Switches induced by quorum sensing in a model of enzyme-loaded microparticles

Abstract

Quorum sensing refers to the ability of bacteria and other single-celled organisms to respond to changes in cell density or number with population-wide changes in behaviour. Here, simulations were performed to investigate quorum sensing in groups of diffusively coupled enzyme microparticles using a well-characterized autocatalytic reaction which raises the pH of the medium: hydrolysis of urea by urease. The enzyme urease is found in both plants and microorganisms, and has been widely exploited in engineering processes. We demonstrate how increases in group size can be used to achieve a sigmoidal switch in pH at high enzyme loading, oscillations in pH at intermediate enzyme loading and a bistable, hysteretic switch at low enzyme loading. Thus, quorum sensing can be exploited to obtain different types of response in the same system, depending on the enzyme concentration. The implications for microorganisms in colonies are discussed, and the results could help in the design of synthetic quorum sensing for biotechnology applications such as drug delivery.

1. Introduction

The remarkable ability of cellular biological systems to coordinate activity has fascinated scientist for decades. The term quorum sensing was first applied to bacteria that displayed a population-wide change in behaviour above a critical density of cells, driven by production and release of a small diffusible molecule, the autoinducer, into the environment [1,2]. In bacteria, increases in cell density can induce bioluminescence and biofilm formation that protects the cells from antibiotics [3]. Other microorganisms such as yeast and the slime mould, Dictyostelium discoideum, display density-dependent dynamics including synchronized chemical oscillations above a critical cell density [4,5]. These oscillations play an important part in the life cycle of D. discoideum as they result in travelling waves of cyclic AMP used to direct the motion of cells and formation of multicellular slugs when individual cells are starving [6].

More recently, quorum sensing has inspired the investigation of synchronous behaviour in various systems including inorganic catalytic microparticles [7,8], electronic circuits [9,10], laser arrays [11] and genetically modified organisms [12]. Although diverse in their underlying mechanisms, common to these systems is some internal means of amplifying a signal (positive feedback) and communication of the signal via a common surround. Combined, these factors drive a sudden sharp change in state across the whole population. Switch-like, ultrasensitive responses can arise in cellular systems through a number of mechanisms; positive feedback is generally required for bistability and oscillations [13]. Applications are beginning to emerge, for example, a synthetic quorum sensing circuit in genetically modified bacteria has been exploited for pulsatile drug delivery in vivo [14]. Enzyme-loaded particles or vesicles also have potential applications in medicine [15] and are excellent candidates for synthetic quorum sensing, but evidence of this behaviour has not been reported to date.

Mathematical modelling and simulations have provided insights into quorum sensing in both natural and synthetic systems [16–20]. Here, simulations were performed in order to determine the behaviour of groups of enzyme-loaded microparticles in a bath of substrate solution. An enzyme-catalysed reaction was chosen that displays positive feedback through the pH; the urea–urease reaction. This reaction is well characterized and occurs across a variety of plants and cellular organisms [21,22]. The enzyme urease is a virulence factor produced by certain bacteria and, conversely, has been used in engineering applications [23], materials synthesis [24,25] and self-propelled micro- or nanomotors [26,27]. Our goal was to determine the types of response that might be obtained with changes in group size under reaction–diffusion conditions.

The model was inspired by ureolytic bacteria such as Helicobacter pylori in the acidic, non-buffered environment of the stomach and Proteus mirabilis which colonizes the urinary tract and devices such as catheters [28]. Both of these microorganisms produce urease to break down urea and make ammonia thereby raising the pH of their environment. They also form biofilms—communities of microorganisms attached to a surface (e.g. catheter wall) and embedded in glue-like extracellular polymeric substances. Small molecules, such as acid and urea, diffuse between the biofilm and the external solution, whereas enzymes are typically confined to the biofilm. Here, we used simulations to determine the collective behaviour of urease-loaded cells under similar conditions.

We show that different transitions can be obtained with quorum sensing in the same system. Three sharp transitions in state, given by the pH, were obtained by increasing the number of diffusively coupled urease beads: a sigmoidal switch (a buzzer), oscillations (blinker) and a bistable switch (toggle)—dynamical responses that all play an important role in the functioning of cells [29,30]. The generic features are likely to be observed in numerous confined enzyme-catalysed reactions that show feedback. The implications of the results are discussed with regard to cellular organisms, as well as for applications in biotechnology.

