Fig 1.
(A-C) A top view of the same experimental raft is illustrated at the (A) start, (B) middle, and (C) end of a roughly 106 min duration. To visually illustrate treadmilling, a set of structural ants at the perimeter is selected every 22 minutes. These ants are then image-tracked as they flow inwards due to network contraction and the geometry defined by these ants is traced by a distinctly colored and numbered outline. The set of ants labeled “2” in (B), for example, corresponds to the same set of ants labeled “2” in (A), but roughly 53 min later. The label “1” represents the oldest set of ants while “6” represents the newest. The shrinking of these contours indicates retraction of the raft structure, while the existence of new layers indicates outwards expansion. Periods of raft expansion and coinciding protrusion emergence (A,C) were interrupted by interstitial spans of decreased activity and less eccentric morphologies (B). All scale bars represent 10 ℓ where 0 mm is the approximate average body length of 1 ant. See S1 Movie for unannotated video. (D) A schematic visually illustrates the four concurrent mechanisms of treadmilling: (1) structural raft contraction at a global rate , (2) transition of structural ants to freely active ants in the bulk at a nominal rate δ, (3) transport of the free ants on top of the raft with a mean persistence length lp, and (4) binding of free ants back into the structural network at the edges of the raft at nominal rate α. The schematic is taken from Wagner, et al. (2021) [14]. The freely active layer is offset from the structural layer for illustrative purposes only, as it resides directly on top of the structural network in real ant rafts. Furthermore, note that the freely active layer, while shaded continuously, is comprised of dispersed ants while the structural layer is relatively homogeneous and condensed.
Fig 2.
(A) Two free ants of interest (red) are schematically illustrated on a structural section of raft (shaded cyan) in continuous space. Other free ants are shaded grey. The direction of motion () of the ant far from the edge of the raft (left) is predicted entirely by the Vicsek model. In contrast, whether the ant encountering the edge of the raft (right) moves into the water, depends not only on
, but also on the competition between active force
and the effective edge repulsion force
. Each of these forces is governed by the motion of free ants and relative position of water within detection distance R of the ants. (B) A corresponding schematic envisions how these continuous scenarios are coarse-grained into the lattice-based framework of the numerical model. The motion of the free agents of interest (red) remains governed by the direction of travel (white arrows) of neighboring free agents, and effective pairwise repulsion (black arrows) from neighboring water nodes within distance R. However, free agent movement is updated by stepping the free agents to the adjacent structural agents (cyan) or water nodes (white) whose relative direction most closely matches the preferred direction, θi. Nodes are displayed in a hexagonal, close-packed lattice for illustrative purposes only, but are initially offset in both directions of the horizontal plane by some amount in the rangeζ and are further randomized by stochastic structural unbinding events as the simulation progresses. (C-E) The shape evolution of a simulated raft over a duration of 20 min (of virtual time), illustrates the implementation of the lattice-based conceptualization from (B) into the numerical model. Shape change is governed by the transition of free agents (red) into the structural network (cyan) at the raft’s edge. The raft depicted was initiated as a circle and all scale bars represent ℓ. (F-G) Agents encountering water in regions of (F) high and (G) low edge curvature are depicted. These respective agents experience high and low values of FΓ due to the pairwise contributions of repulsion force from detected water nodes (black arrows). The agent in (F) has no freely active neighbors such that the only contribution to its value of Fa is its own self-propulsion force (white arrow), whereas the agent in (G) has many freely active neighbors moving in similar directions towards the water such that it has a high value of Fa oriented off the raft. (F-G) Insets display the vectorial sums that define the effective forces Fa (red) and FΓ (blue) for the respective agent configurations, thus illustrating how the agent in (G) is more likely to edge-deposit based on Inequality 2.
Fig 3.
Comparing Treadmilling Dynamics.
(A-B) The gradient of contractile speed, , towards the anchor point of the rafts (red dot) is illustrated via heat maps within defined regions of interest (ROIs) for both (A) an experimental and (B) simulated raft.
was computed as the component of speed moving towards the stationary reference frame (i.e., the acrylic rod) and was measured for every point within these 2D ROIs, then averaged over durations exceeding 13 minutes. Scale bars represent 10 ℓ. (C)
is plotted with respect to distance from the anchor point, r, for both the experiment (discrete red squares) and simulation (solid black curve). The slopes of the least-squares regression lines are taken as the average contractile strain rate
. The experimental strain rate (
) agrees with the numerical value (
). (D-E) The growth zones of both (D) an experimental and (E) simulated raft after roughly 50 min are shaded in cyan. Scale bars represent 15 ℓ. The bound ants that occupied the perimeter of the raft at reference time, t0 = 0, are outlined in red and were traced through time. (F) The time-evolution of the edge binding rate, α, and bulk unbinding rate, δ, as a percentage per unit raft area are shown for two sets of experiments (squares for α and triangles for δ; red and black for two different experiments) along with the averaged results of 12 simulations (continuous black curves). Note that the initial drops in both α and δ for the simulation data occur since the raft was not initiated at steady state, whereas experimental data was only sampled at pseudo-steady state. (A,C,D,F) Experimental results are courtesy of Wagner, et al. (2021) [14]. All simulated rafts were initiated as circles and shape was allowed to evolve stochastically.
