Oregonator generalization as a minimal model of quorum sensing in Belousov–Zhabotinsky reaction with catalyst confinement in large populations of particles

Abstract

We consider the case of the Belousov–Zhabotinsky (BZ) oscillating reaction where the metal catalyst is confined in a discrete phase of cation-exchange resin particles. We show that modelling of the sudden transitions from the steady state to fully synchronized oscillations, which were previously described only in terms of quorum sensing, does not necessarily require fine-graining of the models into individual particles. As a minimal representation of the system, we propose a generalized version of the Oregonator with only one additional parameter φ that corresponds to the total volume fraction of the catalytic phase in the system. We show that the kinetic consequences of catalyst confinement can be interpreted through φ acting as the second variable stoichiometric factor in the Oregonator mechanism that decreases the order of autocatalysis and simultaneously weakens the negative feedback. Based on the proposed model, an analysis of the so-called quorum sensing is provided in terms of the standard concepts of Hopf bifurcation. The model reproduces correctly that (1) oscillations are only exhibited above a certain critical threshold of φ; (2) as φ decreases, so do the amplitudes of oscillations; whereas (3) the periods of oscillations are almost independent of φ. The model thus appears to be a reasonable minimal representation of uniform states in the BZ reaction with catalyst confinement, and therefore might be a convenient starting point for further refinements representing more complicated transitions in this system.

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1. Introduction

Progress in our understanding of oscillating chemical systems in terms of their internal dynamics has also stimulated considerable interest in how they behave in external interactions. Since the pioneering report of synchronization between two Belousov–Zhabotinsky (BZ) oscillators by Marek and Stuchl,¹ investigations of chemical oscillations in reactors coupled by mass transfer inspired many approaches,^2–16 whether using reactors with common walls with perforations, apertures, canal ports or with active transport between reactors using pumps. In addition, coupling between oscillating reactions was also achieved without direct mass transfer, e.g. using bridged electrodes,¹⁷ reactors that mutually controlled each other's feed flow,^18,19 or with common cooling circuits.²⁰ These efforts yielded a variety of phenomena, including rhythm splitting,^1,3 phase-difference locking,^{2,8–10,14–17} oscillator death^4,8,17,19,20 and rhythmogenesis,^5,19,20 entrained responses to pulsed forcing,^6,7,11,16 presynchronization,¹⁴ quasiperiodicity, bursting and chaos,^{10–13,15,17} or coupled chaotic states.^12,13 Moreover, investigations were also extended to three^21–26 or more interacting reactors,^27–32 offering further new insights into how these systems react to external stimuli in terms of their propagation^{21,22,27–30} or even information encoding and decoding.^31,32

Several approaches exist for studying the collective dynamics of even larger numbers of oscillators. A remarkable body of experimental evidence has been collected for arrays of globally coupled electrodes³³ that can exhibit chaotic synchronization,^34–38 Kuramoto phase synchronization,^39–43 noise coherence,^44–46 amplitude death,⁴⁷ resonant clusters,⁴⁸ desynchronization,⁴⁹ effects of non-isochronicity,^50,51 and even genuine chimera symmetry breaking.⁵² In another line of work, observations of BZ reaction droplets dispersed in oil⁵³ were extended by the use of microfluidics. This approach afforded continuous droplet chains⁵⁴ and droplet stacks⁵⁵ that could synchronize into in-phase and anti-phase oscillations, Turing patterns, or phase clusters.

1.1. Investigation of large populations of BZ oscillators using catalyst confinement

A very straightforward method of preparing large populations of BZ oscillators was developed based on the possibility of confining the metal catalyst in particles of cation-exchange resin. Such catalyst-doped particles were immediately noted to be very interesting because of their ability to exhibit characteristic oscillations individually,⁵⁶ as well as in bulk.^57,58 Some authors focused on refining the mechanism of oscillations, in particular the dependence of their natural frequencies on the radii of the resin beads,^59–61 but the majority directed their attention to how the increase in the number of these oscillators gave rise to synchronized activity, from small groups^62–66 to larger and larger monolayers.^67–69 More recently, unstirred setups were also used for observations of phase clusters and chimera-like patterns that arise from specific coupling of the oscillators under selective illumination^70,71 and even for observations of the stochastic synchronization effects of noise.^72,73

There are, however, only a few studies of these oscillators in stirred systems,^74–77 perhaps due to the obstacles that stirring poses to accurate imaging. Information from such measurements is, in principle, only statistical. So far, the data was not able to directly specify individual histories for each oscillator, although the presence of phase clusters, for example, has been recently supported by indirect evidence.⁷⁷ Nevertheless, the most prominent experimental observation in this line of work remains in the fact that a critical density of catalytic particles is required to make the system oscillate in bulk.

