# Synthetic quorum sensing in model microcapsule colonies

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Edited by Andrea J. Liu, University of Pennsylvania, Philadelphia, PA, and approved June 30, 2017 (received for review February 10, 2017)

## Significance

Many organisms, from bacteria to humans, can sense their local population density and modify their behavior in crowded environments. This density-dependent behavior, known as quorum sensing, is a highly desirable attribute for synthetic systems because it permits novel self-recognition and self-regulating functionality. Using theory and simulation, we design chemical-producing microcapsules that display quorum sensing. When the chemical production is regulated by a biomimetic feedback loop, the capsules exhibit tunable transitions between steady (“off”) and oscillatory (“on”) states as a function of the number and number density of microcapsules in a colony. Such a system can behave as a mechanoresponsive material, modulating chemical activity when applied stresses alter the spatial configuration of the capsules.

## Abstract

Biological quorum sensing refers to the ability of cells to gauge their population density and collectively initiate a new behavior once a critical density is reached. Designing synthetic materials systems that exhibit quorum sensing-like behavior could enable the fabrication of devices with both self-recognition and self-regulating functionality. Herein, we develop models for a colony of synthetic microcapsules that communicate by producing and releasing signaling molecules. Production of the chemicals is regulated by a biomimetic negative feedback loop, the “repressilator” network. Through theory and simulation, we show that the chemical behavior of such capsules is sensitive to both the density and number of capsules in the colony. For example, decreasing the spacing between a fixed number of capsules can trigger a transition in chemical activity from the steady, repressed state to large-amplitude oscillations in chemical production. Alternatively, for a fixed density, an increase in the number of capsules in the colony can also promote a transition into the oscillatory state. This configuration-dependent behavior of the capsule colony exemplifies quorum-sensing behavior. Using our theoretical model, we predict the transitions from the steady state to oscillatory behavior as a function of the colony size and capsule density.

Quorum sensing (QS) refers to the ability of organisms in a population to assess the number and density of individuals present, allowing a specific behavior to be initiated only when a critical threshold in the population size and density is reached. QS plays a vital role in the life cycle of bacteria (1, 2), yeast (3, 4), and slime molds (5, 6), as well as social insects (7). In microorganisms, QS is based on chemical signaling among individuals in a colony. Bacteria (1, 8), for example, produce and secrete signaling molecules, which diffuse into the surrounding medium where they can be detected by other cells in the population. Through regulatory networks, the signaling molecule acts as an autoinducer; the rate of production of the autoinducer increases with its concentration. When the cell density is low, the concentration of the signaling molecule is low, and production is maintained at a low, basal rate. The cells are considered to be in the “off” QS state. When the population density is high, each cell detects a high concentration of the signaling molecule resulting from the collective production of many nearby neighbors. The cells are switched “on,” increasing production of the signaling molecule and activating further metabolic pathways that are triggered by QS. Hence, spatially separated cells interact through regulatory networks, which coordinate the collective behavior of the colony.

An intriguing challenge in both synthetic biology and materials science is to design man-made systems that exhibit behavior analogous to QS, i.e., sensing and responding to the system size and density. Such synthetic QS abilities could provide a route to creating materials and devices with novel self-recognition and self-regulating functionality. Namely, elements in these materials systems would effectively “count” and only operate when the number of neighboring elements becomes sufficiently high. For example, a soft material with embedded QS elements could switch on a certain chemical behavior when compressed and switch off when stretched.

Several mathematical models of the autoinducer system have been developed to describe how the switch from the off to the on state is controlled in bacterial QS (9⇓–11). Common to these models is the existence of multiple steady states such that a change in the concentration of external signaling molecules leads to a sudden jump from a low production state to a high production state. The resulting QS transition is hysteretic; the cell density at which the population switches on is higher than the density at which the population switches off. This hysteresis prevents the colony from rapidly alternating between on and off states in the presence of small fluctuations, thus stabilizing the population behavior.

To accurately detect the population density in a synthetic materials system, however, the existence of two stable states leads to ambiguity, as the behavior of the system would depend on the history of past states. A potential solution to this problem, allowing self-regulating materials to uniquely determine the present density, is to use a different regulatory network known as the “repressilator,” which exhibits a single steady state. The repressilator circuit (12) consists of a cycle of three genes that code for the repressors of the next gene in the cycle, and thus the whole system operates through an engineered negative feedback loop. Under the repressilator, chemical “c_{1}” suppresses the production of chemical “c_{2},” which suppresses the production of chemical “c_{3},” which in turn suppresses the production of “c_{1}.” The repressilator was engineered as an artificial gene network in *Escherichia coli* cells, resulting in oscillations in the production of a specific protein (12).

