Bacteria and other microbes can communicate with each other using chemical languages. They release small signaling molecules called autoinducers into their surroundings and sense the levels of the autoinducers in the environment. The response to these autoinducers – known as quorum sensing – can regulate how whole communities of microbes grow and behave; for example, autoinducers can alter the ability of microbes to infect humans or enable the microbes to collectively switch on light production.

Recent experiments suggest that, in a population of genetically identical microbes, some individuals may produce autoinducers while others do not. The coexistence of these different “phenotypes” in one population may enable different individuals to perform different roles, or act as a “bet-hedging” strategy that helps the population to survive if it is later exposed to a stressful situation.

It is not clear how microbes regulate autoinducer production so that only some individuals produce these molecules. Bauer, Knebel et al. developed a theoretical model to address this question. In the model, the microbes shape their environment by producing autoinducers and can respond to this self-shaped environment by changing their level of autoinducer production. Bauer, Knebel et al. found that this establishes a feedback loop that can result in autoinducers being produced by some individuals but not others.

The next step following on from this work is to carry out experiments to test the assumptions and predictions made by the theoretical model. These findings may help to understand how the coexistence of different phenotypes affects collective behaviors, and vice versa, in populations of microbes that use quorum-sensing.

Autoinducers are small molecules that are produced by microbes, secreted into the environment, and sensed by the cells in the population (Keller and Surette, 2006; Hense and Schuster, 2015). Autoinducers can trigger a collective behavior of all cells in a population, which is called quorum sensing. For example, quorum sensing regulates the transcription of virulence genes in the Gram-positive bacterium Listeria monocytogenes (Gray et al., 2006; Garmyn et al., 2011; da Silva and De Martinis, 2013) and the transcription of bioluminescence genes in the Gram-negative bacterium Vibrio harveyi (Xavier and Bassler, 2003; Anetzberger et al., 2009), and it may also autoregulate the transcription of autoinducer synthase genes (Fuqua and Greenberg, 2002; Waters and Bassler, 2005). When the concentration of autoinducers reaches a threshold value, a coordinated and homogeneous expression of target genes may be initiated in all cells of the population (Waters and Bassler, 2005; Hense and Schuster, 2015; Papenfort and Bassler, 2016), or a heterogeneous gene expression in the population may be triggered at low concentrations (Anetzberger et al., 2009; Williams et al., 2008; Boedicker et al., 2009; Garmyn et al., 2011; Pérez and Hagen, 2010; Ackermann, 2015; Grote et al., 2014, Grote et al., 2015; Papenfort and Bassler, 2016; Pradhan and Chatterjee, 2014). To implement all of these functions and behaviors, a microbial population needs to dynamically self-regulate the average autoinducer production.

Within a given population, the promoter activity of autoinducer synthase genes may vary between genetically identical cells (Garmyn et al., 2011; Grote et al., 2014; Anetzberger et al., 2012; Plener et al., 2015; Cárcamo-Oyarce et al., 2015; Grote et al., 2015). For example, during the growth of L. monocytogenes under well-mixed conditions two subpopulations were observed, one of which expressed autoinducer synthase genes, while the other did not (Garmyn et al., 2011). Such a phenotypic heterogeneity was associated with biofilm formation (Garmyn et al., 2011; da Silva and De Martinis, 2013; Hense and Schuster, 2015; Cárcamo-Oyarce et al., 2015). The stable coexistence of different phenotypes in one population may serve the division of labor or act as a bet-hedging strategy and, thus, may be beneficial for the survival and resilience of a microbial species on long time scales (Ackermann, 2015).

