Dynamical Quorum Sensing and Synchronization in Large Populations of Chemical Oscillators

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Science  30 Jan 2009:
Vol. 323, Issue 5914, pp. 614-617
DOI: 10.1126/science.1166253


Populations of certain unicellular organisms, such as suspensions of yeast in nutrient solutions, undergo transitions to coordinated activity with increasing cell density. The collective behavior is believed to arise through communication by chemical signaling via the extracellular solution. We studied large, heterogeneous populations of discrete chemical oscillators (∼100,000) with well-defined kinetics to characterize two different types of density-dependent transitions to synchronized oscillatory behavior. For different chemical exchange rates between the oscillators and the surrounding solution, increasing oscillator density led to (i) the gradual synchronization of oscillatory activity, or (ii) the sudden “switching on” of synchronized oscillatory activity. We analyze the roles of oscillator density and exchange rate of signaling species in these transitions with a mathematical model of the interacting chemical oscillators.

From the periodic firing of neurons to the flashing of fireflies, the synchronization of rhythmic activity plays a vital role in the functioning of biological systems (13). The mechanisms by which single cells or whole organisms coordinate their activity continue to inspire research over a range of disciplines (46). Synchronization may occur by global coupling, where each oscillator is connected to every other oscillator through a common (mean) field. With this mechanism, mathematically formalized by Kuramoto (7), oscillators are regulated by the average activity of the population (the mean field) and a collective rhythm emerges above a critical coupling strength. A number of oscillatory systems are thought to synchronize by this mechanism (8, 9), such as stirred suspensions of the cellular slime mold Dictyostelium discoidium (10), but it has been experimentally characterized only recently in a system of coupled electrochemical oscillators (11).

A distinctly different type of transition to synchronized oscillatory behavior has been observed in suspensions of yeast cells (12). The density-dependent transition, discovered by Aldridge and Pye (13) more than 30 years ago, recently has been studied in a stirred-flow reactor configuration (14). Relaxation experiments reveal that slightly below the critical cell density, the system is made up of a collection of quiescent cells rather than unsynchronized oscillatory cells, whereas slightly above the critical density, the cells oscillate in nearly complete synchrony (12). This type of transition is much like quorum-sensing transitions in bacteria populations, where each member of a population undergoes a sudden change in behavior with a supercritical increase in the concentration of a signaling molecule (autoinducer) in the extracellular solution (15). Many examples of quorum-sensing transitions have been found, such as the appearance of bioluminescence in populations of Vibrio fischeri (1618) and biofilm formation in Pseudomonas aeruginosa (19).

We studied a population of chemical oscillators to characterize the transition to synchronization as a function of population density and transport rate of signaling species to the surrounding solution. We used porous catalytic particles (∼100 μm in radius) that were suspended in a fixed volume of catalyst-free Belousov-Zhabotinsky (BZ) reaction mixture (20, 21). The catalyst for the reaction, Fe(phen)2+3 (ferroin), is immobilized on the cation exchange particles (22, 23). Reagents in the solution react with the ferroin to produce an activator, HBrO2, which catalyzes its own production, and an inhibitor, Br, which inhibits autocatalysis. A catalyst-loaded particle changes from red to blue as ferroin is oxidized and HBrO2 is produced. The oxidized metal catalyst reacts with solution reagents to regenerate the reduced form of the catalyst and Br. The cycle repeats when the inhibitor level falls sufficiently. Each catalyst-loaded particle has its own oscillatory period, on the order of 1 min (45 ± 13 s), which depends on the catalyst loading and the particle size. The period distribution (fig. S1) is obtained by monitoring the color change associated with the oxidation of the ferroin catalyst on each particle in unstirred solutions (21, 24).

A sketch of the experimental setup is shown in Fig. 1. Both the activator HBrO2 and inhibitor Br are exchanged between the catalyst particles and the surrounding solution, with the exchange rate depending on the stirring rate (21). The surrounding solution is monitored with a Pt electrode, where the potential increases with an increasing concentration of HBrO2 relative to Br, and the oxidation state of the particles is monitored by high-speed video (21). The first time series in Fig. 1 illustrates the change in amplitude of the potential as the oscillator density is increased. The second time series shows the corresponding average intensity obtained from the mean of the individual particle intensities in each image (scaled by the average intensity of fully oxidized particles). When all particles are simultaneously oxidized, the scaled average intensity increases from 0 (all particles red) to 1 (all particles blue), and the value is less than 1 when a fraction of the population is oxidized. The coherence of the population can be seen in images of the stirred particles (Fig. 1).

Fig. 1.

Experimental setup (21). Catalytic microparticles are globally coupled by exchange of species with the surrounding catalyst-free BZ reaction solution. Electrochemical time series illustrates the change in oscillatory amplitude and period with increasing particle density (red line) for a stirring rate of 600 rpm. A typical series of images obtained during one oscillation is shown, from which the (normalized) average intensity of the particles is calculated as a function of time. The associated time series illustrates the change in oscillatory amplitude and period with increasing particle density (red line) for a stirring rate of 600 rpm. A density of 0.02 g cm–3 corresponds to ∼1.3 × 104 particles cm–3.

