Introduction
Quorum sensing (QS) is the mechanism using which many bacteria modulate gene expression depending on changes in their density. This modulation is accomplished by the production and sensing of small signal molecules that, at a sufficiently high concentration, activate specific target functions (Keller and Surette, 2006; Williams et al, 2007). A QS system can be divided into a sensor module that houses the signal synthesis, secretion and detection systems and an effector module that carries out the targeted function when induced.
For the sensors, a wide variety of signal molecules have been identified. Gram-negative bacteria often use acyl homoserine lactones (AHLs) as signals (Lazdunski et al, 2004; Williams et al, 2007). These AHLs are typically synthesized by LuxI-type enzymes from fatty acids, in which LuxI is the canonical AHL synthase from bacterium Vibrio fischeri. Gram-positive bacteria often use small peptides as QS signals (Kleerebezem et al, 1997; Sturme et al, 2002). These peptides differ in size and in complexity of post-translational modifications. In all QS sensors, signals are produced intracellularly and transported to the extracellular environment. Small AHLs diffuse freely across bacterial cell membranes (Kaplan and Greenberg, 1985), whereas large AHLs seem to be actively transported by pumps, such as the multidrug efflux (mex) pumps in Pseudomonas aeruginosa (Pearson et al, 1999). The peptides are typically too large for diffusion across membranes and are transported by dedicated ATP-binding cassette (ABC) transporters (Sturme et al, 2002; Lyon and Novick, 2004).
Different strategies are in place to detect these signals. AHL signals often lead to the activation of cytoplasmic regulator proteins, such as LuxR in V. fischeri (Kaplan and Greenberg, 1985; Lazdunski et al, 2004), which then activates target gene expression. The peptide signals and even some AHLs, are typically sensed by membrane-associated receptors to initiate a phosphorylation cascade that leads to target gene expression (Kleerebezem et al, 1997; Sturme et al, 2002). In Vibrio harveyi, an AHL (HAI-1) and a furanone (AI-2) are detected by different surface receptors (LuxN and LuxP/Q, respectively) (Henke and Bassler, 2004). An additional level of complexity arises in that bacteria often house multiple QS systems. In V. harveyi, three QS sensors work in parallel to control luminescence of which only one is confirmed to be an AHL-based system (Henke and Bassler, 2004). In V. fischeri, two AHL-based QS systems, ain and lux, are involved in the control of luminescence and symbiotic growth in the squid host (Lupp and Ruby, 2004, 2005) but in the ain system the extracellular AHL signal is detected by a membrane-based sensor, whereas in the lux system the receptor (LuxR) is cytoplasmic.
The list of bacterial functions under QS control has expanded tremendously from its initial discovery for bioluminescence in V. harveyi (Nealson et al, 1970) and competence regulation in Streptococcus pneumoniae (Tomasz, 1965) to diverse functions, such as exoenzyme secretion in P. aeruginosa and other plant pathogens (Smith and Iglewski, 2003; Von Bodman et al, 2003), conjugation in Agrobacterium tumefaciencs (Fuqua and Winans, 1994), sporulation control in Myxococcus xanthus (Kuspa et al, 1992), virulence in Staphylococcus aureus (Winzer and Williams, 2001) and eukaryote host detection in Enterococcus faecalis (Coburn et al, 2004). The QS systems behind these functions are all drastically different and as such the link between QS characteristics and the function regulated in terms of benefit to the host bacterium is unclear (Redfield, 2002; Bassler and Losick, 2006).
Here we show that despite the diversity in structure and function, the essential properties of QS can be captured by a simple generic metric, 'sensing potential'. The metric is based on a universally conserved 'core' signaling module that consists of signal synthesis, transport, detection, and degradation as the fundamental parameters; one that appears across diverse QS systems. We exploit its universality to model this core module and derive sensing potential as a general measure of QS. This sensing potential conveys the ability of a QS bacterium to measure the size of its enclosure. We validate our model using experimental observations of diverse QS bacteria reported in literature. In doing so, we also provide a comprehensive survey of the available quantitative information on the kinetics of these QS systems (see Supplementary Text 2). We find that, in addition to providing a concise, integrated description of the sensing property of a QS module, the sensing potential also captures the dominant trend of sensing characteristics of different QS systems. Thus, capitalizing on these properties, we focus on the effector modules under QS regulation and study how QS characteristics affect the effector regulation. Starting with a common QS-controlled effector, exoenzyme synthesis (Smith and Iglewski, 2003; Von Bodman et al, 2003), we study how and when QS regulation of an effector benefits the host bacterium. The analysis shows how QS characteristics may be tuned to maximize the host bacterial fitness by providing effective regulation.