2. Model

The model is based on experiments [31], in which polymer beads were loaded with the enzyme urease and placed in a solution of acid (pH 4) and urea in a Petri dish. In earlier work [31,32], we explored the behaviour of individual beads. Now, we consider the situation, illustrated in figure 1a, in which a group of beads are placed in close proximity and loaded with a pH indicator to show the change in pH when the reaction occurs. Complete conversion of urea to ammonia takes several days; hence, the concentrations in the bulk solution are approximately constant over several hours. The experimental set-up is a grossly simplified version of the biofilm scenario described in the introduction; however, it demonstrates the feasibility of observing the behaviours in vitro.

Figure 1.

(a) Illustration of an experimental set-up with urease-loaded polymer beads placed on the base of a 5 cm diameter Petri dish containing 50 ml solution of urea (0.1 M) and acid (pH 4). The enzyme beads contained a pH indicator which is yellow when acidic and purple when basic (pH >7.5). (b) The computational domain used in simulations showing in this case a single urease-loaded cell as a sphere (number of enzyme beads N_T = 1) and the surrounding solution cells as circles (N_S = 94). The enzyme bead is directly coupled to the solution cells in red; green cells are next neighbours and blue cells show third neighbours. The blue (dark) solution cells at the edge of the domain are coupled to the bulk solution with constant concentrations of urea and acid. (c) A two-dimensional (2D) slice of the domain showing a group of hexagonally packed enzyme beads with the number of rows N_r = 2 and N_T = 19. The length scale of cells in simulations was l = 100 µm. (Online version in colour.)

The urea–urease reaction results in production of ammonia and an increase in pH through the following overall processes:

2.1

2.2

Following on from our earlier work, the full model of the reaction was reduced to two variables, preserving the generic behaviour of the system (see electronic supplementary material for more detail). The rate of change of substrate concentration, S, and acid concentration, H⁺, was given by:

2.3

where denotes the discrete Laplacian for hexagonal close packing, D_s and D_H are diffusion constants of substrate urea: D_S = 1.4 × 10^–5 cm² s^–1 and acid: D_H = 9 × 10^–5 cm² s^–1, K_w is the water ion product from the equilibrium:

2.4

and R is the rate of the enzyme-catalysed step:

2.5

This is a modified Michaelis–Menten expression [33,34] that takes into account the bell-shaped enzyme rate dependence on acid concentration where K_ES1 and K_ES2 are the binding constants of the enzyme to acid: K_ES1 = 5 × 10^–6 M, K_ES2 = 2 × 10^–9 M. The maximum enzyme rate V_max was given by V_max = k_eE where k_e = 3.7 × 10^–6 M s⁻¹ U⁻¹ ml and E = [enzyme] in U ml⁻¹: the enzyme concentration was in units ml⁻¹ (M min⁻¹ ml⁻¹) in order to compare with experimental data for urease [22]; and the Michaelis constant was K_m = 3 × 10^–3 M.

Reaction–diffusion simulations were performed on a three-dimensional hexagonal close packed (hcp) coarse grid of spatial step size l = 100 µm. The hexagonal packing and length scale were chosen to be amenable to future experimental investigations involving approximately 100 µm-sized enzyme-loaded beads submerged in a solution of acid and urea. The coarse grid approach allowed us to obtain data from multiple runs in order to map out behaviour in phase space with both homogeneous and heterogeneous distributions in enzyme loading. It also allowed us to simulate over a thousand enzyme-loaded beads with a total length scale greater than 1 cm on a reasonable timescale. A similar approach has been taken for modelling heterogeneous biofilms in three dimensions [35].

The computational domain consisted of two types of cell: enzyme-loaded cells (domain Ω_E) on an inert surface (i.e. the base of the Petri dish) and solution cells (domain Ω_s) containing urea and acid but no enzyme to represent a thin solution layer at the interface of enzyme-loaded beads and the bulk solution (figure 1b). The total number of cells in a given simulation was given by the sum of the enzyme cells and the neighbouring solution cells. Urea and acid diffuse in from the bulk solution through domain Ω_s and are consumed in the beads resulting in a gradient of these species. We found that with at least three neighbours of solution cells around the enzyme beads, the concentrations of acid and substrate approached the constant, bulk solution values smoothly.