We observed two distinct types of transitions to synchronous activity, depending on the stirring rate. One type occurs at low stirring rates, as shown in Fig. 2A for 300 rpm. At low particle densities, the global electrochemical signal is noisy with no regular oscillations. There are also no oscillations in the average intensity of the particle images; however, an approximately constant fraction of oxidized catalyst particles is observed (∼20%), representing the fraction of the oscillatory cycle in the oxidized state. As the density is increased, small-amplitude oscillations emerge, and there is a gradual growth in the amplitude of the global electrochemical signal. Corresponding oscillations are also observed in the average image intensity. With a density of 0.02 g cm–3, ∼50% of the particles are simultaneously oxidized in an oscillation, and this approaches 100% for densities greater than 0.04 g cm–3, as shown by the maximum average intensity as a function of particle density. Also shown is the period of the oscillations decreasing slightly with increasing particle density.

Fig. 2.

Dynamical transitions with increasing particle density at low stirring rate (A) and high stirring rate (B). From top to bottom: Amplitude of oscillation in the surrounding solution from the electrochemical signal; maximum of the average particle intensity, corresponding to the degree of coherence of the oscillators; and period of oscillation.

The other type of transition to synchronous activity occurs at high stirring rates. At low particle density, there is no oscillatory signal in the electrochemical potential and there are no oxidized catalyst particles in the associated images, as shown in Fig. 2B for 600 rpm. As the density is increased beyond a threshold value, large-amplitude oscillations in the global signal suddenly appear, and analysis of the average image intensity shows that ∼80% of the particles are simultaneously oxidized each oscillation. As the density is increased further, the maximum fraction of oxidized particles during an oscillation approaches a constant value. The period of oscillation, which is greater than in the low stirring rate case, decreases with increasing density.

Insights into these transitions to synchronized oscillatory behavior can be gained by examining a model of the oscillatory particle system (20, 21). A sketch of the exchange between the particles and the surrounding solution is shown in Fig. 3. The model is based on the three-variable ZBKE scheme (25) for the ferroin-catalyzed BZ reaction (26), with variables X for the autocatalyst, Y for the inhibitor, and Z for the oxidized form of the metal ion catalyst. The concentration of the autocatalyst for the ith particle (i = 1,... N) is given by Math(1) and in the surrounding solution Math(2) where f(Xi, Yi, Zi) represents the chemical reaction on the particle, and g(Xs, Ys) represents the reaction in the surrounding solution (21). The exchange rate of the autocatalyst between the particle and the surrounding solution is given by the term –kex(XiXs), where the value of the exchange rate constant kex increases with increasing stirring rate (21). The sum of the contributions from each particle gives the concentration in the surrounding solution, with the dilution factor /Vs, where Vs is the total volume of solution and is the average volume of a particle. The particle dynamics is periodic for kex = 0 and depends on the number density and the exchange rate for nonzero kex. There is no mechanism for oscillation in the surrounding solution in the absence of particles; however, in the presence of oscillatory particles, oscillations in Xs and Ys arise from the exchange between the particles and the solution.

Fig. 3.

Behavior of BZ particle population model (20, 21) with oscillators coupled by exchange of species with the surrounding solution (see table S1 for parameter values). (A) The amplitude of the oscillations in autocatalyst in the surrounding solution as a function of number density of oscillators and exchange rate constant. (B) The order parameter K as a function of n for low exchange rates. Lower plots show the distribution of oscillator periods at indicated densities (i to iii). (C) The order parameter K at high exchange rates and corresponding oscillator periods.

The coherence of the population of oscillators is determined for a sample of N = 1000 using the order parameter K introduced by Shinomoto and Kuramoto (27): Embedded Image(3) where θj is the phase of the jth oscillator and angle brackets indicate the time average. This coherence measure is 0 when all particles oscillate out of phase with each other (or when the particles are not oscillating) and 1 when all particles oscillate in perfect synchrony. For kex = 0, the particles have a broad distribution of natural periods (38 ± 6 s).

The surface plot in Fig. 3A shows the amplitude of the activator species Xs in the surrounding solution as a function of the number density of particles n, where n = N/Vs, and the exchange rate constant kex. The growth in the amplitude of Xs with increasing n depends on the value of kex. For kex = 0.3, there is a distribution in the natural period for low n (Fig. 3B). There is a corresponding gradual increase in the coherence parameter K with increasing particle density as the individual oscillators gradually align their frequencies and phases. For kex = 3.0, the catalyst particles are not oscillatory for low values of n (Fig. 3C); however, when n is increased beyond a threshold value, the particles suddenly begin to oscillate in perfect synchrony. The coherence parameter switches from 0 to 1. The gradual transition from unsynchronized to synchronized oscillations at low values of kex, and the sharp switching transition from steady-state behavior to synchronized oscillations at high values of kex can be seen in the surface plot of the global signal Xs (Fig. 3A).