1.2. Understanding the quorum sensing of BZ oscillators

In analogy to similar behavior in populations of bacteria or yeast,^78–81 the fact that the system does not oscillate unless the threshold amount of oscillators is exceeded was named quorum sensing.⁷⁵ Accordingly, a mechanistic explanation of this phenomenon was also formulated in a way that heavily focused on representing each oscillator individually, and the global dynamics of their synchornization was examined through detailed statistics of their more local interactions.

Such fine-graining is certainly justified in situations where the phenomena that are examined are local in character. Detailed consideration of phase information from individual oscillators is indispensable in studying any state of a system preceding full synchronization, e.g. transitions that occur gradually through the formation of clusters, in studying chimera symmetry breaking, etc. However, it should be an inevitable question whether there might be a more efficient approach in cases where the given collective behavior has a prominent component that could be modelled as globally uniform and whether the so-called quorum sensing might be such a case.

This follows a convenient approach for investigating coupled oscillators – starting with a first-order description of the dynamics that is globally uniform, e.g. based on the mean-field approximation, and only then adding the finer details, which are more localized, in further refinements. Even with populations of BZ oscillators, it should be of interest to distinguish different orders in the dynamics of the system. Of course, some features will only be understood through fine-grained approaches, with full resolution of the phase information. However, this might be more efficient if we first formulate an approximate framework for describing those aspects that can be interpreted more globally, e.g. in the form of a modified overall stoichiometry of the system.

Certainly, the so-called quorum sensing in populations of BZ oscillators does appear to have a globally uniform aspect: based on the total content of catalyst-confining particles, the system can behave as a globally uniform oscillator or can be globally in a steady state. Although the existing experimental data^74–77 is not sufficient for mapping out a complete phase diagram and assert definite conclusions about the abundance of mixed states, it is without doubt that in many experiments the transition is sudden.^74–76 This makes it seem likely that even in the models of these sudden transitions there might be an approximate first-order representation, where the local details of the transition are equally reduced. Perhaps, on this level, the corresponding threshold of catalytic particles might not even need to be treated differently from the fact that some threshold concentrations of molecules are required with respect to any other reagent in the solution, e.g. malonic acid or bromate.

How the concentrations of BZ reagents determine whether oscillations are exhibited is traditionally understood in terms of Hopf bifurcation. This is based on the global mass action of species, without any need for fine-graining of models by considering every single molecule individually. This is obviously in dramatic contrast to current interpretations of how the presence of oscillations in the BZ reaction is determined if the catalyst is confined in a discrete population of catalytic resin beads. The purpose of this study is to analyze models of such catalyst confinement in BZ oscillators and attempt to clarify whether the particulate character of the catalytic phase plays a critical role in the correct representation of its global behavior or, conversely, whether the so-called quorum sensing might be represented to some extent without fine-graining and understood similarly, in much simpler terms of Hopf bifurcation.

2. Fine-grained modelling particle by particle

Despite the complex chemistry of the BZ reaction, for many purposes the essence of its oscillations is conveyed, with good agreement, by the classic Oregonator model,⁸² which consists of five reaction steps:

Applying mass balance to the intermediates leads to three differential equations that describe the dynamics through concentrations of HBrO₂, Br⁻ and Ce^IV, denoted as X, Y and Z, respectively:

Of course, in the case where the catalyst is confined in particles of cation-exchange resin, all the steps where species Z is involved are also confined exclusively to this phase. As has already been pointed out, various levels of detail are possible in approaching a mathematical representation of such confinement. For example, in one possible approach, Showalter et al.^74–77 chose to model the situation by considering each catalytic particle individually. Inside each particle, the chemical composition was considered to be homogeneous, represented by the triplet of concentrations X_i, Y_i and Z_i. An extra pair of concentrations X_s and Y_s represented the composition of the surrounding solution. Therefore, for a system of N particles, this approach yielded simulations based on 3N + 2 differential equations.