Results
Modeling framework and definitions
We note that every QS sensor falls into one of the two categories (which we refer to as Type I or Type II) of the core module depending on where the signal concentration (A) is detected. In a Type I system (Figure 1A), the extracellular signal concentration is sensed whereas in a Type II system, the intracellular signal concentration is sensed (Figure 1B). To describe the dynamics of each Type of sensing, we assume that the signal (A) is synthesized at a constant rate k inside the bacterium (of volume Vc) and is lost by degradation and transport to its microenvironment (of volume Ve). The signal concentration inside the cell (Ai) and that outside (Ae) is assumed to be uniform and transport across the bacterial membrane is assumed to be rate limiting (see Materials and methods and Supplementary Text 1). In the microenvironment, the exported signal is diluted by a factor of Vc/Ve and is again subject to degradation. With these assumptions, we can derive the dependence of Ai and Ae on Ve at steady state (see Materials and methods). For both Type I and Type II systems, Ai and Ae increase with a decreasing Ve (Figure 1C). For a sufficiently small Ve (<Ve,c), the signal concentration would exceed a threshold (K) required for phenotypic expression (Figure 1C).
Figure 1
Sensing framework. (A) Type I: the extracellular signal concentration (Ae, red box) is sensed. (B) Type II: the intracellular concentration (Ai, red box) is sensed. Arrows represent reactions. Stippled arrows represent transport. (C) The steady-state signal concentration (A) decreases with Ve. The parameters of the lux system of V. fischeri are used to plot the curve (see Supplementary Text 2). At a sufficiently small Ve, A will exceed the threshold level K to elicit effector response. Inset: by definition, K is the signal concentration to induce an effector (brown line). Experimentally, a QS system often shows a graded response (blue curve) and K is determined as the half-activation signal concentration. (D) Sensing potential (
) as a function of basic physical and biochemical parameters of QS modules for Type I and Type II systems (see Materials and methods).
We define the dimensionless ratio Ve,c/Vc as the sensing potential () of the sensor for the host bacterium. The value for is determined using the key QS parameters (Figure 1D), such as signal synthesis rate (k), activation threshold of a target function (K), and rate constants for degradation (da) and transport (D). These parameters can be rearranged into two dimensionless variables by appropriately scaling with respect to da: 
=k/Kda and 
=D/da. These dimensionless forms were chosen such that conveys how fast a signal is transported from a cell once it is made at a rate conveyed in . As the ratio of signal synthesis rate to the activation threshold and signal degradation rate constant, quantifies the efficiency of signal synthesis. is the ratio of the transport rate constant and degradation rate constant. As such, 1/ is analogous to the Thiele modulus seen in reaction–diffusion processes (Bird et al, 1960; Truskey et al, 2004) and developmental processes, involving diffusion (Goentoro et al, 2006), where it measures the relative rates of reactive and diffusive processes. The above analysis can be readily generalized to account for variations, such as feedback, two-way signal transport in the Type I case (Type Ia) and the use of specialized pumps for signal transport (Type IIa) (see Supplementary Text 1).
By definition conveys the size of the microenvironment required for effector activation, in multiples of the bacterium volume. To interpret sensing potential, we note that is 0 for a QS system that is never activated. In contrast, approaches infinity if target function is always active, as under a constitutive promoter. Note that is drawn from the simple core-module depiction of QS, which, in reality, can have additional complex regulation. Despite this (as we shall show later), it is still applicable and provides a convenient integrated measure for the more complex cases.
As an analogy, consider the following; under appropriate assumptions, the kinetic theory of gases gives rise to a simple gas law that relates different gas properties (state variables), pressure (P), volume (V) and temperature (T) (Bird et al, 1960). A 'real' gas would still have the pressure property P, though its dependence on T would be more complex, depending on the specific molecular properties of the real gas. Extending this analogy a little further, the microscopic movement of individual molecules averages to a mean free path for collisions and leads to the combined gas pressure (Bird et al, 1960). Similarly, for a population of cells (n) in a large volume (V), Ve represents the average enclosure volume (Ve=V/n) for each cell (Supplementary Figure S9A). Here, signal sensing by individual cells averages to a critical enclosure size Ve,c for activation that corresponds to their sensing potential.
Measuring the sensing potential of QS bacteria
In population-level experiments, Ve,c can be approximately calculated as 1/dcrit, where dcrit is the population density observed to trigger a QS phenotype (Henke and Bassler, 2004). Thus, the observed sensing potential observed is 1/(dcritVc). This observed reflects the phenotype (potential) of the actual QS system, comprising of all its regulatory interactions in addition to the core module whereas calculated represents the estimated potential based only on the core module. To test the applicability of our framework, we compare the sensing potential calculated (calculated) using our framework with observed (Figure 2).