The total number of enzyme beads was given by N_T = 3N_r(N_r + 1) + 1, where N_r indicated the number of rows of enzyme cells after the central cell; an example with N_r = 2 is shown in figure 1c. Changes in group size were achieved by increasing the number of rows of beads. The initial conditions for enzyme beads (domain Ω_E) at t = 0 were given by:

The solution cells contained no enzyme thus R = 0. The initial conditions for solution cells (domain Ω_s) at t = 0 were given by:

where S₀ and H₀ are the bulk solution concentrations. The value of H₀ in all simulations was 1 × 10⁻⁴ M (pH₀ = 4) while S₀ and E₀ were varied.

A Dirichlet boundary condition was applied at the sides and top of the solution cells (boundary ∂Ω_s) to provide the cells with a constant supply of substrate and acid from the bulk solution (i.e. the solution in the rest of the Petri dish):

No-flux Neumann boundary conditions were applied at the base of the domain (boundary ∂Ω_b) to simulate the diffusion barrier at the base of the Petri dish:

For heterogeneous loadings, simulations were performed using a normal (Gaussian) random number generator for enzyme concentration, with mean μ_E and coefficient of variation σ = 10–30%. Data from 11 runs with different initial spatial distributions of enzyme were collected for each value of μ_E and σ, and the number of times a resultant behaviour (high pH steady state, oscillatory and low pH steady state) occurred was recorded relative to the total number of runs.

3. Results

3.1. Switches with the substrate

The behaviour of a single enzyme bead in substrate solution was mapped out in enzyme–substrate space (figure 2a). Positive feedback in the urea–urease reaction is driven by the product, ammonia and the bell-shaped rate–pH curve (inset, figure 2a). The maximum enzyme rate is at pH 7, correlated with a maximum in the active form of the enzyme. If the enzyme is in a solution of acid, the rate is initially low. The production of ammonia raises the pH and the rate of reaction accelerates. Negative feedback was provided by the constant supply of acid by diffusion from the surrounding solution [32].

Figure 2.

Phase diagram as a function of enzyme and substrate concentrations mapping dynamic behaviour of (a) a single 100 µm bead and (b) a group of 100 µm beads with N_r = 10 and low pH state (SS_L), high pH state (SS_H), bistable (BS) and oscillatory (OSC). With N_r = 6: (c) sigmoidal switch in pH with E₀ = 8000 U ml⁻¹; (d) oscillations in time with E₀ = 1000 U ml⁻¹; (e) bistable switch with E₀ = 200 U ml⁻¹. In (c–e), central bead pH (thick line) and average pH of the group (dotted line). (Online version in colour.)

Although ammonia was not explicitly included in the two-variable model, its concentration is correlated with the pH. A cross-shaped phase diagram was obtained, where at low substrate, S₀, and enzyme concentration, E₀, the ammonia was produced at an insufficient rate compared with the influx of acid from the surround resulting in an unreacted, low pH steady state (SS_L) in the bead. At high S₀ and E₀, the rate of reaction is high enough to overcome the influx of acid and the bead switched to a reacted, high pH state (SS_H). Separating these two states are regions of oscillations (OSC) or bistability (BS) in pH. The results qualitatively agree with previous findings obtained from two- and eight-variable compartment models of the urea–urease reaction [32].

The phase diagram for a hexagonal array of beads with number of rows N_r = 10 is shown in figure 2b. Increasing the number of beads shifted the high pH states to lower enzyme and substrate concentrations, but the same general features were preserved. Three different types of transition were obtained with an increasing substrate: a sigmoidal switch in pH at high enzyme (figure 2c), switch to oscillations at an intermediate enzyme (figure 2d) and a bistable switch at low enzyme concentrations (figure 2e). The beads at the edge of the group typically had a lower pH than the central beads resulting in a reduced average pH over the entire domain (dotted line). These transitions may be considered switches in the sense that there is a sharp change from an ‘off’ (low pH) state to an ‘on’ (high pH) state.

