Noise-Induced and Control of Collective Behavior ... - ACS Publications

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Noise-Induced and Control of Collective Behavior in a Population of Coupled Chemical Oscillators Alberto P. Muñuzuri*,†,‡ and Juan Pérez-Mercader*,†,§ †

Department of Earth and Planetary Sciences, Harvard University. Cambridge, Massachusetts 02138-1204, United States University of Santiago de Compostela, 15706 Santiago de Compostela, Spain § Santa Fe Institute, Santa Fe, New Mexico 87501, United States ‡

S Supporting Information *

ABSTRACT: Synchronization of intercommunicating individual oscillators is an important form of collective behavior used in nature as a mechanism to face dangers, act collectively, and communicate. The involvement of the medium where oscillators exist is an important ingredient. Because of their nature and their multiple different components, the medium and the environment are often perceived as stochastic relative to the deterministic nature of the individuals on some scale. This injects energy/matter into the system in ways that can enhance or de-enhance communication in a stochastic manner. Here we experimentally consider a large number of coupled nonlinear-chemical oscillators under the effect of a controlled normally distributed noise. Experiments show that the collective behavior of the oscillator is triggered by this stochastic perturbation, and we observe the dependence on the noise parameters. Our results point to the potential use of environmental fluctuations in determining the emergence and properties of collective behaviors in complex systems.

1. INTRODUCTION Many of the oscillatory behaviors observed in living systems are the consequence of underlying complex nonlinear mechanisms.1 A subclass of these phenomena involves the synchronization of simpler independent oscillators whose collective behaviors are the synchronized states we observe. The collective behavior per se involves some form of signal communication among the individual oscillators, which, given the appropriate state for their (internal and/or external) environments, share a common collective state.2 Noise, however, is often present in many-component systems and, even more so, for open systems in contact with a dynamic or fluctuating environment.3 Hence the communication between individual oscillators can be affected by the presence of fluctuating signals to which the oscillators or their medium are sensitive. These random signals we call “noise”. In many-body phenomena3 noise is well known to lead to order and organization. This is, of course, the case for noiseinduced phase transitions4 or in stochastic resonance (see ref 5 and references therein). Conceptually, noise in a system may be thought of as a fluctuating pattern of energy or matter that is “detected” by some system components and affects system kinetics, evolution, or performance. In living systems noise could arise from fluctuations within the living system itself or from its environment. Clearly the transfer and use of the energy contained in the noise signal depends on the statistical properties of the noise, the receiver, and the specific stage of the dynamical state at which the individual component of the complex system receiving the signal is poised when the © XXXX American Chemical Society

information contained in the noise signal reaches that particular component. We uncover experimentally and numerically some basic, but generic, features of the interaction between a noisy signal and a well-known many-oscillator chemical system.6−8 The chemical system in our experiments consists of a reactor comprising a light-sensitive version of the Belousov− Zhabotinsky (BZ) chemical reaction9−11 where all the chemicals, except for the photosensitive catalyst (a ruthenium complex) that is attached to the surface of inactive and insoluble resin beads, are in the well-stirred solution to which catalyst-loaded beads were also added.6,8,12−16 Given the appropriate concentrations for the BZ ingredients, a redox cycle occurs via the oxidization of the ruthenium complex by reagents in the solution. According to the well-known Field− Körös−Noyes (FKN) mechanism of the BZ reaction,7 this results in the production of an autocatalytic activator, HBrO2, and an inhibitor, Br−. The oxidized ruthenium complex reacts with the solution and regenerates the reduced form of the catalyst while producing the inhibitor. The net result of the above is that a pattern emerges and repeats itself in the form of chemical oscillations as the inhibitor falls below a threshold determined by the BZ recipe in use. In the above configuration, the BZ reaction takes place only at the surface of the beads, but all other reaction components, intermediates, and side products (including, of course, the inhibitor and activator species) diffuse Received: August 19, 2016 Revised: February 15, 2017 Published: February 16, 2017 A