Figure 2
Calculated and observed sensing potentials for 15 well-characterized QS systems. Squares represent Type I sensing; triangles Type II sensing. Each dot represents a different module. Details of the parameters and equations to estimate calculated as well as the calculation of observed from experimental observations are provided in Supplementary Text 2. See Supplementary Tables S1 and S2 for a synopsis of the data used to plot this figure. For each system, the equation for the base model (either Type I or Type II) shown in Figure 1D or the variation that best represents the system (Supplementary Figure S2 and Supplementary information) is used to estimate calculated.
Several factors affect this comparison, in addition to the uncertainty in measurements of biological parameters. The first is the estimation of a threshold K. The experimental measurement of K as A at half maximal activation is an approximation for when phenotypic expression can be considered ON (Figure 1C, inset). Second, owing to lack of reliable data to quantify positive feedback, we do not include this effect while estimating calculated. The positive feedback increases (see Materials and methods) and many, but not all, of the QS systems considered here show this regulatory phenomenon. Thus, calculated will probably under predict observed. Third, observed is calculated from continuously growing cultures, in which species concentrations may differ from steady-state values.
Despite these issues, we find a strong correlation between calculated and observed (n=15, P<0.05), indicating that calculated captures the dominant characteristics of the diverse QS systems listed (see additional notes in Supplementary information and Supplementary Figure S10). A linear regression of calculated against observed gives a slope slightly greater than one (1.1
0.05), which is consistent with our expectation that calculated would tend to under predict observed. In addition, six of the sensor modules shown in Figure 2 occur together in pairs in QS bacteria, where they form a hierarchical structure of phenotypic activation. For each of the three pairs from V. fischeri (ain, lux) (Lupp et al, 2003), V. harveyi (HAI-1, luxS) (Henke and Bassler, 2004) and P. aeruginosa (las, rhl) (Latifi et al, 1996), calculated correctly predicts the order of activation (see Supplementary Text 2).
Modulation of sensing potential
We use the model to study the interplay between signal syntheses, its transport and sensing and its effect on activation by looking at the effect of and on . In Type I sensing, in which the extracellular signal is detected, an increase in helps speed up the extracellular-signal accumulation leading to an increase in (Figure 3A). This increase, however, is limited by , representing the amount of signal being made (Figure 3A). Thus, in Type I sensing, increases with both and but the increase in with saturates at a level depending on .
Figure 3
Sensing potential plotted as a function of and from equations in Figure 1D. (A) Type I sensing, in which the extracellular signal is detected (Figure 1A). Thus an increase in increases the extracellular signal accumulation leading to an increase in . The increase is limited by the amount of signal being made given by . (B) Type II sensing, in which the intracellular signal is sensed (Figure 1B). Faster export (larger ) removes the intracellular signal leading to a reduction in . As the signal is produced intracellularly, low signal transport, in comparison to the signal production, could lead to intracellular signal accumulation to above threshold levels, irrespective of . This appears as a steep rise in ( 
) for particular combinations of and and represents 'self-activation' of the QS host. The interplay of and leading to self-activation can be seen as follows; for =10, the vertical line Q marks a critical (=-1), below which approaches infinity (effector self-activation). Line R does the same for =100. Consider point M on =10 and low . If is increased to 100 while keeping constant, the change results in self-activation.
In Type II sensing, gene expression is triggered by the intracellular signal. Although increasing increases , faster export (larger ) tends to remove the intracellular signal and reduce (Figure 3B). The dependence of the two Types of sensing on is hence opposite. Importantly, as the signal is both produced and detected intracellularly in Type II systems, for a given , transport across the bacterial membrane places an upper limit on (Figure 3B). An increase in beyond this limit results in a discontinuity in ( ). To restate, if signal synthesis is fast (large ) and its export rate is sufficiently small (small ), its intracellular concentration would always exceed the activation threshold (K), regardless of the microenvironment size. This can also be seen mathematically by considering the case, in which Ve so that the extracellular signal is infinitely diluted, Ae 0. Putting Ae=0 in equation (1.3) shows that Ai can still exceed the threshold K if synthesis k is sufficiently large, and/or D+da is sufficiently small. Hence, fast signal synthesis or slow signal turnover, or both, could lead to 'self-activation' of the effector (irrespective of Ve). This predicted Type II 'self-activation' seems to occur in nature under appropriate conditions. In P. aeruginosa, starvation can cause increased signal synthesis leading to effector activation irrespective of cell density (van Delden et al, 2001). In A. tumefaciens, TraM sequesters TraR from the TraR–3OC8-HSL complex (Hwang et al, 1995; Swiderska et al, 2001). As TraR induces the QS phenotype on binding with 3OC8-HSL, deletion of TraM can give rise to a lower K (higher ) such that (see Supplementary information) leading to constitutive activation (Hwang et al, 1995; Swiderska et al, 2001). As in Type I, becomes insensitive to in Type II systems for sufficiently fast transport, and is limited by (Figure 3B).