The mechanism by which a heterogeneous expression of autoinducer synthase genes is established when their expression is autoregulated by quorum sensing has remained elusive. For example, expression of the above mentioned autoinducer synthase genes in L. monocytogenes is up-regulated through quorum-sensing in single cells (Garmyn et al., 2011, Garmyn et al., 2009; Waters and Bassler, 2005). From an experimental point of view it is often not known, however, whether autoinducer synthesis is up-regulated for all autoinducer levels or only above a threshold level. To explain phenotypic heterogeneity of autoinducer production, currently favored threshold models of quorum sensing typically assume a bistable gene regulation function (Fujimoto and Sawai, 2013; Pérez-Velázquez et al., 2016; Goryachev et al., 2005; Dockery and Keener, 2001). For bistable regulation, cellular autoinducer synthesis is up-regulated above a threshold value of the autoinducer concentration in the population, whereas it is down-regulated below the threshold ('all-or-none' expression); see Figure 1B. Stochastic gene expression at the cellular level then explains the coexistence of different phenotypes in one population. If, however, cellular autoinducer synthesis is up-regulated for all autoinducer concentrations (monostable up-regulation), the mechanism by which phenotypic heterogeneity can arise and is controlled has not been explained.

Figure 1

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Here we show that the coupling between ecological and population dynamics through quorum sensing can control a heterogeneous production of autoinducers in quorum-sensing microbial populations. At the same time, the overall autoinducer level in the environment is robustly self-regulated, so that further quorum-sensing functions such as virulence or bioluminescence can be triggered. We studied the collective behavior of a stochastic many-particle model of quorum sensing, in which cells produce autoinducers to different degrees and secrete them into the well-mixed environment. Production of large autoinducer molecules (for example oligopeptides) and accompanied gene expression are assumed to reduce fitness such that non-producers reproduce faster than producing cells. Moreover, it is assumed that quorum sensing enables up-regulation of autoinducer production, that is, individuals can increase their production in response to the sensed average production level in the population (Figure 1). As a central result, we found that the population may split into two subpopulations: one with a low, and a second with a high production rate of autoinducers. This phenotypic heterogeneity in the autoinducer production is stable for many generations and the autoinducer concentration in the population is tightly controlled by how production is up-regulated. If cellular response to the environment is absent or too frequent, phase transitions occur from heterogeneous to homogeneous populations in which all individuals produce autoinducers to the same degree. To capture these emergent dynamics, we derived the macroscopic mean-field equation (1) from the microscopic stochastic many-particle process in the spirit of the kinetic theory in statistical physics, which we refer to as the autoinducer equation. The analysis of the autoinducer equation explains both phenotypic heterogeneity through quorum sensing and the phase transitions to homogeneity.

The key aspect of our work is how the composition of a population changes in time when its constituents respond to an environment that is being shaped by their own activities (see Box 1). This ecological feedback is mediated by quorum sensing and creates an effective global coupling between the individuals in the population. Such a global coupling is reminiscent of long-range interactions in models of statistical mechanics, such as in the classical XY spin model with infinite range interactions (Antoni and Ruffo, 1995; Yamaguchi et al., 2004; Barré et al., 2002; Choi and Choi, 2003; de Buyl et al., 2010; Campa et al., 2009; Pakter and Levin, 2013). Our analysis suggests that quorum sensing in microbial populations can induce and control phenotypic heterogeneity as a collective behavior through such a global coupling and, notably, does not rely on a bistable gene regulatory circuit (see Discussion).

We now introduce the quorum-sensing model for a well-mixed population of $N$ individuals (Figure 1). The phenotype of each individual $i=1,\mathrm{\dots },N$ is characterized by its production degree ${\displaystyle p_i\in [0,1]}$, that is, the extent to which it produces and secretes autoinducers. In an experiment with microbes, the promoter activity of autoinducer synthase genes or their enzymatic activity could be a proxy for the production degree. The limiting case ${\displaystyle p_i=0}$ denotes a non-producer, and ${\displaystyle p_i=1}$ denotes a full producer.

The state of the population ${\displaystyle \mathbf{p}=(p_1,\dots ,p_N)}$ changes stochastically (Figure 1A): An individual $i$ reproduces with rate ${\varphi }_i$, which we refer to as the individual’s fitness. We assume that fitness decreases with incurring metabolic costs of induction and synthesis of autoinducers, and with other metabolic burdens in the cell’s phenotypic state (Ruparell et al., 2016; Diggle et al., 2007; He et al., 2003). For simplicity, we choose ${\displaystyle {\varphi }_i=\varphi (p_i)=1-s{p}_i}$. The selection strength ${\displaystyle 0\le s<1}$ scales the fitness difference with respect to the non-producing phenotype ($\varphi (0)=1$). Thus, the larger an individual’s production, the smaller its reproduction rate. This assumption is discussed in detail further below (see Discussion).

