The propagation of active-passive interfaces in bacterial swarms

Propagating interfaces are ubiquitous in nature, underlying instabilities and pattern formation in biology and material science. Physical principles governing interface growth are well understood in passive settings; however, our understanding of interfaces in active systems is still in its infancy. Here, we study the evolution of an active-passive interface using a model active matter system, bacterial swarms. We use ultra-violet light exposure to create compact domains of passive bacteria within Serratia marcescens swarms, thereby creating interfaces separating motile and immotile cells. Post-exposure, the boundary re-shapes and erodes due to self-emergent collective flows. We demonstrate that the active-passive boundary acts as a diffuse interface with mechanical properties set by the flow. Intriguingly, interfacial velocity couples to local swarm speed and interface curvature, raising the possibility that an active analogue to classic Gibbs-Thomson-Stefan conditions may control this boundary propagation.


Supplementary
. Characterization of the bacterial velocity field. (a) A sample snapshot the colony edge overlaid by the bacterial velocity field. The velocity fields are measured using particle image velocimetry (PIV). Experiments are conducted close to the colony edge -in the area inside the white boxwhere collective motion is observed to be strongest. (b) Spatial and (c) temporal autocorrelations of the bacterial velocity field in its initial state (before exposure to high-intensity light). Measurements are made in the region surrounded by the white box in (a). The data shows that the collective flows in the swarm are correlated over a characteristic length ⇡ 20 µm and time ⇡ 0.24 s. Figure 2. Velocity fields from the half-space aperture geometry. (a) An image of the swarm during exposure shows a sub-region of the swarm exposed to a half-plane of high-intensity light. (b) The bacterial velocity field (t = 10 s past exposure) shows that at the top (where the swarm was exposed) the bacteria are immobile while at the bottom bacteria continue swarming. (c) A time sequence of the bacterial vorticity field ! and corresponding streamlines near the interface (overlaid black line). Snapshots start t = 10 s past exposure. The vortices change over time and clockwise and counterclockwise vortices are often observed aligned next to each other along the interface. To compare these phase interface profiles with those from the intensity fluctuations ( Fig. 2 main text), we plot the mean boundary positions (ii) and interface width (iii) from each over time. The notation d corresponds to (red) and d v from v (blue). Likewise, w(t) corresponds to and w v (t) from v . The data is extracted from fits to the phase profiles (i, Eq'n 5 & 6), and results from four independent experiments are combined to obtain the mean, minimum, and maximum values as shown. We find that the interface positions measured from v and are approximately the same whereas estimated widths are larger for v (bacterial speed) than for (intensity fluctuations).  Figure 5. Interface erosion dynamics depends on the direction of the expanding colony front. (a) Schematic of a rectangular aperture geometry that creates two active-passive interfaces, one that is parallel to the colony edge and one that is perpendicular. (b) As shown by the active-passive interface (left) and colony edge (right) positions tracked together, the active-passive interface moves more than the expanding colony edge over the 70 second time interval shown. We measure the speed v of the interfaces at the five labeled locations r. (c) The average interface speed is greatest in the bulk of the swarm (r 3 , r 4 , and r 5 ) away from the colony edge. Error bars are standard deviation (from gathering over time). (d) Traces of the interface velocity over time show that the interface velocity is approximately constant for locations along the perpendicular interface (r 3 & r 4 ), while the speed increases for the parallel interface (r 5 ).