Cost and benefit of QS regulation for exoenzyme secretion
As an integrated measure of QS characteristics, represents a collective QS phenotype, irrespective of the parameters that lead to it. For example, two QS systems could have the same potential but resulting from different synthesis and transport-rate parameters. This framework can then be conveniently applied to study the phenotypic consequences of differing QS characteristics. As one such application, we study the potential benefit that QS regulation offers its host and how this benefit depends on the sensor's characteristics.
For this, we first define a QS-associated change in host fitness (
f) as the benefit gained minus the cost incurred upon effector activation. Assuming the cost of sensor operation to be negligible compared with effector cost (Haas, 2006), we examine f due to effector activation by different sensors with varying values. We note that effector activation (E) by QS for a bacterium in an enclosure of size Ve can be approximately modeled with a Hill equation in terms of : 
, where a is the Hill coefficient and Emax represents maximal activation (see Supplementary Text 1). The cost and benefit of the effector (which are functions of E) then determine whether QS regulation is beneficial to the host bacterium in a given scenario and, if so, how tuning affects the host fitness.
To elucidate this, we model a typical biological target function regulated by QS: the secretion of exoenzymes (Redfield, 2002; Von Bodman et al, 2003; Diggle et al, 2007). Here, QS controls the synthesis of the enzyme (P), which is secreted to the extracellular microenvironment (Figure 4A). In this context, E represents the synthesis rate of P. In the microenvironment, P degrades a substrate (S) to produce a nutrient N (Figure 4A). We assume that diffusion of enzymes across the cell membrane is negligible because of their large size and that they are actively secreted by pumps. Furthermore, we assume that diffusion and mixing in the environment are much faster than cell growth so that all species (enzyme and generated nutrient) are uniformly distributed in the microenvironment. Consider a batch culture, in which, starting from a low density, the bacteria grow in number (n) for a time span (T) in a constant culture volume V with unlimited S. The cost of effector activation to each individual cell depends on E, whereas the benefit depends on the amount of N reaching the cell (Figure 4A). Both in turn depend on the per cell enclosure volume Ve (Ve=V/n). Using this extended model, we first derive f for the host cell as a function of E and Ve. By further assuming that the bacterial growth rate g is an increasing function of f (see Materials and methods) (Koch, 1983; Dekel and Alon, 2005), we can analyze the overall benefit of QS-mediated effector regulation during T.
Figure 4
Effector model and optimal sensor tuning. (A) A QS-mediated synthesis of a costly but beneficial exoenzyme for nutrient foraging. Subscripts i and e refer to concentrations inside the cell and in the enclosure, respectively. Enzyme synthesis under QS control is modeled as 
where Emax is maximal enzyme-synthesis rate. (B) Fitness increase (f) due to effector activation controlled by QS sensors with distinct sensing potentials. Typical early (large ), intermediate and late (small ) inducing sensors are shown. (C) Collective fitness nT as a function . opt here marks the sensing potential for maximum nT. Colored circles mark nT values for corresponding fitness curves for the three sensors shown in B. Note that nT under QS regulation (finite positive ) is greater than with effector shut off (=0) and effector constitutively activated ( ) showing QS regulation is advantageous. The following parameters were used for generating the figures: Cost–benefit: bn=1000, bnm=104 nM, cp=0.4, cpm=10-4 nM-1 hr. Effector: Dp=100 hr-1, dp=0.01 hr-1. Nutrient: Dn=100 hr-1, dn=0.01 hr-1. Reaction: kn=103 nM hr-1, Km=100 nM. Induction: Emax=103 nM hr-1. Growth: Constitutive rate g0=1 hr-1, fitness f (unitless) was scaled by m=0.01 hr-1 before adding to g0, n0=100, nm=109 cell per ml. T=15 h. See Materials and methods for modeling details. (D) Optimal QS characteristics (opt) for a general beneficial effector. nT calculated for one parameter set of the general benefit function is shown for the case, in which QS regulation is beneficial. Typical QS activation characteristics corresponding to cell density (early or late induction) are marked. nT from QS regulation is higher than with effector shutoff (nT at =0) or constitutive activation (nT at ) and is maximal at a unique finite opt. Inset: collective fitness curves for other parameter sets, in which QS proves advantageous. The opt for each curve is determined by the effector characteristics. Curves shown are from at least 100 combinations of the cost and benefit parameters bn (between 10–103) and cp (between 0.1–1). Similar results are generated when the other parameters, such as bnm, cpm, bnv, x, y are changed (data not shown). The following specific parameters were used to generate D: x, y=1, bn=200, bnm=1, bnv=10-3, cp=0.4 and cpm=10-4.