Mathematical models for chemical production regulated by the repressilator network have shown that the long-term behavior of the network is either stationary or oscillatory, depending on system parameters (12⇓⇓–15). Associating the stationary behavior with an off state and oscillatory behavior with an on state, the repressilator network could support QS ability if the emergent state depended on the population density. Prior models (12⇓⇓–15), however, described the dynamics of small systems in which spatial variations in concentrations are unimportant, such as within a single cell or tightly packed cluster of a few cells. Hence, these models cannot reveal how changes in colony size and number density could lead to oscillations and, ultimately, QS.

It is known that oscillatory reactions can lead to complex dynamical behavior, including the emergence of traveling waves, in large colonies of chemically communicating cells (16) or continuous reactive media (17). Moreover, studies of systems at intermediate sizes, such as a growing colony of bacteria (18) and clusters of ∼100 catalytic particles in an excitable medium (19), have demonstrated that oscillatory networks can endow a system with QS characteristics; oscillations emerge and become synchronized only when a critical density and number of agents is reached. Theoretical understanding of the conditions required for oscillations in such systems, however, is lacking. Specifically, it remains unclear what combinations of numbers and densities of agents allow a “quorum” to be reached (as indicated by the presence of sustained oscillations) for a given regulatory network and set of physical and chemical parameters.

As a route to addressing these questions and thereby articulating necessary conditions for QS in a synthetic system, we model the dynamics in a colony of immobile microcapsules that release diffusible signaling chemicals into the surrounding fluid. We consider the production of these signaling chemicals to be regulated by an abstraction of the repressilator feedback network; namely, we do not model the biochemical pathways of a genetic repressilator circuit but simply consider three chemical species that follow the cycle of inhibition prescribed by the repressilator network. Adopting the generalized definition of QS as a qualitative switch in the behavior of the system with a change in the size and density of a colony, we numerically determine conditions under which colonies of capsules achieve a quorum, characterized by sustained chemical oscillations in the repressilator network. As described in *Results and Discussion*, we find that there is a critical colony radius below which a quorum is not reached. By tuning the reaction kinetics, it is also possible to quench oscillations in large colonies, so that only colonies within a specific range of sizes become activated. Finally, at the end of *Conclusions*, we allude to recent experimental studies that can facilitate the physical realization of this synthetic QS system.

## Results and Discussion

### The Discrete Capsule Model.

We model the colony of microcapsules as a set of regions in space where the production of chemicals is regulated by the repressilator feedback network. The number of capsules in the colony is *A*. Each capsule is a 3D sphere of radius *B*). The capsule’s shell is selectively permeable; it prevents the enzymes from leaving the capsule but allows the reactants and products of the enzymatic reaction to freely diffuse into and out of the shell. Assuming that the reactants are always in excess, the rates of reaction can be taken to be independent of reactant concentrations. Hence, we do not explicitly model nor further discuss the reactants. It is the products that are of interest, because they act as signaling species in our system.

With the repressilator (12⇓–14) modulating the production of signaling species in the capsule, species *B*). (The indices referring to chemical species are interpreted modulo 3.) Treating the enzyme as a uniformly distributed continuum within the capsules, the rates of production per unit volume *SI Text* for details of nondimensionalization), the Hill function reads

In addition to production, each signaling species undergoes first-order degradation and diffuses throughout a laterally (

We numerically solve the reaction−diffusion equations using a finite difference method (*Methods*). There are five model parameters: Hill coefficient *SI Text*. Fixing the Hill coefficient to *A*). The amplitudes of these oscillations are all comparable, and the three species oscillate out of phase with each other.

Conversely, when the colony size is decreased while keeping the same number density of capsules (*B*). The change in capsule activity between sustained oscillatory chemical production in large colonies and steady, or quiescent, production in small colonies exemplifies the QS characteristic of our system.

To determine whether the production capacity affects the behavior of the system, we increase the maximum production rate per unit volume of the capsules from *C*). This finding is consistent with our theoretical understanding of the corresponding well-mixed repressilator system, which is known to exhibit oscillations when *C*, but increase the number density by reducing the radius of the colony from *D*). Even though the capsule density is higher, oscillations cannot be sustained, due to the smaller span of the colony. These results indicate that the colony behavior depends on the combination of colony size and capsule density.

