here are explained by the existence of analogous bacterial edge currents. At the boundary of an isolated flattened droplet of a bacterial suspension, a single layer of cells—an edge current—can be observed swimming against the bulk circulation^{15,16}. This narrow cell layer is key to the suspension dynamics: the hydrodynamics of the edge current circulating in one direction advects nearby cells in the opposite direction, which in turn dictate the bulk circulation by flow continuity through steric and hydrodynamic interactions^{16,27}. Identical edge currents are present in our lattices (Supplementary Movie 3) and explain both order regimes as follows. In the antiferromagnetic regime, when w < w_{crit}, the bacterial edge current driving a particular vortex will pass over the gap without leaving the cavity (Fig. 1c). Interaction with a neighbouring edge current through the gap favours parallel flow, inducing countercirculation of neighbouring vortices and therefore driving antiferromagnetic order (Fig. 1d). However, when w > w_{crit}, the edge currents can no longer pass over the gaps and instead wind around the starshaped pillars dividing the cavities (Fig. 1g). A clockwise (resp. anticlockwise) bacterial edge current about a pillar induces anticlockwise (resp. clockwise) fluid circulation about the pillar in a thin region near its boundary. Flow continuity then induces clockwise (resp. anticlockwise) flow in all cavities adjacent to the pillar, resulting in ferromagnetic order (Fig. 1h). Thus by viewing the system as an anticooperative Union Jack lattice^{28,29} of both bulk vortex spins V_{i} and nearpillar circulations P_{j}, we accommodate both order regimes: antiferromagnetism as indefinite circulations P_{j} = 0 and alternating spins V_{i} = ±V (Fig. 1d), and ferromagnetism as definite circulations P_{j} = −P < 0 and uniform spins V_{i} = V > 0 (Fig. 1h). To verify these considerations, we determined the net nearpillar circulation P_{j}(t) using PIV (Methods) and found that the r.m.s. circulation 〈P_{j}(t)^{2}〉^{1/2} shows the expected monotonic increase as the intercavity gap widens (Fig. 1k).
Competition between the vortex–vortex and vortex–pillar interactions determines the resultant order regime. Their relative strengths can be inferred by mapping each experiment onto a continuousspin Union Jack lattice (Fig. 1d, h). In this model, the interaction energy of the timedependent vortex spins V = {V_{i}} and pillar circulations P = {P_{j}} is defined by the LFT Hamiltonian
The first two sums are vortex–vortex and vortex–pillar interactions with strengths J_{v}, J_{p} < 0, where ∼ denotes adjacent lattice pairs. The last two sums are individual vortex and pillar circulation potentials. Vortices must be subject to a quartic potential function with b_{v} > 0 to allow for a potentially doublewelled potential if a_{v} < 0, encoding the observed symmetry breaking into spontaneous circulation in the absence of other interactions^{15,27}. In contrast, our data analysis implies that pillar circulations are sufficiently described by a quadratic potential of strength a_{p} > 0 (Supplementary Fig. 4 and Supplementary Section 4). To account for the experimentally observed spin fluctuations (Fig. 1i and Supplementary Fig. 1), we model the dynamics of the lattice fields V and P through the coupled stochastic differential equations (SDEs)
where W_{v} and W_{p} are vectors of uncorrelated Wiener processes representing intrinsic and thermal fluctuations. The overdamped dynamics in equations (2) and (3) neglects dissipative Onsagertype crosscouplings, as the dominant contribution to friction stems from the nearby noslip PDMS boundaries (Supplementary Section 7). The parameters T_{v} and T_{p} set the strength of random perturbations from energyminimizing behaviour. In the equilibrium limit when T_{v} = T_{p} = T, the stationary statistics of the solutions of equations (2) and (3) obey the Boltzmann distribution ∝e^{−H/T}. We inferred all seven parameters of the full SDE model for each experiment by linear regression on a discretization of the SDEs (Supplementary Fig. 2 and Supplementary Section 2). The differing sublattice temperatures T_{v} ≠ T_{p} found show that the system is not in thermodynamic equilibrium owing to its active microscopic constituents (Supplementary Fig. 2). Instead, the system is in a pseudoequilibrium statistical steady state (Supplementary Section 1), which we will soon show can be reduced to an equilibriumlike description. As a crossvalidation, we fitted appropriate functions of gap width w to these estimates and simulated the resulting SDE model over a range of w on a 6 × 6 lattice concordant with the observations (Supplementary Section 3). The agreement between experimental data and the numerically obtained vortex–vortex correlation χ(w) supports the validity of the doublelattice model and its underlying approximations (Fig. 1j).