Fig 4.
Comparing Protrusion Dynamics.
(A-B) Probability mass functions are shown for (A) the average protrusion widths, W, and (B) growths rates, V of more than 400 experimental observations (grey) and numerical observations (light red) each. Here, R = 0.9 ℓ, η = 0.2 and . (C-D) The direction of motion of free ants on experimental sections of (C) a protrusion and (D) the bulk of a raft are visually illustrated with the color of a free agent representing its direction of travel during one frame-to-frame observation. (E-F) The same visual analysis is made for sections of (E) a protrusion and (F) the bulk of a simulated raft, where the direction of travel is measured between one timestep. (C-F) Colors are assigned according to orientation based on the color wheel depicted in (E). (G-J) 2D velocity distributions are shown, courtesy of Wagner, et al. (2021) [14]. (G-H) correspond to (C-D), respectively, while (I-J) are the ensembled results from 11 in silico protrusions and on the order of 100,000 discrete velocity observations, each. A simulated protrusion at the start (K) and end (L) of a roughly 21 min duration exhibits how directional motion on protrusions culminates in clustering of freely active agents (black circles) at the tip and rapid, anisotropic growth. (A-B,C-D,G-H) Experimental results are courtesy of Wagner, et al. (2021) [14]. Scale bars in (C,E,K,L) represent 10 ℓ. All simulated rafts were initiated as circles such that the in silico protrusion growths depicted (and from which data were collected) occurred stochastically.
Fig 5.
Dynamic Effects of Activity Level.
Snapshots of modelled rafts at different simulation times (t) and activity levels () are depicted to illustrate the effect of
on raft development. The raft on the far left depicts the initial conditions which were the same for each simulation throughout this work (a circular raft with ϕ = 1). On the right each row depicts a single raft as it evolves in time (moving from left to right along the horizontal axis). For all simulations shown, η = 0.2 and R = 0.9 ℓ. Structural agents are depicted in cyan, while dispersed free agents are depicted in red. The scale bar in the top right is universal to all snapshots and represents 14 ℓ.
Fig 6.
(A) Snapshots of initially circular, simulated rafts are shown after 1.5 hours of simulation time. Here, η = 0.2, R = 0.9 ℓ and . (B) Snapshots of protrusions, each representing the minimum observed radius of tip curvature from its raft at final simulation time, are depicted for each of the respective values of
. The black line cropping each snapshot at its bottom is an open border to the remainder of the raft. The values of
yielding each morphology for (A-B) are denoted beneath each snapshot in (B). (C) Mean freely active agent packing fraction, ⟨ϕ⟩, (blue) and maximum surface excess, Smax, (red) are plotted with respect to
, and averaged over 5 simulations at each value of
, with error bars presenting standard error of the mean. Horizontal dotted lines in (C) represent the experimentally measured values of ϕ = 0.24 and Smax≈1.8. The bounds of the parameter space that matches experiments are marked where these respective lines intersect the numerical data (see “Exp. Match Zone” between
and 1.47). There exists a zone between roughly
and 2.0 of continuous phase transition between rafts with minimal-to-no growth whatsoever (ϕ≈1 and S~1.2) at low activity levels and frequent protrusion growth (low ϕ and S>2) at high activity levels. (D) Mean protrusion tip radius (Rκ) is plotted with respect to
. Anywhere from four (in the case of no growth) to forty-one observations were ensemble averaged depending on protrusion frequency. Where no protrusions were available (
) the mean convex edge radius is reported instead. The top dotted line represents the initial raft diameter of 10 ℓ, while the bottom dotted line represents the limit of Rκ→0.5 ℓ, corresponding to the radius of one agent. (C-D) share a horizontal axis. (E-G) Three chronological snapshots of an experimental ant raft exhibiting different phases of protrusion growth are compared to (H-J) three chronological snapshots of a simulated raft when
was modulated between 1.1 and 1.6. (K) The time evolution of surface excess as measured from one experiment (red circles) and ensemble averaged over 28 numerical simulations (black curve with a negligible shaded region representing standard error of the mean) are displayed. Note that the simulations start close to S = 1 given the initially circular raft shape. Time, t*, is normalized by the experiment duration for a more direct comparison. Structural agents are depicted in cyan, while dispersed free agents are depicted in red. All scale bars represent 10 ℓ. All simulated rafts displayed were initiated as circles such that protrusions emerged stochastically.
Table 1.
Commonly referenced values.
Table 2.
Free parameters of model.