The gradual synchronization of particles for low kex can be understood within the framework of the Kuramoto model, with Xs playing the role of the mean-field coupling. The time variation of Xi, Xs, and the exchange rate –kex(XiXs) for a particle are shown in Fig. 4A for one cycle, with kex = 0.3 and n = 4000. The value of Xs increases when several of the oscillators fire together (Fig. 4A, top). This increase in turn influences the individual oscillators through the exchange rate, which becomes positive for a particle when the concentration is higher in the solution (Fig. 4A, middle), leading to an increase in Xi. When Xi crosses a threshold value, the particle fires in synchrony with the others (Fig. 4A, bottom). For low n, particles with frequencies that differ from that of the global oscillation are little affected by the weak signal in Xs; however, as n is increased, the magnitude of the Xs signal increases, and more of the oscillators join the rhythm. In this way, the particles are synchronized by an internally generated signal.

Fig. 4.

Influence of exchange rate –kex(XiXs) on model dynamics. (A) Variation of autocatalyst in solution (blue line), on a particle (black line), and the exchange rate (red line) during one oscillation, with n = 1.8 × 104 cm–3 and kex = 0.3 s–1. (B) Transition from oscillations to steady state with increasing kex (n = 1.8 × 104 cm–3). Time-averaged autocatalyst on the particles (black line), in solution (blue line), and loss rate of autocatalyst from the particles Embedded Image. (C) Appearance and desynchronization of oscillations in autocatalyst in solution (blue line) and electrode potential (black line) with decreasing kex (n = 1.8 × 104 cm–3) and stirring rate (density = 0.0162 g cm–3) shown as red lines. (D) Transition from steady state to oscillations with increasing n (kex = 3.0 s–1). Color coding is same as in (A).

The exchange rate for all particles averaged over one cycle is negative and therefore constitutes an overall loss rate in Xi, shown as a function of kex in Fig. 4B. As kex increases for fixed n, the average concentration of activator in the surrounding solution Xs increases (blue line), but the loss rate also increases (red line), and hence the average concentration of Xi on a particle decreases (black line). When it decreases below a threshold value, oscillations are no longer supported and a sudden transition to the steady state is observed. The Xi loss rate increases slightly as kex is further increased.

The model predicts that both a dynamical quorum-sensing transition and a desynchronization transition will be observed upon decreasing the stirring rate (Fig. 4C, upper panel). The oscillations suddenly appear and then decrease in amplitude with decreasing kex because of the loss of synchronization until a noisy nonoscillatory signal corresponding to out-of-phase oscillators is observed. The transition to the desynchronized state is best observed at low particle densities, as can be seen in Fig. 3A for n = 1.8 × 104 cm–3. The transition from steady-state to oscillatory behavior followed by a gradual desynchronization is also observed experimentally (Fig. 4C, lower panel).

For high kex and low n, the catalyst particles are quiescent, as oscillations are not supported because of the high loss rate of Xi. As the number density is increased, the concentration both in the surrounding solution Xs and on the individual particles Xi increases, with a corresponding slight decrease in the average loss rate of Xi on the particles (Fig. 4D). At a critical density n, the value of Xi on the individual particles reaches the threshold for the transition from steady-state behavior to oscillatory behavior. Although there is a considerable spread in oscillator frequencies, the transition is sharp, with all particles oscillating synchronously, indicating cooperative behavior in the transition (21). The time series in Fig. 1 show experimental measurements of the dynamical quorum-sensing transition from steady-state to oscillatory behavior with increasing particle density, with a Pt electrode giving the global signal and video images giving the average particle intensity.

We have shown that there are two distinct types of transitions to synchronized oscillatory behavior with increasing oscillator density, where coupling occurs by exchange of signaling species through the surrounding solution. For low exchange rates, the oscillators gradually synchronize their rhythms via a weak coupling through the low solution concentration of activator. For high exchange rates, the oscillators are quiescent below a critical density, as the activator concentration on the particles is not sufficient to support oscillatory behavior. Higher exchange rates give rise to lower activator concentrations on individual particles, because only activator-consuming processes occur in the surrounding solution. As the density is increased, the activator concentration in the surrounding solution and on each particle increases until a threshold concentration is reached, and synchronous oscillations suddenly appear at the critical density. The sudden transition has features that are suggestive of an “oscillator death” transition (28); however, the coupling between particles occurs through the surrounding solution in which there is an additional decay of the activator species.

The exchange rate of the activator HBrO2 and the inhibitor Br determines whether the transition to synchronized oscillatory behavior occurs by a synchronization mechanism or a dynamical quorum-sensing mechanism. It is possible that suspensions of cellular organisms, such as yeast or D. discoidium, may also undergo both types of transitions with different efficacies of molecular signaling. Our system of chemical oscillators suspended in a stirred solution is an idealized configuration of oscillators interacting by global coupling.

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