Whenever an individual divides into two offspring individuals in the stochastic process, another individual from the population is selected at random to die such that the population size $N$ remains constant. Qualitative results of our model remain valid if only the average population size is constant, which may be assumed, for example, for the stationary phase of microbial growth in batch culture. One recovers the mathematical set-up of frequency-dependent Moran models for Darwinian selection (Moran, 1958; Ewens, 2004; Blythe and McKane, 2007; Nowak et al., 2004) if one restricts the production degrees to a discrete set, for example, to full producers or non-producers only, ${\displaystyle p_i\in \{0,1\}}$. The mathematical set-up of the well-known Prisoner’s dilemma in evolutionary game theory is recovered if, in addition, the secreted molecules would confer a fitness benefit on the population (Nowak et al., 2004; Traulsen et al., 2005; Melbinger et al., 2010; Assaf et al., 2013). Since we are interested in the mechanism by which heterogeneous production of autoinducers might be induced and do not study the context under which it might have evolved, we do not include any fitness benefits through signaling, for example at the population level, into the modeling here (see Discussion).

A central feature of our model is the fact that individuals may adjust their production degree via a sense-and-response mechanism through quorum sensing, which is implemented as follows. After reproduction, both offspring individuals sense the average production level of autoinducers ${\displaystyle ⟨p⟩=1/N{\sum }_i{p}_i}$ in the well-mixed population. With probability $\lambda$, they independently adopt the value $R(⟨p⟩)\in [0,1]$ as their production degree in response to the sensed environmental cue $⟨p⟩$, whereas they retain the ancestor’s production degree with probability $1-\lambda$ through non-genetic inheritance. In an experimental setting, the response probability $\lambda$ relates to the rate with which cells respond to the environment (Kussell and Leibler, 2005; Acar et al., 2008; Axelrod et al., 2015) and regulate their production through quorum sensing. We refer to the function $R(⟨p⟩)$ as the response function, which is the same for all individuals. The response function encapsulates all biochemical steps involved in the autoinducer production between perception of the average production level $⟨p⟩$ and adjustment of the individual production degree to $R(⟨p⟩)$ in response (He et al., 2003; Williams et al., 2008; Drees et al., 2014; Hense and Schuster, 2015; Maire and Youk, 2015); see Figure 1B. For example, it may be a bistable step or bistable Hill function, which is often effectively assumed in threshold models of phenotypic heterogeneity (Fujimoto and Sawai, 2013; Pérez-Velázquez et al., 2016; Goryachev et al., 2005; Dockery and Keener, 2001). For a bistable response function, cellular production is up-regulated above a threshold value of $⟨p⟩$, whereas it is down-regulated below the threshold. For the bistable response function sketched in Figure 1B, both values $⟨p⟩=0$ and $⟨p⟩=1$ are stable fixed points. In this work, however, we particularly focus on monostable response functions $R(⟨p⟩)$ to model microbial quorum-sensing systems in which autoinducer synthesis is up-regulated at all autoinducer production levels in the population (Garmyn et al., 2009; Waters and Bassler, 2005). In other words, cellular production always increases with respect to the sensed production level in the population (stable fixed point at $⟨p⟩=1$ and unstable fixed point at $⟨p⟩=0$). The sense-and-response mechanism is further discussed in the Discussion section.