Although Showalter et al. chose to employ the ZBKE model⁸³ of the BZ reaction for their calculations, the overall rationale of their approach may be implemented with any model. By adopting the outlined representation with the more traditional option, the Oregonator, the corresponding 3N + 2 differential equations are

where S_i and V_i denote the individual surface area and volume of the i^th particle, V_s denotes the total volume of the solution, and k_ex is the mass transfer coefficient, which specifies the mass exchange between the catalytic particles and the solution in the form of an intensive quantity, expressing the number of moles of an intermediate that cross the interface per unit of time per unit of interface area and per unit of concentration difference that causes the mass transport. (Specifically, k_ex is assumed to be identical for all species, X, Y and Z. For further details on how the abovementioned system of equations was derived and how k_ex should be understood refer to the more elaborate treatment available in the ESI, Sections S2–S4†).

3. From a population of oscillators to a uniform catalytic phase

In search for a representation that would distinguish the global (bulk) dynamics of the system from the more fine-grained (phase-dependent) features, model 3 is greatly simplified by considering the case where the catalyst-carrying particles are monodisperse. With respect to size, this was, in fact, already assumed by Showalter et al.; nevertheless, their simulations retained the individual character of each particle in assumptions of random variations in their initial states, as well as in the stoichiometric factors f that described the organic subset of the reaction, which were chosen for each particle individually. However, if we consider all the particles to be identical in every respect, including those that Showalter et al. varied, the whole population may be represented at any one time by one triplet of “unison” variables X_b, Y_b and Z_b. By denoting the radius of all particles as r and their total volume as ∑V_i = V_b,tot, we arrive (as shown in the ESI, Section S5†) at the following description of the system:

Having replaced 3N + 2 variables by only five, the size of the problem is now independent of the number of catalyst-carrying particles, and their combined (collective) effect is lumped in the “bulk” variables X_b, Y_b and Z_b.

4. The simplest model of BZ system with catalyst confinement

If the catalyst particles are small enough, the model may be reduced by one step further by assuming that X and Y are exchanged between the particles and the solution very rapidly (in system 4, for instance, r → 0 yields 3k_ex/r → ∞). Under these circumstances, only Z remains with an inhomogeneous distribution, because it is captured inside the catalytic particles. On the other hand, due to rapid mass transport, the concentrations of X and Y must become the same throughout the system, inside or outside the particles (for a discussion showing that this assumption is, in fact, no less realistic than the assumptions that are already implied by other authors in all existing models of this system, see the ESI, Section S6†). By denoting the volume fraction of the system that corresponds to the catalyst-carrying phase as V_b/(V_b + V_s) = φ and emphasizing the confinement of Z (and all its reactions) exclusively to this fraction by denoting it as Z_φ, system 4 is ultimately reformulated into three equations (for a detailed procedure of how this was achieved, see the ESI, Section S6†):

Obviously, this representation of the BZ reaction with the catalyst confined in a discrete fraction of the system is a generalized version of the original Oregonator with only one additional parameter φ, which represents the total volume fraction of the catalyst-confining phase. In fact, the system of eqn (5) corresponds to dynamical equations that could be derived directly from a modified version of the Oregonator, if φ is not considered only as the volume fraction that the catalytic particles occupy in the system, but also as an additional variable stoichiometric factor that modifies the production and consumption of X and Y in reactions involving the catalyst:

It is interesting to note that, based on the mechanism 6, the overall kinetic effect of catalyst confinement may be interpreted as a simultaneous modification of the two steps that lie at the heart of its nonlinearity. Decreasing the volume of the system with the catalyst places more emphasis on the reactions that are catalyst-independent. This formally decreases the order of autocatalysis from 2 to 1 + φ, but at the same time also weakens the negative feedback loop, replacing f by φf.

5. Analysis of the proposed model

In the form given by the system of eqn (5), the effect of catalyst confinement on the reaction may be understood through the same analysis that was originally shown for the Oregonator.^82,84,85 First, after scaling to dimensionless form and with the steady-state assumption applied to Y, we arrive at a reduced system of two ODEs:

In the specific case where φ = 1, the nullcline diagram will be that of the original Oregonator. However, as the value of φ decreases, the parabolic portion of the x-nullcline becomes, according to system of eqn (7), z ≈ x(φ − x)/fφ. Interestingly, this change is at the same time significant and subtle, as illustrated by the nullcline diagrams for various values of φ that are depicted in Fig. 1.