To reconnect with the classical 2D Ising model and understand better the experimentally observed phase transition, we project the Hamiltonian (1) onto an effective square lattice model by making a meanfield assumption for the pillar circulations. In the experiments, P_{i} is linearly correlated with the average spin of its vortex neighbours [P_{i}]_{V} = (1/4){\Sigma}_{j:{V}_{j}\sim {P}_{i}}V_{j}, with a constant of proportionality −α < 0 only weakly dependent on gap width (Supplementary Fig. 4 and Supplementary Section 4). Replacing effectively P_{i} → − α[P_{i}]_{V} as a meanfield variable in the model eliminates all pillar circulations, yielding a standard quartic LFT for V (see Supplementary Section 4 for a detailed derivation). The meanfield dynamics are then governed by the reduced SDE with effective temperature T ≈ T_{v} + 4T_{p}J_{p}^{2}/a_{p}^{2} and energy
which has steadystate probability density with β = 1/T, and where a = a_{v} − 4J_{p}^{2}/a_{p} and b = b_{v}. Note that in the limit a → −∞ and b → +∞ with a/b fixed, the classical twostate Ising model is recovered by identifying . The reduced coupling constant J relates to those of the doublelattice model (J_{v}, J_{p}) in the thermodynamic limit as J ≈ J_{v} − (1/2)αJ_{p} (Supplementary Section 4), making manifest how competition between J_{v} and J_{p} can result in both antiferromagnetic (J_{v} > (1/2)α J_{p}) or ferromagnetic (J_{v} < (1/2)α J_{p}) behaviour. We estimated βJ, βa and βb for each experiment by directly fitting the effective onespin potential via the loglikelihood (Fig. 2, Supplementary Fig. 5 and Supplementary Section 5). These estimates match those obtained independently using SDE regression methods (Fig. 2a–c and Supplementary Section 5), and show the transition from antiferromagnetic interaction (βJ < 0) to ferromagnetic interaction (βJ > 0) at w_{crit} (Fig. 2a). As the gap width increases, the energy barrier to spin change falls (Fig. 2b) and the magnitude of the lowest energy spin decreases (Fig. 2c) as a result of weakening confinement within each cavity, visible as a flattening of the onespin effective potential (Fig. 2d–f and Supplementary Fig. 5).
Experiments on lattices of different symmetry groups lend further insight into the competition between edge currents and bulk flow. Unlike their square counterparts, triangular lattices cannot support antiferromagnetic states without frustration. Therefore, ferromagnetic order should be enhanced in a triangular bacterial spin lattice. This is indeed observed in our experiments: at moderate gap size w ≲ 18 μm, we found exclusively a highly robust ferromagnetic phase of either handedness (Fig. 3a, b, d and Supplementary Movie 4), reminiscent of quantum vortex lattices in Bose–Einstein condensates^{30}. At comparable gap size, the spin correlation is approximately four to eight times larger than in the square lattice. Increasing the gap size beyond 20 μm eventually destroys the spontaneous circulation within the cavities and a disordered state prevails (Fig. 3c, d), with a sharper transition than for the square lattices (Fig. 1j). Conversely, a 1D line lattice exclusively exhibits antiferromagnetic order as the suspension is unable to maintain the very long uniform edge currents that would be necessary to sustain a ferromagnetic state (Supplementary Fig. 6 and Supplementary Section 6). These results manifest the importance of lattice geometry and dimensionality for vortex ordering in bacterial spin lattices, in close analogy with their electromagnetic counterparts.
Understanding the ordering principles of microbial matter is a key challenge in active materials design^{13}, quantitative biology and biomedical research. Improved prevention strategies for pathogenic biofilm formation, for example, will require detailed knowledge of how bacterial flows interact with complex porous surface structures to create the stagnation points at which biofilms can nucleate. Our study shows that collective excitations in geometrically confined bacterial suspensions can spontaneously organize in phases of magnetic order that can be robustly controlled by edge currents. These results demonstrate fundamental similarities with a broad class of widely studied quantum systems^{17,19,30}, suggesting that theoretical concepts originally developed to describe magnetism in disordered media could potentially capture microbial behaviours in complex environments. Future studies may try to explore further the range and limits of this promising analogy.
Methods
Experiments.
Wildtype Bacillus subtilis cells (strain 168) were grown in Terrific Broth (Sigma). A monoclonal colony was transferred from an agar plate to 25 ml of medium and left to grow overnight at 35 °C on a shaker. The culture was diluted 200fold into fresh medium and harvested after approximately 5 h, when more than 90% of the bacteria were swimming, as visually verified on a microscope. 10 ml of the suspension was then concentrated by centrifugation at 1,500g for 10 min, resulting in a pellet with volume fraction approximately 20% which was used without further dilution.
The microchambers were made of polydimethyl siloxane (PDMS) bound to a glass coverslip by oxygen plasma etching. These comprised a square, triangular or linear lattice of ∼18μmdeep circular cavities with 60 μm between centres, each of diameter ∼50 μm, connected by 4–25μmwide gaps for linear and square lattices (Fig. 1a, e and Supplementary Fig. 6) and 10–25μmwide gaps for triangular lattices (Fig. 3a–c). The smallest possible gap size was limited by the fidelity of the etching.