DOI: 10.1021/acs.jpca.6b12489 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

every 5 s (sampling time), except where otherwise indicated. The probability distribution function (PDF) for the stochastic values of the light intensity was truncated to ensure that no negative powering voltage values were applied to the LED and that the symmetry of the distribution was preserved. Typical experiments begin with no illumination for ∼2 h until collective oscillations start. Then, the specific configuration of light intensity and noise is applied to the reactor for at least 30 min before we start analyzing the data. Note that the behavior of the system remains stationary for almost 2 h before the oscillations significantly change due to reactants aging. This can be seen in control experiments (see the SI for details). The control parameters in our experiments are the dispersion of the luminic noise and the sampling time. Each experimental realization here presented was repeated at least two to three times.

into the liquid medium and mix with the rest of the BZ compounds. Depending on the density of beads in the reactor or the magnitude of the coupling between them (related to their mixing and the stirring rate in the reactor and as such represented in our simulations) different collective behaviors have been observed6,13−15,17 (a summary is shown in the Supporting Information (SI), Table S2). They are intrinsically different from the behaviors observed in the homogeneous BZ reaction. The effect of stochastic fluctuations on the homogeneous BZ reaction has been object of research, and actually it has been demonstrated that it can induce the desired behavior given the appropriate values of the noise parameters.18 The situation considered in the present manuscript is completely different due to the discrete nature of our system. Here we will be discussing how noise affects the synchronization state of thousands of interacting chemical oscillators. In our experiments the particular catalyst that we chose is photosensitive; therefore, light can be used as another control parameter17 by changing its intensity or frequency. In fact, the noise will be injected in the form of luminous fluctuations that interact with the catalyst and, most importantly, affect the concentration of the BZ inhibitor species, Br−, present in the reactor and, therefore, the oscillatory behavior of the BZ reaction around each of the individual beads. Illumination enhances the production of inhibitor and consequently arrests the oscillations, whereas the absence of illumination allows the standard production of inhibitor and therefore the set-in of the standard BZ regime for sustained chemical relaxation oscillations (see Figure S1 in the SI). We report on the use of noise as a means to control the mechanism of synchronization of oscillators and explore the influence of the different parameters characterizing the noisy signal. We will use normally distributed noise with amplitude characterized by a standard deviation (σ). (Note that the average intensity of the noisy light was chosen so as to suppress oscillations in the system (see Figure S2 in the SI).)

3. RESULTS AND DISCUSSION In Figure 1 we show the evolution of the redox potential measured inside the reactor in a typical experiment. At t = 205

2. METHODS We work with a large population of resin beads (DOWEX50WX4, 100 μm in radius approx.) loaded with ruthenium (Ru(bpy)32+) as a catalyst ([Ru(bpy)32+] = 3.5 μmol/g) and immersed in a catalyst-free Belousov−Zhabotinsky (BZ) solution9 ([CH2(COOH)2] = 0.135 M, [NaBrO3] = 0.41 M, [NaBr] = 0.0875 M, [H2SO4] = 0.65 M). Typical experiments here presented contained 1.8 × 104 beads/mL (for a bead concentration of 0.03 g/mL). We conduct the experiments in a cylindrical closed reactor (as described in the SI). A platinum electrode immersed in the reactor provided with redox recordings of the system and an optical window in the reactor allowed us to modify the amount of illumination reaching the reactor as well as the visual inspection of the system. Details of the experimental configuration are in the SI. In the absence of illumination, with the concentrations and parameters we selected, the system exhibits a collective synchronized oscillatory behavior. As the value of the constant illumination is increased, the collective synchronous state is lost for values of ϕ (the experimental luminic flux) larger than those obtained by powering the LED with 1 V (see the SI for details). To implement the noisy luminous signal, random numerical sequences were first generated in a computer and then translated into the LED irradiation intensity using the above experimental setup. The LED irradiation intensity was updated