First, two scenarios emerge in which regulation of exoenzyme synthesis is unnecessary. If the cell density during T is never sufficient for benefit to outweigh the cost of synthesis and secretion (f<0), the best strategy is not to activate the effector (=0). However, if the benefit always overwhelms the cost during T, the best strategy is constitutive enzyme synthesis at the maximal rate ( ).
Excluding these two scenarios, QS regulation of exoenzyme synthesis is advantageous and needs to be optimally tuned to maximize bacterial fitness. As bacteria grow in the culture, there is a gradual reduction in the average enclosure volume Ve per bacterium (Supplementary Figure S9B) from (V/(n0Vc)) to (V/(ntVc)), where n0 and nt are the numbers of bacteria at t=0 and t=T, respectively. This leads to an increase in E by QS-controlled activation. Overall, this results in a continual change in f and its path depends on the value of sensor's (Figure 4B). The best QS strategy, such as early activation (with large ) or late activation (with small ) of enzyme production during growth can be determined by the accrued f during T. This 'collective' fitness can be measured by nT, the final number of bacteria at the end of T. The numerical calculation indicates that nT is a biphasic function of (Figure 4C) with a distinct potential (opt) at which nT is maximal. opt represents the optimal QS sensor characteristics for each set of physical and biological parameters that define this cost–benefit scenario.
Studying the properties of the function itself shows the underlying mechanism for QS regulation to be advantageous. The available benefit from enzyme secretion depends on Ve as the secreted enzyme gets diluted in the microenvironment. Thus, for maximum fitness during bacterial growth (changing Ve), enzyme secretion needs to be continually changed with Ve (see Supplementary Text 1 and Supplementary Figure S4). QS provides just such Ve-dependent regulation that is critical for it to become beneficial (see Supplementary Text 1).
Generality of unique optimal sensing potential
The above analysis shows that: (a) the cost and benefit of exoenzyme secretion determines whether QS regulation is advantageous to the host in a given scenario and (b) when advantageous, a unique tuning of the QS characteristics (opt) provides the maximum nT (see Figure 4C). How applicable are these results for other QS-regulated target functions?
To address this question, we extend the analysis to a general effector that is costly but beneficial. Moreover, nT depends on the specific cost (C) and benefit (B) functions for the effector. C used in exoenzyme analysis is based on the measurements of the effect of gene expression on growth rate (Dekel and Alon, 2005) and can be assumed to remain qualitatively unchanged for different effectors. Learning from the exoenzyme study, B will depend on the extent of effector activation E and enclosure volume Ve. In particular, we note that B can be assumed to be an increasing but saturating function in terms of E and 1/Ve (equation (2.7)), which can capture the effects of a wide range of effectors.
Similar to exoenzyme regulation, QS regulation is unnecessary when either the cost or benefit of effector activation overwhelms the other (Supplementary Figures S6A and B). In case the cost of effector activation is much larger than its benefit, the best strategy is to keep the cost minimal by shutting off the effector (=0). However, if benefit from effector activation is overwhelming, the best strategy is to simply maximize possible benefit (and hence f) by always operating the effector at full activation ( ). Both scenarios require no regulation. Otherwise, QS regulation is advantageous over constitutive effector control and its characteristics need to be uniquely tuned (opt) to maximize nT (Figure 4D). A large opt indicates that early activation during growth is optimal, whereas a small opt indicates that activation at a high density is optimal. opt, in each case, is uniquely determined by the parameters of the effector (Figure 4D, inset). This result shows that distinct functions require QS systems with appropriately tuned characteristics for optimal regulation.
Discussion
Here we develop a simple metric 'sensing potential' to quantify the ability of a bacterium to sense the confinement of its microenvironment. The metric emerges from a 'core module' seen in all QS systems so that it is a generic measure; can be measured for any given QS system. We have made a number of simplifying assumptions in our analysis to derive from the core module properties, such as homogeneous distribution of reactive species in a cell and in its microenvironment and negligible impact of a periplasm. These assumptions are based on experimental observations of QS signal diffusion and mathematical analysis (see Supplementary information). They allow us to reduce the complex nature of QS-regulated activation, which typically involves multiple steps and many regulatory species (Hwang et al, 1995; Tu and Bassler, 2007), down to four fundamental measurable parameters governing signal synthesis, transport, degradation and detection.
Despite the simplicity of the metric, our analysis indicates that of the core module can capture the 'dominant' trend of sensing properties across the highly diverse QS systems (Figure 2). The same analysis also shows cases of deviations between the actual potential of a QS system (observed) and the estimated potential (calculated) that is based on the simple core module. This deviation indicates additional regulatory interactions that act over and above that captured by the minimal core module. For example, positive feedback on signal synthesis—positive feedback was not included in the estimations in Figure 2— would increase calculated (see Materials and methods) and could account for many of the deviations (see additional notes in Supplementary Text 2).