3.2. Switches with group size

The same three types of transition were obtained if instead of increasing substrate, the concentration of substrate was fixed and the number of rows of beads (N_r) was increased. An E₀ – N_r phase diagram is plotted in figure 3 showing regions of low pH steady state (blue, SS_L), oscillations (purple, OSC) and high pH steady state (orange, SS_H). Increasing N_r had a similar effect to increasing substrate. When the N_r is less than 4, acid diffused in from the surround keeping the concentration of ammonia and the pH low in the beads. The inset shows the number of beads at the edges or the array compared with the total number of beads [N_edge/N_T = 3/(3(N_r + 1) + 1/N_r)]. As N_r was increased, a smaller fraction of the beads was in contact with the acid at the edges of the array and a sharp transition to a high pH state or oscillations was obtained.

Figure 3.

Phase diagram as a function of enzyme concentration and number of rows of enzyme beads where blue (dark) = SS_L, purple = OSC and orange = SS_H. The tiles show the pH in time of the central bead and the inset shows the ratio of edge beads to total beads with increasing N_r. The value of S₀ = 0.4 mM. (Online version in colour.)

With S₀ = 0.4 mM and E₀ = 600 U ml⁻¹, a switch from low to high pH was obtained (figure 4a). When E₀ was decreased, oscillations were observed above a threshold number of cells (figure 4b). A bistable switch could not be obtained with reasonable values of N_r with S₀ = 0.4 mM; the fraction of edge beads approached zero with N_r = 20, and thus, little change was observed in the dynamics for a larger number of rows. However, bistability was obtained within the range N_r = 1–20 when S = 0.5 mM, as shown in figure 4c.

Figure 4.

Switches in pH with the number of rows of beads. (a) Sigmoidal switch, (b) switch to oscillations (maximum pH shown), (c) bistable switch and pH of beads in the array with N_r = 7 (SS_L) and 8 (SS_H). The central bead pH (thick line) and average pH of the group (dotted line) are shown and S₀ = 0.4 mM in (a,b) and S₀ = 0.5 mM in (c). (Online version in colour.)

Quorum sensing is typically associated with increases in the autoinducer concentration in solution and/or the rate of loss of autoinducer initiating autocatalysis as the number of cells is increased (one does not necessarily imply the other) [8]. Here, the transition was correlated with the change in pH and hence acid concentration. The pH profile across a central slice of the array is shown in figure 5a for the same conditions as in figure 4c. For N_r < 8, the pH across the group was low (less than 5), and a gradient in pH can be seen from the centre bead outwards. The pH was lowest at the edges of the group where the beads were in contact with the acid solution from both the side and above. As N_r was increased, the pH of all the beads increased as a result of the decrease in the fraction of edge beads. At N_r = 8, there was a large amplitude increase in pH across the group. Correspondingly, the acid concentration fell to a threshold level in both the edge beads and the adjacent solution cells up to N_r = 7 (figure 5b). Note, however, the difference between the edge beads and solution acid concentration increased. So, the increased reaction rate with increasing pH must overcome the increased influx rate of acid to initiate autocatalysis.

Figure 5.

Switch in pH with the increasing number of rows from 1 to 12 for conditions in figure 4c. (a) pH profiles and (b) concentration of acid in beads at the edge of the group (lower curve, circles) and adjacent solution cells (upper curve, squares). Dashed lines indicate the transition to the high pH state.

With the substrate and acid concentrations fixed, the critical N_r for a change in state and the outcome of the transition, whether to oscillations or high pH steady state, were determined by the enzyme concentration. The threshold increased with decreasing E₀ (figure 3).

3.2.1. Sigmoidal switch with N_r

A sigmoidal switch in pH was obtained for sufficiently high enzyme (E₀ > 600 U ml⁻¹ in figure 3) with increasing the number of rows of beads. Sigmoidal switches are reversible: in figure 4a, a large amplitude increase in pH occurred as N_r was increased from 3 to 4 and decreasing N_r back to 3 resulted in a drop back to low pH. This switch is referred to as a buzzer [29] because the ‘on’ state is reached whenever a parameter, here N_r, is raised above a single threshold value.

3.2.2. Bistable switch with N_r

Bistable switches in pH were obtained with a low concentration of enzyme (E₀ = 300 U ml⁻¹ in figure 4c). The value of the pH was dependent on the system history. So, if N_r was increased, the beads remained in a low pH state until a threshold was reached at N_r = 8, then the pH switched to high. If N_r was then decreased, the pH remained in a high state until the lower limit of N_r = 3 when it dropped back down. Between these values, the low and high pH state coexisted. This is an example of a toggle switch [29] in the sense that it can be flipped between the ‘on’ (high pH) and ‘off’ (low pH) states.