Figure 1. Noise-induced collective behavior in a set of nonlinear chemical oscillators. (A) Evolution of the light intensity in an experiment with redox potential as in panel B. Fluctuations around the mean value are suppressed during the central part of the experiment (period of time between the two vertical dashed lines), and the collective behavior is also suppressed. (C) Evolution of the light intensity in an experiment shown in panel D. Normal noise is applied during the central part of the experiment and oscillations are induced. (E) Distribution of periods for the oscillations induced by noise. Bead density in the reactor 0.03 g/mL, sampling time = 5 s, ϕmax = 110 mW/cm2. B


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The Journal of Physical Chemistry A min the intensity of the LED was made for 20 min (sampling time = 5 s) to follow a normal distribution with a mean value = 55 mW/cm2 (that corresponds to an LED feed voltage of 2.5 V) and σ = 44 mW/cm2. During these 20 min, oscillations were observed and recorded, demonstrating the existence of collective activity in the system. Note that each peak corresponds with a collective oscillation of a significant number of the beads in the system. The nonregularity of the oscillations reflects the random nature of the process. At t = 225 min, fluctuations in the light intensity were stopped: Only constant light reached the reactor (ϕ = 55 mW/cm2, σ = 0 mW/cm2, sampling time = 5 s), and the collective oscillations immediately disappeared. When 40 min had elapsed, the previous noisy signal was restarted ( = 55 mW/cm2, σ = 44 mW/cm2, sampling time = 5 s) and oscillations reappeared in the system. Figure 1A shows the evolution of the light intensity with time, while panel B presents the values measured for the system redox potential. Note that the average light intensity in the system is well above the threshold needed to inhibit the collective oscillatory behavior (see the SI for details). Each of the peaks in Figure 1B corresponds to a collective oscillation involving most of the beads in the system. Figure 1C,D shows the results of an experiment complementary to the previous one. Here light was kept constant with ϕ = 55 mW/cm2 until time t = 130 min. Then, stochastic fluctuations in the light following a normal distribution ( = 55 mW/cm2, σ = 44 mW/cm2, sampling time = 5 s) were introduced for 75 min. After that, only constant light reached the reactor. Figure 1D shows the evolution of the redox potential in the reactor. Note that collective behavior was only observed when luminous fluctuations were applied to the system independently of the history of the experiment. A direct inspection of the collective behavior recorded by the redox potential seems to be aperiodic. In Figure 1E, the distribution of intervals of time between two consecutive peaks is plotted for one experiment. This distribution resembles a Gaussian (although statistics is not good enough as to ensure it) with the lower values of the independent variable truncated, a mean value around 80 s, and a large standard deviation (±100 s). This is a clear consequence of the stochastic nature of the noisy perturbation: High values of the light intensity tend to produce larger oscillatory periods or even suppress oscillations. Much shorter periods are not accessible because the chemistry needs a minimum recovery time to be able to produce another oscillation, and this translates into a truncated distribution for the periods. To check system sensitivity to the specific properties of the noise, we carried out experiments with different noise-standard deviation, σ, but with the same mean value for the light intensity. A summary of the results is shown in Figure 2, where we plot the average number of oscillations per minute versus σ (keeping constant = 33 mW/cm2). For values of σ < 17.5 mW/cm2 (approx.), no collective behavior was observed. Panels on the left-hand side of the Figure show (upper left corner) the PDF for a σ = 2.2 mW/cm2 and (lower left corner) the typical evolution of the redox potential in this case. For values of σ > 17.5 mW/cm2 (approx.), the behavior recorded with the redox potential is completely different. Now collective oscillations with significant amplitude (∼20 mV in redox potential) are observed. Panels on the right-hand side of the Figure show (upper right corner) the PDF for σ = 44 mW/cm2 and (lower right corner) the typical evolution of the redox potential in this case. Note that as σ is increased, the stochastic