In addition to providing an intuitive classification of QS modules (Figure 1), the framework also helps show the commonality and difference between Type I and Type II sensing. For sufficiently fast signal transport ( ), the sensing potential for both types is uniquely determined by and approximately proportional to , suggesting a common strategy to modulate sensing potential. We see several examples of this strategy. In the plant pathogen A. tumefaciens, plant-produced compounds called opines act as primary regulators of the tra QS (Type II) system by controlling signal synthesis (Piper et al, 1999; Farrand et al, 2002). Without opines, transcription of the signal synthase traI is repressed to a low basal level (low as is low), whereas the presence of opines, indicating the presence of the plant host, relieves the repression and leads to normal expression of traI and virulence at high density (high , see Figure 2). The staphylococcal accessory regulator SarA has a similar function for the agr QS (Type I) system in S. aureus (Heinrichs et al, 1996; Novick, 2003). SarA acts as a global virulence-factor regulator in S. aureus and, under certain conditions, is shown to modulate QS-controlled phenotypic induction by directly controlling transcription at the agr locus.
However, when is small, it has opposite, significant effects on the two types: increasing decreases in Type II systems but increases in Type I systems. In this case, manipulation of is an effective strategy to modulate . Consistent with this idea, Type I systems typically use peptide signals that are generally too large to diffuse freely across bacterial membranes, and the use of specialized pumps (e.g. ABC transporters) for signal export (Havarstein et al, 1995; Lazazzera and Grossman, 1998; Dunny and Winans, 1999) will effectively modulate the sensing potential. The dependence of on (for small ) is more complex in Type II sensing. As approaches a critical threshold (-1), drastically increases to approach infinity. Below the threshold, a Type II system can activate its effector irrespective of cell density (Figure 3B), which is impossible in Type I systems. This control strategy seems to be adopted by some bacteria. In P. aeruginosa (van Delden et al, 2001), starvation causes faster signal synthesis in both the las and rhl QS systems. In A. tumefaciens (Hwang et al, 1995), deletion of a repressor element in the tra system lowers the signal-sensing threshold. In both cases, the change in biochemical parameters causes a large increase in driving to infinity (Figure 3B), leading to the activation of the QS-regulated effector independent of cell density. Thus, Type II systems seem to have an additional layer of effector control over QS so as to subvert it under certain scenarios. On the same note, we observe that Type II systems commonly control exoenzyme secretion in free-living bacteria, such as Pseudomonas, Erwinias and in Rhizobia, such as Rhizobium leguminosarum and Sinorhizobium meliloti (Gonzalez and Marketon, 2003; Von Bodman et al, 2003). These bacteria may frequently encounter nutrient exhaustion, not necessarily caused by their own growth; in such cases density-independent secretion of exoenzymes enables nutrient foraging at low densities (van Delden et al, 2001).
Overall, sensing potential provides a concise, integrated description of the sensing characteristics of a QS module, even if its underlying mechanism is more complex than the core module. Thus, can be used as a single modulated (reduced) variable to study how QS characteristics affect the downstream regulation. To illustrate its application, we use our framework to study the scenarios in which QS regulation of functions proves advantageous. This additionally provides an insight into the evolution of QS as a regulation strategy, as seen in the analysis of other evolutionary strategies (Kussell and Leibler, 2005; Wolf et al, 2005a, 2005b).
By modeling exoenzyme control and then generalizing the conclusion to other effectors, we show that QS regulation is advantageous when the cost of effector synthesis is comparable to its resulting benefit (Figure 4). A closer look shows that an underlying requirement is that the benefit from such an effector's activation depends on the environment size. For enzyme secretion, benefit decreases with increased dilution of the enzyme in the microenvironment (increasing Ve). In this case, QS regulation allows effector synthesis to be kept low for large Ve, in which benefit is low compared with cost, and increases it gradually with Ve (Figure 4B), hence providing optimal control (see Supplementary Figure S4 and related text in Supplementary information). In the absence of such benefit dependence on Ve, QS regulation, regardless of any cost–benefit parameter combination, will probably not be advantageous (see Supplementary Text 1 for more details). This conclusion is quite general as many QS-controlled effectors show similar (Ve dependent) benefit function as described in equation (2.7), wherein benefit is an increasing but saturating function of effector activation and density (Figure 4D). Consider QS-controlled colonization and luminescence by V. fischeri in its symbiotic host squid (Lupp et al, 2003; Lupp and Ruby, 2005). The colonization requires synthesis of several costly aggregation factors (Visick and Ruby, 2006), whose effects (and thus benefit) increase with concentration (Yip et al, 2005) and saturate at sufficiently high concentrations. Similarly, luminescence is costly but is suggested to benefit the bacterium by consuming the O2 in the crypt region after colonization takes place (Ruby and McFall-Ngai, 1999; Visick et al, 2000; Stabb, 2005). This benefit will be limited to the total amount of O2 present in the region.