From a mathematical point of view, the introduced sense-and-response mechanism through quorum sensing constitutes a source of innovation in the space of production degrees because an individual may adopt a production degree that was not previously present in the population. Thus, a continuous production space with ${\displaystyle p_i\in [0,1]}$ as opposed to a discrete production space is a technical necessity for the implementation of the quorum-sensing model. The coupling of ecological dynamics (given by the average production level of autoinducers $⟨p⟩$) with population dynamics (determined by fitness differences between the phenotypes) through quorum sensing results in interesting collective behavior, as we show next. We emphasize that, as long as this coupling is present, the effects of the quorum-sensing model that we found and report next are qualitatively robust against noise at all steps; see below.

Box 1

An ecological feedback can control phenotypic heterogeneity in quorum-sensing microbial populations.

Our work demonstrates that the coupling of ecological and population dynamics through quorum sensing cannot only lead to homogeneously producing populations, but can also control a heterogeneous production of autoinducers in microbial populations. Phenotypic heterogeneity becomes manifest in the quorum-sensing model as long-lived, bimodal states of the population that are dynamically stable; see sketch below and Equation (2).

In the quorum-sensing model, ecological dynamics are determined by the average production level of autoinducers, while population dynamical changes are determined by fitness differences between non-producers and producers of autoinducers. Because individuals sense and respond to autoinducers in the environment, the ecological dynamics are coupled with the population dynamics. In other words, an ecological feedback loop is established when cells respond to an environment that is being shaped by their own activities. When fitness differences between non-producers and producers of autoinducers balance with cellular response to autoinducers in the environment, separated production degrees stably coexist in one population. Therefore, we expect that a heterogeneous production of autoinducers may be induced and controlled by such an ecological feedback in real microbial populations, suggesting an alternative mechanism to stochastic gene expression in bistable gene-regulatory circuits to control phenotypic heterogeneity (see Discussion).

Box 1—figure 1

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The quorum-sensing model was numerically simulated by employing Gillespie’s stochastic simulation algorithm (Gillespie, 1976, 1977) for a population size of $N={10}^4$ individuals and an exemplary selection strength $s=0.2$, such that $sN\gg 1$. In this regime, demographic fluctuations are subordinate (Nowak et al., 2004; Wild and Traulsen, 2007; Blythe and McKane, 2007). Within the scope of our quorum-sensing model, the precise value of the selection strength $s$ that scales the fitness differences is not important for the reported mechanism by which phenotypic heterogeneity can be induced, see below. We tracked the state of the population $\mathbf{𝐩}$ over time, and depict the histogram of production degrees and the population average in Figure 2.

Figure 2

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Video 1

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First, we studied the stochastic many-particle process without sense-and-response (${\displaystyle \lambda =0}$); see Figure 2A,D and Video 1. In this case, non-producers always proliferate because they reproduce at the highest rate in the population, which is well-studied in evolutionary game theory (Taylor and Jonker, 1978; Maynard Smith, 1982; Hofbauer and Sigmund, 1998). Thus, the initially uniform distribution in the population shifts to a peaked distribution at low production degrees. Ultimately, a homogeneous (unimodal) stationary state is reached in which all individuals produce autoinducers to the same low degree ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}\simeq 0}$. Such a stationary state is absorbing (Hinrichsen, 2000), that is, the stochastic process offers no possibility of escape from this state of the population.

With quorum sensing (${\displaystyle \lambda >0}$), absorbing states are reached if, again, all individuals produce to the same degree $p^{*}$ and, in addition, the value of this production degree is a fixed point of the response function ($R(p^{*})=p^{*}$); see Figure 2B,E and Video 2. In such a homogeneous absorbing state with ${⟨p⟩}_{\mathrm{\infty }}=p^{*}$, an offspring individual can no longer alter its production degree. It either takes over the production degree $p^{*}$ from its ancestor or it adopts that same degree $R({⟨p⟩}_{\mathrm{\infty }})={⟨p⟩}_{\mathrm{\infty }}=p^{*}$ through sense-and-response. Thus, all individuals continue to produce with degree $p^{*}$ and the state of the population remains homogeneous (unimodal).