For insight into the internal dynamics of the colony, we map the concentration fields and spatial distributions of chemical production at various times over a period of oscillation (Fig. 3; also see Movie S1) for the simulation presented in Fig. 2*C*. All capsules oscillate in phase with each other and have similar production rates at any given time. Although chemical production only occurs in the discrete capsules, the concentrations vary smoothly throughout the colony, and the discrete nature of the sources is difficult to discern. The concentration field is approximately radially symmetric and is highest at the center of the colony.

As indicated by the in-phase oscillations of capsules in the colony, our system behaves as a collection of oscillators that undergo synchronization (17⇓–19, 22). In our colony, however, a single capsule in isolation does not exhibit oscillations; oscillations require coupling among many capsules. Notably, similar phenomena have been reported for synchronized glycolytic oscillations in yeast (23).

The above numerical approach is feasible in a limited range of parameter space, because it is computationally demanding to simulate large, 3D domains with each capsule spatially resolved over time scales that are sufficient to evaluate the long-term behavior. To achieve a broad, qualitative understanding of the system, we develop a simpler, coarse-grained model of the colony that allows rapid characterization over a large region of parameter space.

### The Continuum Colony Model.

To reduce the complexity of our colony model, we replace the collection of discrete capsules with a spatially continuous distribution throughout the volume occupied by the colony. In this continuum model, we do not resolve inhomogeneities on the length scale of the capsule radius but rather consider average effects over macroscopic scales. This simplification is valid assuming the capsules are both small and lie close together relative to the diffusion length of the chemicals. (If the capsules were far apart from each other, then they would effectively behave independently because the signaling chemicals would decay before diffusing to neighbors.)

The continuum model is described by the same reaction diffusion equations, Eq. **2**, boundary conditions and initial concentrations, Eq. **3**, as the discrete capsule model. The continuum approximation replaces the sum of discrete production terms with a single term,

We further simplify the model by assuming that the production rates per unit volume are spatially uniform throughout the colony at any given time. This assumption is motivated by observations of in-phase synchronized oscillations of all capsules in our discrete simulations (Fig. 3) and is justified if the inhibitor concentrations are approximately homogeneous throughout the colony. We choose to evaluate the homogeneous production rates based on the chemical concentrations at the center of the colony, denoted by

We first consider the case *SI Text* for details). A unique pair of steady-state concentration and production rate (identical for all three species *A* shows the respective regions of parameter space with linearly stable and unstable steady states. We also performed numerical solution of the reaction−diffusion equations, relaxing the assumption of uniform chemical production (*SI Text* for details). The simulations were consistent with the uniform colony stability analysis; oscillatory behavior emerges when the steady state is linearly unstable, and steady behavior emerges in the linearly stable regime (Fig. 4*B*). This comparison validates our simplification of considering spatially uniform production rates within the colony.

Noting that the maximum production rate per unit volume *A*. The horizontal line denotes a critical value of the production capacity

The vertical portion of the transition curve corresponds to a critical colony radius

The QS characteristics we observe for this colony of synthetic capsules (with Hill coefficient

In the case

The characteristic that oscillations cease as the colony radius increases (for *Supporting Information*, we also compare the periods and amplitudes of oscillation for the well-mixed and heterogeneous systems (Figs. S1 and S2).]

In the above discussion, we focused on analysis of a 2D colony of capsules because this is potentially the simplest arrangement to realize experimentally; unless the fluid and microcapsule mass densities are carefully matched, the microcapsules would naturally settle into a 2D layer in a microfluidic chamber, as in experiments with porous catalytic particles (19). Nevertheless, 3D arrangements of capsules are experimentally possible, as are 1D arrangements in a capillary tube or other channel. It is straightforward to adapt both the discrete and the continuum colony models to determine the behavior in such systems, although numerical solution becomes much slower as the dimensionality increases. In *SI Text*, we present a QS phase map for spherical colonies in unbounded, 3D space (Fig. S3). The results are similar to the 2D case shown in Fig. 5; oscillations are favored by large colony sizes, high production capacities, and large values of the Hill coefficient.

## Conclusions

We developed a theoretical model for the chemical activity of a colony of microcapsules that produce and release chemical species. Our approach extends standard mathematical models for reaction kinetics in well-mixed systems to a spatially heterogeneous description by considering capsules to be diffusively coupled, localized sources. Simulations showed that the colonies regulated by the repressilator feedback network exhibited QS behavior. For example, small colonies gave steady, low production of chemicals (off state), whereas large, dense colonies generated large-amplitude oscillations in chemical production rates and concentrations (on state). Using a simplified model, we characterized the dependence of the emergent behavior on model parameters, enabling systems with particular transitions to be designed. For instance, colonies are generally switched on when the colony radius is sufficiently large. With the Hill coefficient

The models and simulation techniques we have implemented can be extended and applied to arbitrary regulatory networks to investigate the coordinated behavior in other systems of synthetic or biological cells. Notably, these findings show that QS can emerge from regulatory networks that are not used for QS in biological systems and hence reveal that a variety of feedback loops could enable sensitivity to the number and density of elements in a colony.