We further show that when QS-mediated control is advantageous, each effector will require a QS sensor with a unique opt that maximizes host fitness (Figure 4D, inset). This is consistent with observations in bacteria with multiple QS systems, each controlling a distinct effector. Furthermore, consider the typical V. fischeri life cycle that starts with colonization of a juvenile host squid's crypt regions followed by growth and light production at high cell densities (Lupp and Ruby, 2005). Colonization and luminescence thus represent distinct effectors to be sequentially induced. Consistent with our analysis, colonization is regulated by the ain sensor with a larger calculated (early activation) than the lux sensor (late activation), which controls luminescence (Figure 2 and see Supplementary Text 2). A similar situation is seen in the pathogenesis of P. aeruginosa, in which virulence involves secretion of exoenzymes (Smith and Iglewski, 2003; Von Bodman et al, 2003) as well as the formation of biofilms (Kirisits and Parsek, 2006). The exoenzyme secretion is controlled by the las sensor whereas secretion of the rhamnolipid involved in biofilm formation is largely controlled by the rhl sensor (Pearson et al, 1997). The las and rhl sensors show distinct calculated values (Figure 2) and are activated hierarchically during growth (Latifi et al, 1996). Lastly, we note that Bacillus subtilis uses two distinct QS sensors (Grossman, 1995; Schauder and Bassler, 2001) with vastly different potentials (Figure 2) to tightly regulate competence development and sporulation (Grossman, 1995).
A number of hypotheses have been proposed for the nature of information that a QS system conveys to its host. By its traditional definition, QS measures cell density (Fuqua et al, 1994; Bassler, 2002), and its benefit lies in coordination of gene expression by a population of cells. In contrast, the hypothesis of 'diffusion sensing' (DS) (Redfield, 2002) proposes that QS measures the mass-transfer characteristics of the environment surrounding an individual bacterium. In DS, the accumulation or dispersal of a QS signal reflects how a secreted effector would also be distributed. The DS can then avoid costly exoenzyme secretion under conditions in which it would be lost by dispersal. To reconcile DS with the traditional QS definition, the 'efficiency sensing' (ES) hypothesis was recently proposed (Hense et al, 2007). The ES argues that QS cells measure the combined effects of density, mass-transfer properties and their own spatial distribution. It also suggests that the benefit of QS may lie in conveying the efficiency of secreting extracellular effectors. Our analysis is based on the signaling dynamics of a single QS cell and is analogous to the approach suggested by DS, but here we do not consider diffusion limitations in the environment. Instead, we model the environment as an enclosure within which the signal concentration is uniform. This simple framework allows a quantitative analysis of QS that can be understood in terms of the different hypotheses. In particular, sensing potential provides an intuitive and measurable connection between an individual QS cell and the population-level phenotype (Supplementary Figure S9A). In addition, the benefit of QS regulation emerges naturally by analyzing the effector controlled by sensors of different potentials (Figure 4C). Taken together, our analysis combines sensing with regulation benefit so that it can be understood and quantified in terms of both a single QS cell and a population of QS cells (Supplementary Figure S9B).
Our analysis has limitations, which arise naturally from the assumptions made to provide a simple, yet generic framework. When the distribution of species in the environment is not homogenous—this could happen due to diffusion limitations or other mass-transfer phenomena—sensing potential may not accurately predict activation characteristics (see Supplementary Text 1). Moreover, our framework is based on an individual QS cell or, equivalently, a population of identical QS cells and does not capture the properties of a heterogeneous population. For example, we do not account for the presence of cheaters in a population that do not signal but respond to it (Diggle et al, 2007). The sensing potential measured from such a heterogeneous population containing cheaters would underestimate an individual QS cell's (non-cheater) potential.
Lastly, we do not explicitly account for cross talk between different sensors when multiple QS systems are present in the same host. Here, signals from one QS system could weakly activate the effector of another system (Holden et al, 1999; Collins et al, 2005), as well as other regulatory parameters and hence affect the sensing potential. Such cross talk could be accounted for by simultaneously modeling the effects of all the functional QS systems present in the host. Nevertheless, sensing potential does provide a standard measure for QS systems in scenarios, in which the model assumptions are justified, and it can readily be extended. On the effector side, we note that the 'one-effector-per-sensor' model used is a simplification. A QS sensor usually controls multiple effectors (Antunes et al, 2007), and multiple sensors may coordinate to control a common effector (Henke and Bassler, 2004). The analysis of these systems will follow the same method but will require the estimation of a combined fitness contribution from each effector under each sensor. Similarly, if the QS sensor itself is found to be significantly costly, this cost needs to be included in the fitness calculation. Taken together, our work provides the theoretical framework as well as an experimental method to study QS regulation, its benefit and hence its evolution. The analysis presented may also help guide experimental efforts in engineering new synthetic gene circuits (Sprinzak and Elowitz, 2005; Marguet et al, 2007; Keasling, 2008; Tanouchi et al, 2009).