Surprisingly, for small response probabilities $\lambda$, we found that the population may get trapped in heterogeneous (bimodal) states for long times before a homogeneous absorbing state is reached. The temporal evolution of such a heterogeneous state is shown in Figure 2C,F and Video 3 for $\lambda =0.05$. A monostable response function was chosen with ${\displaystyle R(⟨p⟩)>⟨p⟩}$ for all $⟨p⟩\in (0,1)$ (unstable fixed point at $0$, and stable fixed point at $1$) such that the production degree is always up-regulated through quorum sensing; see sketch in Figure 1B. After some time has elapsed, the population is composed of two subpopulations: one in which individuals produce autoinducers to a low degree ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}}$, and a second in which individuals produce to a higher degree ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}$ that is separated from ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}}$ by a gap in the space of production degrees. Only through strong demographic fluctuations can the population reach one of the homogeneous absorbing states (${⟨p⟩}_{\mathrm{\infty }}=0$ or $1$ for the response function chosen above). The time taken to reach a homogeneous absorbing state grows exponentially with $N$ (Figure 3A). Therefore, states of phenotypic heterogeneity are quasi-stationary and long-lived. These heterogeneous states arise for a broad class of response functions and initial distributions (Appendix 1—figure 1), and they are robust against demographic noise that is always present in populations of finite size (Figure 3A); see our mathematical analysis below. We demonstrated that states of phenotypic heterogeneity are also robust against changes of the model set-up, which might account for more biological details (see, for example, Papenfort and Bassler, 2016 and references therein). Upon including, for example, noisy inheritance of the production degree, noisy perception of the environment, and noisy response to the environment into the quorum-sensing model, heterogeneous states still arise; see Appendix 1—figure 2. Furthermore, the average production in the heterogeneous state is finely adjusted by the interplay between the response probability $\lambda$ and the selection strength $s$ (Figure 2F).

Figure 3

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The establishment of long-lived, heterogeneous states induced by quorum sensing is one central finding of our study. We interpret this phenotypic heterogeneity as the result of the robust balance between population and ecological dynamics coupled through quorum sensing (see Box 1). On the one hand, fitness differences due to costly production favor non-producers. On the other hand, sensing the population average and accordingly up-regulating individual production enables producers to persist. Remarkably, fitness differences and sense-and-response balance such that separated production degrees may stably coexist in one population; the population does not become homogeneous at an intermediate production degree as one might naively expect. Heterogeneity of the autoinducer production is a robust outcome of the dynamics (and not a fine-tuned effect), and the average production level in the population is adjusted by the interplay of the response probability $\lambda$ and the selection strength $s$. Phenotypic heterogeneity does not rely on a bistable response function, but arises due to the global intercellular coupling of ecological and population dynamics through quorum sensing, as we show next. The relevance of quorum sensing for phenotypic heterogeneity in microbial populations is further explored below (see Discussion).

In the following, the observed long-lived states of phenotypic heterogeneity in the quorum-sensing model are explained. First, we derived the macroscopic mean-field equation (the autoinducer equation (1)) from the microscopic dynamics of the quorum-sensing model. Second, we analyzed this mean-field equation and characterized phenotypic heterogeneity of autoinducer production.

The microscopic dynamics of the quorum-sensing model are captured by a memoryless stochastic birth-death process as sketched in Figure 1. Starting from the microscopic many-particle stochastic process, we derived a mean-field equation for the probability distribution of finding any individual at a specified production degree $p$ at time $t$ in the spirit of the kinetic theory in statistical physics (Kadar, 2007). We call this one-particle probability distribution the production distribution $\rho$; Figure 2 shows the corresponding histogram numerically obtained from the stochastic many-particle process. The mean-field equation for $\rho$, which we refer to as the autoinducer equation, is obtained as:

where ${\overline{\cdot }}_t$ denotes averaging with respect to $\rho$ at time $t$. The details of the derivation of the autoinducer equation from the microscopic dynamics are given in the Methods and materials section and in Appendix 2.