Supplementary
Supplementary Figure 6. Probability distribution functions of (a) the bacterial speed v, (b) the bacterial velocity orientation cos(✓) and (c) the flow vorticity ! for varying distances from the active-passive interface (half-space [H] aperture geometry). Figure 7. Linking the bacterial velocity field to the motion and curvature of the active-passive interface. (a,b) We discern local variations in the interface position and bacterial velocity fields by considering discrete time intervals of t = 1.7 s, which is large enough to distinguish displacements in the interface position h(x, t). (c) For this sample time step, the interface speed ranges from -0.5 to 2 µm/s along the interface; the location of largest interface velocity (x ⇡ 75 µm) corresponds to a location of negative interface curvature C. The bacterial velocity profiles are gathered approximately parallel to the interface at a distance of Y = 10 ± 3 µm away from the mean interface position in the moving coordinate frame Y = y d(t) (dashed line in (b)); we find that the normal velocity components v N are correlated over the t = 1.7 s time step. (d) Overlay of the energy spectrum of the bacterial flow E(q) and the static structure factor of the interface |h 2 q |. The planes are obtained by least-squares fits of the data to a linear form v int = a + b C + cX , where X corresponds to v, v T , v N , or !, respectively. (e-h) Two-dimensional projections of the v int -C-v N plane for the half-space and octagon geometries. The slope from a linear-regression test is presented for each plot and the corresponding p-value from a two-tail t-test with a confidence interval of 95%.

Supplementary
Supplementary Figure 9. Temporal correlations characterizing the interface height and bacterial flow fields. The decay of the interface (⌧ = 16 s) is significantly slower than that of the flow (<1 sec.).
Supplementary Figure 10. (a) An image of the advancing bacterial colony front embedded with 2 µm polystyrene spheres: we use the tracers as probes of the active swarming collective flows in the region of the swarm enclosed by the black box. The particle trajectories are obtained from standard particle tracking techniques. The sample trajectories are shown for a 4 second time interval. (b) We determine the particle speed distribution p(v) by pooling together the particle speed over time for hundreds of particles; the particle speed is defined as the two-dimensional particle displacement over a 1 second time interval, which is long enough to allow tracers to sample multiple vortex structures (characteristic lifetimes of ⇠ 0.1 second, main text Fig. 3). The particle speed distribution measurement (blue circles) seems to follow a 2D Maxwell-Boltmann distribution (red dashed line), where m is the mass of the polystyrene particle, k B is the Boltzmann constant, and T eff ⇡ 2.2 ⇥ 10 5 K, approximately 700 times the thermal temperature (293 K). This effective temperature is to be interpreted as a mixture temperature purely due to the energy in the swarming collective flows.

Supplementary Note 1: Details of high intensity light exposure
The effects of high intensity light exposure on the swarming motility of Serratia marcescens was explored for a range of exposure times and light intensities. The light source was an unfiltered wide-spectrum mercury lamp. We used standard fluorescence microscope optics to focus the light on selected areas of the bacterial swarm. The shape of the selected area was controlled by the aperture geometry, which was either an octagonal aperture [O] or a half-plane aperture [H]. We varied the light intensity using neutral density filters, and the intensity of light incident on the sample was measured using a spectrophotometer (Thorlabs, PM100D). The response of the bacteria to the light depends on the exposure time and light intensity. We observed three types of bacterial response to the light: (i) always active, (ii) temporarily passive, or (iii) always passive. In case (i), the cells remain motile during light exposure and do not stop moving. In case (ii), the cells stop moving but regain their motility suddenly, typically in the first 5 seconds after the light is switched off. In case (iii), the exposed cells stop moving and do not regain their motility for the duration of the experiment (60 -300 seconds). We found that for small exposure times (20 -40 seconds) and weak intensities (I < 220 mW at 535 nm) the cells remain always active or are temporarily passivated. For long exposure times (> 60 seconds) and sufficiently high intensities (I > 220mW at 535 nm), the bacteria are rendered permanently passive. For our experiments, we choose an exposure time of 60 seconds at intensity I = 370 µW (535 nm) to ensure that the exposed bacteria are passive throughout the experiment.