Materials and methods
The sensor
We model the signaling dynamics by accounting for the signal synthesis, transport and degradation. We assume that: (1) signal transport (D, hr-1) and degradation (da, hr-1) are proportional to the signal concentration (A, nM); (2) the signal is synthesized at a constant rate (k, nM hr-1). For a Type I system (Figure 1A), the rate of change of the intracellular and extracellular signal respectively is hence given by
For Type II systems (Figure 1B), we have:
where D (Ai-Ae) now accounts for the two-way transport. Here, Ve represents the volume of the bacterial microenvironment and Vc represents the cell volume of an average bacterium.
For Type I system, equations (1.1) and (1.2) are solved simultaneously to get the steady-state Ae as a function of Ve:
For Type II system, equations (1.3) and (1.4) give:
According to equations (1.5) and (1.6), both Ae and Ai increase with decreasing Ve (Figure 1C). The critical Ve,c and hence the sensing potential , for which Ae (Type I) and Ai (Type II) cross the threshold K, is calculated by solving equations (1.5) and (1.6) for at Ae, Ai=K respectively. For Type 1 system, we get: 
where =k/Kda and =D/da. For Type II system, we get: 
.
Positive feedback
We model positive feedback in signal synthesis by assuming that the signal synthesis rate increases linearly with its own concentration with rate constant ka (see Supplementary Text 1 for more details). This leads to the addition of a term kaAi in equations (1.1) and (1.3). By solving these modified equations for steady-state signal concentration and then explicitly for at which A=K as done earlier, we get
where 
=ka/da, is the dimensionless parameter for feedback scaled using da. From these equations we see that for both Type I and Type II systems, positive feedback (as ) acts to effectively increase , which corresponds to increased signal synthesis. The effect of on can thus be studied equivalently as the effect of on .
The effector
The effector activation E under QS control is given by
where Emax is the maximal synthesis rate and a the hill coefficient depending on the cooperativity of the signal-induced activation (see Supplementary Text 1).
For the exoenzyme case, E represents the enzyme synthesis rate. We model the exoenzyme dynamics using the following equations:
where i and e are the concentrations inside the cell and in the microenvironment, respectively; Dp and dp the transport rate constant and the degradation rate constant of P, respectively.
Enzyme–substrate kinetics and nutrient
Following a model of bacterial foraging (Vetter et al, 1998), in which the enzyme absorbed to the substrate catalyzes the production of nutrient (Rubinov, 1975), the rate of production of nutrient in the environment (dNe/dt) is given by 
where kn and Km are appropriate reaction rate and binding constants, respectively. Using nutrient transport and degradation rate constants, Dn and dn, respectively, the mass balance equations for N are:
Cost, benefit and fitness
For any enzyme synthesis rate E and enclosure size Ve, equations (2.2), (2.3), (2.4) and (2.5) are solved simultaneously for steady-state concentrations of enzyme and nutrient. The benefit provided by N is then calculated (Dekel and Alon, 2005) as B=bnNi/(bnm+Ni), where Ni is intracellular nutrient concentration. The cost of effector activation can be modeled (Monod, 1949; Dekel and Alon, 2005) as C=cpE/(1-cpmE). bn, bnm (nM), cp, cpm (nM-1 hr), are benefit and cost function parameters such that B and C unitless. Fitness
Growth rate g (hr-1) is modeled as a linear combination of the growth seen in the absence of an effector (g0 (hr-1)) and in its presence. Without any loss of generality of our conclusions, we assume: g=g0+f. The collective fitness nT is given by 
, where cell growth during T is modeled by a logistic equation 
, with nm as carrying capacity. nT from a QS sensor of given potential is obtained by numerical integration of the logistic growth equation where growth rate at each time point is calculated based on f. To obtain opt the procedure is repeated for a range of 's and a nT versus graph is plotted to find the at which nT is maximal (Figure 4C and D).
We use the following equation to represent the benefit function for a general QS-controlled effector:
This equation captures the characteristics of a wide range of beneficial effectors depending on choice of parameters (bn, bnm and bnv) and hill coefficients (x, y). Note that B increases with 1/Ve and E, but saturates eventually. The calculation of nT for the general function is repeated as above with equation (2.7) being used in equation (2.6) to calculate f.
All equations were solved analytically using Mathematica (Wolfram Research) whereas simulations and plots were done using MATLAB 7.1 (MathWorks).