3.2.3. Oscillations with N_r

A transition to oscillatory behaviour occurred at intermediate enzyme levels (E₀ = 400–600 U ml⁻¹ in figure 3). A time series of the oscillatory state, referred to as a blinker [29], is shown in figure 6. The array did not oscillate uniformly; there was a phase lag between the centre and the outer edge of the array.

Figure 6.

Illustration of oscillatory behaviour in an array of urease beads with S₀ = 0.4 mM, E₀ = 600 U ml⁻¹ and the number of rows of beads N_r = 6 (surrounding solution cells not shown). Movie available at https://digitalmedia.sheffield.ac.uk/media/figure6/1_v6vhpyg2. (Online version in colour.)

Different types of oscillatory dynamics were observed depending on the size of the group and enzyme concentrations. For N_r = 9 and E₀ = 500 U ml⁻¹, the whole group oscillated with a front spreading from the centre of the array outwards and then contracting in from the edges (figure 7a). The length scale of the array (23 * 100 µm = 2.3 mm) was small and a reaction–diffusion wave was not observed; the acid diffused in from the edge quenching the high pH state before recovery of the central beads could take place. For larger enzyme, E₀ = 600 U ml⁻¹ and N_r = 11, the central cells remained in the high pH state while the outer cells oscillated (figure 7b). This is a mixed high pH-binker state. A dual frequency state was also observed as the number of rows was increased to N_r = 19, where the edge beads oscillated with double the frequency of the centre beads (figure 7c).

Figure 7.

Oscillatory dynamics of bead arrays with S₀ = 0.4 mM and pH–time traces show central bead pH (thick line) and average pH of array (dotted line). (a,d) Blinker with E₀ = 500 U ml⁻¹ and N_r = 9. (b,e) Mixed high pH-blinker state with E = 600 U ml⁻¹ and N_r = 11; (c,f) dual frequency state E₀ = 500 U ml⁻¹ and N_r = 19. Movies available at (a) https://digitalmedia.sheffield.ac.uk/media/figure7a/1_rlm18nvc, (b) https://digitalmedia.sheffield.ac.uk/media/figure7b/1_gf26kloi and (c) https://digitalmedia.sheffield.ac.uk/media/figure7c/1_68yh1s8p. (Online version in colour.)

3.3. Transitions with numbers of beads: heterogeneous distributions of enzyme

The influence of heterogeneity in enzyme loading on these dynamical transitions was also investigated with values of E selected randomly from a normal distribution. Multiple runs were performed to estimate the probability of a high pH steady state (SS_H), oscillatory (OSC) or low pH steady state (SS_L) for a given mean enzyme activity, μ_E and coefficient of variation, σ. The resulting μ_E–N_r phase diagram was similar in form to figure 3, even with σ = 30% (figure 8a). The sharp transitions with N_r were still obtained, resulting in a sigmoidal switch at μ_E = 700 U ml⁻¹ and a transition to oscillations at μ_E = 500 U ml⁻¹ (figure 8b,c). However, with μ_E = 500 U ml⁻¹, there was an increased probability of the high pH steady state as N_r and σ were increased.

Figure 8.

Switches in heterogeneous enzyme arrays with S₀ = 0.4 mM, pH₀ 4 and μ_E = mean enzyme activity. (a) Phase diagram with σ = 30% where each bar shows fraction of SS_H (orange), SS_L (blue, dark) or OSC (purple) obtained from multiple runs. (b,c) Probability of either SS_H or OSC (active, black squares) and probability of SS_H (orange line) with (b) E = 500 U ml⁻¹ and (c) E = 700 U ml⁻¹. (Online version in colour.)

The spatial distribution of the enzyme played an important role in the selection of dynamical behaviour; SS_H or OSC. In two separate runs with μ_E = 500 U ml⁻¹ and σ = 10%, an oscillatory state was obtained with total enzyme of E_T = 3.6162 × 10⁵ U ml⁻¹ and a high pH steady state was obtained with lower total enzyme E_T = 3.6004 × 10⁵ U ml⁻¹. The formation of the steady state and an oscillatory state are shown in figure 9, as well as the spatial configuration of the enzyme concentration (greyscale) and normal distribution of enzyme (figure 9c). Waves propagated asymmetrically across the domain in case b, resulting in travelling structures that gave rise to aperiodic average pH–time traces. The time-average pH over the array was 8 in the case of the steady state and less than that in the oscillatory case; however, individual spikes reached up to pH 9.