The autoinducer equation (1) involves two contributions: the sense-and-response term with prefactor $2\lambda$, and the replicator term with prefactor $1-2\lambda$. Through the replicator term, probability weight at production degree $p$ changes if the fitness $\varphi (p)$ is different from the mean fitness in the population ${\overline{\varphi }}_t$ (here $\varphi (p)-{\overline{\varphi }}_t=-s(p-{\overline{p}}_t)$). Without quorum sensing ($\lambda =0$), Equation (1) reduces to the well-known replicator equation of the continuous Prisoner’s dilemma (Bomze, 1990; Oechssler and Riedel, 2001; Hofbauer and Sigmund, 2003; Cressman, 2005; McGill and Brown, 2007). The sense-and-response term, on the other hand, encodes the global feedback by which individuals adopt the production degree $R({\overline{p}}_t)$ upon sensing the average ${\overline{p}}_t$ through quorum sensing at rate $2\lambda$. The difference between the current state $\rho$ and the state in which all individuals have this production degree $R({\overline{p}}_t)$ determines the change in $\rho$ at every production degree. Through the replicator term and the sense-and-response term, the ecological dynamics (average production level ${\overline{p}}_t$) are coupled with the dynamics of $\rho$.

We now present our results for the long-time behavior of the autoinducer equation (1). First, the autoinducer equation (1) admits homogeneous stationary distributions. Without quorum sensing ($\lambda =0$), the initially lowest production degree in the population, ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}}$, constitutes the homogeneous stationary distribution ${\displaystyle {\rho }_{\mathrm{\infty }}(p)=\delta (p-p_{\mathrm{l}\mathrm{o}\mathrm{w}})}$, which is attractive for generic initial conditions. With quorum sensing (${\displaystyle \lambda >0}$), fixed points of the response function $p^{*}=R(p^{*})$ yield homogeneous stationary distributions as ${\rho }_{\mathrm{\infty }}(p)=\delta (p-p^{*})$, which are attractors of the quorum-sensing dynamics (1) for all initial distributions if ${\displaystyle \lambda >s/2}$; see analysis below. These homogeneous stationary distributions confirm our observations of homogeneous absorbing states in the quorum-sensing model, in which all individuals produce to the same degree; see Figure 2A,B. Time scales at which stationarity is approached are discussed in the Methods and materials section.

Second, to analytically characterize long-lived heterogeneous states of the population, we decomposed $\rho$ into a distribution at low production degrees and a remainder distribution at higher degrees. We found that such a decomposition yields the bimodal, heterogeneous, stationary distribution of the autoinducer equation (1):

if the conditions ${\displaystyle 0<p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}\le 1}$ and ${\displaystyle 0<y<1}$ are fulfilled; see Box 1 for an illustration and Appendix 3 for the derivation. The parameter $\beta =2\lambda /s$ quantifies the balance between fitness differences and sense-and-response mechanism through quorum sensing. Heterogeneous stationary distributions (2) are constituted of a probability mass $y$ at the low-producing degree ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}=0}$ and a coexisting $\delta$-peak with stationary value $1-y$ at a high-producing degree ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}$ separated from ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}}$ by a gap. Such heterogeneous stationary distributions have mean ${\overline{p}}_{\mathrm{\infty }}=\beta$ and variance $\mathrm{Var}{(p)}_{\mathrm{\infty }}=\beta (R(\beta )-\beta )$. Therefore, the interplay between selection strength $s$ and response probability $\lambda$ adjusts the average production of autoinducers in the population (Figure 2F). For simplicity, we assumed in Equation (2) that the initially lowest production degree in the population is ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}=0}$; generalized bimodal distributions for arbitrary initial distributions ${\rho }_0$ are given in Appendix 3.