Our findings have important implications for designing novel mechanoresponsive materials. In particular, the capsules can be embedded within and chemically linked to a polymer gel, or the enzymes can be directly immobilized in hydrogels (25). Mechanical deformation (extension or compression) can be harnessed to tailor the separation between the capsules and thus create materials that controllably emit oscillatory signals.

In systems of free (unbound) capsules, chemical concentration gradients can cause the capsules to move due to diffusiophoresis, for example. The signaling molecules, or other secreted chemicals, would control the motion of the capsules in addition to regulating reactions. Such systems could display complex self-organizing behavior, as there is bidirectional feedback between the spatial arrangement of capsules and the chemical activity. Previous simulations demonstrated that a group of three capsules, each producing a different component of the repressilator network, could dynamically regulate chemical concentration gradients to induce rapid aggregation of the capsules (24). Applying a similar mechanism for gradient-driven motion of QS capsules could enable large populations to aggregate when the population density locally reaches a critical threshold, mimicking the coordinated aggregation of slime mold cells. It is also possible to envision engineering other collective phenomena, such as oscillating contraction and expansion of colonies, by modifying the diffusiophoretic response of capsules to each chemical species.

Finally, based on recent experimental studies (26⇓⇓⇓–30), we propose potential routes for physically realizing our QS microcapsules. A key component for QS in our system is a regulated production of chemicals. In our model, this is achieved through the repressilator network, which enables self-sustained oscillations under specific conditions. The original realization of the repressilator was based on regulation of a synthetic gene network in living cells (12). The repressilator motif can, however, be implemented in cell-free alternatives, which use biochemical mixtures to execute the regulation scheme (26, 29). In particular, in vitro transcriptional circuits can be systematically assembled to form arbitrary networks in an experimentally simple manner (31); oscillators, including the repressilator, have been successfully constructed using this technique (26, 27).

The three chemical species in the repressilator network are inhibitor species of RNA (26, 27, 31). The kinetics following this mechanism of inhibition were experimentally shown to be approximated by the Hill function, as was assumed in our model (31). The value of the effective Hill coefficient *n* varied between 3 and 6, depending on controllable experimental conditions (31). Thus, our predictions on variations in the system behavior with changes in *n* could be tested experimentally.

In addition to the regulated production of chemicals, to experimentally realize our system, we need to enclose the regulating network within microcontainers, while still allowing chemical communication among these containers. Encapsulation of cell-free biochemical networks has been achieved using phospholipid vesicles (30) and water-in-oil microemulsion droplets (27, 29). Certain small molecules used in bacterial QS, such as *N*-(3-oxo-hexanoyl)-l-homoserine lactone, could pass through vesicle membranes (30) and the oil phase surrounding emulsion droplets (29), thus facilitating intercellular communication. Channel proteins could be used to allow transport of other signaling species across vesicle membranes (32), and stomatocytes with large openings would also freely exchange a wide range of materials with the external fluid (33). Hence, the necessary components for experimentally realizing our system are available.

As an alternative route, the necessary enzymes can be confined in membraneless compartments by complex coacervation (34, 35). Under specific conditions of temperature, pH, and ionic strength, polyelectrolytes can undergo phase separation and form coacervate droplets that are enriched in protein molecules (34). Lacking membranes, signaling species could freely diffuse into and out of the droplets. Intriguingly, the condition dependence of coacervation provides a mechanism for dynamically aggregating and dispersing the enzymes. Because our model indicates that oscillations occur when there is a large, concentrated aggregate of chemical-producing material, we could design systems that oscillate when a certain temperature is reached, for example, and phase separation occurs.