Figure 9.

Dynamic behaviour in heterogeneous enzyme arrays with μ_E = 500 U ml⁻¹, σ = 10%, S₀ = 0.4 mM and N_r = 15. First image shows spatial enzyme distribution and subsequent evolution of the array is shown in a series of images where blue = low pH, orange = high pH. (a) Steady state and (b) oscillations. (c) Normal distribution of enzyme loading. (d) and (e) Average pH in the array in time for (a) and (b), respectively. Movie for (b) available at https://digitalmedia.sheffield.ac.uk/media/figure9/1_vkbhfxpa.

4. Discussion

Here, we examined transitions in the behaviour of groups of enzyme-loaded microparticles (beads) that displayed feedback and exchanged chemicals with a common surround via passive diffusion. Our coarse grid approach with the two-variable model allowed us to explore parameter space and identify some generic features that may aid in the implementation of synthetic quorum sensing in applications. The simulations were inspired by quorum sensing in bacteria and other single-celled organisms and may provide some insights into some of the dynamic behaviours observed in growing colonies of microorganisms.

The term quorum sensing refers to a population-wide change in behaviour above a threshold number or density of cells [1,3]. In line with other work, we considered quorum sensing transitions in a uniform layer of cells with constant local density but growing in size [16]. However, the spatial proximity of cells within a colony may play a role in such transitions, as well as other processes that influence the mass transfer such as advection. This has led to the introduction of the terms ‘diffusion sensing’ and later ‘efficiency sensing’ in order to take into account these factors [36]. Simulations were performed with heterogeneities in enzyme loading which likely play a similar role in clustering effects although this warrants further investigation. Nevertheless, we found that the sharp changes in state with increasing group size were robust.

Quorum sensing in cells involves the production and release of small diffusible molecules into the extracellular solution. Changes in state are generally associated with a build-up of autoinducer in the surrounding solution to some threshold level or a decrease in the loss rate of autoinducer from cells by diffusion [8,16]. One or more autoinducers may be involved in a complex network of reactions. For example, Dictyostelium cells use the molecules PSF and cAMP as intercellular signals [37]. PSF accumulates in the extracellular solution with increasing cell density. When PSF reaches a threshold level and food is in short supply, this glycoprotein initiates a series of processes resulting in activation of the enzyme required for cAMP synthesis. The cAMP catalyses its own production and is emitted in pulses, propagating as waves through the colony that direct the motion of cells.

Here, a well-characterized enzyme-catalysed reaction was selected that is both present in microorganisms and accessible in vitro: the urea–urease reaction [22]. In a simple analogy to a biological quorum sensing circuit, the enzyme, urease, was confined to a microparticle and cell-to-cell communication was achieved through diffusion of acid and substrate. The enzyme reaction raised the pH and feedback occurred as an increase in pH led to an increase in rate. An individual bead was unable to raise the pH sufficiently to overcome the influx of acid from the surrounding solution. However, in a group of beads, there was a lower fraction of beads in contact with acid at the edge of the array and the pH increased in both the beads and the adjacent solution cells to some threshold level, initiating autocatalysis.

The signal–response curves obtained here are switch-like in the sense that there is a change from a low pH ‘off’ state to a high pH ‘on’ state with increases in group size. The nature of the switch was found to depend on the enzyme concentration of the beads: at high enzyme a sigmoidal switch (buzzer) was obtained; at intermediate enzyme oscillations were observed (blinker) and at low enzyme a bistable (toggle) switch resulted. These are all important dynamical responses that arise in cellular systems [29], and the results have some interesting implications for growing colonies of microorganisms.

If cells contain sufficient enzyme, then a sharp transition to a high autoinducer ‘on’ state is obtained with increasing group size. This switch, the buzzer, is not robust as small changes in parameters in the vicinity of the transition point result in collapse of the behaviour; however, the cells are producing large amounts of enzyme. If the cells contain intermediate amounts of enzyme, an increase in the group size results in an oscillating state, a blinker. The amount of autoinducer produced in time is lower than for the sigmoidal switch, but conversion is still achieved, and pulses of autoinducer can be used to direct the motion of cells.