Supplementary Note 2: Bacterial velocity fields from PIV
Bacterial velocity fields were extracted from videos using particle image velocimetry (PIV, PIVLab [1]). Velocity vectors were computed by calculating spatial correlations between successive images of bacteria; the video frame rates were 60 or 125 frames per second, and the velocity fields were sampled at 3 µm spatial intervals. Since we do not visualize the fluid suspending the bacteria, the bacterial velocity fields we extract will generally differ from fluid velocity fields. A sample snapshot of the swarm edge and associated velocity fields before exposure is shown in Supplementary Figure 1 We characterize the initial (pre-exposure) bacterial flow in a 400 µm 2 area, a distance of 100 µm from the edge of the expanding colony (white box, Supplementary Figure 1(a). We calculate the spatial correlation function of the bacterial velocity C v ( r) and the temporal correlation C t ( t) from the PIV data using the In supplementary equations 1 and 2, the angular brackets denote averages over reference positions r 0 (within 400 µm 2 white box, Supplementary Figure 1  The active bi-phasic system we study differs from classical bi-phasic passive systems, such as a solidifying melt or melting ice, in many ways. Perhaps the most noteworthy of these is the way in which the interfacial region is continuously eroded and remodeled by the extraction and convective redistribution of paralyzed, exposed bacteria from the passive phase by emergent self-organized flows generated in the active phase. Note that while passive bacteria enter the active phase as they are convected away from the interface, their fraction far from the interface is expected to be small since the overall (initial) size of the passive phase is small compared to the total swarming area.
Nevertheless, direct observation of the interface between the active and passive phases and results from PIV illustrating a mixing region between the two phases suggests that the inter-phase boundary may be represented as a diffuse interface with a finite thickness with the density of active motile bacteria varying very sharply across a boundary layer thickness. A mathematically defined diffuse boundary may then be obtained from the phase field profiles using order parameters as done in previous investigations involving interface phenomena [2]. To test if such a description may prove useful in our active system, we utilize two independent scalar order parameters, one based on intensity fluctuations and the other based on the bacterial velocity, to track and characterize the interface.

Order parameter from intensity fluctuations
We first use a dynamic order parameter based on pixel intensity fluctuations measured from successive images of the swarm. This approach assumes that fluctuations in image intensity correlate with fluctuations in bacterial density. This assumption forms the basis of differential dynamic microscopy (DDM) [3][4][5][6][7][8][9], which is an image analysis technique also used to characterize the motility of particles or cells from video images.
Here, we define intensity fluctuations as | I(r, t, t)| = |I(r, t + t) I(r, t)|, where I(r, t) is the image intensity at pixel position r (in 2 dimensions) at time t and t is the time step [4,6]. We To reduce noise in the system (due to pixel resolution, short-range fluctuations, and background light fluctuations), we filter the pixel-wise calculated order parameter by smoothing the data over 3 x 3 µm 2 areas.
The averaged -field satisfies 1   +1, with -1 corresponding to a completely passive phase and 1 corresponding to a completely active phase (Fig. 2, main text). The locus of points set by = 0 is defined as the interface separating the active and passive phases.

Order parameter from PIV velocity fields
Alternately, active and passive domains of the swarm can also be identified by tracking the bacterial velocity fields v(r, t). As evident from Supplementary Figure 2(b), the passive phase is readily identified as the region where the macroscopic PIV-derived bacterial speeds are zero. We note that there are locations in the active phase that have instantaneous velocities near zero; however spatial averages (> 10 µm) of the velocities or temporal averages (over ⇠ 1 s) discern between the fully active and fully passive phases. We use the velocity fields to define a second order parameter v map as where v A and v P are the average velocity of the active and passive phases far from the interface and r is the position of the velocity vectors (sampled at 3 µm intervals). Again, 1  v  +1, with -1 corresponding to a completely passive phase and 1 corresponding to the active phase.