Approximately 5 μl of the concentrated suspension was manually injected into the chamber using a syringe. Both inlets were then sealed to prevent external flow. We imaged the suspension on an inverted microscope (Zeiss, Axio Observer Z1) under brightfield illumination, through a 40× oilimmersion objective. Movies 10 s in length were recorded at 60 f.p.s. on a highspeed camera (Photron Fastcam SA3) at 4 and 8 min after injection. Although the PDMS lattices were typically ∼15 cavities across, to avoid boundary effects and to attain the pixel density necessary for PIV we imaged a central subregion spanning 6 × 6 cavities for square lattices, 7 × 6 cavities for triangular lattices, and 7 cavities for linear lattices (multiple of which were captured on a single slide).
Fluorescence in Supplementary Movie 3 was achieved by labelling the membranes of a cell subpopulation with fluorophore FM464 following the protocol of Lushi et al. ^{16} The suspension was injected into an identical triangular lattice as in the primary experiments and imaged at 5.6 f.p.s. on a spinningdisc confocal microscope through a 63× oilimmersion objective.
Analysis.
For each frame of each movie, the bacterial suspension flow field u(x, y, t) was measured by standard particle image velocimetry (PIV) without time averaging, using a customized version of mPIV (http://www.oceanwave.jp/softwares/mpiv). PIV subwindows were 16 × 16 pixels with 50% overlap, yielding ∼150 vectors per cavity per frame. Cavity regions were identified in each movie by manually placing the centre and radius of the bottom left cavity, measuring vectors to its immediate neighbours, and repeatedly translating to generate the full grid. Pillar edges were then calculated from the cavity grid and the gap width (measured as the minimum distance between adjacent pillars).
The spin V_{i}(t) of each cavity i at time t is defined as the normalized planar angular momentum
where r_{i}(x, y) is the vector from the cavity centre to (x, y), and sums run over all PIV grid points (x, y)_{i} inside cavity i. For each movie, we normalize velocities by the rootmeansquare (r.m.s.) suspension velocity , where the average is over all grid points (x, y) and all times t, to account for the effects of variable oxygenation on motility^{9}; we found an ensemble average μm s^{−1} with s.d. 3.6 μm s^{−1} over all experiments. This definition has V_{i}(t) > 0 for anticlockwise spin and V_{i}(t) < 0 for clockwise spin. A vortex of radially independent speed—that is, , where is the azimuthal unit vector—has V_{i}(t) = ±1; conversely, randomly oriented flow has V_{i}(t) = 0. The average spin–spin correlation χ of a movie is then defined as
where Σ_{i∼j} denotes a sum over pairs {i, j} of adjacent cavities and 〈⋅〉 denotes an average over all frames. If all vortices share the same sign, then χ = 1 (ferromagnetism); if each vortex is of opposite sign to its neighbours, then χ = −1 (antiferromagnetism); if the vortices are uniformly random, then χ = 0. Similarly, the circulation P_{j}(t) about pillar j at time t is defined as the normalized average tangential velocity
where is the unit vector tangential to the pillar, and sums run over PIV grid points (x, y)_{j} closer than 5 μm to the pillar j.
Results presented are typically averaged in bins of fixed gap width. All plots with error bars use 3 μm bins, calculated every 1.5 μm (50% overlap), and bins with fewer than five movies were excluded. Error bars denote standard error. Bin counts for square lattices (Figs 1j, k and 2a–c and Supplementary Figs 4 and 7) are 8, 8, 13, 14, 21, 27, 27, 22, 18, 22, 20, 11, 7, 13, 7; bin counts for triangular lattices (Fig. 3d) are 5, 14, 16, 13, 16, 15, 5, 5, 10, 7; and bin counts for linear lattices (Supplementary Fig. 6) are 5, 7, 8, 8, 9, 9, 6, 5, 6, 7, 6, 8, 9, 5, 6, 5, 6.
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Acknowledgements
We thank V. Kantsler and E. Lushi for assistance and discussions. This work was supported by European Research Council Advanced Investigator Grant 247333 (H.W. and R.E.G.), EPSRC (H.W. and R.E.G.), an MIT Solomon Buchsbaum Fund Award (J.D.) and an Alfred P. Sloan Research Fellowship (J.D.).
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All authors designed the research and collaborated on theory. H.W. performed experiments and PIV analysis. H.W. and F.G.W. analysed PIV data and performed parameter inference. F.G.W. and J.D. wrote simulation code. All authors wrote the paper.
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Wioland, H., Woodhouse, F., Dunkel, J. et al. Ferromagnetic and antiferromagnetic order in bacterial vortex lattices. Nature Phys 12, 341–345 (2016). https://doi.org/10.1038/nphys3607
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DOI: https://doi.org/10.1038/nphys3607
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