For low enzyme concentrations, the system displays a bistable toggle switch with increasing group size. This is useful, because the cells are producing small amounts of enzyme but the switch is robust to noise under these conditions. Once the transition to the ‘on’ state is made, small changes in group size or substrate concentration do not lead to a return to the low autoinducer state.

Although it is not implicated in quorum sensing, the enzyme urease is a virulence factor exploited by bacteria such as H. pylori and P. mirabilis in the non-buffered environment of the stomach or urinary tract. The increase in pH associated with the reaction is believed to protect H. pylori against the acidic environment of stomach [38]. Proteus mirabilis forms rafts—small groups of cells linked together—that allow the bacteria to rapidly colonize catheters. Cells produce a particularly potent urease that drives an increase in urine pH and precipitation of phosphates leading to the formation of kidney stones and catheter encrustations [28]. We have shown how sharp switches in pH can be obtained when a group reaches a critical size. There may be gradients in pH in time and space, but with feedback the maximum pH obtained locally can result in sufficiently high values to trigger rapid biomineralization, even if the average pH of the whole system is lower. It would be of interest to couple the enzyme processes included here with cell motion in order to better understand how feedback through pH may influence pathogenic behaviour [39].

The main reason for our choice of urease was that the system reported here is implementable in experiments. Urease has been used in sensing, crack repair (by inducing calcium carbonate precipitation) [40] and polymer synthesis [24], and has been immobilized on numerous solid supports including alginate [41]. In earlier work, it was demonstrated how features such as waves and oscillations obtained in the two-variable urea–urease model are also possible in the full model including all chemical processes and enzyme inhibition, albeit over a smaller region of parameter space [32,42]. Propagating waves of pH and bistable switches have been obtained in the gel beads, but oscillations were not observed, probably because a key requirement is the differential transport (D_H > D_S) of acid and substrate, and diffusion constants for acid in gels may be lower than for dilute solutions [43]. Evidence of collective behaviour has not yet been reported in urease beads.

The cross-shaped phase diagram is a universal map, spanning many different mechanisms of autocatalysis, that has been used to find oscillations and patterns in chemical systems [44]. The same general topology was obtained here in enzyme–substrate and enzyme–group size phase space. The feedback mechanism exploited involves coupling the bell-shaped rate–pH curve with production of an acid or base. Originally proposed in simulations with an esterase [45], this method is widely applicable because most enzymes display similar rate–pH curves [46]. It seems likely that a similar diagram will be obtained for other enzyme-catalysed reactions, as well as other autocatalytic processes.

Synthetic quorum sensing might be exploited in biotechnology to induce a sharp change in state in response to a change in a density or group size in, for example, targeted drug delivery. Collective behaviour has been extensively investigated in inorganic catalytic particles and more complex behaviours than reported here are possible [47,48]. However, for applications in medicine biocompatible feedback is required. Feedback itself is widely used for complex information processing in biological cells. There is increasing interest in the design of biocompatible reaction networks involving organic molecules [49,50], peptides, enzymes [51,52] and even DNA [53,54] that might be used to generate bioinspired emergent behaviour in synthetic systems [55]. Enzyme-loaded microparticles or vesicles remain the best candidates for obtaining collective behaviour inspired by bacteria such as quorum sensing.

5. Conclusion

We have demonstrated in reaction–diffusion simulations how three different transitions can be achieved with increasing numbers of enzyme-loaded microparticles: a sigmoidal buzzer, an oscillatory blinker and bistable toggle switch. The simulations exploited the use of a single enzyme, urease, found in numerous plants and microorganisms, that raises the pH of the medium and experimental implementation of the results is feasible. The combination of cell-to-cell communication and feedback might be exploited to generate more complex collective behaviours and spatial organization in enzyme catalytic particles for bioinspired dynamic materials or devices.

Data accessibility

Electronic supplementary material, movies and code are available at the University of Sheffield repository ORDA: 10.15131/shef.data.5357494; 10.15131/shef.data.5357503 and 10.15131/shef.data.535750610.15131/.

Authors' contributions

T.B. and A.F.T. conceived the study. T.B. wrote the code and collected the data. A.F.T. and T.B. performed data analysis and wrote the manuscript.

Competing interests

We declare we have no competing interests.

Funding

The research was supported by Engineering and Physical Science Research Council grant no. EP/K030574/2.