Figure 2. Dependence of collective behavior on the noise amplitude dispersion. The number of oscillations per unit of time is plotted versus the noise amplitude (σ). A minimum value of σ is required to activate the mechanism. We note the existence of a noise-induced transition between collective regimes around a light intensity dispersion of σ = 17 mW/cm2. The dashed vertical line should be taken only as a rough indication of the range of values of σ where the transition takes place. Each bullet in the plot summarizes the results of several different experiments (two to three typically). The left upper panel shows the probability distribution function (PDF) of the fluctuations when σ = 2.2 mW/cm2 (horizontal axis is light intensity, ϕ, in mW/cm2; vertical axis is the statistical probability of this value); the corresponding redox potential is plotted in the lower left panel (horizontal axis is time in min; vertical axis is the redox potential in volts.). The upper right panel shows the PDF for σ = 44 mW/cm2 (horizontal axis is light intensity, ϕ, in mW/cm2; vertical axis is the statistical probability of this value) and the corresponding redox potential evolution (lower right panel: horizontal axis is time in min; vertical axis is the redox potential in volts.). Density of beads = 0.035 g/mL, = 33 mW/cm2, and sampling time = 5 s.

distribution of the noise signal becomes flatter and that for σ = 44 mW/cm 2 it almost is a uniform distribution. In consequence, for large enough values of σ, the observed behavior, and thus the number of oscillations per minute, should be constant. So far, the sampling time was kept constant and equal to 5 s in each experiment. This sampling time is a parametric feature in our noise and can also be used as a control parameter. To investigate the sensitivity of the system to changes in the characteristic time of the stochastic signal, we carried out experiments with different sampling times (keeping σ = 44 mW/cm2). Figure 3 shows the average number of oscillations per minute versus sampling time. For low values of the sampling time the effect of noise is almost negligible: The system only “feels” the average value of the light intensity (which was selected to be just enough to suppress any oscillation). As sampling time is increased, we can see the emergence of collective behavior that therefore suggests the existence of an optimal value for the sampling time at which the collective behavior is more effective. For much larger values of the sampling time, the system remains exposed to each particular value of the light intensity for a period of time large enough as to accommodate to that particular value, so the dynamical behavior of the whole system is just a succession of stationary states. The above system has previously been modeled numerically.12,19,20 We used this qualitative model (briefly described in the SI) to reproduce the above experimental results. (Note that the stochastic parameter describing the illumination of the system is denoted by Φ in the equations.) In Figure 4A we C


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show a phase diagram summarizing the effect of constant light on the system for different values of Kex. Here Kex is a parameter controlling the strength of the coupling between the different beads (the equivalent to the stirring rate in experiments). For a constant value of Φ = 0.06, no oscillations are observed for any value of the coupling parameter, Kex. We took this value as the mean value for the stochastic light intensity in the system ( = 0.06) and considered fluctuations with a normal probability distribution in the same way as in the experiments. We also study the effect of the noise standard deviation (σ) in Figure 4B. This Kex−σ phase portrait shows the values of the parameters compatible with collective behavior of the system. Figure 4C shows the values of σ and sampling time that are compatible with the existence of collective behavior in the system when = 0.06 and Kex = 4. (A summary of the collective signals simulated for each of the cases we considered is shown in the SI.) Note that the model used for simulations is only qualitative, and thus there is no direct equivalency with experimental parameters. Our results here show that the same transitions observed in experiments

Figure 3. Dependence of the collective behavior on the sampling time. The number of oscillations per unit of time is plotted versus the sampling time. (density of beads = 0.035 g/mL, light intensity varied between 0 and 66 mW/cm2). The dotted line is meant only to guide the eye. Each point in the plot averages over more than two different experiments. The large dispersion of the experiments for a sampling time of 5 s reflects the existence of a transition around that value.