Phase fields and v provide interface position and width
To compare the bacterial velocity phase fields with those from intensity fluctuations, we determine spatially-average phase profiles from each. The two-dimensional phase fields ( and v ) may be expressed in Cartesian (x, y) or polar (r, ✓) coordinates. For experiments using the [H] aperture, we average (x, y, t) over x, the appropriate arc-length coordinate for this geometry to obtain a one-dimensional phase profile ⇤ (y, t). For experiments using the [O] aperture, we exploit the initially discrete symmetry and azimuthally average the order parameter to obtain the one-dimensional radially dependent field ⇤ (r, t). The same procedure extended to the velocity based fields yields the one dimensional descriptions ⇤ v (y, t) and ⇤ v (r, t). We find that profiles of ⇤ v (Supplementary Figure 4) are qualitatively similar to ⇤ (Fig. 2, main text). In particular, both phase-fields follow a hyperbolic tangent form over a significant part of the time the interfaces are tracked. We therefore obtain the (mean) interface position (d(t), d v (t)) and (average) interface width (w(t), w v (t)) by fitting the phase profiles at each time t to either for experiments exposed using the [H] aperture geometries or 4. Choosing initial alignment of exposed region relative to colony edge Given that the bacterial colony expands radially out from the inoculation cite, we also investigated how the location and alignment of the active-passive interface relative to the bacterial colony edge impacts the dissolution trends. We focus on two questions: (1) how does the active-passive interface velocity depend on the distance from the colony edge and (2) how does the direction in which the colony expands (relative to the interface) influence the active/passive interface velocity?
To answer these questions, we used an aperture that resulted in an exposure pattern with two mutually perpendicular straight edges as shown in Supplementary Figure 5 5(c)). This behavior may be due to changes in the swarming velocity of the cells observed in Fig. 1 (main text): cells at the edge of the expanding colony move slower compared to bacteria within the swarm (50 to 500 µm from the colony edge). Previous investigations involving swarming bacteria [12] showed that cells at the edge of the colony frequently stall and even stop moving, resulting in an overabundance of slow-moving cells at the edge. Our measured dependence of interface speed with colony edge distance suggests that the active-passive interface speed may be limited by the mean bacterial swarming velocity.
We next consider the orientation of the active-passive interface relative to the colony edge. We find that the temporal dynamics of the interface speed depends on this orientation (Supplementary Figure 5(d)). The perpendicular interface speed is reasonably constant over time (as shown for r 3 and r 4 ); however the parallel interface (r 5 ) has a time-dependent erosion rate, the speed increasing from 0.3 µm/s to 1.2 µm/s over an interval of 70 seconds past exposure. We conjecture that the parallel interface may block the expansion of the swarm, and the observed increase in velocity over time could be due to an accumulation of actively swarming bacteria at the interface; if an increase in bacterial density increases the interface speed, then an accumulation of bacteria at this interface could lead to an increasing interface speed over time. Based on the results, we conducted our experiments in the range of 100 to 500 µm from the colony edge to avoid spatial variations in bacterial swarming velocity and we align the flat active-passive interfaces perpendicularly to the swarm edge so that the interface velocity is constant over time. h(x, t) using standard geometry relating the curvature to variations in the arc-length in two dimensions.
Note that this is possible since to leading order the mean interface is flat, and overhangs are neglected since we observe them very rarely. To smooth over individual bacteria, we average the interface position over a typical bacterial length, 5 µm. The local interface velocity v int at each time t quantifies the erosion rate of the interface and is normal to the tangent vector. Here, we define the local interface velocity as µm as a measure of the flow along the interface, noting that this region is less than a vortex size from the interface (Fig. 2, main text). We also note that to leading order the mean interface geometry (at macroscopic Supplementary Figure 8 contains sample scatter plots of v eff -C-X data, which is shown for an experiment using an [H] aperture geometry. The data is gathered at 3 µm intervals along the interface and at 1 second time intervals for 10 s < t < 40 s past the light exposure to ensure that the interface dynamics are quasi-steady and to avoid transient states at the beginning of the experiment. We find that the 3D scatter plots collapse reasonably well onto a plane, consistent with equation (9)