Fig. 1 Nullcline diagram and positions of boundary points A–D of the ODE system of eqn (7) for values of f = 2/3, q = 8 × 10⁻⁴ and φ = 1.0 (red), 0.8 (yellow), 0.6 (green), 0.4 (blue) and 0.2 (violet).

Even though the new parameter φ affects the absolute position of the steady state (and the magnitude of the limit cycle, for that matter), the relative geometry of the nullclines, on the other hand, remains qualitatively the same. For example, if f and q are such that the z-nullcline intersects the parabola of the x-nullcline at its vertex in the case of φ = 1, the same will be true (approximately) for all other values of φ. Varying the stoichiometric factor f leads to the same type of bifurcation behavior as in the original Oregonator, and the bounds on the values of f that correspond to oscillations remain the same for all the values of φ. Nevertheless, φ does affect the stability of the steady state as a function of ε, because the Jacobian of the system is given as follows:

The effect of φ on the locus of points in the ε–f plane, where the trace of this Jacobian vanishes is demonstrated in Fig. 2.

Fig. 2 Locus of Hopf bifurcation points in the f–ε plane, which delimit the oscillatory region of the ODE system of eqn (7), for values of f = 2/3, q = 8 × 10⁻⁴ and φ = 1.0 (red), 0.8 (yellow), 0.6 (green), 0.4 (blue) and 0.2 (violet).

6. Explaining the quorum sensing effect of the catalytic phase content

As can be seen in Fig. 2, as well as proven mathematically, the values of ε that correspond to Hopf bifurcation points change almost proportionally with φ. The maxima of ε that allow oscillations still occur at f = 1, but their values decrease by a factor of φ and the area of the oscillatory region of the system shrinks accordingly. As a consequence, at any fixed value of ε (as given by the ratio of the concentrations of bromate and organic substrate), oscillations may only appear if φ exceeds a certain threshold value that corresponds to a critical content of catalytic particles, and below this threshold, the system will evolve towards a stable steady state. Obviously, this reproduces the essence of observations that led to denoting this phenomenon as quorum sensing, only now in much simpler terms of the classical concepts of Hopf bifurcation.

To further confirm that this model is a suitable minimal representation of the global dynamics of the catalyst-confined modification of the BZ reaction and the corresponding quorum sensing, we can also investigate how the parameter φ enters the expressions for the approximate coordinates of the boundary points of the limit cycle of oscillations (shown in Fig. 1 as A–D). This is summarized in Table 1, and a few examples of oscillations around the critical value of φ are shown in Fig. 3.

Table 1 The values of x, y, and z for the extreme points of the limit cycle (denoted as A–D, referring to Fig. 1), expressed in terms of f, q and φ

	x	y	z
A
B	φ/2	φ/2	φ/4f
C	q + 8q^2/φ	φ^2/8q	φ/4f
D

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Fig. 3 Limit cycle oscillations (red to blue) and the steady state (violet) as values of x, as obtained by integration of the ODE system of eqn (7) for values of f = 2/3, q = 8 × 10⁻⁴, ε = 4 × 10⁻², and φ = 0.20 (red), 0.18 (yellow), 0.16 (green), 0.14 (blue) and 0.12 (violet).

As clearly demonstrated by the coordinates in Table 1, as well as the examples of oscillations in Fig. 3, a decrease in the value of φ also decreases the amplitudes of oscillations. Conversely, the periods of oscillations appear to be affected by the value of φ to a much lesser extent. In fact, if we approximate the period by the time spent in the limit cycle between points A and B (referring to Fig. 1), the period may be estimated as follows:

If approximate values of x_A ≈ φ and x_B ≈ φ/2 are used to evaluate this expression, it turns out that the parameter φ does not appear in the formula for the period of oscillation, and thus the period of oscillation is approximately the same for all the values of φ. The period is only a function of f in unchanged form as reported for the original Oregonator by Tyson.^84,85 These predictions about the amplitudes of oscillations and their periods offer further agreement of the proposed model with experimental data that reports the so-called quorum sensing among catalyst-carrying particles in the BZ reaction.