Figure 4. Numerical phase diagrams with regions of noise-induced collective behavior. (A) Constant value of illumination versus the coupling parameter (Kex). (B) Coupling (Kex) versus noise amplitude (σ) ( = 0.06, S.T. = 0.1 t.u.). (C) Sampling rate versus noise amplitude (Kex = 4 and = 0.06). Red region (Δ) corresponds to the existence of collective behavior; for the green region (●), each oscillator independently oscillates but not synchronously; the white region (×) is for the amplitude death. Model parameters: ε = 0.01, ε′ = 0.015, q = 0.002, h = 0.7 ± 0.03, p1 = 1, p2 = 1, Vbead = 1 × 10−4, and Vtot = 0.2. (D) Number of oscillations per unit time as a function of the standard deviation (σ) for three different values of the luminic noise signal sampling time (0.05, 0.15, and 0.25 t.u.) (E) Number of oscillations per unit time as a function of the sampling time (σ = 0.06). D


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Power-law-correlated time-dependent noise and other sampling time probability distributions functions, where theoretical work is also possible, together with the extension to biological systems, are clear paths to follow in the future and where many interesting results await.

can be achieved in simulations given the proximity of a Hopf bifurcation. Note that the system is sensitive to the amplitude of noise as reported in the experiments. Figure 4D shows the number of oscillations per unit time as a function of the standard deviation (σ) for different values of the luminic noise signal sampling time. This Figure qualitatively reproduces the experimental transition described in Figure 2. Below some threshold, which depends on the sampling time, no collective behavior is observed. Figure 4E is the numerical equivalent of the experimental results plotted in Figure 3. Here we also observe sensitivity of the system to the noise characteristic time (given in our case by the sampling time).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b12489. Details of the experimental setup as well as the description of the system under constant illumination. Description of the numerical model used for the simulations presented in this manuscript and the numerical description of the behaviors observed under constant illumination. (PDF)

4. CONCLUSIONS Controlled luminic fluctuations with a normal PDF were used to trigger cooperative behavior in a system of coupled nonlinear photosensitive chemical oscillators. For the appropriate noise parameters, fluctuations induce not only oscillations of the individual beads but also the global synchronized oscillation. The equivalent deterministic situation (i.e., without fluctuations) results in the amplitude death of any oscillation. Under noise, the induced collective state is both qualitatively and quantitatively dependent on the standard deviation of the noise PDF (σ, which is related to the noise amplitude). Under our experimental conditions, for σ > 17.5 mW/cm2 (approx.) the system is driven into an oscillatory regime, while for σ < 17.5 mW/cm2 (approx.) the system does not oscillate. We can account for the above results by the straightforward generalization with an additive stochastic forcing of a simple previous model. Different experimental parameters such as bead density may lead to different synchronization states; here one can also expect noise-induced transitions for the appropriate noise parameters. The above behaviors are expected on a theoretical basis because the presence of external noise is known to modify the system parameters, as predicted by the renormalization group equations for the system parameters.21,22 Condensed matter physics teaches that the presence of any kind of fluctuations to which a system is sensitive induces a scale dependence in the system whose detailed structure depends on the nature of the fluctuations and their statistical properties and on whether the noise acts additively or multiplicatively. This is a general phenomenon that in a chemical system implies that noise modifies the values of reaction constants and other relevant parameters such as diffusion constants: Their effective values become noise dependent. Hence, Lyapunov exponents, and consequently the stability of the system, inherit this noise dependence and make the states of the system also become noise-dependent. We also note a nontrivial dependence of the specific nature of the cooperative behavior of the system on the noise sampling time and have explored this effect, also expected on theoretical grounds, for a normal noise with σ = 44 mW/cm2. We finally conclude from our experiments and simulations that external noise can be used as a tool for selection and control of the dynamical cooperative state of a population of interacting chemical oscillators. On the basis of already available theoretical and numerical work, we expect that the statistical properties of the noise will influence both the nature and details of the transition found here as well as contribute to better understand the nature of the effects of noise in biochemical and biological phenomena.

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AUTHOR INFORMATION

Corresponding Authors

*A.P.M.: E-mail: [email protected]. Tel: +34 677 947799. *J.P.-M.: E-mail: [email protected]. Tel: +1 617 496 9315. ORCID

Alberto P. Muñuzuri: 0000-0002-0579-9347 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Repsol S. A. for supporting this research. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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REFERENCES

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