Emergence of Collective Behavior in Groups of Excitable Catalyst-Loaded Particles: Spatiotemporal Dynamical Quorum Sensing

Spontaneous spatiotemporal wave activity occurs in groups of excitable particles for groups larger than a critical size. Experiments are carried out with particles loaded with the catalyst of the Belousov-Zhabotinsky reaction that are immersed in catalyst-free reaction mixture. The particles diffusively exchange activator and inhibitor species with the surrounding solution. All particles are nonoscillatory when separated from the other particles; however, target and spiral waves are exhibited in sufficiently large groups. A cellular particle model of the system also exhibits transitions from excitable steady state behavior to spatiotemporal wave activity with increasing group size.

Figure 1

(a), (b) Images separated by 12 s of a target wave in a group of 89 catalyst-loaded particles; (c), (d) images separated by 8 s of spiral wave behavior in a group of 106 catalyst-loaded particles. Pacemakers and target waves were seen in $\sim 70\%$ of experiments showing wave activity, with spiral wave behavior observed otherwise. Solution concentrations: $[{\mathrm{BrO}}_3^{-}]=0.30\mathrm{M}$, $[{\mathrm{H}}_2{\mathrm{SO}}_4]=0.60\mathrm{M}$, $[\mathrm{NaBr}]=0.10\mathrm{M}$, $[{\mathrm{CH}}_2(\mathrm{COOH}{)}_2]=0.20\mathrm{M}$. Scale bar length is 1.0 mm.

Figure 2

(a) Probability of wave activity in a group as a function of the number of particles $n$ for $[{\mathrm{BrO}}_3^{-}]=0.30\mathrm{M}$ (red solid line), 0.27 M (blue dashed line), and 0.24 M (green dotted line). Other concentrations as in Fig. 1. Each point ($+$) corresponds to the average of $\sim 8$ experiments. (b) Schematic 2D representation of a 3D cell model ($13\times 13\times 4$) with a $5\times 5\times 1$ group of catalyst particle cells, $n=25$. Particles are indicated by a cell containing a (red) dot and the surrounding solution is indicated by empty cells. (c) The probability of spatiotemporal wave activity in 3D cell model for identical particles as a function of group size $n$ for $A=0.2430\mathrm{M}$ (red solid line), 0.2425 M (blue dashed line), and 0.2420 M (green dotted line). (d) Steady state concentrations of activator $X$ (blue) and inhibitor $Y$ (red) for $A=0.2420\mathrm{M}$ along a cross section (between x’s) of the array of cells in (b). The $x$ axis corresponds to the cell number from left to right through the cross section.

Figure 3

Behavior of particle model as a function of the group size $n$ of identical particles with $C_i=0.0053\mathrm{M}$ and $A=0.242\mathrm{M}$. (a) Average steady state concentration of ${\mathrm{HBrO}}_2$ on group particles (solid line) with increasing $n$ and in the solution cells directly above the particle cells (dashed line, scaled by factor of 2.4). (b) Average loss rate of ${\mathrm{HBrO}}_2$ from $n$ particles to the surrounding solution. The average loss rate is determined as the average of the diffusive loss from each particle cell. (c) Steady state concentration of ${\mathrm{HBrO}}_2$ on the central particle (solid line) with increasing $n$ and in the solution cell directly above the particle cell (dashed line, scaled by factor of 2.4). (d) Loss rate of ${\mathrm{HBrO}}_2$ from the central particle. The vertical dashed line in each plot represents the value of $n_{\mathrm{crit}}$, above which spatiotemporal wave activity occurs. For steady state group or particle behavior, $n\le n_{\mathrm{crit}}$, each point corresponds to a square or nearly square group of $n$ particles (1, $2\times 2$, $3\times 3$, etc.). For spatiotemporal behavior, $n>n_{\mathrm{crit}}$, each point corresponds to a time average for the group or particle in such groups of $n$ particles. The curves for $n>n_{\mathrm{crit}}$ are scaled by a factor of 4.4.

Figure 4

(a) Probability of wave activity with increasing group size $n$. Each point is based on 50 simulations, each with a Gaussian distribution of particle cell catalyst concentrations: $C_{\mathrm{mean}}=0.0053\mathrm{M}$, $C_{\sigma }=0.0006\mathrm{M}$. The three curves correspond to $A=0.2430\mathrm{M}$ (red solid line), 0.2425 M (blue dashed line), and 0.2420 M (green dotted line). (b) Probability of wave activity with increasing $C_{\sigma }$, where each point is calculated as in (a) with $C_{\mathrm{mean}}=0.0053\mathrm{M}$, $A=0.2420\mathrm{M}$, and $n=49$.