From the conditions on ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}$ and $y$ below Equation (2), one can derive the following conditions on the response function and the value of the response probability $\lambda$ (for given selection strength $s$) for the existence of heterogeneous stationary distributions: (i) The response function needs to be nonlinear with ${\displaystyle R({\overline{p}}_{\mathrm{\infty }})=p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}>{\overline{p}}_{\mathrm{\infty }}}$; that is, quorum sensing needs to up-regulate the cellular production in some regime of the average production level. Therefore, both monostable and bistable response functions depicted in Figure 1B may induce heterogeneous stationary distributions through the ecological feedback. (ii) The response probability needs to be small with ${\displaystyle \lambda <{\lambda }_{\mathrm{u}\mathrm{p}}=s/2}$; that is, to induce phenotypic heterogeneity, cells must respond only rarely to the environmental cue $\overline{p}$. This estimate of an upper bound on $\lambda$ is confirmed by our numerical results of the stochastic process (Figure 3A–C). Vice versa, for a given response probability, the selection strength needs to be big enough to induce heterogeneous stationary distributions. As we show in the Methods and materials section, phase transitions in the space of stationary probability distributions govern the long-time dynamics of the autoinducer equation (1) from heterogeneity to homogeneity as the response probability changes ($\lambda \to 0$ and ${\displaystyle \lambda \to {\lambda }_{\mathrm{u}\mathrm{p}}}$); see Figure 3C.

For small $\lambda$, the coexistence of the low-producing and the high-producing peaks in solution (2) is stable due to the balance of fitness differences and sense-and-response through quorum sensing. In Appendix 3 we show that the heterogeneous stationary distributions (2) are stable up to linear order in perturbations around stationarity. As our numerical simulations show, these bimodal distributions are the attractor of the mean-field dynamics (1) for a broad range of initial distributions when $\lambda$ is small; see Appendix 1—figure 1 for some examples. They are also robust against noisy inheritance, noisy perception, and noisy response as demonstrated in Appendix 1—figure 2. We interpret the stability of the bimodal stationary distributions (2) as follows (see also Box 1). Fitness differences quantified by the selection strength $s$ increase probability mass at production degree ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}}$, whereas nonlinear response to the environment with probability $\lambda$ pushes probability mass towards the up-regulated production degree ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}=R({\overline{p}}_{\mathrm{\infty }})}$. The gap ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}-p_{\mathrm{l}\mathrm{o}\mathrm{w}}>0}$ ensures that the exponential time scales of selection and sense-and-response stably balance the coexistence of both peaks; see Methods and materials. Because heterogeneous stationary distributions (2) are attractive and stable, heterogeneous states of the stochastic many-particle process arise and are quasi-stationary. Consequently, the time to reach a homogeneous absorbing state in the stochastic process through demographic fluctuations scales exponentially with the population size $N$ (Elgart and Kamenev, 2004; Kessler and Shnerb, 2007; Assaf and Meerson, 2010; Frey, 2010; Hanggi, 1986); see Figure 3A. Thus, phenotypic heterogeneity is long-lived.

In summary, our mathematical analysis explains how phenotypic heterogeneity in the autoinducer production arises when quorum sensing up-regulates the autoinducer production in microbial populations (Box 1). As an emergent phenomenon, the population may split into two subpopulations: one in which cells do not produce autoinducers (‘off’ state, ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}=0}$) and a second in which cells produce autoinducers (‘on’ state, ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}=R(2\lambda /s)>0}$), but grow slower. The fraction of individuals in the ‘off’ state is given by the value of $y$ in Equation (2). If quorum sensing is absent ($\lambda =0$), the whole population is in the ‘off’ state ($y=1$), whereas all individuals are in the ‘on’ state ($y=0$) if quorum sensing is frequent (${\displaystyle \lambda \ge {\lambda }_{\text{up}}}$). Only when response to the environment is rare (${\displaystyle 0<\lambda <{\lambda }_{\text{up}}}$) can the two phenotypic states, ${\displaystyle p_{\mathrm{l}\mathrm{o}\mathrm{w}}}$ and ${\displaystyle p_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}$, coexist in the population (${\displaystyle 0<y<1}$). The transitions from heterogeneous to homogeneous populations are governed by nonequilibrium phase transitions when the response probability changes ($\lambda \to 0$ and ${\displaystyle \lambda \to {\lambda }_{\mathrm{u}\mathrm{p}}}$). Our mathematical analysis shows that phenotypic heterogeneity arises dynamically, is robust against perturbations of the autoinducer production in the population, and is robust against noise at the level of inheritance, sense, and response.