## Methods

We use a finite difference approach to numerically solve an approximation to the model described by Eqs. **2**−**4**. The forward-time centered-space scheme is implemented for the temporal and spatial derivatives (36, 37). The computational domain is the 3D box defined by *x* and *y* boundaries, and impermeable boundary conditions are used for the z boundaries,

Restricting our studies to 2D arrangements of capsules, we consider a thin domain

The domain is discretized using a regular square grid with spacing *B* and *C*. The time step size used with the finite difference scheme is

The centers of all capsules are confined to the plane *x*,*y*) coordinates are chosen randomly from a uniform distribution. The position is rejected and redrawn from the uniform distribution if the capsule would be placed outside the colony (**1** is applied to all grid cells within a distance

## SI Text

## Nondimensionalization of Governing Equations

Before nondimensionalization, the reaction−diffusion equations governing the concentrations of the chemical species in our repressilator system are*Results and Discussion*. Repression is modeled using the Hill function, which has the dimensional form**S1** and **S2** are transformed into the dimensionless forms in *Results and Discussion* by introducing the dimensionless variables

## Physical Interpretation of Dimensionless Lengths in the Model

The time scale for nondimensionalization in our model is the inverse decay constant for first-order degradation of the chemicals,

Given a spatial arrangement of capsules, the dimensionless colony radius can be altered by changing the diffusion coefficient

## Stability Analysis of Uniform, Continuous, 2D Colony Model

As discussed in *Results and Discussion*, we assume that the production rates per unit volume inside the colony are spatially homogeneous for each chemical species at any given time. In the current analysis, we assume a 2D (circular) colony in an unbounded domain. Results from applying the same procedure to 3D (spherical) colonies are briefly presented in *Stability Analysis of Uniform, Continuous, 3D Colony Model*. We first obtain a steady-state solution for our model and then use a linear expansion to determine the stability of the steady state. If the steady state is stable, then we expect the colony to relax to constant, nonoscillatory dynamics. If the steady state is unstable, then sustained oscillations in the chemical production rates and concentrations will emerge.

For convenience, we repeat the reaction−diffusion equations from Eq. **2**,

The production terms are given by

Although the chemical concentrations are spatially varying in this model, the behavior of the system is completely determined by the concentrations at the center of the colony. Once we obtain consistent solutions for **S3** without the production term is

Suppose that a steady state exists with equilibrium concentrations **S5** to the steady state,

Noting that the Hill function in Eq. **S4** relating **S7** can only hold for all

We now linearize about the steady state,

From Eq. **S11**, we conclude that the eigenvalue

## Stability Analysis of Uniform, Continuous, 3D Colony Model

A procedure analogous to that in *Stability Analysis of Uniform, Continuous, 2D Colony Model* can be performed to determine the stability of steady states for a spherical colony in a 3D domain. The Green’s function Eq. **S6** is replaced by the 3D form,

## Comparison of Oscillation Amplitudes and Periods in Finite Capsule Colonies and Well-Mixed Systems

We have demonstrated that colonies of large enough radius adopt the qualitative behavior expected for well-mixed repressilator systems modeled by the set of ordinary differential equations,*A*. Here, we have taken the initial concentrations to be *A*. For comparison, the time series of average production rates in the discrete colony are reproduced in Fig. S1*B*. The period of oscillations in the discrete colony is

To ascertain whether the differences in oscillation period and amplitude are due to the finite colony size or some other effect, such as the spatial heterogeneity within the colony, it would be ideal to systematically vary the colony radius from the small to the large limit. The colony radius is, however, limited by constraints on computational resources for detailed discrete capsule simulations. Instead, we use our coarse-grained model assuming uniform, continuous chemical production throughout the colony, as used to efficiently construct the phase maps in Fig. 5. Under the equivalent conditions as with the discrete capsule case above, the coarse-grained colony (Fig. S1*C*) has a larger period *D*), oscillations of the coarse-grained colony become indistinguishable from the well-mixed case (Fig. S1*A*).

We plot the dependences of *B*) and the coarse-grained model (Fig. S1*C*). Above

## Continuum Model with Radially Varying Chemical Production

For the continuum colony model, the chemical source terms are homogeneous throughout the disk-shaped colony, as described by Eq. **5**. We prescribe uniform initial concentration fields for each chemical species and, hence, expect the chemical concentration fields to remain independent of *z* and azimuthally symmetric about the center of the colony. Enforcing these symmetry considerations, the reaction−diffusion system (Eqs. **2** and **3**) can be expressed in cylindrical coordinates as

## Acknowledgments

This work was supported as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0000989.

## Footnotes

↵

^{1}Present address: Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1.- ↵
^{2}To whom correspondence should be addressed. Email: balazs{at}pitt.edu.

Author contributions: H.S. and A.C.B. designed research; H.S. performed research; H.S. and A.C.B. analyzed data; and H.S. and A.C.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1702288114/-/DCSupplemental.

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