7. Onsets of synchronization

Finally, it is worth noting that the proposed model of confined-catalyst BZ oscillations also allows a very straightforward expression of synchronization in the case where the system contains two initially distinct populations of catalyst-carrying particles. If their contents in the system are φ₁ and φ₂ and initially each of them is in a different state, i.e. with different initial values of Z, denoted as Z₁ and Z₂, the proposed model describes the system by a set of four equations:

After scaling and application of the steady-state approximation for Y, analogously to the abovementioned case, we find

This system of equations becomes particularly visual after a suitable change of variables. Instead of z₁ and z₂ we reformulate the equations in terms of just one of them, e.g. z = z₂, and their difference, ζ = z₁ − z₂. And instead of the two contents of individual populations, φ₁ and φ₂, we use just one, e.g. φ = φ₁, and their sum φ_tot = φ₁ + φ₂. In terms of these variables we obtain

In this system of three equations, the first two are analogous to system of eqn (7), but there is one more equation that describes the difference between the two populations of catalytic particles. According to this last equation, this difference between the values of z in the two catalytic phases must decay exponentially to ζ → 0 until the system is governed again by only the first two equations, in the same way as system of eqn (7), but with the content of the catalytic phase given by φ_tot.

Certainly, the situation can be more complicated in a case with more than two populations. However, the system of eqn (12) demonstrates that under the conditions that were assumed in developing the proposed model (corresponding to very strong coupling based on very fast mass transfer between the catalytic phase and the solution) two populations of catalyst-carrying particles will always synchronize and the synchronized state will be approached exponentially. Whether this synchronized state will be oscillatory or a steady state is determined only by the total content of the catalyst phase φ_tot. This is illustrated in Fig. 4, which shows two populations of catalyst particles that would not be capable of producing oscillations on their own, but after synchronization, their combined effect yields an oscillatory behaviour.

Fig. 4 Two populations of catalyst-carrying particles, each evolving into a steady state when on its own (pale lines) but synchronized into oscillations when present together (dark lines), as obtained by integration of the ODE systems of eqn (7) and 11 for values of f = 2/3, q = 8 × 10⁻⁴, ε = 4 × 10⁻², x[0] = 0.047 (for all) and φ = 0.12, z[0] = 0.047 (green) and φ = 0.04, z[0] = 0.0053 (violet).

8. Conclusions

In conclusion, the proposed model appears to describe the essential features of the BZ reaction with a particle-confined catalyst and the sudden transitions between its globally steady state and globally synchronized oscillations. These were previously interpreted only by the concept of quorum sensing and by deriving the global behavior from a complicated analysis of how the individual particles interact locally. Certainly, the proposed model does not aim to interpret all aspects of the dynamics of this system, e.g. it cannot provide any local details of the intermediate states involved in the transitions. These will be observed only with more elaborate models that consider each catalytic particle individually. However, the predictions of the model are in agreement with how the fully synchronized states are observed to occur experimentally. This suggests that the sudden transitions from a non-oscillatory regime to uniform limit cycle oscillations at a specific threshold of the catalytic particle content may be understood in terms of the classical concepts of Hopf bifurcation.

As a consequence, it appears that the sudden quorum sensing of catalyst-carrying particles in the BZ reaction is, in essence, not very different from other transitions of the system, as observed when any other reagent crosses the threshold that is required for oscillations. Underlying the quorum sensing of the catalyst-confining particles, the system appears to have important features that are of a global nature, contributing to its dynamics through the global mass balance of chemicals modifying the overall stoichiometry rather than through phase details of individual interactions. Fine-graining of models to the level of individual particles for the purpose of representing this global behavior thus appears to be superfluous, and the proposed simple model appears to be a reasonable macroscopic limit for the representation of the global aspect in the collective dynamics of BZ reaction with catalyst confinement in large populations of catalytic particles, including the phenomenon of their sudden quorum sensing. As such, we believe that this model might provide a convenient approximate first-order representation of the system that is well-suited for adding localized refinements of phase information.

Acknowledgements

The author wishes to thank Juan Pérez-Mercader for mentoring and support, Alberto P. Muñuzuri for critical reading of the manuscript and committed discussions about alternative interpretations of collective phenomena observed in the BZ reaction with catalyst confinement, J.-S. Gagnon for his many valuable advices on improving the clarity of the manuscript, and all other members of Pérez-Mercader's team for countless inspiring and motivating exchanges.

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Footnotes

† Electronic supplementary information (ESI) available: Detailed commentary to eqn (1)–(5) and (11) and (12). See DOI: 10.1039/c5ra12841b

‡ Current address: Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University in Bratislava, Mlynská dolina, 84215 Bratislava, Slovakia. E-mail: E-mail: erik.szabo@fns.uniba.sk

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